It is known that increased reproductions leads to reduced longevity for female fruitflies. Patridge and Farquhar (1981) studied whether the same phenomenon held for male fruitflies. The experiment set up five treatment groups, each consisting of 25 randomly assigned male fruitflies. The males in one treatment were housed with eight virgin females per day. In another treatment, the males were housed with one virgin female day. There were three control groups: males housed with eight newly impregnated females, housed with one newly impregnated female, and housed alone. (Newly inseminated females will not usually mate within two days). It is also believed that the males' thorax length at birth has a correlation with their lifespan.

Treatments: 8 virgin female fruitflies, 1 virgin female fruitflies, 8 pregnant female fruitflies, 1 pregnant female fruitflies and "none" (housed alone).

Response: Longevity (lifespan in days).

Covariate: Thorax length of male fruitflies at birth (in mm).

In the Lab we analyzed the longevity on the original scale. However, some may argue that relative change in lifespan is more important than absolute change because the longevity of fruitflies vary a lot. Statistically, this amounts to using the logarithm of lifespan instead of lifespan. We will also see that using log-lifespan improves the diagnostics (which we did not discuss in the lecture).

Questions:

1.[4 points] Fit an additive model of longevity on the treatment and thorax length (no interaction) and show the diagnostic plots (residual vs fitted and QQ-plot of the residuals). Discuss any concern and suggest a fix if necessary.

2.[4 points] Fit the model on the log-scale, i.e. log-longevity on treatments and thorax lengths (without interaction). Show the ANOVA table of effects tests and diagnostic plots.

We will work with the model in 2. to answer the following questions.

3.[2 points] Show the estimate and SE of the slope coefficient (of Thorax) and provide an interpretation in the given context. [Remember to interpret on the original scale as ratio of median lifespans].

4.[3 points] Perform an all possible pairwise comparison of the treatments after adjusting for the thorax length at birth, using a suitable adjustment for multiple testing. Show either the pairwise differences table or a CLD.

Is there any evidence for a difference among the control groups?

5.[6 points] Regardless to your answer to 4, estimate and test the effects of each treatment (8-virgin and 1-virgin), with a multiplicity adjustment, pooling the three control groups (i.e. E.g. 8-virgin minus the average of three control groups etc). Interpret these estimates in the given context.

[4 points] Find 95% prediction intervals for the median lifespan of a male fruitfly with 0.6mm long thorax when born, if housed with 8 virgin female fruitflies everyday. Interpret this interval in the given context.

A B C D 1 Treatment Thorax Longevity 2 8-pregnant 0.64 35 3 8-pregnant 0.68 37 4 8-pregnant 0.68 49 5 8-pregnant 0.72 46 6 8-pregnant 0.72 63 7 8-pregnant 0.76 39 8 8-pregnant 0.76 46 9 8-pregnant 0.76 56 10 8-pregnant 0.76 63 11 8-pregnant 0.76 65 12 8-pregnant 0.8 56 13 8-pregnant 0.8 65 14 8-pregnant 0.8 70 15 8-pregnant 0.84 63 16 8-pregnant 0.84 65 17 8-pregnant 0.84 70 18 8-pregnant 0.84 77 19 8-pregnant 0.84 81 20 8-pregnant 0.84 86 21 8-pregnant 0.88 70 22 8-pregnant 0.88 70 23 8-pregnant 0.92 77 24 8-pregnant 0.92 77 25 8-pregnant 0.92 81 26 8-pregnant 0.94 77 27 none 0.64 40 28 none 0.7 37 29 none 0.72 4430 none 0.72 47 31 none 0.72 47 32 none 0.76 47 33 none 0.78 68 34 none 0.8 47 35 none 0.84 54 36 none 0.84 61 37 none 0.84 71 38 none 0.84 75 39 none 0.84 89 40 none 0.88 58 41 none 0.88 59 42 none 0.88 62 43 none 0.88 79 44 none 0.88 96 45 none 0.92 58 46 none 0.92 62 47 none 0.92 70 48 none 0.92 72 49 none 0.92 75 50 none 0.92 96 51 none 0.94 75 52 1-pregnant 0.64 46 53 1-pregnant 0.68 42 54 1-pregnant 0.72 65 55 1-pregnant 0.76 46 56 1-pregnant 0.76 58 57 1-pregnant 0.8 42 58 1-pregnant 0.8 48\f87 1-virgin 0.84 60 88 1-virgin 0.84 60 89 1-virgin 0.84 65 90 1-virgin 0.84 68 91 1-virgin 0.88 60 92 1-virgin 0.88 81 93 1-virgin 0.88 81 94 1-virgin 0.9 48 95 1-virgin 0.9 48 96 1-virgin 0.9 56 97 1-virgin 0.9 68 98 1-virgin 0.9 75 99 1-virgin 0.9 81 100 1-virgin 0.92 48 101 1-virgin 0.92 68 102 8-virgin 0.64 16 103 8-virgin 0.64 19 104 8-virgin 0.68 19 105 8-virgin 0.72 32 106 8-virgin 0.72 33 107 8-virgin 0.74 33 108 8-virgin 0.76 30 109 8-virgin 0.76 42 110 8-virgin 0.76 42 111 8-virgin 0.78 33 112 8-virgin 0.8 26 113 8-virgin 0.8 30 114 8-virgin 0.82 40115 8-virgin 0.82 54 116 8-virgin 0.84 34 117 8-virgin 0.84 34 118 8-virgin 0.84 47 119 8-virgin 0.84 47 120 8-virgin 0.88 42 121 8-virgin 0.88 47 122 8-virgin 0.88 54 123 8-virgin 0.88 54 124 8-virgin 0.88 56 125 8-virgin 0.88 60 126 8-virgin 0.92 44