***Predicting the probability of getting into graduate school (admit=1) versus not getting in (admit=0)
(Predicting the probability of Y=1 vs Y=0) proc logistic data=data.binary descending;
class rank / param=ref ;
model admit = gre gpa rank;
run;
|
| Model Information | ||
|---|---|---|
| Data Set | DATA.BINARY | Written by SAS |
| Response Variable | ADMIT | |
| Number of Response Levels | 2 | |
| Model | binary logit | |
| Optimization Technique | Fisher's scoring | |
| Number of Observations Read | 400 |
|---|---|
| Number of Observations Used | 400 |
| Response Profile | ||
|---|---|---|
| Ordered Value |
ADMIT | Total Frequency |
| 1 | 1 | 127 |
| 2 | 0 | 273 |
Probability modeled is ADMIT=1.
| Class Level Information | ||||
|---|---|---|---|---|
| Class | Value | Design Variables | ||
| RANK | 1 | 1 | 0 | 0 |
| 2 | 0 | 1 | 0 | |
| 3 | 0 | 0 | 1 | |
| 4 | 0 | 0 | 0 | |
| Model Convergence Status |
|---|
| Convergence criterion (GCONV=1E-8) satisfied. |
| Model Fit Statistics | ||
|---|---|---|
| Criterion | Intercept Only | Intercept and Covariates |
| AIC | 501.977 | 470.517 |
| SC | 505.968 | 494.466 |
| -2 Log L | 499.977 | 458.517 |
The portion of the output labeled Model Fit Statistics describes and tests the overall fit of the model. The -2 Log L (499.977) can be used in comparisons of nested models.
| Testing Global Null Hypothesis: BETA=0 | |||
|---|---|---|---|
| Test | Chi-Square | DF | Pr > ChiSq |
| Likelihood Ratio | 41.4590 | 5 | <.0001 |
| Score | 40.1603 | 5 | <.0001 |
| Wald | 36.1390 | 5 | <.0001 |
The likelihood ratio chi-square of 41.4590 with a p-value of 0.0001 tells us that our model as a whole fits significantly better than an empty model.
The Score and Wald tests are asymptotically equivalent tests of the same hypothesis tested by the likelihood ratio test, not surprisingly, these tests also indicate that the model is statistically significant.
| Type 3 Analysis of Effects | |||
|---|---|---|---|
| Effect | DF | Wald Chi-Square |
Pr > ChiSq |
| GRE | 1 | 4.2842 | 0.0385 |
| GPA | 1 | 5.8714 | 0.0154 |
| RANK | 3 | 20.8949 | 0.0001 |
| Analysis of Maximum Likelihood Estimates | ||||||
|---|---|---|---|---|---|---|
| Parameter | DF | Estimate | Standard Error |
Wald Chi-Square |
Pr > ChiSq | |
| Intercept | 1 | -5.5414 | 1.1381 | 23.7081 | <.0001 | |
| GRE | 1 | 0.00226 | 0.00109 | 4.2842 | 0.0385 | |
| GPA | 1 | 0.8040 | 0.3318 | 5.8714 | 0.0154 | |
| RANK | 1 | 1 | 1.5514 | 0.4178 | 13.7870 | 0.0002 |
| RANK | 2 | 1 | 0.8760 | 0.3667 | 5.7056 | 0.0169 |
| RANK | 3 | 1 | 0.2112 | 0.3929 | 0.2891 | 0.5908 |
This shows the coefficients (labeled Estimate), their standard errors (error), the Wald Chi-Square statistic, and associated p-values. The coefficients for gre, and gpa are statistically significant, as are the terms for rank=1 and rank=2 (versus the omitted category rank=4). The logistic regression coefficients give the change in the log odds of the outcome for a one unit increase in the predictor variable.
- For every one unit change in gre, the log odds of admission (versus non-admission) increases by 0.002.
- For a one unit increase in gpa, the log odds of being admitted to graduate school increases by 0.804.
- The coefficients for the categories of rank have a slightly different interpretation. For example, having attended an undergraduate institution with arank of 1, versus an institution with a rank of 4, increases the log odds of admission by 1.55.
| Odds Ratio Estimates | |||
|---|---|---|---|
| Effect | Point Estimate | 95% Wald Confidence Limits | |
| GRE | 1.002 | 1.000 | 1.004 |
| GPA | 2.235 | 1.166 | 4.282 |
| RANK 1 vs 4 | 4.718 | 2.080 | 10.701 |
| RANK 2 vs 4 | 2.401 | 1.170 | 4.927 |
| RANK 3 vs 4 | 1.235 | 0.572 | 2.668 |
This gives the coefficients as odds ratios. An odds ratio is the exponentiated coefficient, and can be interpreted as the multiplicative change in the odds for a one unit change in the predictor variable.
For a one unit increase in gpa, the odds of being admitted to graduate school (versus not being admitted) increase by a factor of 2.24.
| Association of Predicted Probabilities and Observed Responses | |||
|---|---|---|---|
| Percent Concordant | 69.1 | Somers' D | 0.386 |
| Percent Discordant | 30.6 | Gamma | 0.387 |
| Percent Tied | 0.3 | Tau-a | 0.168 |
| Pairs | 34671 | c | 0.693 |