The General Linear Model

The General Linear Model is general (also linear and a model, but that is another topic). What is general about it? What does it do? One way to try to understand a statistic is by the underlying mathematics. I read a paper today where most of it was written in Greek. Seriously. There were a lot of equations where the natural logarithm of the probability of event A was multiplied by the probability of event B from which was subtracted the log of the probability of events other than A.

If you had included all of the Greek letters and formula it was even less comprehensible than it sounds. I did finally understand what they were doing after I read it three times. At the end, I looked like this.

I think a more useful way for understanding statistics, for most people, is to look at the types of questions you are able to answer.

Almost all questions that can be stated:

Is there a relationship between this thing and this other thing?

.. can be answered using the General Linear Model. Another way this type of question can be put,

Is the difference in scores of variable X, between this group and some other group, greater than one would expect to find purely at random?

This is really just another way of saying the first question, that is, “Is there a relationship between group membership and X?” Some examples of where the General Linear Model can be used:

• Testing for the significance of differences between the mean scores of two different groups. For example, if one wanted to test to see whether the difference in average salaries of men and women is greater than one would expect by chance. If you are familiar with statistics, you may realize right away that this could be done with an independent t-test. The t-test is a specific case of the general linear model.
• Testing the difference between the mean scores of the same group taken at two different times. For example, we might want to determine whether the amount of time devoted to leisure activities declines after a child is born, or is this just a myth. We could survey people in the year before a child was born and the year after. You may recognize this as a dependent t-test. This is another specific case of the general linear model.
• Predicting scores on one variable from another variable can be done using the general linear model. For example, I might want to know whether it is possible to predict marital happiness a year after marriage from the number of months a couple dated before marriage.

Not every question can be answered with the General Linear Model. If you have two categorical variables, such as, gender and being a jerk and you want to answer the question, does one gender have a higher proportion of jerks than the other, you would not use the General Linear Model. You would use a chi-square for this question. Or, you could just ask me. (Short answer – yes. )