Smoking status that was missing for Wave1 was imputed from later waves of the
panel where possible. There were 510 observations with incomplete data that were
omitted from the model (5.5% of the balanced panel). The number of complete cases
in the final multinomial logit model was 8,686.
The final multinomial logit model failed the test for independence of irrelevant
alternatives (IIA) (Small-Hsiao test, p<.001). Therefore we investigated models that
relaxed the IIA assumption using the “mdc” procedure in SAS V9.1. We chose a
multinomial probit model that assumes the error term εij for each alternative is
normally distributed, but allows error terms to be heteroskedastic and correlated
across alternatives. Two multinomial probit models were fitted and compared:
1. An approximation to multinomial logit with restrictions on the error terms to
be homoskedastic and uncorrelated across alternatives.
2. The unrestricted multinomial probit model that allowed the error terms to be
heteroskedastic and freely correlated across alternatives.
The fit of the unrestricted model was compared to the fit of the model with
homoskedastic variance and uncorrelated error terms, using the likelihood ratio test.
The final preferred model was used to simulate predicted probabilities for each
alternative for each respondent. A dataset was created with hypothetical observations
to observe the effect of changing levels of each explanatory variable on the estimated
probability of the alternative outcomes.
A series of index individuals were created, one for each outcome alternative, as a base
to examine the effects of each explanatory variable on the probability of that
particular outcome. The model coefficients were used to select levels of the
explanatory to create an individual with a high probability for a particular outcome.
The explanatory variables were then varied one level at a time to estimate their effects
on the probability of the alternative of interest, keeping all other variables at the level
of the index individual. Index individuals were created for the three alternatives of
most interest; purchasing hospital cover because of lifetime health cover, joining after
2000, and leaving after 2000.
The effects of age were estimated holding all other explanatory variables at the level
of the sample mean.
4. Results
The goodness of fit of the multinomial logit and multinomial probit models are
summarised in Table 3. The unrestricted multinomial probit fitted the data better than
the multinomial probit model with homoskedastic independent error terms (LR chisq
(64, 14) p < .0001). We therefore proceeded with the unrestricted multinomial probit
as the preferred model for the analysis.
TABLE 3 NEAR HERE
The characteristics of the three index individuals are summarised in Table 4 along
with their predicted probabilities for each choice alternative.