A Note on the Heterogeneous Choice Model

The heterogeneous choice model (HCM) has been proposed as an extension of the standard logit and probit models (Williams 2009). In this note, I show that in an important special case, this model is just another way to specify an interaction effect.

For developing the argument, I refer to a logit model. Let Y denote a binary variable. The logit model makes the distribution of Y dependent on values of explanatory variables:

P r Y = 1 | X = x , Z = z = L α + x β x + z β z ,

1

where X is a vector of explanatory variables with corresponding parameter vector β _x , Z is a further explanatory variable with parameter β _z , α is a constant, and L ( x ) = e x p ( x ) / ( 1 + e x p ( x ) ) is the standard logistic distribution function.

In equation (1), L is not used as a distribution function for a corresponding random variable but as a function for linking covariates to the distribution of Y, that is, a binomial distribution characterized by a single parameter. The HCM starts from a different model that relates to a latent variable, say Y*, which is connected with Y through Y = 1 ⇔ Y ∗ > 0 :

Y ∗ = α + x β x + z β z + ε .

2

Assuming that ε has a standard logistic distribution, equation (1) can be derived from equation (2). However, in model (2), there is now a newly created random variable, ε , and one can refer to its variance. While this variance cannot be estimated with data on Y, X, and Z, one nevertheless can set up a model that makes this variance dependent on some of the covariates. This is the HCM which, for example, can be specified as

Y ∗ = α + x β x + z β z + ε e x p z γ ,

3

where it is now assumed that the residual variance depends on the value of Z ( e x p ( . ) is used to guarantee a positive value). Dividing by e x p z γ , one gets the corresponding logit formulation:

P r Y = 1 | X = x , Z = z = L α / e x p z γ + x β x / e x p z γ + z β z / e x p z γ .

4

I now consider the special case where Z is a binary variable. Instead of equation (4), one can consider a logit model with an interaction term xz (being a vector of interaction terms if x is a vector):

P r Y = 1 | X = x , Z = z = L α ∗ + x β x ∗ + z β z ∗ + x z β x z ∗ .

5

Both models are equivalent, meaning that they entail the same conditional probabilities. This can be achieved by setting

α ∗ = α , β x ∗ = β x β z ∗ = α 1 / e x p γ − 1 + β z / e x p γ β x z ∗ = β x 1 / e x p γ − 1 .

6

In fact, these relationships are implicitly satisfied when estimating the models with maximum likelihood. They also entail that one cannot add the e x p ( z γ ) term to a logit model that already includes an interaction term.

How to interpret the equivalence of equations (4) and (5) depends on the intended analysis. I suppose that one is interested in investigating how a binary variable, Y, depends on another binary variable (representing, e.g., two groups). The interest concerns the dependence of the probabilities P r Y = 1 | X = x , Z = z on values of Z, or on values of X conditional on values of Z. One should then start from model (5), which includes a parameter, β x z ∗ , for a possible interaction between X and Z (in addition to interactions entailed by the nonlinearity of the logit link function).

Given a significant interaction parameter, one might ask whether the HCM suggests a different interpretation. The answer depends on the understanding of the HCM. I first assume that the HCM is taken as a model for the binary variable Y as formulated in equation (4). Then, if X and Z interact in model (5), the same is true for model (4). Of course, both models provide different parameter values, but this is just a consequence of a different parameterization of the same model. For the research interest mentioned previously, it is only important that both parameterizations entail identical conditional probabilities.

Now I assume that the HCM is taken as a model for the latent variable Y* as formulated in equation (3). Referring to the conditional expectation of Y*, that is

E ( Y ∗ | x , z ) = α + x β x + z β z ,

suggests the conclusion that the effect of X on Y* is independent of Z. However, the interest eventually concerns the effect of X on Y, not on Y*, and the effect of X on Y depends not only on the expectation of Y* but also on its variance. Consequently, if the variance of Y* depends on Z, as assumed by the HCM, one immediately gets an interaction between X and Z (as already shown by the equivalence of equations (4) and (5)).