Past exam of the systems biology course of the University of Cambridge / 2024 / Paper 3 / 1

The estimated intercept means that a female mouse with weight zero would have predicted mean testosterone concentration $2.56486ng/dL$ under this linear regression; zero weight is an extrapolation without a meaningful adult-mouse interpretation.

Holding sex fixed, each additional gram of weight is associated with an estimated $0.03378ng/dL$ increase in mean testosterone concentration, according to the fitted multiple linear regression.

At the same weight, male mice have an estimated mean testosterone concentration $7.12162ng/dL$ higher than female mice; the sex regression coefficient is an adjusted difference between the groups.

At a conventional 5% significance level, there is evidence for an association with sex after adjusting for weight, because its two-sided p-value is $0.00131<0.05$. There is insufficient evidence for a weight association after adjusting for sex, because its p-value is $0.62392>0.05$.

These Student t-tests test whether the corresponding regression coefficients are zero within the additive multiple linear regression. Failure to reject the weight hypothesis does not establish an exactly zero effect; these associations alone also do not justify a causal inference. The later model with a statistical interaction asks a different question, allowing the weight association to differ by sex.

Two assumptions of the usual normal linear model used for the reported coefficient Student t-tests are:

The second statement includes homoscedasticity. Normality concerns the conditional errors, rather than requiring the weights themselves to have a normal distribution.

Include both main effects and their interaction term:

fit_interaction <- lm(testosterone ~ weight * sex, data = mice)

Here mice denotes the data frame containing the named variables. In this R formula, weight * sex expands to weight + sex + weight:sex; it does not omit the main effects.

The interaction term estimates that the weight-associated change in mean testosterone concentration is $0.40688ng/dL$ per gram greater in males than females: the female slope is $-0.04518$ and the male slope is $-0.04518+0.40688=0.36170ng/dL$ per gram.

Use Poisson regression for the offspring count, with testosterone as a predictor. It models a nonnegative integer response through its conditional mean. If the conditional variance substantially exceeds that mean, a negative binomial regression is a possible alternative for overdispersion.

Use the log link:where $N_i$ is the offspring count and $T_i$ is testosterone concentration. This link function keeps the fitted conditional mean positive.

A one-unit increase in testosterone concentration, here $1ng/dL$, multiplies the expected offspring count by $\boxed{e^{{\beta }_T}}$ under the Poisson regression with a log link. Thus the exact percentage change in the conditional mean is $\boxed{100(e^{{\beta }_T}-1)\%}$. The regression coefficient ${\beta }_T$ itself is the change in the logarithm of the expected count, not an additive change in offspring number.