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

(e) Make a graphical display, showing the fitted relationship between the proportion of hard seed remaining for each species and the time elapsed since the start of exposure, and showing the scatter of the observed values around this relationship.
The spreadsheet in Table 1.8 gives data on the greyhounds that ran in the Kanyana Stake(2) inWestern Australia in December 2005. (Data reproduced by kind permission of David Shortte, Western Australian Greyhound Racing Association.)
(a) Calculate the average speed of each animal in each of its recent races. Plot the speeds against the age of each animal.
(b) The first value of speed for ‘Squeaky Cheeks’ (in row 31 of the spreadsheet) is an outlier: it is much lower than the other speeds achieved by this animal. Consider the arguments for and against excluding this value from the analysis of the data.For the remainder of this exercise, exclude
(c) Perform a regression analysis with speed as the response variable and age as the explanatory variable, treating each observation as independent. Obtain the equation of the line of best fit and draw the line on your plot of the data.
(d) Specify a more appropriate regression model for these data, making use of the fact that a group of observations was made on each animal. Fit your model to the data by the ordinary methods of regression analysis. Obtain the accumulated analysis of variance from your analysis.
(e) Which is the appropriate term against which to test the significance of the effect of age:(i) if ‘name’ is specified as a fixed-effect term?(ii) if ‘name’ is specified as a random-effect term?Obtain the F statistic for age using both approaches and obtain the corresponding p-values.
(f) Re-analyse the data by mixed modelling, fitting a model with the same terms but specifying ‘name’ as a random-effect term. Use the F statistic to test the significance of the effect of age. Also obtain the Wald statistic.
(g) Obtain the equation of the line of best fit from your mixed-model analysis. Draw the line on your plot of the data and compare it with that obtained when every observation was treated as independent.
(h) Obtain a subset of the data comprising only the last two observations on each animal.Repeat your analysis on this subset and confirm that the F statistic for the effect of age obtained by mixed modelling now has the same value as that obtained by hand in Part (e)-ii.
6.1 In the experiment to compare four commercial brands of ravioli described in Chapter 2, it may be argued that ‘assessor’ should be specified as a random-effect factor.(a) Consider the case for this decision, and present the arguments for and against.
(b) If ‘brand’ is specified as a fixed-effect term and ‘assessor’ as a random-effect term, how should ‘brand.assessor’ be specified?
(c) Make these changes to the mixed model fitted to these data. Perform the new analysis and interpret the results. Explain the effect of the changes on the SEs of differences between brands.
(d) Perform appropriate tests to determine whether the new random-effect terms are significant.
6.2 Return to the data set concerning the effect of oil type on the amount of wear suffered by piston rings, introduced in Exercise 2.2.(a) For each oil type, plot the value of wear against the ring number (1–4). Omit the values from the ‘oil ring’ from this plot. Is there evidence of a trend
(b) Repeat the mixed-model analysis performed on these data previously, but exclude the values from the ‘oil ring’.
(c) Now specify ‘ring’ as a variate instead of a factor. Fit this new model to the data by mixed modelling. Do the results confirm that there is a linear trend from Ring 1 to Ring 4? Does this trend vary significantly depending on the oil type?Note that when this change is made to the model,
(d) What source of variation is included in these terms in the previous model, but excluded in the present model?Now suppose that the three types of oil tested are considered to comprise an exchangeable set.
(e) When this change is made, which parts of the expression ‘oil*ring’ should be regarded as fixed-effect terms, and which as random-effect terms?(f) Fit this new model to the data by mixed modelling. Does the new model indicate that the linear trend from Ring 1 to Ring 4 varies significantly

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