Introduction To Mixed Modelling 2nd Edition N. W. Galwey - Solutions

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9.1 Seven experimental treatments, A, B, C, D, E, F and G, are to be compared in a replicated experiment. However, the experimental units form natural groups of six, so one treatment must be omitted from each group.
(a) Devise a balanced incomplete block design within these constraints. How many replications are required to achieve balance?
(b) Invent values for the response variable from this experiment and perform the appropriate analysis of variance.
(c) Confirm that the efficiency factor in this analysis agrees with Equation 9.1 and that SEDifference for the treatment means agrees with Equation 9.3.
(d) Show that if one block is omitted from the experiment, it can no longer be simply analysed by analysis of variance, but that it can still be analysed by mixed modelling.
(e) Obtain the treatment means from the analysis of variance of the complete experiment:(i) with recovery of inter-block information(ii) without recovery of inter-block information.(f) Obtain the same two sets of means from mixed-model analyses.
9.2 (a) Produce an alpha design with 37 treatments in four replicates, each replicate comprising seven blocks.
(b) Determine the number of blocks in which each pair-wise combination of treatments occurs. Confirm that each combination occurs either in one block or in none.
(c) Determine the number of treatments with which each treatment is combined. How closely do these values approach to the ideal outcome?
(d) Invent values for the response variable from this experiment and perform the appropriate mixed-model analysis.
9.3 If you are using the software R to solve these exercises, use the method given in Section 2.8 to obtain SEs for the following results from the incomplete block design with Day 10 omitted:
(a) the difference between the means for treatments ‘TT3’ and ‘TT2’
(b) the difference between the means for treatments ‘TT3’ and ‘TT1’.
9.4 If you are using the software R to solve these exercises, use the method given in Section 2.8, adapted as indicated in Section 9.7, to obtain SEs for the following results from the alphalpha design:(a) the difference between the means for genotypes 3 and 2(b) the difference between the means
2.1 A field experiment to investigate the effect of four levels of nitrogen fertilizer on the yield of three varieties of oats was laid out in a split plot design (Yates, 1937). The experiment comprised six blocks of land, each divided into three main plots. Each main plot was sown with a single
(a) Make a sketch of the field layout of Block 1. N.B. It is not possible to determine the orientation of the sub-plots relative to the main plots from the information provided:make a sensible assumption about this.
(b) Analyse the experiment by the methods of analysis of variance. Determine whether each of the following treatment terms is significant according to the F test:(i) variety(ii) nitrogen level(iii) variety × nitrogen level interaction.
(c) Analyse the experiment by mixed modelling. Confirm that theWald statistic for each treatment term has the expected relationship to the corresponding F statistic. Explain the difference in the p-values between each F statistic and the corresponding Wald statistic.
2.2 An experiment was conducted to compare the effects of three types of oil on the amount of wear suffered by piston rings in an engine (Bennett and Franklin, 1954, Section 8.62, pp. 542–543). Five piston rings in the engine were weighed at the beginning and end of each 12-hour test run, and the
(a) The experiment has a split plot design. Identify the block, main-plot and sub-plot factors. Identify the treatment factor that varies only among main plots, and the treatment factor that varies among sub-plots within each main plot.
(b) What steps should have been taken during the planning of this experiment to ensure that treatment effects are not confounded with any other trends?
(c) Arrange the data for analysis by GenStat, R or SAS.
(d) Analyse the experiment by the methods of analysis of variance. Identify the three terms that belong to the treatment structure, and determine whether each is significant according to the F test.
(e) Analyse the experiment by mixed modelling. Confirm that theWald statistic for each term in the treatment structure has the expected relationship to the corresponding F statistic.
(f) The mean log-transformed weight loss is smallest during test runs with Oil B. Is it safe to conclude that this type of oil causes less wear than the other two?N.B. The sums of squares obtained in the analysis of these data do not agree exactly with those presented by Bennett and Franklin.
3.1 In a research on artificial insemination of cows, a series of semen samples from six bulls was tested for the ability to conceive (Snedecor and Cochran, 1989, Section 13.7, pp.245–247). The results are presented in Table 3.11.
(a) Arrange these data for analysis by GenStat, R or SAS.
(b) Analyse the data by the analysis of variance and by mixed modelling, making the assumption that the bulls studied have been chosen at random from a population of similar animals. Perform the best available test to determine whether the variation among bulls is significant.The data are
(c) Obtain diagnostic plots of the residuals and investigate whether there is evidence of a serious breach of the assumptions on which your analyses are based.
(d) Estimate the following variance components:(i) among bulls(ii) among samples within each bull.Suppose that the results from this experiment are to be used to design an assay procedure to estimate the fertility of similar bulls in the future. Suppose also that the cost of including an additional
(e) How many samples should be tested from each bull included in the assay?N.B. The sums of squares and MSs in the anova of these data do not agree with those presented by Snedecor and Cochran.
3.2 An experiment using 36 samples of Portland cement is described by Davies and Goldsmith(1984, Section 6.75, pp. 154–158). The samples were ‘gauged’ (i.e. mixed with water and worked) by three gaugers, each one gauging 12 samples. After the samples had set, their compressive strength was
Pearson Education Ltd.each of the three gaugers. The results are presented in Table 3.12. (Data reproduced by the kind permission of Pearson Education Ltd.)
(a) Arrange these data for analysis by GenStat, R or SAS.‘Gauger’ and ‘Breaker’ are both to be specified as random-effect terms.
(b) Justify these decisions.
(c) Analyse these data by the analysis of variance and by mixed modelling.
(d) Perform F tests for the significance of the following model terms:(i) Gauger(ii) Breaker(iii) Gauger × Breaker interaction.
(e) Obtain estimates of the variance components for the same model terms and for the residual term, from the anova. Confirm your answers from the mixed modelling results.
(f) According to the evidence from this experiment, which of these sources of variation needs to be taken into account when obtaining an estimate of the mean strength of samples of Portland cement?The breakers in this experiment were human assistants who operated the testing machine, and the
(g) Suppose that the cost of employing an additional breaker is 10 times the cost of getting the current breaker to test an additional sample. Determine the number of samples that each breaker should test in order to obtain the most accurate estimate possible of the strength of the cement, at a
3.3 Two inbred cultivars of wheat were hybridized, and the seed of 48 F2-derived F3 families(that is, families in the third progeny generation, each derived by inbreeding from a single plant in the second progeny generation) was obtained. The seed was sown in field plots in competition with
c) Obtain estimates of the variance components for the following model terms:(i) residual(ii) family(iii) block.
(d) Compare the estimate of the variance component for the term ‘family’ with its SE.Does their relative magnitude suggest that this term might reasonably be dropped from the model?
(e) Test the significance of the term ‘family’ by comparing the deviances obtained with and without the inclusion of this term in the model.
(f) Obtain diagnostic plots of the residuals from your analysis.
(g) Compare the results of the significance tests conducted in Parts (b) and (e) with each other, and with the informal evaluation conducted in Part (d). Consider how fully the assumptions of each test are likely to be met. Considered together, do these tests indicate that the term ‘family’
(h) Estimate the heritability of yield in this population of families. (N.B. The estimate obtained using the methods described in this chapter is slightly biased downward, as some of the residual variance is due to genetic differences among plants of the same family.)
(i) Estimate the genetic advance that is expected if the highest-yielding 10% of the families are selected for further evaluation. Find the expected mean yield of the selected families and compare it with the observed mean yield of the same families.Account for the difference between these values.
(j) When these estimates of genetic advance and expected mean yield are obtained, what assumptions are being made about the future environment in which the families are evaluated?These data have also been analysed and interpreted by Mokhtari et al. (2002).
4.1 Refer to your results from Exercise 1.1, in which effects on the speed of greyhounds were modelled. Obtain SEs for the estimates of the constant and of the effect of age that you obtained from your mixed model.
4.2 Refer to your results from Exercise 2.1, in which effects on the yield of oats were modelled.Examine the output produced by the analysis of variance to determine the effects of variety and nitrogen level in a split-plot design.
(a) Obtain the SE for the difference between the mean yields for the following factor levels or combinations of levels:(i) nitrogen Level 1 vs. nitrogen Level 3(ii) variety ‘Victory’ at nitrogen Level 1 vs. variety ‘Victory’ at nitrogen Level 3(iii) variety ‘Victory’ at nitrogen Level 1
(b) Obtain the LSD at the 5% level between variety means. Are there any two varieties that are significantly different according to this criterion?
(c) Obtain the SE for nitrogen-level means on the following bases:(i) conditional on the blocks and main plots(ii) taking into account the random effects of blocks and main plots.
4.3 (a) If you are using the software R to solve these exercises, use the method given in Section 2.8 to obtain SEs for the following results from the split-plot experiment to compare commercial brands of ravioli:(i) the difference between the means for treatments ‘Brand C, Assessor FAB’
(b) For each result in Part (a), and for the difference between the means for treatments‘Brand C, Assessor FAB’ and ‘Brand B, Assessor ANA’ considered in Section 2.8, state which other results have the same SE.
(a) Fit the following model to these data:Response variate: available chlorine Fixed-effect model: linear, quadratic and cubic effects of time Random-effect model: deviation of mean available chlorine at each time from the value predicted by the fixed-effects model.
(b) Investigate whether any terms can reasonably be dropped from this model.
(c) Obtain diagnostic plots of the residuals and consider whether the assumptions on which the analysis is based are reasonable.
(d) Plot the estimate of each random effect against the corresponding time and consider whether there is any evidence of a trend over time that is not accounted for by the fixed-effect model.N.B. As the level of available chlorine declines, the rate of decline becomes lower, and this exponential
7.2 In an experiment to determine the effect of four experimental diets on the growth of pigs of both sexes, 32 pigs were allocated to four randomized blocks, with eight pigs of each sex in each block.Within each block, one pig of each sex, chosen at random, was allocated to each diet. The weight
(a) Produce a plot of the values of Weight against those of Time, connecting the points representing each pig with a line. Note any trends common to all pigs and any patterns of variation among the pigs.
(b) Consider whether the weight of each pig at Time 1 should be used as a baseline value in the analysis of these data. Give reasons for your judgement.For the rest of this exercise, do not use a baseline value.
(c) Specify an appropriate mixed model for these data. Your model should include the specification of a covariance structure between random-effect terms, where appropriate.Fit your model to the data.From the results of your mixed-model analysis, answer the following questions:
(d) What is the correlation between the estimated weight of each pig when Time=0 and its rate of growth over time?
(e) Is there a significant main effect of Sex or of Diet, or a significant Sex × Diet interaction?What interpretation should be placed on significant effects among these terms?
(f) How should the main effect of Time be interpreted? Do the sexes differ significantly in their growth rate over time? Do the diets have significantly different effects on Table 7.9 Weight of pigs of both sexes on experimental diets, recorded on nine occasions.A B C D E F 1 Pig! Block! Diet! Sex!
(g) Do the results of your mixed-model analysis indicate that any of the model terms are unimportant with regard to their influence on weight? If so, drop these terms from the model and fit the resulting model to the data. From your new model, obtain the predicted values of weight at each time for
7.3 An experimentwas performed to determine the effect of the enzyme lactase (which hydrolyses the sugar lactose) on the composition of the milk in a sheep’s udder. In each side of the udder in each of eight lactating sheep, the levels of fat and of lactose in the milk (%)were measured during a
(a) Identify the block and treatment terms in this experiment. (‘Period’ might be placed in either category: specify it as a treatment term.) Specify the block-structure and treatment-structure models.
(b) Determine the effects of the block and treatment terms on each response variable, by analysis of variance and by mixed modelling. For each significant relationship observed, state the nature of the effect – for example, does lactase raise or lower the level of lactose?In each mixed-model
(c) Should these terms be placed in the fixed-effect or the random-effect model?
7.4 An experiment was performed to investigate the factors that influence the predation of seeds lying on the ground in an area on which a crop has been grown. Seeds of three species were placed on the ground in cages of four designs that excluded different types of predators, in areas on which
(a) Identify any treatment factors for which randomization cannot be performed Table 7.12 Percentage of the predation of seeds of three species at different levels of three experimental variables.A B C D E F 1 distance block residue cage_type species %pred 2 0 1 2 1 wo 100.0 3 0 1 2 2 wo 95.0 4 0 1
(b) Determine the block and treatment structures of this experiment.
(c) Analyse the data by analysis of variance.
(d) Perform an equivalent analysis on the data by mixed modelling and obtain diagnostic plots of the residuals. Note any indications that the assumptions underlying the analysis may not be fulfilled and consider why this may be the case.
(e) Create a variate holding the same values as the factor ‘distance’, and add this to the mixed model. Consider whether the factor ‘distance’ should now be specified as a fixed-effect term or a random-effect term. Consider also the consequences of your decision with regard to the
(f) Assuming that any departures from the assumptions underlying analysis of variance and mixed modelling are not so serious as to render these analyses invalid, interpret the results of the analysis performed in Section (e). In particular, consider the following points:(i) Is there a linear trend
(g) Obtain predicted values of predation at representative distances from the bushland for each species, crop residue and cage type. Make a graphical display of the relationship between these variables.A refinement of this analysis, taking account of the problems identified in the diagnostic plots,
7.5 An experiment was conducted to determine the effect of anoxia (lack of oxygen) on the porosity of roots in nine genotypes of wheat. At an early stage of development, control plants of each genotype were grown in a well-aerated solution, while treated plants of each genotype were grown in
(a) Determine the block and treatment structures of this experiment.
(b) Analyse the response variable ‘porosity_final’ by analysis of variance and by mixed modelling.The term ‘porosity_initial’ can be added to the mixed model.
(c) Should this term be placed in the fixed-effect or the random-effect model?
(d) Make this change to the mixed model and repeat the analysis. Is the final value of porosity related to its initial value? Does the adjustment for the initial value give any improvement in the precision with which the effects of the treatments are estimated?
(e) Interpret the results of your analysis. In particular, consider the following points:Table 7.13 Porosity of roots of a range of wheat genotypes with and without exposure to anoxia.A B C D E F G 1 block plant genotype treatment root_type porosity_initial porosity_final 2 1 1 1 anoxic seminal
(i) Is the porosity of roots affected by anoxia, and does it vary among genotypes and root types? If so, what is the direction of the effects of anoxia and root type?Which genotype has the most porous roots, and which the least?
(ii) Is there evidence of two-way interactions between anoxia, root type and genotype?If so, what is the nature of these interactions?
(iii) Is there evidence of a three-way interaction among these factors?These data have also been analysed and interpreted by McDonald et al. (2001).
7.6 An experiment was performed to determine the effect of nutritionally inadequate diets on the quality of sheep’s wool. The experiment consisted of a 21-day pre-experimental period (ending on 22 October 1997), a 12-week restricted-intake period (ending on 14 January 1998) and a 6-week recovery
(a) Assign an appropriate block number to each row of the data.
(b) Analysing measurements made on each occasion separately, and using the block factor and the treatment factor as model terms, perform analyses of variance on the measurements of fibre diameter made during the restricted-intake period and the recovery period. Interpret the results.
(c) Stack the measurements made on these three occasions into a single variate and perform a newanalysis by mixed modelling. Include the following variables in your initial model:(i) The time at which the measurement was made.(ii) The fibre diameter from the same sheep during the pre-experimental
(d) Interpret the results of your analysis. In particular, consider the following points:(i) Is the fibre diameter influenced by the sheep’s pre-experiment live weight?(ii) Is the fibre diameter during the experiment related to the fibre diameter in the pre-experimental period?(iii) Is the fibre
(e) Obtain predicted values of fibre diameter for each treatment at representative times and make a graphical display of the relationship between these variables.
7.7 The data on the yield of F3 families of wheat presented in Exercise 3.3 in Chapter 3 are a subset from a larger investigation, in which two crosses were studied, and the F3 families were grown with and without competition from ryegrass. The experiment had a split-split plot design, with the
(a) Analyse the data according to the experimental design, both by analysis of variance and by mixed modelling.
(b) Modify your mixed model so that family-within-cross is specified as a randomeffect term.
(c) Interpret the results of your analysis. In particular, consider the following points:(i) Does the presence of ryegrass affect the yield of the wheat plants? If so, in which direction?(ii) Is there a difference in mean yield between the crosses?(iii) Do the families within each cross vary in
(d) Compare the results with those that you obtained from the subset of the data in Chapter 3.7.8 Return to the data set concerning the vernalization of F3 chickpea families, introduced in Exercise 5.2 in Chapter 5.

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Introduction To Mixed Modelling 2nd Edition N. W. Galwey - Solutions

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