Questions and Answers of Introduction To Mixed Modelling

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
(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
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
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
(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
(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)
(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
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).
(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
(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
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
(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
(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
(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
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
(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
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
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
(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
(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
(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
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
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
(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
(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
(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
(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
(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
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
(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
(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
(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
(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
(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
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
(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
(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
(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
(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
(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
(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
(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
(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
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
(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
(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
(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
(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
(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
(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
(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
(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
(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
(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

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