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Ordinal Logistic Regression
Version Info: Code for this page was tested in Mplus version 6.12.
Please note: The purpose of this page is to show how to use various data analysis commands. It does not cover all aspects of the research process which researchers are expected to do. In particular, it does not cover data cleaning and checking, verification of assumptions, model diagnostics and potential follow-up analyses.
Examples of ordered logistic regression
Example 1: A marketing research firm wants to investigate what factors influence the size of soda (small, medium, large or extra large) that people order at a fast-food chain. These factors may include what type of sandwich is ordered (burger or chicken), whether or not fries are also ordered, and age of the consumer. While the outcome variable, size of soda, is obviously ordered, the difference between the various sizes is not consistent. The differece between small and medium is 10 ounces, between medium and large 8, and between large and extra large 12.
Example 2: A researcher is interested in what factors influence medaling in Olympic swimming. Relevant predictors include at training hours, diet, age, and popularity of swimming in the athlete's home country. The researcher believes that the distance between gold and silver is larger than the distance between silver and bronze.
Example 3: A study looks at factors that influence the decision of whether to apply to graduate school. College juniors are asked if they are unlikely, somewhat likely, or very likely to apply to graduate school. Hence, our outcome variable has three categories. Data on parental educational status, whether the undergraduate institution is public or private, and current GPA is also collected. The researchers have reason to believe that the "distances" between these three points are not equal. For example, the "distance" between "unlikely" and "somewhat likely" may be shorter than the distance between "somewhat likely" and "very likely".
Description of the Data
For our data analysis below, we are going to expand on Example 3 about applying to graduate school. We have generated hypothetical data, which can be obtained here.
This hypothetical data set has a thee level variable called apply (coded 0, 1, 2), that we will use as our response (i.e., outcome, dependent) variable. We also have three variables that we will use as predictors: pared, which is a 0/1 variable indicating whether at least one parent has a graduate degree; public, which is a 0/1 variable where 1 indicates that the undergraduate institution is a public university and 0 indicates that it is a private university, and gpa, which is the student's grade point average. Let's start with some descriptive statistics for the variables of interest.
Title: Ordinal logistic regression in Mplus;
Data:
File is D:\documents\ologit in Mplus DAE\ologit.dat ;
Variable:
Names are apply pared public gpa;
categorical are apply;
Analysis:
type = basic;
Plot:
type = plot1;For this output only, we will display all of the information in the output. You will want to look at this carefully to be sure that the data were read into Mplus correctly. You will want to make sure that you have the correct number of observations, and that the categorical and continuous variables have been correctly specified. We have not used a missing statement because we have no missing data in this data set. If any of our variables had missing data we would have specified "missing = #" in the variable statement, where # is the numeric value given to missing values (e.g. -9999). Below the output are histograms for each of our four variables, these were produced using the plotting function in Mplus. In order to be able to do this, we included the plot statement and specified "type = plot1" which tells Mplus to create the auxiliary files necessary for the plotting function.
INPUT READING TERMINATED NORMALLY
Ordinal logistic regression in Mplus;
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 400
Number of dependent variables 4
Number of independent variables 0
Number of continuous latent variables 0
Observed dependent variables
Continuous
PARED PUBLIC GPA
Binary and ordered categorical (ordinal)
APPLY
Estimator WLSMV
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Parameterization DELTA
Input data file(s)
D:\documents\ologit in Mplus DAE\ologit.dat
Input data format FREE
SUMMARY OF CATEGORICAL DATA PROPORTIONS
APPLY
Category 1 0.550
Category 2 0.350
Category 3 0.100
RESULTS FOR BASIC ANALYSIS
ESTIMATED SAMPLE STATISTICS
MEANS/INTERCEPTS/THRESHOLDS
APPLY$1 APPLY$2 PARED PUBLIC GPA
________ ________ ________ ________ ________
1 0.126 1.282 0.157 0.143 2.999
CORRELATION MATRIX (WITH VARIANCES ON THE DIAGONAL)
APPLY PARED PUBLIC GPA
________ ________ ________ ________
APPLY
PARED 0.234 0.133
PUBLIC 0.052 0.079 0.122
GPA 0.179 0.186 0.227 0.158
STANDARD ERRORS FOR ESTIMATED SAMPLE STATISTICS
S.E. FOR MEANS/INTERCEPTS/THRESHOLDS
APPLY$1 APPLY$2 PARED PUBLIC GPA
________ ________ ________ ________ ________
1 0.063 0.085 16970.182 17629.901 0.020
S.E. FOR CORRELATION MATRIX (WITH VARIANCES ON THE DIAGONAL)
APPLY PARED PUBLIC GPA
________ ________ ________ ________
APPLY
PARED 0.053 6574.693
PUBLIC 0.054 0.044 6025.901
GPA 0.060 0.047 0.046 0.013
Analysis methods you might consider
Below is a list of some analysis methods you may have encountered. Some of the methods listed are quite reasonable while others have either fallen out of favor or have limitations.
- Ordered logistic regression: the focus of this page.
- OLS regression: This analysis is problematic because the assumptions of OLS are violated when it is used with a non-interval outcome variable.
- ANOVA: If you use only one continuous predictor, you could "flip" the model around so that, say, gpa was the outcome variable and apply was the predictor variable. Then you could run a one-way ANOVA. This isn't a bad thing to do if you only have one predictor variable (from the logistic model), and it is continuous.
- Multinomial logistic regression: This is similar to doing ordered logistic regression, except that it is assumed that there is no order to the categories of the outcome variable (i.e., the categories are nominal). The downside of this approach is that the information contained in the ordering is lost.
- Ordered probit regression: This is very, very similar to running an ordered logistic regression. The main difference is in the interpretation of the coefficients.
Ordinal logistic regression
Before we run our ordinal logistic model, we will see if any cells (created by the crosstab of our categorical and response variables) are empty or extremely small. If any are, we may have difficulty running our model. We cannot do this in Mplus, so the tables below come from Stata. You can use whatever statistics package you prefer to do this.
| pared
apply | 0 1 | Total
-----------+----------------------+----------
0 | 200 20 | 220
1 | 110 30 | 140
2 | 27 13 | 40
-----------+----------------------+----------
Total | 337 63 | 400
| public
apply | 0 1 | Total
-----------+----------------------+----------
0 | 189 31 | 220
1 | 124 16 | 140
2 | 30 10 | 40
-----------+----------------------+----------
Total | 343 57 | 400
None of the cells is too small or empty (has no cases), so we will run our model in Mplus. The syntax in bold below contains our model. Under analysis we have specified "estimator = ml". Had we not specified that the estimator should be ml, Mplus would have performed a probit regression model using weighted least squares, specifying "estimator = ml" instructs Mplus to estimate an ordinal logit model and to estimate it using maximum likelihood. Notice that we specify that the dependent variable, apply, is categorical.
Title: Ordinal logistic regression in Mplus,
Descriptive statistics;
Data:
File is D:\documents\ologit in Mplus DAE\ologit.dat ;
Variable:
Names are
apply pared public gpa;
categorical are apply;
Analysis:
Type = general ;
estimator = ml;
Model:
apply on pared public gpa;
MODEL FIT INFORMATION
Number of Free Parameters 5
Loglikelihood
H0 Value -358.512
Information Criteria
Akaike (AIC) 727.025
Bayesian (BIC) 746.982
Sample-Size Adjusted BIC 731.117
(n* = (n + 2) / 24)
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
APPLY ON
PARED 1.048 0.266 3.942 0.000
PUBLIC -0.059 0.298 -0.197 0.844
GPA 0.616 0.261 2.363 0.018
Thresholds
APPLY$1 2.203 0.780 2.826 0.005
APPLY$2 4.299 0.804 5.345 0.000
LOGISTIC REGRESSION ODDS RATIO RESULTS
APPLY ON
PARED 2.851
PUBLIC 0.943
GPA 1.851
- Towards the top of the output is the final log likelihood (-358.512), which can be used in comparisons of nested models.
- Under the heading "Information Criteria" we see the Akaike and Bayesian information criterion values. Both the AIC and the BIC are measures of fit with some correction for the complexity of the model, but the BIC has a stronger correction for parsimony. In both cases, lower values indicate better fit of the model.
- Under the heading "MODEL RESULTS" we see the coefficients, their standard errors, and the z-test (Est./S.E.)
- Both pared and gpa are statistically significant; public is not. The estimates in the output are given in units of ordered logits, or ordered log odds.
- So for pared, we would say that for a one unit increase in pared (i.e., going from 0 to 1), we expect a 1.048 increase in the log odds of moving from a given level of apply to any higher category, given all of the other variables in the model are held constant.
- For gpa, we would say that for a one unit increase in gpa, we would expect a 0.616 increase in the expected value of apply in the log odds scale, given that all of the other variables in the model are held constant.
- Below the table of coefficients are the Thresholds. The thresholds shown at the bottom of the output indicate where the latent variable is cut to make the three groups that we observe in our data. Note that this latent variable is continuous. In general, these are not used in the interpretation of the results. Note that different statistics packages use different formulations for thresholds, and that some packages label these as cutpoints rather than thresholds. Mplus produces thresholds which match the cutpoints Stata produces. For further information on the different formulations, and for help in converting from one to the other, please see the Mplus technical appendices or the Stata FAQ
- Finally, we see the results in terms of proportional odds ratios. We would interpret these pretty much as we would odds ratios from a binary logistic regression.
- For pared, we would say that for a one unit increase in pared, i.e., going from 0 to 1, the odds of high apply versus the combined middle and low categories are 2.851 times greater, given that all of the other variables in the model are held constant.
- Likewise, the odds of the combined middle and high categories versus low apply is 2.85 times greater, given that all of the other variables in the model are held constant.
- For a one unit increase in gpa, the odds of the low and middle categories of apply versus the high category of apply are 1.85 times greater, given that the other variables in the model are held constant. Because of the proportional odds assumption (see below for more explanation), the same increase, 1.85 times, is found between low apply and the combined categories of middle and high apply.
One of the assumptions underlying ordinal logistic (and ordinal probit) regression is that the relationship between each pair of outcome groups is the same. In other words, ordinal logistic regression assumes that the coefficients that describe the relationship between, say, the lowest versus all higher categories of the response variable are the same as those that describe the relationship between the next lowest category and all higher categories, etc. This is called the proportional odds assumption or the parallel regression assumption. Because the relationship between all pairs of groups is the same, there is only one set of coefficients (only one model). If this was not the case, we would need different models to describe the relationship between each pair of outcome groups. Mplus does not have a formal test for the proportional odds assumption. One way to asses whether the proportional odds assumption is reasonable is to turn your ordered dependent variable into a series of binary variables that are equal to one if y is greater than or equal to a given value, and zero otherwise. You will need k-1 of these binary variables, where k is the number of values your dependent variable takes on. You will then want to perform a series of binary logistic regression analyses, using each of these new variables as the outcome. If the proportional odds assumption is reasonable, the coefficients should be similar across each of these binary logistic regression models.
Things to consider
- Perfect prediction: Perfect prediction means that one value of a predictor variable is associated with only one value of the response variable.
- Sample size: Both ordered logistic and ordered probit, using maximum likelihood estimates, require sufficient sample size. How big is big is a topic of some debate, but they almost always require more cases than OLS regression.
- Empty cells or small cells: You should check for empty or small cells by doing a crosstab between categorical predictors and the outcome variable. If a cell has very few cases, the model may become unstable or it might not run at all.
- Pseudo-R-squared: There is no exact analog of the R-squared found in OLS. There are many versions of pseudo-R-squares. Please see Long and Freese 2005 for more details and explanations of various pseudo-R-squares.
- Diagnostics: Doing diagnostics for non-linear models is difficult, and ordered logit/probit models are even more difficult than binary models.
See also
- Annotated output for the ologit command
- Logistic Regression Troubleshooting and Ologit Interpretation
References
- Agresti, A. (1996) An Introduction to Categorical Data Analysis. New York: John Wiley & Sons, Inc
- Agresti, A. (2002) Categorical Data Analysis, Second Edition. Hoboken, New Jersey: John Wiley & Sons, Inc.
- Liao, T. F. (1994) Interpreting Probability Models: Logit, Probit, and Other Generalized Linear Models. Thousand Oaks, CA: Sage Publications, Inc.
- Powers, D. and Xie, Yu. Statistical Methods for Categorical Data Analysis. Bingley, UK: Emerald Group Publishing Limited.
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