End of chapter questions for elementary statistical modelling, chapter 6

C6.Q1 Write a full model description of the model characterising the function relating birth mass to horn type and sex from section.

Recall that the model was this:

round(summary(hornSexModel)$coefficients,4)

##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   2.2377     0.0369 60.5744   0.0000
## HornP        -0.0476     0.0543 -0.8755   0.3815
## HornS        -0.0514     0.0457 -1.1234   0.2615
## SexM          0.0808     0.0423  1.9071   0.0568

That output provides a bit of a reminder of what the contrasts were, and, by process of elimination, reminds us that the default level of sex was females, and the default level of horn type was scurrs. So the answer to the question is:

The model of the partial effects of horn type and sex on birth mass took the form
$y_i = \alpha + \delta_{poll|sex}\phi_{poll,i} + \delta_{scur|sex}\phi_{normal,i} + \delta_{male|horn}\phi_{male,i} + e_i$ .
$y_i$ is the birth mass (kg) of the $i^{\text{th}}$ individual. $\alpha$ is the model intercept, representing the mean birth mass of female unicorns with scurred horns. $\phi_{poll,i}$ and $\phi_{normal,i}$ are indicator variables that take values of 1 when the horn type of individual $i$ is polled or normal, respectively, and zero otherwise. $\delta_{poll|sex}$ and $\delta_{normal|sex}$ are contrasts, describing how much heavier (or lighter) polled and normal horned individuals are than scurred individuals, given their sex. In other words $\delta_{poll|sex}$ and $\delta_{normal|sex}$ represent the partial effects of polled and normal horns, respectively, accounting for differences in horn size attributable to sex. $\phi_{male,i}$ is an indicator variable taking a value of 1 for individuals that are male, and a value of 0 for individuals that are female. $\delta_{male|horn}$ is a contrast representing the partial effect of being male, as opposed to female, accounting for effects of horn type. $e_i$ are residuals.

C6.Q2 Construct a table and some sensible text reporting the results of the multiple regression of birth mass on horn type and sex, such as would be appropriate to the results section of a scientific paper.

Table 1. Coefficient estimates for the partial effects of horn type and sex on birth mass in unicorns (Monoceros europus). Units for all model terms are kg. Model terms are as defined in the equation given to answer question 1.

	coefficient estimate	standard error	$p$
$\alpha$	2.238	0.037	0.000
$\delta_{poll\|sex}$	-0.048	0.054	0.382
$\delta_{scur\|sex}$	-0.051	0.046	0.262
$\delta_{male\|horn}$	0.081	0.042	0.057

Note that a table for a scientific paper would typically have heavy or double horizontal lines at the top and bottom, and a lighter or single line separating the column names from the data. A scientific table would not normally have vertical lines.

Some sensible text might be something like this:

“Conditional on sex, estimated effects of horn type on birth mass are modest and are not statistically significant. Accounting for horn size, males are approximately 80 g heavier than females (SE = 0.042, $p$ = 0.057).”

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