18.2: Ordinal Dependent Variables

Another variety of categorical dependent variables is ordinal. A variable is ordinal if it is both categorical and the categories have an underlying order to them. Examples include movie ratings (number of stars), hurricane intensity, and so forth.

There are actually at least four ways of handling ordinal dependent variables:

1. Treat them as nominal. This allows us to fit ordinal data using previous techniques. Unfortunately, it is inefficient as it ignores important aspects of the data itself.
2. Treat their cumulative level as nominal. If the ordinal variable takes on values 1 – 5, then create nominal variables corresponding to Level 1, Levels 1 and 2, Levels 1–3, Levels 1–4, and Levels 1–5. This preserves much of the underlying information and allows us to fit it using a previous method.
3. Assume that there is an underlying continuous process that you wish to fit. The ordinal nature is just several threshold values along the possible values. This reduces to a pseudo-OLS, where you also need to fit the threshold values, not just the slopes and intercepts. Using Maximum Likelihood methods, this is trivial to solve.
4. Pretend that the ordinal values are continuous and fit it using ordinary least squares or one of its offsprings. This has the advantage of being easily fit.

Three of these ways have already been discussed, and you are quite adept at using them (Options 1, 2, and 4). Only the third option is completely new to you. This chapter focuses on how to fit Option Three.

Option Three

Let us assume that there is an underlying continuous process. We only experience (observe) this process through the ordinal variable. This is very similar to how we first looked at binary variables: underlying process exhibited only in the 0/1 outcomes. Here, there is more than just the one threshold (which traditionally defaulted to \(0.500\)). Thus, we have two sets of parameters to fit. The first is the parameters which describe the process (the \(\beta\)s). The second is the position of those threshold values (the \(\tau\)s).

Figure \(\PageIndex{1}\): Schematic diagram of the thresholding process. The line represents the linear continuous process. The \(\tau\)s represent the threshold values. A, B, C, and D represent the ordinal outcomes.

Without going into the details, we will use Maximum Likelihood Estimation as our fitting method because it has many nice properties. Thus, our underlying process is
\begin{equation}
\eta = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_k x_k
\end{equation}

Our thresholding process is illustrated in the figure above. The line represents the underlying continuous process that you are trying to model. The A, B, C, and D represent the observed ordinal values. The threshold values, \(\tau_1\), \(\tau_2\), and \(\tau_3\) are the values of \(\eta\) that separate the observed ordinal values.

This model is very straight forward and understandable. Using R to obtain the fitting is also straight forward. The results presented are also relatively straight forward.

Example \(\PageIndex{1}\): Warmth for Obama

Let us use some more data from the GSS. This data explores the "warmth of feeling" the respondent has for President Obama. The demographic information is the gender (male), the race (white), the age, and the number of years of education (ed). The response variable has four ordered levels: Strongly Disagree (SD), Disagree (D), Agree (A), and Strongly Agree (SA). Our goal is to explain a person's feelings toward the president based solely on demographic information.

Solution.
Let us fit this data with ordinal regression. The function in R is polr, which stands for "proportional odds logistic regression" (although the probit is an option as a link function). This function requires the MASS package. Thankfully, since MASS also comes with the base distribution of R, there is no need to install it, only to load it via the library(MASS) command.

The actual command to fit this model using ordinal regression is

polr( warm~male + white + age + ed )

Table \(\PageIndex{1}\): Result of ordinal regression in R. Note that the women tend to view President Obama in a more favorable light; whites, less; older, less; and higher educated, more. All of these agree with multiple surveys throughout his tenure as President.

		Value	Std. Error	t-value
Variables:
	Woman	0.743	0.078	9.50
	White	-0.400	0.118	-3.39
	Age	-0.020	0.0024	-8.17
	Years of Education	0.098	0.013	7.52
Thresholds:
	SD | D	-1.700	0.237	-7.18
	D | A	0.111	0.233	0.48
	A | SA	1.979	0.236	8.37

This command will give the coefficients of the underlying linear regression and the threshold values separating the four categories. From the table, we see that the equation for the underlying linear process is
\begin{equation}
\eta = 0.743 \texttt{Woman} - 0.400 \texttt{white} - 0.020 \texttt{age} + 0.098 \texttt{ed}
\end{equation}

The thresholds are also listed. The threshold between Strongly Disagree and Disagree is at \(\tau_1 = -1.700\). The threshold between Disagree and Agree is \(\tau_2 = 0.111\). The threshold between Agree and Strongly Agree is \(\tau_3 = 1.979\). Thus, to calculate our prediction, we calculate the prediction based on the linear model, \(\eta\), and compare that value to the intervals described by the thresholds. Thus, for Bob, who is Male, White, 40 years old and has 20 years of education, we have \[ \eta = 0.740 \times 0 - 0.400 \times 1 - 0.020 \times 40 + 0.098 \times 20 = 0.76 \] As \(\eta = 0.76\), we have our prediction that Bob agrees with the president. If we actually want probabilities that Bob Strongly Disagrees, Disagrees, Agrees, or Strongly Agrees, we would have to back-transform using the inverse of the logit function and calculate each probability using integral calculus… or we could just ask the computer to do it for us:

BOB = data.frame(male="Men", white="White", age=40, ed=20)
predict(model.ol1, newdata=BOB, type="probs")

This gives the probabilities as

    SD       D       A      SA
0.0785   0.263   0.429   0.229

Thus, it is far from certain that Bob agrees (or strongly agrees) with the president, although that probability is rather high: \(0.429 + 0.229 = 0.658\).

Accuracy

Finally, let us look at the accuracy of the model. I leave it as an exercise to show that the relative accuracy is \(1.105\), which indicates that the model is about 10.5% better than the null model (the modal category is "Agree"). This is not a fantastic increase in accuracy, but we do know how certain demographics feel about the president: Whites tend to disagree, Males tend to disagree, older people tend to disagree, and lesser educated people tend to disagree.

Of course, we could have added in a quadratic education term to the model to see if both the more-educated and the less-educated both support the president. I also leave this as an exercise to show that there is no evidence of this. Thus, we have no evidence that the relationship between education and presidential support is anything other than linear.