Week 9 - linear models: multiple factors

either or both

analyses replicate values of Y for all combinations of groups in two or more factors (categorical Xs) Might have three null hypotheses - main effects (two hypotheses) and the third is the interaction

if lines are parallel, then each factor’s effect on Y is the same across groups in the other factor (i.e., factors don’t interact) if lines aren’t parallel, then each factor’s effect on Y differs across groups in the other factor (i.e., factors interact) Interaction with A and B (cross): but they can make tests of main effects invalid… so we must always consider the validity of main effects if factors interact Tests of main effects may not be invalid but we need to think about it

different categories of a fixed factor are predetermined of direct interest and repeatable.

a variable whose groups are not predetermined but instead are randomly sampled from a population of groups.

ANCOVA Investigate if linear regression fitted to data from multiple groups have the same slope.

A linear model with one numerical and one categorical explanatory variable fits seperate regression lines to each group of the categorical variable A test of the interaction term between the two variables is a test of whether the slopes of the regression lines is the same for all groups of the categorical variable (focuses on interaction) or to adjust the factor’s effect on Y to account for the numerical variable (assumes no interaction) Three null hypotheses

If we can assume that there is no difference between the regression slopes then we can fit a model without an interaction term Do this with great caution