We assessed endogeneity of smoking in 2
ways: testing the q coefficient obtained from
the maximum likelihood estimation and the
Wald statistics from the 2-step estimation.
The q coefficient represented the correlation
between the error terms of the 2 jointly
modeled regressions: the regression of average
number of cigarettes smoked on the instrumental
variables and the probit regression of
new-onset mental disorders on the predicted
smoking variable from the first model. A large
and statistically significant q coefficient and
a statistically significant Wald statistic suggest
that the estimator for the relationship of exposure
and outcome obtained from the naive
regression models would not be consistent
because of endogeneity of the exposure variable.
31 In this case, estimators from instrumental
variable probit analyses are more likely
to be consistent. A nonsignificant q coefficient
and Wald statistic would suggest that
exposure is not endogenous in the model
predicting the outcome and that the results of
the naive regression model without instrumental
variables are consistent and can be
interpreted