Conditional Logit Regression
The conditional logit model has the form:
In this model, subjects are presented with choice alternatives and asked to choose the most preferred alternative. The set of alternatives is typically the same for all subjects and the explanatory variables are all choice specific. Unlike in the multinomial logit model, the parameters are not specific to the choice.
PHREG Procedure
The SAS PHREG procedure performs regression analysis of survival data based on the Cox proportional hazards model. Its likelihood function is similar to that of the conditional logit model.
To fit a conditional logit model with PROC PHREG, you need to rearrange the data set in a form that is consistent with survival analysis data. The most preferred choice is said to occur at time 1 and all other choices are said to occur at later times or to be censored. You also need to create a status variable to denote whether the observation was censored or not, i.e., whether the alternative was chosen or not. The censoring indicator variable has the value of 0 if the alternative was censored (not chosen) and 1 if not censored (chosen). The basic syntax is:
proc phreg; strata strata_varname; model time_varname*status_varname(0) = x1 x2; run;
where strata_varname is the name of variable to specify the variable that determines the stratification, time_varname is the name of failure time variable (the smaller value means the alternative was chosen), status_varname is the name of the censoring indicator variable, of which 0 is the value to indicate censoring, and X1 and X2 are explanatory variables.
This example is from SAS (SAS, 1995, Logistic Regression Examples Using the SAS System, pp. 2-3). Chocolate candy data are generated in which 10 subjects are presented with eight different chocolate candies. The subjects choose one preferred candy from among the eight types. The eight candies consist of eight combinations of dark(1) or milk(0) chocolate, soft(1) or hard(0) center, and nuts(1) or no nuts(0). The following data step creates the data set CHOCO:
data choco; input subject choose dark soft nuts @@; t=2-choose; cards; 1 0 0 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 2 0 0 0 0 2 0 0 0 1 2 0 0 1 0 2 0 0 1 1 2 0 1 0 0 2 1 1 0 1 2 0 1 1 0 2 0 1 1 1 3 0 0 0 0 3 0 0 0 1 3 0 0 1 0 3 0 0 1 1 3 0 1 0 0 3 0 1 0 1 3 1 1 1 0 3 0 1 1 1 4 0 0 0 0 4 0 0 0 1 4 0 0 1 0 4 0 0 1 1 4 1 1 0 0 4 0 1 0 1 4 0 1 1 0 4 0 1 1 1 5 0 0 0 0 5 1 0 0 1 5 0 0 1 0 5 0 0 1 1 5 0 1 0 0 5 0 1 0 1 5 0 1 1 0 5 0 1 1 1 6 0 0 0 0 6 0 0 0 1 6 0 0 1 0 6 0 0 1 1 6 0 1 0 0 6 1 1 0 1 6 0 1 1 0 6 0 1 1 1 7 0 0 0 0 7 1 0 0 1 7 0 0 1 0 7 0 0 1 1 7 0 1 0 0 7 0 1 0 1 7 0 1 1 0 7 0 1 1 1 8 0 0 0 0 8 0 0 0 1 8 0 0 1 0 8 0 0 1 1 8 0 1 0 0 8 1 1 0 1 8 0 1 1 0 8 0 1 1 1 9 0 0 0 0 9 0 0 0 1 9 0 0 1 0 9 0 0 1 1 9 0 1 0 0 9 1 1 0 1 9 0 1 1 0 9 0 1 1 1 10 0 0 0 0 10 0 0 0 1 10 0 0 1 0 10 0 0 1 1 10 0 1 0 0 10 1 1 0 1 10 0 1 1 0 10 0 1 1 1 ;
where SUBJECT is the subject number, CHOOSE is the status variable, and T is the time variable. Because this data set is arranged in a survival analysis form you can use the PROC PHREG. You can use the syntax:
proc phreg data=choco; strata subject; model t*choose(0)=dark soft nuts; run;
As a result, you will have:
Sample Program: Conditional Logit Regression
The PHREG Procedure
Data Set: WORK.CHOCO
Dependent Variable: T
Censoring Variable: CHOOSE
Censoring Value(s): 0
Ties Handling: BRESLOW
Summary of the Number of Event and Censored Values
Percent
Stratum SUBJECT Total Event Censored Censored
1 1 8 1 7 87.50
2 2 8 1 7 87.50
3 3 8 1 7 87.50
4 4 8 1 7 87.50
5 5 8 1 7 87.50
6 6 8 1 7 87.50
7 7 8 1 7 87.50
8 8 8 1 7 87.50
9 9 8 1 7 87.50
10 10 8 1 7 87.50
Total 80 10 70 87.50
Testing Global Null Hypothesis: BETA=0
Without With
Criterion Covariates Covariates Model Chi-Square
-2 LOG L 41.589 28.727 12.862 with 3 DF (p=0.0049)
Score . . 11.600 with 3 DF (p=0.0089)
Wald . . 8.928 with 3 DF (p=0.0303)
Analysis of Maximum Likelihood Estimates
Parameter Standard Wald Pr > Risk
Variable DF Estimate Error Chi-Square Chi-Square Ratio
DARK 1 1.386294 0.79057 3.07490 0.0795 4.000
SOFT 1 -2.197225 1.05409 4.34502 0.0371 0.111
NUTS 1 0.847298 0.69007 1.50762 0.2195 2.333
The result shows the estimation result as:
The positive parameter estimates of DARK and NUTS mean that dark and nuts each increases the preference. The negative parameter estimate of SOFT denotes soft center decreases the preference.
For each of eight types of candies, the predicted probabilities can be computed as follows:
This shows that the most preferred type of candy is the dark chocolate with a hard center and nuts.
COXREG Procedure
With SPSS, you can use the COXREG procedure to fit a conditional logit model. The basic syntax is:
coxreg time_varname with X1 X2 /status=status_varname(1) /strata=strata_varname.
where time_varname is the name of the failure time variable (the smaller value means the alternative was chosen), status_varname is the name of the censoring indicator variable, of which 1 is the value to indicate the event has occurred (not censored), strata_varname is the name of variable to specify the variable that determines the stratification, and X1 and X2 are explanatory variables.
Using the data in Example 28, if you use:
coxreg t with dark soft nuts /status=choose(1) /strata=subject.
you will have the following SPSS output:
C O X R E G R E S S I O N
80 Total cases read
0 Cases with missing values
0 Valid cases with non-positive times
0 Censored cases before the earliest event in a stratum
0 Total cases dropped
80 Cases available for the analysis
Dependent Variable: T
SUBJECT Events Censored
1.00 1 7 (87.5%)
2.00 1 7 (87.5%)
3.00 1 7 (87.5%)
4.00 1 7 (87.5%)
5.00 1 7 (87.5%)
6.00 1 7 (87.5%)
7.00 1 7 (87.5%)
8.00 1 7 (87.5%)
9.00 1 7 (87.5%)
10.00 1 7 (87.5%)
Total 10 70 (87.5%)
Beginning Block Number 0. Initial Log Likelihood Function
-2 Log Likelihood 41.589
Beginning Block Number 1. Method: Enter
Variable(s) Entered at Step Number 1..
DARK
NUTS
SOFT
Coefficients converged after 5 iterations.
-2 Log Likelihood 28.727
Chi-Square df Sig
Overall (score) 11.600 3 .0089
Change (-2LL) from
Previous Block 12.862 3 .0049
Previous Step 12.862 3 .0049
-------------------- Variables in the Equation ---------------------
Variable B S.E. Wald df Sig R Exp(B)
DARK 1.3863 .7906 3.0749 1 .0795 .1608 4.0000
NUTS .8473 .6901 1.5076 1 .2195 .0000 2.3333
SOFT -2.1972 1.0541 4.3450 1 .0371 -.2375 .1111
Covariate Means
Variable Mean
DARK .5000
NUTS .5000
SOFT .5000
The estimation result is exactly the same as what you obtained with SAS.
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