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I . .. .. .1352 Vtfi 0 EA: . x U ..th httpflwklg: utéQltDJ.nvflhf1U-u+y u | en... r 9.0.“. ....Vuxbrtigbfiahuk... ‘urgu. . tang? tbflfiYB .1540? 21. ..r .9!!! ..-... N 2. .. 1...}? .....-......K:3.o.tnvl l. ..< .....r . u . . 9 .1. ...ll Eg‘g} ...: kixi‘.‘ .t‘il .. w Mic'mgan btate ._ University This is to certify that the thesis entitled AN INQUIRY INTO CONSTRUCTION OF ACCOUNTING PREDICTORS OF FUTURE MUNICIPAL FINANCIAL INSOLVENCY presented by David Richard Lee Gabhart has been accepted towards fulfillment of the requirements for Ph.D. Accounting degree in . L ‘ /EM W V Major professor 4 /' I ' / / Date ._'__L 0-7639 "Was: Ill\lllllllillflfllllllllmllllfllflllul , AN INQUIRY INTO CONSTRUCTION OF ACCOUNTING PREDICTORS OF FUTURE MUNICIPAL FINANCIAL INSOLVENCY By David Richard Lee Gabhart A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Accounting and Financial Administration 1977 Supported by a grant of the American Accounting Association. \ GSZsz / - Dd)". ABSTRACT AN INQUIRY INTO CONSTRUCTION OF ACCOUNTING PREDICTORS OF FUTURE MUNICIPAL FINANCIAL INSOLVENCY By David Richard Lee Gabhart The primary objective of this research effort was to find,in municipal financial statements, reliable predictors of future finan— cial insolvency of municipalities in the State of Michigan. The con— strained financial condition of some municipalities is of great con- cern to state and local officials, and therefore an "early warning" system is desired to head off conditions of insolvency before they have a chance to develop. Since under federal bankruptcy law, cities cannot feasibly declare bankruptcy, a surrogate for bankruptcy is used; that is, technical financial insolvency, which in turn is defined, for measurement purposes, as having a June 30 cash balance which is less than 10% of total General Fund assets. The predictors for this study were drawn from accounting data only, presented in the audited financial statements of Michigan munici— palities, prepared under uniform accounting procedures prescribed in Public Act 2, 1968. Data used were for the period of 1971 through 1975. Data were from the General Fund only reflecting the revenues David Richard Lee Gabhart and expenses of the general operation of the municipality, the admin- istration to, protection of, and service to the public. Data were gathered for 86 municipalities; missing data and year ends other than June 30 reduced the eligible cities to 60. Some 32 potential independent variables for each year were drawn from the financial statements which might serve as prior—year indicators of subsequent—year insolvency. By judgmental review, then successive application of factor analysis, multiple linear regression, and quantal response (a methodology similar to multiple discriminant analysis) the 32 potential indicators were reduced to sixteen which appeared to be significant in §§_antg_classification of cities into one of three classes: insolvent (cash—poor), cash-adequate, and cash—rich for a subsequent year. When these predictors were applied, using 1971—2-3-4 data to classify cities according to their June 30, 1975 cash position, 87.5% of the 60 cities were properly classified. In the combination of independent predictor variables which gave the highest (87.5%) rate of correct classifications, earlier-year expenditures for Administra- tion and for Parks and Recreation contributed the most to correct classification. In other high-yielding predictor variable combina- tions, Property Tax Revenues, Interfund Borrowings Due to the General Fund, and Past Due Taxes Receivable were significant measures for classifying cities into subsequent-year Insolvent, Cash-Adequate, and Cash-Rich categories. Various ratios and balances were tried as indicators, using both Balance Sheet and Operating Statement numbers, with disappointing fiw David Richard Lee Gabhart results. The universe size was too small to have a hold—out group for validation of the findings. However, subsequent validation can be tested by using the predictors found here, to apply to 1972-3-4-5 data to classify cities ggflantg_as of June 30, 1976; then to compare the predictions with the actual cash status of those cities at June 30, 1976. The pragmatic usefulness of our research may be tested by finding out whether these predictors 25 ante classify cities more or less reliably than do the more intuitive means relied upon by state supervisory officials, and on which they act. The conclusion was drawn that the investigation partially achieved its goals in that it is possible through this methodology to classify certain Michigan municipalities as to the percentage of cash they will hold. Greater time spans and other significant statistical characteristics of the accounting information of municipalities should be investigated. PREFACE This preface allows the inclusion of three items in this paper which are quite unrelated, but will, nevertheless give the reader some insight into the study. The three items are, (l) the means by which the study originated; (2) some observations noted in the course of gathering information, and (3) the acknowledgments to those who contributed to the paper. The idea for this study originated while attending a gather— ing for a couple who was moving out to the West coast. During the idle chatter, someone made the chance inquiry as to how they had picked the town where they were going to live. As one might guess, the conversation then turned to the subject of how one should properly evaluate a municipality, whether for choice of a home or other reasons. Somewhat spontaneously, we offered the thought that the fi— nancial statements of a municipality contained one item that might disclose three significant generalizations about the city. It was the amount of money a city spent on the library, of course relative to amounts spent on other factors and relative to amounts Spent in previous years on the library. Those three conclusions which might be reached were that expenditures on the library show: 1. The amount of future planning that a city does, for the expenditures on a library are certainly of a long-term nature as opposed to expenditures for something like a swimming pool. ii 2. The educational aspirations of the community, for even though the number of books on the shelves of the library do not directly influ- ence the intellectuality of the citizens, they certainly do reflect the interests. 3. The financial solvency of the city, to a de— gree, for the library will probably be one of the first of discretionary expenditures to suffer in a finanoial "crunch". The line of reasoning that was then taken was that the activi- ties which are supported (or not) are assumed to be reflective of the needs and wants of the standing citizenry and those activities are reflected in the financial data of the municipality. If such data was readily available, it might serve as a mirror of the community. From this, it was felt that the financial data could be a portent of the future of the municipality. Thus, the motivation for the study. The second point to be covered in this preface is that during the span of the effort of gathering data, two peculiarities regarding municipal solvency were noted. Although they are not verifiable at this point, they were sufficiently interesting to warrant mention. First, it became apparent that a city which is financially solvent has a readily identifiable, positive personality. The inverse is equally apparent--those cities which are insolvent lack a distinct personality or may even have a negative one. Detroit's one—time positive image as the automobile capital of the world is quickly slip— ping into oblivion, for operative producing plants in Detroit are few. Other less commendable characteristics of the city have become more prominent in the public's mind. Similarly, few refer to New York City any longer as the "Big Apple" or "Fun City". At the same time, Atlanta and Dallas retain positive images of stability and progress-—"nice iii places to live." Second, it seemed that those cities with apparent financial difficulties frequently filed their financial statements with the appropriate regulatory agencies in a manner that was not timely. In many cases, an insolvent condition could be anticipated if financial statements were "not in file" at the regulatory agency. Finally, a portion of papers such as this is usually given to some degree of reflection. We do not wish to be an exception to such a tradition and therefore offer the following thoughts. The students entering doctoral programs bring a wide variety of skills and knowl- edge, but one necessary characteristic for all is that of commitment. This is a recognition that full accomplishment of one's objectives is possible only through sacrifice of time and effort in varying degrees. As with his other educational experiences, the student usually realizes that the rewards of the entire experience are surely as pleasurable as the achieving Of the end in view. However, certain portions of an ad— vanced degree program differ significantly from the other educational experiences which the candidate might have had. Here, the candidate depends upon the effort of many in the earning of his degree. Conse— quently, the sense achievement falls upon all who lent their knowledge to the task. Perhaps it is this unique characteristic which impelled the poet and song writer Percy French to memorialize the Ennis and West Clare Railroad in song. The road climbed seemingly ever upward through extremely mountainous country in Ireland to the town of En— nistymon and then down to West Clare. The railroad had a long history of never making its schedule, seldom completing its run, and at all times imperiling its passengers. But the customers did not abandon iv this institution. Even when the locomotive quit completely, the passengers joined the crew in seeing that the train continued to move. As French recorded it: Uphill the old engine is climbin, While the passenger pushed with a will. You're in luck when you reach Ennistymon For all the way home is downhill. To those who have made this journey with me, I want to express grateful appreciation. Faculty, fellow candidates and friends will all find evidence of their contributions. The list reciting their names and ways of participation would probably be longer than this paper. Nevertheless, recognition of specific contributions is a plea- sure afforded writers of papers such as this and is not to be denied here. The Chairman of the Committee, Dr. Gardner M. Jones, is to be thanked for his magnificent patience, the rewriting of bits and pieces of thoughts so that the thread of logic was somewhat visible, his sage guidance to a goal of practicality and most of all, for his constant friendship throughout the entirety of the experience. Dr. William Schmidt provided the statistical model and guidance through the maelstrom of the ”numbers". His spirit of accommodation was al— ways present in spite of a most pressing schedule. Dr. Robert Ander— son of the Institute for Community Development is to be thanked for his encouragement and criticisms and that ever—present reminder that the products of our research are best if they make the world a better place in which to live. Programming for the Quantal Response model was done by Rita Grant. Her hours were evidence of a friendship beyond belief. Our wishes are that her success in this vein might be realized quickly. In addition, we wish to acknowledge all the help of Mr. Douglas Ar— nold, Local Audit Bureau, State Treasury Department. His assistance made the data gathering an education experience in and of itself. Without his cooperation and guidance, the project would have been im— possible. Finally, and perhaps most important of all, Mrs. Jo McKen— zie deserves applause loud and long. In her role of typist, she assumed the function of editor, counselor, the one who not only gent- ly reminded me of deadlines but prodded me into meeting them and both she and I know the extent of her effort. With apologies to John Crecine, a graduate student at the Uni- versity of Michigan who made a similar conclusion to the preface of his dissertation, we add this postscript. We acknowledge for the last time all those who asked innumerable times, "How's it going Dave?" To them we say: "now we have reached Ennistymon--all the way home is downhill." vi TABLE OF CONTENTS PREFACE. . . . . . . . . . . . . . . . LIST OF TABLES . . . . . . . . . . LIST OF FIGURES. . . . . . . . . . . . . Chapter I INTRODUCTION. . . . . . . . . Statement of purpose. . . . Statement of the problem. Present means of evaluating performance and status. Bond ratings. . . . . . . . Legal measures. . . . . . . Other studies . . . . . . . Relevant writings . . . . . Further considerations. . . II DATAr-SOURCE, NATURE AND COLLECTION Source of data. . . . . . . Fund accounting . . . . . . General types of funds. . . Specific types of funds . . Capital assets. . . . . . . Basis of accounting . . . . Modified accrual basis. . . municipal The municipal accounting system . . . Special Revenue Funds . Debt Service Funds. . . . . Capital Projects Funds. . . Enterprise Funds. . . . . . Trust and Agency Funds. . . c o o o o o o o n c c o o o Expendable/nonexpendable Funds. . . . Intragovernmental Service Funds . . Special Assessment Funds. . Format of Act 2 financial statements. The General Fund. . . . . . Observations. . . . . . Nature of the observed variables. Data gathering. . . . . . . vii Page ii ix xiii Chapter III IV APPENDIX C STATISTICAL METHODOLOGY . . . . . . . . . . . . Dichotomous situation--a basis for the design Discriminant Analysis . . . . . . . . . . . Maximum Likelihood. . . . . . . . . . . . . . Development of the concept. . . . . . . . . . Probits and logits. . . . . . . . . . . . . . Parameter estimation. . . . . . . . . . . . . Parameters under Maximum Likelihood . . . . . Likelihood ratio test . . . . . . . . . . . . Quantal response techniques . . . . . . . . . Two models for two situations . . . . . . . . Probability of the first model. . . . . . . . Description of the second situation . . . . . Probability of the second model . . . . . . . Further issues. . . . . . . . . . . . . . . . Direct causal linkage . . . . . . . . . . Indirect causal linkage . . . . . . . . . . . Mathematical distinction between models . . . Single predictor, dichotomous criterion . Multiple predictors, dichotomous criterion. . Multiple Predictors and Polychotomous Criteria. . . . . . . . . . . . . . . Assumptions and effects of departure from those assumptions. . . . . . . . . . FINDINGS AND CONCLUSIONS. . . . . . . . . . . . Introduction. . . . . . . . . . . . . . . . . Specification of the model. . . . . . . . . Quantal response. . . . . . . . . . . . . . . Range of analysis-—criterion variable . Criterion variables--final runs . . . . . . . Range of analysis--predictor variables. Data output . . . . . . . . . . . . . . . . Runs of data. . . . . . . . . . . . . . . Empirical results . . . . . . . . . . . . . . Format 10 . . . . . . . . . . . . . . . . Formats 8 and 9 . . . . . . . . . . . . . Formats 1 and 2 . . . . . . . . . . . . . . . Formats l6 and 17 . . . . . . . . . . . . . Common dollar statements. . . . . . . . . . . Minicipal Data Bank Questionnaire . . . . Alphabetical Listing of Cities in the State of Michigan Having Populations Exceeding 10,000 in 1970 . . . . . . . . . . . . . . . . . . Data Gathering Form . . . . . . BIBLIOGRAPHY . . . . . . . . . . . . . Page 38 38 39 43 44 46 48 50 52 54 55 56 57 57 58 59 60 61 61 66 68 71 73 73 73 74 75 77 78 80 81 81 94 103 117 133 151 155 158 160 161 LIST OF TABLES Table Page 1 Classifications from Format 11 . . . . . . . . . . . . . 82 2 Estimated Probabilities of Classification Under Format 11 O O O O O O O O O O O O O O O O O O O O 0 82 3 Incorrect Classifications Under Format 11. . . . . . . . 83 4 Estimations of Vectors for 82 AND 83 Under Format 11. . . . . . . . . . . . . . . . . . . . . 88 5 Computation of Relative Weights of Predictor Variables of Category Two Under Format 11. . . . . . . . 89 6 Computation of Relative Weights of Predictor Variables of Category Three Under Format 11. . . . . . . 90 7 Ranking of Bi Under Format 11. . . . . . . . . . . . . . 91 8 Ranking of Variables Under Format 11 . . . . . . . . . . 92 9 Ranking of Administrative, and Parks and Recreation Variables . . . . . . . . . . . . . . . . . . 93 10 Classifications from Format 13 . . . . . . . . . . . . . 94 ll Classifications from Format 10 . . . . . . . . . . . . . 95 12 Estimated Probabilities of Classification Under Format 10. . . . . . . . . . . . . . . . . . . . . 96 13 Estimations of Vectors for 82 and 83 Under Format 10.. . . . . . . . . . . . . . . . . . . 98 14 Computation of Relative Weights of Predictor Variables of Category Two Under Format 10. . . . . . . . 99 15 Computation of Relative Weights of Predictor Variables of Category Three Under Format 10. . . . . . . 100 16 Ranking of éi Under Format 10. . . . . . . . . . . . . . 101 17 Ranking of Variables Under Format 10 . . . . . . . . . . 102 ix Table Page 37 Computation of Relative Weights of Predictor Variables of Category Three Under Format 1 . . . . . . . 123 38 Ranking of 821 Under Format 1. . . . . . . . . . . . . . 124 . m 39 Ranking of Variables Under Format 1. . . . . . . . . . . 125 40 Classifications from Format 2. . . . . . . . . . . . . . 126 41 Estimated Probabilities of Classification Under Format 2 . . . . . . . . . . . . . . . . . . . . . 126 42 Estimation of Vectors for 82 and 83 Under Format 2 . . . . . . . . . . . . . . . . . . . 129 43 Computation of Relative Weights of Predictor Variables of Category Two Under Format 2 . . . . . . . . 130 44 Computation of Relative Weights of Predictor Variables of Category Three Under Format 2 . . . . . . . 131 45 Ranking of Eii Under Format 2. . . . . . . . . . . . . . 132 46 Ranking of Variables Under Format 2. . . . . . . . . . . 133 47 Classifications from Format 16 . . . . . . . . . . . . . 134 48 Estimated Probabilities of Classification Under Format 19. . . . . . . . . . . . . . . . . . . . . 135 49 Estimation of Vectors for 82 and 83 Under Format 16. . . . . . . . . . . . . . . . . . . . . 138 50 Computation of Relative Weights of Predictor Variables of Category Two Under Format 16. . . . . . . . 139 51 Computation of Relative Weights of Predictor Variables of Category Three Under Format 16. . . . . . . 140 52 Ranking of gii Under Format 16 . . . . . . . . . . . . . 141 53 Ranking of Variables Under Format 16 . . . . . . . . . . 142 54 Classifications from Format 17 . . . . . . . . . . . . . 143 55 Estimated Probabilities of Classification Under Format 17. . . . . . . . . . . . . . . . . . . . . 144 56 Estimations of Vectors for 82 and 83 Under Format 17. . . . . . . . . . . . . . . . . . . . . 146 xi Table 57 58 59 60 61 62 Page Computation of Relative Weights of Predictor Variables of Category Two Under Format 17. . . . . . . . 147 Computation of Relative Weights of Predictor Variables of Category Three Under Format 17. . . . . . . 148 Ranking of E21 Under Format 17 . . . . . . . . . . . . . 149 Ranking of Variables Under Format 17 . . . . . . . . . . 150 Common Dollar Balance Sheets of Municipalities . . . . . 151 Common Dollar Operating Statement of Municipalities - 1975. . . . . . . . . . . . . . . . . . 152 Figure l 2 LIST OF FIGURES Single Entity versus Multiple Entity Accounting . Types of State and Local Government Funds and Account Groups. . . . . . . . . . . . . Typical Resource Flow Pattern—-Expendab1e Funds . Comparison of Curves. . . . . . . . . . . . . . Direct Causal Linkage . . . . . . . . . . . . . Indirect Causal Linkage . . . . . . . . . . . . xiii Page 17 30 30 47 59 60 CHAPTER I INTRODUCTION Statement of purpose . . . This dissertation is about an inquiry into construction of an accounting predictor of future municipal financial insolvency. Primary objectives of the study were: 1. That the prediction model to be established will identify municipalities of population in excess of 10,000 in the State of Michigan which are likely to be financially insolvent; and to predict the probability thereof. 2. That the model will set forth the characteristic differences between that group of municipalities which are likely to become insolvent and that group which are not likely to do so. 3. That, based on the above findings, inferences may be made and probabilities expressed regarding the future financial solvency of specific municipalities. 4. That a financial "profile of a healthy city" and a corresponding financial "profile of a weak city" will be generated. In addition, the accumulation of this body of data will be useful to both the Treasury Department and the Municipal Finance Com- mission in their monitoring of the fiscal health of Michigan's cities. Plans are being made to extend this study into one of national scope. Statement of the problem . For sundry reasons, numerous cities in the State of Michigan (and other states for that matter) have encountered seemingly insoluable 1 2 financial problems recently. Within a relatively short span of years, the cities of Ann Arbor, Ecorse, River Rouge and others have faced financial insolvency. In 1974 the City of Hamtramck was so short of cash it was unable to make pension payments to retirees; workers scheduled for retirement refused to leave their jobs because of fear of being without income. The fiscal problems of the City of Detroit have existed for some time and are now chronic. Currently, the plight is so bad that continuous services are now being funded with discon- tinuous revenues. The present dilemma could be amplified significantly in the future for it is predicted that Detroit's unfunded accrued pension liability will exceed the legal limit for bonded debtand that every dollar of tax revenue could be obligated to that liability.1 With increased severity of the situation in many cities has gone an increased awareness of that severity. In fact, ". it is now fashionable among American intellectuals to express tender concern for the city's future, to hope that its decay may be arrested, and to offer plans for its revitalization."2 Often such plans call for sig— nificant disruption of existing living patterns, reallocation of expenditures, or more likely, a heavy infusion of federal monies such as Detroit's Mayor Coleman Young's request for $2.6 billion.3 These and other proposed solutions could be described most aptly as "major lAfter revision of actuarial methods, it is now estimated that the unfunded accrued pension liability of the city amounts to $1,166,000,000; See: Financial Statements, City of Detroit, June 30, 1975. 2Morton White and Lucia White, The Intellectual Versus the City (Boston: The President and Fellows of Harvard College and the Massachusetts Institute of Technology, 1972), p. 13. 3Detroit Free Press (May 1, 1975), p. l. 3 corrective surgery"—-what in fact usually takes place is a "mercuro— chrome and band-aid" treatment. The most significant characteristic of such a palliative effort is that they are exgpost facto (which perhaps translates colloquially as "too little, too late"). The State of Michigan has recognized a responsibility to local units of government. That recognition is pronounced in the preamble . . . 4 to the Final Report Part 1, State Superv131on of Local Finances, pre- pared by the Sub-Committee on Fiscal Powers of Local Government which begins: While the state has the responsibility to provide local units of government with the necessary fiscal authority to finance essential local public services, the state has a concurrent responsibility to insure that local units main- tain sound and proper financial practices. As the state becomes increasingly involved in the financing of local units of government, state exercise of its responsibility to insure sound local government fiscal practices becomes even more important.5 The analysis performed in this research effort was an attempt to assess financial insolvency in a qualitative manner. It was desired that this be done in a timely and meaningful context. Here the word "timely" should connote that the assessment can occur before the incidence of financial insolvency; "meaningful" should connote that statistical meaningfulness is present, but more importantly, inferen— tial meaningfulness is present. 4Sub-Committee on Fiscal Powers of Local Government, Governor's Special Commission on Local Government, Final Report, Part 1, State Supervision of Local Finance (Lansing, Michigan: 1971). 5Ibid., p. l. 4 Present means of evaluating municipal performance and status . . . Currently, two means of evaluating the financial performance and status of a municipality are employed. The first method——of which there are unnumerable variations--could be described as an intuitive process. Such a method is utilized in the development of municipal bond ratings by agencies such as Standard & Poors or Moody's. The second method—-equally arbitrary but less subjective—-includes the exercise (in some states) of supervisory control analogous to receivorship. Bond ratings . . . Mr. James Marling, former Deputy Treasurer of the State of Michigan, detailed the development of a bond rating for a city or an issue previously unrated. He stated that the city would provide the following to the rating agency: Budgets, current and for two or three preceding years; Audited (or unaudited) financial statements for prior years; Information on the tax base for the past five years This would be broken down by industrial, commercial and residential classifications. It would include an array of the ten to fifteen largest taxpayers and copies of the 1960 and 1970 censuses. Present indebtedness-—bond issues and contracts outstanding; Brief 'public relations' type descritpions of the community discussing history, stability, etc.; Brief biographical sketch of local politicians, councilmen, and civic leaders; Details on anticipated bond issues discussing security, revenues, etc.; and, Overlapping debt. (Overlapping debt is that debt for which the residents of the city are obligated via another governmental body such as the county. The opposite is called underlying.) In addition there are discussions between the rating agent and the city manager, the municipal finance director, the auditors, etc. In 5 the case of a larger issue, the rating agent might visit the community. The questionnaire of one of the large rating agencies is shown as Appendix A. 4 The eventual rating is simply the sum of value judgments—— subjective considerations by learned people of what are deemed to be significant variables. Recommendations have been made to develop models which would employ quantifiable objective factors while still retaining the opportunity for expression of those subjective conclu- sions. "Under such an approach, ratings would clearly be a product of both objectively measurable and subjective impressionistic con— siderations."6 Other individuals with whom we have met have expressed prefer— ence for different indicators of a municipality's financial difficulty. Mr. William Carter of the Citizens' Research Council (Detroit) prefers a singular discrete measure: an unbalanced budget (i.e., forecast expenditures exceed forecast revenues).7 Carter believes that this antedates innumerable other measures in signalling financial problems. Likewise he summarized financial difficulty per se quite succinctly: a deficit cash position. Mr. Phillip Dearborn, formerly of the Advisory Commission on Intergovernmental Relations (ACIR) and now with D. C. Municipal 6John E. Petersen, The Rating Game (New York: The Twentieth Century Fund, Inc., 1974), p. 149. 7Carter cites one potential difficulty of using this criterion. Presently the State of Michigan does not require cities to budget, but budgets must be balanced. In order to avoid an unbalanced budget the City of Detroit simply did not submit a budget. Ergo, they did not violate the state law requiring that budgets be balanced. 6 Research Bureau, posed several alternatives which might indicate finan— cial difficulty: General operating fund deficit for two consecutive years; A ratio less than a certain percentage of cash to total assets; The ratio of floating (current) debt to total revenues; The ratio of short—term borrowing to property tax revenue; That ratio of interfund borrowing to total revenue; An accumulated deficit in excess of total revenue; The ratio of total taxes to total revenue; The ratio of total pension expenditure to total revenue; The ratio of unfunded pension liability to total revenue; The amount of delinquent taxes; and, The percentage of increase in total annual expenditures. Dearborn's ultimate test is comparable to that of Carter: " . . . the degree of likelihood that a government will have the cash available in the future to pay debt service commitments when due. The key element 8 by this definition is cash." Legal measures . . . The second method, used by some states in a regulatory manner, is dependent upon the occurrence of one or more specific conditions before the state imposes direct management. Review of the literature discloses that there is no single event or measure which the states have adopted as a universal measure for evaluating performance and status. Rather, there appears to be diversity in regulatory measures used, varying from some states in which numerous standards are used to other states which have no measures of financial performance or status of a municipality. An example of the latter is found in the State of Michigan 8Municipal Finance Officers' Association, Proceedings of the Municipal Credit Information and Credit Quality User/Research Seminar (Washington, D.C.: October, 1974), p. 3. 7 which has no legal criteria for determination of insolvency of a municipality. Rather, insolvency is determined through judicial pro— ceedings. As an example of the other extreme, the State of New Jersey assumes fiscal control of a city upon occurrence of one or more of the five following conditions: 1. Municipal default of debt principal or interest; 2. Over-due payments of taxes to state or other agencies; 3 A budget deficit for two years in excess of 5 percent of the tax levy; 4. Excessive floating debt, measured as a percent of budget; and, 5. Excessive tax delinquency measured as a percent of taxes levied.9 The State of Maine uses a tripartite measure: When a municipality becomes one year and six months in arrears in the payment of its taxes to the state in full or in part or defaults on any bond issue or payment of interest thereon or refuses or neglects to pay school or other salaries due . . Both of these examples might be described as weak laws for determining the point at which state control should be exercised in that a condi- tion of financial difficulty probably existed for some time prior to the occurrence of any of the criterion events cited. Consequently the problem is more complex and demanding than it should be. 9Advisory Commission on Intragovernmental Relations, Ci y Financial Emergencies (Washington, D.C.: Government Printing Office, 1973), p. 155. 101bid., p. 155. Other studies . . Bibliographic researchll disclosed no study similar in goal or technique to this effort, specifically, predicting future municipal financial insolvency from accounting data only. On the periphery to this study, there is a plethora of theoretical and empirical works. The literature regarding urban problems is not only vast but also longwithstanding; the recent fiscal crises in local units of government have given impetus to publishing in this topical area. It has occurred frequently as a topic in widely read publications such as Business Week and the New York Times. In my search for background for this study, due attention was devoted to the various levels of the related liter— ature . Relevant writings . . . Any attempt to develop and present an exhaustive listing of relevant works might be just that bo both researcher and reader-— exhausting. An extensive and detailed bibliography of related mate- rials is found in the Index of Economic Journals.12 Although neither as contemporary or comprehensive as the "Index,” an excellent guide to the specific area of Public Finance is found in Mitchell and Walter's State and Local Finance.l3 11The search began with employment of "Datrix II,” and the adjunct publications: Comprehensive Dissertation Index, Business and Economics, Supplements for 1973, 1974 and 1975 (Ann Arbor: Xerox University Microfilms), The "Datrix II" disclosed no similar disser- tations. 12Index of Economic Journals (Homewood, Ill., Richard D. Irwin, Inc.). 13William E. Mitchell and Ingo Walter (ed.), State and Local Finance (New York: The Ronald Press Company, 1970). 9 A logical beginning point for an investigation of municipal insolvency is with a general viewing of the public finance literature. Although these works do not constitute a reasonable description of the positive model, they do constitute a foundation of information. Musgrave and Musgrave14 provide an established text which analyzes the empirical and theoretical material of the public sector. Local govnermental units are not treated specifically, but rather throughout the text. Equally acceptable as an alternative is Herber's elementary text.15 Moving from introductory texts to those dealing specifically with urban problems, the work of Millsl6 must be recognized as out- standing. He provides a unique analysis of urban phenomena and problems. The short work is generally non-quantitative in its approach (Chapter 5 uses calculus), but perceptive in insight. Two readings books are stimulating enough to warrant mention. The first, edited by Schreiber et al,17 though somewhat out of date, presents the readings in such a way that they are complemented through their 14Richard A. Musgrave and Peggy B. Musgrave, Public Finance in Theory and Practice, 2nd ed. (New York: McGraw-Hill Book Company, 1976). 15Bernard P. Herber, Modern Public Finance: The Study of Public Sector Economics, 3rd ed. (Homewood, 111.: Richard D. Irwin, Inc., 1975). 16Edwin S. Mills, Urban Economics (Glenview, 111.: Scott, Foresman and Company, 1972). 17Arthur F. Schreiber, Paul K. Gatons, and Richard B. Chamber (ed.) Economics of Urban Problems Selected Readings (Boston: Houghton Mifflin Company, 1971). 10 . . . . . . l8 , interrelationships. Hochman s collection of readings includes some of the most provocative ranging from the highly illuminating "The Cost of Disease of the Personal Services and the Quality of Life" by Baumol and Oates to an excerpt from The Unheavenly City19 in which Banfield states that there is no solution to urban problems! Two works concentrating on the fiscal aspects of the urban economy are outstanding. The first, by Hirsch et a120 looks first at the broad question of fiscal health and then specific problems of commuters, nonwhites and overlapping governments. Unfortunately, the populations included in the statistical studies (cities) are not com- parable making conclusions less convincing than they might have been. The work of Greene et a121 is more comprehensive in its viewing of the aspects of the fiscal problems. However, the study dealt with only one city-—Washington D.C.--and therefore the validity of infer- ential generalizations are constrained by the reader's appraisal of the quality of the authors' measurements and logic. Innumerable other economic works propose innumerable explanations and solutions, but few authors approached the subject with the candor of Pettengill and Uppal.22 18Harold M. Hochman (ed.), The Urban Economy (New York: W. W. Norton & Company, Inc., 1976). 19Edward C. Banfield, The Unheavenly City (Boston: Little, Brown and Company, 1970). 20Ertnrt Z. Hirsch, Phillip E. Vincent, Henry S. Terrell, Donald C. Shoup and Arthur Rosett, Fiscal Pressures on the Central City (New York: Praeger Publishers, 1971). 21Kenneth V. Greene, William B. Neenan, Claudia D. Scott, Fiscal Interactions in a Metropolitan Area (Lexington, Mass.: D.C. Heath and Company, 1974). 22 Robert B. Pettengill and Jogindar S. Uppal, Can Cities Survive? The Fiscal Plight of American Cities (New York: St. Martin's Press, Inc., 1974). 11 The foregoing description and analysis show that the question should really be 'How Can Cities Survive?‘ That they do survive, that they have survived many fiscal crises is clear. But cities are basically people and people have their discontents. They are not satisfied with this or that aspect of their lives. One group of dissatisfactions centers around the political-economic-social-geographic entities we call cities. The urban situations that people don't like and want to improve range all the way from too high taxes to too few services. Some people deplore this aspect, and some deplore that. The dissatisfied people range from the powerful to the impotent, from the wealthy and the incumbents to the poor and the leaders of the party out of power. At any moment each sees the survival of his city somewhat differ— ently. Each seeks to improve a different facet of the situation. Struggling against opposition and inertia, each may wonder whether his city as he sees it can survive the apathy, the greed, the shortsightedness,the tax burdens, the neglect, the deprivation, the whatever, that are the particular objects of his despair. 80, too, the methods differ that people follow in trying to ensure the healthier survival of their city, healthier when measured by their value scales. A 'solution' from one point of view may be seen as a serious problem from another. One man's meat is another man's poison. Therefore no uni— versal prescription is possible. The authors have tried to show that situations differ as well as goals and that all benefits have cost. Each concerned citizen should be aware of the manifold alterna— tives and, weighing each carefully, push that one which seems best in the long run for himself, for his group, for his city as he sees it. Then in the give and take of the political process, the pulls and pressures of contending forces, our cities will move from one crises to another, continuing to survive while forever changing and being changed.23 The publications of the Advisory Commission on Intergovern- mental Relations have been well received. City Financial Emergencies: The Intergovernmental Dimension, mentioned earlier,24 is colloquially captioned "The Bible" by those involved with municipal fiscal prob— lems. Seventeen other works of ACIR are listed in the bibliography. 231bid., pp. 147—148. 24See Supra, fn. 9. 12 One important horizon of current research is the application of data processing to the budgetary process of the municipality. The viewing of the budget as a problem in resource allocation with solu- tions subject to a series of constraints was posed in a significant manner by Wildavsky.25 Since then there have been forecasting and budgeting efforts in Cleveland,26 Detroit,27 Los Angeles,28 Mobile,29 New Haven,30 Philadelphia,31 and Pittsburgh.32 These models are in part an attempt to avert financial difficulties through sound budgeting and forecasting. As additional capability is gained such efforts will surely become more successful. Then there should be greater employment of sophisticated techniques in the maximization of available resources. Further considerations . Certain extensions of the findings are properly excluded from consideration. These are by necessity, considering the data involved. 25Aaron Wildavsky, The Politics of the Budgetary Process (Boston: Little, Brown and Company, 1964). 26John P. Crecine, Governmental Problem-Solvigg (Chicago: Rand McNally & Company, 1969). 27Ibid. 28Hall and Licari, Journal of Regional Science, 1974. 29Semoon Chang, "Forecasting Revenues to Municipal Government: The Case of Mobile, Alabama," Governmental Finance ( February, 1976), pp. 16-20. 30Claudia Devita Scott, Forecasting Local Government Spending (Washington, D.C.: The Urban Institute), 1972. 31Glickman, Journal of Regional Science, 1971. 32Crecine, Problem-Solvigg. 13 The first, and perhaps most significant of these exclusions, is found in the objective33 in that the study refers only to cities in Michigan having population greater than 10,000. Therefore in absence of further research efforts generalizations to cities in other states could be justified only by the reader ”bridging" from this study to another population. Furthermore, certain cities (having populations greater than 10,000) were considered unacceptable for the project, in that data was missing or a different year—end was used. Missing from the study is any attempt to explain the variance in the amount of cash a city would hold. No investigation in the causal link has been investigated, as this effort was viewed in the eyes of the researcher as being the first approach to the problem of urban financial strain. Other variables to be considered in future studies might be concerned with the type of city government and management, the effect of political parties, the location of the municipality, its size, its sympathetic relationship to an adjunct,its sources of revenues, compo— sition of standing citizenry, base valuation of property, and other innumerable factors. In the interest of eventually bringing this study to a conclusion, these will be deferred to the future. 33See supra, p. l. CHAPTER II DATA—-SOURCE, NATURE AND COLLECTION Source of data . The data to be analyzed were drawn from the financial reports of municipalities in the State of Michigan. These municipal reports differ significantly from the more familiar financial statements of profit-seeking ventures or those of numerous other governmental units in that the form of the financial report is prescribed by law. In 1968 the State of Michigan enacted legislation1 which had as its pri- mary objective the providing of ". . . a means for the accumulation of financial information which will be uniform for all local units2 and of similar size."3 The uniformity decreed in Act 2 has three general aspects; statements are to be: 1. Prepared in accord with a designated chart of accounts4 1Act 2, Public Acts of 1968, State of Michigan. (This law is hereafter referred to as Act 2). 2In reality, the uniform accounting legislated in Act 2 covers all lower governmental units (e.g., counties, municipalities, drain districts, villages, etc.); for our purposes, the term "local unit" will be construed to mean ”municipality." 3Systems and Procedures Staff, Local Audit Division, Depart- ment of Treasury, Uniform Accounting and Procedures Manual for Local Units of Government (Lansing, Michigan: State of Michigan, 1975), p. l. 4Ibid., pp. F-l and F—26. l4 15 2. Audited by certified public acountants5 3. Submitted annually to the Local Audit Division of the State Treasury Department. Filings were to have begun with the fiscal year ending in 1968; how— ever, the fiscal year 1970 is generally regarded as the first year for which filings are complete. As of this writing reports have been filed forthuzyears. The study originally focused on the financial statements of 86 cities in the State of Michigan each having a popula— tion of over l0,000——an arbitrarily set lower limit.6 Some consideration was given to the possibility that (for research purposes) the time period of six years might be viewed as being insufficient. An alternative population was available; it con- sisted of those cities which had had audits for years prior to 1968. In such a case, the audited statement of that city would have been prepared on a basis (modified accrual, which is explained later) which is generally held to be comparable to the statement prepared in accord with the requirements of Act 2. The adversities of such a choice were clear. First, because less than half of the 86 cities had annual audits before 1968 by certified public accountants, such a choice would have resulted in a significant dimunition of the population. Even more important was a constraint of generalizability of findings. Conclusions could then 5The City of Detroit enjoys de jure relief from the filing of annual financial statements which have been audited by certified public accountants; unaudited statements are acceptable. This is because cities of over 1,000,000 population are required to file such statements only every fifth year. 6Appendix B lists those 86 cities. 16 be imputed only to those cities in the State of Michigan having popu- lation over 10,000 which previously prepared audited financial state- ments on the modified accrual basis. In addition to that applica- bility considerations, there was the practical aspect of laxity in filings and inaccessibility of those statements as they are already stored in archives. The conclusion was reached that the benefits to be gained--if any—~from the extra years were not worth the constraint upon the con— clusions. Although it is not directly related to an understanding of the accounting prescribed for municipalities in Act 2, the following discussion should lend itself to an understanding of the nature of the relevant data variables. Fund accounting . Although there are several similarities in the accounting of profit-seeking ventures and that of municipalities, there are also several differences. The most salient difference is that the accounting for the latter is organized around funds. One definition of a fund is: An independent fiscal and accounting entity with a self- balancing set of accounts recording cash and/or other resources together with all related liabilities, obli— gation reserves, and equities which are segregated for the purpose of carrying on specific activities or attain— ing certain objectives in accordance with specual regu— lations, restrictions or limitations.7 7National Council on Governmental Accounting, Governmental Accounting, Auditing and Financial Reporting (Chicago: Municipal Finance Officers Association, 1968), pp. 161-62. In subsequent citations this work will be referred to by its acronym, GAAFR. 17 The word "fund" should have a connotation greater than cash alone. A fund could consist of assets, liabilities and an equity balance. Those assets could be cash, receivables, inventories, fixed assets, and prepaid items at times; liabilities would consist of accounts and notes payable. In summary, each fund may be viewed as a separate self-con— tained reporting entity, with the total of all the funds seldom pre— sented in municipal reporting. In contrast, profit—seeking ventures seek to present the economy entity in full. This contrast between the accounting of the profit—seeking venture and the municipality is emphasized in the following figure: FIGURE 1* SINGLE ENTITY VERSUS MULTIPLE ENTITY ACCOUNTING FB Fund Balance (of the individual fund) The not-ior-proiit organization as a whole- for which statements are generally not prepared SINGLE ENTITY MULTIPLE ENTITY (PROFIT-SEEKING ENTERPRISE) (NOT-FOR-PROFIT ORGANIZATION) r _________________________ . i I I : FUNDI FUND2 FUND3 I : A=L+FB A=L+FB A=L+FBI I I I I I I I I I I I FUND 4 FUND 5 FUND n = I I A 'INW : A=L+FB A=L+FB A=L+FB: I I I I I I : FIXED LONG-TERM I . ASSETS DEBT I : (Original (Principal : I Cost) Owed) l L _________________________ .1 Legend: A = Assets I. = Liabilities NW = Net Worth (of the enterprise) *After Lynn and Freeman, Fund Accounting, p. 9. 18 The commercial enterprise will have a single unified set of accounting records which will summarize all of the financial trans- actions while the city will have a set of self-balancing accounts for gggh fund that has been set up. Each fund will have its own budget; usually the financial statements of that fund will display with particular emphasis the comparison between budgeted and actual revenues and expenses. General types of funds . The two general types of funds reflect one way of classifying the types of municipal operations. The primary criterion for this categorization is whether the ". . . resources of the fund may be expended or are to be maintained on a self—sustaining basis.”8 These two types of funds are frequently captioned "Expendable" and "Non— Expendable." The former would be utilized in accounting for recurring operations which supply basic services to the general populace. Such a fund is usually under stringent budget control. Resources are expendable. The budget is prepared under the assumption that resources will be replenished and expended each year. The latter covers those self-sustaining functions of a municipality which are operated as an entity on a basis similar to that of a commercial enterprise. In such a case, resources of the fund are not expendable. 8Edward S. Lynn and Robert Freeman, Fund Accounting Theory and Practice (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1974), p. 32. 19 Specific types of funds . The National Council on Government Accounting recommends eight different types of funds: (1) (2) (3) (4) (5) (6) (7) (8) The General Fund to account for all financial .\ transactions not properly accounted for in another fund;9 Special Revenue Funds to account for the proceeds of specific revenue sources (other than special assessments) or to finance specific activities required by law or administrative regulation; Debt Service Funds to account for the payment of interest and principal on long term debt other than special assessment and revenue bonds; Capital Projects Funds to account for the receipt and disbursement of moneys used for the acquisi— tion of capital facilities other than those financed by special assessment and enterprise funds; Enterprise Funds to account for the financing of services to the general public where all or most of the costs involved are paid for in the form of charges by users of such services; Trust and Agency Funds to account for assets held by a governmental unit as trustee or agent for individuals, private organizations and other governmental units; Intragovernmental Servicg Funds to account for the financing of special activities and services performed by a designated organization unit within a governmental jurisdiction for other organization units within the same governmental jurisdiction; Special Assessment Funds to account for special assessments levied to finance public improvements or services deemed to benefit the properties against which the assessments are levied.10 9Such a negative, global definition lacks informational content. The General Fund will be explained in a positive manner on page 10National Council, GAAFR, pp. 161-62. (?) 20 Those funds which are generally considered as Expendable Funds are 1,2,3,4, and 8; those funds which would be classified as Non-Expend— able Funds would be the Enterprise (5) and Intragovernmental Services (7) Funds. Trust and Agency Funds (6) is often classified as either expendable or non-expendable depending on the intended use of resources; in the State of Michigan, it is expendable. As a further elaboration on the particulars of these funds, it should be noted that there is only one General Fund but there can be many of the seven other types of funds. Each fund would have its own accounts necessary to reflect the transactions, assets, liabilities, revenues and expenditures. Each fund would be self—balancing. Capital assets . . . A further distinction between the accounting used by munici— palities and commercial enterprises exists in regard to the accounting for expenditures for capital assets by municipalities. Capital assets of the expendable funds are not recorded in those funds, but rather in an account entitled "General Fixed Assets."11 The accounting treatment for expenditures for capital assets to be used in the operations of the Enterprise Funds and Intragovern- mental Service Funds is similar to the treatment of such expenditures in the accounting records of the commercial enterprise in that they are capitalized at historical cost. Depreciation--using the straight line method—~is recorded on such assets by municipalities. 11Therefore, there exists a third type of accounts. These are called the "Non-Fund Group of Accounts" and two primary groups are required in Michigan statements. They are the General Fixed Assets and General Long Term Debt. Although they are self—balancing they function more as a schedule than as an account. 21 Basis of accounting . . . The third and final difference, not as significant as the other two, is found in the basis of accounting for certain of the funds of a municipality. Generally speaking, the accounts of the Enterprise Funds, Intragovernmental Service Funds, Special Assessment Funds and the Trust and Agency Funds are maintained and reported on an accrual basis comparable to the basis employed by most commercial enterprises. The difference is found not in those accounts but rather in the General Fund, the Special Revenue Funds, and the Debt Service Funds which are maintained and reported on the Modified Accrual Basis. Modified accrual basis . . . Under the Modified Accrual Basis of accounting those revenues which are measurable and available are accrued because they are resources which may then be appropriated. Those revenues not suscep- tible to accurate estimation are recorded when they are received. Revenues from fees for services and income tax would be two examples of revenues which do not meet a test of reasonable certainty. Expenditures under the Modified Accrual Basis are recorded when the goods or services are received excepting for interest on general obligation long—term debt which is accrued. An additional modification to the Madified Accrual Basis of accounting can be made in the form of an encumbrance; this will be explained in the para- graph on the format of the Act 2 statements. Having cited the dif- ferences between the accounting system for a municipality and that for a commercial enterprise, we present a summary view of how the 22 accounting system is related to the general operations of the munici- pality. We shall also explain the general transactions to be found in the various funds. The municipal accounting system . . . This overview of the typical municipal accounting system follows the order given earlier in the recommendations of the National Council on Governmental Accounting. The first of the funds is the General Fund, the prime focus of this research effort. In this dis- cussion it should be remembered that the General Fund is singular in number and that it is expendable, that is, the resources are expended and replenished annually. In this category the general fund are funds established to account for resources devoted to financing the general services which the governmental unit performs for its citizens. These include general administration, protec- tion of life and property, sanitation and similar broad services.12 In the State of Michigan, the typical sources of revenue for the General Fund are principally the real and personal property tax, state shared revenues and to a lesser degree, fees for services rendered by various departments. State and Federal grants are often present. In some municipalities another major source of revenue has been attained through the levying of a personal income tax. Expenditures of the General Fund are primarily concerned with the operation of the municipality: administration, police, fire, parks and recreation, public works and several minor activities. The 12R. M. Mikesell and Leon E. Hay, Governmental Accounting (Homewood, Ill.: Richard D. Irwin, Inc., 1969), p. 4. 23 General Fund also functions as a clearing account for the distribu— tion of revenues (e.g. property taxes) to other funds or governmental units. Special Revenue Funds . . . Revenues of a Special Revenue Fund are similar to those of the General Fund except that they are self—imposed for specific pur— poses or so to speak are "earmarked." The expenditures for this type of fund are for specific purposes required by law or contract and are of a continuing nature, usually. Titles of some special revenue funds which can be established by municipalities will aid an understanding of this type of fund: Cemetery Fund Ambulance Fund Drain Fund Street Lighting Fund Parking Meter Fund Comprehensive Employment and Training Act Fund 13 If any capital assets are acquired, the expenditure from this fund is treated as an expenditure for any other expense, such as that for wages or materials; the capital asset acquired would be carried as the asset of some other entity within the governmental unit or would be shown in the non-fund account, General Fixed Assets. Debt Service Fund . . . This fund is rigidly controlled; it is used ". . . to account for the payment of interest and principal on long term debt other than . l4 . spec1al assessment and revenue bonds." Furthermore, the Debt Serv1ce 13Local Audit Division, Uniform Accounting Manual, pp. 5—39. 14National Council, GAAFR, p. 161. 24 Fund ". . . is necessary to maintain the separate identity and charac- ter of general debt operations by governmental units and to permit the proper disclosure thereof in financial statements and reports."15 The expenditures of this fund are for principal, interest and any service charges adjunct to the general long—term debt. Revenues are derived from numerous sources related to the General Fund revenues, typically property tax. In the case of a refunding of an outstanding issue with another issue, the Debt Service Fund would be used as a clearing account for the transaction. Capital Projects Funds . . . The revenues16 of the Capital Projects Funds come from trans— fers from the General Fund, other governmental units (such as county, state or federal) and the sale of certain bonds; in the case of bonds, the proceeds would be designated for a capital project and the liability would subsequently be transferred to the Debt Service Funds. The Capital Projects Funds account for the purchase of new facilities and equipment or the undertaking of such projects as street paving, building additions or improvements, etc. It does not include the acquisition of capital facilities financed by a Special Assessment or Enterprise Fund. As examples, the State of Michigan provides the following fund titles: lSIbid., p. 37. 6Some accountants question the use of the term "Revenues" in connection with Capital Project Funds. Such "revenues" are inflows to the unit as a whole but rather transfers and appropriations from other funds. See: Lynn and Freeman, Fund Accountipg, p. 282. 25 Hospital Building Fund Industrial Complex Construction Fund Animal Shelter Construction Fund Mental Health Construction Fund Library Building Construction Fund Medical Care Facility Building Fund Park System Construction Fund MVH Act 175 Major Street Construction Fund MVH Act 175 Local Street Construction Fund Airport Fund Sewage Disposal Fund17 In addition to these titles listed above (in order of their number in the prescribed chart of accounts) other titles could be set up as needed in the "open" numbers provided. For each major project there would be a fund and the accounts in that fund would be used in the recording of expenditures related to that project and the assets retained. The individual funds are presented on a summary statement, classified as to whether they are completed or not. The expenditures would typically consist of land, building materials, labor and other related costs for the project for which the fund was established. Such expenditures could even include indirect costs if the municipality is acting as general contractor; overhead is usually not included because of some rather notorious instances of fraud in the past; but if it is included it will be for well-defined items. The fund could be abolished upon completion of the project or it might be of an on—going nature such as street paving. Enterprise Funds . . . The charges to the users for the operations and sale of products or services are the principal revenues of an Enterprise Fund. l7Local Audit Division, Uniform Accounting Manual, pp. 59-75. 26 A familiar example of such a fund would be a municipally owned and operated electric power company or a water system. The City of Detroit's water system has both bulk and individual customers and revenues would be so classified on the Statement of Revenues and Expenditures. It is to the revenues that one looks for determination of whether a fund should be classified as an Enterprise Fund or not: . . .If a substantial amount of the revenues used to finance an activity or series of related activities in a single fund is derived from user charges, the fund can be appropriately classified and accounted for as an Enterprise Fund.18 Expenditures of an Enterprise Fund are quite comparable to those of a profit-seeking venture. Included are all of the expenses of producing the product or rendering the service. Contrary to the traditional not-for-profit accounting, depreciation may be recognized as an expense by an Enterprise Fund. From time to time some munici- palities have an Enterprise Fund make remittances to the General Fund. This action is another departure from the usual governmental accounting, for these distributions are treated in a manner analogous to the divi— dends of a commercial enterprise to its shareholders. Some of the titles of Enterprise Funds could be: Abstract Fund Ambulance Fund Mobile Home Park System Fund Markets Fund Fair Board Fund Airport Fund Golf Course Fund Auto Ferry Fund Civic Auditorium Fund Marina Fund 18National Council, GAAFR, p. 50. 19Local Audit Division, Uniform Accounting Manual, pp. 77-107. 27 Trust and Agency Funds . Trust and Agency Funds are used for moneys which are being held by the municipality (as a trustee or agent) to be distributed later. Because of the custodial nature of such funds, assets will exactly equal liabilities and there will be no equity balance. Revenues are supplied by tax collections, payroll deductions, transfers and shared revenues from other governmental units. Expenditures are in accord with the prescribed purpose of the particular fund. Some of the titles suggested by the State of Michigan illus- trate uses of Trust and Agency Funds: Emergency Employment Act Program Cemetery Trust Fund Employees Death Benefit Fund Employees Sick Pay Fund Police and Fire Retirement System Fund Urban Renewal Escrow Fund20 Trust and Agency Funds in the State of Michigan are expendable; this type of fund is explained in the subsequent section. Expendable/nonexpendable Funds . In theory, the classification of a fund as expendable or non— expendable is determined by whether the resources of that fund are expendable or not. The constraints (or lack of constraints) upon con— sumption of those resources could emanate from law, contract or perhaps action of an administrative or regulatory body. While most funds are expendable, those which are nonexpendable might be constrained in total or in part (e.g., being able to spend only the interest from certain investments). By nature then, it follows that an expendable fund 201bid., pp. 145—66. 28 would hold only "liquid" assets-—those which may be expended——con- sisting of cash, receivables and short—term investments (and perhaps limited supplies); assets which would not be held would include long— term investments, property, plant and equipment. By definition, one would conclude that assets such as these are not readily expendable. The "other side of the coin" is then found in the nature of the capital of the expendable fund. It would be of a non—permanent nature and, in general, would rise and fall in sympathy to the assets of that fund. In recognition of the fact that resources could be dis- sipated, expendable funds come under budgetary control. This element of budgetary control, stated simply, is another of the characteristics of the expendable fund. Although some non-expendable funds might also be subject to budgetary control, most expendable funds are annually budgeted. Finally, it remains true only in theory (and seldom in practice) that the resources of the expendable fund should be expended completely within a twelve month period. Intragovernmental Service Funds . . . The Intragovernmental Service Funds are: . . used to account for the financing of special activities and services performed by a designated activity or department within a governmental jurisdiction for other units or depart— ments within the same governmental jurisdiction.21 These funds are operated as self-supporting enterprises. They derive their revenues from billing other departments for the services to those departments. Expenditures might include salaries and wages, interest on long-term debt, supplies, administrative fees, etc. Here 21National Council, GAAFR, p. 162. 29 again, depreciation is recorded as an expense. Gains or losses on sale of equipment are recognized. These funds do have equity balances. Both fixed assets and long—term liabilities are recognized on the balance sheet of an Intragovernmental Service Fund. Some of the typical services for which intragovernmental charges could be made would be building and grounds maintenance, data processing, mailing, radio com— munications, vehicles, office equipment, electricity, gas and water. Special Assessment Funds . . . The final type of fund is the Special Assessment Funds. These are used to account for construction or improvement projects which benefit only a particular group of real property owners rather than the general populace. Examples of such projects would be streets, roads, sidewalks, lighting, sewers and watermains. The owners who benefitted from the improvements are charged pro-rata shares of the cost of the improvements. These funds are a hybrid of two other funds, Capital Projects and Debt Service Funds. Special Assessment Funds combine the functions of both but only as related to special assessments. Revenues are from General Fund appropriations, sale of bonds, special assessments against the property owners who benefitted from the improvement, interest received from investing excess cash, and in some cases, from sale of delinquent special assessment receivables. Expenditures, in general, are for payment of contracts, bond principal and interest, and any direct and/or indirect costs relating to the construction of the project. The eight types of funds are summarized in Figure 2 on page 30 and Resource Flows are illustrated in Figure 3, page 30. 30 FIGURE 2** TYPES OI" STATE AND LOCAL GOVERNMENT FUNDS AND ACCOUNT GROUPS (NCGA RECOMMENDATIONS) I NONEXPENDABLE EXPENDABLE (GovernmenseI-TVPOI (Commercial-Type) f I I SPECIAL GENERAL ASSESSMENT AGENCY ENTERPRISE ‘\“ ’I/V DEBT \‘ ” FU-—N°-S ‘r’ SERVICE INTRA- SPECIAL ' CAPITAL ' TRUST I I GOVERNMENTAL REVENUE : PROJECTS : (Some) SERVICE I— j\‘ : ’I' : ‘s I 1’ I ‘\ | I, . ~¥l ' l l t I I I I I muss : : (Sam) I I I I I I I I I I I I NONFUNO GENERAL LEGEND ACCOUNT FISEEin'SSETS LONG-TERM [— - A gavernment hos Only one. GROUPS 05" - A government may have one, f'f none, or many - as required. FIGURE 3** TYPICAL RESOURCE FLOW PATTERN—EXPENDABLE FUNDS“ Proceeds from Property Taxes, - Except Enterprises, etc. Licenses, Fines, Charges for Special Service, Income Taxes, Other Revenues Proceeds of Long-Term Debt Issuances Contributions to and Earnings of Pension, Disability and Similar Trusts, Bail Bonds, etc. Collections for Other Governments, Persons, or Organizations I I T L—T F°I New FOI' RCIUN“ Assessments Levied against F“ Proiects 5"9 P9030595 Specific Properties Bene- General fited by Special Praiects Purposes [—Far Specified Pbrposes J °' 5°""°‘ I I Current Capital Debt Operations ‘ Outlay Service Th .I IL I [I [r1 e SPECIAL CAPITAL DEBT SPECIAL TRUST AGENCY “Paul REVENUE PROJECTS SERVICE ASSESSMENT Funds Fund‘ ' und Funds Funds Funds ' Funds (Expendable) Provision of General Government Acquisition of Payment of Manning Payment of Transmittal Services-Police and Fire Pro- General Fix“ Assets Interest and Principal Pentioni: to Others tection, Recreation. Sanitation, of Long-Term General Refund of Administration, Inspection, Zoning, Obligation Debt Bail Bonds; etc. etc. ' Flaws to and from nonexpendable funds are excluded. I General Long-term Debt may be serviced directly from other expendable funds. As indicated by ----, resources may be transferred to the Debt Service Fund from other funds; also, **After Lynn and Freeman, Fund Accounting, pp. 32—33. 31 Format of Act 2 financial statements . The minimum financial statements and schedules to be furnished by each city (regardless of size) to the state in accord with the for- mat of uniform financial reporting under Act 2 are: GENERAL FUND (With or Without Encumbrances) Balance Sheet Analysis of Changes in Fund Balance Statement of Revenues—-Estimated and Actual Statement of Appropriations and Expenditures MAJOR STREET FUND (Without Encumbrances) Balance Sheet Analysis of Changes in Fund Balance Statement of Revenues--Estimated and Actual Statement of Appropriations and Expenditures DEBT SERVICE FUND Balance Sheet Statement of Revenues, Expenditures and Fund Balance CAPITAL PROJECTS FUND Balance Sheet Analysis of Changes in Fund Balance WATER AND SEWER FUND Balance Sheet Statement of Retained Earnings Statement of Income Analysis of Income Available for Debt Retirement Schedule of Operating Statistics INTRAGOVERNMENTAL SERVICE FUND Balance Sheet Analysis of Changes in Contributions Statement of Retained Earnings Statement of Operations EMPLOYEES RETIREMENT SYSTEM Balance Sheet Analysis of Changes in Reserves TRUST AND AGENCY FUNDS Balance Sheet Statement of Revenues, Expenditures and Fund Balance GENERAL FIXED ASSET GROUP OF ACCOUNTS Schedule of Changes in Fixed Assets 32 LONG-TERM DEBT GROUP OF ACCOUNTS Statement of General and Special Revenue Long-Term Debt Schedule of Indebtedness22 Other statements, schedules and statistics are frequently furnished. An encumbrance is defined as Obligations in the form of purchase orders, contracts or salary commitments which are chargeable to an appropriation and for which a part of the appropriation is reserved. They cease to be an encumbrance when paid or when the actual liability is set up.23 The choice then is between mutually exclusive alternatives, the General Fund Without Encumbrances or the General Fund With Encumbrances. In the opinion of Mr. James Marling, former Deputy Treasurer of the State of Michigan, the difference between these two types of General Fund is not significant enough to make data from the two systems non-compar- able; therefore, we include cities having both reporting formats. The General Fund . . . Additional comments regarding the General Fund are in order, particularly because it is the focus of our attention. The importance of this fund is found in the fact that the accounting for most of the current activities and operations of a municipality are contained within this fund. The city's main budget is prepared and administered through this fund. The important economic factors and the financial status of a municipality should be reflected in the revenues of this fund. The level of the choices of the various expenditures should not only indicate the needs of the city, but also to a degree, its 22Department of Treasury, Uniform Reporting Format for Finan- cial Statements for Local Government Units in Michigan (Lansing, Mich.: State of Michigan, 1971), pp. 1-2. 23Lynn and Freeman, Fund Accounting, p. 985. 33 desires. In addition to being encompassing, the revenues and expendi- tures are also recurring on a regular basis. Where some funds could span several years or a period as short as a month, the General Fund is distinctly related to a single year. It is this fund from which the majority of the observations are drawn. Observations . . The following data points were to be gathered for each city for each year: Cash and Certificates of Deposit in the General Fund Unpaid delinquent taxes receivable Interfund borrowing due to the General Fund Total of the General Fund Total current liabilities of the General Fund (often captioned "Floating Debt") not including encumbrances or bonded debt Unencumbered, unappropriated balance (equity) of the General Fund Total revenues applicable to the General Fund Revenues from current property taxes applicable to the General Fund Revenues from local income tax applicable to the General Fund Revenues from. state shared revenues applicable to the General Fund Revenue from Federal Revenue Sharing Fund included in the total General Fund revenue Total expenditures applicable to the General Fund Administrative expenditures applicable to the General Fund Police expenditures applicable to the General Fund Fire expenditures applicable to the General Fund Park and recreation expenditures applicable to the General Fund General obligation bonds outstanding Unfunded accrued pension liability Cash and Certificates of Deposit in Federal Revenue Sharing Fund Total of Federal Revenue Sharing Fund Capital Expenditures of Federal Revenue Sharing Fund Other expenditures of Federal Revenue Sharing Fund In general, most observations were readily available. In some instances, however, items were missing. For example, the City of 34 Farmington combines police and fire protection costs in a single figure captioned "Public Safety." The figure most frequently missed was that of unfunded accrued pension liability. Where figures were missing either that data point was then omitted from the analysis (as in the case of the unfunded accrued pension liability) or the par— ticular city was considered ineligible (as in the case of Farmington). Another cause of disqualification of a particular municipality from study was that of having a year—end other than June 30, the typical fiscal year-end for not-for-profit organizations, and espe- cially governmental units.24 Such cities have been marked with an asterisk in Appendix B. Revenues and expenditures would have been comparable regardless of the year-end. The difference occurs with the balance sheet observations--particularly cash. The primary revenue of most municipalities is the property tax; this flows in at a specific time unlike other revenues. Nature of the observed variables . . In this section we intend to comment regarding two aspects of the data points. First, we shall view the accounting content of some of these variables and then we shall comment on the predictive ability hypothesized for some of the variables at the time of selection. The first is Cash and Certificates of Deposit in the General Fund. This observation serves as both criterion and predictor vari- able, depending upon the year—end balance used. As discussed in Chapter I, the balance of cash at the end of the most recent year will 24Currently there is a movement underway for legislation that would standardize all municipalities to the same year—end, June 30. 35 serve as a surrogate for liquidity (or its counterpart, financial insolvency). The cash balance of the other funds was not considered relevant because it was not readily available for satisfying opera- tions or liabilities of the General Fund. The Unpaid Delinquent Property Taxes Receivable are for both real and personal property taxes from prior years which remain unpaid. They are traditionally shown net of an allowance for uncollectibles. This treatment is comparable to that found in the accounting of most commercial enterprises. In a few instances, the balance was totally reserved. Upon the advice of Mr. James Bolthouse, then Deputy Treasurer of the State of Michigan, the balance of the outstanding delinquent taxes was added back, increasing that account, and the total of the General Fund, and the Unencumbered, Unappropriated (equity) balance of the General Fund. It was postulated that this observation would serve as an excellent predictor of insolvency, reflecting perhaps unemployment, a declining average income, abandoned property, etc. The next variable, Interfund Borrowing Due to the General Fund, should be considered relative to the account Interfund Borrowing Due From the General Fund. These two accounts are considered to be so relevant that if a consolidated balance sheet for all the funds of the city is prepared, these balances cannot be eliminated in consolidation but (by law) must be shown. The former account, an asset of the General Fund, arises from the lending of moneys by the General Fund to other funds, the rendering of services, moneys due to be remitted by some other fund to the General Fund, etc. The latter account, of course, would arise from an opposite transaction taking place. In particular though, there can be borrowings of cash from 36 other funds if the revenues of the General Fund are insufficient to keep the city solvent. Such borrowings are generally held to be indicative of overwhelming financial problems which might be accom— panied by deceptive budgeting and possibly poor fiscal management. An example of insufficient revenues is found in the case of the City of Royal Oak. The financial statements for the year 1973 were held up for over twelve months while the auditors deliberated with the Local Audit Division of the State Treasury Department over the disposition of long—outstanding interfund borrowings by the General Fund from the Water System Fund. The one million dollars had been outstanding for several years. The State maintained that repayment was not probable and therefore taxes were imposed to repay the amount within a reasonably short time. The unwritten rule of thumb among financial managers was found to be that neither asset nor liability should be outstanding for a long period of time nor should the liability exceed the asset by a significant amount. It was assumed that the presence of revenues from a local income tax applicable to the General Fund would be an excellent indi- cator of financial problems. If a municipality levied such a tax, they had then used up a significant portion of their ”revenue capacity." A similar reasoning was applied in the case of revenues from the Federal Revenues Sharing Fund included in the revenue of the General Fund. It was presumed that this was a use of non-continuous revenues (assuming that Federal Revenue Sharing had a stated expira— tion and might not be renewed) to meet continuing expenses. As a corollary to that, it was concluded on an intuitive basis that the more solvent municipality could use its Federal Revenue Sharing Funds 37 for capital expenditures rather than traditional expenses. 0f the four expenditures considered all were presumed to be strong indicators of financial solvency. It was hypothesized that as a municipality became financially troubled, it would be less effi- cient and that administrative costs would rise relative to total expenditures. It was assumed that the need for police expenditures could be highly related to financial difficulties. The fire expendi- tures were thought to be reflective of the decline in the value of the property, property having poor wiring, being used beyond its capacity--that is, because of deterioration of the community. The fourth expenditure, parks and recreation, was considered to be related to financial insolvency in a negative manner. That is that the municipality with insufficient cash would reduce its park and recre— ation expenditures. Data gathering . . . Data were gathered directly from the annual financial state— ments of each municipality. Data were recorded on the ten digit, 124 field form shown as Appendix C. To minimize errors and employ consistent interpretation the data were gathered only by this researcher. Upon receipt of the statement, a review was made of the auditor's opinion and relevant footnotes. Scoresheets were then marked, optically scanned and cards produced. Cards were occasion— ally verified to statements on a rather random basis. CHAPTER III STATISTICAL METHODOLOGY Dichotomous situation--a basis for the design. Much of the statistical analysis in business is concerned with observations that take one of two possible mutually exclusive out- comes such as: A borrower pays his loan or defaults A purchase order is deemed to be properly prepared or is not The balance of an account receivable is confirmed by the debtor or is not An interviewee is hired or is not A part ordered from a supplier meets specifications or it is found to be defective. At times these "two outcome" situations and the subsequent observations are described as "zero-one" with the zero (0) utilized to record a failure or a "miss" while the one (1) represents a success or a "hit." An equally descriptive caption is "all or nothing at all." Such observations are also called "binary." Throughout the field of biological sciences (which is respon— sible for much of the methodology employed in analysis of binary data) an older and somewhat more obscure term, quantal, is frequently used to describe these mutually exclusive observations. This term emanates from the dosal or quantal response to a measured dose in the controlled treatment experiment. When such a dose is administered to a subject a binary outcome can be the result--alive or dead, cured or still ill, 38 39 etc. This term, quantal, now connotes a meaning beyond merely binary observation in that methods under this label--such as the one utilized in this research-~can be applied to the polychotomous outcome. Discriminant Analysis . . . The statistical design originally proposed for this study was to utilize two methods of multivariate analysis, factor analysis and Discriminant Analysis (DA). Although it is an interruption at this point, comments regarding DA are enlightening to the discussion in view of the large amount of research which is allied to this project which has employed DA. Discriminant Analysis can be used either in a descriptive or a predictive manner. In the latter situation, ". . . we seek linear combinations of a set of variables that best differentiate among several (two or more) groups."1 Specifically in view of the data at hand, DA could be used to find linear combinations of those sundry predictor variables which would show significant differences between those municipalities which will experience financial difficulties and those which will not experience financial difficulties with the smallest possible proportion of misclassification. Since first developed by R. A. Fisher, this technique, which was originally applied in the biological sciences, has seen use in numerous areas. Specifically, in the analysis of problems related to 1Maurice M. Tatsuoka, Multivariate Analysis (New York: John Wiley & Sons, Inc., 1971), p. 5. 40 finance and business, DA was first used by Durand2 in 1941 to dif- ferentiate between "good" and "bad" consumer loan applications, by Walter3 in 1959 in classifying firms into high and low price—earnings ratio groups, in 1963 as one of several methods in developing a numer- ical credit evaluation system by Myers and Forgy,4 by Smith5 in 1965 in classifying firms into standard investment categories, in 1968 by Altman6 in predicting corporate bankruptcy and again in 19717 and by Pinches and Mingo8 in 1973 in evaluating industrial bond ratings. The 1968 study of Altman was considered a "landmark" article in the field of finance, but criticisms were pronounced in that the data suffered from the malady of non-homogeneity. Altman accepted the criticisms as valid and corrected the deficiency by employing data 2D. D. Durand, "Risk Elements in Consumer Installment Financing," Studies in Consumer Installment Financing (New York: National Bureau of Economic Research, 1941), pp. 105-42. 3J. E. Walter, "A Discriminate Function for Earnings Price Ratios of Large Industrial Corporations," Review of Economic and Statistics, Vol. XLI (February, 1959), pp. 44-52. 4H. Myers and E. W. Forgy, "Development of Numerical Credit Evaluation Systems," Journal of American Statistical Association, Vol. 50 (September, 1963), pp. 797-806. 5K. V. Smith, Classification of Investment Securities Using .MQA, Institute Paper #101 (Lafayette, Indiana: Purdue University, Institute for Research in the Behavioral, Economic and Management Sciences, 1965). 6Edward Altman, "Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy," Journal of Finance, Vol. XXIII (September, 1968), pp. 589-609. 7Edward Altman, "Railroad Bankruptcy Propensity," Journal of Finance, Vol. XXVI (May, 1971), pp. 333-45. 8George E. Pinches and Kent A. Mingo, "A Multivariate Analysis of Industrial Bond Ratings," Journal of Finance, Vol. XXVII (March, 1973), PP. 1-15 41 related to only one industry, railroads. It is to that earlier study that we look to find an acceptable and easily understood explanation of Multiple Discriminant Analysis (MDA): MDA is a statistical technique used to classify an obser- vation into one of several a priori groupings dependent upon the observation's individual characteristics. It is used primarily to classify and/or make predictions in problems where the dependent variable appears in a qualitative form, e.g., male or female, bankrupt or non-bankrupt. Therefore the first step is to establish explicit group classifications. The number of original groups can be two or more. After the groups are established, data are collected for the objects in the groups; MDA then attempts to derive a linear combination of these characteristics which "best" discriminates between the groups. If a particular object, for instance a corporation, has characteristics ( financial ratios) which can be quantified for all of the companies in the analysis, the MDA dEEEFmifiés a set of discriminant coefficients. When these coefficients are applied to the actual ratio, a basis for classifications into one of the mutually exclusive groupings exists. The MDA technique has the advantage of considering an entire profile of character— istics common to the relevant firms, as well as the inter- action of these properties. A univariate study, on the other hand, can consider the measurements used for group assignment only one at a time. Another advantage of MDA is the reduction of the analyst's space dimensionality, i.e., from the number of different independent variables to g — 1 dimension(s), where g equals the number of original a priori groups. This paper is con- cerned with two groups, consisting of bankrupt firms on the one hand, and of non—bankrupt firms on the other. There— fore, the analysis is transformed into its smallest form: one dimension. The discriminant function of the form Z = v x + v x + . . . + v x transforms individual variables into a single discrimingnE score or Z value which is then used to classify the object where v1, v2, . . . , vn = discriminant coefficients and x , x , . . ., xn = independent variables. The MDA computes the discriminant coefficients, v. while the independent variables, x., are the actual values w ere j = 1, 2, . . . , n. J When utilizing a comprehensive list of financial ratios in assessing a firm's bankruptcy potential there is reason to believe that some of the measurements will have a high degree of correlation or collinearity with each other. While this aspect necessitates careful selection of predic- tive variables (ratios), it also has the advantage of yielding a model with a relatively small number of selected measure- ments which has the potential of conveying a great deal of information. This information might very well indicate 42 differences between groups but whether or not these differ- ences are significant and meaningful is a more important aspect of the analysis. To be sure, there are differences between bankrupt firms and healthy ones; but are these dif— ferences of a magnitude to facilitate the development of an accurate prediction model? Perhaps the primary advantage of MDA in dealing with classification problems is the potential of analyzing the entire profile of the object simultaneously rather than sequentially examining its individual characteristics. Just as linear and integer programming have improVed upon traditional techniques in capital budgeting the MDA approach to traditional ratio analysis has the potential to reformu- late the problem correctly. Specifically, combinations of ratios can be analyzed together in order to remove the possible ambiguities and misclassifications observed in earlier traditional studies.9 Then the advantages of MDA over other methods are clear; all characteristics common to the subjects can be considered concommi— tantly while at the same time viewing interaction thereby avoiding redundancy and reducing the researcher's scope of view. Careful selection of variables can eliminate those which are highly correlated; this means that the researcher's view is reduced even more. What Altman does fail to discuss is the prime inherent deficiency of MDA. That principal shortcoming is the need for rela- tively equal size classificatory groups (e.g., the number of bankrupt firms, in Altman's study should have qualled the number of non-bank- rupt firms). Morrison10 states: In summary, when one group is much larger than the other, almost all individuals are classified as the larger group. This means several will automatically be correctly classified. When we allow the posterior odds to classify the individuals--see 5——we usually get even fewer classified in the smaller group than actually belong to it. There is 9Altman, "Corporate Bankruptcy," pp. 591-93. 10Donald G. Marrison, "On the Interpretation of Discriminant Analysis," Journal of Marketing Research (May, 1969), pp. 156-63. 43 often more interest in the smaller group and classifica— tion tables like the preceding two are not the best way to assess the discrimination power of the independent variables.11 There does exist, however, an analogue to MDA which not only offers all the advantages claimed for MDA but overcomes the noted deficiency. In addition, it is also possible to gain the advantage of parameter estimation and calculation of probability of correct classification in either the dichotomous or polychotomous situation. For these reasons, a more applicable and versatile technique under the Maximum Likelihood concept has been chosen. Maximum Likelihood . . . The concept of Maximum Likelihood provides a means of deter- mining estimations of population parameters which have to a substan- tial degree, those desired characteristics of efficiency, consistency and sufficiency.12 Essentially the Maximum Likelihood concept may be summarized as being the estimation of a population parameter, 0, from the data of the actual samples. The concept can be stated in a more formal manner: If a population parameter, 0, is a variable with many possible values, Maximum Likelihood methods will lead to the choice of that one 0, if in fact it does exist, which renders the likelihood (i.e., the probability of occurrence) of randomly obtaining the observed sample outcome as great as possible. llIbid., pp. 160—61. 12Properties of estimators are treated in numerous texts on statistics. For example, see: Gene V. Glass and Julian C. Stanley, Statistical Methods in Education and Psychology (Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1970), pp. 250-56. 44 Here the logic of the Maximum Likelihood concept is inverse to the logic employed in the more familiar research situation in that assignment of likelihood is not to the parameter, but rather to the sample in estimating the probability density of that parameter. Simply stated, the logic of this inversion can be summarized as: "Choose as the estimate the value which the parameter must have in order to maxi- mize the likelihood to the observations obtained."13 In particular, there is a certain enhancement in the applica- tion of Maximum Likelihood to the data at hand. Without previous research to guide, estimation of parameters through this technique seems intuitively attractive. To reach a more encompassing conclusion based upon such a fixed sample could be considered a fault in the design. Development of the concept . . . The concept of Maximum Likelihood is intimately woven into the fabric of the history and development of statistics. A point of orien— tation to the origination of this concept is furnished by Ashton:14 "One might begin by recalling that Gauss, in a letter to Bessel, specifically repudiated the principle of Maximum Likelihood in favor "15,16 of least squares. The concept remained dormant from the 13Ann Hughes and Dennis Grawoig, Statistics: A Foundation for Analysis (Reading, Mass.: Addison Wessley Publishing Company, 1971), p. 167. 14Winifred D. Ashton, The Lpgit Transformation with Special Reference to Its Uses in Bioassay (New York: Hafner Publishing Com— pany, 1972). 15Ibid., p. 34. 16Ashton does not cite the date of the letter, but a general interval of time is found in the fact that Karl Gauss was born in 1777 and died in 1855. 45 nineteenth century until 1922 when Fisherl7 revived interest in the concept. At that time he was the first to explicate a method of Maximum Likelihood. Fisher,18 in an earlier work that same year had startled the "world of statistics" by challenging concepts regarding x2 which had been laid down by the "master," Karl Pearson. It is not only interesting to consider Fisher's thoughts in view of the descrip- tion of Maximum Likelihood given above but also to note their relation to the data at hand: Any opinion put forward by Professor Pearson is worthy of respect, but it is impossible to agree with his state- ment that 'This result cannot be taken as obvious, as the size of the array in the sample varies.‘ The fact, however, Pearson has verified for large samples as far as the third order of approximation. The difference in principle is of some importance, since the simplicity of many of the results here obtained is a consequence of the fact that we have not attempted to eliminate known quantities, given by the sample studied, but only the unknown quantities-—parameters of the population from which the sample is drawn. . . .19 (Emphasis added.) Since that time, several specific methods dependent upon the Maximum Likelihood concept have been developed. The one which is most familiar is linear regression. Numerous other methods have been developed, primarily in the field of biological sciences. The common characteristic of these methods is that they permit the researcher to 17R. A. Fisher, "The Goodness of Fit of Regression Formulae and the Distribution of Regression Coefficients," Journal of the Royal Statistical Society, Vol. LXXXV (1922), pp. 597-612. Reprinted in: R. A. Fisher, Contributions to Mathematical Statitics, ed. Walter Shewhart (New York: John Wiley & Sons, Inc., 1950). 18R. A. Fisher, "On the Interpretation of Chi Square from Con- tingency Tables and the Calculation of P," Journal of the pral Statis- tical Society, Vol. LXXXV (1922), pp. 87-94. Reprinted in: R. A. Fisher, Contributions to Mathematical Statistics, ed. Walter Shewart (New York: John Wiley & Sons, Inc., 1950). 19 Fisher, "Goodness of Fit," p. 598. 46 estimate parameters which fit various functions. Attention is directed to two methods utilizing curvilinear functions. Probits and logits . . The first of two methods under the Maximum Likelihood concept is the method of probits.20 This method is attributed to J. H. Gaddum and C. I. Bliss and it is based on the integrated Normal curve. Gaddum and Bliss' work was summarized and extended by D.J. Finney,21 who is considered by many to be the prime developer of probit analysis. The second method, that of "logits,"22 was initiated in 1944 by Joseph Berkson who continued his work through the 1950's. This method is based on the logistic or dosal curve conceived by Pearl and Reed23 in 1920 to describe the population growth in theUnited States of America beginning in 1790. Although both curves are frequently mentioned in the field of bioassay, the method we have chosen best utilizes the logistic curve. In comparing the two functions, it can be said that the inte- grated Normal curve and the logistic curve are congruent throughout much of their range. Ashton illustrates the two curves with the following considerations having been made. Set a = 0, so that the 20The term "probit" is a contraction of the words "probability unit" and is used to express the deviation of a unit. 21D. J. Finney, Probit Analysis (Cambridge: Cambridge Univer— sity Press, 1947). 22An Analogy to Bliss' "probit"; a contraction of the words, logistic unit." 23R. Pearl and L. J. Reed, "On the Rate of Growth of the Popu— lation of the U. 8. since 1790 and its Mathematical Representation," Proceedings of the National Academy of Science, Vol. 6 (1920). 47 curves are skew-symmetric about the lines x = 0 and P = 1/2. To com- plete the traditional linear function, a + Bx, B's are chosen so that the curves agree at some point; values chosen were 0.6 for the inte- grated Normal curve and 0.988 for the logistic curve. The agreement of the curves is shown as Figure 1 below: FIGURE 4* COMPARISON OF CURVES At Logistic 4‘1 ‘ rrrr Normal 2A a 8 'o 0.5 / 1.0 8° 0 / > g Probability of -24 response, P -440 *After Ashton: The Logit, p. 12. Although the curves are congruent throughout much of their range, the point of inflection away from the limit occurs earlier in the logistic curve than in the integrated Normal curve. The general shape of these curves being quite similar makes it necessary to look once again to Ashton; she notes: The logistic estimates--both those obtained by the method of Maximum Likelihood and those obtained by Minimum Logit x2--are sufficient as well as asymptotically efficient. Those obtained by using Maximum Likelihood and the inte— grated Normal curve are not sufficient. General agreement 48 has it that 'sufficient statistics' are the best that can be had.24 The form of the logistic curve is l = —oo< >00 P l + e—(a + 8x) , x Once again, the utilization of Maximum Likelihood permits the choosing of those parameters, a and B, so that the given sample has the highest probability of occurring. The view is now directed to parameter esti— mation, first in a general sense and then specifically as concerns Maximum Likelihood. Parameter estimation . . . The statistician frequently employs inferential reasoning, moving from that which he observes (his sample) to that which he can- not or chooses not to observe (the population). A traditional means of reasoning from the particular to the general is by utilizing esti- mating procedures. Specifically, it is that "A statistic computed on a sample can be regarded as estimating a parameter in a population."25 It is well known that the population parameter p is best estimated with the sample mean, R, and that r, the correlation coefficient for a sample having two variables, is wisely used to estimate p, the corre- lation between two variables having a bivariate normal distribution in 2 the population, and that o is best estimated by $2. 24Ashton, The Logit, p. 75. 25Glass and Stanley, Statistical Methods, p. 242. 49 . 26 . . . Kirk prov1des an elementary but useful exp031tion on para- meter estimation: Associated with every experimental design is a mathe- matical model that purports to include all sources of variability affecting individual scores. To the extent that the model accurately represents these sources of variability, the experimenter can evaluate the effects of a treatment. The linear model for a completely randomized design is: X.. = + B. + 6.. 13 H J 13 According to this model, an individual score is equal to the population mean u, Plus a treatment effect 8 plus an error effect c. , which is unique for each individual subject. In a parEicular experiment, the parameters u, Bj, and 51 are unknown, but sample estimates of these parameters are given by u, - and e ., respectively. It can be shown by maximum likelihoo methods that un- biased estimates of the required parameters are provided by the statistics I] = X.. -——> p Bj = (Xoj — Xee) .T Bj .. = X.. - 7.. —> .. €13 ( 13 X J) €13 The symbol -+ indicates that the term on the left is an estimator of the term on the right. According to the maximum-likelihood method, the best estimate is one that gives the highest probability of obtaining the observed data. It should be noted that a maximum-likelihood esti— mator is not necessarily unbiased, although the center of its distribution is generally close to the value of the parameter estimated. 26Roger E. Kirk, Experimental Design: Procedures for the Behavioral Sciences (Belmont, California: Brooks/Cole Publishing Company, 1968). 27Kirk, Design, pp. 13-14. 50 Parameters under Maximum Likelihood . . . Before proceeding further, two matters of notation must be attended to; as above let 0 represent the parameter to be estimated. Further, as in traditional matrix notation, let the underlining of.g indicate a vector of parameters to be estimated and the addition of a caret, @, shall signify a vector of statistics to estimate those parameters. The goal of the researcher can be stated as a logical question "What is the best 82" That §_shall be deemed best which when substi- tuted for 0 maximizes the likelihood of the joint density of the sample obtained, or, maximizes the likelihood function. Therefore, the desired outcome is that §_which gives the maximum likelihood of the sample's occurrence. The basic principle may be stated in functional form max L (Q) = Q L(®) or, take that estimate which is the Maximum Likelihood of the parameter of interest. It is to be understood that the Likelihood function, L(f), is the joint density (probability, H, of the observations, where obser- vations are taken as given and the parameters are considered as mathe- matical variables. If the observations are independent and if {(y) represent the density function from which all observations were taken, then the joint density of the sample may be expressed as I I 7c (y) 51 which states that the joint density of a sample may be achieved by taking the joint product of the probability of the observations. The Likelihood function is a simple restatement of this joint density state— ment The Maximum Likelihood parameters are estimated mathematically from the function which has been defined above. However, the task of seeking a Maximum Likelihood parameter is simplified through the utilization of the natural logarithm of the Likelihood function for logeL and L are monotonically related and will achieve their maximum at the same point. To obtain the estimates of a and B, one first obtains the logarithm of the expression for the probability of all the observations occurring jointly or the joint density statement shown above. Then partial derivatives with respect to the desired parameters are computed. Then if logeL is differentiable, the maximum point will be that point at which the partials equal zero, or 3 logeL = 0 a 9 These are the solutions to the Maximum Likelihood equations (in a general sense) and provide the desired estimates. In many situations, solutions to Maximum Likelihood equations are familiar and obvious. For example, the Maximum Likelihood estimate of u is X. In other cases, the Maximum Likelihood estimates do not result in simple solutions. This situation could be the case, to use 52 the previous example of graphic portrayal, where a maximhm point and a similar but only near maximum point exist on a surface in two or three dimensional space. Here an iterative program (such as LISREL) is appropriate. Likelihood ratio test . . . The discussion of Maximum Likelihood concept is concluded with a further elaboration of the concept, the introduction of a Likelihood ratio test. This test provides a test of fit showing consistency between the data and the model to which it is being fitted, or more specifically the fit of the data to the logistic curve in this case. A concomitantly realized benefit is that this is also a test against constraints. That is, if the model has fewer parameters than the true data, there are constraints imposed by the model and the Likelihood ratio test provides a convenient test of these constraints. In the case of Maximum Likelihood testing, the null hypothesis is (usually) stated in some expression indicating that the "model does not fit" or approximate the logistic functional curve. The alternative, therefore, would be that the "model fits." As is true in the more familiar hypothesis testing situation, to reject the null hypothesis in the Maximum Likelihood test does not prove the alternative hypothe— sis. Rather, it only proves, in the case of the Maximum Likelihood test, that the observed data are consistent with the model--they fit the logistic functional curve--and that it may be aSsumed reasonable only within the confines of that model. Nor should it be forgotten that other models equally reasonable could exist. The null hypothesis could be stated in notational form 53 H : 0 cm 0 ._ In words, this states that the vector of parameters is in some restricted space m. This then sets the form for the alternative hypothesis 0 CO or, in words, the alternative hypothesis is that the vector of para- meters is in some larger (and therefore less restricted) space. The Maximum Likelihood ratio test is then, with a slight modi- fication, the relationship of these two hypotheses. It results in the ratio which we shall designate A, max _9 em L l = max _9 CO L which is the ratio of two Likelihood functions, each for their own maximum. The numerator indicates the Maximum Value the Likelihood can produce in a restricted space (the maximum being designated by the "max" superscripted above the_g. The space in the denominator is the less restricted space. The ratio A indicates whether the constraining of the function to a subspace had much effect or not. The distribution of x2 = - 210g 1 is asymptotically x2. In this Likelihood ratio test, the denominator should be larger than the numerator for it should be easier to obtain a higher likelihood with— out restrictions than with restrictions. If A = 1, then the constraint of a restricted space did not reduce the likelihood significantly and x2 = - 210g (1) = 0. Finally, the degrees of freedom (df) are equal 54 to the number of parameters free to vary in unrestricted space less the number of parameters estimated to the model. It was not until 1935 that an exact method for solving the estimation problem was developed by Fisher.28 His solution was with the integrated Normal curve in the context of the probit method. Fisher's solution has since been extended to the logistic function; one of those solutions is now presented. Quantal response techniques . . In a paper entitled "Quantal Response Techniques for Random "29 McSweeney and Schmidt develop and present a Predictor Variables versatile alternative to the oft-used multiple linear regression and the previously discussed Multiple Discriminant Analysis, specifically extensions of Quantal Response techniques.30 Earlier in this paper, the term "quantal" was described as the dosal or quantal response measurement so familiar in the biological sciences. The quantal response technique is one of numerous bioassay techniques for the analysis of the relationship between one or more quantitative predictor variables and the quantal response which takes the form of a qualitative criterion (usually referred to as the 28R. A. Fisher, "The Case of Zero Survivors," Appendix to C. I. Bliss, "The Calculation of the Dosage-Mortality Curve," Annals of Applied Biolggy, Vol. 22 (1935), pp. 164-67. 29M'aryellen McSweeney and William H. Schmidt, "Qualtal Response Techniques for Random Predictor Variables," (Paper presented at the Annual Convention of American Educational Research Association, 1974). 30The thoughts expressed here draw heavily upon the work of McSweeney and Schmidt as summarized in their paper and reflected in their computer programs. Appreciation is expressed for their generosity. 55 "criterion variable" but also recognizable as the "dependent variable"). The predictors may be captioned as treatments (at various levels); the outcomes may be dichotomous (alive or dead) or polychotomous (alive, moribund, or dead). Recently, the usage of Quantal Response has not been limited to research in the field of biological for In the social sciences, quantal response techniques have been used primarily in calibration of subjective estimates of weight, pitch and loudness in psychophysics, in latent trait analysis in psychometrics and in the determination of receiver-operating characteristic curves in signal detection.3 The sustentacular relationship of the Maximum Likelihood concept is implicit in the discussion of the four Quantal Response models.32 Two models for two situations . . . Although many familiar statistical models pose no constraint as to the nature of predictor variables, such a distinction is inherent in Quantal Response techniques. The two general Quantal Response models are classified as to whether predictor variables are mathematical or stochastic. The contrast between the situations and the variables is forthright. In one situation the researcher will have discretionary choice as to the amount of treatment (or doses) to be administered to the subject. Here the quantitative classification variable is con- sidered to be mathematical. In the second situation the researcher has no control over the predictor variables. 31McSweeney and Schmidt, "Quantal Response," p. 1. 32See supra, pp. 7-16. 56 Probability of the first model . . . The resultant probability estimates elaborate the nature of the first model. Cornfield et a1.33 comment For the first model (A) we consider an experiment situ— ation in which the probability that an observational unit is characterized by a dose between X and X + dX is f(X)dX, where £ may depend on one or more parameters. The condi- tional probability that the quantal variate for a unit will will take on the value one, given that it is exposed to dose X is P(X). The function P may also depend on one or more unknown parameters, say 01, but it is also assumed that (2.1) 8f(X)/BO = 0 As a situation to which this model might apply, consider a population of animals exposed to food containing a poison. The actual dose, X, ingested depends on the amount of food eaten and is a random variable with p. d. f. {(X). Having eaten amount X, the probability that the animal will die is p(X,01...). The Oi are determined by physiological charac- teristics of the animal and the assumption (2.1) is not un- reasonable for such a situation. From this, four relevant thoughts may be arrayed: l. The quantitative classification variable can be used as a stratification dimension in the model. 2. "The researcher can determine a priori how many subjects will be exposed to amounts (X1, X2 . . -Xk) of the K quantitative predictors;. . ."35 3. An alternative design would permit the researcher to determine how many of the subjects at each level, (X1, X2, . . 'Xk) of the quantitative predictor variable will be chosen for his study. 4. Independence between subjects is a necessary assumption. 33Jerome Cornfield, Tavia Gordon, and Willie W. Smith, "Quantal Response Curves for Experimentally Uncontrolled Variables," Bulletin of the International Statistical Institute, Vol. 38 (1960), pp. 97-115. 34Cornfield, et al., "Quantal Curves," p. 98. 35McSweeney and Schmidt, "Quantal Response," pp. 1—2. 57 Description of the second situation . . . If the first situation is classified as a controlled case, the second involving stochastic variables may be viewed as the uncontrolled case. "The researcher records and classifies according to the observed values of the predictors but does not have the liberty of determining the sample composition with respect to values for those variables."36 Stated simply, the particular observed element is not under the control of the researcher and it is therefore properly considered to be a random variable. Probability of the second model . . Once again, the statement of probability extends the descrip- tion of the model applicable to the second situation. Cornfield et a1. state Consider now Model B. We have a population, each element of which is characterized by values for two sets of vari- ables, a quantal variate and a quantitative classification variable, X. We shall refer to units for which the quantal variate takes value one or zero as responding or non- responding units respectively. Denote the probability that the quantal variate takes on the value one by p. Denote the conditional probability density functions with respect to X of responding and non-responding units by ¥1(X) and £O(X) respectively. The probability that an element selected at random has a dose between X and X + dX is then (2-2) [qfo (X) + pfl (X)] where q = l - p. The joint probability that an element selected at random has a dose between X and X + dX and a response of one is (2.3) pfl (X)dx 6McSweeney and Schmidt, "Quantal Response," p. 2. 58 The conditional probability that an element with dose between X and X + dX will have a response of one is then the ratio of (2.3) to (2.2). Thus, (2.4) P(X) = l / [l + qfo(X) / pf1(X)Io This yields a dose-response curve whose functional form is dependent upon -0(X). Furthermore, if (2.4) depends upon parameters 0, so does (2.2). Then assumption (2.1) cannot be made; the distribution of the number of observations by dose level is dependent upon the parameters of the dose- response curve, and the traditional estimation procedures do not apply. As an example of a situation which might be appropriately described in this way, let X be the squared difference between two measures on a pair of twins and let p be the probability that a pair selected at random is monozygotic. Then fl(X) and }O(X) might be chi-quared distributions each with one degree of freedom, but with E(X) much larger for dizygotics. The probability that a twin pair with squared difference X is monozygotic is then given by P(X).37 Further issues Thus far the appropriate model to apply seems to be dictated in to by consideration of whether the predictor variables are mathe- matical or stochastic. This criterion leads to the logical conclusion that in most cases models MAl38 and MA3 are appropriate when the rela- tionship of the predictor variable is close to the criterion variable could be considered to be a link in a causal chain. McSweeney and Schmidt maintain that the use of models MAl and MA3 are less clear in two situations. First, in that situation in which the predictor variable is not manipulated but is used as a 37Cornfield et a1. "Quantal Curves," pp. 98-99. 38As is often true, the models assume different captions from different authors. We shall follow the McSweeney-Schmidt mode of letting "A" and "B" indicate mathematical and stochastic respectively; for each model, the lower number indicates dichotomous criterion and the higher number polychotomous criterion. 59 stratification device; second, that situation in which the researcher employs a classification procedure based upon the observed values of the predictor and the criterion variables of a sample randomly drawn from a particular population. Here both predictor and criterion variables would be random with joint distribution and correlation coefficients determinable. The applicability of the MA models is dependent upon the directness of the linkage between predictor and criterion variables. If such linkage is direct, then MA models are appropriate and the characteristic of the stochastic variable is ignored. Direct causal linkage . McSweeney and Schmidt represent the direct and indirect causal linkage between variables graphically in the following manner. Letting Xi represent the predictor variables, Y the criterion variables and Z, a factor which is causally related to the predictors Xi’ then FIGURE 5 DIRECT CAUSAL LINKAGE (z 7. .Z) l’ 2' ' L /\7 P(lel, X2. . . xk) (X1’ X2. . . XK) Here the linkage is direct and the predictors are not influenced by the Z factors and the functional relationship is reasonable. The factors (Z1, Z2, . . . , ZL) may be thought of as latent or unobserved variables not disturbing the functional relationship between predictor and criterion variables. 60 Indirect causal linkage . The indirect causal linkage is contrasted with the direct in this manner FIGURE 6 INDIRECT CAUSAL LINKAGE (21, 22, . , 2L) —-> (x1, x2, . . . , Xx)”; P(YIXl, x2. . . XK) This is described by McSweeney and Schmidt as "Indirect causal linkage 39 e H to Y mediated through (X1, X2, . . . , XK). In the above figure, the P(Yle, X2, . . . , XK) is another of the desired results of the analysis through Quantal Response--the model of the probability of the ". . .occurrence of each level of the criterion to the predictor variables."40 The linkage between predictor and criterion variables is direct but the factors (21’ Z2, . . . , ZL) are not only direct but also in— direct. Therefore influence will be direct but also mediated through the predictors. McSweeney and Schmidt state that the "parameters governing P(Yle, X2, . . . , XK) are not influenced by the same factors that the parameters of f(X1, X2, . . . XK) would be."41 Consequently, the MB models would be considered applicable in that this relationship is not functional but rather predictive. 9McSweeney and Schmidt, "Quantal Response," 40Ibid., p. 4. 41Ibid., p. 4. p. 4. 61 Mathematical distinction between models . . . Although the design of this research focused on a model con- sisting of multiple random predictor variables and polychotomous criterion variables, distinctions can be advantageously drawn between this model and others. Specifically these distinctions are related to the particulars of estimation of parameters, Maximum Likelihood estimators, probabilities, etc. Beginning with the simplest, the presentation takes the following form Single predictor, dichotomous criterion Multiple predictors, dichotomous criterion Multiple predictors, polychotomous criterion for both the MA and MB models where applicable. Single predictor, dichotomous criterion . . . Let X represent the quantitative predictor variable and Y the criterion variable. For this model the predictor is single and the totality of the outcome of is represented in Y = 0 or 1. Then P, the probability of the occurrence of the desired outcome given a specified amount of treatment, may be represented as P = (Prob (Y = lIX). Further, the Likelihood takes the form of the familiar binomial prob- ability function, that is, the probability of obtaining exactly x successes in n independent trials. If Ni represents the number of subjects receiving the specified treatment Xi’ if ni represents that number having the desired outcome and if Q is computed in the tradi- tional manner as l - P and signifies the probability of the outcome which is not desired, then Likelihood is computed N. 1 nv N- - n- L: IiI n PlQl l i 62 The second model is viewed in the context of random predictor variables. The conditional probability is still represented as P, while p signifies the unconditional probability of the outcome of interest. Then let nO signify those respondents showing the alterna— tive outcome and n1 those having the desired outcome. The denominator will be simply the sum of the outcomes, or those outcomes which will appear and the numerator is the outcome of interest, in this case, then P = gp_f (XLY = 1) p.£(xTY=1)+qf(X[Y=0) letting f (XIY = l) and f (XIY = 0) represent the conditional distri- butions of the predictors of the two outcomes.44 In words, this formula states that the conditional probability, P, is equal to the unconditional probability of the function of X given that Y = l, the outcome of interest, divided by the sum of unconditional probability of the function of X given that Y 1 plus the unconditional probability of the function of X given that Y 0, the latter part of the denomin— ator being that outcome which is the alternative to the outcome of interest. The model is extended by Cornfield et al.: In many problems it is natural to assume that $0 and f1 are both normal density functions, but with differing means and variances, i.e., that 1 row = Z‘“ a: exp - 2 cg 0‘ “ W2 44For a discussion of Bayes Theorem.with regards to this appli— cation, see: Samuel A. Schmitt, Measuring Uncertainty an Elementary Introduction to Bayesian Statistics (Reading, Mass: Addison-Wesley Publishing Company, 1969. 63 (2.5) 1 l f1(X) = 01 /§_F exp _— 2 o (X ‘ U1)2 t—‘N for this case (2.4) becomes45 2 . - qu p00 exp - 2 00 01 This is a dose-response curve which for 00 # 01 reverses directions at _ 2 2 2 246 (2.7) X - IJo/o0 - IJI/Ol (l/oo) - (1/01) Certain assumptions simplify Cornfield's formula (2.7) to a degree. In many cases the dose-response curve can be assumed to be monotonic. This in turn leads to the simplifying assumption that the conditional distributions have the characteristic of equal variance i.e. of = 0%, (not an unusual assumption in statistical analysis), which then implies the additional assumption 01 = 00 (standard devia- tions are equal) and then the conditional probability expressed in Cornfield's (2.6) reduces to This states that the conditional probability is equal to one divided by the sum of one plus the fraction of the unconditional probability of the alternative outcome over the unconditional probability of out- come of interest (hereafter referred to as the "ratio of probability 4SFormula (2.4) was presented on p. 46Cornfield et al., "Quantal Curves," p. 90. 64 of outcomes") times the exponent less the product of the mean differ- ences divided by the variance times the observation less the average of the two means. This is the logistic dose-response curve of the form: P = 1 / {1 + exp - (8+BX)} with the parameters 8 = (U1 - uo)/02 , a = 1n (g) - B ilfllfijili In words this states that the conditional probability is equal to one over the sum of one plus the exponent to the negative power of alpha plus beta x; beta is equal to the difference between the means divided by the variance, while alpha is equal to the difference between one times the number of observations times the ratio of the unconditional probabilities of outcomes and beta (as previously defined) times the average of the two means. McSweeney and Schmidt present the Likelihood function for the dichotomous case as: n + n 0 l n E“ II n n n1 n0 1 _ 0 _ p (q) f(X.Y—l) :F(X.IY—0) l1 = ll 1| Ii = ll 1 l and the effective part of the logarithm of the Likelihood is 2 n0 - n1 2 n1 (Xi — U1) £nL' = n an + n in (q) ----- zno - - X _—_“j;““ l 0 2 i=1 0 n1 (x. - uo)2 _ _1_ z 1 47 2 i=1 02 47 McSweeney and Schmidt, "Quantal Response,‘ p. 8. 65 That Likelihood function formula may be expressed in narrative form as the product of the combination of the number of observations taken in the number of the outcomes of interest times the unconditional prob— ability of the outcome of interest raised to the power of the number of the outcomes of interest times the unconditional probability of the alternative outcome raised to the power of the number of alternative outcomes times the Likelihood of the function of X given that Y equals one times the Likelihood of the function of X given that Y equals zero. n Then, as is traditional, . . .the parameters of the dosage-response curve are estimated by substituting sample proportions, means and variances for the corresponding population parameters."48 9 = nl/(nO + “1) “0 = Z Xo/no = X0 “1 = Z Xl/nl = X1 0 no + n1 2 (X0 X0) + 2 (X1 X1) 3 In the same sequence as above, these state that the estimated uncondi- tional probability of the outcome of interest is equal to the ratio of the number of the occurrences of such observations in the sample to the number of all observations in the sample; the estimated mean of the alternative and the desired outcomes are computed in traditional manner, i.e., the sum of the observations of that type of outcome divided by the number of those outcomes; the estimate of variance is equal to one over the number of observations times the sum of squares. . . 2 Use of the unbiased estimator, 3 leads to, 48Cornfield, et al., "Quantal Curves," p. 101. 66 - - — 2 B = (X1 — X0) / s , and, &=-1n3 g u(W131 p 2 Although the point has been established earlier, it seems worthy of repetition in view that some might consider linear regression to be an analogue to Quantal Response in the context of the MB models. Cornfield et a1. note that approximate ( 1 - a)% confidence intervals for B can be constructed by referring to the t- distribution with n1 + n2 - 2 degrees of freedom. These intervals differ from traditional linear regression inter- vals for the slope because the quantitative variable is a random variable under model two assumptions, but a mathe— matical variable under model one or traditional linear regression assumptions.49 Multiple predictors, dichotomous criterion . . The previously given model having single predictor and dichot— omous criterion is easily amplified for inclusion of multiple predictors by stating the model in matrix form. The conditional distribution of the multiple predictors having outcomes Y = l and Y = O are indicated as f (Xil Y = l and I (Xil Y = 0). Further, the vector of K random predictor variables (X1, X2, . . . , XK) shall be indicated as XI where the small superscript "T" is construed to mean a transpose. Necessary assumptions are that the conditional distributions of the predictor variables, f (X.IY): ml 1. Are multi-variate normal 2. Have identical variance—covariance matrices (Z) 3. Have mean vectors “1 and “0 respectively. 49McSweeney and Schmidt, "Quantal Response,’ p. 9. 67 McSweeney and Schmidt state that ". . . the conditional probability of occurrence of response Y = 1 can be expressed as a cumulative compound logistic distribution function: P = 1/ [1 +-1 exp — 1/2 [(x - u )T z'lo(x - p )T 2'1 (X u )q "50 p m 0 m ml m ml where 2-1 (sigma inverse) is understood to be the inverse of the "grand” variance covariance matrix. The formula may be stated in this manner: the conditional probability is equal to one over one plus the ratio of unconditional probabilities of outcomes times the exponent less one- half of the difference between the product of the transpose of the matrix of mean differences of alternative outcomes times the inverse of the variance-covariance matrix times the matrix of mean differences of alternative outcomes and the product of a similar term for the desired outcomes. In this case of multiple predictor variables, P is in form similar to that presented for the single predictor51 excepting for the expression of B and X as vectors: T P = l/(l + exp — (a B X). A, ’b X being in one class and not the other. The parameters a and B are found to be: T -1 R n(q/p) - 1/2 (pl 2 p1 - £0 2 “0) and Q II _ ‘1 _ B ‘ 2 (£1 £0) 501bid., p. 10. 51See supra, p. 26. 68 the latter bearing distinct similarity to the formula presented in the case of the single predictor variable excepting of course that the variance-covariance matrix is now substituted for 02 and the expression of the means pl and “0 as vectors. These parameters are conditioned on class zero. Appropriate changes to the Likelihood function are the expression of predictor variables in vector form: 0 + nl n n The effective part of the logarithm of the Likelihood is: n I (no + n1) '1 O T BnL = n an + n £n(q) + in IE I - 1/2 E (X, - u ) l 0 2 . 10 0 i=1 m n z— (x - ) — 1/2 20 (x - )T 2'1 (x — ) “’10 '10 i=1 il klll mil 51 . Multiple Predictors and Polychotomous Criteria . Because the research effort utilized three and at times four and five criterion variables and multiple predictors it was necessary to make certain modifications to the original McSweeney-Schmidt pro- gram. Furthermore, it seemed more logical to present the findings of the maximum Likelihood estimators in this section rather than the preceding discussion. X D T 0 Once again, X = (X . , XK) is taken as a vector of ’\1 l’ 2’ K random predictor variables, but now {(Xilyj) shall denote the condi- (\1 tional distribution of those predictor variables having outcome yj, and such conditional distributions are assumed to be multi-variate 69 normal with identical covariance matrices having mean vectors uj. In the theoretical solution of the probability of occurrence of response yk it is necessary to condition on one of the categories. In the practical application of the program, conditioning is always on category 1. Letting subscript "j" represent the category being conditioned upon and subscript "k” denote the category being worked with, the necessary constraint arises that j # k as subsequently noted beneath the summation sigma. Then, the computational formula for the conditional probability of the occurrence of outcome yk becomes: J P. _ P = 1/ 1 + 2 .ml exp - 1/2 (X - u.)T Z 1 (X - u.) - k . = m «,3 'b m3 3 1 Pk j #k T —1 X - Z X - (b 5k) (b 5k) The earlier expressed Likelihood function is modified even more: J-l -3 _ ' J-l ' J-l ' J-l n. J-l n_ z n.Lz_|_g_ L — N./-T-T n. (n - Z n.). I I P. j (1 = Z P.) '=1 3 (2 )nk . j=l 3 j=1 J j=1 J J I 2 J=l “j Z T ex - 1/2 . X. - . P 3:1 (“J1 uJ) 2"1 (X. - .) m m1 U "OJ where once again the vertical lines enclosing the variance-covariance matrix 2 indicate not the absolute value but rather the determinant of that matrix. The effective part of the logarithm of the Likelihood is: J J-l J-l Z n.2n p. + (n - Z n.)£n (l - 2 :1 J J j:l J j: E. _1 _ 22ml: 1 RnL P.+ J) j l J-l nj T 1/2 2 z (x. - p.) 2‘1 (x. - p.) ”I j=1 i=1 ml m3 m3 when we "let Xj denote the vector of means on the predictors for . . KxK subjects whose responses fall in category yj, let Sj denote the cor- responding sample matrix of sums of squares and cross-product deviations 52 about the respective means. Let u = 2"." The cumulative logistic distribution function is represented as Ph = l/(l + exp — (a + 8T 5)) ’b The parameters then become: T —1 = Q) P . .... a n J/Pk 1/2 [gk 2 where J is being conditioned upon, and = Z-1 -— . B (flk £3) The difficulties encountered in expanding the program to permit expres- sion of a greater number of B was paramount to the success of the research. As Cornfield, et al. note: For applications in which model B applies, such as those of sections 8 and 9, one wishes to know whether the classi- ficatory variable is associated with the quantal variate, and, if so, the magnitude of the association. For such a question, 8, and the confidence limits (4.7) are of more interest than a or confidence limits about it.53 Letting (X - Xj) represent the input vector and remembering (\J 'b that the constraint of j # k will be met by "skipping the computation" 52McSweeney and Schmidt, "Quantal Response," p. 13. 53Cornfield et al., "Quantal Curves," p. 102. if > 71 ' = k, then sample estimates are: 1/[1 + J P./P exp - [(x - R.)T S“1 (x — x.) - (x - xk)T S—l jél J k n, mJ m mJ n. % jrk (36 - 3519]] A A -T 1 -T -l - -£n P,/P - 1/2 X S - X, S X and k [wk £18 «,3 «J ] _1 _ S - X (ER NJ) where S is substituted for the biased maximum likelihood estimator Z and X is an unbiased linear estimator of u. Assumptions and effects of departure from those assumptions . . . The brevity of this section is dictated by lack of established consideration of these aspects. Two assumptions previously cited were that the conditional distributions of the predictor variables are multi-variate normal and that those predictor variables have identical covariances. Without substantiation it has been held by some that the method of Quantal Response is robust to violation of these assumptions. An additional assumption, that of correct classification, has been investigated by Cornfield: The previous section has established that if model B is applicable and model A assumptions are made, there may be considerable loss in precision. This arises from dis- regarding the additional information contained in the pdf¢(X). The estimates of the parameters of the dose— response are clearly consistent, however, so long as the dose-response curve has the form assumed. If model A is applicable, however, but model B assump- tions are made, the estimates obtained are inconsistent.54 54Ibid., p. 102. 72 Cornfield investigates the effect of incorrect classification by viewing how far the P(X) obtained by assuming i (X) and {1 (X) to be normal departs from the P(X) which, in Eact applies when they are not. We have considered the following case: 6.1 *0 (X) = XI1 -1 exp - X/F (n), *1 (X) = £1 (X + c) and have contrasted the P(X) implied by (6.1) with that by assuming f1 and {0 normal, with the same means and variances as those given by (6.1). We contrast in Table l numerical values for P(X) obtained from the normal assumption with the true ones for p = 1/5 and the two cases n = 4, c = l, and n = 9, c = 2. In the first case the means are 4 and 5 respectively, with standard deviations of 2 in both cases. In the second case the means are 9 and 11, with standard deviations of 3. It will be seen that in the interval mean 1 two sigma, true and approximate curves are in reasonably good agreement in both cases. The agreement becomes poorer farther out in the tails, where of course, there are many fewer observations. More extreme departures from normality than those assumed would, of course, entail larger dis- crepancies, although appropriate transformations might in some cases prevent this from becoming a major problem. 551bid., p. 102. CHAPTER IV FINDINGS AND CONCLUSIONS Introduction . . . The purpose of this chapter is to report the findings of this research project and from those findings draw appropriate conclusions. The presentation of findings should be preceded with somewhat divergent comments about the dilemma of completeness of problem description versus simplicity of model specification. The brevity and non—technical approach should not be construed to be an intended dimunition of the subject; rather the thoughts are presented in this manner so that they may serve as a fitting caveat. Specification of the model . . Generally, the researcher approaches his investigation with two desires which might be at times diametrically opposed. The desired model will be: 1. As realistic as possible 2. As mathematically simple as possible. The first of these two desired characteristics relates to the number of variables to be included in the analysis. The difficulty inherent to the determination of the proper number is of itself paradoxical; if the researcher were to attempt to capture every nuance of the observed event in his research model, he would have replicated the happening 73 74 which he seeks to explain. This could result in such a complex model that it might be uninterpretable if not impossible to manipulate. Accordingly the researcher may well approach his investigation by specifying a model that will not fully explain the facet of life he views, sacrificing completeness for operability. Wisely, the researcher accepts less than a complete description, by settling for those predictor variables which may be considered as strongly associ- ated as possible with the dependent variable(s). Another way of describing this-—our goal--is to build the model in as parsimonious fashion as possible, but still representative of the observed events. With these thoughts in hand we return to the findings. Quantal response . . The computer program for the Logistic Polychotomous Technique was written in fortran IV for the Control Data 6500 computer. By modifying the original McSweeney-Schmidt program it was possible to "pack" the matrix thus permitting the employment of as many as fifteen predictor variables (drawn from the financial data for the years 1971— 74) in an analysis. This afforded computation of probabilities of classification of each of the 60 cities into three groups utilizing as criterion variable the amount of cash in the general fund June 30, 1975 expressed as a percentage of total general fund assets, June 30, 1975. Because it seemed reasonable to assume that other underlying variables in the causal chain may be jointly influential on that amount of cash held by a municipality, Model B Quantal Response Tech- nique was utilized.1 1For the means by which linkage effects model choice, see Supra, pp. 59-60. 75 Range of analysis—-criterion variable . Various analyses were made to enhance the classificatory ability of the model because of the lack of prior information regarding crite- rion and predictor variables. Recalling that the criterion (dependent) variable is that which will be the basis of classification of the municipality into one actual specific group or another, it should be noted that the Quantal Response technique provides the researcher with great flexibility and ease of actual classification of observations. Specifically, an observation is designated as actually belonging to one group or another by simply changing a "header" card which precedes the data deck for that observation from one group's designation to another. The ulti- mate impact of such a change is in reality a change in the dimensions or ranges of the groups. An example will serve to clarify the dis- cussion. Suppose that a researcher was investigating test results which were expressed in the form of a decimal ratio having potential out— comes between zero and one. Using four groups he might logically choose the dimensions for classification as: Range of Groups Group 1 0 .250 Group 2 .251 to .500 Group 3 .501 to .750 Group 4 .751 to 1.000 Suppose now that the researcher decided that the analysis should reflect a traditional grading scale, A = 90 to 100%, etc., and that he decided that he would still keep four groups calling his last, "D and lower." He could change the dimensions of the groups by 76 redefining certain levels of the criterion variables: Range of Gronps Group 1 (D's and lower) 0 to 69% Group 2 (C's) 70 to 79% Group 3 (B's) 80 to 89% Group 4 (A's) 90 to 100% In this particular example, the dimensions would change significantly. The first two groups and part of the third in the quartile classifi- cation above would be collapsed into the Group 1 on the grading scale (immediately above). We can amplify the process and make it more analogous to the situation at hand by looking at an additional example. Assume that a researcher was investigating the various "causes" of why a city would lose population, retain its existing levels of population or increase its population, a three group model. Clearly the researcher can change the range of the classificatory variable for a particular group-— in this case, some percentage change of population from previous years-- by simply redefining what he means when he says "losing population," etc. To extend this reasoning to our problem, we varied our defini- tions of our three groups by varying the level of the classificatory variable, the amount of cash, 1975, expressed as a percentage of total general fund assets, 1975 as shown on the financial statements. The ranges investigated were arbitrarily chosen. For example, the ranges investigated began with: Range of Cash2 Group 1 (Cash Poor) 0 to 25% Group 2 (Cash Satisfactory) 26 to 75% Group 3 (Cash Rich) 75 to 100% 2The variable, cash, is computed in this way: $ Cash, 1975/ Total General fund assets, 1975. 77 Group 1, as discussed shortly, became 0 to 10%, for at this level the greatest classificatory ability was found. Obviously, the change in the level of the classificatory variable causes a shift in the number of cities in a particular group. For example, lowering the upper limit of Group 1 (above) from 25% to say 10% would shift approximately ten cities from Group 1 to Group 2. Criterion variables——final runs . The majority of runs were based on a triad of levels of the criterion variable, percentage of cash, for classification. Once again, the implication of a variation in the percentage level chosen as criterion is a variation in the classification of specific munici- palities, ergo, variation in the number in a specific group. The levels chosen for final analysis were: Category 1 (captioned "Cash Poor" or "Cash Short") This category included ten municipalities having cash ranging from zero percent to ten percent of total fund assets, on the Balance Sheet, June 30, 1975. Category 2 (captioned "Cash Satisfactory") This category included 41 municipalities having cash ranging from eleven percent to seventy-four percent of total general fund assets on the Balance Sheet, June 30, 1975. Category 3 (captioned "Cash Rich" or "Cash Excess") This category included nine municipalities having cash ranging from seventy-four percent to one hundred percent of total general fund assets on the Balance Sheet, June 30, 1975. Reiteration of already established points regarding our choice of cash as our criterion variable, the focus of our attention, seems apropos at this point. Earlier, we reasoned that although over 200 cities in this country have declared bankruptcy, there have been no 78 recent declarations nor has there been any of significant size. Such an action seem unlikely if at all feasible because of the constraining requirement of Chapter IX of the Federal Bankruptcy Act necessitating that the plan of settlement have approval of over 50 percent of the creditors.3 Although declaration of bankruptcy provides a critical event for research in private sector solvency, it is essentially a vestigial in studies of public sector solvency. It became necessary, therefore, to establish another item as an acceptable surrogate for bankruptcy. Although various states have governing measures of insolvency or poor financial condition, we felt that a readily discernible measure would be desirable. After extensive conversations with knowledgeable individuals, it was concluded that the amount of cash a municipality held would be an acceptable sub- stitute for insolvency. Range of analysis--predictor variables . In addition to varying the levels of the criterion variables for the three groups, enhancement of the model was achieved through variations in the predictor variables. Specifically, three aspects of the predictor variables were changed: the combinations of the vari— ables used, the metric of the variables used, and the time frame of the variables used. There is little practicality in discussing all of the 3In the recent fiscal crises of New York City, the individual who was aware of the implications of Chapter IX understood that although the bonds might go "flat" bankruptcy was remote and highly improbable. One need only imagine the vast amount and number of bearer bonds issued by the City and held throughout the crises and he becomes relatively certain that consensus among such lenders——to a plan of arrangement—-if their identity could ever even be deter— mined-—might be a greater task than resolving the fiscal dilemma. ‘— 79 variations, so we shall describe them narratively. Others are dis- cussed on later pages dealing with the findings. The first variation is found in the combinations of variables utilized. First, variables were segregated by the statement on which they appeared, the Balance Sheet or the Income Statement. Each of the elements (in "raw" dollar amounts) from both statements was analyzed for its classificatory power. Some of these, including the element "Cash," will be discussed on later pages. Greater classificatory ability was realized through two approaches. First, the using of totals, such as Total General Fund Assets, Total Revenues, Total expenditures, etc. improved the predictive ability of the model over that of one made up of an intuitively appealing combination of elements. Additional classificatory ability was realized through the utilization of totals and the elements composing that total. For example, one of the more successful runs had as predictor variables the Total Expendi— tures and the four subsets of expenditures, Administration, Police, Fire and Parks and Recreation Expenditures for various years. Another utilized Total Revenues and its subsets for various years. The second variation is in the metric utilized. Initial runs employed predictor variables expressed in "raw" dollars. Later runs were based on the development of ratios, such as Administration Expenditures Total Expenditures It was felt that this would minimize any potential distortion because of disparity in size of the units observed. A probable insufficient investigation was made into trends through the computation of the per— centage change in some of the elements. It is stated that this is 80 probably insufficient because there is the feeling that the data possesses time lags worthy of exploration. Finally, the predictor variables were drawn from different years. This is also discussed on the following pages. The base year of the classificatory variable, l975--that to which the prediction is made—-remained the same throughout the study. It can be generally concluded that the model improved with a lag of at least one year. That is, in general, greater classificatory ability was realized utilizing predictor variables from the years 1971 through 1973 than from 1971 through 1974 or 1972 through 1974. As additional years of data are accumulated, this phenomenonological behavior can be explored further. Data output . . . Output for Model B Quantal Response Technique presents the estimated (percentage) probabilities of classification of a particular municipality into each of the three groups, that is an estimated prob— ability that the city is "cash poor, that it is "cash satisfactory" and that it is "cash rich." These three estimations can be considered as a probabilistic profile of each city, given a particular set of pre- dictor variables. Obviously, the category having the highest estimated probability associated with it is the category into which that city would be classified. Correct classification (a "hit") should connote that through fitting of the logistic polychotomous model and proper viewing of the estimated probabilities, a municipality was classified in the same category as the experienced actual percentage of cash on June 30, 1975 did actually place it. That is predicted percentage 81 cash level 1975 was the same as actual percentage cash level, 1975. Runs of data . . . The original list of predictor variables was reduced through numerous computer runs to the following optimal combination of 15 variables: Total Expenditures 1973, 1972, 1971 Administrative Expenses 1973, 1972, 1971 Police Expense 1973, 1972, 1971 Fire Expense 1973, 1972, 1971 Park and Recreation Expense 1973, 1972, 1971 Items are arranged solely in order of appearance in the data bank and such an arraying should not be construed as to imply significance either as related to the variables or to the factor of time. Further— more, it is only the classificatory ability of the entire combination which is evaluated--although we do offer calculations of the weights of individual variables. In addition, we make no claim that the resultant logistic function (format 11) is the most optimal; rather, it is the most optimal we evaluated. Empirical results . A total of 50 correct classifications of cities and ten incor— rect classifications was realized. This results in a "hit" ratio of 87.5%. These classifications can be viewed in Table 1. These tables should be interpreted in this manner. The analytical results are shown in vertical columns. The actual classifications are shown horizontally. The diagonal running from the upper left to the lower right should be interpreted as the number of correct classifications. The total number in the group is the sum of the (horizontal) row. 82 TABLE 1 CLASSIFICATIONS FROM FORMAT ll Classification as predicted from the fitted model Poor Sat.* Rich Poor 6 3 1 ACTUAL * CLASSIFICATION 53" l 39 1 Rich 0 4 5 *Satisfactory The estimated probabilities for format 11 are shown in Table 2: TABLE 2 ESTIMATED PROBABILITIES OF CLASSIFICATION UNDER FORMAT 11 NEEEZr P1 P2 P3 First Category 1 .4431 .5189 .0380 2 .1968 .0594 .7438 3 .7342 .2510 .0148 4 .9964 .0006 .0029 5 .7680 .2310 .0010 6 1.0000 .0000 .0000 7 .1540 .7985 .0474 8 .0829 .8541 .0630 9 .5513 .4475 .0012 10 .7675 .2154 .0171 Second Category 11 .1007 .8177 .0816 12 .0479 .8962 .0289 83 TABLE 2--Continued City . ~ . Number P1 P2 P3 13 .4098 .4868 .1034 14 .0209 .8792 .0999 15 .0454 .9052 .0494 16 .1235 .8288 .0477 17 .0698 .9259 .0043 18 .1155 .8256 .0589 19 .0995 .6418 .2587 20 .0205 .9708 .0087 21 .0820 .8867 .0312 22 .0733 .3774 .5493 23 .0909 .9053 .0038 24 .2376 .7421 .0202 25 .1295 .8524 .0181 26 .0965 .8731 .0304 27 .0229 .9637 .0134 28 .0227 .9617 .0156 29 .1869 .5424 .2707 30 .1655 .7648 .0696 31 .0244 .9413 .0343 32 .0352 .9470 .0178 33 .0636 .8781 .0583 34 .0228 .8892 .0880 35 .0322 .9526 .0152 36 .2644 .6771 .0585 37 .1281 .8563 .0156 38 .1918 .7080 .1002 39 .0304 .5736 .3960 40 .0044 .9912 .0044 41 .0850 .8740 .0409 42 .1074 .7986 .0940 84 TABLE 2--Continued NEiEZ. P1 P2 P3 43 .0332 .9424 .0244 44 .3168 .6761 .0071 45 .0842 .7248 .1910 46 .2537 .7277 .0186 47 .0938 .8489 .0572 48 .1122 .8825 .0053 49 .4085 .5722 .0193 50 .6599 .3363 .0038 51 .0295 .9415 .0290 Third Category 52 .0002 .0059 .9939 53 .0952 .6906 .2142 54 .1283 .0734 .7983 55 .0807 .8801 .3312 56 .0320 .7837 .1843 57 .0110 .0539 .9361 58 .1204 .3231 .5565 59 .0384 .0255 .9361 60 .1400 .7934 .0666 As is true in all probability functions, the sum of the esti— mated probabilities for a particular city must equal one: Pl + P2 + P3 = l The occasional rounding error evidences itself in 2P = .9999. Additional insight into the optimality of a particular run of Quantal Response can be gained by observing the distance of groups. 85 TABLE 3 INCORRECT CLASSIFICATIONS UNDER FORMAT ll City Actual Classified as Number Classified From the in Group Fitted Model 1 l 2 2 l 3 7 l 2 8 l 2 22 2 3 51 2 l 53 3 2 55 3 2 56 3 2 6O 3 2 In only one case (City Number 2) was the category into which the city was classified incorrectly further than the adjacent category further inspection of the data of Table 1 shows that the misclassification errors had associated conditional probabilities in the range .03 < R < .45 or a range of 42 percent. In reality, this "importance" of the predictor variables is reflected as variability of the criterion variable. This relationship may be compared to the simple linear regression model, which is ex- pressed as: Y1 = a + Bxi That expression of a simple linear model may be expanded to recognize that variability can be partitioned into two distinct parts, systematic 86 and random (error): Y1 = a + Bxu + pi In this case, the a, as is traditional, represents the intercept of the regression line and the Y axis; the 8 represents the systematic variability (the slope of the regression line); the n represents the random variability or that which will not be explained by the model. The less the size of the random error the greater the explanation of total variability as systematic variability. In practice, it is most obvious that the outcome of any par- ticular Y is dependent upon several factors. Liquidity preference as evidenced by the amount of cash held by a municipality is the function of innumerable input variables such as revenues, expenditures, the partitioning of those revenues and expenditures, demands of the come munity and higher levels of governments, etc.r The description of the linear model having a dependent variable Y which is related to two or more independent variables X and X is shown in the following 2 3 model: Y1 = 81 + 82X2 + 83x3 + ... + Ban + “i In this case, the 82, B 8n still denote the slope of the regression 3 ... line. They are usually captioned "partial regression coefficients." Having dealt with the connotation of the various elements in both a simple and multiple linear model, there is no difficulty of a theoretical nature in extending our analysis to the Quantal Response model. Recalling that the formula for the estimation of probability under Quantal Response is 87 - (9+BX) P = l/[l + exp ] with the meaning of the B comparable to that stated in the discussion of the linear model, excepting that the appraisal of the particular variable in classification is seen as an indirect relation to the size of 8, that is, the smaller the estimation of B, the greater the prob- ability of correct classification. Thus the means of estimating the relative importance of predictor variables for Model B is to view the vectors of estimations of B (8) for the categories (as shown in Table 4, for example). With one degree of freedom and, as stated above, estimated probabilities for the three categories summing to l, g is given for only two categories. The estimate of B for the remaining category would be obtainable by conditioning on a different category. However, as structured, the program conditions on the first category and consequently vectors of the estimations are for 82 and 83. It is quite relevant to realize that similar probability estimations were obtained regardless of which category was conditioned upon. The vectors obtained are shown below as Table 4. The variables relevant to the array of estimators would be in reverse chronological order, 1973 through 1971. This ordering possesses no significance other than that of the arbitrary programming of the computer, putting the most recent first. The following order of appearance in each year prevails. Total Expenditures Administrative Expenditures Police Expenditures Fire Expenditures Park and Recreation Expenditures TFhe.relative weight of the é's is the product of those estimated para- nleters and the arithmetical average of each of the fifteen observations 88 TABLE 4 ESTIMATIONS OF VECTORS FOR 82 and B3 UNDER FORMAT 11 £2 Variables E3 .00074 .00051 -.00070 -.00082 -.00534 —.OO624 .00368 .00781 .00600 .00569 -.00083 .00018 .00022 .00452 .01496 .01266 -.02104 -.03456 -.00362 .00017 .00035 -.00117 -.00031 —.00107 —.01334 —.00478 .02139 .02292 .00009 —.00002 (the predictor variables) over the sixty cities. These are arrayed in Table 5 for category 2. Similar computations for category 3 are shown in Table 6. 89 TABLE 5 COMPUTATION OF RELATIVE WEIGHTS OF PREDICTOR VARIABLES OF CATEGORY TWO UNDER FORMAT ll $253233 22 x X. 825‘. Tot Exp 73 .00074 16835.80 12.45849 Adm Exp 73 .00070 1604.13 - 1.12289 Pol Exp 73 .00534 4365.08 -23.30953 Fire Exp 73 .00368 1883.18 6.93010 P&R Exp 73 .00600 873.98 5.24388 Tot Exp 72 .00083 15210.45 -12.62467 Adm Exp 72 .00022 1367.20 .30078 Pol Exp 72 .01496 3867.65 57.86004 Fire Exp 72 .02104 1628.30 -34.25943 98R Exp 72 .00362 837.45 - 3.03157 Tot Exp 71 .00035 14744.88 5.16071 Adm Exp 71 .00031 1257.43 — .38980 Pol Exp 71 .01334 3288.15 -43.86390 Fire Exp 71 .02139 1447.68 30.96590 P&R Exp 71 .00009 1104.38 .09939 90 TABLE 6 COMPUTATION OF RELATIVE WEIGHTS OF PREDICTOR VARIABLES OF CATEGORY THREE UNDER FORMAT ll $§§i§§1§r 83 x i' 83.1 Tot Exp 73 .00051 7132.00 3.63732 Adm Exp 73 .00082 1370.44 -1.12376 P01 Exp 73 .00624 1274.11 -7.95045 Fire Exp 73 .00781 744.56 5.81501 PER Exp 73 .00569 534.67 3.04227 Tot Exp 72 .00018 6566.56 1.18198 Adm Exp 72 .00452 1301.78 5.88404 P01 Exp 72 .01266 1164.89 14.74751 Fire Exp 72 .03456 699.11 -24.16124 PER Exp 72 .00017 603.56 1.02605 Tot Exp 71 .00117 5477.67 —6.40887 Adm Exp 71 .00107 1148.89 —1.22931 P01 Exp 71 .00478 1066.33 —5.09706 Fire Exp 71 .02292 643.22 14.74260 PER Exp 71 .0002 620.22 .01240 91 Although these arrays display the relative weight, a logical extension of the analysis is found in displaying the estimated 82 in order of their effect. In order of absolute size, beginning with the smallest (that one having the greater effect). TABLE 7 RANKING 0F 8 i UNDER FORMAT 11 82 83 P&R Exp 71 P&R Exp 71 Adm Exp 72 P&R Exp 72 Adm Exp 71 Adm Exp 73 Adm Exp 73 Tot Exp 72 P&R Exp 72 Adm Exp 71 Tot Exp 71 P&R Exp 73 P&R Exp 73 Tot Exp 73 Fire Exp 73 P01 Exp 71 Tot Exp 73 Fire Exp 73 Tot Exp 72 Adm Exp 72 P01 Exp 73 Tot Exp 71 Fire Exp 71 Pol Exp 73 Fire Exp 72 Fire Exp 71 P01 Exp 71 P01 Exp 72 P01 Exp 72 Fire Exp 72 92 Because of the relationship of the four expenditures to the total expenditures and the total relationship of the financial items it seems reasonable to conclude that multicollinearity could be present. Consequently, it is logical to look at the application of the entire analysis rather than the parts. Nevertheless insight into the analysis can be gained in another way, looking at the relative importance of the variables: TABLE 8 RANKING OF VARIABLES UNDER FORMAT 11 £32 £53 Tot Exp 73 9 7 Adm Exp 73 4 3 P01 Exp 73 11 12 Fire Exp 73 8 9 P&R Exp 73 7 6 Tot Exp 72 10 4 Adm Exp 72 2 10 P01 Exp 72 15 14 Fire Exp 72 13 15 P&R Exp 5 2 Tot Exp 6 ll Adm Exp 3 5 P01 Exp 71 14 8 Fire Exp 12 13 PER Exp 71 1 1 93 Viewing Table 7 once again discloses that of the first five variables appearing in each columns, four are similar. In addition, it is ob— served, on an intuitive basis, that the expenses having the greatest impact in classification were those which might be described as more discretionary, Administrative and Parks and Recreation; the Police and Fire are more closely aligned with the concept of a fixed cost because of the usual union contracts. An additional casual impression can be realized by observing the rankings of Table 8. By way of emphasis we create Table 9 which is simply the extraction of Administration and Parks and Recreation Expenditures from Table 8 to amplify the importance of the earlier years: TABLE 9 RANKING OF ADMINISTRATIVE, AND PARKS AND RECREATION VARIABLES 43.2 $3 Adm Exp 73 4 3 P&R Exp 73 7 6 Adm Exp 72 2 10 P&R Exp 72 5 2 Adm Exp 71 3 5 P&R Exp 71 1 1 The implication which may be drawn is that there is, simply stated, a time lag. Not only is this lag significant here but the variables of Total Expenditures, Administrative Expenditures, Police Expenditures, Fire Expenditures and Park and Recreation Expenditures 94 for the three years 1972-1974, format 10, one year later than format 11 had a "hit" ratio of only 77 percent. In addition other runs evidenced similar findings. Format 13 utilized the same variables from similar periods as format 11 excepting that Total Expenditures was omitted. That omission resulted in a reduction in the "hit" ratio to 75 percent. More critical was the weakness in classification in Category 1 as illustrated in Table 10. TABLE 10 CLASSIFICATIONS FROM FORMAT 13 Classification as predicted from the fitted model Poor Sat. Rich Poor 1 7 2 ACTUAL CLASSIFICATION sat' 0 41 0 Rich 0 6 3 A similar description of findings will be presented for the other elements in the operating statement. Because of the evidenced weakness in the classificatory ability in the Poor category, further discussion of the results will not be entered into. Format 10 . Affirmation of the belief of time lag is found in the analysis of the results of Format 10 runs. The combination of predictor vari- ables is similar to that of Format 11, excepting that Format 10 covers a period one year later: 95 Total Expenditures 1974, 1973, 1972 Administrative Expenses 1974, 1973, 1972 Police Expense 1974, 1973, 1972 Fire Expense 1974, 1973, 1972 Park and Recreation Expense 1974, 1973, 1972 The criterion variable, as in all other runs, was the percentage of cash to total assets in the general fund, June 30, 1975. Correct classifications decreased from the fifty of Format 11 to forty—four, with the most significant declination in the Poor category. Results are summarized in Table 11: TABLE 11 CLASSIFICATIONS FROM FORMAT 10 Classification as predicted from the fitted model Poor Sat. Rich Poor 0 3 6 ACTUAL CLASSIFICATION sat' 0 40 1 Rich 0 5 4 Another way of viewing the results is to form a "fraction" expressing the "hit" ratio for each class. This involves utilizing the number of "hits" as the numerator and the actual total in the class as the denominator. For Format 10, the diagonal fractions of the matrix read: 0/10, 40/41, 4/9, : 44/60, or, a "hit" ratio of 73.3 percent. The estimated probabilities are shown as Table 12. 96 TABLE 12 ESTIMATED PROBABILITIES OF CLASSIFICATION UNDER FORMAT 10 NEEEZr P1 P2 P3 First Category 1 .0000 .0000 1.0000 2 .0001 .0013 .9986 3 .0000 .0000 1.0000 4 .0000 .0000 1.0000 5 .0000 .0000 1.0000 6 .0142 .1366 .8492 7 .0768 .7491 .1741 8 .2595 .7404 .0000 9 .1354 .8524 .0122 10 .0777 .9011 .0212 Second Category 11 .1297 .8544 .0159 12 .0741 .8680 .0579 13 .0334 .8575 .1091 14 .1625 .8371 .0005 15 .0947 .9008 .0045 16 .1848 .7923 .0229 17 .0418 .9519 .0063 18 .0944 .8567 .0490 19 .0816 .5353 .3830 20 .0273 .9696 .0031 21 .1181 .8805 .0014 22 .1034 .6811 .2155 23 .1036 .8955 .0010 24 .1352 .8585 .0063 25 .1457 .8490 .0052 26 .1167 .8583 .0250 27 .1175 .8786 .0039 97 TABLE 12——Continued NEEEZr P1 P2 P3 28 .1560 .8325 .0115 29 .0541 .4287 .5172 30 .2390 .7520 .0090 31 .0659 .8107 .1234 32 .1280 .8574 .0147 33 .0573 .7660 .1767 34 .2948 .7046 .0006 35 .0842 .9111 .0046 36 .0837 .9098 .0065 37 .1900 .8050 .0050 38 .2799 .7186 .0015 39 .1438 .8248 .0314 40 .0531 .9468 .0001 41 .1150 .8456 .0394 42 .2015 .7850 .0135 43 .1005 .8956 .0039 44 .3049 .6875 .0076 45 .0378 .9598 .0023 46 .1416 .8450 .0134 47 .0542 .6771 .2686 48 .3576 .6414 .0010 49 .1498 .8334 .0167 50 .0046 .9415 .0538 51 .0310 .9465 .0225 Third Category 52 .0000 .0021 .9979 53 .0704 .8261 .1035 54 .0889 .8195 .0916 55 .0717 .7784 .1499 56 .0599 .5880 .3521 98 TABLE 12——Continued City A A A Number P1 P2 P3 57 .0003 .0025 .9972 58 .0143 .0327 .9530 59 .0007 .0072 .9921 60 .1126 .8437 .0437 Then, as previously done, we show an array of the estimated VBC tors 3 TABLE 13 ESTIMATIONS 0F VECTORS FOR 82 AND 83 UNDER FORMAT 10 E2 Variables £3 -.00032 -.00500 .00086 .00309 .00075 .00424 -.00106 -.00982 .00313 .01087 .00033 .00142 .00069 -.00169 -.00498 -.00317 .01055 .03535 .00201 —.00079 .00014 .00458 -.00191 .00321 .00408 .00222 -.00136 -.04150 .00229 .00479 99 TABLE 14 COMPUTATION OF RELATIVE WEIGHTS 0F PREDICTOR VARIABLES OF CATEGORY TWO UNDER FORMAT 10 Predictor A _ x _ Variable 32 x Xi szi Tot Exp 74 -.00032 18063.65 -5.78018 Adm Exp 74 .00086 1832.93 1.57632 P01 Exp 74 .00075 2434.88 1.82616 Fire Exp 74 —.00106 2017.95 -2 13903 PER Exp 74 .00313 1071.66 3.35430 Tot Exp 73 .00033 16500.61 5.44520 Adm Exp 73 .00069 1540.46 1.06292 P01 Exp 73 —.00498 4269.63 -21.26276 Fire Exp 73 .01055 1846.00 19.47530 PER Exp 73 .00201 856.12 1.72080 Tot Exp 72 .00014 14891.61 2.08483 Adm Exp 72 —.00191 1340.44 -2 56024 P01 Exp 72 .00408 3783.41 15.43631 Fire Exp 72 —.01136 1597.22 -18.14442 PER Exp 72 .00229 822.46 1.88343 100 TABLE 15 COMPUTATION OF RELATIVE WEIGHTS 0F PREDICTOR VARIABLES 0F CATEGORY THREE UNDER FORMAT 10 5:53:5in .133 x >'<- 233?. Tot Exp 74 .00500 7406.67 -37.03335 Adm Exp 74 .00309 1524.78 4.71157 Pol Exp 74 .00424 1453.67 6.16356 Fire Exp 74 .00982 799.44 -7.85050 P&R Exp 74 .01087 481.44 5.23325 Tot Exp 73 .00142 7132.00 10.12744 Adm Exp 73 .00169 1370.44 -2.3l604 Pol Exp 73 .00317 1274.11 -4.03893 Fire Exp 73 .03535 744.56 26.32020 P&R Exp 73 .00079 534.67 —0.42239 Tot Exp 72 .00458 6566.56 30.07484 Adm Exp 72 .00321 1301.78 4.17871 Pol Exp 72 .00222 1164.89 2.58606 Fire Exp 72 .04150 699.11 —29.0l307 P&R Exp 72 .00479 603.56 2.89105 101 TABLE 16 RANKING OF 8 i ’b UNDER FORMAT 10 $2-1 §3ii Adm Exp 73 P&R Exp 73 Adm Exp 74 Adm Exp 73 P&R Exp 73 P01 Exp 72 P01 Exp 74 P&R Exp 72 P&R Exp 72 Adm Exp 72 Tot Exp 72 Pol Exp 73 Fire Exp 74 Adm Exp 74 Adm Exp 72 P&R Exp 74 P&R Exp 74 Pol Exp 74 Tot Exp 73 Fire Exp 74 Tot Exp 74 Tot Exp 73 Pol Exp 72 Fire Exp 72 Fire Exp 72 Fire Exp 73 Fire Exp 73 Tot Exp 72 P01 Exp 73 Tot Exp 74 Once again it is possible to take this data and present them in a format comparable to Table 8. However this presentation will involve an additional computation, the averaging of the two rankings. This is shown in Table 17. Because of lack of recognition of 102 differences in group sizes, it should be realized that this is not a weighted average. TABLE 17 RANKING 0F VARIABLES UNDER FORMAT 10 EZRi E3Xi Average Tot Exp 74 11 15 13,0 Adm Exp 74 2 7 4.5 Pol Exp 74 4 9 6.5 Fire Exp 74 7 10 8.5 P&R Exp 74 9 8 8.5 Tot Exp 73 10 11 10.5 Adm Exp 73 1 2 1.5 Pol Exp 73 15 6 10.5 Fire Exp 73 14 12 13.0 P&R Exp 73 3 1 2.0 Tot Exp 72 6 14 10.0 Adm Exp 72 8 5 6.5 Pol Exp 72 12 3 7.5 Fire Exp 72 l3 13 13.0 P&R Exp 72 5 4 4.5 Recognizing that the variables having the lower numbers contribute most to the correct classification, a visual review of Table 17 leads to the appraisal that it seldom matters how much is spent (Total Expenditures) rather, the discretionary amounts (Administration and 103 Parks and Recreation) have greater impact on the statistical outcome than the expenditures which approach fixed costs by nature of union contracts (Police and Fire). The lag effect, discussed as it was recognized in analysis of format 11 results, is present in format 10 although not as pronounced. An interesting combination of the ranking is shown in Table 18 which presents the averages of the four year span of formats 10 and 11. It should be stated that although both had the same criterion variable, results are sensitive to the combination of predictor variables. Table 18 is shown on page 104. Formats 8 and 9 . . . The two primary sources of revenue for the General Fund of a municipality in the State of Michigan are Property Tax Income and those items categorized under the general caption of State Shared Revenues. As commented on previously, this latter amount will con— sist of those tax revenue items collected by the state and then dis— tributed on the basis of a "sharing" formula. These items consist of, but are not limited to, sales and use tax, state income tax, liquor licenses, and intangibles tax. The two items generally comprise approximately 60 to 70 percent of total revenues, however if a personal income tax is present, the percentage drops significantly to 40 to 50 percent or less. Property Tax Income, derived from local taxes on both real and personal property, is to some extent a measure of the base value of taxable property in that it is standardized to a degree through State Equalized Values. Essentially then, both property tax income and State Shared Revenues are somewhat beyond the immediate discretion of the administration of the municipality. 104 TABLE 18 AVERAGE OF RANKINGS 0F VARIABLES 0F FORMATS 10 AND 11 Format 10 Format 11 Average Average Total Exp 74 13.0 - Adm Exp 74 4.5 - Pol Exp 74 6.5 - Fire Exp 74 8.5 - P&R Exp 74 .5 - Tot Exp 73 10.5 8.0 Adm Exp 73 1.5 .5 P01 Exp 73 10.5 11.5 Fire Exp 73 13.0 8.5 P&R Exp 73 2.0 6.5 Total Exp 72 10.0 7.0 Adm Exp 72 6.5 6.0 Pol Exp 72 7.5 14.5 Fire Exp 72 13.0 14.0 P&R Exp 72 4.5 3.5 Tot Exp 71 - 8.5 Adm Exp 71 - 4.0 Pol Exp 71 - 11.0 Fire Exp 71 - 12.5 P&R Exp 71 - 1.0 105 Formats 8 and 9 investigated these two particular revenues combined with the total revenues of the municipality. In summary, two things could be stated. First, the lag effect was present once again but not as strong as in the expenditures analysis. Second, there was a slight deterioration in correct classification in comparison to the analysis of expenditures under formats 10 and 11. The erosion which was most noticed is found in the Poor category. Format 8 utilized the nine variables listed below: Total Revenues 1973, 1972, 1971 Property Tax Income 1973, 1972, 1971 State Shared Revenues 1973, 1972, 1971 There is no way to evaluate the effect of six less variables other than the classification table. The classification results for format 8 are presented as Table 19: TABLE 19 CLASSIFICATIONS FROM FORMAT 8 Classification as predicted from the fitted model Poor Sat. Rich Poor 4 2 1 ACTUAL CLASSIFICATION sat' 6 39 6 Rich 0 0 2 The "hit" rates can be derived from this table: 4/10, 39/41, 2/9 : 45/60 which is then stated as a 75% correct classification. Probability estimations are arrayed in Table 20. 106 TABLE 20 ESTIMATED PROBABILITIES OF CLASSIFICATION UNDER FORMAT 8 Ngifiir P1 P2 P3 First Category 1 .4439 .5448 .0213 2 .1652 .7503 .0845 3 .8294 .1533 .0173 4 .3217 .5631 .1152 5 .4804 .4508 .0688 6 .0925 .8208 .0867 7 .0544 .8193 .1263 8 .0741 .7790 .1469 9 .8673 .1162 .0165 10 .4790 .4568 .0642 Second Category 11 .0855 .7709 .1436 12 .0756 .7985 .1259 13 .0282 .9387 .0331 14 .0541 .8678 .0781 15 .1397 .7787 .0816 16 .0644 .8467 .0889 17 .1139 .7450 .1411 18 .0537 .8396 .1067 19 .0747 .8270 .0982 20 .0250 .9427 .0323 21 .0809 .8374 .0816 22 .0499 .7677 .1824 23 .0555 .8309 .1137 24 .0346 .8734 .0920 25 .0408 .8210 .1382 26 .0654 .7932 .1413 27 .1699 .7340 .0961 107 TABLE 20--Continued N31111:; P1 P2 1’3 28 .1830 .7567 .0603 29 .0931 .7479 .1590 30 .2895 .6141 .0963 31 .0808 .8316 .0876 32 .1793 .7001 .1206 33 .0525 .8015 .1460 34 .0380 .6348 .3272 35 .0630 .8850 .0520 36 .1071 .8014 .0915 37 .0590 .8118 .1292 38 .5431 .8366 .1091 39 .2780 .6514 .0706 40 .9362 .0566 .0072 41 .0261 .7909 .1831 42 .0401 .8230 .1369 43 .4256 .4772 .0972 44 .0076 .7392 .2532 45 .0210 .8931 .0859 46 .0372 .7924 .1704 47 .0230 .6369 .3401 48 .1477 .8260 .0263 49 .0711 .8341 .0948 50 .5709 .3194 .1097 51 .0977 .7233 .1790 Third Category 52 .0006 .0241 .9753 53 .6071 .3373 .0555 54 .3193 .8770 .0910 55 .0615 .8167 .1218 56 .1223 .7779 .0998 108 TABLE 20--Continued City A A 3 Number P1 P2 P3 57 .0185 .1897 .7919 58 .0784 .8442 .0774 59 .0706 .8421 .0873 60 .0372 .8542 .1086 Once again, observation of misclassifications shows only 15 misses were recorded; of those 15, only one was not in the next group. TABLE 21 INCORRECT CLASSIFICATIONS UNDER FORMAT 8 City Actual Classified as Number Classified From the in Group Fitted Model 1 1 2 2 l 2 4 l 2 6 l 2 7 1 2 8 l 2 38 2 1 4O 2 l 53 3 l 54 3 2 55 3 2 56 3 2 58 3 2 59 3 2 6O 3 2 109 The estimation of the E's for the nine variables is shown in Table 22. TABLE 22 ESTIMATIONS 0F VECTORS FOR 82 AND 83 UNDER FORMAT 8 A E2 Variables £3 .00010 -.00094 -.00053 .00005 —.00996 —.00749 .00118 .00140 -.00017 .00023 -.00209 -.00528 -.00078 .00045 .00041 -.00071 .01020 .01012 The means of the variables of the first category when multi- plied by the appropriate vectors Inxxhxxa the relative weight of each variable. TABLE 23 110 COMPUTATION OF RELATIVE WEIGHTS 0F PREDICTOR VARIABLES 0F CATEGORY TWO UNDER FORMAT 8 52:1:812r 82 x Xi EZXI Tot Rev 73 .00010 17325.07 1.73251 Pry Tax 73 -.00053 4950.12 —2.62356 SS Rev 73 -.00996 2330.49 -23.21168 Tot Rev 72 .00118 17184.37 20.27756 Pry Tax 72 -.00017 5880.39 0.99967 SS Rev 72 -.00209 2475.73 5.17428 Tot Rev 71 -.00078 14551.76 11.35037 Pry Tax 71 .00041 5678.46 2.32817 SS Rev 71 .01020 1772.12 18.07562 Table 24 presents COMPUTATION OF RELATIVE WEIGHTS 0F PREDICTOR VARIABLES similar computations for Category Three. TABLE 24 0F CATEGORY THREE UNDER FORMAT 8 5::IigIZr E3 X Xi §3Xi Tot Rev 73 .00094 7806.44 —7.33805 Pry Tax 73 .00005 4061.00 0.20305 SS Rev 73 .00749 1121.22 -8.39738 Tot Rev 72 .00140 7518.78 10.52629 Pry Tax 72 .00023 4067.56 0.93554 SS Rev 72 .00528 1032.22 -5.45012 Tot Rev 71 .00048 6852.67 3.28928 Pry Tax 71 .00071 3781.11 —2.68459 SS Rev 71 .01012 928.56 9.39703 These rankings are arrayed in Table 25: 111 TABLE 25 RANKING OF VARIABLES UNDER FORMAT 8 g i E E. Average Tot Rev 73 2 6 4.0 Pry Tax 73 4 l 2.5 SS Rev 73 9 7 8.0 Tot Rev 72 8 9 8.5 Pry Tax 72 1 2 1.5 SS Rev 72 5 5 5.0 Tot Rev 71 6 4 5.0 Pry Tax 71 3 3 3.0 SS Rev 71 7 8 7.5 By reformulating the ranking we can emphasize the variables contributing the most to the classification beginning with the smallest, that which contributes most. TABLE 26 RANKING OF éii UNDER FORMAT 8 szi E3Xi Pry Tax 72 Pry Tax 73 Tot Rev 73 Pry Tax 72 Pry Tax 71 Pry Tax 71 Pry Tax 73 Tot Rev 71 SS Rev 72 SS Rev 72 Tot Rev 71 Tot Rev 73 SS Rev 71 SS Rev 73 Tot Rev 72 SS Rev 71 SS Rev 73 Tot Rev 72 112 Comments will be reserved until following format 9. In format 9 nine similar variables are used with an advance of one year: Total Revenues 1974, 1973, 1972 Property Tax Income 1974, 1973, 1972 State Shared Revenues 1974, 1973, 1972 The presentation of classifications is in Table 27: TABLE 27 CLASSIFICATIONS FROM FORMAT 9 Classification as predicted from the fitted model Poor Sat. Rich Poor 3 7 ACTUAL CLASSIFICATION 33" 2 39 0 Rich 0 8 1 The 43 "hits" (for a percentage of 71.7 correct classification) can be summarized in fractions: 3/10, 39/41, 1/9, : 43/60 However two of the probabilities were sufficiently close so as to war— rant mention. For the first city in Category One the classification was: P1 = .4330, P2 = .4668 for the sixth city in Category Three, P1 = .4879 and P2 = .4698 Correct classification would have improved both "tails", the areas of difficult classification. 113 In accord with previous presentations estimated B's are shown: TABLE 28 ESTIMATION OF VECTORS FOR 82 AND 83 UNDER FORMAT 9 £2 Variables é3 -.00026 -.00051 -.00056 -.00009 -.00793 -.00420 .00136 .00224 -.00175 -.00116 .00897 .00550 -.00050 -.00105 .00286 .00083 -.00569 -.00601 The estimated vectors are multiplied against the mean of the variable to produce the relative weight: 114 TABLE 29 COMPUTATION OF RELATIVE WEIGHTS OF PREDICTOR VARIABLES OF CATEGORY TWO UNDER FORMAT 9 Predictor Variable E2 x Xi §2X1 Tot Rev 74 -.00026 17184.37 -4.46794 Pry Tax 74 —.00056 5880.39 -3.29302 SS Rev 74 —.00793 2475.73 -19.63254 Tot Rev 73 .00136 14551.76 19.79039 Pry Tax 73 -.00175 5678.46 —9.9373l SS Rev 73 .00897 1772.12 15.89592 Tot Rev 72 -.00050 14852.80 -7.42640 Pry Tax 72 .00286 5352.34 15.30769 SS Rev 72 -.00569 -6.39l98 The approach is now applied to Category Three: TABLE 30 COMPUTATION OF RELATIVE WEIGHTS OF PREDICTOR VARIABLES 0F CATEGORY THREE UNDER FORMAT 9 Predictor “ -, A - Variable E3 x X1 83x1 Tot Rev 74 -.00051 7518.78 -3.83458 Pry Tax 74 -.00009 4067.56 -0.36608 SS Rev 74 —.00420 1032.22 -4.33532 Tot Rev 73 .00224 6852.67 15.35000 Pry Tax 73 -.00116 3781.11 -4.38609 SS Rev 73 .00550 928.56 5.10708 Tot Rev 72 -.00105 5461.78 -5.70337 Pry Tax 72 .00083 2659.00 2.20697 SS Rev 72 -.00601 774.78 -4.65643 115 The combination of these two rankins in Table 31 will assist in formulation of general statements: TABLE 31 RANKING OF VARIABLES UNDER FORMAT 9 82-1 83—1 Average Tot Rev 74 2 3 2.5 Pry Tax 74 l 1 1.0 SS Rev 74 8 4 6.0 Tot Rev 73 9 9 9.0 Pry Tax 73 5 5 5.0 88 Rev 73 7 7 7.0 Tot Rev 72 4 8 6.0 Pry Tax 72 6 2 4.0 SS Rev 72 3 6 4.5 Because of the smaller group, the averages will not have as great a visual impact. However, as might be anticipated, those variables of a wider ranking will become less influential in the classification. The comparison for the four—year period is achieved through the com- bining of formats 8 and 9 in Table 32. 116 TABLE 32 AVERAGE OF RANKINGS OF VARIABLES UNDER FORMATS 8 AND 9 Format 9 Format 8 Average Average Tot Rev 74 Pry Tax 74 SS Rev 74 Tot Rev 73 Pry Tax 73 SS Rev 73 Tot Rev 72 Pry Tax 72 SS Rev 72 Tot Rev 71 — Pry Tax 71 - SS Rev 71. - O | J-‘J-‘Chwkflkoc‘sl-‘N MOOOU‘IU'OUIO MOOOOO \IUQU'IU'II-‘mCDN-L‘ If the variables are partitioned into years, it may be noted that the Property Tax is consistently the lowest in each year, thus contribut- ing the most to the proper classification of observations. One inter- pretation of this prominence of Property Tax Revenue might be that the Total Revenues, on a comparative basis, are not as important for estimation as are the sources of those revenues, particularly the amount generated by property tax. One difficulty in perfecting the analysis to any considerable degree is found in the lack of information in the financial reports as to whether tax revenues are from residential property, or non— residential property and their proportion to the total base value 117 of the property in the municipality. Trends in the relative amounts contributed by each category of property might also potentially im— prove the analysis. Finally, it should be noted that although the previous analysis dealt with expenditures which were discretionary to a degree, revenues are much less controllable by the city management. Formats 1 and 2 . . . The first two runs investigated the classificatory ability of the Total General Fund and the Assets, Cash, Past Due Taxes Receivable and the receivable, Interfund Borrowings Due from (other funds). For- mat 1 utilized twelve variables: Total General Fund 1974, 1973, 1972 Cash 1974, 1973, 1972 Past Due Taxes Receivable 1974, 1973, 1972 Interfund Borrowing Due from 1974, 1973, 1972 The program assigned the municipalities correctly in 76.7 percent of the cases: TABLE 33 CLASSIFICATIONS FROM FORMAT 1 Classification as predicted from the fitted model Poor Sat. Rich Poor 4 6 0 ACTUAL CLASSIFICATION Sat° 1 39 1 Rich 0 6 3 These results are expressed in fractional from: 4/10, 30/41, 3/9 : 46/60 118 The weakness in tail classification appears moderate. Estimations of probabilities are shown in Table 34 so that the reader may observe the classification strength. TABLE 34 ESTIMATED PROBABILITIES OF CLASSIFICATION UNDER FORMAT 1 NEiBZr P1 P2 P3 First Category 1 .1444 .7868 .0688 2 .0413 .8115 .1472 3 .0190 .6759 .3051 4 .9992 .0007 .0000 5 .9944 .0056 .0000 6 .0716 .8484 .0800 7 .0519 .8319 .1162 8 .8535 .1390 .0075 9 .1914 .8071 .0015 10 .5390 .4595 .0015 Second Category 11 .0192 .8544 .1264 12 .1098 .7618 .1284 13 .4483 .4051 .1466 14 .0441 .9496 .0063 15 .0288 .8990 .0722 16 .0356 .8736 .0908 17 .0704 .8485 .0811 18 .1457 .8304 .0239 19 .0480 .9128 .0392 20 .0133 .9759 .0108 21 .0143 .9330 .0527 22 .0075 .9653 .0272 23 .1372 .7676 .0952 119 TABLE 34--Continued N:;5:r P1 P2 P3 24 .1054 .8615 .0331 25 .0443 .8907 .0650 26 .0362 .8489 .1149 27 .0276 .9330 .0394 28 .0502 .8980 .0519 29 .1053 .8452 .0494 30 .1191 .8505 .0304 31 .0265 .9727 .0008 32 .0726 .8788 .0486 33 .0731 .9198 .0071 34 .0021 .7274 .2705 35 .0501 .9269 .0230 36 .0848 .8110 .1042 37 .1083 .8777 .0140 38 .0166 .9758 .0076 39 .0298 .7488 .2215 40 .0103 .9738 .0160 41 .0286 .8714 .1000 42 .0252 .8026 .1722 43 .2377 .7410 .0213 44 .0095 .4342 .5564 45 .0729 .9024 .0247 46 .0425 .9175 .0401 47 .0272 .8811 .0917 48 .0521 .7674 .1805 49 .0802 .8256 .0942 50 .1531 .8235 .0235 51 .3287 .6073 .0640 Third Category 52 .0001 .0046 .9953 120 TABLE 34-—Continued xiiiir P1 P2 P3 53 .0463 .8818 .0719 54 .0362 .8854 .0784 55 .0733 .7982 .1285 56 .0904 .8217 .0880 57 .0002 .0219 .9779 58 .0571 .7680 .1749 59 .0005 .1595 .8399 60 .0300 .8292 .1412 Of those sixty classifications, three were sufficiently narrow to warrant citation: City 10: Pl City 13: P1 City 44: P2 Of these three, City ful classification could be difficult, or questionable. = .5390, P2 = .4595 = .4483, P2 = .4051 = .4342, P3 = .5564 13 is most likely the only one for which meaning- The vectors were estimated and are shown in the following table. 121 TABLE 35 ESTIMATION OF VECTORS FOR 82 AND 83 UNDER FORMAT l 8 Variables B m m -.00078 -.00208 .00866 .02205 .00135 .00423 -.12096 .00462 —.01155 —.02665 -.00048 .00219 .25494 .16672 -.00246 -.00287 -.00290 —.00100 -.00367 —.00470 -.12044 -.06293 The relative weight of the asset predictor variables for format 1 can be calculated in this manner: 122 TABLE 36 COMPUTATION OF RELATIVE WEIGHTS 0F PREDICTOR VARIABLES OF CATEGORY TWO UNDER FORMAT l iiiiifiiir 52 x Xi = , EZXi Tot GF 74 -.00078 1537.95 —1.19960 Cash 74 .00866 389.61 3.37402 PD Tax 74 .00135 909.12 1.22731 IFBDF 74 -.12096 46.07 -5.57262 Tot GF 73 .00310 1104.49 3.42392 Cash 73 -.o1155 378.59 -4.37271 PD Tax 73 —.00048 770.17 -O.36968 IFBDF 73 .25494 45.61 11.62781 Tot GF 72 —.00246 806.44 —1.98384 Cash 72 .00290 331.56 0.96152 PD Tax 72 —.00367 415.32 -1.52422 IFBDF 72 -.12044 33.88 -4.08051 Similar computations are now made for the third category. 123 TABLE 37 COMPUTATION OF RELATIVE WEIGHTS OF PREDICTOR VARIABLES 0F CATEGORY THREE UNDER FORMAT l Predictor Variable EB x Xi E3XI Tot GF 74 .00208 2261.67 4.70427 Cash 74 .02205 117.56 2.59220 PD Tax 74 .00423 102.33 0.43286 IFBDF 74 .00351 30.78 0.10804 Tot GF 73 .00462 2207.67 10.19944 Cash 73 .02665 103.00 2.74495 PD Tax 73 .00219 113.78 0.24918 IFBDF 73 .16672 26.22 4.37140 Tot GF 72 .00287 1794.89 5.15133 Cash 72 .00100 115.44 0.11544 PD Tax 72 .00470 114.33 0.53735 IFBDF 72 .06293 21.78 1.37062 These rankings result in Table 38 which presents the variables listed in order of most important through least important. 124 TABLE 38 RANKING OF gii UNDER FORMAT 1 szi 83 1 PD Tax 73 IFBDF 74 Cash 72 Cash 72 Tot Gf 74 PD Tax 73 PD Tax 74 PD Tax 74 PD Tax 72 PD Tax 72 Tot GF 72 IFBDF 72 Cash 74 Cash 74 Tot GF 73 Cash 73 IFBDF 72 IFBDF 73 Cash 73 Tot GF 74 IFBDF 74 Tot GF 72 IFBDF 73 Tot GF 73 This array can be presented as in previous viewings. Table 39 shows the ordering and averages of that ordering. 125 TABLE 39 RANKING OF VARIABLES UNDER FORMAT l E2 1 B321 Average Tot GF 73 3 10 6.5 Cash 7 7 7.0 PD Tax 74 4 4 4.0 IFBDF 74 ll 1 6.0 Tot GF 73 8 12 10.0 Cash 73 10 8 9.0 PD Tax 73 l 3 2.0 IFBDF 73 12 9 10.5 Tot GF 72 6 11 8.5 Cash 72 2 2 2.0 PD Tax 72 5 5 5.0 IFBDF 72 9 6 7.5 The consistently high ranking of the three Past Due Taxes Receivable as a classifying variable is in accord with perceptions prior to the analysis. It was assumed that the amount of taxes which were past due would provide a good predictor for intuitively there should be pronounced differences. Those residents in the cash rich cities will be less subject to the situations in which taxes will become past due. The introduction of format 2 finds the time lag missing. Table 40 shows the classifications: 126 TABLE 40 CLASSIFICATIONS FROM FORMAT 2 Classification as predicted from the fitted model Poor Sat. Rich Poor 3 7 0 ACTUAL CLASSIFICATION sat' 0 39 3 Rich 0 7 2 The fractions then are: 3/10, 39/41, 2/9 : 44/60 which results in a "hit" ratio of 73.3 percent. The probabilities comprising those fractions are: TABLE 41 ESTIMATED PROBABILITIES OF CLASSIFICATION UNDER.FORMAT 2 xfiifiir P1 P2 P3 First Category 1 .1214 .8275 .0511 2 .2518 .7278 .0004 3 .2832 .3827 .3341 4 .9864 .0136 .0001 5 .9669 .0326 .0004 6 .0543 .8006 .1451 7 .0710 .8685 .0605 8 .5351 .4610 .0031 9 .3874 .6116 .0010 10 .1625 .8321 .0054 127 TABLE 4l--Continued City g '11) Number 1 2 P3 Second Category 11 .0442 .9061 .0498 12 .9757 .8021 .1222 13 .0446 .9214 .0340 14 .4254 .5499 .0247 15 .9787 .8577 .0636 16 .9551 .8805 .0644 17 .0652 .6563 .2786 18 .1155 .8202 .0643 19 .0785 .8751 .0464 20 .0282 .9680 .0112 21 .0458 .9173 .0369 22 .1965 .7857 .0178 23 .1384 .7647 .0970 24 .0712 .8876 .0412 25 .0606 .8767 .0628 26 .0565 .8826 .0609 27 .0613 .9130 .0258 28 .0728 .8338 .0934 29 .2695 .7183 .0122 30 .1061 .8804 .0135 31 .1953 .7972 .0075 32 .0854 .8167 .0979 33 .1339 .8494 .0167 34 .0153 .5332 .4514 35 .1130 .8668 .0202 36 .0997 .8900 .0103 37 .0959 .8827 .0214 38 .1716 .8173 .0111 39 .3766 .4202 .5421 40 .0803 .8501 .0696 128 TABLE 41-—Continued NEAEZ. P1 P2 P3 41 .0419 .9147 .0434 42 .0431 .8879 .0690 43 .2974 .6920 .0106 44 .0010 .4513 .5477 45 .4370 .4529 .1100 46 .0604 .9212 .0184 47 .0614 .8178 .1209 48 .0435 .7746 .1819 49 .0698 .8374 .0928 50 .4205 .5713 .0082 51 .2013 .7110 .0877 Third Category 52 .0001 .0023 .9977 53 .0471 .8616 .0913 54 .0479 .8732 .0788 55 .0159 .6250 .3591 56 .0569 .7973 .1458 57 .0004 .0762 .9233 58 .0399 .7553 .2049 59 .0697 .6619 .2684 60 .0433 .7899 .1668 Review of Table 41 indicates predominately precise classifi— cation excepting for the city numbered 3 which is spread over the three categories: '11) *U) I F = .2832, - .3341 1 = .3827, 2 3 129 In addition the city numbered 45 did not have precise classi— fication: P1 = .4370, P2 = .4529 The presentation of the 8's is made below: 'b TABLE 42 ESTIMATION OF VECTORS FOR 82 AND 83 UNDER FORMAT 2 éZ Variables £3 .00197 .00170 .00043 .00780 .00103 .00019 -.07085 —.20574 -.00268 .00060 -.00981 -.02197 -.00530 -.00661 .13378 .11832 .00003 -.00205 .00746 .00205 -.00018 -.00400 .01224 .32215 As in prior analyses, the relative weights are calculated: 130 TABLE 43 COMPUTATION OF RELATIVE WEIGHTS OF PREDICTOR VARIABLES OF CATEGORY TWO UNDER FORMAT 2 52:1:8122 32 x Xi szi Tot GF 73 .00197 1104.49 2.17584 Cash 73 .00043 378.59 1.62794 PD Tax 73 .00103 770.17 0.79328 IFBDF 73 —.O7085 45.61 -3.23147 Tot GF —.00268 806.44 -2.16126 Cash 72 —.OO981 331.56 —3.25260 PD Tax -.00530 415.32 —2.20120 IFBDF 72 .13378 33.88 4.53247 Tot GF 71 .00003 591.80 0.01775 Cash 71 .00746 296.41 2.21122 PD Tax 71 —.00018 290.85 -0.05235 IFBDF 71 .01224 32.93 0.40306 Relative weights are shown for Category Three in Table 44: 131 TABLE 44 COMPUTATION OF RELATIVE WEIGHTS OF PREDICTOR VARIABLES OF CATEGORY THREE UNDER FORMAT 2 Predictor Variables §3xi x Xi E3Xi Tot GF 73 .00170 2207.67 3.75303 Cash 73 .00780 103.00 0.80340 PD Tax 73 .00019 113.78 0.02162 IFBDF 73 -.20574 26.22 -5.39450 Tot GF 72 .00060 1794.89 1.07693 Cash 72 -.02197 115.44 -2.53622 PD Tax 72 -.00661 114.33 -0.75572 IFBDF 72 .11832 21.78 2.57701 Tot GF 71 —.00205 1713.67 -3.5l302 Cash 71 .00205 138.00 0.28290 PD Tax 71 -.00400 163.78 -0.65512 IFBDF 71 .32215 20.89 6.72971 These computations may be ranked for the following order: 132 TABLE 45 RANKING OF Sii UNDER FORMAT 2 ’\1 A A- szi §3Xi Tot GF 71 PD Tax 73 PD Tax 71 Cash 71 IFBDF 71 PD Tax 73 PD Tax 73 PD Tax 72 Cash 73 Cash 73 Tot GF 72 Tot GF 72 Tot GF 73 Cash 72 PD Tax 72 IFBDF 72 Cash 71 Tot GF 71 IFBDF 73 Tot GF 73 Cash 72 IFBDF 73 IFBDF 72 IFBDF 71 preliminary to intuitive comments: From this ranking the following additional array can be devised as a RANKING OF VARIABLES UNDER FORMAT 2 133 TABLE 46 éZii 83—1 Average Tot GF 73 6 10 8.5 Cash 73 5 5 5.0 PD Tax 4 l 2.5 IFBDF 73 10 11 10.5 Tot GF 6 6 6.0 Cash 72 ll 7 8.5 PD Tax 72 8 4 6.0 IFBDF 72 12 8 10.0 Tot GF 71 l 9 5.0 Cash 71 9 2 5.5 PD Tax 71 2 3 2.5 IFBDF 71 3 12 7.5 From this listing two variables appear to have consistent strength, Past Due Taxes Receivable 1973 and 1971. Table 39 shows the 1973 Past Due Taxes Receivable of a like consistent strength, but compar- able in average to Cash 72. Formats l6 and 17 . . . The final runs to be presented are an inquiry into the utiliza- tion of ratios in the classification of municipalities. Ratios were drawn from combinations of elements from both statements. For format 16 three ratios spanned two years while the other two covered four years; this total of fourteen was one less than the capacity of the 134 model. The variables utilized were: Cash/Total Liabilities 1974, 1973 Cash/Total Revenues 1974, 1973, 1972, 1971 Total Liabilities/Total General Fund 1974, 1973 Interfund Borrowing Due From/Total General Fund 1974, 1973, 1972, 1971 Interfund Borrowing Due To/Total General Fund 1974, 1973 The ratios utilized numerator and denominator drawn from the year's data indicated in the caption. For example, the first ratio was Cash at the end of 1974 divided by total liabilities at the end of 1974. The other variables in order were Cash and Total Revenue, Total Liabilities as a percentage of the Total General Fund, Interfund Borrowing due from the other funds as a percentage of the Total General Fund and Interfund Borrowing (by the General Fund) due to other funds as a percentage of the Total General Fund. Selection of these vari- ables was based upon results of runs preceding format 16. The analysis resulted in the correct classification of 49 municipalities: TABLE 47 CLASSIFICATIONS FROM FORMAT l6 Classification as predicted from the fitted model Poor Sat. Rich Poor 6 4 0 ACTUAL CLASSIFICATION sat' 0 38 3 Rich 0 4 5 The hit ratios derived from this table are: 6/10, 38/41, 5/9 : 49/60 135 The probability estimation from application of these ratios. TABLE 48 ESTIMATED PROBABILITIES OF CLASSIFICATION UNDER FORMAT 10 City *6) P *U) Number 1 2 3 First Category 1 .8579 .1420 .0001 2 .9236 .0757 .0007 3 .3479 .4412 .2109 4 .9954 .0044 .0000 5 .6138 .3776 .0086 6 .2466 .7376 .0157 7 .0607 .9361 .0032 8 .9875 .0125 .0000 9 .9837 .0163 .0000 10 .2916 .7003 .0080 Second Category 11 .0079 .9715 .0207 12 .0053 .7587 .2360 13 .1480 .8405 .0115 14 .0008 .9259 .0733 15 .0324 .8762 .0914 16 .0045 .9001 .0953 17 .0110 .8880 .1010 18 .3148 .6849 .0003 19 .3639 .6297 .0064 20 .0924 .8982 .0093 21 .0002 .9956 .0042 22 .0278 .9694 .0028 23 .0744 .8623 .0633 24 .0749 .9079 .0172 25 .0836 .8875 .0288 TABLE 48--Continued NEEEZI P1 P2 P3 26 .2246 .7332 .0421 27 .0001 .9969 .0030 28 .0023 .9849 .0127 29 .0373 .9509 .0118 30 .1388 .8562 .0050 31 .0007 .9991 .0001 32 .3154 .6777 .0069 33 .0060 .9930 .0010 34 .0001 .1905 .8094 35 .1492 .8325 .0184 36 .0256 .9708 .0036 37 .0035 .9725 .0239 38 .0006 .9975 .0019 39 .0027 .4884 .5090 40 .0030 .8914 .1056 41 .0275 .9295 .0430 42 .0814 .7664 .2254 43 .0071 .6773 .3157 44 .0002 .6703 .3295 45 .1462 .8475 .0062 46 .0324 .8553 .1123 47 .2721 .7243 .0037 48 .0209 .7033 .2759 49 .1776 .8220 .0004 50 .1304 .7599 .1097 51 .0002 .0238 .9759 Third Category 52 .0039 .6406 .3554 53 .0287 .9302 .0411 54 .0294 .4965 .5005 137 TABLE 48-—Continued NSEHZr P1 P2 P3 55 .0014 .4034 .5953 56 .0016 ] .5876 .4108 57 .0000 .0060 .9940 58 .0692 .2047 .7261 59 .0001 .0504 .9496 60 .0020 .5143 .4837 Of these cities, three classifications were with less than clear definition. City A A Number P2 P3 39 .4884 .5090 54 .4965 .5005 60 .5143 .4837 Of these three, one (60) was incorrectly classified by the model. Classification was always into the next adjacent class if wrong. Vectors were estimated as follows: ESTIMATION OF VECTORS FOR 82 AND B 138 TABLE 49 3 UNDER FORMAT 16 E2 Variables £3 .35843 .43800 .15805 .80548 2.62740 2.99873 19.94017 18.76468 ~12.77597 19.65129 -3.81666 3.53225 3.48423 4.31339 -3.82480 -5.93604 4.25447 -l.51457 5.23904 9.20041 —3.85439 -9.21081 -4.53639 -5.31218 -12.08583 -9.76234 .52254 2.74759 These vectors, when multiplied against the means of Category Two and Three give the relative weight of the variables. 139 TABLE 50 COMPUTATION OF RELATIVE WEIGHTS OF PREDICTOR VARIABLES OF CATEGORY TWO UNDER FORMAT 16 52:88:: P32 = Cash/Tot Liab 74 .35843 1.79 .64159 Cash/Tot Liab 73 .15805 1.93 .30504 Cash/Tot Rev 74 2.62740 .13 .34156 Cash/Tot Rev 73 19.94017 .14 2.79162 Cash/Tot Rev 72 —12.77597 .14 —1.78864 Cash/Tot Rev 71 -3.81666 .13 - .49617 Tot Liab/Tot Gen Fund 74 3.48423 .44 1.53306 Tot Liab/Tot Gen Fund 73 —3.82480 .39 —l.49167 IB from/Tot Gen Fund 74 4.25447 .30 1.27634 IB from/Tot Gen Fund 73 5.23904 .24 1.25737 IB from/Tot Gen Fund 72 -3.85439 .23 -l.15632 IB from/Tot Gen Fund 71 —4.53639 .17 .77119 IB to/Tot Gen Fund 74 -12.08583 .09 1.08773 13 to/Tot Gen Fund 73 .52254 .08 .04180 The computation is then repeated for Category Three. 140 TABLE 51 COMPUTATION OF RELATIVE WEIGHTS OF PREDICTOR VARIABLES OF CATEGORY THREE UNDER FORMAT l6 32:83:: A x = 5.. Cash/Tot Liab 74 .43830 4.35 1.90530 Cash/Tot Liab 73 .80548 4.41 3.55217 Cash/Tot Rev 74 2.99873 .33 .98958 Cash/Tot Rev 73 18.76468 .29 5.44176 Cash/Tot Rev 72 19.65129 .27 5.30584 Cash/Tot Rev 71 3.53225 .26 .91839 Tot Liab/Tot Gen Fund 74 4.31339 .21 .90581 Tot Liab/Tot Gen Fund 73 -5.93604 .25 1.48401 IB from/Tot Gen Fund 74 -1.51457 .02 '.03029 IB from/Tot Gen Fund 73 9.20041 .06 .55202 IB from/Tot Gen Fund 72 -9.21081 .04 .36843 IB from/Tot Gen Fund 71 -5.31218 .05 .26561 IB to/Tot Gen Fund 74 -9.76234 .03 .29297 IB tO/Tot Gen Fund 73 2.74759 .03 .08243 These weighted variables can then be arrayed to emphasize their ranking: 141 TABLE 52 RANKING OF éiiUNDER FORMAT l6 EZXi E3Xi IB to/Tot Gen Fund 73 IB from/Tot Gen Fund 74 Cash/Tot Liab 73 IB to/Tot Gen Fund 73 Cash/Tot Rev 74 IB from/Tot Gen Fund 71 Cash/Tot Rev 71 IB to/Tot Gen Fund 74 Cash/Tot Liab 74 IB from/Tot Gen Fund 72 IB from/Tot Gen Fund 71 IB from/Tot Gen Fund 73 IB to/Tot Gen Fund 74 Tot Liab/Tot Gen Fund 74 IB from/Tot Gen Fund 72 Cash/Tot Rev 71 IB from/Tot Gen Fund 73 Cash/Tot Rev 74 IB from/Tot Gen Fund 74 Tot Liab/Tot Gen Fund 73 Tot Liab/Tot Gen Fund 73 Cash/Tot Liab 74 Tot Liab/Tot Gen Fund 74 Cash/Tot Liab 73 Cash/Tot Rev 72 Cash/Tot Rev 72 Cash/Tot Rev 73 Cash/Tot Rev 73 142 TABLE 5 3 RANKING OF VARIABLES UNDER FORMAT 16 éZii 83K. Average Cash/Tot Liab 74 11 8.0 Cash/Tot Liab 73 2 12 7.0 Cash/Tot Rev 74 3 9 6.0 Cash/Tot Rev 73 14 14 14.0 Cash/Tot Rev 72 l3 13 13.0 Cash/Tot Rev 71 4 8 6.0 Tot Liab/Tot Gen Fund 74 12 7 8.5 Tot Liab/Tot Gen Fund 73 11 10 10.5 IB from/Tot Gen Fund 74 10 1 5.5 IB from/Tot Gen Fund 73 9 6 7.5 IB from/Tot Gen Fund 72 8 5 6.5 IB from/Tot Gen Fund 71 6 3 4.5 IB to/Tot Gen Fund 74 7 4 5.5 IB to/Tot Gen Fund 73 l 2 1.5 Here, no clear pattern may be discerned. Those predictor variables of a low ranking for Category Two ranked high for Category Three, and vice versa. In part, the inability to critically evaluate mixed results may be due to the lack of a rule of thumb or test of significance for this methodology applied to this kind of data at this point. Therefore, we look to our final run of ratios, format 17. Format 17 was a somewhat intuitive combination of ratios, based like format 16 on prior analyses. The unique characteristic of 143 this analysis is that although it evidenced a slight decrease in total classificatory ability, it displayed a significant increase the lower class classification: 8/10, 35/41, 5/9 : 48/60 for a "hit" ratio of 80%. The fourteen predictor variables utilized in format 17 were: Cash/Tot Liab 1974, 1973 Cash/Tot Rev 1974, 1973, 1972, 1971 Tot Liab/Tot Gen Fund 1974, 1973, 1972, 1971 IF from/Tot Gen Fund 1974, 1973, 1972, 1971 The classification results are shown in TABLE 54 CLASSIFICATIONS FROM FORMAT 17 Classification as predicted from the fitted model Poor Sat. Rich Poor 8 2 0 ACTUAL CLASSIFICATION sat° 3 35 3 Rich 0 4 5 For an explanation of the ratios and caption see: Supra, p. 134. 144 TABLE 55 ESTIMATED PROBABILITIES OF CLASSIFICATION UNDER FORMAT l7 xiiiir P1 P2 P3 First Category 1 .9907 .0093 .0000 2 .1782 .8191 .0027 3 .4303 .3439 .2258 4 .9950 .0050 .0000 5 .8981 .0980 .0039 6 .5819 .4017 .0164 7 .8982 .1006 .0012 8 .9244 .0754 .0002 9 .2546 .7451 .0003 10 .7728 .2232 .0040 Second Category 11 .0204 .9508 .2887 12 .0070 .7550 .2379 13 .0722 .9094 .0184 14 .0004 .9224 .0772 15 .0293 .9023 .0685 16 .0650 .7822 .1527 17 .0131 .8706 .1163 18 .5733 .4264 .0003 19 .1708 .8238 .0054 20 .4627 .5279 .0094 21 .0016 .9898 .0085 22 .0140 .9842 .0018 23 .0323 .8968 .0690 24 .1094 .8718 .0189 25 .0394 .9278 .0329 26 .0822 .8892 .0286 27 .0006 .9934 .0060 145 TABLE 55--Continued NHEIEZr P1 P2 P3 28 .0002 .9915 .0082 29 .0279 .9569 .0152 30 .2161 .7793 .0046 31 .0007 .9992 .0001 32 .0009 .9974 .0017 33 .0924 .9892 .0015 34 .0000 .2588 .7412 35 .0729 .9161 .0110 36 .6270 .3683 .0047 37 .0025 .9700 .0275 38 .0067 .9887 .0046 39 .0112 .3934 .5953 40 .0077 .8609 .1315 41 .0175 .9237 .0588 42 .0028 .8387 .1585 43 .0019 .6953 .3028 44 .0006 .5375 .4619 45 .0002 .9984 .0014 46 .0002 .9710 .0288 47 .1617 .8345 .0038 48 .0316 .6824 .2861 49 .9003 .0995 .0002 50 .0036 .9512 .0452 51 .0002 .0256 .9743 Third Category 52 .0030 .6335 .3634 53 .0369 .9210 .0421 54 .0059 .4085 .5856 55 .0045 .2815 .7140 56 .0016 .5755 .4228 146 TABLE 55—-Continued City A ) Pd) Number P1 P2 3 57 .0000 .0059 .9941 58 .0123 .3900 .5977 59 .0000 .0543 .9457 60 .0012 .5365 .4623 Vectors were then estimated for 82 and B3. TABLE 56 ESTIMATIONS OF VECTORS FOR 82 AND 83 UNDER FORMAT 17 82 Variables E3 .41779 .49773 .32045 .90799 -6.64581 -.37541 28.17662 24.53208 —13.338990 -l8.79221 -5.82788 1.78895 -l.32388 .43239 4.98293 4.53155 -12.87480 -12.72850 6.41366 4.42183 .50460 —4.22338 7.59569 10.67230 -l.30348 -6.59703 -3.07429 -4.08173 147 The relative weights are attained by the multiplication of the estimated vector by the mean of the variable: TABLE 57 COMPUTATION OF RELATIVE WEIGHTS OF PREDICTOR VARIABLES 0F CATEGORY TWO UNDER FORMAT l7 Predictor Variable 32 x Xi = $2 1 Cash/Tot Liab 74 .41779 1.79 .07478 Cash/Tot Liab 73 .32045 1.93 .61847 Cash/Tot Rev 74 —6.64581 .13 —.86395 Cash/Tot Rev 73 28.17662 .14 3.94473 Cash/Tot Rev 72 ~13.33890 .14 -1.86745 Cash/Tot Rev 71 -5.82788 .13 —.75762 Tot Liab/Tot Gen Fund 74 -l.32388 .44 .58251 Tot Liab/Tot Gen Fund 73 4.98293 .36 1.79385 Tot Liab/Tot Gen Fund 72 -12.87480 .39 -5.02117 Tot Liab/Tot Gen Fund 71 6.41366 .49 3.14269 IB from/Tot Gen Fund 74 .50460 .30 .15138 IB from/Tot Gen Fund 73 7.59569 .24 1.82297 IB from/Tot Gen Fund 72 -l.30348 .23 -.29980 IB from/Tot Gen Fund 71 -3.07429 .17 -.52263 This algorithm is then performed for Category 3: 148 TABLE 58 COMPUTATION OF RELATIVE WEIGHTS OF PREDICTOR VARIABLES OF CATEGORY THREE UNDER FORMAT 17 Predictor A Variable £2 Xi $2 1 Cash/Tot Liab 74 .39773 4.35 2.16531 Cash/TOt Liab 73 .90799 4.41 4.00424 Cash/Tot Rev 74 -.37541 .33 -.01239 Cash/Tot Rev 73 24.53208 .29 4.90642 Cash/Tot Rev 72 -18.79221 .27 ~5.07390 Cash/Tot Rev 71 1.78895 .26 .46513 Tot Liab/Tot Gen Fund 74 .43239 .21 .09080 Tot Liab/Tot Gen Fund 73 4.53155 .19 .86099 Tot Liab/Tot Gen Fund 72 -12.72850 .25 -3.18213 Tot Liab/Tot Gen Fund 71 4.42183 .25 1.10546 IB from/Tot Gen Fund 74 -4.22338 .02 —.08447 IB from/Tot Gen Fund 73 10.67230 .06 .64034 IB from/Tot Gen Fund 72 -6.59703 .04 -.26239 IB from/Tot Gen Fund 71 -4.08173 .05 -.20409 These predictor variables were then ranked: TABLE 59 RANKING OF £21 UNDER FORMAT l7 EZXI Cash/Tot Liab 74 IB from/Tot Gen Fund 74 IB from/Tot Gen Fund 72 IB from/Tot Gen Fund 71 Tot Liab/Tot Gen Fund 74 Cash/Tot Liab 73 Cash/Tot Rev 71 Cash/Tot Rev 74 Tot Liab/Tot Gen Fund 73 IB from/Tot Gen Fund 73 Cash/Tot Rev 72 Tot Liab/Tot Gen Fund 71 Cash/Tot Rev 73 Tot Liab/Tot Gen Fund 14 §3Xi Cash/Tot Rev 74 IB from/Tot Gen Fund 74 Tot Liab/Tot Gen Fund 74 IB from/Tot Gen Fund 71 Cash/Tot Liab 74 IB from/Tot Gen Fund 72 Cash/Tot Rev 71 IB from/Tot Gen Fund 73 Cash/Tot Rev 71 (Tot Liab/Tot Gen Fund 73 Tot Liab/Tot Gen Fund 72 Cash/Tot Liab 73 Cash/Tot Rev 73 Cash/Tot Rev 72 By applying ordinal ranking we can appraise the variables. 150 TABLE 60 RANKING OF VARIABLES UNDER FORMAT 17 £2 1 §3ii Average [Cash/Tot Liab 74 l 5 3.0 Cash/Tot Liab 73 6 12 9.0 Cash/Tot Rev 74 8 1 4.5 Cash/Tot Rev 73 13 13 13.0 Cash/Tot Rev 72 11 14 12.5 Cash/Tot Rev 71 7 7 7.0 Tot Liab/Tot Gen Fund 74 5 3 4.0 Tot Liab/Tot Gen Fund 73 9 9 9.0 Tot Liab/Tot Gen Fund 72 14 11 12.5 Tot Liab/Tot Gen Fund 71 12 10 11.0 IB from/Tot Gen Fund 74 2 2 2.0 IB from/Tot Gen Fund 73 10 8 9.0 IB from/Tot Gen Fund 72 3 6 4.5 IB from/Tot Gen Fund 71 4 4 4.0 In contrast to the immediately preceding analysis, the ratios of format 17 display consistency in certain aspects, particularly the ratios of Interfund Borrowing due from Other Funds/Total General Fund. The relatively low and consistent ranking possibly holds this ratio forth in this combination as one of the more discerning characteristics of municipalities. Furthermore, the deletion of the ratio for Interfund Borrowing Due to/Total General Funds resulted in significant improve- ment in classification of municipalities with low levels of cash. The conclusion may also be drawn in regard to ratio analysis that it 151 resulted in less correct classifications overall than other formats but greater strength in correctly classifying the ”Cash Poor" munici— palities. Common dollar statements . . . Another of the objectives of this effort was to prepare finan— cial statements representative of the ”Cash Poor" and ”Cash Rich" municipalities. These statements were prepared by averaging raw dollar balances of accounts for both categories of municipalities: TABLE 61 COMMON DOLLAR BALANCE SHEETS OF MUNICIPALITIES Cash Poor Cash Rich Z Z Assets Cash 3.1 86.1 Past Due Taxes Receivable 9.6 2.7 Interfund Borrowing Due from 41.1 2.7 Other Assets 46.2 8.5 Total Assets 100.0 100.0 Liabilities and Fund Balance Interfund Borrowing Due to 26.7 6.6 Other Current Liabilities 36.2 14.8 Total Current Liabilities 62.9 21.4 Fund Balances Reserves 14.9 28.5 Unappropriated Fund Balance 22.2 50.1 152 The Operating Statement for the Cash Poor and Cash Rich municipalities is presented next as Table 62. TABLE 62 COMMON DOLLAR OPERATING STATEMENT OF MUNICIPALITIES 1975 Cash Poor Cash Rich % % Revenues Property Tax Income 47.7 49.7 Personal Income Tax Income 3.2 3.4 State Shared Revenues 22.5 20.7 Other Revenues 26.6 26.2 Total Revenues 100.0 100.0 Expenditures (as a % of Total Revenues) Administration 19.9 19.4 Police 26.4 24.0 Fire 16.3 8.8 Parks & Recreation 6.0 7.2 Other 30.7 35.9 Total Expenditures 99.3 95.3 Summary and conclusions In summary, the primary objective of this research effort was to make an inquiry into the development of a predictor of future mu— nicipal financial insolvency and to attempt to set forth differences in those cities which display indications of financial insolvency and those cities which do not. The predictor was based upon information that is readily available without great cost, accounting information taken from the uniform financial reports which are filed annually by the municipalities with the appropriate regulatory agencies of the State of Michigan. 153 The reader should be aware of particular factors underlying those conclusions and his inferences from those conclusions should be so savored. First, the cities investigated were in the State of Michigan. Consequently, these cities are affected by laws and conditions peculiar to that state alone. To the extent that other states have similar laws and conditions, inferences might be drawn, if the reader so desires. Second, the accounting information was taken from financial statements prepared under the uniform accounting procedures prescribed in Public Act 2, 1968. Reporting requirements likewise are prescribed. To the extent that accounting and reporting practices of other states do not differ significantly, the statements of those municipalities might be considered similar and inferences could be drawn to the degree desired. Finally, it was decided at the beginning of the study to not make use of validation techniques such as a holdout group or a random iterative process such as a "Midas Two-Way" test. Opinions expressed by Dr. McSweeney were that the group of sixty units was too small to partition into sample body and holdout group. Validation would have been (and will be) best achieved by extension of the study to other areas and other time frames. The model utilized a classification technique similar to that of multiple discriminant analysis. Several runs were made utilizing various combinations of predictor variables to classify cities into the three groups, Cash Poor, Cash Satisfactory and Cash Rich. The criterion variable that was found to be most effective was the amount of cash of the General Fund at the end of 1975 expressed as a percent- age of the total assets of the General Fund at the end of 1975. The numerous runs resulted in ranges of ability to classify 154 the cities correctly. In the analysis displaying the most correct classifications, (87.5%), the predictor variables making the greatest contribution to those results were the Administrative Expenditures and the Park and Recreation Expenditures. It was inferred that these dis- cretionary expenses would exhibit the the following trends: that as the Cash Poor city was able to retain less cash it would also find Administrative Expenditures increasing as the city's governing body attempted to serve the changing body of standing citizenry. Further— more, the Parks and Recreation Expenditures would be contracted in a direct relationship to the amount of cash held. Three other analyses displayed a significant number of correct classifications and supportive strength in the Cash Poor class. The prominent variables taken from those runs were: Past Due Taxes Receivable (Formats 1 and 2) Property Tax Revenues (Format 8) Interfund Borrowings Due to the General Fund as a percentage of the General Fund Total Assets (Format 17) A proper extension of the study should involve devlopment of coeffi- cients and expression of them in formula form. Other extensions of the study would involve investigation of the prominent lag factor. This lag factor was deduced from runs us- ing the same variables over different time periods; those runs of the earlier years gnerally exhibited greater strentgh than the subse- quent years. In addition, a greater time span will be possible as additional data becomes available. In these subsequent investigations it is felt that the real value of this study will be realized in that a longer time horizon will permit the state and other interested parties the opportunity for corrective action, if possible. 155 .APTHNHDEKJA §1ANDARD & POOR'S CORPORATION MUNICIPAL DATA BANK.QUESTIONNAIRE Official Government Unit Name Fiscal Year Ends Have any areas obeen annexed State since 1970? If yes - Population Source of Data A.V. Pop, Agea (Sq. Mi. 1974 1973 '1972 1971 1970 1969. Fiscal Year Assessed Valgation of Taxable Basis of Assessed Value finding Property (Real & Personal) in § (fi of A.V. to Mkt. Value) 1974 1973 1972 1971 1970 1969 unto or last revaluation Dollar ($) loss to A.V. due to exemptions (current year) Is exemption reimbursed by State? Percent of Prgperty Tax Collected Percentage Composition of AN. (by type of preperty) Current Leg! Total Collected Latest Year 1974 76 96 Residential 96 1973 X .5 Commercial )5 1972 )6 5 Industrial ' l 1971 J )6 Tax-Exempt _g 1970 i A 1969 % fi Other 7‘ 1.56 Tax Rate (§l,000 of Agv.) Tax Rate Limit 1974 Operations Only 1973 Operations 8 Debt Svc. 1972 Debt Svc. Only 1971 1970 1969 Debt Limit List other taxing units overlapping your unit: Debt Information ngg-Termg6.0. Debt Out- [9.0. Debt Supported.by Other standing at Pigcal Year End Than Property Taxes 1974 1973 1972 1971 1970 1969 6.0. debt presently authorized but unissued: note any change in State or Federal Air'programs affecting your unit: 157 List the Ten Top Taxpayers flame.o£ Tgxpaygr 2:29 0; Business Assessed value lo nt Municipal.0££icial to Contact: Name Title Address Telephone 158 APPENDIX B Alphabetical Listing of Cities in the State of Michigan Having Populations Exceeding 10,000 in 1970 Source: Adrian Albion* Allen Park Alpena Battle Creek Bay City Benton Harbor Berkley Big Rapids Birmingham Burton Centerline Clawson Dearborn Dearborn Heights Detroit East Detroit East Grand Rapids East Lansing Ecorse Escanaba Farmington Ferndale Flint Fraser Garden City Grand Haven Grand Rapids Grandville Grosse Pointe Grosse Pointe Grosse Pointe Hamtramck Harper Woods* Hazel Park Highland Park Holland Inkster Jackson Kalamazoo* Farms Park Woods U.S. Census, 1970 1 1,5 1 1 20,382 12,112 40,747 13,805 38,931 49,449 16,481 21,878 11,995 26,170 32,540 10,379 17,617 04,199 80,069 11,482 45,920 12,565 47,540 17,515 15,368 10,329 30,850 93,371 11,868 41,864 11,844 97,649 10,764 11,701 15,585 21,878 27,245 20,186 23,784 35,444 26,337 38,595 45,484 85,555 159 APPENDIX B——Continued Kentwood 20,310 Lansing 131,546 Lincoln Park 52,984 Livonia* 110,109 Madison Heights 38,599 Marquette 21,967 Melvindale* 13,862 Menominee* 10,748 Midland 35,176 Monroe 23,894 Mt. Clemens 20,476 Mt. Pleasant* 20,504 Muskegon* 44,631 Muskegon Heights 17,304 Niles* 12,988 Norton Shores 22,271 Oak Park 36,762 Owosso 17,179 Plymouth 11,758 Pontiac* 85,279 Portage 33,590 Port Huron 35,794 River Rouge 15,947 Riverview 11,342 Romulus 22,879 Roseville 60,529 Royal Oak 86,238 Saginaw 91,849 St. Clair Shores 88,093 St. Joseph 11,042 Sault Ste. Marie 15,136 Southfield 69,285 Southgate* 33,909 Sterling Heights 61,365 Taylor 70,020 Traverse City 18,048 Trenton 24,127 Troy 39,419 Walker 11,492 Warren 179,260 Wayne 21,054 Westland 86,749 wyandotte 41,061 Wyoming 56,560 Ypsilanti 29,538 *Has year—end other than June 30; data being collected only for benefit of State Treasury Department. APPENDIX C DATA GATHERING FORM 160 m u. _ i b 8 n9.” Vn9u .05 ..94 .9: n9u nu... n0... n9” n9u n9u v n9u q9u n9u n9u “9.. n9u me... med "9“. n9c Fn9u n9u -9”. n9u 19o “no: .95 ..9u no.” ....9... nlu knlu reu :5 rlu nst nsu nau moon nnu nOuh ecu nag 2.3 Can nau nan nan n80 nau nBu Moan -Bu nan man new. PC: n8”. ..B._ ”In Geo W n7u 8n7o ”7 ..7a n7” an... n7o n7u n73 ..7u ml. 25 ad n7“. n7u nvu n10 n7u n70 470 n7o n7u "7” n7“ n70 n7..fin7o ..7u nvu .../u .17D 0 r . .m 98 mass use Duo ”60 mneu nsu Tau 93 me... man.“ nan 03 as“ Don 06qu Too nan “so as... ....3 end... n6... me. men net 693 n64 men we” 93 . .n .1 and Tnsu n5“ 23 Dan ensu 05.. C3 95.. TB nsum 23 us“. 93 ass 03 $93 ea” n5u n5- n8 tnsu 93 .23 nsu n50 4.51. ms... r53 .5 n3” WES &n4u n40 n40 n4u MTG n4“ Can. .33 n4.. naus n4u nah n3 n4... nACMndu 13 n4”. n3 ...4.“ Oman n3 ..4.. n4u 4. duds .4. r4- n... ..a. 9.6 mna... nan 9:... 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"On the Rate of Growth of the Population of the U.S. since 1790 and its Mathematical Representation," Proceedings of the National Academy of Science, Vol. 6, 1920. Pinches, George E. and Mingo, Kent A. "A Multivariate Analysis of Industrial Bond Ratingsfi' Journal of Finance, March, 1973, pp. 1-15. Revsine, Lawrence. ”Predictive Ability, Market Prices, and Operating Flows," The Accountinngeview, July, 1971, pp. 480-89. 164 Smith, K. V. "Classification of Investment Securities Using MDA," Institute Paper #101 Purdue University Institute for Research in the Behavioral Economic and Management Sciences, 1965. Walter, J. E. "A Discriminant Function for Earnings Price Ratios of Large Industrial Corporations," Review of Economics and Statistics, February, 1959, pp. 44-52. Wiggins, Nancy and Hoffman, Paul J. "Three Models of Clinical Judgment," Journal of Abnormal Psychology, Vol. 73, No. 1, pp. 70-77. Zellner, Arnold and Lee, Tong Hun. 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