7.4 How the Model Makes Predictions

Here’s the best-fitting model of thumb length using sex as an explanatory variable:

\[\text{Thumb}_i=b_0+b_1\text{Sexmale}_i+e_i\]

\[\text{Thumb}_i=58.3+6.4\text{Sexmale}_i+e_i\]

Armed with this understanding of how \(\text{Sexmale}_i\) (a.k.a. \(X_i\)) works, you can now use the model to make predictions. If \(X_i=1\), then the student is male. In this case, the predicted thumb length would be \(b_0+b_1*1\). (A reminder: The asterisk represents multiplication in R so we’re using it as a way of notating multiplication in GLM.)

If \(X_i = 0\), meaning the student is not male, then the second parameter estimate won’t get added in, because \(b_1\) times 0 is equal to 0. And if the second parameter drops out, the prediction would simply be the \(b_0\), which is the mean of female thumb lengths.

\(X_i\) is a variable, meaning it can take a different value for different students in the data frame. We show that this is a variable by putting a subscript \(i\) after the \(X\). Each student can either have the value of 0 or 1 for \(X_i\) because each student in this data set is or is not male.

How Does R Know Which Sex to Represent with \(X_i\)?

The answer to this question is: R doesn’t know; it’s just taking whatever group comes first alphabetically (in this case, female) and making it the reference group. The mean of the reference group is the first parameter estimate (\(b_0\) or the Intercept in the lm() output).

R then takes the second group (in this case, male) and represents it with the dummy variable \(X_i\). If \(X_i\) is coded 1 then the student is male. If it is coded 0, then the student is not male.

Let’s say, just for fun, that you changed the label for male to man and the label for female to woman in the data frame. Because man comes before woman alphabetically, man becomes the reference group, and its mean is now the estimate for the intercept (\(b_0\)).

As long as R knows that a variable is categorical (e.g., a factor), it doesn’t really care how you code it. You can code the categories under Sex with any characters you choose (e.g., female or woman), or with any numbers you choose (e.g, 1 and 2, 0 and 1, or 1 and 500). However you code it, R will take the category that comes first as the intercept (\(b_0\)) and then the next one as the dummy explanatory variable, which R will code 0 or 1.