7. The Conditional Logit Regression Model
Imagine a choice of the travel modes among air flight, train, bus, and car. The data set and model here are adopted from Greene (2003). The model examines how the generalized cost measure (cost), terminal waiting time (time), and household income (income) affect the choice.
These independent variables are not characteristics of subjects (individuals), but attributes of the alternatives. Thus, the data arrangement of the conditional logit model is different from that of the multinomial logit model (Figure 2).
Figure 2. Data Arrangement for the Conditional Logit Model
+------------------------------------------------------------------------------+
| subject mode choice air train bus cost time income air_inc |
|------------------------------------------------------------------------------|
| 1 1 0 1 0 0 70 69 35 35 |
| 1 2 0 0 1 0 71 34 35 0 |
| 1 3 0 0 0 1 70 35 35 0 |
| 1 4 1 0 0 0 30 0 35 0 |
| 2 1 0 1 0 0 68 64 30 30 |
|------------------------------------------------------------------------------|
| 2 2 0 0 1 0 84 44 30 0 |
| 2 3 0 0 0 1 85 53 30 0 |
| 2 4 1 0 0 0 50 0 30 0 |
| 3 1 0 1 0 0 129 69 40 40 |
| 3 2 0 0 1 0 195 34 40 0 |
| subject mode choice air train bus cost time income air_inc |
|------------------------------------------------------------------------------|
| 1 1 0 1 0 0 70 69 35 35 |
| 1 2 0 0 1 0 71 34 35 0 |
| 1 3 0 0 0 1 70 35 35 0 |
| 1 4 1 0 0 0 30 0 35 0 |
| 2 1 0 1 0 0 68 64 30 30 |
|------------------------------------------------------------------------------|
| 2 2 0 0 1 0 84 44 30 0 |
| 2 3 0 0 0 1 85 53 30 0 |
| 2 4 1 0 0 0 50 0 30 0 |
| 3 1 0 1 0 0 129 69 40 40 |
| 3 2 0 0 1 0 195 34 40 0 |
The example data set has four observations per subject, each of which contains attributes of using air flight, train, bus, and car. The dependent variable choice is coded 1 only if a subject chooses that travel mode. The four dummy variables, air, train, bus, and car, are flagging the corresponding modes of transportation. See the appendix for details about the data set.
7.1 Conditional Logit in STATA (.clogit)
STATA has the .clogit command to estimate the condition logit model. The group() option specifies the variable (e.g., identification number) that identifies unique individuals.
. clogit choice air train bus cost time air_inc, group(subject)
Iteration
0: log likelihood = -205.8187
Iteration 1: log likelihood = -199.23679
Iteration 2: log likelihood = -199.12851
Iteration 3: log likelihood = -199.12837
Iteration 4: log likelihood = -199.12837
Conditional (fixed-effects) logistic regression Number of obs = 840
LR chi2(6) = 183.99
Prob > chi2 = 0.0000
Log likelihood = -199.12837 Pseudo R2 = 0.3160
------------------------------------------------------------------------------
choice | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
air | 5.207443 .7790551 6.68 0.000 3.680523 6.734363
train | 3.869043 .4431269 8.73 0.000 3.00053 4.737555
bus | 3.163194 .4502659 7.03 0.000 2.280689 4.045699
cost | -.0155015 .004408 -3.52 0.000 -.024141 -.006862
time | -.0961248 .0104398 -9.21 0.000 -.1165865 -.0756631
air_inc | .013287 .0102624 1.29 0.195 -.0068269 .033401
--------------------------------------------------------------------------------------
Iteration 1: log likelihood = -199.23679
Iteration 2: log likelihood = -199.12851
Iteration 3: log likelihood = -199.12837
Iteration 4: log likelihood = -199.12837
Conditional (fixed-effects) logistic regression Number of obs = 840
LR chi2(6) = 183.99
Prob > chi2 = 0.0000
Log likelihood = -199.12837 Pseudo R2 = 0.3160
------------------------------------------------------------------------------
choice | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
air | 5.207443 .7790551 6.68 0.000 3.680523 6.734363
train | 3.869043 .4431269 8.73 0.000 3.00053 4.737555
bus | 3.163194 .4502659 7.03 0.000 2.280689 4.045699
cost | -.0155015 .004408 -3.52 0.000 -.024141 -.006862
time | -.0961248 .0104398 -9.21 0.000 -.1165865 -.0756631
air_inc | .013287 .0102624 1.29 0.195 -.0068269 .033401
--------------------------------------------------------------------------------------
Let us run the .listcoef command to compute factor changes in odds. For a one unit increase in the waiting time for a given travel mode, for example, we can expect a decrease in the odds of using that travel by 9 percent (or a factor of .9084), holding other variables constant.
. listcoef
clogit
(N=840): Factor Change in Odds
Odds of: 1 vs 0
--------------------------------------------------
choice | b z P>|z| e^b
-------------+------------------------------------
air | 5.20744 6.684 0.000 182.6265
train | 3.86904 8.731 0.000 47.8965
bus | 3.16319 7.025 0.000 23.6460
cost | -0.01550 -3.517 0.000 0.9846
time | -0.09612 -9.207 0.000 0.9084
air_inc | 0.01329 1.295 0.195 1.0134
--------------------------------------------------
Odds of: 1 vs 0
--------------------------------------------------
choice | b z P>|z| e^b
-------------+------------------------------------
air | 5.20744 6.684 0.000 182.6265
train | 3.86904 8.731 0.000 47.8965
bus | 3.16319 7.025 0.000 23.6460
cost | -0.01550 -3.517 0.000 0.9846
time | -0.09612 -9.207 0.000 0.9084
air_inc | 0.01329 1.295 0.195 1.0134
--------------------------------------------------
SAS has the MDC procedure to fit the conditional logit model. The TYPE=CLOGIT indicates the conditional logit model; the ID statement specifies the identification variable; and the NCHOICE=4 tells that there are four choices of the travel mode.
PROC MDC DATA=masil.travel;
MODEL choice = air train bus cost time air_inc /TYPE=CLOGIT NCHOICE=4;
ID subject;
RUN;
The MDC Procedure
Conditional Logit Estimates
Algorithm converged.
Model Fit Summary
Dependent Variable choice
Number of Observations 210
Number of Cases 840
Log Likelihood -199.12837
Maximum Absolute Gradient 2.73152E-8
Number of Iterations 5
Optimization Method Newton-Raphson
AIC 410.25674
Schwarz Criterion 430.33938
Discrete Response Profile
Index CHOICE Frequency Percent
0 1 58 27.62
1 2 63 30.00
2 3 30 14.29
3 4 59 28.10
Goodness-of-Fit Measures
Measure Value Formula
Likelihood Ratio (R) 183.99 2 * (LogL - LogL0)
Upper Bound of R (U) 582.24 - 2 * LogL0
Aldrich-Nelson 0.467 R / (R+N)
Cragg-Uhler 1 0.5836 1 - exp(-R/N)
Cragg-Uhler 2 0.6225 (1-exp(-R/N)) / (1-exp(-U/N))
Estrella 0.6511 1 - (1-R/U)^(U/N)
Adjusted Estrella 0.6212 1 - ((LogL-K)/LogL0)^(-2/N*LogL0)
McFadden's LRI 0.316 R / U
Veall-Zimmermann 0.6354 (R * (U+N)) / (U * (R+N))
N = # of observations, K = # of regressors
Conditional Logit Estimates
Parameter Estimates
Standard Approx
Parameter DF Estimate Error t Value Pr > |t|
air 1 5.2074 0.7791 6.68 <.0001
train 1 3.8690 0.4431 8.73 <.0001
bus 1 3.1632 0.4503 7.03 <.0001
cost 1 -0.0155 0.004408 -3.52 0.0004
time 1 -0.0961 0.0104 -9.21 <.0001
air_inc 1 0.0133 0.0103 1.29 0.1954
Conditional Logit Estimates
Algorithm converged.
Model Fit Summary
Dependent Variable choice
Number of Observations 210
Number of Cases 840
Log Likelihood -199.12837
Maximum Absolute Gradient 2.73152E-8
Number of Iterations 5
Optimization Method Newton-Raphson
AIC 410.25674
Schwarz Criterion 430.33938
Discrete Response Profile
Index CHOICE Frequency Percent
0 1 58 27.62
1 2 63 30.00
2 3 30 14.29
3 4 59 28.10
Goodness-of-Fit Measures
Measure Value Formula
Likelihood Ratio (R) 183.99 2 * (LogL - LogL0)
Upper Bound of R (U) 582.24 - 2 * LogL0
Aldrich-Nelson 0.467 R / (R+N)
Cragg-Uhler 1 0.5836 1 - exp(-R/N)
Cragg-Uhler 2 0.6225 (1-exp(-R/N)) / (1-exp(-U/N))
Estrella 0.6511 1 - (1-R/U)^(U/N)
Adjusted Estrella 0.6212 1 - ((LogL-K)/LogL0)^(-2/N*LogL0)
McFadden's LRI 0.316 R / U
Veall-Zimmermann 0.6354 (R * (U+N)) / (U * (R+N))
N = # of observations, K = # of regressors
Conditional Logit Estimates
Parameter Estimates
Standard Approx
Parameter DF Estimate Error t Value Pr > |t|
air 1 5.2074 0.7791 6.68 <.0001
train 1 3.8690 0.4431 8.73 <.0001
bus 1 3.1632 0.4503 7.03 <.0001
cost 1 -0.0155 0.004408 -3.52 0.0004
time 1 -0.0961 0.0104 -9.21 <.0001
air_inc 1 0.0133 0.0103 1.29 0.1954
Alternatively, you may use the PHREG procedure that estimates the Cox proportional hazards model for survival data and the conditional logit model.
In order to make the data set consistent with the survival analysis data, you need to create a failure time variable, failure=1choice. The identification variable is specified in the STRATA statement. The NOSUMMARY option suppresses the display of the event and censored observation frequencies.
PROC PHREG DATA=masil.travel NOSUMMARY;
STRATA subject;
MODEL failure*choice(0)=air train bus cost time air_inc;
RUN;
The PHREG Procedure
Model Information
Data Set MASIL.TRAVEL
Dependent Variable failure
Censoring Variable choice
Censoring Value(s) 0
Ties Handling BRESLOW
Number of Observations Read 840
Number of Observations Used 840
Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Without With
Criterion Covariates Covariates
-2 LOG L 582.244 398.257
AIC 582.244 410.257
SBC 582.244 430.339
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 183.9869 6 <.0001
Score 173.4374 6 <.0001
Wald 103.7695 6 <.0001
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
air 1 5.20743 0.77905 44.6799 <.0001 182.625
train 1 3.86904 0.44313 76.2343 <.0001 47.896
bus 1 3.16319 0.45027 49.3530 <.0001 23.646
cost 1 -0.01550 0.00441 12.3671 0.0004 0.985
time 1 -0.09612 0.01044 84.7778 <.0001 0.908
air_inc 1 0.01329 0.01026 1.6763 0.1954 1.013
Model Information
Data Set MASIL.TRAVEL
Dependent Variable failure
Censoring Variable choice
Censoring Value(s) 0
Ties Handling BRESLOW
Number of Observations Read 840
Number of Observations Used 840
Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Without With
Criterion Covariates Covariates
-2 LOG L 582.244 398.257
AIC 582.244 410.257
SBC 582.244 430.339
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 183.9869 6 <.0001
Score 173.4374 6 <.0001
Wald 103.7695 6 <.0001
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
air 1 5.20743 0.77905 44.6799 <.0001 182.625
train 1 3.86904 0.44313 76.2343 <.0001 47.896
bus 1 3.16319 0.45027 49.3530 <.0001 23.646
cost 1 -0.01550 0.00441 12.3671 0.0004 0.985
time 1 -0.09612 0.01044 84.7778 <.0001 0.908
air_inc 1 0.01329 0.01026 1.6763 0.1954 1.013
While the MDC procedure reports t statistics, the PHREG procedure computes chi-squared (e.g., 12.3671=-3.52^2). The PHREG presents the hazard ratio at the last column of the output, which is equivalent to the factor changes under the e^b column of the SPost .listcoef command.
7.3 Conditional Logit in LIMDEP (Clogit$)
LIMDEP fits the conditional logit model using either the Clogit$ or the Logit$ command. The Clogit$ command has the Choices$ subcommand to list the choices available.
CLOGIT;
Lhs=choice;
Rhs=air,train,bus,cost,time,air_inc;
Choices=air,train,bus,car$
Normal
exit from iterations. Exit status=0.
+---------------------------------------------+
| Discrete choice (multinomial logit) model |
| Maximum Likelihood Estimates |
| Model estimated: Sep 19, 2005 at 09:20:39PM.|
| Dependent variable Choice |
| Weighting variable None |
| Number of observations 210 |
| Iterations completed 6 |
| Log likelihood function -199.1284 |
| Log-L for Choice model = -199.12837 |
| R2=1-LogL/LogL* Log-L fncn R-sqrd RsqAdj |
| Constants only -283.7588 .29825 .29150 |
| Response data are given as ind. choice. |
| Number of obs.= 210, skipped 0 bad obs. |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
AIR 5.207443299 .77905514 6.684 .0000
TRAIN 3.869042702 .44312685 8.731 .0000
BUS 3.163194212 .45026593 7.025 .0000
COST -.1550152532E-01 .44079931E-02 -3.517 .0004
TIME -.9612479610E-01 .10439847E-01 -9.207 .0000
AIR_INC .1328702625E-01 .10262407E-01 1.295 .1954
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
+---------------------------------------------+
| Discrete choice (multinomial logit) model |
| Maximum Likelihood Estimates |
| Model estimated: Sep 19, 2005 at 09:20:39PM.|
| Dependent variable Choice |
| Weighting variable None |
| Number of observations 210 |
| Iterations completed 6 |
| Log likelihood function -199.1284 |
| Log-L for Choice model = -199.12837 |
| R2=1-LogL/LogL* Log-L fncn R-sqrd RsqAdj |
| Constants only -283.7588 .29825 .29150 |
| Response data are given as ind. choice. |
| Number of obs.= 210, skipped 0 bad obs. |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
AIR 5.207443299 .77905514 6.684 .0000
TRAIN 3.869042702 .44312685 8.731 .0000
BUS 3.163194212 .45026593 7.025 .0000
COST -.1550152532E-01 .44079931E-02 -3.517 .0004
TIME -.9612479610E-01 .10439847E-01 -9.207 .0000
AIR_INC .1328702625E-01 .10262407E-01 1.295 .1954
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
The Clogit$ command has the Ias$ subcommand to conduct the Hausman test for the IIA assumption (e.g., Ias=air,bus$). Unfortunately, the subcommand does not work in this model because the Hessian is not positive definite.
The Logit$ command takes the panel data analysis approach. The Pds$ subcommand specifies the number of time periods. The two commands produce the same result.
LOGIT;
Lhs=choice;
Rhs=air,train,bus,cost,time,air_inc;
Pds=4$
+--------------------------------------------------+
| Panel Data Binomial Logit Model |
| Number of individuals = 210 |
| Number of periods = 4 |
| Conditioning event is the sum of CHOICE |
| Distribution of sums over the 4 periods: |
| Sum 0 1 2 3 4 5 6 |
| Number 0 210 0 0 0 5 10 |
| Pct. .00100.00 .00 .00 .00 .00 .00 |
+--------------------------------------------------+
Normal exit from iterations. Exit status=0.
+---------------------------------------------+
| Logit Model for Panel Data |
| Maximum Likelihood Estimates |
| Model estimated: Sep 19, 2005 at 09:21:58PM.|
| Dependent variable CHOICE |
| Weighting variable None |
| Number of observations 840 |
| Iterations completed 6 |
| Log likelihood function -199.1284 |
| Hosmer-Lemeshow chi-squared = 251.24482 |
| P-value= .00000 with deg.fr. = 8 |
| Fixed Effects Logit Model for Panel Data |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
AIR 5.207443299 .77905514 6.684 .0000
TRAIN 3.869042702 .44312685 8.731 .0000
BUS 3.163194212 .45026593 7.025 .0000
COST -.1550152532E-01 .44079931E-02 -3.517 .0004
TIME -.9612479610E-01 .10439847E-01 -9.207 .0000
AIR_INC .1328702625E-01 .10262407E-01 1.295 .1954
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
| Panel Data Binomial Logit Model |
| Number of individuals = 210 |
| Number of periods = 4 |
| Conditioning event is the sum of CHOICE |
| Distribution of sums over the 4 periods: |
| Sum 0 1 2 3 4 5 6 |
| Number 0 210 0 0 0 5 10 |
| Pct. .00100.00 .00 .00 .00 .00 .00 |
+--------------------------------------------------+
Normal exit from iterations. Exit status=0.
+---------------------------------------------+
| Logit Model for Panel Data |
| Maximum Likelihood Estimates |
| Model estimated: Sep 19, 2005 at 09:21:58PM.|
| Dependent variable CHOICE |
| Weighting variable None |
| Number of observations 840 |
| Iterations completed 6 |
| Log likelihood function -199.1284 |
| Hosmer-Lemeshow chi-squared = 251.24482 |
| P-value= .00000 with deg.fr. = 8 |
| Fixed Effects Logit Model for Panel Data |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
AIR 5.207443299 .77905514 6.684 .0000
TRAIN 3.869042702 .44312685 8.731 .0000
BUS 3.163194212 .45026593 7.025 .0000
COST -.1550152532E-01 .44079931E-02 -3.517 .0004
TIME -.9612479610E-01 .10439847E-01 -9.207 .0000
AIR_INC .1328702625E-01 .10262407E-01 1.295 .1954
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
Like the SAS PHREG procedure, the SPSS Coxreg command, which was designed for survival analysis data, provides a backdoor way of estimating the conditional logit model.
COXREG failure WITH air train bus cost time air_inc
/STATUS=choice(1)
/STRATA=subject.
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