18.4: The State University of Ruritania

A second example will try to model the level of the student given some information about the student. Again, this may be interesting for imputation.

Example: Eliska in School

Previously, we modeled the grade point average of students at the State University of Ruritania (Státní Univerzita v Ruritánii). Let us turn this around and model the student's class (Freshman, Sophomore, Junior, Senior) given only their reported gender and the current GPA of the student.

Let us also predict the class of Eliska, a female student with a 3.33 GPA.

Solution.

As usual, the first step is to import the data and look at a summary, including a cross-tabulation of our categorical independent variable and the dependent variable:

library(MASS)

suvrData = read.csv("http://rur.kvasaheim.com/data/suvr.csv")
summary(suvrData)

Let us pause here. Note that the class variable is an ordinal variable.

We need to let R know this:

suvrData$class = ordered(suvrData$class, levels=c("Non-Matriculated", "Freshman", "Sophomore", "Junior", "Senior") )
summary(suvrData)

There we go, the levels for the class variable are in the right order. Let's continue.

attach(suvrData)
table(gender, class)

Note that none of the non-matriculated students are female. This is just something to know and remember as we get results.

Now, we can fit our model and look at the summary results:

suvrModel = polr(class ~ gender + gpa, data=suvrData)
summary(suvrModel)

A quick check that you have ordered the levels correctly is to look at the second table in the summary output. The rows should describe subsequent levels.

The AIC of this model is 1547. The AIC of the null model

suvrNullModel = polr(class~1, data=suvrData)
summary(suvrNullModel)

is 1566. Thus, our model is an improvement.

The model predicts that Eliska is a Junior (44.7%) or a Senior (36.5%):

eliska = data.frame(gender="Female", gpa=3.33)
predict(suvrModel, eliska, type="prob")

Here are the (abbreviated) raw results

Non-Mat    Fresh      Soph       Junior     Senior
0.00211    0.02356    0.16293    0.44656    0.36483

Thus, we do have an estimate for Eliska's class level, but there is a second option which is rather close. I'm not sure I would bet any money on where to put Eliska. Regardless, it is highly unlikely for Eliska to be either non-matriculated or a Freshman. Those probabilities, while non-zero, are very low.

Beware!

Remember that the data are not representative of the population. The distribution of the classes is quite similar to the probabilities predicted for Eliska. This is not surprising. The effect of the independent variables on the dependent variable are not statistically significant. Thus, these probabilities are essentially the relative proportions of the classes in the sample.

Graphic: Class against GPA

As for a graphic, we need our dependent variable to be the probability of each class. Since there is only one numeric independent variable, GPA, that will be the variable we graph along the x-axis.

The ultimate question is:

What do we do with the gender variable?

One option is to plot the effect of gender on the same graphic. That means we will have \(5 \times 2\) curves on the same plot (the number of levels by the number of genders recorded). That may be problematic as it may overwhelm the graphic. The figure below is this figure. Note that it does allow us to compare everything at once. However, you may find it overwhelming… or not.

Figure \(\PageIndex{1}\): Graphic of the probability for each class level for each gender. Note that the non-matriculated and the Freshman levels uniformly have low probability. This is due to the nature of the data; only 2 non-matriculated and 22 Freshmen are in the sample of size \(n=661\). This limits what we can say about the population, unless the level distributions are similar to the population.

For the higher GPA values, it is most likely that the student is a Junior, regardless of the gender. At no place is it likely the student is either a Freshman or non-matriculated. This is supported by the data, as the number of non-matriculated students is just \(2\) and the number of Freshman is just \(22\) — out of a sample size of \(n=661\).

We can also use this graphic to estimate the various probabilities for Eliska. Remember she has a GPA of \(3.33\). Since Eliska is female, we look at the dashed lines. Going to \(3.33\) on the x-axis and move vertically, we see that Eliska is most likely a member of the cyan level — Junior — with a close second being the magenta level — Senior. This conclusion agrees with our prediction above.