Help the Stat Consulting Group by giving a gift

Ordered Logistic Regression

This page shows an example of an ordered logistic regression analysis with footnotes explaining the output. The data were collected on 200 high school
students and are scores on various tests, including science, math, reading and social studies. The outcome measure in this analysis is
socio-economic status (**ses**)- low, medium and high- from which we are going to see what relationships exist with science test scores (**science**),
social science test scores (**socst**) and gender (**female**). Our response variable, **ses**, is going to be treated as ordinal
under the assumption that the levels of **ses** status have a natural ordering
(low to high), but the distances between adjacent levels are unknown. The first half of this page
interprets the coefficients in terms of ordered log-odds (logits) and the second half interprets the coefficients in terms of proportional odds.

use http://www.ats.ucla.edu/stat/stata/notes/hsb2, clear ologit ses science socst femaleIteration 0: log likelihood = -210.58254 Iteration 1: log likelihood = -195.01878 Iteration 2: log likelihood = -194.80294 Iteration 3: log likelihood = -194.80235 Ordered logit estimates Number of obs = 200 LR chi2(3) = 31.56 Prob > chi2 = 0.0000 Log likelihood = -194.80235 Pseudo R2 = 0.0749 ------------------------------------------------------------------------------ ses | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- science | .0300201 .0165861 1.81 0.070 -.0024882 .0625283 socst | .0531819 .015271 3.48 0.000 .0232513 .0831126 female | -.4823977 .2796939 -1.72 0.085 -1.030588 .0657922 -------------+---------------------------------------------------------------- _cut1 | 2.754675 .869481 (Ancillary parameters) _cut2 | 5.10548 .9295388 ------------------------------------------------------------------------------

Iteration 0: log likelihood = -210.58254 Iteration 1: log likelihood = -195.01878 Iteration 2: log likelihood = -194.80294 Iteration 3: log likelihood = -194.80235

a. This is a listing of the log likelihoods at each iteration. Remember that ordered logistic regression, like binary and multinomial logistic regression, uses maximum likelihood estimation, which is an iterative procedure. The first iteration (called iteration 0) is the log likelihood of the "null" or "empty" model; that is, a model with no predictors. At the next iteration, the predictor(s) are included in the model. At each iteration, the log likelihood increases because the goal is to maximize the log likelihood. When the difference between successive iterations is very small, the model is said to have "converged", the iterating stops, and the results are displayed. For more information on this process for binary outcomes, see Regression Models for Categorical and Limited Dependent Variables by J. Scott Long (pages 52-61).

Ordered logit estimates Number of obs^{c}= 200 LR chi2(3)^{d}= 31.56 Prob > chi2^{e}= 0.0000 Log likelihood = -194.80235^{b}Pseudo R2^{f}= 0.0749

b.** Log Likelihood** - This is the log likelihood of the fitted model. It is used in the Likelihood Ratio Chi-Square test of whether all predictors'
regression coefficients in the model are simultaneously zero and in tests of nested models.

c.** Number of obs** - This is the number of observations used in the ordered logistic regression.
It may be less than the number of cases in the dataset if there are missing
values for some variables in the equation. By default, Stata does a listwise
deletion of incomplete cases.

d.** LR chi2(3)** - This is the Likelihood Ratio (LR) Chi-Square test that at least one of the predictors' regression coefficient is not equal to zero in
the model. The number in the parenthesis indicates the degrees of freedom of the Chi-Square distribution used to test the LR Chi-Square statistic and is
defined by the number of predictors in the model. The LR Chi-Square statistic can be calculated by -2*( L(null model) - L(fitted model)) = -2*((-210.583) -
(-194.802)) = 31.560, where *L(null model)* is from the log likelihood with just the response variable in the model (Iteration 0) and *L(fitted model)*
is the log likelihood from the final iteration (assuming the model converged) with all the parameters.

e.** Prob > chi2** - This is the probability of getting a LR test statistic as extreme as, or more so, than the observed under the null
hypothesis; the null hypothesis is that all of the regression coefficients in the model are equal to zero. In other words, this is the probability of obtaining this
chi-square statistic (31.56) if there is in fact no effect of the predictor variables. This p-value is compared to a specified alpha level, our willingness
to accept a type I error, which is typically set at 0.05 or 0.01. The small p-value from the LR test, <0.00001, would lead us to conclude that at least
one of the regression coefficients in the model is not equal to zero. The parameter of the Chi-Square distribution used to test the null hypothesis is defined
by the degrees of freedom in the prior line, ** chi2(3)**.

f.** Pseudo R2** - This is McFadden's pseudo R-squared. Logistic regression does not have an equivalent to the R-squared that is found in OLS regression;
however, many people have tried to come up with one. There are a wide variety of pseudo R-squared statistics
which can give contradictory conclusions. Because this statistic does not mean what
R-squared means in OLS regression (the proportion of variance for the response variable explained by the predictors), we suggest interpreting this statistic with great
caution.

------------------------------------------------------------------------------ ses^{g}| Coef.^{h}Std. Err.^{i}z^{j}P>|z|^{j}[95% Conf. Interval]^{k}-------------+---------------------------------------------------------------- science | .0300201 .0165861 1.81 0.070 -.0024882 .0625283 socst | .0531819 .015271 3.48 0.000 .0232513 .0831126 female | -.4823977 .2796939 -1.72 0.085 -1.030588 .0657922 -------------+---------------------------------------------------------------- _cut1 | 2.754675 .869481 (Ancillary parameters) _cut2 | 5.10548 .9295388 ------------------------------------------------------------------------------

g. **ses** - This is the response variable in the ordered logistic regression. Underneath **ses** are the predictors in the models and the
cut points for the adjacent levels of the response variable.

h. **Coef.** - These are the ordered log-odds (logit) regression coefficients.
Standard interpretation of the
ordered logit coefficient is that for a one
unit increase in the predictor, the response variable level is expected to change by its respective regression coefficient in the
ordered log-odds scale while the
other variables in the model are held constant. Interpretation of the ordered logit estimates
is not dependent on the ancillary parameters; the ancillary parameters are used to differentiate the adjacent levels of the response variable.
However, since the ordered logit model estimates one equation over all levels of
the dependent variable, a concern is whether our one-equation model is valid or
a more flexible model is required. We can test this hypothesis with the test for
proportional odds test (a.k.a. Brant test of parallel regression assumption).
This test can be downloaded by typing **findit spost13_ado** in the command line
and using the **brant** command (see
How can I use the findit command to search for programs and get additional
help? for more information about using **findit**).

** science** - This is the ordered log-odds estimate for a one unit increase in **science** score on the expected
**ses** level given the other variables are held constant in the model. If a
subject were to increase his **science** score by
one point, his ordered log-odds of being in a higher **ses** category would increase by 0.03 while the other variables
in the model are held constant.

** socst** - This is the ordered log-odds estimate for a one unit increase in **socst** score on the expected **ses** level given the other
variables are held constant in the model. A one unit increase in **socst** test scores would result in a 0.0532 unit increase in the
ordered log-odds
of being in a higher **ses** category while the other variables in the model are held constant.

** female** - This is the ordered log-odds estimate of comparing females to males on expected **ses** given the other variables are held
constant in the model. The ordered logit for females being in a higher **ses** category is -0.4824 less than males
when the other variables in the model are held constant.

** Ancillary parameters** - These refer to the cutpoints
(a.k.a. thresholds) used to differentiate the adjacent levels of the response variable.
A threshold can then be defined to be points on the latent variable, a
continuous unobservable mechanism/phenomena, that result in the different
observed values on the proxy variable (the levels of our dependent variable used
to measure the latent variable).

** _cut1 - **This is the estimated cutpoint
on the latent variable used to
differentiate low **ses** from middle and high **ses** when values of the
predictor variables are evaluated at zero. Subjects that had a value of 2.75 or less on the underlying latent
variable that gave rise to our **ses** variable would be classified as low **ses**
given they were male (the variable **female** evaluated at zero) and had zero **
science** and **socst** test scores.

** _cut2 - **
This is the estimated cutpoint on the latent variable used to
differentiate low and middle ses from high ses when values of the predictor
variables are evaluated at zero. Subjects that had a value of 5.11 or greater on the underlying latent
variable that gave rise to our **ses** variable would be classified as
high **ses** given they were male and had zero **science** and **socst**
test scores. Subjects that had a value between 2.75 and 5.11 on the underlying latent
variable would be classified as middle **ses**.

i. **Std. Err.** - These are the standard errors of the individual regression coefficients. They are used in both the calculation of the **z **test
statistic, superscript j, and the confidence interval of the regression coefficient, superscript k.

j. **z** and **P>|z|** - These are the test statistics and p-value, respectively, for the
null hypothesis that an individual predictor's regression
coefficient is zero given that the rest of the predictors are in the model. The test statistic **z** is the ratio of the **Coef.** to the
**Std. Err.** of the respective predictor. The **z** value follows a standard normal distribution which is used to test against a two-sided
alternative hypothesis that the **Coef.** is not equal to zero. The probability that a particular **z** test statistic is as extreme as, or more
so, than what has been observed under the null hypothesis is defined by **P>|z|**.

The **z** test statistic for the predictor **science** (0.030/0.017) is 1.81 with an associated p-value of 0.070. If we set our
alpha level to 0.05, we would fail to reject the null hypothesis and conclude that the regression coefficient for **science** has not been found to be
statistically different from zero in estimating **ses **given **socst** and **female** are in the model.

The **z** test statistic for the predictor **socst** (0.053/0.015) is 3.48 with an associated p-value
of <0.0001. If we again set our alpha level to 0.05, we would reject the null hypothesis and conclude that the regression coefficient for **socst** has
been found to be statistically different from zero in estimating **ses** given
that **science** and **female** are in the model.
The interpretation for
a dichotomous variable such as **female**, parallels that of a continuous variable: the observed
difference between males and females on **ses** status was not found to be
statistically significant at the 0.05 level when controlling for **socst**
and **science **(p=0.085).

k. **[95% Conf. Interval]** - This is the Confidence Interval (CI) for an individual regression coefficient given the other predictors are in the model.
For a given predictor with a level of 95% confidence, we'd say that we are 95% confident that the "true" population regression coefficient lies
in between the lower and upper limit of the interval. It is calculated as the **Coef.** ± (z_{α/2})*(**Std.Err.**), where z_{α/2}
is a critical value on the standard normal distribution. The CI is equivalent to the **z** test statistic: if the CI includes zero, we'd fail to
reject the null hypothesis that a particular regression coefficient is zero given the other predictors are in the model.
An advantage of a CI is that it is illustrative; it provides a range where the "true" parameter may lie.

The following is the interpretation of the ordered logistic regression in terms of
proportional odds ratios and can be obtained by
specifying the **or** option. This part of the interpretation applies to the output below.

ologit ses science socst female, orIteration 0: log likelihood = -210.58254 Iteration 1: log likelihood = -195.01878 Iteration 2: log likelihood = -194.80294 Iteration 3: log likelihood = -194.80235 Ordered logit estimates Number of obs = 200 LR chi2(3) = 31.56 Prob > chi2 = 0.0000 Log likelihood = -194.80235 Pseudo R2 = 0.0749 ------------------------------------------------------------------------------ ses | Odds Ratio^{a}Std. Err. z P>|z| [95% Conf. Interval]^{b}-------------+---------------------------------------------------------------- science | 1.030475 .0170916 1.81 0.070 .9975149 1.064525 socst | 1.054622 .0161052 3.48 0.000 1.023524 1.086664 female | .6173015 .1726554 -1.72 0.085 .3567972 1.068005 ------------------------------------------------------------------------------

a. **Odds Ratio** - These are the proportional odds ratios for the ordered
logit model (a.k.a. proportional odds model) shown earlier. They can be obtained by exponentiating the
ordered logit coefficients, e^{coef.}, or by specifying the **or** option.
Recall that ordered logit model estimates a single equation (regression
coefficients) over the levels of the dependent variable. Now, if we view the change in levels in a cumulative sense and interpret the coefficients in odds, we are comparing the people who are in
groups greater than *k* versus those who are in groups less than or equal to
*k*, where *k* is the level of the response variable.
The interpretation would be that for a one unit change in the predictor variable, the odds for cases in
a group that is greater than *k* versus less than or equal to *k*
are the proportional odds times larger. For a general discussion of OR, we refer to the following
Stata FAQ
for binary logistic regression: How do I interpret odds ratios in
logistic regression?

** science** - This is the proportional odds ratio for a one unit increase in **science** score on **ses** level given that the
other variables in the model are held constant. Thus, for a one unit increase in
**science** test score, the odds of high **ses**
versus the combined middle and low **ses** categories are 1.03 times greater, given the other variables are held constant
in the model.
Likewise, for a one unit increase in **science** test score, the odds of
the combined high and middle **ses** versus low **ses** are 1.03 times
greater, given the other variables are held constant.

** socst** - This is the proportional odds ratio for a one unit increase in **socst** score on **ses** level given that the
other variables in the model are held constant. Thus, for a one unit increase in **socst** test score, the odds of high **ses**
versus the combined middle and low **ses** are 1.05 times greater, given the other variables are held constant
in the model.
Likewise, for a one unit increase in **socst** test score, the odds of the
combined high and middle **ses** versus low **ses** are 1.05 times
greater, given the other variables are held constant.

** female** - This is the proportional odds ratio of comparing females to males on **ses** given the other variables in the model are held
constant. For females, the odds of high **ses** versus the combined middle
and low **ses** are 0.6173
times lower than for males, given the other variables are held constant. Likewise, the odds of
the combined categories of high and middle **
ses** versus low **ses** is 0.6173 times lower for females compared to males, given the other variables are held constant
in the model.

b.** [95% Conf. Interval]** - This is the CI for the proportional odds ratio given the other predictors are in the model.
For a given predictor with a level of 95% confidence, we'd say that we are 95% confident that the "true" population proportional odds ratio lies
between the lower and upper limit of the interval. The CI is
equivalent to the **z** test statistic: if the CI includes one
(not zero, because we are working with odds ratios), we'd fail to
reject the null hypothesis that a particular regression coefficient is one given the other predictors are in the model. An advantage of a CI is that it is
illustrative; it provides a range where the "true" proportional odds ratio may lie.

The content of this web site should not be construed as an endorsement of any particular web site, book, or software product by the University of California.