MSU LIBRARIES “ RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. @2290" 5X .L 01 7 MULTIFACTOR ANALYSIS OF ENVIRONMENTAL PRE-CONDITIONING OF TOMATO SEEDLINGS ON PROTOPLAST CULTURE AND DEVELOPMENT BY RANDALL PAUL NIEDZ A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Plant Breeding and Genetics Inter-Departmental Program 1987 :8 1r ABSTRACT MULTIFACTOR ANALYSIS OF ENVIRONMENTAL PRE-CONDITIONING OF TOMATO SEEDLINGS ON PROTOPLAST CULTURE AND DEVELOPMENT BY Randall Paul Niedz Leaf mesophyll protoplasts were enzymatically isolated from 24-30 day old seedlings of Lycopersicon esculentum P.I. 367942 grown either in a controlled environment chamber (CEO) or i_ giggg (IV). Prior to protoplast isolation, donor seedlings were pre—conditioned in a 3x3 factorial arrangement of 0, 12 or 24 hours at 10°C and O, 24 or 48 hours in the dark. The variables measured included protoplast yield, 244 hour viability, 30—day microcalli diameter and shoot morphogenesis. Multifactor fixed effect linear models were constructed to determine the influence of seedling source and cold/dark pre-conditioning on the observed variables. Protoplasts were cultured in the dark (27°C) in a liquid modified Kao and Micahyluk (KM) medium over a semi-solid modified Murashige and Skoog (MS) medium. Both media contained the growth regulators (mg/L): 2,4- dichlorophenoxyacetic acid (1AM, alpha—naphthaleneacetic acid (1.0) and 6-benzylaminopurine (01”. To stimulate shoot morphogenesis protoplast—derived calli were cultured in the light (cool white fluorescent lamps; 30 uEm'zs'l) on modified MS medium with 2 mg/L zeatin. Protoplasts from CEC donor seedlings pre—conditioned with RANDALL PAUL NIEDZ 0 hours dark/12 and 24 hours cold died in 24-48 hours. Protoplast yield from CEC seedlings preconditioned with 12 and 24 hours of cold was reduced an average of 27.4% (compared to 0 hours cold over all levels of dark). However, the subclass 12 hours cold/48 hours dark was not significantly different from 0 hours cold/all levels of dark. In.contrast, yield from IV seedlings was not significantly reduced by cold preconditioning, but decreased an average 43.9% by a 48 hour dark treatment over all levels of cold. Protoplast viability was improved an average of 14.1% of CEO seedlings receiving a 48 hour dark treatment, and was 0% for the 2 treatments 12 and 24 hours cold/O hours dark. For IV seedlings preconditioned with 12 and 24 hours cold/0 hours dark protoplast viability was 6.9% greater than compared to 0 hours cold. CEC microcalli diameter of 12 and 24 hour cold were 12.5% less than 0 hour cold, In contrast, IV microcalli from 12 hour cold/0 hour dark:were~11.8% larger than microcalli from all other IV preconditioning treatments. Shoot morphogenesis from CEC seedlings was 10.3% less from 0 hours dark compared to 24 and 48 hours dark, and 25.4% less from 24 hour cold compared to O and 12 hour cold preconditioning of IV seedlings. TABLE OF CONTENTS Page LIST OF TABLES ---------------------------------------- v LIST OF FIGURES --------------------------------------- vi ABBREVIATIONS ----------------------------------------- vii INTRODUCTION ------------------------------------------ 1 LITERATURE REVIEW ————————————————————————————————————— 4 I. History of the tomato -------------------------- 5 II. In vitro culture ------------------------------- 5 A. Tissue culture ------------------------------- 5 B. Genetic control of shoot morphogenesis ------- 8 C. Protoplast culture --------------------------- 11 D. Anther culture ------------------------------- 13 E. Uses ----------------------------------------- 14 III. Preconditioning of donor tissue ---------------- 18 A. Genotype ------------------------------------- 18 B. Greenhouse, CEC, _I_n_ y_ tr_q ------------------ 19 C. Light ---------------------------------------- 20 D. Nutrition ------------------------------------ 21 E. Tissue age and source ------------------------ 22 F. Senescence retardents ------------------------ 23 G. Temperature and humidity --------------------- 23 MATERIALS AND METHODS --------------------------------- 24 I. Protoplast isolation and culture ---------------- 24 A. Source Material ------------------------------- 24 B. Seedling Preparation -------------------------- 24 1. CEC grown seedlings ------------------------ 24 2. In vitro grown seedlings ------------------- 24 3. Cold and dark pretreatment of donor tissue — 25 ii C. Protoplast Isolation -------------------------- 25 1. CEC grown seedlings ------------------------ 25 2. In vitro grown seedlings ------------------- 27 D. Protoplast Culture and Shoot morphogenesis -—-— 27 II. Experimental design and data analysis ———————— ~—— 28 RESULTS ----------------------------------------------- 29 I. Yield ------------------------------------------- 29 A. Test of interaction effect -------------------- 29 B. Goodness of fit ------------------------------- 29 C. Contrasts ------------------------------------- 30 II. Viability --------------------------------------- 33 A. Test of interaction effect -------------------- 33 B. Goodness of fit ------------------------------- 33 C. Contrasts ------------------------------------- 33 III. Microcalli Diameter ------------------------------ 36 A. Test of interaction effect -------------------- 36 B. Goodness of fit ------------------------------- 36 C. Contrasts ------------------------------------- 36 IV. Shoot Morphogenesis ------------------------------ 39 A. Test of interaction effect -------------------- 39 B. Goodness of fit ------------------------------- 39 C. Contrasts ------------------------------------- 40 DISCUSSION -------------------------------------------- 44 I. Protoplast Yield ------------------------------- 44 II. Protoplast Viability --------------------------- 45 III. Microcalli Diameter ---------------------------- 47 IV. Shoot Morphogenesis ---------------------------- 49 SUMMARY AND CONCLUSIONS -------------------------------- 51 APPENDIX A Summary Statistics and Data Analysis ------- 53 APPENDIX B Least Squares Approach --------------------- 72 1. General fixed model ------------------------- 73 2. Estimation ---------------------------------- 76 3. The normal equations ------------------------- 77 4. Solving for b ------------------------------- 80 5. Estimable functions ------------------------- 81 6. Other calculations -------------------------- 85 111 7. MEthOd of calculation ——————————————————————— LIST OF REFERENCES iv Table A1. A2. A3. A4. A5. A6. A7. A8. A9. A10. A11. A12. A13. A14. A15. A16. A17. A18. A19. Bl. 82. 83. 84. LIST OF TABLES Page Reductions in sum of squares for yield ------------ 53 Reductions in sum of squares for viability -------- 54 Reductions in sum of squares for microcalli diameter ------------------------------------------ 55 Reductions in sum of squares for shoot morphogenesis ------------------------------------- 56 Single factor evaluation -------------------------- 57 Goodness of fit for 2-way interaction model ——————— 58 Goodness of fit for 3-way interaction model ------- 59 Smallest subclass means for yield ----------------- 60 Contrasts for yield ------------------------------- 61 Smallest subclass means for viability ------------- 62 Contrasts for viability --------------------------- 63 Smallest subclass means for microcalli diameter --- 64 Interaction effects ------------------------------- 65 Contrasts for microcalli diameter ----------------- 66 Smallest subclass means for shoot morphogenesis --- 67 Contrasts for shoot morphogenesis ----------------- 68 Pre-conditioning treatments with the highest means for protoplast yield, viability, microcalli diameter and.shoot morphogenesis --------------- 69 Tomato protoplast media --------------------------- 70 Purification of agar ------------------------------ 71 Protoplast yield (x106) of dark pre-conditioned seedlings ----------------------------------------- 74 x matrix as a 2-way table ------------------------- 78 Four solutions for b0 ----------------------------- 82 Estimates of four linear functions ---------------- 84 LIST OF FIGURES Figure 1. 2. Effect of cold and dark pre-conditioning on protoplast yield. (a) CEC grown seedlings. (b) IV grown seedlings. ----------------------------- Effect of cold and dark pre-conditioning on protoplast viability. (a) CEC grown seedlings. (b) IV grown seedlingS. _________________________ Effect of cold and dark pre-conditioning on microcalli diameter. (a) CEC grown seedlings. (b) IV grown seedlings. ------------------------- Effect of cold and dark pre-conditioning on shoot morphogenesis. (a) CEC grown seedlings. (b) IV grown seedlings. -------------------------------- vi Page 31 34 37 42 ABBREVIATIONS BAP — 6—benzylaminopurine NAA - alpha-naphthalenacetic acid 2,4-D - 2,4-dichlorophenoxyacetic acid CEC - controlled environment chamber IV - in vitro C — factor cold D - factor dark 8 - factor source CD - cold x dark interaction CS - cold x source interaction DS - dark x source interaction CDS - cold x dark x source interaction uEm"28-1 - Micro Einstein per square meter and second vii INTRODUCTION Most plant breeding programs follow a process of recombination and selection within an elite germplasm pool to produce superior cultivars. Techniques such as induced polyploidy, wide hybridization, mutant induction and i gitro methods supplement primary breeding operations by providing geneth: variability difficult and/or impossible to obtain otherwise. ;_ yitrg methods can be classified into three general catagories; 1) the micropropagation of shoot tips to produce viruS*free stock, multiply hard to propagate species and expensive F1 hybrids, 2) the culture and manipulation of protoplasts, cells and tissue to produce genetically useful individuals and, 3) vector introduced genes. Plant regeneration from protoplasts has been demonstrated in over 70 species, half of which are represented by the Solanaceae. Successful application of in yitgg techniques such as exogenous DNA and chromosome uptake by protoplasts, somatic cell fusion and cell selection to improve a species economically, requires a base technology of efficient plant regeneration from protoplasts. Factors important in successfully regenerating plants from cultured protoplasts include genotype, growth conditions of the donor plant, pre-conditioning of donor tissue, types and 1 2 concentrations of isolation enzymes, composition of the culture medium and the environmental conditions during isolation and culture. In tomato and potato the physiological condition of the donor plant is of critical importance in achieving successful isolations of viable protoplasts capable of sustained divisions (160,164,168). Tabaeizadeh et al. (175) found that a cold/dark treatment of whole plants and detached leaves significantly increased the number of protoplast-derived calli though the effect was not as great with detached leaves. Shahin (160) reported the necessity of dark and low temperature preconditioning of donor tomato tissue in isolating viable protoplasts. Likewise, Shepard and Totten (165) used a short photoperiod and low'temperature preconditioning of donor plants. Without such preconditioning, protoplasts failed to undergo division regardless of the culture medium used. How these factors influence the various stages of growth from initial cell viability to shoot formation from.protoplast-derived calli is little understood as much of the research to date is empirical and/or anecdotal. This study attempts to identify those stages of protoplast development most influenced by dark and cold preconditioning of the donor plant by preconditioning donor tissue and measuring protoplast yield, viability, rate of cell colony growth and shoot morphogenesis. The extent that such treatments are necessary in achieving shoot regeneration can then be examined by determining the contribution of cold 1" 3 and dark on each measured variable. The tomato, Lycqpersicon esculentum, is used in this study as it is the worldws leading vegetable crop and is also a prominent species for somatic cell genetic manipulations. LITERATURE REVIEW I. History of the Tomato The cultivated tomato, Lyggpersicon escglentgm, originated in western South America and is a member of the family Solanaceae. All ten species in the genus Lycqpersicon are 2g-2g-24 (146). Domestication is thought to have first occurred in Mexico. When the tomato was first taken to Europe in the 16th century it was thought to be poisonous due to deadly glycoalkaloids present in the other members of the nightshade family such as belladonna and henbane. The tomato currently ranks second to potato in economic importance among vegetables, with a combined value for processing and fresh market fruit exceeding 800 million dollars per year in the United States (111). In addition to its culinary appeal the tomato is one of the best known crop plants in terms of its established genetics and cytogenetics (147). A favorable biology such as a short life cycle, ecological versatility, high seed yields (up to 20,000 seeds per plant), diploid genome, self—pollinating and amenability to hybridization are responsible for this species appeal to researchers. Plant breeders have made significant improvements during the past 50 years as follows: 1. Increased yields - larger and greater numbers of fruit per plant, improved flower set and a more concentrated fruit set 2. Plant habit changed to accomodate cultural and harvesting operations - the gp gene for determinate growth is mainly responsible for such improvements 4 s ' v . . l . \t. I. . .J l : x O I. : .. . v . t I -a r .1, v y- a» ‘ r a . . l e 4: . f h . I ‘ 4 .~ . m. . . v. k . ._ a ‘. . . x. ‘ I ,.... on. , 1. .. nJ . v. . . 2 . o J . 4 . A l. . I w J r. . . - .. o \ . . . O V. .. .\ I ,) x. J .. 4 . a r . . a. . . . 4 . . ‘. s . .. I g I P n . o .. . ., I v Io \. 3,4 . , V- A I ,g a .. . _ . ... , p . 4A . . .. . .4 . . . . 3. Improved handling and storage 4. Pest resistance. II. In Vitro Culture A. Tissue culture The first successful 1 yitro culture of tomato was reported by White in 1934 (199). He succeeded in culturing isolated root tips in liquid suspension culture. The roots grew indefinitely for over a year. This indicated to White that root growtthas not dependent on the top of the plant. Twenty years later root cultures were again utilized, but from the wild.species Lycgpersicon peruvianum (127). Some of the cultured roots formed shoots that were transferred to soil. It was not until 1973 that routine shoot morphogenesis was first reported using cultured internodes of’;u_esculentum (42). Numerous reports followed detailing rapid shoot morphogeneous from a variety of tissues: leaves (8,35,74,80,87,106,129,133,137,17B), stem internodes (41,103), apical meristems (86), cotyledons (66,92,134,195) and hypocotyls (66,71,103,131,210). The general methodology consisted of placing explants from in gitgg or greenhouse seedling explants onto a semisolid medium containing inorganic salts, macronutrients and micronutrients, usually those of Murashige and Skoog (120) , vitamins of the B group (thiamine, pyridoxine and nicotinic acid), glycine, myo- inositol, sucrose and an auxin and/or cytokinin. 6 Using 64 treatment combinations of indole—3-acetic acid (IAA) and 6-furfurylaminopurine (kinetin), Padmanabhan (133) classified tomato as one of the plant species with morphogenetic expression regulated by the auxin/cytokinin ratio as first discovered by Skoog and Miller (169). When an auxin is used, IAA is reported to be optimum for achieving shoot morphogenesis (80,87), with alpha-naphthaleneacetic acid (NAA) and 2,4-dichlorophenoxyacetic acid (2,4-D) almost always inhibitory to shoot morphogenesis (7,131,195). Moreover, 2,4-D is generally most useful when initiating or maintaining callus and suspension cultures. The cytokinins zeatin, 6-benzlyaminopurine (6-BAP), kinetin and 6- (gamma,gamma-dimethylallylamino)-purine (21?), in increasing efficiency, all seem to initiate shoots of tomato with or without an accompanying auxin. Although there are numerous reports on shoot morphogenesis in the cultivated tomato, these experiments achieved shoot formation from primary leaf, stem or hypocotyl explants (8,80,87,131,133,178) and not from secondary or older callus. DeLanghe and DeBruiJne (41) studied shoot regeneration in L; esculentum and L. peruvianum. They hypothesized that the lower shoot—forming capacity of L; gggglggggg was due to a higher endogenous level of gibberellic acid (CA), as addition of GA to the culture medium resulted in a high production of callus and fewer shoots. Adding 1000 ppm chlormequat (CCC), a GA inhibitor, to the culture medium had no effect on the shoot-forming I. . . 1 . u . . r . a .J .n _ .J . a. , . . 1.. , . . L. s . .l . . . .J . . _ ‘0 .. . I .4 o ‘ 4 4(- l . . . 1 I . N Is . e . . I . c .. . . 4. s. : w \m . — p I 4. 0 I. p... A . - 1. . . . , . . .. _ . - 1.. , . .4. . . . . . I . r . I! . l . 4-. . . . . . . . , . . , , u ‘ ‘1' . I . o a I A u 1 . .. x J , . . 1 I I r . . s . a I H: . . vk . f c I \ e, p .- 1.. u ‘ s . 4k . .._ . . . , . . . J . 4 O O :- . .. . s . t I e e A r . n .. . I I . . a . . a . 4 . . . _ - . . . . . .. . .. _ x. . . O . . . a I . . . , , 1.. VI . 7 ability of the explants. However, if the donor seedlings were sprayed daily with 2000 ppm CCC callus derived from these seedlings maintained a undiminished shoot-forming capacity after 2 years. Meredith (109) studied shoot regeneration from 17—28 month old callus cultures of L2. eggulgntgm and LA pimpinellifiglm. L_._ pimpinellifiglm callus did not regenerate shoots, but callus from L; esculentum 'VFNT Cherry' formed shoots, however the shoots were morphologically'abnormal and could not be rooted. Plantlets regenerated from younger, 4-month-old callus, were normal and rooted easily. Behki and Lesley (7) studied the response of secondary leaf callus to the nitrogen source used in the culture medium and the growth regulators used in the induction and differentiation media. They found that decreasing the amount of nitrogen (MS salts used) inhibited shoot regeneration. A similar inhibition of shoot regeneration occurred when the NR4+ concentration equalled or exceeded the N03‘ concentration.(NH4+:N03 is 1:2 in MS). Noa', however, could be substituted for NH4+ without any deleterious effect on shoot formation. They also found in both the callus induction and the differentiation media the duration of the induction period and the presence of light during the induction period was important. I . . , .7 . .' v . - I v o s ' u 8 B. Genetic control of shoot morphogenesis In addition to media components, a strong genetic effect on shoot morphogenesis is found among L; esculentug genotypes (56,64,71,96,97,131,134,186,210). The wild Lycopersicon species L; peruvianum is in general superior to LLH span x oaoo .m.c .m.: x” *0 omo .0.9 &H xfi xfi xnmn x Unoo x monaom mammcomonmnos nuumam«n >uuflwn0w> Ufimww mcofiuOMMOch uoonm «Hamoonofiz .3003. 60308635 63 032. 66 Table A14. Contrasts for microcalli diameter. Source ho P value Sign. CEC 012 ' C24 = 0 <1 n.s. 024 ’ D‘s = 0 <1 n.s. Co ‘ C24 3 C48 58.528 1% Do 3 024 3 D48 2.329 n.s. IV 2CD12.0 - (CD0,o + CDo,24) = 0 13.182 1% (090,24 + CD12,24’ - ZCD24'24 = 0 4.231 n.s. CD12,‘8 ’ CD0,48 a O 5.407 5% 67 Table A15. Smallest subclass means for shoot morphogenesis. Smallest Subclass Means CEC IV u + co + ”0 33.460 1 1.271 33.436 1 0.966 u + 00 + 024 36.041 1 1.169 33.757 1 1.040 u + co + 048 39.576 1 1.316 34.091 1 1.176 n + C12 + no - 34.450 1 1.180 u + 012 + 024 32.133 1 1.332 34.771 1 1.115 u + 012 + D48 35.670 1 1.254 35.105 i 1.138 u + 02‘ + Do - 27.206 1 1.211 u + 024 + 024 32.915 1 1.259 27.526 1 1.226 u + 024 + 048 36.453 1 1.239 27.661 1 1.310 68 Table A16. Contrasts for shoot morphogenesis. Source ho F value Sign. CEC co=012=c24 7.665 1% 130=024==D48 14.991 1% C24 — 012 = 0 2.782 n.s 20o - (012 + 024) a 0 15.334 1% C12 - C24 = 0 <1 n.s. 2048 - (00 + 024) = 0 29.963 1% no - 02‘ a 0 4.679 1% IV (012 + Co) - 2024 = 0 >1000 1% c0 - 012 = 0 17.660 1% 012 - 024 = 0 701.365 1% 0o - 024 = 0 1.749 n.s 024 - 048 = 0 1.654 n.s 69 Table A17. Pre—conditioning treatments with highest means for protoplast yield, viability, microcalli diameter and shoot morphogenesis. Source Variable CEC IV Yield 090.0' CD0.24 CD0.24 CD24.0 CD0.46 CD12,46 CD12.0 Viability CD0,43 CD12,0 Microcalli Co + 924 CD12,0 diameter 00 + D48 Shoot Co + D43 Co + D0,24 & 48 morphogenesis C12 + D0,24 & 48 70 Table A18. Tomato protoplast culture media Components KM-T A4 KN03 1875 1875 MS salts w/o NH4N03 myo-Inositol 2000 2000 Thiamine HCl 10 10 Nicotinic acid 1 0.5 Pyridoxine HCl 1 0.5 L—glutamine 100 100 D-Ca Pantothenate 1 - Folic acid 0.4 - p—ABA 0.02 - 1/2 V-KM organic acids2 + Biotin 0.01 — Choline Chloride 1 - Ascorbic acid 2 - V-KM sugars and sugar alcohols2 + - V-KM supplements2 + — NAA 1 1 6-BAP 0.5 0.5 Sucrose 17100 17100 Mannitol 63700 31850 Glucose — 31500 Purified agar1 - 6000 pH 5.8 5.8 1 See Table 2. 2 ref. 17. 71 Table A19. Purification of agar 5. Suspend 1/2 pound of Bacto Agar in ddH20 and wash by drawing of the supernatent. Resusgend agar in ddeo and allow it to stand overnight at 4 C. The resultant cake is resuspended in acetone and filtered through 2 ply cheesecloth. The resultant cake is resuspended in 100% ethanol and filtered through 2 ply cheesecloth. The cake is dried. APPENDIX B 72 Least Squares Approach In cell and tissue culture experiments the data collected is commonly unbalanced. In other words, there are unequal numbers of observations in the individual treatment classes. The problem with unbalanced data is the sums of squares due to the individual factors do not sum to the total sum of squares due to all of the factors. Hence, the factors are not independent and so must be considered simultaniously. The calculations involved in this type of analysis are usually more complicated than those of traditional analysis of variance for balanced data. Prior to the era of computers only limited demand existed for analyzing unbalanced data. Now, however, because of the availability of computers, analysis of unbalanced data is increasingly more common. Analysis of unbalanced data cannot be made Just by means of some minor adjustments to the traditional analysis of variance of balanced data. Unbalanced data analyses have their own analysis of variance techniques, and those for balanced data are merely special cases of the techniques for unbalanced data. Unbalanced data analyses use matrix expressions, many of which do not simplify in terms of summation formulas. When the data is balanced or, in other words, the number of observations are all the same, these matrix expressions simplify considerably. They reduce, in fact, to the well-known summation formula of traditional analysis of variance of designed experiments. It is easy then to think of balanced y- 73 designed experiments as special cases of the more basic analysis of variance for unbalanced data. The method of analysis used in these experiments is the Least Squares Approach, also known as the method of fitting constants of the linear model. 1. General fixed model A simplified example will best introduce the idea of the general fixed model used in these experiments and the matrices necessary to manipulate and solve them. Example. Numbers of protoplasts isolated per gram of fresh tissue was obtained for 47 individual protoplast isolation experiments. Protoplasts were isolated from seedlings preconditioned by 0, 24 or 48 hours in the dark prior to protoplast isolation. I will consider just 9 isolation experiments for illustrative purposes, 4 from 0 hours, 3 from 24 hours and 2 from 48 hours dark preconditioning. For the entries in TableBi let Yij represent the protoplast yield of the 3th plant of the ith dark treatment, i having a value of 1, 2 or 3 for 0, 24 and 48 hours respectivelyu and j=1,2,. ..,n1, where n1 is the number of observations in the ith dark treatment. The problem is to estimate the effect of dark preconditioning on protoplast yield. To do this the observation Yij is considered as the sum of three parts Yij 3 u + d1 + e13, (1) C} l a O LIE" fit .. r. . . x *O r u .. .. . \ r. . L ... ,2 1. . n . 9 . . I . . . ‘ I m r. _ 74 Table Bl. Protoplast yield (x106) of dark precondition- ed seedlings. 9 hrs. _1 hrs 18 hrs 1.073 1.002 0.884 0.992 1.202 0.973 1.053 1.136 1.023 Totals 4.106 3.393 1.857 75 where u represents the population mean or the constant common to all observations, d1 is the effect of dark preconditioning i on protoplast yield, and e13 is the random error associated with each Yij' The eij's are assumed to be independent and distributed around a mean of zero. Also, it is assumed each e13 has the same variance,<32, so that the variance-covariance matrix of the vector of eij's is €21. r — F — 620....0 10...0 2 (1‘2...0 201...0 C I a = (T 2 00 q. 0 1 a a L A The problem is to estimate u and the three d1 terms, and also <72, the variance of the error terms. To provide an estimation of these parameters each observation is written in terms of the above a-part equation (1): 1.038 = Yll g u + ldl + ell 0.992 = Y12 3 u + 1d1 + 312 1.053 ' Y13 : u + ldl + 313 1.023 = Y14 = u + 1d1 + 814 1.055 = Y21 = u + 1d2 + 921 1.202 a Y22 = u + 1d2 + 922 1.136 a Y23 a u + 1d2 + 823 0.88‘ B Y31 g u + 1d3 + 831 0.973 3 Y32 B u + Ida + 832 The equations are easily converted into matrix forms 76 1.038 VII 1 1 O 0 ell 0.992 y12 1 1 0 0 u e12 1.053 y13 1 1 0 0 d1 + e13 1.023 = Y14 = 1 1 O 0 d2 614 1.055 Y21 1 0 1 0 daj 821 1.202 Y22 1 O 1 0 _ 822 1.136 Y23 1 0 1 O 823 0.884 Y31 1 0 O 1 331 0.973 Y32 1 O O 1 832 L a h d L J b a representing the general form of the fixed model y = Xb + e where y is the vector of individual observations, the X is termed the incidence matrix, because the presence of 0's and 1's describes the incidence of the terms in the model (u, d1, d2 and d3), and b is the vector of parameters to be estimated, and e is the vector of error terms. 2. Estimation The problem is to estimate b as all computational procedures for‘estimation and hypothesis testing requires estimates of b. Applying the method of "Least Squares" results in an estimator of b that minimizes the squared errors (i.e. sum of squared deviations) Differentiating e'e with respect to b minimizes e'e. The resultant equations are termed the normal equations. A 3'!!!) - X'y A A where the "A" represents the solutions of b. Therefore, b is solved for algebraically from: A b - (x'X)'1x'x where (X'X)"1 is the inverse of the coefficient matrix (X'X). 3. The normal equations The coefficient matrix, mat, in the normal equations is obtained from the incidence matrix x in the model, y - Xb + e. The incidence matrix, 1:, can be thought of as a 2-way table with the parameters as headings to the columns and the observations as headings to the rows (Table 82). From the example, ,“A‘ 78 Table 82. x matrix as a 2-way table. Parameters Observations u d1 d2 d3 y12 1 1 0 0 y14 1 1 0 0 23 1 O 1 0 31 1 0 0 1 O 79 — — — - 1 1 1 1 1 1 1 1 1 1 1 0 0 9 4 3 2 1 1 1 1 O 0 0 O O 1 1 O 0 4 0 0 0 O O O O 1 1 1 O 1 O 2 L... .1 ._ _J l | 000011100 1010 3.030 I I 1 O O 1 L. .3 The top left-hand element, 9, is the total number of observations with the next three elements, 4, 3 and 2 being the number of observations in those particular classes, 0, 24 and 48 hours dark preconditioning, repectively. The submatrix is composed of the number of observations in each subclass. So, the intersection of 24 hrs. and 48 hrs. is 0 as the seedlings were preconditioned in one of the three treatments. Likewise, the 1!"! matrix is obtained from the incidence matrix X and the y observation vector from the model y - X'b + e. In the )1"! 9.356 is the top element and is the sum of all nine observations, and the following elements are the corresponding sums of the individual classes 0, 24 and 48 hrs dark ( d1, d2 and d3 ), respectively. 80 1.038 0.992 1 1 1 1 1 1 1 0 0 1.053 9.356 1 1 1 1 0 0 0 0 0 1.023 4.106 0 0 0 0 1 1 1 0 0 1.055 3.393 000000011 1.202 1.857 1.136 0.884 0.973 Writing the normal equations from the example out in full is the fol lowing: 9 I 4 3 2 u 9.356 4 : 4 0 0 d1 4.106 3 | 0 3 0 d2 3 393 I 2 | 0 0 2 d3 1.657 The normal equations are constructed using numbers of observations in the 3': and their sums in the X'y. 4. Solving for b Unlike the regression model the fixed classification model is a non—full rank model. The result is the inverse of X'X, (X'X)'1, does not exist and, estimates cannot therefore be calculated by b - (X'X)"1X'y. For the fixed 81 classification model a generalized inverse, (XUU') is used. One of the properties of a generalized inverse is that solutions can be found for b but those solutions are not unique as with a regression model. In fact there are an infinite number of solutions for b. Therefore, the normal equations are now written as (x'X)b° - 1m: with b° signifying these equations have no single solution for b°. None of these solutions estimates b, the solutions only solve the normal equations. So, for the example Table 83 lists four of the many solutions for b. 5. Estimable functions As they stand none of the solutions for b° in Table 83 estimate b, but certain linear functions of the elements of b° can be used. Suitable linear functions yield the same estimate regardless of the solution for b°. This set of linear functions is called "estimable”. Table B4 lists four linear functions and their respective solutions using the solutions for b° from Table 83. Linear functions 1 and 2 give a different solution for each b° solution, but linear functions 3 and 4 are invariant to the solutions for b0. Hence, linear functions 3 and 4 are estimable linear functions and can be used to estimate the parameters of the linear model. To determine the estimability of a given linear function it is not necessary to test its invariance with several bo u 0 hrs (d1) 24 hrs (d2) ‘8 hrs ((13) 82 Table 83. Four solutions for b°. b1 b2 b3 b‘ 0 .929 1.131 1.027 1.027 .098 -.105 0 1.131 .203 0 .1045 .929 0 -.203 -.098 83 solutions. There are two methods to determine the estimability of a function. 1) The smallest subclass means are estimable as are any linear combination of them. In the example, the smallest subclass means are (u + d1), (u + d2) and (u + d3). So, linear function 3 in Table 84 is estimable as it is a linear combination of the smallest subclass means (u + d1) and (u + d2). (u+d1)-(u+d3) =u+d1-u—d3 “d1’d3 2) A linear function k'b is estimable if it satisfies the following equation: k'(X'X)'X'x a k' where k is a vector containing the values of the interested contrast. So, linear function 3, (d1 - d3), would be written as follows: k'b=[0101]u and, 84 Table 84. Estimates of four linear functions. Solutions linear function b1 b2 b3 b4 1) 1/2 (d1 + d3) .978 .049 -.154 -.049 2) (u + d1 + d2 + d3)/3 1.029 .410 .274 1.034 3) d1 — da .098 .098 .098 .098 4) (d1 ‘1" d2) "' 2d3 .3 .3 .3 .3 k(X'X)-X'X = [ 0 BIG =k' 1 1 85 0 1 ] 0 0 0'1/4 0'0 0:0 1 1 1 0 1 0 0 1 ] hence, d1 - d3 is estimable. 6. Other calculations With solutions for b° and estimable functions suitable to answer the research questions of interest sums of squares are calculated and hypotheses tested. equations used in this study: 0 0 0 0 1/3 0 0 1/2 0 0 0 1 9 4 3 2 4 '4 .0—0- 3 :0 3 0 2 :0 0 2 L _. The following are the 1) Sums of squares (SS) of the full model a R(full) _ bOsxlx 2) Sums of squares minus the mean = R(factorslu) = bOlex _ ny2 3) 4) 5) 6) 7) 8) 9) 10) 11) 86 Residual SS ,3 YIY _ bOIXIx = residual SS/(n-r) where, n is the number of observations and r is the rank of the X'X. Proportion of variation explained by the model - 32 - R(factors|u)/(y'y - ny2) Contribution of individual factors 3 32: - R(factorlother)/[y'y - R(others)] Confidence intervals for an estimable function, k'b k'b° 16th,“ “/2 k'(X'X)"k Test for goodness of model . r -= R(factorslu)/(r-1)3'2 Test for partial contribution of factors a F a R(factors|other)/(r-rothers)2 Numerator sum of squares (Q-value) for specific hypotheses. 0 = (k'b° - m)[k'(X'X)’k]’1(k'b° - m) where, m is a vector of specified constants of order s x 1 and s is the number of simultaneous hypotheses being considered. Test for specific hypotheses (F-ratio). ‘“2 F(H) = Q/SQ’ 87 7. Method of calculation Matrix manipulations were preformed on an IBM Personal Computer using software written in BASIC. Matrix inversion was done by a modified Cause-Jordan Elimination Method. 8E3 (X'X) for colony diameter with dependencies removed. C322 0532 0322 0332 CD32 CD33 19 13 29 27 104 52 S2 12 19 24 21 42 36 10 13 28 24 52 42 112 0 10 14 36 46 52 36 143 36 46 52 36 19 13 CS 22 36 0 12 10 CS 32 0 46 19 13 19 13 19 13 DS 22 12 19 DS 32 10 13 CD 32 (X'XY for oolmy dimer. C0 33 05 CO CD CD C0 C0 C0 C0 C0 6 6666 6 32 23 11 12 13 21 22 32 13 32 11 21 31 12 22 11 21 31 12 22 6666 6 S 2 U 8833!!! iii! ‘6'. 89 33g? 3&5: 33. 333333: 3333313. 3333333ig-ifis-23:1 I I II 23 a: 8 8° $8 53 I I II 33°3 333 33 333333 333333 333333333 3393 333 33 333333 333333 333333333 3363 '333 33 333333 333333 333333333 3333 333 33 333333 333333 333333333 3393 333 33 333333 333333 333333333 3393 333 33 333333 333333 333333333 °°3° 333 33 333333 333333 333333333 3 ‘5" $3 3 '- 8° 33g'3: 333 83 333383. 3333;; 333333333. ‘ II I II II 3388 383 3_ 333338 3333—.v-_ 333333338 ' II I II II 3393 333 33 -333333 333333 333333333 3393 333 33 333333 333333 333333333 °°3° 333 33 333333 333333 333333333 °°3° 333 33 333333 333333 333333333 .-l I II 8 2 '- 3 g 31’ 33%3. 333 32. 333333 33333§ 33333338.; ,°I I II 3303 333 33 333333 333333 333333333 3393 333 33 333333 333333 333333333 3363 333 33 333333 333333 333333333 °°3° 333 33 333333 333333 333333333 8 9'- 5; 83 853 3393. 3§§_ 33 33338.: 333333 333333333 I I II II 33 3 fig 3303 388 83 333333 3333 8 333333333 II [~- 23 3'- '— 8b 252 .°l I II II 3: N: '- 251' 8; g": oogg :33 3g 333333 999933 9999:9983 .'I II II I 3303 333 33 333333 333333 333333333 N" 3: 8 2'5 2.- 8g 33%,: 333 33 33333:. 333385. 33333333.? ' II I II II 3 '— 28 2 II II I 3333 333 33 333333 333333 333333333 L3333 333 33 333333 333333 333333333] an at use.» not.» Qflflflflih o-Nm v-wn v-N 3:385:33 3:889:33 2532538338 3333 333 mm 888888 888888 888888888 (X'Y) for oo1cny diameter. 000:: CO d C511 $21 C331 C812 C322 $32 $11 $21 $31 $12 $22 $32 CD11 CD12 CD13 0021 CD22 CD32 CD13 CD23 CD33 02.911" 4151.9 2100.8 2014.4 2275.3 3476.3 2514.5 4458.9 3808.2 2576.1 1020.9 861.9 1575.8 1079.9 1152.5 745.0 2096.2 1617.7 1530.3 1380.1 897.8 1438.9 1785.9 927.1 475.3 858.6 766.9 361.1 831.8 L_821.5_J 90 (KW-(X7) = b 91 .4 88 ‘38 “8 ° :3 2‘" as a; §8 33 :8 3% L99fi‘9 9 7 9a 999999 999990? 9999999 F4 II F902;! "aw": a0: 902026.02“! 969-19909 cave—«29.029991 .géaé §§§ §§ §§§§§§ §§§§§§ :éfiea§§§§. F783 2 a 2 gm 8M 9989. 933 93. 9999§8 999993 9999999293- I I II 9 "N 21° 8 “a. '3 8‘8 99§§ 988 98 999933. 999933. 99999992.: I I II 99°9 999 99 999999 999999 999999999 99°9 999 99 999999 999999 999999999 9999 999 99 999999 999999 999999999 9999 999 99 999999 999999 999999999 99°9 999 99 999'999 999999 999999999 99°9 999 99 999999 999999 999999999 99°9 999 99 999999 999999 999999999 8: a" 32 ~ ~ 8° 993.3 98§ 33 99993.3 9999;; 999999933 II I II II 9 "'5 338 g 1‘3 '- '— 9938 988 ES: 9999§8 9999§§ 999999933 II I II II 99°9 999 99 999999 999999 999999999 99°9 999 99 999999 999999 999999999 9999 999 99 999999 999999 999999999 9969 999 99 999999 999999 999999999 "'"’ 2‘." a" 2 a“; 99%;. 933 93. 99998.3: 99998.3 999999938 I I II 9923. 983 93 99993.3. 9999§§ 999999933 I I II 9909 999 99 999999 999999 999999999 9999 999 99 999999 999999 999999999 9999 999 99 999999 999999 999999999 9999 999 99 999999 999999 999999999 8 __ '- 8“ 3 9998. 9§§_ 93 999933 99993§ 99999993§ I I II II 3 9999 933 §9 999999 9999§§ 999999999 II :8 0" 3 8" 3:3 993% 982 fig 99993.3: 999932 99999998.; |'9 I 8| II 2 “2 S'— 251' 3 eggs 9&3 3§ 999933 eoeogg 9999999§§ I'I II II I 9999 999 99 999999 999999 999999999 II I II II II II I 9999 999 99 999999 999999 999999999 [3999 999 99 999999 999999 999999999] Far CEC data: (X'X) ufith dependencies removed. 01 P50 0 0110 20 | C2 020 0'014 03 0 0241010 0 1 18 D 0 : 18 0 02 30 14 10 I0 52 (m-(X'th '0' 0 0 0 0 .0139 .015 .0125 0 .015 .041 .0094 0 .0125 .0094 .0495 0 -.0439 -.015 -.0125 0 -.0301 -.0225 -.0100 _0 0 0 0 -.0439 -.015 -.0125 .0994 .0301 -.0301 -.0226 -.0188 .0301 .0451 £312 °J "0050.9” 2575.1 1020.9 861.9 745.0 2096.2 @171 _ 1 0 43.4204 36.3278 35.8018 -2.0315 .2658 0 -.0302 -.0274 -.0257 .0465 .0279 ForIVdata: (X'X)I11thdepende101'a reroved. c c c 0 0 1 2 3 1 2 01 ”6'1 0 0127 21‘ l 02 0 33 0.11 12 03 0 0 15:11 19 01 27 11 11:55 0 02 g 12 19: 0 52 (X'X)-(X'Y)=b 0 0 0 0 0 .0395 .0213 .021 0 .0213 .0173 .0193 0 .021 .0193 .0111 0 -.0302 -.0271 -.0251 0 -.0200 -.0235 -.0279 _0_ 0 0 0 0 5313 -.0286 -.0265 -.0279 .0279 .0471 1575.8 1079.9 1152.5 1530.3 1380.1 3800.27 897.8 _ q 23.9124 28.2542 23.4151 2.9326 1.8077 _ d LIST OF REFERENCES 10. 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