The spreadsheet in Table 1.8 gives data on the greyhounds that ran in the Kanyana Stake (2) in Western 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. Table 1.8 Data on greyhounds that ran in the Kanyana Stake (2) in December 2005. box = box from which the animal started this race; time = time taken to complete recent races (seconds); distance = distance run in recent races (metres). A B C D E 1 name box birthdate time distance 2 Leprechaun Kate 1 Jan-04 515 3 Leprechaun Kate 1 Jan-04 525 4 Leprechaun Kate 1 Jan-04 30.89 530 5 Leprechaun Kate 1 Jan-04 31.31 530 6 Leprechaun Kate 1 Jan-04 31.23 530 7 Mystifier 2 Dec-03 430 8 Mystifier 2 Dec-03 450 9 Mystifier 2 Dec-03 457 10 Mystifier 2 Dec-03 457 11 Mystifier 2 Dec-03 430 12 Mystifier 2 Dec-03 24.43 410 13 Mystifier 2 Dec-03 24.35 410 14 Mystifier 2 Dec-03 24.33 410 15 Proudly Agro 3 Feb-03 31.72 530 16 Proudly Agro 3 Feb-03 31.23 530 17 Proudly Agro 3 Feb-03 31.72 530 18 Proudly Agro 3 Feb-03 31.30 530 19 Proudly Agro 3 Feb-03 31.55 530 20 Proudly Agro 3 Feb-03 31.65 530 21 Proudly Agro 3 Feb-03 31.15 530 22 Proudly Agro 3 Feb-03 31.37 530 23 Desperado Lover 4 Jan-02 30.50 509 24 Desperado Lover 4 Jan-02 30.80 509 25 Desperado Lover 4 Jan-02 30.92 509 26 Desperado Lover 4 Jan-02 30.84 509 27 Desperado Lover 4 Jan-02 31.39 509 28 Desperado Lover 4 Jan-02 31.40 530 29 Desperado Lover 4 Jan-02 31.68 530 30 Desperado Lover 4 Jan-02 31.81 530 31 Squeaky Cheeks 5 Nov-03 34.82 530 32 Squeaky Cheeks 5 Nov-03 31.72 530 33 Squeaky Cheeks 5 Nov-03 32.34 530 34 Squeaky Cheeks 5 Nov-03 31.32 530 35 Squeaky Cheeks 5 Nov-03 31.85 530 (Continued overleaf ) 42 The need for random-effect terms when fitting a regression line Table 1.8 (continued) A B C D E 36 Squeaky Cheeks 5 Nov-03 31.22 530 37 Squeaky Cheeks 5 Nov-03 31.55 530 38 Squeaky Cheeks 5 Nov-03 31.52 530 39 Beyond the Sea 6 Apr-04 31.37 530 40 Beyond the Sea 6 Apr-04 30.75 530 41 Keith Kaos 7 Feb-03 30.56 509 42 Keith Kaos 7 Feb-03 31.31 530 43 Keith Kaos 7 Feb-03 31.81 530 44 Keith Kaos 7 Feb-03 31.71 530 45 Keith Kaos 7 Feb-03 31.58 530 46 Keith Kaos 7 Feb-03 31.04 530 47 Keith Kaos 7 Feb-03 31.61 530 48 Keith Kaos 7 Feb-03 31.59 530 49 Elza Prince 8 Nov-02 530 50 Elza Prince 8 Nov-02 31.30 530 51 Elza Prince 8 Nov-02 31.55 530 52 Elza Prince 8 Nov-02 31.92 530 53 Elza Prince 8 Nov-02 31.26 530 54 Elza Prince 8 Nov-02 31.73 530 55 Elza Prince 8 Nov-02 31.86 530 56 Elza Prince 8 Nov-02 31.52 530 57 Jarnat Boy 9 Apr-03 31.30 530 58 Jarnat Boy 9 Apr-03 31.90 530 59 Jarnat Boy 9 Apr-03 31.59 530 60 Jarnat Boy 9 Apr-03 31.55 530 61 Jarnat Boy 9 Apr-03 31.28 530 62 Jarnat Boy 9 Apr-03 31.12 530 63 Jarnat Boy 9 Apr-03 32.44 530 64 Jarnat Boy 9 Apr-03 31.64 530 65 Shilo Mist 10 Apr-03 31.71 530 66 Shilo Mist 10 Apr-03 32.30 530 67 Shilo Mist 10 Apr-03 31.49 530 68 Shilo Mist 10 Apr-03 32.17 530 69 Shilo Mist 10 Apr-03 32.14 530 70 Shilo Mist 10 Apr-03 31.74 530 71 Shilo Mist 10 Apr-03 31.98 530 72 Shilo Mist 10 Apr-03 31.82 530 Exercises 43 Table 1.9 Levels of available chlorine in batches of a chemical product manufactured at two-week intervals, after a period of storage. Length of time since production (weeks) Available chlorine 8 0.49 0.49 10 0.48 0.47 0.48 0.47 12 0.46 0.46 0.45 0.43 14 0.45 0.43 0.43 16 0.44 0.43 0.43 18 0.46 0.45 20 0.42 0.42 0.43 22 0.41 0.41 0.40 24 0.42 0.40 0.40 26 0.41 0.40 0.41 28 0.41 0.40 30 0.40 0.40 0.38 32 0.41 0.40 34 0.40 36 0.41 0.38 38 0.40 0.40 40 0.39 42 0.39

(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 this outlier from the data.

(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 regarded as a fixed-effect term? (ii) if ‘name’ is regarded as a random-effect term? Obtain the F statistic for age using both approaches, and obtain the corresponding P values. Note which test gives the higher level of significance, and explain why. 44 The need for random-effect terms when fitting a regression line

(f) Reanalyse the data by mixed modelling, fitting a model with the same terms but regarding ‘name’ as a random-effect term. Use the Wald statistic to test the significance of the effect of age. (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 Wald statistic for the effect of age now has the same value as the corresponding F statistic.