The datafile 'butterflies_2023.txt' contains a subset of the butterfly data from Table 1.3 in the text book by Manly (2005). Sixteen colonies of butterflies were sampled and the data set contains information related to 4 environmental variables and 3 Colony Alt annualprec maxtemp mintemp 0.6 0.8 1 SS 500 43 98 17 3 22 57 SB 808 20 92 32 16 20 38 WSB 570 28 98 26 6 28 46 JRC 550 28 98 26 4 19 47 JRH 550 28 98 26 1 8 50 SJ 380 15 99 28 2 19 44 CR 930 21 99 28 0 15 50 UO 650 10 101 27 21 40 25 LO 600 10 101 27 26 32 28 DP 1500 19 99 23 1 6 80 PZ 1750 22 101 27 4 34 33 MC 2000 58 100 18 7 14 66 IF 2500 34 102 16 9 15 47 AF 2000 21 105 20 7 17 32 GH 7850 42 84 5 5 7 84 GL 10500 50 81 -12 3 1 92

The datafile 'butterflies_2023.txt' contains a subset of the butterfly data from Table 1.3 in the text book by Manly (2005). Sixteen colonies of butterflies were sampled and the data set contains information related to 4 environmental variables and 3 gene frequency variables. Assume the data meets assumptions of MVN.

a). Produce (1 mark) and comment (3 marks) on a pairwise correlation between the 7 measurement variables of interest. Do these correlation matrices suggest that canonical correlation would be an appropriate form of analysis and why (1 mark)? (5 marks total)

b). Perform a canonical correlation using the set of Environment variables (standardised) as the X variables and the Genetic variables (standardised) as the Y variables. Provide appropriate output (1 mark each), definition (1 mark each) and interpretation (1 mark each) for: (15 marks total) canonical correlations (3 marks) eigenvalues (3 marks) redundancy coefficients for the variance in the Y set of variables explained by the variance in the X set. (3 marks) Rao's F approximation significance test (3 marks). Bartlett's chi-square test of significance (3 marks)

4 [Note 1: 'appropriate output' requires you to select the appropriate parts of the output from your analysis to address each dot-point - do not include all R output]. [Note 2: If you are using Rmarkdown and cannot isolate the output for Bartlett's test please screen shot and add as a figure or recreate the output table in markdown].

c). Why are there 3 canonical functions (1 mark) and why do they become successively weaker but do not add up to one (1 mark). Comment about how our assumption of MVN (1 mark) may influence these results. (3 marks total)

d). Provide the coefficient equations that describe the first canonical function (2 marks). Also, provide output of the canonical loadings (1 mark) and interpret (3 marks). (6 marks total)

e). Why is canonical correlation an appropriate technique for this analysis and not multiple regression or MANOVA? (3 marks total)

f). What are the limitations associated with canonical correlation analysis? (2 marks total)