«I..- ragw- 1'H“ “Iv ' IIII IIIIII II IIII III IIIIII IIII IIII II III II III IIIIII II IIII L , ‘; 1'5 R A R ‘1 712 I Michigan ‘K... This is to certify that the thesis entitled Comparison of Four Sequential Sampling Plans Applied to Forest Tent Caterpillar Eggs on Sugar Maple Branches presented by Jan P. Nyrop has been accepted towards fulfillment of the requirements for I Master degree in Entomology Mfimnuo Date August 10, 1979 0-7639 OVERDUE FINES: 25¢ per day per item RETURNIMS LIBRARY MATERIALS: N Place in book return to remve charge from circulation records IL COMPARISON OF FOUR SEQUENTIAL SAMPLING PLANS APPLIED TO FOREST TENT CATERPILLAR EGGS ON SUGAR MAPLE BRANCHES By Jan P. Nyrop A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Entomology 1979 ABSTRACT COMPARISON OF FOUR SEQUENTIAL SAMPLING PLANS APPLIED TO FOREST TENT CATERPILLAR EGGS ON SUGAR MAPLE BRANCHES By Jan P. Nyrop Sequential sampling is a valuable tool for classifying population density. To date, all applications in insect sampling have used Wald's Sequential Probability Ratio Test (SPRT). Use of the procedure necessitates knowing the distribution of the underlying population and requires that the distribution be constant in time and space. When these assumptions are not met sequential t-tests and a new sequential test proposed by Iwao provide alternatives. Four sequential procedures were compared through simulation. These were the SPRT, Iwao's test and two sequential t-tests proposed by Barnard and Fowler and O'Regan. Forest tent caterpillar (Malacosoma disstria) egg band sampling was used as a test case. Though not universally true, in this instance SPRT was robust to changes in k of the negative binomial distribution. Fowler's and O'Regan's t-test and Iwao's procedure were comparable to the SPRT. Fowler's and O'Regan's t-test required the least informa- tion about the population distribution for construction. ACKNOWLEDGMENTS Many people have contributed both directly and indirectly to my thesis and I extend my thanks to them: To my advisor, Dr. Gary Simmons, who first fired my interest in entomology and has since provided much support, professionalism and friendship. To my committee, Drs. Tom Edens, Dean Haynes and Henry Webster who contributed greatly during my graduate work. To Jeri, for her encouragement. This work was supported in part by a grant from the Michigan Department of Natural Resources, Forest Management Division. 11 TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . INTRODUCTION . . . . . . . . . . . . . . . . . . Wald's Sequential Probability Ratio Test . . . . . . A. Maximum Likelihood Estimation . . . . . . . . B. Operating Characteristic Function and the Average Sample Number Function . . . . . . . . . . . . . C. Applications Using the Negative Binomial Barnard's Sequential Procedure . . Fowler's Sequential Procedure . . . . . . . . . . . . . . Iwao's Sequential Procedure . . . . . METHODS . . . . . . . . . . . . . . RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean Crowding and Mean Density and Comparison of OC and ASN Effects of Changing Population Distribution on SPRT . . Effects of Different Sample Unit Sizes on FTTEST . . . . DISCUSSION AND CONCLUSIONS . . . . . . . . . . . . . . . . . . APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . Nominal Values of Decision Boundaries - . - . . . . - - - Description of Computer Routines Used . - . . - . - - . REFERENCES CITED 0 o o o o o o o o o o o o o o ' ° ' ° ° ’ ' ° iii PAGE iv 10 11 13 16 16 17 18 29 35 35 39 SS FIGURE 10. 11. 12. LIST OF FIGURES PAGE Regression of mean crowding on mean density of forest tent caterpillar egg masses on 76 cm branch samples from 19 trees. The model is; Y = A + BX + e where Y = mean crowding, X = mean density, A = -0.0052, B = 1.695, R2 . 0.8704 ............................................. 20 Adjusted OC curves for FTTEST, BTTEST, ITEST and SPRT ... 21 Average sample number in relation to population mean for FTTEST, BTTEST, ITEST and SPRT .......................... 22 Proportion of no decisions after fifty observations in relation to mean density for BTTEST, ITEST and SPRT ..... 23 Probability of accepting H relative to the population mean for three hypothetica? mean - k relationships employ1ng SPRT 0.0000000000000000oooooooooooooooooooooooo 24 Effect of changes in k of negative binomial distribution on frequency distribution of forest tent caterpillar egg masses. Class refers to number of egg masses per branch sample ......OOOOOOOOOOCOOOOIOO..........OOOCOOOOOOOOOOOO 25 Probability of accepting H relative to the population 0 mean for five sample surfaces represented by branches per observation employing FTTEST ............................ 26 Distribution changes in forest tent caterpillar egg mass distribution associated with changes in sample surface. Sample surface is represented by number of 76 cm branches in an observation. Class refers to number of eggs per observation ............................................. 27 Average sample number curves for FTTEST for five sample surfaces represented by branches per observation ........ 28 Probability of accepting H adjusted for no decision cases relative to the popuTation mean for SPRT and FTTEST (ac and B = 0.05, FTTEST sample surface = 7) ............ 32 Average sample number in relation to population mean per fifty samples for FTTEST and SPRT (x'and fi’= 0.05, FTTEST sample surface87) 0.0.0.0.........IOOIOOOOOOOOO00...... 33 Proportion of no decisions after fifty observations in relation to population mean for SPRT (drandIG = 0.05) ... 34 iv FIGURE PAGE 1a. Decision boundaries, OC and ASN curves of SPRT for forest tent caterpillar egg mass sampling (see text for details) O......O..O...O..........ICCOOCOOIOOO0...... 35 Za. Decision boundaries of BTTEST for forest tent caterpillar egg mass sampling (see text for details) ................ 36 3a. Decision boundaries of FTTEST for forest tent caterpillar egg mass sampling (see text for details) ................ 37 4a. Decision boundaries of ITEST for forest tent caterpillar egg mass sampling (see text for details) ................ 38 INTRODUCTION Biological monitoring is an important component of any successful pest management program. Often, it is only necessary to classify an arthropod population as above or below a specified density. In such cases sequential hypothesis testing provides an attractive alternative to sampling schemes which employ a fixed number of observations. Fixed sample size methods are invariably inadequate at low population densi- ties and excessive at high densities. With sequential procedures sample size is dependent on the outcome of each successive observation. These tests, on the average, require fewer observations than do equally reliable tests based on fixed sample size procedures. For this reason, they are attractive sampling schemes when cost and time efficiency are important. To date, all applications of sequential hypothesis testing in in- sect sampling have employed Wald's Sequential Probability Ratio Test (SPRT) (Wald i947). Pieters (1978) provides an extensive list of insect species for which such sampling schemes have been developed. Use of the SPRT requires that the underlying population distribution and variance be known. However, these parameters are often unknown and the population distribution upon which an SPRT is developed may change with changes in the density, quality and age of the population and spatial and temporal changes in the environment. Such changes will affect the results ob- tained using SPRT. Alternatives to the SPRT exist. These are sequential t-tests and a new sequential procedure based on the regression of Lloyd's mean crowding on mean density (Iwao 1975). The latter requires an estimation of the population distribution based on the aforementioned regression, however, this relationship is reported to be stable in time and space 1 (Iwao and Kuno 1971). Sequential t-tests assume the population is approximately normally distributed. The purpose of this paper is to compare, through simulation, four different sequential procedures. Forest tent caterpillar (Malacasoma disstria) (FTC) egg band sampling is used as a test case. The four sequential procedures investigated are the SPRT, Barnard's sequential t-test (BTTEST) (Barnard 1952), Fowler's and O'Regan's truncated sequen~ tial t-test (FTTEST) (Fowler and O'Regan 1974) and Iwao's sequential test (ITEST) (Iwao 1975). Simulation, the process of conducting experi- ments on a model, as opposed to attempting the experiment with the real system, provides an ideal tool for this evaluation. The result provides the reader with a method to weigh relative merits of each test and facilitate a choice between tests. Development follows four parts: 1) A detailed description of the SPRT and a less intense outline of the other sequential procedures and problems in their use; 2) a description of the model used to investigate these problems and compare tests; 3) experiments conducted with the model and their results; 4) discussion and conclusion. Wald's Sequential Probability Ratio Test IAL_Maximum Likelihood Estimation The SPRT is defined as the ratio of the probabiltiy of obtaining a given set of observations if an alternate hypothesis is true to the probability of obtaining the same set of observations if the null hypoth- esis is true. Initially, two hypotheses (H0, H1) of the actual popur lation parameter (9) are established. Consecutive samples are then examined and evaluated until cumulative results dictate acceptance of one of the hypotheses. The SPRT is based on the maximum likelihood which is now introduced through illustration. Assume x is distributed binomially with probability density function (p.d.f.); x l-x f(x,p) = p (l-p) where x = 0,1 and 0 s p s 1. We wish to find an estimation u(X1,X2,X3,...,Xn) such that u(xl,x2,x3, ...,xh) is a good estimate of p where x1, x2, x ., xn are observed. 3,.. The probability that X X X .., Xh takes on these particular values 1’ 2’ 3" is; P(Xl=xl, X2=x2, X3=x3,..., Xn=xn) which equals; xi l-xi in n- in fir) (l-p) = p (l-p) i=1 This is the joint p.d.f. of x1, x2, x xn. An estimate of p may be 3,..., arrived at by regarding the p.d.f. as a function of p and finding the value of p which maximizes it. In other words, we wish to find the p value most likely to have produced these values. This is the likeli- hood function L(p; xl,x2,x3,...,xn) and will henceforth be designated as L(O). Instead of finding the value p which maximizes L(O) it is easier to find p which maximizes 1n L(Q). The maximum is found by taking the first derivative and setting it equal to zero. Calculation will show the solution is; 1/n (2:Xi) or the mean as expected. Neyman and Pearson (1928) suggested that a useful criterion for testing hypotheses is provided by the likelihood ratio; L(O) when 9 = 90 A: L(O) when 9 = 91 9 = 9 The hypothesis H 1; l O; 9 = 90 will be accepted when A is large and H accepted when.K is small. Values of.k may be selected to assure a specified level of a; the probability of accepting H1 when H0 is true and G; the probability of accepting HO when H1 is true (Type I and II errors respectively). Wald (1947) applied this criterion in developing the SPRT. Approximate values of %.are calculated using the following logic. Suppose a large number of sequential tests are made. Those which terminate with acceptance H have a likelihood ratio equal to or slightly 0 greater than A0. Those terminating with acceptance of H1 have a like- lihood ratio slightly less than )1. Consider the group of samples with likelihood ratio greater than )0. The probability that the sample really originated from the population with parameter 9 is A times as great as 0 O the probability that it originated from a population with parameter 91. This is true for every sample, thus the total chance of obtaining a sample of this sort is A0 times as large when H0 is true as when H1 is true. We desire the chance of getting such a sample from a population where 9 = 90 to be 1 -oCand the chance to be 6 for 9 = 91. Consequently, in order to satisfy both conditions A. must equal (l-dDAS. Using similar 0 reasoning A1 must equal «7(1-8). In practice, the likelihood function is solved for the total units found in Q observations. At each sample this total (T) is compared to a function of A0 and A1 and decisions made: 2 T f(xo) accept H0 < . T - f(xl) reject H0 £00) 2 T 2 fosl) continue sampling An illustration is given assuming a normal distribution: The probability of a single observation from a normal distribution is; (x -u 2 or: Iexp [___i_0__] (W) 20‘2 The probability of a sample of n independent observations x1, x2, x3, ..., xn is the product of n such expressions; 2 1 am [-Z(x1-u0) l n n/2 a' (2”) 2,2 Therefore, the likelihood ratio is; 2 epr- Z m2] 2 exp I- xxi-ul) ml] Taking the natural log and rearranging; _ _ 2 Xxi T - Iv/(n>~>I/11nt<1-«1) ml] [-a2/(u1-u0) lln[(1-¢)/6l [-a'z/ (ul-uo) llnl (1-a)kexp((xi-u)2/2v2> Whichimiy be stated as; L(x.s/u, 0') because x. and s are jointly sufficient statistics for u and<7. Con- sidering the ratio t = x.(n/s)%, on H0 t has a non-central t distribution with (n—l) degrees of freedom and parameter 8. On H1 the non-centrality parameter is 8;. With the probability density indicated by f the likeli- hood ratio criteria, considering only the distribution of t is; >rt/s, 8') = f(t/al, n)/f(t/8, n) Evaluating >(t/8, 81) as a function of t for each value of n requires a difficult series of approximations. Tables are available for comparing the test statistic U = Z(x--u0)/Z(x-u0)2 for various values of 0C, 0 and difference in 8 which it is important to detect. Little is known about the OC and ASN for these tests. The test is not linear and therefore usual ASN formulas are not applicable. Rushton (1950) provides means for obtaining the lower bound to the mean sample size. Fowler's Squential Procedure Fowler and O'Regan (1974) present a truncated sequential t-test derived by employing Monte Carlo procedures to approximate the distribution of the conditional test statistic at each stage of the test. A truncated test ensures that a decision will be reached before or at a specified number of observations. Their test is constructed by specifyinng, the truncation point nO and a probability boundary pattern. This boundary pattern establishes the probability of accepting and rejecting HO (HO 11 being true) at each stage of the test such that the overall probability of rejecting H0 with HO true is an The test statistic is defined as; d = (; - u0)/ 21x1 - ;)2/n(n-1) n which has a t distribution with one degree of freedom for d1 and an unknown conditional distribution for dn (n>1). Decision points for re- jection and acceptance of HO based on dn were approximated using Monte Carlo procedures. Null and alternate hypotheses are stated in terms of 8; HO: u=uoor 8a 50 =:0 1 0C and ASN functions of each test are approximated with simulation tech- H: u- “1 or 8: 81, (81>0) niques. The sequential t procedures assume that the population being sampled is approximately normal. Divergence from this assumption is known to alter the power of a test employing this statistic (Pearson and Please 1975). Iwao's Sequential Procedure Iwao (1968) proposed the regression of Lloyd's mean crowding on mean density as a method for analyzing aggregation patterns. Lloyd (1967) established the parameter mean crowding as the mean number per individual of other individuals in the same quadrat. The parameter is defined as; swimmfi: 3:1 J 1‘1 where Q is the total number of quadrats and x is the number of indivi- i duals in the jth quadrat. Mean density is related to it through; 3= u + (v2/(u-l)) 12 The parameter is estimated by substituting the sample mean and variance E; 32 for u, v?. The relationship; 3 a A + Bu has been shown to be linear in a wide variety of applications (Iwao 1968, Iwao and Kuno 1971). If a population distribution follows a Poisson series the regression line passes through the origin (A=0) and its slope B is equal to unity. For a negative binomial with a common k, A - 0 and B a 1 + (1/k). Underdispursed or completely uniform distributions are characterized by 3 taking zero up to u - 1 and increasing along the linear regression of slope B é 1. Through extrapolation of the regres- sion line from u z 1, A is -l. The value of the intercept A may be interpreted as the number of organisms which would live together with A other individuals at some infinitessimally small density. Iwao (1968) categorized this as the "Index of Basic Contagion". The slope B is an index of the spatial pattern of habitat use by individuals or groups of individuals in rela- tion to their density. This is called the "Density - Contagiousness Coefficient". Both indices are necessary to describe a distribution. In relation to the problem currently under investigation, a linear rela- tionship has been shown to hold even when k varies with population den- sity in a negative binomial (Iwao and Kuno 1971). As previously indicated, mean crowding, 3, is related to population variance (v2) by; 3= u + (Wm-1)) Therefore; V2 = u(U-u+l) Substituting the mean crowding to mean density relationship; 13 v2 - (A+1)u + (B-l)u2 The half width of a confidence interval is given by d = tSE-where t is the value of the normal deviate corresponding to a desired confidence probability (Student's t). Employing the variance relationship Iwao and Kuno (1968) calculated d as; d . t(((A+1)u + (3—1)u2/n)5) where n is the number of samples. As illustrated by Iwao (1975) this relationship may be used to cal- culate a sequential sampling procedure based on whether observations fall within desired confidence limits based on a hypothesized mean (uo). The upper limit is defined as; Tu = nuo + t(n((A+1)u0 + (B-1)ug)%) and the lower limit as; T a nu - t(n((A+l)u + (B-l)uz)k) l 0 0 0 Decisions are made with the assigned confidence limit that Q > uo when T > Tu and § < u0 when T < T1 where T is the total number of individuals. As in other sequential procedures when i is very close to u a large 0 number of samples may be required for termination. However, the maximum number of observations required for a desired confidence limit, d, is; nmax = (tZ/d2)((A+l)uo + (3-1)u3) This method is intuitively appealing because of the reported stability of the mean crowding - mean density relationship. This clearly circum- vents problems encountered due to variable distribution parameters and deviations from the normality assumption. METHODS We are now in a position to investigate the properties of the afore- mentioned tests. Specificly; l) the power (0C) defined as the probability 14 of correctly choosing between two hypotheses about a true population parameter and average number of samples (ASN) required for each test and, 2) the effect of deviations from assumptions inherent to tests are in- vestigated. In brief, the experimental design was to sample a population and then use this sample as a population with which to simulate sampling. The sampled population consisted of FTC egg masses located on 76 cm (30 inch) branch samples. All possible samples were taken from the upper crown of sugar maple (Acer saccharum) trees by felling the trees and then carefully pruning off the branches. The mean number of branch samples per tree was 148.7 (s = 41.3). Data collection took place in August and September 1978 in an area surrounding Pellston, Michigan. In total, nineteen complete trees were enumerated. Mean crowding and mean density were calculated for each tree and a regression of these parameters formulated to describe the egg mass dis- tribution. For the simulation, six routines were developed. These are des- cribed below. A listing of each routine may be found in respective appendices. 1) STAT calculates statistics; mean, variance and estimate of k through the maximum likelihood method of the population (Appendix pg. 39). 2) NEGBIN distributes egg masses among 500 samples according to a negative binomial. The mean and k are specified exogenous from the routine (Appendix pg, 42), 3) SEQUAN randomly samples the population 1000 times and makes decisions regarding the mean with SPRT. The null hypothesis is given as x s 0.2 egg masses per branch and the alternate hypothesis as i z 0.5. The latter density corresponds to the reported level at which heavy 15 defoliation by FTC on sugar maple can be expected at the start of an infestation (Connola e£_§I_1957, Marshall and Hoffard 1976). The proba- bility of Type I and II errors are 0.1. Decision boundaries and nominal values of the OC and ASN based on all data points are given in appendix figure 1. Because the procedure is Open ended, during the simulation sampling was arbitrarily truncated at 50 observations. If no decision had been made after 50 observations, output indicated such (Appendix 98- 45)- 4) BTTEST randomly samples the population 1000 times and makes decisions regarding the mean with BTTEST. In order to facilitate use of tables, error levels were set at 0.05. The null and alternate hypothe- ses are given as i - 0.5 and x > 0.5 egg masses per branch respectively. In addition, the difference, stated in terms of standard deviations D, (8:= Dv), which it is important to detect, was set at 0.5. Decision boundaries are given in appendix figure 2. As in SEQUAN sampling is truncated at 50 observations (Appendix pg. 47). 5) TTEST randomly samples the population 1000 times and makes de- cisions regarding the mean with FTTEST. The null and alternate hypothe- ses and error levels are identical to those in BTTEST. Because this is a truncated test a decision is assured at the termination point of 10 observations. Decision boundaries are given in appendix figure 3. The probability boundary pattern employed specifies «.at each decision stage to increase at a constantly increasing rate (Appendix pg. 50) 6) SEQRT randomly samples the papulation 1000 times and makes de- cisions regarding the mean with ITEST. The null and alternate hypotheses are 2 < 0.5 and i 7 0.5 egg masses per branch respectively. Type I and II errors are set at 0.1. Decision boundaries are illustrated in appendix figure 4. Sampling was again terminated at 50 observations 16 (Appendix pg. 53). Three simulations were executed using the described routines: 1) A comparison of the OC and ASN of the four sequential procedures assuming the population distribution was constant regardless of popula- tion density; 2) the effect of distribution varying with density on DC of SPRT; and 3) a comparison of sample unit sizes as applied to FTTEST. RESULTS Mean Crowdi3g_and Mean Density and Comparison 9§_gg_§gg_§§§ Regression of mean crowding on mean density (fig. 1) provided the relationship; §= -o.0052 + 1.69562) While NBD is characterized with A = 0, A of -0.0052 in this instance meets this requirement considering the variability about the regression line. Therefore, the data is described by NBD with k = 1.449. From the original sample a subsample of 500 with x = 0.222, 32 = .273 and k of 1.0768 was randomly selected for use in the simulation. No loss in accuracy of the results is incured by using a population of n = 500, however, computations and costs are substantially reduced. Simulation results are presented in figures 2, 3 and 4. 0C curves were adjusted for no decision cases. Only SPRT and ITEST possessed use- ful 0C curves (fig. 2). That BTTEST and FTTEST were inferior in this regard is not surprising considering the skewed distribution from which samples were drawn. A complete picture of the usefulness of a sequential procedure must also include the ASN function. In the simulation this is comprised of two components; the average number of samples per decision (fig. 3) and the proportion of no decisions after the truncation point of 50 17 samples (fig. 4). Although the sequential procedure may not have reached a decision stage prior to the truncation point, gathered information is still usefull. Both SPRT and ITEST exhibited similar variations about an estimated mean, however, the number of times this estimate is used differed between tests. At the critical density the 95 percent confi- dence interval half width is approximately 0.2. Assuming a constant distribution this is calculated by knowing the actual papulation variance at a density of 0.5 and calculating the standard deviation of the mean with a sample size of 50. An estimate with sample size of 50 is there- fore 0.5 i 0.2 egg masses. SPRT must be considered superior in this regard as this estimate is used less frequently than with ITEST. Effects gf_Changing_Population Distribution As previously stated the assumption of invariant population distri- bution in relation to pOpulation density is often not upheld. The second simulation was designed to investigate the effect of divergence from this assumption on SPRT. Linear relationships between k and the population mean were established. Maximum divergence are given by; k - 1.077 + 10(Mean) k a 1.077 - 0.5(Mean) The first resulted in the population becoming more random while the second produced a more aggregated distribution. The relationships do not imply a realistic representation and are presented merely as examples. However, this has no effect on conclusions drawn from the results. The simulation employed NEGBIN to update the population distribution with each change in mean density. Changes in OC associated with the largest change of k in relation to the mean are presented in figure 5. A more random distribution 18 actually improved the power of a test considering it important to detect population levels exceeding the critical density. A tendancy toward greater aggregation in relation to the mean had an opposite effect. Differences in OC due to changes in k must be considered in light of respective changes in population distribution. From figure 6 it is evi- dent that variable k did not induce large changes in the distribution. Therefore, changes in OC were likewise not great. Divergence in OC may be much greater for other NBD. Effects g£_Different Sample Unit Sizes 22_FTTEST Obtaining estimates of parameters required to describe a population distribution is often tedious. Also, these parameters are rarely invar- iant with density. For these reasons sequential t procedures are appealing. However, we have already demonstrated their ineffectiveness when the distribution is highly skewed. More specificly we are dealing in this case with a population containing a high percentage of zero values. This situation commonly presents itself when sampling insect populations. A solution to this problem is now presented. Population distribution is largely a function of the size of the sample unit. By varying the sample unit, the distribution and associated mean and variance change (Elliot 1977, Pielou 1978). Employing this principle, sample surface can be varied by considering 1, 2,..., n branches as an observation. A sequential procedure may then be consid- ered by increasing the densities defined under the null and alternate hypotheses by the same change in factor as the sample surface. Thus, if the null hypothesis is; x - 0.5 with a sample surface of 1 it will be increased to i - 1.0 with a sample surface of 2. 19 The effectiveness of the strategy was investigated with FTTEST and this comprised the third simulation. The OC and ASN functions for sample surfaces of 1 to 9 branches are given in figures 7 and 8 respectively. ASN is still expressed in terms of number of branches and not observa- tions. Additionally, the effect of change in sample surface on distri- bution is portrayed in figure 9. As sample surface increased, the OC became more favorable. Though a sample surface of 9 has a superior CC to sample surface 7, the cost in terms of ASN is large. It is also evident that increasing sample surface to 9 did not normalize the distri- bution. Thus, FTTEST must be considered robust to some deviation from the assumption of normality. 20 Figure 1. Regression of mean crowding on mean density of forest tent caterpillar egg masses on 76 cm. branch samples from 19 trees. The model is: Y = A + BX + e where Y = mean crowding, X = mean density, A = —o.0052, B = 1.695, R2 = 0.8704. 0.5 0.6 I Mean Crowding 0.3 0.2 0.05 0.1 - 0.15 0.2 0.25 0.3 Mean Density 21 Figure 2. Adjusted OC curves for FTTEST, BTTEST, ITEST and SPRT. 1.0. .8- O :1: CD a '3 a..64 Q) U U <3 u... 0 Ba FTTEST ~H FI.4~ -H ..Q :13 ...C) O H 9-: BTTEST .2~ l I I *** := ... I .322 .522 .722 .922 1.122 Population Mean 22 Figure 3. Average sample number in relation to population mean for FTTEST, BTTEST, ITEST and SPRT. 401 a’ ‘x ’ \ 4’ ‘x 35 . / BTTEST 30. G O 'v-l (O '3 g 25 «4 H 0) 9-: CD CD Fl 3‘ .2 20. ‘H O H .8 E :1 ‘z 15. 0 DO 3 “:3 <1: ITEST 10‘ SPRT 5 . W FTTEST .322 .522 .722 .922 1.i22 Population Mean 23 Figure 4. Proportion of no decisions after fifty observations in rela- tion to mean density for BTTEST, ITEST and SPRT. .6 . I "1 .5 . . I \. I \. I \. " \ I \, / \ m 4 . I \ / ‘\ g . / \ 8 I \. / \ Q I \I \BTTEST O . Z 3. I I). 0 e ’ I ‘ 2% / I ‘ S I / \ 8‘ 2 \ F: " I I . I \. . 1 . \ ~ITEST .322 .522 .722 .922 1.122 POpulation Mean 24 Figure 5. Probability of accepting H relative to the population mean for three hypothetical meag - k relationships employing SPRT. 1.0, O (I) A O 0‘ J 1.077 1.077 - .5 x Mean 1.077 + 10 x Mean WWW II Probability of Accepting HO c: 41> 0.2- .322 .522 .722 .922 Population Mean 25 Figure 6. Effect of changes in k of negative binomial distribution on frequency distribution of forest tent caterpillar egg masses. Class refers to number of egg masses per branch sample. Mean = .222 Mean = .422 5. 4 4 c>‘ I O H O ”3. _k=1.077 23. """k= 5.297 2 N ........ k: .865 E .., oz. 81 I": :3 0) Co a: > __ '81. $1. ...... 0) .0 L—"u'-==-—-~ O L_"‘“g‘j=r— or1r2 3 4 0 1 2 3 4 5 Class Class Mean = .622 Mean = .822 4 41 O O S """ """"k-- 7.297 S; ....... ““"k= 9.297 x I--- ........ k: .765 x ' ........ k: .665 a a ---I 0 02¢ ...; H U __ u ———1 E ‘ E 3 ...... 314 .o .o o o 26 Figure 7. Probability of accepting Ho relative to the population mean for five sample surfaces represented by branches per observa- tion employing FTTEST. 1.01 ‘<.~.:§I W. \\‘<.. 0.89 O m 0 g s 210.6d 3\ 8 "1\\ 0 °. <0 '.' \ ...\ n5 ..-\ \ >~ Branches .3' \ 3 0 44 '°.\. \ d ‘ *— 1 '-,\. \ :1 --- a -\ \ O _'—'- 5 \ \ E - \ .......... 7 . \. \ 9 -. \. \ 0.2q ' .0. \ \ \ . ’. \. o \ -—-. “-— .3‘22 .522 .7'22 .922 1.122 Population Mean 27 Percentage Occurance H.o I'll mecnm m. uwmnnwccnwon armbmmm H: monmmn noon omnmanHHmH 6mm ammo awmnnHUano: mmmoowmnmm sun: nwmpmmm H: mmsppm mcnmmnm. mmspHm mcemmnm Hm Hmpnmmmonmm we assume om No as. camoormm H: m: ocmmn where D equals silx and Tn is the total of n observations. When the underlying distribution of a population is unknown, 31 sequential t procedures may be used. As demonstrated, complete ignor- ance of the population distribution is unacceptable. A large percentage of zero counts or a highly skewed distribution invalidates the normality assumption and destroys the test. However, with far less effort than is needed to quantify a distribution, information may be gathered to assure that the sample surface is of proper size to insure a viable DC. A comparison of 0C for FTTEST and SPRT with error levels of both tests set at 0.05 is presented in figure 10. Previously a and G of SPRT were each set at 0.1 to insure reasonable ASN. FTTEST has a better DC at low densities, however, beyond the critical density of 0.5 egg masses SPRT is superior. ASN curves for both tests are compared in figures 11 and 12. Because SPRT is an Open test some cases will result in no decisions and this factor must be taken into account when comparing ASN curves. In this regard, FTTEST is superior at low densities and the reverse is true at higher population levels. Finally, FTTEST will be largely invariant to changes in population distribution. Clearly, neither test is optimal in all instances. Users must weigh these attri- butes and base a decision for test use on criteria specific to each case. From this study the following generalized conclusions may be drawn: 1) Attributes of SPRT are dependent on the underlying population distri- bution and vary with changes in this distribution. These changes may or may not be significant. 2) ITEST and FTTEST offer likely alternatives to SPRT and may be superior when confronted with a changing population distribution. 3) FTTEST requires the least information when applicable for construction. 4) Simulation provides a quick and inexpensive method for analyzing different sequential schemes. 32 Figure 10. Probability of accepting H adjusted for no decision cases relative to the population mean for SPRT and FTTEST (a: and (3 - 0.05, FTTEST sample surface = 7). 1.0, ‘ ‘ \ \ \ \ 0.8. \ \ o \ m as \ I; 0.6- \ w \ 8 <9 \ 1H SPRT FTTEST o \ £7 \ a 0.4. \ jg \ 5’: \\ 0.2« \ \ \ \ \ ~ .- V V I I ~ ‘1 .322 .522 .722 .922 1.122 Population Mean 33 Figure 11. Average sample number in relation to population mean per fifty samples for FTTEST and SPRT (cc and G - 0.05, FTTEST 40. sample surface - 7). 354 304 N U1 1 204 15J Average Number of Samples Per Decision 104 59 .322 .522 .722 .922 iflzz Population Mean 34 Figure 12. Proportion of no decisions after fifty observations in relation to population mean for SPRT (cc and 6 - 0.05). .6“ 05. U) 04‘ t: o H U) 01-1 0 Q) Q 2 .34 LH 0 C1 0 H u H O 8 u .2 m 1 .l-I v I I I ' .322 .522 .722 .922 1.122 Population Mean APPENDIX 35 Figure 1a. Decision boundaries, 0C and ASN curves of SPRT for forest Total Egg Masses Probability of Accepting H0 N O ...: U1 A p... O A tent caterpillar egg mass sampling (see text for details). Decision Boundaries R t H ejec 0 Continue Sampling SJ Accept HO O I fl W 30 40 50 -5' Number of Samples 0C 30‘ ASN 1A) 6 £3 “’20-: .5. o v-l S m U) o no 310‘ ‘5’ 0 ‘ fi I I j < .1 .2 .3 .4 .5 Population Mean 0 I I V .2 4 .6 .8 Population Mean 36 Figure 2a. Decision boundaries of BTTEST for forest tent caterpillar egg mass sampling (see text for details). 3 Reject HO \ / fl 2: Continue Sampling 1: A H ccept 0 0 v I f r a 10 20 30 40 50 -14 -21 Figure 3a. 1.7‘ 1.5- 1.2- .91 .6‘I .3d Continue Sampling 37 Reject H 0 Accept HO Decision boundaries of FTTEST for forest tent caterpillar egg mass sampling (see text for details). .Sample Number 10 Figure 4a. Total Egg Masses 50' 40‘ 30' 201 104 38 Decision boundaries of ITEST for forest tent caterpillar egg mass sampling (see text for details). Reject H 0 Continue Sampling Accept H U I j 20 30 40 50 Sample Number 39 Subroutine STAT STAT calculates the mean (MEANEG), variance (VAREGG) and total num- ber of egg masses (TOTEGG), and an estimate of k (KEST) of the negative binomial. Input into the routine consists of the population of 500 branch samples (EGG). A maximum likelihood estimate of k is calculated using an iterative approach until the difference between two successive estimates is 0.001 or twenty five iterations have been processed. Finally, a frequency distribution of egg masses is created. 4O SUBROUTINE STAT (EGG,MEANEG,KEST,TOTEGG) DIMENSION EGG(500), A(6) REAL LRB,MEANEG,KEST,MLHL,MLHR F0=F1=F2=F3=F4=F5=TOTEGG=TOTEGZ=MLHR=0 C CALCULATE THE FREQUENCY DISTRIBUTION DO 100 I=1.500 EGGTT = EGG(I) IF (EGGTT.EQ.0.) F0 =F0+1 IF (EGGTT.EQ.1.) F1 = F1+1 IF (EGGTT.EQ.2.) F2 = F2+1 IF (EGGTT.EQ.3.) F3 = F3+1 IF (EGGTT.EQ.4.) F4 = F4+1 IF (EGGTT.EQ.5.) F5 = F5+1 TOTEGG = TOTEGG +EGG(I) TOTEGZ = TOTEGZ + (EGG(I)**2) 100 CONTINUE C CALCULATE MEAN AND VARIANCE MEANEG - TOTEGG/SOO VAREGG (TOTEGZ - ((TOTEGG**2)/500))/499 C CALCULATE K ESTIMATE C MOMENT ESTIMATE 0F K KEST = (MEANEG**2)/(VAREGG-MEANEG) C CALCULATE K WITH MAXIMUM LIKELIHOOD METHOD A(1) = 500-F0 A(2) = A(1) - F1 A(3) = A(2) —F2 A(4) = A(3) -F3 A(S) = A(4) ~F4 A(6) = A(5) -F5 C C CALCULATE LEFT PORTION OF MLH EQUATION. DO 110 J=1,25 ADD = .1/J MLHL = 500*(AL0010(1+(MEANEG/KEST)))*2.30259 C CALCULATE RIGHT PORTION OF MLH EQUATION MLHR = 0 DO 120 N=1,6 MLHR = MLHR + (A(N)/(KEST+(N-1))) 120 CONTINUE C CALCULATE THIS DIFFERENCE BETWEEN LEFT AND RIGHT PORTIONS DIFF = MLHL-MLHR IF (ABS(DIFF).LE. .001) GO TO 200 C ADD 0R SUBTRACT (ADD) TO ESTIMATE AND REPEAT IF (DIFF.LT.0.) KEST KEST + ADD IF (DIFF.GT.0.) KEST KEST -A00 110 CONTINUE 200 CONTINUE PRINT 220 220 FORMAT ("0MEAN, VARIANCE,K, ITERATIONS, DIFFERENCE") PRINT 230, (MEANEG,VAREGG,KEST,J,DIFF) 230 FORMAT (1X,F5.3,4X,F6.4,1X,F8.4,3X,12,2X,F8.4) PRINT 240 II "7R 41 240 FORMAT (" EGGS IN CLASSES 0 THROUGH 5 RESPECTIVELY") PRINT 250, (FO,F1,F2,F3,F4,F5) 250 FORMAT (12X,6(F4.0,3X)) RETURN END 42 Subroutine NEGBIN NEGBIN distributes egg masses according to a negative binomial distri- bution with exogenously specified mean (MEANEG) and k (KEST). Change in k relative to the mean is given by; KEST = (estimated k) + (BETA * MEANEG) where BETA is specified. With each iteration the mean increases linearly through the function; MEANEG = MEANEG + CHANGE where CHANGE is Specified. The probability of occurrance of egg masses per branch with a range of zero to five egg masses is given P(X) where X is equal to one through six. Egg masses are allocated over a sample of 500. 43 SUBROUTINE NEGBIN (KEST,EGG,MEANEG,CHANGE, BETA ) DIMENSION EGG (500), PX(6), SAMP(6) REAL KEST,MEANEG F0=F1=F2=F3=F4=F5=0 C CALCULATE CHANGE IN THE MEAN AND KEST MEANEG = MEANEG + CHANGE KEST=1.0768 + (BETA*MEANEG) C CALCULATE P(X=O) = PX(l) PX(1) = 1/((1+(MEANEG/KEST))**KEST) C CALCULATE P(X(GT)0) MITH GENERAL FORM: C P(X= J)= P(X=J-1)*((K+J-1)/J)*(XBAR/(XBAR+K) OO 10 N=2,6 M=N-1 PX(N)= PX X(M)*((KEST + N-2)/(N-1))*(MEANEG/(MEANEG+KEST)) 10 CONTINUE C CALCULATE NUMBER OF SAMPLES WITH X EGGS PER SAMPLE C AND TOTAL SAMPLES IN DISTRIBUTION COUNT = O OO 20 J=1,5 M=7-O SAMP(N) = PX(N) * 500 I = (SAMP(N) + .5) DO 30 K=I,I M = K+COUNT EGG(M) = 30 CONTINUE COUNT = COUNT + SAMP(N) + .5 20 COUNT SAMP(I) + PX(I) * 500 J = COUNT DO 40 I=J.5OO EGG(I) = 0 CONTINUE 00-9 0 PRINT 50 50 FORMAT ("OSAMPLES DEFINED IN DISTRIBUTION FOR X; 0-5”) PRINT 60, (SAMP(J), J=I,6) 6O FORMAT (1X,5(3X,F7.3)) PRINT 70 70 FORMAT (" MEAN 0F SAMPLE, KEST 0F SAMPLE”) PRINT 80, MEANEG,KEST 80 FORMAT (2X,F7.4,2X,F7.4) DO 100 I=1.500 EGGTT = EGG(I) IF (EGGTT.EQ.0) F0 = FO+1 IF (EGGTT.EQ.1.) = F1+1 IF (EGGTT.EQ.2.) = F2+1 IF (EGGTT.EQ.3.) F3 = F3+1 IF (EGGTT.EQ.4.) = F4+1 IF (EGGTT.EQ.5.) = F5+I 100 CONTINUE PRINT 200 44 200 FORMAT (" SAMPLES IN CLASSES 0 THROUGH 5 RESPECTIVELY") PRINT 210, FO,F1,F2,F3,F4,F5 210 FORMAT (12X,6(F4.0,3X)) RETURN END 45 Subroutine SEQUAN SEQUAN randomly samples the egg mass papulation 1000 times with a truncation point of 50 observations per sample and classifies the population according to the Sequential Probability Ratio Test (Wald 1947). Input consists of the egg mass pOpulation (EGG). Output consists of the number of rejections (NREJ) and acceptances (NACEPT) of the null hypothe- sis, no decision cases (NODEC) and the average sample size (AVGSAM). The upper and lower rejection boundaries (URB and LRB) are given by; URB 3.125 + (.323*N) LRB -3.125 + (.323*N) for a and 6 equal to 0.10 where N is the sample size. For a and 0 equal to 0.05 the equations take the form; URB 4.193 + (.323*N) LRB -4.193 + (.323*N) 46 SUBROUTINE SEQUAN (EGG) DIMENSION EGG (500) REAL LRB NREJ = NACEPT = TOTSAM = O 00 500 K= 1,1000 EGGT = 0 C WITH AN OPEN TEST A MAX OF 50 SAMPLES ARE TAKEN DO 60 N=1.50 TOTSAM = TOTSAM +1 C GENERATE A RANDOM SAMPLE I= 500*RANF(O.) +1 EGGT = EGGT + EGG(I) C CALCULATE URB AND LRB C URB = IU + B(N) URB = 3.125 + (.323*N) C LRB = IC + B(N) LRB = -3.125 + (.323*N) C DECISION MAKING IF (EGGT .GT. URB) GO TO 100 IF (EGGT .LT. LRB) GO TO 110 60 CONTINUE GO TO 500 100 NREJ = NREJ +1 GO TO 500 110 NACEPT = NACEPT +1 500 CONTINUE NODEC = 1000 - NREJ - NACEPT AVGSAM = TOTSAM/1000 C PRINT RESULTS PRINT 200 200 FORMAT (" SEQUAN SUMMARYzACCEPTANCES, REJECTIONS, NO DECISIONS PRINT 250, (NACEPT,NREJ,NODEC) 250 FORMAT (2X,3(I3,8X)) PRINT 300 300 FORMAT (" AVERAGE NUMBER OF SAMPLES PER DECISION") PRINT 350, AVGSAM 350 FORMAT (3X,F7.4) RETURN END 47 Subroutine BTTEST BTTEST randomly samples the egg mass population 1000 times with a truncation point of 50 observations per sample and classifies the population according to Barnard's sequential t-test (Barnard 1952). In- put consists of the egg mass population (EGG). Upper and lower rejec- tion boundaries (UNR, UNA) are initialized as these were determined from published tables (National Bureau of Standards t-test Tables). The test initially requires seven random samples. Following these samples the decision statistic (DESTAT) is calculated and recalculated for each subsequent single sample. The first and subsequent test statis- tics are calculated as; DESTAT = TDEv/(TDEVSQ)15 where; 1) TDEV = ZDEV and DEV is the deviation from the critical density of each observation, 2) TDEVSQ = ZKDEV)2. Output consists of the number of acceptances (NACEPT) and rejections (NREJ) of the null hypothesis, no decision cases (NODEC) and average sample number (AVGSAM). 48 SUBROUTINE BTTEST (EGG) DIMENSION UNR(43), UNA(43), EGG(SOO) NREJ = NACEPT = TSAMP = O C ENTER VALUES 0F UNR, UNA, A=B = .05, D=.5 C DATA UNA /-1.51.-1.33.-1.15.-1.034,-.918,-.802,-.686,-.57,-.498 +-.426,-.354,-.282,-.21,-.154,-.098,-.042,.014,.07,.114,.158,.202, +.246,.29,.328,.366,.404,.442,.48,.514,.548,.582,.616,.65,.678, +.706,.734,.762,.79,.816,.842,.868,.894,.92/ DATA URN /2.56,2.51,2.46,2.436,2.412,2.388,2.364,2.34,2.334,2.328, +2.322,2.316,2.31,2.308,2.306,2.304,2.302,2.30,2.304,2.308,2.312, +2.316,2.32,2.328,2.336,2.344,2.352,2.36,2.368,2.376,2.384,2.392, +2.4,2.408,2.416,2.424,2.432,2.44,2.45,2.46,2.47,2.48,2.49/ DO 100 M=1,1000 TDEV = TDEVSQ = O C THE TEST REQUIRES 7 RANDOM SAMPLES INITIALLY DO 110 J=1.7 I= 500*RANF(O.) +1 C CALCULATE DEVIATION, SQUARED,TOTALS,FROM H0 DEV = EGG(I) - .499999 DEVSQ = DEV**2 TDEV = TDEV + DEV TDEVSQ = TDEVSQ + DEVSQ 110 CONTINUE TSAMP = TSAMP + 7 C CALCULATE DECISION STATISTICS FOR N UP TO 50 DO 120 L = 1.43 TSAMP = TSAMP + 1 I = 500*RANF(O.) +1 DEV = EGG(I) - .499999 DEVSQ = DEV**2 TDEV = TDEV + DEV TDEVSQ = TDEVSQ + DEVSQ IF (TDEVSQ .EQ.0.) GO TO 200 C CALCULATE DECISION STATISTIC DESTAT = TDEV/(SQRT(TDEVSQ)) C DECISION MAKING IF (DESTAT .GT. UNR(L)) GO TO 300 IF (DESTAT .LT. UNA(L)) GO TO 310 GO TO 120 200 DESTAT = 0 IF (DESTAT .LT. UNA (L)) GO TO 310 120 CONTINUE GO TO 100 300 NREJ = NREJ +1 GO TO 100 310 NACEPT = NACEPT + 1 100 CONTINUE AVGSAM = TSAMP/1000 NODEC = 1000-NREJ-NACEPT C PRINT RESULTS PRINT 400 400 FORMAT ("-SUMMARRY OF BARNARD'S TTEST DECISIONS") PRINT 410 410 420 430 440 49 FORMAT (" ACCEPTANCES. REJECTIONS, N0 DECISIONS") PRINT 420, NACEPT. NREJ, NODEC FORMAT (5X,I4,5X,I4,8X,I4) PRINT 430 FORMAT (" AVERAGE SAMPLE NUMBER") PRINT 440, AVGSAM FORMAT (4X,F6.2) RETURN END 50 Subroutine TTEST TTEST randomly samples the egg mass population 1000 times and classifies the population according to Fowler's and O'Regan's truncated sequential t-test (Fowler and O'Regan 1974). Input consists of the egg mass population (ECG) and the number of branches (NN) to serve as an observation. This is refered to as sample surface in the text. Upper and lower rejection boundaries (UNR, UNA) are initialized. Two random samples are initially required. The decision statistic (DESTAT) is calculated following this sampling and for each subsequent single observation until a decision is reached through; DESTAT = TOTSUM/TOTSSQ where; 1) TOTSUM = ZSUM and SUM is the deviation from the critical density for each sample, 2) TOTSSQ = EKSUM)2. Output consists of the number of acceptances (NACEPT) and rejections (NREJ) of the null hypothe- sis and average number of branches sampled (AVGBRN). 51 SUBROUTINE TTEST (EGG, NN) DIMENSION UNR(IO), UNA(IO), EGG(SOO) NREJ = NACEPT = 0 DATA UNR / 1.4093,1.544,1,265,1,036,.8958,.7795,.7483,.7149,.7385 DATA UNA / -1.3261,-.9465,-.5178,-.2148,.0199,.2343,.4265,.5772, +.7385 TSAMP = 0 DO 500 M = 1,1000 C THE TEST REQUIRES 2 RANDOM SAMPLES INITIALLv EGGI = EGGZ = 0 DO 20 N = 1,NN I = 500*RANF(O.)+1 EGGI = EGGI + EGG(I) I = 500*RANF(O.)+1 EGGZ = EGG? + EGG(I) 20 CONTINUE TSAMP = TSAMP + 1 C CALCULATE SUM AND SUM SOUARED EGG SUMI EGGI - (.49999*NN) SUM2 EGG? - (.49999*NN) SUMSOl = SUM1**2 SUMSOZ = SUM2**2 C CALCULATE TOTAL SUM, TOTAL SUMSQUARED TOTSUM = SUMI + SUM2 TOTSSQ = SUMSQl + SUMSQZ C CALCULATE DECISION STATISTIC DESTAT = TOTSUM/(SQRT(TOTSSQ)) IF (DESTAT.GT.UNR(1)) GO TO 100 IF (DESTAT.LT.UNA(1)) GO TO 110 C IF NO DECISION WAS MADE CONTINUE SAMPLING, MAX N = 10 DO 10 J=1,8 TSAMP = TSAMP+1 EGGS = 0 DO 30 N=1,NN I = 500*RANF(O.)+1 EGGS = EGGS + EGG(I) 30 CONTINUE L = J+1 SUM = EGGS - (.49999*NN) SUMSQ = SUM**2 TOTSUM TOTSUM + SUM TOTSSQ TOTSSQ + SUMSQ DESTAT - TOTSUM/(SQRT(TOTSSQ)) IF (DESTAT.GT.UNR(L)) GO TO 100 IF (DESTAT.LT.UNA(L)) GO TO 110 10 CONTINUE GO TO 500 100 NREJ = NREG + 1 GO TO 500 110 NACEPT = NACEPT + 1 500 CONTINUE TBRAN = TSAMP * NN 52 AVGBRN = TBRAN/1000 C PRINT RESULTS PRINT 200 200 FORMAT (" SUMMARY OF T-TEST DECISIONS") PRINT 210 210 FORMAT (" REJECTIONS, ACCEPTANCES, SAMPLE SURFACE") PRINT 220, NREJ, NACEPT, NN 220 FORMAT (2X,3(I3,8X)) PRINT 230 230 FORMAT (" AVERAGE NUMBER OF BRANCHES SAMPLED PER DECISION") PRINT 240, AVGBRN 240 FORMAT (5X,F7.4) RETURN END 53 Subroutine SEQRT SEQRT randomly samples the egg mass population 1000 times with a truncation point of SO observations per sample and classifies the pOpu- lation according to Iwao's sequential test (Iwao 1975). Input consists of the egg mass population (ECG) and critical density (CDEN) which is to be exceeded for rejection of the null hypothesis. The upper (URB) and lower (LRB) decision boundaries take the form; URB = (N*CDEN) + A LRB = (N*CDEN) - A where; A = (T * (O.9948*N*CDEN)+(O.6952*N*CDEN2) and T is the desired confidence probability (Student's t). Output consists of the number of rejections (NREJ) and acceptances (NACEPT) of the null hypothesis, no decision cases (NODEC) and average sample size (AVGSAM). 54 SUBROUTINE SEQRT (EGG, CDEN) C CALCULATE DECISION BOUNDARIES; N=1, (1), 50 C DECISION BOUNDARY FORM; NM + T(SQRT(N(A+1)M + (B-l)M**2)) C M=CDEN = CRITICAL DENSITY, T = STUDENT'S T 10 DIMENSION URB(50), LRB (50), EGG(500) REAL LRB T = 1.645 00 IO N=1.50 A = T*(SQRT((0.9948*N*CDEN) + (.6952*N*(CDEN)**2)))) LRB(N) = (N*CDEN) - A URB(N) = (N*CDEN) + A CONTINUE NGTHO = NLTHO = TOTSAM = O C CALCULATE DECISIONS STATISTIC FOR 1000 SAMPLES DO 500 K=1,1000 EGGT = 0 C A MAXIMUM 0F 50 SAMPLES ARE TAKEN DO 60 N=1,50 TOTSAM = TOTSAM+I C GENERATE A RANDOM SAMPLE I = 500*RANF(O.)+1 EGGT = EGGT + EGG(I) C DECISION MAKING 60 100 110 500 T0 100 IF (EGGT.GT.URB(N)) GO N ) GO TO 110 IF (EGGT.LT.LRB( ) CONTINUE GO TO 500 NGTHO = NGTHO + 1 GO TO 500 NLTHO = NLTHO + 1 CONTINUE NODEC = 1000 - NGTHO - NLTHO AVGSAM TOTSAM/1000 AVGDEC (TOTSAM -(NODEC*50))/(1000-NO0EC) C PRINT RESULTS 200 250 300 350 PRINT 200 FORMAT (" SEQRT: GREATER THAN H0, LESS THAN H0, N0 DECISION") PRINT 250, NGTHO,NLTHO,NODEC FORMAT (3(10X,I4)) PRINT 300 FORMAT (" AVERAGE SAMPLES, AVERAGE SAMPLES W/ DECISION") PRINT 350, AVGSAM, AVGDEC FORMAT (4X,F7.3,11X,F7.3) RETURN END REFERENCES C ITED References Cited Abrogast, R. 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