The MDC procedure is similar in use to the other regression model procedures in the SAS System. However, the MDC procedure requires identification and choice variables. For example, consider a random utility function

where the cumulative distribution function of the stochastic component is a Type I extreme value, . You can estimate this conditional logit model with the following statements:

proc mdc; model decision = x1 x2 / type=clogit choice=(mode 1 2 3); id pid; run;

Note that the MDC procedure, unlike other regression procedures, does not include the intercept term automatically. The dependent
variable `decision`

takes the value 1 when a specific alternative is chosen; otherwise, it takes the value 0. Each individual is allowed to choose
one and only one of the possible alternatives. In other words, the variable `decision`

takes the value 1 one time only for each individual. If each individual has three elements (1, 2, and 3) in the choice set,
the NCHOICE=3 option can be specified instead of CHOICE=(`mode`

1 2 3).

Consider the following trinomial data from Daganzo (1979). The original data (`origdata`

) contain travel time (`ttime1–ttime3`

) and choice (`choice`

) variables. The variables `ttime1–ttime3`

are the travel times for three different modes of transportation, and `choice`

indicates which one of the three modes is chosen. The choice variable must have integer values.

data origdata; input ttime1 ttime2 ttime3 choice @@; datalines; 16.481 16.196 23.89 2 15.123 11.373 14.182 2 19.469 8.822 20.819 2 18.847 15.649 21.28 2 12.578 10.671 18.335 2 11.513 20.582 27.838 1 10.651 15.537 17.418 1 8.359 15.675 21.05 1 ... more lines ...

A new data set (`newdata`

) is created because PROC MDC requires that each individual decision maker has one case for each alternative in his choice
set. Note that the ID statement is required for all MDC models. In the following example, there are two public transportation
modes, 1 and 2, and one private transportation mode, 3, and all individuals share the same choice set.

The first nine observations of the raw data set are shown in Figure 18.1.

The following statements transform the data according to MDC procedure requirements:

data newdata(keep=pid decision mode ttime); set origdata; array tvec{3} ttime1 - ttime3; retain pid 0; pid + 1; do i = 1 to 3; mode = i; ttime = tvec{i}; decision = ( choice = i ); output; end; run;

The first nine observations of the transformed data set are shown in Figure 18.2.

The decision variable, `decision`

, must have one nonzero value for each decision maker that corresponds to the actual choice. When the RANK option is specified,
the decision variable must contain rank data. For more details, see the section MODEL Statement. The following SAS statements estimate the conditional logit model by using maximum likelihood:

proc mdc data=newdata; model decision = ttime / type=clogit nchoice=3 optmethod=qn covest=hess; id pid; run;

The MDC procedure enables different individuals to have different choice sets. When all individuals have the same choice set,
the NCHOICE= option can be used instead of the CHOICE= option. However, the NCHOICE= option is not allowed when a nested logit
model is estimated. When the NCHOICE=*number* option is specified, the choices are generated as . For more flexible alternatives (for example, 1, 3, 6, 8), you need to use the CHOICE= option. The choice variable must have
integer values.

The OPTMETHOD=QN option specifies the quasi-Newton optimization technique. The covariance matrix of the parameter estimates is obtained from the Hessian matrix because COVEST=HESS is specified. You can also specify COVEST=OP or COVEST=QML. See the section MODEL Statement for more details.

The MDC procedure produces a summary of model estimation displayed in Figure 18.3. Since there are multiple observations for each individual, the "Number of Cases" (150)—that is, the total number of choices faced by all individuals—is larger than the number of individuals, "Number of Observations" (50).

Figure 18.3: Estimation Summary Table

The MDC Procedure

Conditional Logit Estimates

Model Fit Summary | |
---|---|

Dependent Variable | decision |

Number of Observations | 50 |

Number of Cases | 150 |

Log Likelihood | -33.32132 |

Log Likelihood Null (LogL(0)) | -54.93061 |

Maximum Absolute Gradient | 2.97024E-6 |

Number of Iterations | 6 |

Optimization Method | Dual Quasi-Newton |

AIC | 68.64265 |

Schwarz Criterion | 70.55467 |

Figure 18.4 shows the frequency distribution of the three choice alternatives. In this example, mode 2 is most frequently chosen.

The MDC procedure computes nine goodness-of-fit measures for the discrete choice model. Seven of them are pseudo-R-square measures based on the null hypothesis that all coefficients except for an intercept term are zero (Figure 18.5). McFadden’s likelihood ratio index (LRI) is the smallest in value. For more details, see the section Model Fit and Goodness-of-Fit Statistics.

Figure 18.5: Likelihood Ratio Test and R-Square Measures

Goodness-of-Fit Measures | ||
---|---|---|

Measure | Value | Formula |

Likelihood Ratio (R) | 43.219 | 2 * (LogL - LogL0) |

Upper Bound of R (U) | 109.86 | - 2 * LogL0 |

Aldrich-Nelson | 0.4636 | R / (R+N) |

Cragg-Uhler 1 | 0.5787 | 1 - exp(-R/N) |

Cragg-Uhler 2 | 0.651 | (1-exp(-R/N)) / (1-exp(-U/N)) |

Estrella | 0.6666 | 1 - (1-R/U)^(U/N) |

Adjusted Estrella | 0.6442 | 1 - ((LogL-K)/LogL0)^(-2/N*LogL0) |

McFadden's LRI | 0.3934 | R / U |

Veall-Zimmermann | 0.6746 | (R * (U+N)) / (U * (R+N)) |

N = # of observations, K = # of regressors |

Finally, the parameter estimate is displayed in Figure 18.6.

The predicted choice probabilities are produced using the OUTPUT statement:

output out=probdata pred=p;

The parameter estimates can be used to forecast the choice probability of individuals that are not in the input data set.
To do so, you need to append to the input data set extra observations whose values of the dependent variable `decision`

are missing, since these extra observations are not supposed to be used in the estimation stage. The identification variable
`pid`

must have values that are not used in the existing observations. The output data set, `probdata`

, contains a new variable, `p`

, in addition to input variables in the data set, `extdata`

.

The following statements forecast the choice probability of individuals that are not in the input data set:

data extra; input pid mode decision ttime; datalines; 51 1 . 5.0 51 2 . 15.0 51 3 . 14.0 ; data extdata; set newdata extra; run;

proc mdc data=extdata; model decision = ttime / type=clogit covest=hess nchoice=3; id pid; output out=probdata pred=p; run;

proc print data=probdata( where=( pid >= 49 ) ); var mode decision p ttime; id pid; run;

The last nine observations from the forecast data set (`probdata`

) are displayed in Figure 18.7. It is expected that the decision maker will choose `mode`

"1" based on predicted probabilities for all modes.