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Multinomial Conditional Logistic Regression
Contents
Introduction
Multinomial conditional logistic regression
entails using the conditional logit (CL) model to
estimate a multinomial logistic (MNL) model. A
limitation of the MNL model is that it allows only one response
function (the type of restriction imposed on the dependent
variable) for all independent variables in the model. If more
flexibility is required in the specification of response functions,
then
a CL model can be used to estimate the MNL model.
The CL model can be used to estimate McFadden's choice model or matched
case-control data. In McFadden's choice model, variables characterizing
the choices (i.e. the categories of the dependent variable in the MNL
model) are included. With matching, cases are matched with respect to
certain characteristics. When the model does not include choice
characteristics or matched cases, the likelihood function of the CL
model is equivalent to that of the MNL model.
Under these circumstances, the CL model will produce the same
coefficients, standard errors and log likelihood values as the MNL
model. However, the CL model is much more flexible in allowing
restrictions on the choices (the dependent variable in MNL).
The CL model has the following characteristics:
- A dichotomous dependent variable
- chosen/not chosen is predicted by
choices and the explanatory variables
- Choices and explanatory variables
are independent variables
- if the explanatory variables affect all choices, then
the results are equivalent to an MNL model
- A stratification variable
- values of this variable indicate sets of choices
In the CL model, the main effects of the
choice variables correspond with the intercept term
in
an MNL model. Interactions between these choice
variables and the explanatory variables correspond with the effects
of these variables.
Procedure
To estimate an MNL model as a CL model, the following steps
must be taken for a dependent variable with J categories:
- Create a person-choice file
- create J dummy records for
each respondent
- a stratification variable contains
the original
case numbers
- a response factor associates each
dummy record with a response option
- the dichotomous dependent variable
indicates which dummy record corresponds with the
respondent's actual choice
- The main effects of the response factor
constitute the intercept term
- Interactions between the response factor
and the explanatory variables form the effects of
these
variables
This procedure allows the user to specify a response function
as required for each explanatory variable in the model by imposing
suitable restrictions on the effects of the response factor. If dummy
variables are created for the response factor using the highest
category
as reference category, then a standard MNL model can be obtained.
One use of this flexibility is to include a mobility
model in an MNL model (Logan 1983, Breen 1994). Mobility
models
have been developed for the loglinear analysis of square tables (Hout
1983). Their number of degrees of freedom lies in between an
independence model and a saturated model. To specify them as MNL
models,
a different response function is usually required for each category of
origin.
Extensions
The MCL approach can also be used to estimate certain models
with nonlinear constraints. These models contain both
linear and multiplicative terms. They can be estimated by iteratively
running CL models, treating first one part of the multiplicative term
as
given, then the other. Two models with nonlinear constraints are:
- Stereotyped Ordered Regression (SOR)
- estimates a scaling metric for the
response variable
- effects of covariates are specified with a single
parameter, scaled by this metric
- does not assume a priori ordered categories of the
dependent variable
- requires at least two, preferably more, covariates
- cf. Anderson (1984), DiPrete (1990)
- Row and Columns Model 2 (RC2)
- estimates a scaling metric for the
response variable and a categorical independent
variable
- effect of categorical independent is expressed through
a single parameter
- does not assume a priori ordered categories
- cf. Goodman (1979)
Software
The CL model can be estimated using programs for the Cox
proportional hazard model (event history) as well as programs
for condtional logit models. In the Cox model, a value of 1 (chosen)
for the dependent variable represents failure time, whereas a value of
2 (not
chosen) is treated as censored.
I have written macro programs to facilitate estimation for the
statistical packages STATA and SAS.
MCL models can also be estimated using SPSS, GLIM, and LIMDEP the
following programs, but macros for the SOR and RC2 models are not
available:
References
- Agresti, Alan. (1991).
- Categorical Data Analysis. New York: John Wiley
& Sons.
- Anderson, J.A. (1984).
- Regression and Ordered Categorical Variables.
Journal of the Royal Statistical Society, Series B 46: 1-30.
- Breen, Richard. (1994).
- Individual Level Models for Mobility Tables and Other
Cross-Classifications. Sociological Methods & Research
33: 147-173.
- DiPrete, Thomas A. (1990).
- Adding Covariates to Loglinear Models for the Study of
Social Mobility. American Sociological Review 55: 757-773.
- Goodman, Leo A. (1979).
- Multiplicative models for the analysis of occupational
mobility tables and other kinds of cross-classification tables.
American Journal of Sociology 84: 804-819.
- Hendrickx, John. (1995).
- Multinomial Conditional Logit Models for the Analysis
of Status Attainment and Mobility. ICS Working Papers - 1.
- Hendrickx, John, Ganzeboom, Harry B.G. (1998)
- Occupational Status Attainment in the Netherlands,
1920-1990. A Multinomial Logistic Analysis. European Sociological
Review 14: 387-403.
- Hout, Michael. (1983).
- Mobility Tables. Beverly Hills: Sage Publications.
- Logan, John A. (1983).
- A Multivariate Model for Mobility Tables. American
Journal of Sociology 89: 324-349.
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