Gender Differences in Mental Well-Being: a Decomposition Analysis

Abstract

The General Health Questionnaire (GHQ) is frequently used as a measure of mental well-being. A consistent pattern across countries is that women report lower levels of mental well-being, as measured by the GHQ. This paper applies decomposition techniques to Irish data for 1994 and 2000 to examine the factors lying behind the gender differences in GHQ score. For both 1994 and 2000 about two-thirds of the raw difference is accounted for by differences in characteristics, with employment status the single most important factor.

Appendix: Decomposition Using Ordered Probit

An individual’s GHQ score is an ordered categorical variable. Thus, it may be appropriate to model GHQ using an ordered probit/logit approach. When using such an approach the straightforward Blinder–Oaxaca decomposition outlined in the main text is no longer applicable, as the conditional expectation $E(Y\left| X \right.)$is no longer equal to $\bar{X}\hat{\beta }$. Thus, an alternative way of carrying out the decomposition must be found. In what follows we adapt the procedure outlined in Bauer and Sinning (2006). For the general case of a non-linear decomposition we have the decomposition for the outcome for individual i, Y _i given by

$$ \Updelta_{m}^{\text{NL}} = [E_{{\beta_{m} }} (Y_{im} \left| {X_{im} ) - E_{{\beta_{m} }} (Y_{if} \left| {X_{if} } \right.)] + [E_{{\beta_{m} }} (Y_{if} \left| {X_{if} ) - E_{{\beta_{f} }} (Y_{if} \left| {X_{if} )]} \right.} \right.} \right. $$

where $E_{{\beta_{m} }} (Y_{im} \left| {X_{im} )} \right.$ is the conditional expectation of male outcomes and $E_{{\beta_{m} }} (Y_{if} \left| {X_{if} )} \right.$ is the conditional expectation of female outcomes evaluated with the male parameter vector, β _m . Alternatively, using females as the reference group the decomposition is

$$ \Updelta_{f}^{\text{NL}} = \left[ {E_{{\beta_{f} }} (Y_{im} \left| {X_{im} ) - E_{{\beta_{f} }} (Y_{if} \left| {X_{if} } \right.)] + [E_{{\beta_{m} }} (Y_{im} \left| {X_{im} ) - E_{{\beta_{f} }} (Y_{im} \left| {X_{im} )} \right.} \right.} \right.} \right]. $$

In both cases the first term on the right hand side provides that portion of the difference in conditional expectation arising from differences in characteristics, X _m , X _f and the second term refers to the difference arising from the “returns” to those characteristics, β _m , β _f . Thus, to apply this decomposition it is necessary to obtain the sample counterparts $S(\hat{\beta }_{m} X_{im} )$and $S(\hat{\beta }_{f} X_{im} )$of the conditional expectations, $E_{{\beta_{g} }} (Y_{ig} \left| {X_{ig} )} \right.$ and $E_{{\beta_{h} }} (Y_{ig} \left| {X_{ig} )} \right.$ where (g, h) = (m, f) and m ≠ f. We now apply this decomposition to the case of an ordered model.

An ordered model is based upon a latent regression of the form $Y_{im}^{*} = X_{im} \beta_{m} + \varepsilon_{im}$ where $Y_{im}^{*}$ is unobserved (we give the example here in terms of male outcomes). Instead we observe

$$ \begin{aligned} Y_{im} =\,& 0\quad {\text{if}}\,Y_{im}^{*} \le 0 \\ =\,& 1\quad {\text{if}}\,0 \le Y_{im}^{*} \le \mu_{1} \\ =\,& 2\quad {\text{if}}\,\mu_{1} \le Y_{im}^{*} \le \mu_{2} \\ \ldots \\ =\,& J\quad {\text{if}}\,\mu_{J - 1} \le Y_{im}^{*} . \\ \end{aligned} $$

where the μ _i , the “cut-off points”, are parameters to be estimated along with the vector β _m . The conditional expectation of Y _im evaluated at the parameter vector β _m is:

$$ \begin{array}{l} E_{{\beta_{m} }} (Y_{im} |X_{im} ) = F(\mu_{1} - X_{im} \beta_{m} ) - F( - X_{im} \beta_{m} ) \hfill \\ + 2[F(\mu_{2} - X_{im} \beta_{m} ) - F(\mu_{1} - X_{im} \beta_{m} )] \hfill \\ + \cdots \hfill \\ + J[1 - F(\mu_{J - 1} - X_{im} \beta_{m} )]. \hfill \\ \end{array} $$

If we assume that the error term, ɛ _im , is distributed normally we obtain the ordered probit model and F refers to the cumulative standard normal distribution (if we assume it is distributed logistically we obtain the ordered logit model and F refers to the cumulative logistic distribution).

Given estimation of the parameter vector β _im , the sample counterparts of the components of the decomposition (assuming males to be the reference group) are calculated as follows:

$$\begin{array}{l} S(\hat{\beta }_{m} X_{{im}} ) = N^{{ - 1}} \sum\limits_{{i = 1}}^{N} {\left\{ {[F(\hat{\mu }_{1} - X_{{im}} \hat{\beta }_{m} ) - F( - X_{{im}} \hat{\beta }_{m} )]} \right.} \hfill \\ + 2[F(\hat{\mu }_{2} - X_{{im}} \hat{\beta }_{m} ) - F(\hat{\mu }_{1} - X_{{im}} \hat{\beta }_{m} )] \hfill \\ + \cdots \hfill \\ + \left. {J[1 - F(\hat{\mu }_{{J - 1}} - X_{{im}} \hat{\beta }_{{im}} )]} \right\} \hfill \\ \end{array}$$

The sample counterpart of $E_{{\beta_{m} }} (Y_{if} \left| {X_{if} )} \right.$, $S(\hat{\beta }_{m} X_{if} )$ is obtained by replacing X _im by X _if in the above equation. The sample counterparts are then used to obtain the parts of the decomposition:

$$ \hat{\Updelta } = [S(\hat{\beta }_{m} X_{im} ) - S(\hat{\beta }_{m} X_{if} )] + [S(\hat{\beta }_{m} X_{if} ) - S(\hat{\beta }_{f} X_{if} )]. $$

The case where females are the reference group is the mirror image of above, while the decomposition for the Neumark approach of using the estimates of the pooled sample as the reference group gives:

$$ \hat{\Updelta } = [S(\hat{\beta }_{m} X_{im} ) - S(\hat{\beta }^{*} X_{im} )] + [S(\hat{\beta }^{*} X_{if} ) - S(\hat{\beta }_{f} X_{if} )] + [S(\hat{\beta }^{*} X_{im} - S(\hat{\beta }^{*} X_{if} )]. $$

Table 7 provides the counterpart to Table 3 and gives the results for the ordered probit models for the pooled sample of men and women for 1994 and 2000. As in the case of the linear regression, the gender coefficients are statistically significant. While the interpretation of coefficients for the ordered probit is obviously different from the linear regression case it is noticeable that the pattern of coefficients by sign and significance levels by variable is very similar.

Table 7 Ordered probit regression of GHQ, 1994 and 2000

Table 8 provides the counterpart to Table 4 with the regressions by gender and the pooled regressions (not including the gender variable). Once again the pattern by sign and significance level is very similar, and while not every coefficient takes the same sign as its counterpart in the linear regressions in Table 4, the vast majority do, and in those cases where the sign is different, the coefficient is typically not statistically significant. This gives confidence to our belief that the qualitative results obtained are not sensitive to the choice of OLS or ordered probit.

Table 8 Ordered probit regression of GHQ by gender, 1994 and 2000