Intuitively Interpret Output: Odds ratio, ROC Curve, Concordant, Discordant

In this tutorial, we will use the High School and Beyond data set, hsb2.sas7bdat to describe what a logistic model is, how to perform a logistic regression model analysis and how to interpret the model. Our dependent variable is created as a dichotomous variable indicating if a student's writing score is higher than or equal to 52. We call it

data hsb2; set hsb2; hiwrite = (write >=52); run;

Let's now take a look at a model with both a continuous variable
**math** and a categorical variable **female** as predictors. We will
focus on how to interpret the parameter estimate for the continuous variable.

proc logistic data = hsb2 Desc (the same effect as event='1'); model hiwrite (event='1') = female math /clodds=wald; units math = 5; output out = m2 p = prob xbeta = logit; run; proc template; /*display parameter estimates with more decials */ define table Stat.Logistic.ParameterEstimates; dynamic NRows; column Variable GenericClassValue Response DF Estimate StdErr WaldChiSq ProbChiSq StandardizedEst ExpEst Label; define Estimate; header = "Estimate"; parent = Stat.Logistic.vbest8; format = 20.8 ; end; end ; run ; Proc Logistic Data=A Descending; Model Y=X1 X2 X3 X4; Test X1=0; *Tests H0:Beta1=0; Test X1=X2=0; *Tests H0: Beta1=Beta2=0; Test X1=X2; *Tests H0: Beta1=Beta2; run;

Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq

Intercept 1 -10.3651 1.5535 44.5153 <.0001 FEMALE 1 1.6304 0.4052 16.1922 <.0001 MATH 10.19790.0293 45.5559 <.0001

Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits FEMALE 5.106 2.308 11.298 MATH1.2191.151 1.291

Wald Confidence Interval for Adjusted Odds Ratios Effect Unit Estimate 95% Confidence Limits MATH 5.0000 2.689 2.018 3.584

The interpretation for the parameter estimate of **math** is very
similar to that for the categorical variable **female**. In terms of logit scale, we can say
that for every unit increase in the math score, the logit will increase by
.198, holding everything else constant. We can also say that for a one unit
increase in math score, the odds of scoring 51 or higher in writing test
increases by (1.219-1)*100% = 22%.

Sometimes, a one unit change may not be a desirable scale to use. We can ask SAS to give us odds ratio for different units of change. For example, it may make more sense to talk about change of every 5 units in math score. This can be done using

We also include the option

We can compare the linear predictions and the probabilities in terms of the math scores for the males and females.

proc sort data = m2; by math; run;

Acknowledgement: The tutorial is based on the notes from: www.ats.ucla.edu.symbol1 i = join v=star l=32 c = black; symbol2 i = join v=circle l = 1 c=black; proc gplot data = m2; plot logit*math = female; plot prob*math = female; run; quit;