**NOTE: This page is under construction!!
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So far in this course we have analyzed data in which the response variable has had exactly two
levels, but what about the situation in which there are more than two levels?
In this chapter of the **Logistic Regression with Stata**, we
cover the various commands used for multinomial and ordered logistic regression allowing for more
than two categories.
Multinomial response models have much in common with the logistic regression models
that we have covered so far. However, you will find that there are differences in some of the
assumptions, in the analyses and in the interpretation of these models.

**4.2 Ordered Logistic Regression**

**4.2.1 Example 1**

Let’s begin our discussion of ordered logistic regression with an example that has a binary
outcome variable, **honcomp**, that indicates that a student is enrolled in
an “honors composition” course.
We begin with an ordinary logistic regression.

Next, we will run an ordered logistic regression for the same model using Stata'suse http://www.ats.ucla.edu/stat/stata/webbooks/logistic/hsblog, clear logit honcomp femaleLogit estimates Number of obs = 200 LR chi2(1) = 3.94 Prob > chi2 = 0.0473 Log likelihood = -113.6769 Pseudo R2 = 0.0170 ------------------------------------------------------------------------------ honcomp | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- female | .6513707 .3336752 1.95 0.051 -.0026207 1.305362 _cons | -1.400088 .2631619 -5.32 0.000 -1.915876 -.8842998 ------------------------------------------------------------------------------

**ologit**command.

As you can see, the values of the coefficients and the standard errors are the same, except that, the sign forologit honcomp femaleOrdered logit estimates Number of obs = 200 LR chi2(1) = 3.94 Prob > chi2 = 0.0473 Log likelihood = -113.6769 Pseudo R2 = 0.0170 ------------------------------------------------------------------------------ honcomp | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- female | .6513707 .3336752 1.95 0.051 -.0026207 1.305362 -------------+---------------------------------------------------------------- _cut1 | 1.400088 .2631619 (Ancillary parameter) ------------------------------------------------------------------------------

**_cut1**is reversed from

**_cons**. We will explain shortly what

**_cut1**is although it is already clear that it is related to the constant found in the logistic regression models.

**4.2.2 Example 2**

**ses**as the response variable. It has three ordered categories. Here are the frequencies for each of the categories.

We can also obtain much of the same information using thetabulate sesses | Freq. Percent Cum. ------------+----------------------------------- low | 47 23.50 23.50 middle | 95 47.50 71.00 high | 58 29.00 100.00 ------------+----------------------------------- Total | 200 100.00

**codebook**command.

For a predictor variable we will use the variablecodebook sesses --------------------------------------------------------------- (unlabeled) type: numeric (float) label: sl range: [1,3] units: 1 unique values: 3 coded missing: 0 / 200 tabulation: Freq. Numeric Label 47 1 low 95 2 middle 58 3 high

**academic**which is a dummy variable indicating whether or not students are in an academic program. Here is the ordered logistic model predicting

**ses**using

**academic**.

The format of these results may seem confusing at first. What isn't clear from the output is that logistic regression is a multi-equation model. In this example, there are two equations, each with the same coefficients. This is known as the proportional odds model. Other logistics regression models, which do not assume proportional odds will have one equation, with their own constants and coefficients, for each of the k-1 equations.ologit ses academicOrdered logit estimates Number of obs = 200 LR chi2(1) = 11.83 Prob > chi2 = 0.0006 Log likelihood = -204.66504 Pseudo R2 = 0.0281 ------------------------------------------------------------------------------ ses | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- academic | .9299309 .2745004 3.39 0.001 .39192 1.467942 -------------+---------------------------------------------------------------- _cut1 | -.7643189 .2042487 (Ancillary parameters) _cut2 | 1.41461 .225507 ------------------------------------------------------------------------------

In our example, the results are formatted like a single
equation model when, in fact, this is a two equation model because there are three levels of
**ses**. In ordered logistic regression, Stata sets the constant to zero and estimates the cut
points for separating the various levels of the response variable. Other
programs may parameterize
the model differently by estimating the constant and setting the first cut point to zero.
In order to show the multi-equation nature of this model, we will redisplay the
results in a different format.

/* output showing the multi-equation nature of ordered logistic regression */ ------------------------------------------------------------------------------ ses | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- low | academic | .9299306 .2745004 3.39 0.001 .3919197 1.467941 _cons | .7643188 .2042487 3.74 0.000 .3639987 1.164639 -------------+---------------------------------------------------------------- middle | academic | .9299306 .2745004 3.39 0.001 .3919197 1.467941 _cons | -1.414609 .225507 -6.27 0.000 -1.856595 -.9726238 ------------------------------------------------------------------------------

With ordered logistic regression there are other possible methods that do not involve the
proportional odds assumption. There is a program **omodel** (available from the Stata website)
which can be used to test the proportional odds assumption. You can download
**omodel** from within Stata by
typing **search omodel** (see
How can I use the search command to search for programs and get additional
help? for more information about using **search**).

omodel logit ses academicOrdered logit estimates Number of obs = 200 LR chi2(1) = 11.83 Prob > chi2 = 0.0006 Log likelihood = -204.66504 Pseudo R2 = 0.0281

------------------------------------------------------------------------------ ses | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- academic | .9299309 .2745004 3.39 0.001 .39192 1.467942 -------------+---------------------------------------------------------------- _cut1 | -.7643189 .2042487 (Ancillary parameters) _cut2 | 1.41461 .225507 ------------------------------------------------------------------------------

Approximate likelihood-ratio test of proportionality of odds across response categories: chi2(1) = 2.01 Prob > chi2 = 0.1563

These results suggest that the proportional odds approach is reasonable since the chi-square
test is not significant. If the test of
proportionality had been significant we could have tried the **gologit2** program by Richard
Williams of Notre Dame University. You can download
**gologit2** from within Stata by
typing **search gologit2** (see
How can I use the search command to search for programs and get additional
help? for more information about using **search**). **gologit2** with the
**npl** option
does not assume proportional odds, let's try it just for "fun."

gologit2 ses academic, nplGeneralized Ordered Logit Estimates Number of obs = 200 LR chi2(2) = 13.83 Prob > chi2 = 0.0010 Log likelihood = -203.66708 Pseudo R2 = 0.0328

------------------------------------------------------------------------------ ses | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- low | academic | .6374202 .3389678 1.88 0.060 -.0269444 1.301785 _cons | .8724881 .2250326 3.88 0.000 .4314324 1.313544 -------------+---------------------------------------------------------------- middle | academic | 1.191394 .3388816 3.52 0.000 .5271982 1.85559 _cons | -1.596859 .27415 -5.82 0.000 -2.134183 -1.059535 ------------------------------------------------------------------------------

These results clearly show the multiple equation nature of ordered logistic regression with different constants, coefficients and standard errors.

The **gologit2** command provides us with an alternative method for testing the proportionality
assumption. If the assumption of proportional odds is tenable then there should not be a significant
difference between the coefficients for **academic** in the two equations. The **test**
command computes a Wald test across the two equations.

test [low=middle]( 1) [low]academic - [middle]academic = 0

chi2( 1) = 1.98 Prob > chi2 = 0.1595

The results of this Wald test of proportionality are very similar to those found using the **omodel** command.

Let's rerun the **ologit** command followed by the **listcoef** and **fitstat** commands.

ologit ses academicOrdered logit estimates Number of obs = 200 LR chi2(1) = 11.83 Prob > chi2 = 0.0006 Log likelihood = -204.66504 Pseudo R2 = 0.0281

------------------------------------------------------------------------------ ses | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- academic | .9299309 .2745004 3.39 0.001 .39192 1.467942 -------------+---------------------------------------------------------------- _cut1 | -.7643189 .2042487 (Ancillary parameters) _cut2 | 1.41461 .225507 ------------------------------------------------------------------------------

listcoefologit (N=200): Factor Change in Odds

Odds of: >m vs <=m

---------------------------------------------------------------------- ses | b z P>|z| e^b e^bStdX SDofX -------------+-------------------------------------------------------- academic | 0.92993 3.388 0.001 2.5343 1.5929 0.5006 ----------------------------------------------------------------------

fitstatMeasures of Fit for ologit of ses

Log-Lik Intercept Only: -210.583 Log-Lik Full Model: -204.665 D(197): 409.330 LR(1): 11.835 Prob > LR: 0.000 McFadden's R2: 0.028 McFadden's Adj R2: 0.014 Maximum Likelihood R2: 0.057 Cragg & Uhler's R2: 0.065 McKelvey and Zavoina's R2: 0.062 Variance of y*: 3.507 Variance of error: 3.290 Count R2: 0.475 Adj Count R2: 0.000 AIC: 2.077 AIC*n: 415.330 BIC: -634.438 BIC': -6.537

From the **listcoef**, we see that the relative risk ratio for **academic** is approximately 2.5, which
means that the risk (odds) of being in the high ses versus medium and low ses is 2.5 times greater for students
in the academic program. The same relative risk ratio also applies to the comparison of medium and high ses versus
low ses.

**4.2.3 Example 3**

The variable **academic** that we used in the previous example is a dichotomization of the three category
variable **prog** (program type). Let's look at the frequencies for each of the levels of **prog** and
create dummy coded variables at the same time using the **tabulate** command.

tabulate prog, generate(prog)type of | program | Freq. Percent Cum. ------------+----------------------------------- general | 45 22.50 22.50 academic | 105 52.50 75.00 vocation | 50 25.00 100.00 ------------+----------------------------------- Total | 200 100.00

Now we can use **prog1** and **prog3** in an ordered logistic regression so that the academic group will be
our comparison group.

ologit ses prog1 prog3Ordered logit estimates Number of obs = 200 LR chi2(2) = 12.06 Prob > chi2 = 0.0024 Log likelihood = -204.55398 Pseudo R2 = 0.0286

------------------------------------------------------------------------------ ses | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- prog1 | -1.030315 .3479667 -2.96 0.003 -1.712317 -.3483126 prog3 | -.8500258 .3223129 -2.64 0.008 -1.481747 -.2183042 -------------+---------------------------------------------------------------- _cut1 | -1.695676 .2334022 (Ancillary parameters) _cut2 | .4852592 .195606 ------------------------------------------------------------------------------

Individually, **prog1** and **prog3** are statistically significant and we can determine from the
likelihood ration chi-square (chi2(2) = 12.06) that they are jointly significant, i.e., that the variable
**prog** is significant.

We will follow this analysis with the **omodel** command to check on the proportional odds assumption.

omodel logit ses prog1 prog3Ordered logit estimates Number of obs = 200 LR chi2(2) = 12.06 Prob > chi2 = 0.0024 Log likelihood = -204.55398 Pseudo R2 = 0.0286

------------------------------------------------------------------------------ ses | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- prog1 | -1.030315 .3479667 -2.96 0.003 -1.712317 -.3483126 prog3 | -.8500258 .3223129 -2.64 0.008 -1.481747 -.2183042 -------------+---------------------------------------------------------------- _cut1 | -1.695676 .2334022 (Ancillary parameters) _cut2 | .4852592 .195606 ------------------------------------------------------------------------------

Approximate likelihood-ratio test of proportionality of odds across response categories: chi2(2) = 4.74 Prob > chi2 = 0.0933

The test of proportionality is not significant, thus we can continue looking at the results for the **ologit**
command by following up with **listcoef** and **fitstat**.

listcoefologit (N=200): Factor Change in Odds

Odds of: >m vs <=m

---------------------------------------------------------------------- ses | b z P>|z| e^b e^bStdX SDofX -------------+-------------------------------------------------------- prog1 | -1.03031 -2.961 0.003 0.3569 0.6497 0.4186 prog3 | -0.85003 -2.637 0.008 0.4274 0.6914 0.4341 ----------------------------------------------------------------------

fitstatMeasures of Fit for ologit of ses

Log-Lik Intercept Only: -210.583 Log-Lik Full Model: -204.554 D(196): 409.108 LR(2): 12.057 Prob > LR: 0.002 McFadden's R2: 0.029 McFadden's Adj R2: 0.010 Maximum Likelihood R2: 0.059 Cragg & Uhler's R2: 0.067 McKelvey and Zavoina's R2: 0.064 Variance of y*: 3.513 Variance of error: 3.290 Count R2: 0.475 Adj Count R2: 0.000 AIC: 2.086 AIC*n: 417.108 BIC: -629.362 BIC': -1.460

Note that if the ones and zeros were reversed in both **prog1** and **prog3** then the relative risk ratio for
**prog1** would be 1/.3569 = 2.80 and for **prog3** would be 1/.4274 = 2.34.

The **fitstat** gives a deviance of 409.11 which is lower than the deviance of 409.33 for the
model that
used the dichotomous variable **academic**. This is not a very big change in the deviance. If you look at the
AIC you will see that the value for current model (2.086) is actually larger than for the model with **academic**
(2.077). Again, this is a very small change which suggests that the three category predictor, **prog**, is
not really any better than the dichotomous predictor **academic**.

**4.2.4 Example 4**

Next we will look at a model that has both categorical and continuous predictor variables and their interaction.

generate mathacad = math*academic

ologit ses academic math mathacadOrdered logit estimates Number of obs = 200 LR chi2(3) = 19.02 Prob > chi2 = 0.0003 Log likelihood = -201.07214 Pseudo R2 = 0.0452

------------------------------------------------------------------------------ ses | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- academic | .4449579 1.73113 0.26 0.797 -2.947995 3.837911 math | .0423708 .0243203 1.74 0.081 -.005296 .0900376 mathacad | .0025625 .0327299 0.08 0.938 -.061587 .0667119 -------------+---------------------------------------------------------------- _cut1 | 1.255304 1.181954 (Ancillary parameters) _cut2 | 3.4974 1.21058 ------------------------------------------------------------------------------

We can tell from the test of the individual coefficients that the interaction term is not significant but let's run a likelihood ratio test anyway, just to confirm what we already know.

lrtest, saving(0)

ologit ses academic mathOrdered logit estimates Number of obs = 200 LR chi2(2) = 19.01 Prob > chi2 = 0.0001 Log likelihood = -201.07521 Pseudo R2 = 0.0451

------------------------------------------------------------------------------ ses | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- academic | .578395 .3035933 1.91 0.057 -.0166369 1.173427 math | .0437666 .0165564 2.64 0.008 .0113166 .0762166 -------------+---------------------------------------------------------------- _cut1 | 1.322609 .8117558 (Ancillary parameters) _cut2 | 3.564826 .851694 ------------------------------------------------------------------------------

lrtestOlogit: likelihood-ratio test chi2(1) = 0.01 Prob > chi2 = 0.9376

Now we see that both **math** and **academic** are significant. However, the coefficient for **math**
is for a one point change in the math test score, which is not very meaningful. Let's
create a new variable **math10** which is the math test score divided by ten. A change of ten points on the math
test will be more meaningful than a one point change. The **ologit** will be followed by **listcoef**
and **fitstat**.

generate math10 = math/10

ologit ses academic math10Ordered logit estimates Number of obs = 200 LR chi2(2) = 19.01 Prob > chi2 = 0.0001 Log likelihood = -201.07521 Pseudo R2 = 0.0451

------------------------------------------------------------------------------ ses | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- academic | .578395 .3035933 1.91 0.057 -.0166369 1.173427 math10 | .4376661 .1655641 2.64 0.008 .1131664 .7621657 -------------+---------------------------------------------------------------- _cut1 | 1.322609 .8117558 (Ancillary parameters) _cut2 | 3.564826 .851694 ------------------------------------------------------------------------------

listcoefologit (N=200): Factor Change in Odds

Odds of: >m vs <=m

---------------------------------------------------------------------- ses | b z P>|z| e^b e^bStdX SDofX -------------+-------------------------------------------------------- academic | 0.57840 1.905 0.057 1.7832 1.3358 0.5006 math10 | 0.43767 2.643 0.008 1.5491 1.5069 0.9368 ----------------------------------------------------------------------

fitstatMeasures of Fit for ologit of ses

Log-Lik Intercept Only: -210.583 Log-Lik Full Model: -201.075 D(196): 402.150 LR(2): 19.015 Prob > LR: 0.000 McFadden's R2: 0.045 McFadden's Adj R2: 0.026 Maximum Likelihood R2: 0.091 Cragg & Uhler's R2: 0.103 McKelvey and Zavoina's R2: 0.099 Variance of y*: 3.651 Variance of error: 3.290 Count R2: 0.480 Adj Count R2: 0.010 AIC: 2.051 AIC*n: 410.150 BIC: -636.320 BIC': -8.418

From the **listcoef** results we see that for every ten point increase in math the odds of being in high ses versus
medium and low ses are about 1.5 times greater. The same thing is true for the odds of medium and high ses versus
low ses. The relative risk ratio for **math10** is less than that of **academic** which indicates that the odds are
about 1.8 times greater from students in the academic program.

From the **fitstat** restults we can see that the deviance has dropped to 401.4 and
the AIC is down to 2.05, both of which indicate that this model fits better than the model without **math**.