1. Ordinal Regression Analysis: Fitting the Proportional Odds Model Using Stata and SAS Xing Liu Neag School of Education University of Connecticut

2. Purpose Introduce Ordinal Logistic Regression Analysis Demonstrate the use of the proportional odds (PO) model using Stata (V. 9.0) Compare the results of the proportional odds model using both Stata OLOGIT and SAS LOGISTIC.

3. Why Ordinal Regression Analysis? Ordinal Dependent Variable  Teaching experience  SES (high, middle, low)  Degree of Agreement  Ability level (e.g. literacy, reading) Context is important

4. Why Using STATA and SAS? Both are powerful general statistics software packages Stata are more powerful in the analysis of binary logistic regression and ordinal logistic regression SAS has two options for Ordinal dependent variable  Ascending  Descending

5. Proportional Odds Model (1) One of several possible regression models for the analysis of ordinal data, and also the most common. Model predicts the ln(odds) of being in category j or beyond. Simplifying assumption  Effect of an IV assumed to be invariant across splits

6. Proportional Odds Model (2) Model predicts cumulative logits across K-1 response option categories. For K=6, (here, Y = 0 to 5) these cumulative logits can be used to make predictions for the K-1=five cumulative probabilities, given the collection of explanatory variables: Logit = ln(odds); Probability = odds / (1 + odds) logits  odds  estimated probability

7. A Latent-variable Model (1) It can be expressed as a latent variable model (Agresti, 2002; Greene, 2003; Long, 1997, Long & Freese, 2006; Powers & Xie, 2000; Wooldridge & Jeffrey, 2001) Assuming a latent variable, Y* exists, we can define Y* = xβ + ε Let Y* be divided by some cut points (thresholds): α1, α2, α3… αj, and α1<α2<α3…< αj. The observed child’s literacy proficiency level is the ordinal outcome, y, ranging from 0 to 5

8. A Latent-variable Model (2) We define:

9. A Latent-variable Model (3) We can compute probability of a child attaining each proficiency level.  P(y=0) = P (y* ≤α1) = P(xβ + ε ≤ α1) = F (α1- xβ);  P(y=1) = P (α1<y* ≤α2) = F (α2- xβ)- F (α1- xβ);  …  P(y=4) = P (α4<y* ≤α5) = F (α5- xβ)- F (α4- xβ);  P(y=5) = P (y* >α5) = 1- F (α5- xβ). We can also compute the cumulative probabilities using the form: P(Y≤j) = F (αj - xβ)

10. General Logistic Regression Model In a binary logistic regression model, we predict the log(odds) of success on a set of predictors. In Stata, the ordinal logistic regression model is expressed as: SAS uses a different ordinal logit model for estimating the parameters

11. Methodology Sample: ECLS-K Longitudinal Study (NCES)  n= 3365 from 225 schools  Outcome variable: proficiency in early reading  Eight explanatory variables The PO model with a single explanatory variable was fitted first Full-model with all eight explanatory variables The assumption of the PO models were tested Software packages: STATA and SAS

12. Results

13. Figure 1: Full-Model Analysis of Proportional Odds Using Stata

14. Figure 2: Brant Test of Parallel Regression (Proportional Odds) Assumption

15. Figure 3: Measure of Fit Statistics for Full- Model

16. Conclusions Both packages produce the same or similar results in model fit statistics and the test of the proportional odds assumption Stata produces more detailed information of PO assumption test, and fit statistics The estimated coefficients and cut points (thresholds) are the same in magnitude but may be reversed in sign Compared to Stata, SAS (ascending and descending) does not negate the signs before the logit coefficients in the equations

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