For each health domain, we first fit an ordered probit (OPROBIT) model [
10] to estimate the effect of demographic and SES variables on health. Then, we refit the same specification with a CHOPIT model [
19] that generalises the OPROBIT by allowing cut points or thresholds to be different across individuals.
The CHOPIT model is comprised of two components: the self-assessment and the vignette rating component. In the self-assessment equation, we write the unobserved perceived level of health as:
$$\begin{aligned} y_{i}^{*}\sim N(\mu _{i},1) \end{aligned}$$
(1)
$$\begin{aligned} \mu _{i}= X_{i}\beta \end{aligned}$$
(2)
with subscript
i denotes individuals responding to SRH questionnaire. Individuals’ actual health level
\(\mu _{i}\) varies as a linear function of observed covariates
\(X_{i}\) with parameter vector
\(\beta\). Respondents then turn their perceived level of health
\(y_{i}^{*}\) into reported ordinal category
\(y_{i}\) via the following observation mechanism:
$$y_{i}=k\quad {\mathrm {if}}\; \tau _{i}^{k-1}<y_{i}^{*}<\tau _{i}^{k},\quad k=1,\ldots ,K$$
(3)
where
$$\begin{aligned} -\infty =\tau _{i}^{0}<\tau _{i}^{1}<\tau _{i}^{2}<\cdots <\tau _{i}^{K}=\infty \end{aligned}$$
(4)
To allow for individual-specific response category cut-point shift, thresholds
\(\tau _{i}\) are modelled as a linear function of observed covariates
\(X_{i}\) with parameter vector
\(\gamma\) and are identified in the model using information obtained from the vignette rating exercise.
$$\tau _{i}^{1}= X_{i}\gamma ^{1}$$
(5)
$$\tau _{i}^{k}= \tau _{i}^{k-1}+X_{i}\gamma ^{k},\quad {\mathrm {for}}\quad k=2,\ldots ,K$$
(6)
In the vignette rating equation, we write the perceived level of health of the person described in vignette
j evaluated by survey respondent
i as:
$$z_{ij}^{*}\sim N(\theta _{j},\sigma ^{2})$$
(7)
The actual health level of the person described in the vignette (
\(\theta _{j}\)) is assumed to be identical for every respondent, hence formalising the ‘vignette equivalence’ assumption. As in the self-assessment part of the model, respondents then turn the perceived level of health
\(z_{ij}^{*}\) into the same
K ordinal category via similar mechanism:
$$\begin{aligned} z_{ij}= & {} k\quad {\mathrm {if}}\;\tau _{ij}^{k-1}<z_{ij}^{*}<\tau _{ij}^{k},\quad k=1,\ldots ,K \end{aligned}$$
(8)
Thresholds in the vignette rating equation are determined by the same
\(\gamma\) parameter as in the self-assessment part, but note that the sample used in each model component need not be identical. The appearance of the same
\(\gamma\) parameter vector in both self-assessment and vignette rating components thus formalises the ‘response consistency’ assumption.
For identification and model comparability purposes, the standard ordered probit normalisation restriction (intercept is fixed at zero; variance is set to one) [
37] is imposed upon both OPROBIT and CHOPIT models. Then, formal tests of reporting homogeneity (
\(H_{0}{:}\;{\mathrm {all}}\;\gamma =0\)) and parallel cut-point shift (
\(H_{0}{:}\;\gamma ^{1}=\gamma ^{2}=\cdots =\gamma ^{K-1}\)) [
16] are performed after acquiring the estimate of the CHOPIT model, accompanied by graphical illustrations when necessary. To facilitate interpretation, we also compute the partial effect of relevant variables on the probability of reporting very good health [
16].
Only complete observations are used in the modelling exercise, yielding a sample size of 3069 individuals in the SRH equations (82 % of the original sample) and 939–1130 individuals in the vignette rating equations (75–90 % of the original sample).