A collection of marginalized two-part random-effects models for analyzing medical expenditure panel data: Impact of the New Cooperative Medical Scheme on healthcare expenditures in China

Abstract

Marginalized two-part random-effects generalized Gamma models have been proposed for analyzing medical expenditure panel data with excessive zeros. While these models provide marginal inference on expected healthcare expenditures, the usual unilateral specification of heteroscedastic variance on one of the two shape parameters for the generalized Gamma distribution in these models fails to encompass important special cases within the generalized gamma modeling framework. In this article, we construct marginalized two-part random-effects models that employ the log-normal, log-skew-normal, generalized Gamma, Weibull, Gamma, and inverse Gamma distributions to delineate the spectrum of nonzero healthcare expenditures in the second part of the models. These marginalized models supply additional choices for analyzing healthcare expenditure panel data with excessive zeros. We review the concepts of marginal effect and incremental effect, and summarize how these effects are estimated. For studies whose primary goal is to make inference on marginal effect or incremental effect of an independent variable with respect to healthcare expenditures, we advocate empirical mean square error criterion and information criteria to choose among candidate models. Then, we use the proposed models in an empirical analysis to examine the impact of the New Cooperative Medical Scheme on healthcare expenditures among older adults in rural China.

Keywords

Marginal effects panel data random-effects models healthcare expenditures two-part models marginalized models

1 Introduction

Non-negative semicontinuous healthcare expenditure data^1–3 collected in health economics and health services research studies are right-skewed with a heavy right tail, heteroscedastic in variance with respect to some covariates, and contain excessive zeros observed from the non-users of health facilities. In addition, the data collected in these studies are often panel data, because the studies are designed to repeatedly collect healthcare expenditures from the same cross section of individuals over time. A number of sophisticated statistical models have been proposed in the literature to analyze this type of expenditure data.

Conditional two-part random-effects models that incorporate latent random effects into the analysis of healthcare expenditure panel data were first reported by Olsen and Schafer⁴ and Tooze et al.⁵ Olsen and Schafer⁴ and Tooze et al.⁵ extended the two-part regression models for cross-sectional data to the settings of panel data by including unobserved random coefficients into both parts of the models. Conditional two-part random-effects models are those two-part models with a random-effects logistic or probit model for the probability of observing a panel positive expenditure in Part (I) and a generalized linear random-effects model for the observed panel positive healthcare expenditures in Part (II). The essential assumption on these models is that the random effects from Part (I) and the random effects from Part (II) are correlated. As such, the probability of observing a positive expenditure in Part (I) is correlated to the amount of the positive expenditure at a specific time point. However, Olsen and Schafer⁴ and Tooze et al.⁵ both adhered to a log-normal regression model in Part (II) of the proposed two-part models. Later, Manning et al.⁶ showed that it is more efficient to use the three-parameter generalized Gamma distribution to model the positive healthcare expenditures. The generalized Gamma distribution has been proven superior to the log-normal distribution in Olsen and Schafer⁴ and Tooze et al.,⁵ because it includes the standard Gamma, inverse Gamma, Weibull, exponential, and lognormal distributions as its special cases and consequently can provide much more flexibility in the circumstances where the above distributions cannot adequately characterize the positive expenditures. Manning et al.⁶ also extended the traditional generalized Gamma regression model, so that it allows for the existence of substantial heteroscedasticity among the expenditures. Following Manning et al.,⁶ Liu et al.⁷ and Liu et al.⁸ constructed several conditional two-part random-effects models for healthcare panel expenditures, in which the lognormal random-effects model developed by Olsen and Schafer⁴ and Tooze et al.⁵ in Part (II) of the models was extended to be a log-skew-normal or a generalized Gamma random-effects model with a regression model for a scale parameter that allowed for heteroscedastic variance. Conditional two-part random-effects models in Liu et al.⁷ and Liu et al.⁸ have an inherited drawback that it is inconvenient to make direct marginal inference with respect to the positive or overall panel healthcare expenditures. To overcome this drawback, Zhang et al.⁹ developed two types of marginalized two-part random-effects generalized Gamma models (abbreviated as “generalized Gamma m2REMs”; “m2REMs” stands for “marginalized two-part random-effects models”): the Type I generalized Gamma m2REM was constructed for marginal inference on positive healthcare expenditures and the Type II generalized Gamma m2REM was for marginal inference on overall healthcare expenditures.

This article is devoted to developing the marginalized two-part random-effects models, in which the specification of generalized Gamma distribution in Part (II) is replaced by the specification of log-normal, log-skew-normal, Gamma, and inverse Gamma distribution. The motivation to further develop these models is that, because of the particular setup in the heteroscedastic variance model for one of the two shape parameters in the generalized Gamma m2REMs in Zhang et al.,⁹ the generalized Gamma m2REMs do not include the m2REMs with other distribution specification as their special cases anymore. The generalized Gamma distribution has been reviewed in Zhang et al.⁹ Note that, in the generalized Gamma distribution, there are two shape parameters σ and κ that dominate the variance function of the distribution. In Zhang et al.,⁹ the heteroscedastic variance among positive panel healthcare expenditures is characterized by one of the shape parameters through $log (σ_{it} | Y_{it} > 0, X_{it}) = X_{it}' δ$ whereas the other parameter κ was left constant. In such circumstances, the condition $σ = κ$ or $σ = - κ$ will never be satisfied in the generalized Gamma m2REMs, unless the heteroscedastic variance is omitted. However, these conditions are the necessary conditions for the generalized gamma distribution to degenerate to the Gamma and inverse Gamma distributions, respectively. Thus, it would be beneficial to provide additional choices, the Gamma m2REMs and the inverse Gamma m2REMs, for empirical analysis as the parallel competitors of the generalized Gamma m2REMs. In fact, the conditional, not marginalized, two-part random-effects Gamma models have been employed by Qin and Pan¹⁰ in the empirical analysis in health economics, in which the instrumental variable approach was combined with these conditional models to detect the casual effect of overweight on individual expected medical expenditure. In this article, we proposed the marginalized version of the conditional two-part random-effects Gamma models for the future use when a marginal inference is demanded. Moreover, in the generalized Gamma m2REMs, the shape parameter κ was kept positive. Yet, a necessary condition for the generalized Gamma model to shrink to a log-normal model is that κ goes to zero. Therefore, we proposed the m2REMs with a specification of log-normal distribution in Part (II) as a useful alternative that structurally was not included in the generalized Gamma m2REMs. We also propose the log-skew-normal m2REMs, which will never be a special case of the generalized Gamma m2REMs, and include the Weibull m2REMs, which is a special case of the generalized Gamma models. All the proposed m2REMs are constructed in the format of both Type I and Type II models as in Zhang et al.,⁹ and therefore can achieve the goal of making inference not only on the overall healthcare expenditures but on the positive healthcare expenditures as well.

The scientific contributions of this article are multi-fold. Besides the development of the new m2REMs in Section 2 as the alternatives of the generalized Gamma m2REMs, we also describe in Section 4 how to derive estimates and variance estimates for marginal and incremental effects of a covariate with respect to both overall and positive healthcare panel expenditures within the framework of the m2REMs. Given these multiple choices for modeling healthcare expenditure panel data, we propose an approach to select the best model based upon the observed data; that is, an effect-specific selection approach introduced by Dow and Norton¹¹ is modified in Section 5 to make it suitable to selecting among the m2REMs. The other contribution of this article is our empirical analysis and re-examination on the impact of the New Cooperative Medical Scheme (NCMS)^12,13 on individual healthcare expenditures in China. The New Cooperative Medical Scheme (NCMS) is a health insurance program that was launched in 2003 by the Chinese central government for the rural residents living in China (Chinese citizens are separated into either rural or urban residents by their residency registration). It was reported that the enrollment of the NCMS rose to 97% of rural population in 2011.^14,15 Lei and Lin¹² explored the influence of the NCMS on healthcare expenditures among rural residents in China, and did not find that the NCMS decreases individual healthcare expenditures. Here, we investigate the impact of the NCMS on a subset of rural residents who are older than 60 years old in China. A set of panel expenditure data among the elders is extracted from a carefully designed U.S.–China collaborative large-scale survey, the China Health and Nutrition Survey. The data are analyzed using the proposed m2REMs in Section 2. The results show that, even for the elders who lived in rural areas in China, the NCMS is not a significant factor that can decrease their healthcare expenditures.

2 Modeling framework

2.1 Marginalized two-part random-effects log-normal (log-skew-normal) models

Let Y_it be a semicontinuous dependent variable, representing the amount of healthcare expenditure of the ith individual ( $i = 1, 2, \dots, N$ ) measured at time t ( $t = 1, 2, \dots, T$ ). Let X_it be a K-dimensional vector of dependent variables. Type I marginalized two-part random-effects log-normal model, abbreviated as Type I log-normal m2REM, is constructed as follows. In Part (I), a marginal model for the probability of observing a positive healthcare expenditure (i.e. the probability of observing a use of healthcare facilities or, equivalently, the probability of occurrence of healthcare utilization) from the ith individual at time t is

ℎ_{m_{1}} [P (Y_{it} > 0 | X_{it})] = X_{it}' α_{m}

(1)

in which

ℎ_{m_{1}} (\cdot)

denotes the link function, such as a logit link or a probit link, and α_m is a vector of marginal regression coefficients. In addition to equation (1), a conditional model that characterizes the dependence among the probabilities of positive healthcare expenditures is further specified. It is assumed that

Y_{i 1}, \dots, Y_{iT}

are independent conditioning on

X_{i 1}, \dots, X_{iT}

and the vector of unobserved random effects

a_{i} = (a_{i 1}, \dots, a_{iT})'

. The conditional model for the probability of observing a positive expenditure in Y_it is formulated as

ℎ_{c_{1}} [P (Y_{it} > 0 | X_{it}, a_{it})] = Δ_{it} + a_{it}

(2)

In equation (2), $ℎ_{c_{1}} (\cdot)$ denotes the link function that can be in general different with, but in practice usually chosen to be identical to $ℎ_{m_{1}} (\cdot)$ in equation (1), a_it is a latent random effect, and Δ _it is a function of marginal mean predictor $X_{it}' α_{m}$ and the random effect variance $var (a_{it})$ . In equation (2), a single scalar random effect a_it is used as a general representation. It includes several common models as its special cases, such as the random-intercept model $a_{it} = a_{i 0}$ for $t = 1, 2, \dots, T$ and the mixed-effects model $a_{it} = Z_{it}' a_{i}^{*}$ , in which $a_{i}^{*}$ is a q-dimensional random effect vector and the vector Z_it includes the covariates pertaining to the random effects. It is assumed that the latent random vector a_i follows a normal distribution $a_{i} \sim N (0, A_{i} (θ_{A}))$ , in which θ_A is the vector of variance–covariance parameters. Equations (1) and (2) together constitute the marginally specified model^16,17 for the probabilities of occurrence of healthcare utilization. Equation (1) captures the systematic variation in the mean that is due to X_it, and equation (2) provides measures of random variation both across individuals and over time. Therefore, the marginalized models (1) and (2) separate the model for systematic variation from the model for random variation. Let $f_{a_{it}} (\cdot)$ denote the distribution of a_it, $φ (\cdot)$ denote the probability density function of a standard normal distribution, and $Φ (\cdot)$ denote the cumulative density function of a standard normal distribution. From equation (1), $P (Y_{i t} > 0 | X_{i t}) = ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m})$ , and equation (2) implies that $P (Y_{i t} > 0 | X_{i t}) = \int P (Y_{i t} > 0 | X_{i t}, a_{i t}) f_{a_{i t}} (a_{i t}) d a_{i t} = \int ℏ_{c_{1}}^{- 1} [Δ_{i t} + var {(a_{i t})}^{1 / 2} s] ϕ (s) d s$ . Thus, Δ _it can be obtained as the solution to the convolution equation

ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m}) = \int ℏ_{c_{1}}^{- 1} [Δ_{i t} + var {(a_{i t})}^{1 / 2} s] ϕ (s) d s

This can be solved for Δ _it using numerical integration combined with the Newton–Raphson or quasi-Newton algorithm. A close-form solution may exist for Δ _it in some special cases: when $ℏ_{m_{1}}^{- 1} (\cdot) = ℏ_{c_{1}}^{- 1} (\cdot) = Φ (\cdot)$ , it can be derived that $Δ_{it} = [1 + var (a_{it})]^{1 / 2} X_{it}' α_{m}$ .

In Part (II) of the Type I log-normal m2REM, a marginally specified model for the mean of the positive healthcare expenditure is formulated as

ℎ_{m_{2}} [E (Y_{it} | Y_{it} > 0, X_{it})] = X_{it}' β_{m}

(3)

in which

ℎ_{m_{2}} (\cdot)

is a link function and β_m is the vector of marginal regression coefficients for positive healthcare expenditures. The rationale of equation (3) is to provide direct marginal inference on the average amount of positive healthcare expenditures while using the two-part models for healthcare expenditure panel data. To characterize the dependence among repeated measurements on positive expenditures, we assume that, conditioning on covariates X_it, random effect b_it and Y_it > 0, Y_it follows a log-normal distribution with a density of

f_{LN} (y; μ_{it}, σ_{it})

. For a variable Y, if the transformed value

g (Y) = log (Y)

follows a normal distribution

N (μ, σ^{2})

, then Y follows a log-normal distribution

f_{LN} (y; μ, σ)

or, equivalently,

Y \sim f_{LN} (y; μ, σ)

. Here, we specify

(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{LN} (y; μ_{it}, σ_{it})

and

(μ_{it} | Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} + b_{it}

(4)

in which b_it is a random effect that can be similarly formulated as a_it in equation (2), and the parameter Λ _it is a function of marginal mean predictor

X_{it}' β_{m}

, the random effect variance

var (b_{it})

, and variance parameters σ_it. To characterize heteroscedastic variance of the positive healthcare expenditures, it is assumed that

log (σ_{it} | Y_{it} > 0, X_{it}) = X_{it}' δ

(5)

with a regression coefficient vector δ. As in equation (2), the latent random vector b_i follows a normal distribution

b_{i} \sim N (0, B_{i} (θ_{B}))

, in which θ_B is the vector of variance–covariance parameters. From equation (3),

E (Y_{i t} | Y_{i t} > 0, X_{i t}) = ℏ_{m_{2}}^{- 1} (X_{i t}' β_{m})

. Since

(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{LN} (y; μ_{it}, σ_{it})

, then

E (Y_{it} | Y_{it} > 0, X_{it}, b_{it}) = exp (μ_{it} + σ_{it}^{2} / 2)

, which together with equations (4) and (5) implies that

\begin{matrix} E (Y_{it} | Y_{it} > 0, X_{it}) = \int E (Y_{it} | Y_{it} > 0, X_{it}, b_{it}) f_{b_{it}} (b_{it}) d b_{it} \\ = \int exp [Λ_{it} + var (b_{it})^{1 / 2} s + \frac{exp (2 X_{it}' δ)}{2}] φ (s) d s \\ = exp [Λ_{it} + \frac{exp (2 X_{it}' δ) + var (b_{it})}{2}] \end{matrix}

(6)

Therefore, equations (3) to (5) lead to

Λ_{i t} = \log [ℏ_{m_{2}}^{- 1} (X_{i t}' β_{m})] - \frac{\exp (2 X_{i t}' δ) + var (b_{i t})}{2}

(7)

Equations (1) to (5) together constitute the Type I log-normal m2REM. Here, the random effects a_it in equation (2) and b_it in (4) are assumed to be correlated and follow a multivariate normal distribution so that the two parts are correlated; that is, we assume generally $(a_{it}, b_{it})' \sim N_{2} (0, Σ (θ_{A}, θ_{B}, θ_{AB}))$ , in which θ_AB is a vector that contains the additional unknown parameters in $cov (a_{i}, b_{i})$ other than the ones in θ_A and θ_B. If the correlation coefficients in $cov (a_{i}, b_{i})$ are zero, then the two parts of the model are separated.

The Type I log-normal m2REM aims at providing direct marginal inference with respect to the marginal mean of positive healthcare expenditures over time, but not with respect to the marginal mean of overall healthcare expenditures that includes both the zero expenditures from the non-users of healthcare facilities and the positive expenditures from the users. To achieve the goal of making direct marginal inference on overall healthcare expenditures, we can use the following marginally specified model (8) to replace equation (3) to construct the Type II marginalized two-part random-effects log-normal model (abbreviated as Type II log-normal m2REM). In the Type II log-normal m2REMs, the positive expenditures in Part (II) follow a heteroscedastic log-normal distribution

ℎ_{m_{3}} [E (Y_{it} | X_{it})] = X_{it}' β_{m}

(8)

In equation (8), $ℎ_{m_{3}} (\cdot)$ is a link function and β_m is the vector of marginal regression coefficients for overall healthcare expenditures. Equations (4) and (8) together constitute a marginally specified model for the marginal mean of overall healthcare expenditures over time. With the assumption that the random effects a_it in equation (2) and b_it in equation (4) jointly follow a multivariate normal distribution, equations (1), (2), (4), (5) and (8) together constitute a Type II log-normal m2REM. It emphasizes on the inference of marginal effects of independent variables on the overall mean of panel healthcare expenditures. Due to the replacement of equation (3) by equation (8), the format of Λ _it is altered. From equation (8), $Λ_{i t} = \log [\frac{ℏ_{m_{3}}^{- 1} (X_{i t}' β_{m})}{ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m})}] + \log [1 - Φ^{2} (X'_{i t} δ)] - \frac{var (b_{i t})}{2}$ . It can be derived from equations (1) and (6) that

\begin{matrix} E (Y_{i t} | X_{i t}) = P (Y_{i t} > 0 | X_{i t}) E (Y_{i t} | Y_{i t} > 0, X_{i t}) \\ = ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m}) \exp [Λ_{i t} + \frac{\exp (2 X_{i t}' δ) + var (b_{i t})}{2}] \end{matrix}

(9)

Therefore, in the Type II log-normal m2REM, Λ _it in equation (4) has a form of

Λ_{i t} = \log [\frac{ℏ_{m_{3}}^{- 1} (X_{i t}' β_{m})}{ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m})}] - \frac{\exp (2 X_{i t}' δ) + var (b_{i t})}{2}

(10)

It shows that, in the Type II log-normal m2REM, Λ _it is a function of marginal mean predictor

X_{it}' β_{m}

X_{it}' α_{m}

, and

X_{it}' δ

, and the random effect variance

var (b_{it})

. Model specification of the Type I and Type II log-normal m2REMs is summarized in Table 1.

Table 1.

Model specification of Type I and Type II m2REMs.

The above Type I and Type II log-normal m2REMs can be extended to Type I log-skew-normal m2REM and Type II log-skew-normal m2REM, in which the positive expenditures in Part (II) follow a heteroscedastic log-skew-normal distribution. These models can be constructed by using a log-skew-normal regression model to replace the log-normal regression model in Part (II) of the Type I and Type II log-normal m2REMs. For a variable Y, if Y follows a log-skew-normal distribution, then $Y \sim f_{LSN} (y; μ, σ, κ)$ and

f_{LSN} (y; μ, σ, κ) = \frac{1}{y} \frac{2}{\sqrt{σ^{2} + κ^{2}}} φ (\frac{log y - μ}{\sqrt{σ^{2} + κ^{2}}}) Φ (\frac{κ}{σ} \frac{log y - μ}{\sqrt{σ^{2} + κ^{2}}})

where μ, σ, and κ are the location, scale, and skewness parameters of the distribution, respectively. When

Y \sim f_{LSN} (y; μ, σ, κ), log (Y)

follows a skew-normal distribution.¹⁸ If κ = 0, the log-skew-normal distribution reduces to the log-normal distribution. To construct the Type I log-skew-normal m2REM, we take equations (1) to (5), but assume

(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{LSN} (y; μ_{it}, σ_{it}, κ)

for equations (4) and (5). When

(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{LSN} (y; μ_{it}, σ_{it}, κ)

E (Y_{it} | Y_{it} > 0, X_{it}, b_{it}) = 2 exp [Λ_{it} + b_{it} + \frac{exp (2 X_{it}' δ) + κ^{2}}{2}] Φ (κ)

. Therefore

\begin{matrix} E (Y_{it} | Y_{it} > 0, X_{it}) = \int E (Y_{it} | Y_{it} > 0, X_{it}, b_{it}) f_{b_{it}} (b_{it}) d b_{it} \\ = \int 2 exp [Λ_{it} + var (b_{it})^{1 / 2} s + \frac{exp (2 X_{it}' δ) + κ^{2}}{2}] Φ (κ) φ (s) d s \\ = 2 exp [Λ_{it} + \frac{exp (2 X_{it}' δ) + κ^{2} + var (b_{it})}{2}] Φ (κ) \end{matrix}

(11)

Combined with $E (Y_{i t} | Y_{i t} > 0, X_{i t}) = ℏ_{m_{2}}^{- 1} (X_{i t}' β_{m})$ in equation (3), we conclude that, in the Type I log-skew-normal m2REM

Λ_{i t} = \log [ℏ_{m_{2}}^{- 1} (X_{i t}' β_{m})] - \log [2 Φ (κ)] - \frac{\exp (2 X_{i t}' δ) + κ^{2} + var (b_{i t})}{2}

To construct the Type II log-skew-normal m2REM, we take equations (1), (2), (4), (5) and (8), but assume $(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{LSN} (y; μ_{it}, σ_{it}, κ)$ for equations (4) and (5). It can be derived from equations (1) and (11) that

\begin{matrix} E (Y_{i t} | X_{i t}) = P (Y_{i t} > 0 | X_{i t}) E (Y_{i t} | Y_{i t} > 0, X_{i t}) \\ = 2 ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m}) \exp [Λ_{i t} + \frac{\exp (2 X_{i t}' δ) + κ^{2} + var (b_{i t})}{2}] Φ (κ) \end{matrix}

(12)

Combined with $Λ_{i t} = \log [\frac{ℏ_{m_{3}}^{- 1} (X_{i t}' β_{m})}{ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m})}] + \log [1 - Φ^{2} (X'_{i t} δ)] - \frac{var (b_{i t})}{2}$ from equation (8), we have that, in the Type II log-skew-normal m2REM

Λ_{i t} = \log [\frac{ℏ_{m_{3}}^{- 1} (X_{i t}' β_{m})}{ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m})}] - \log [2 Φ (κ)] - \frac{\exp (2 X_{i t}' δ) + κ^{2} + var (b_{i t})}{2}

Model specification of the Type I and Type II log-skew-normal m2REMs is also summarized in Table 1.

2.2 Marginalized two-part random-effects generalized Gamma (Weibull) models

The generalized Gamma distribution is a continuous probability distribution with one scale parameter and two shape parameters.⁶ For a non-negative response variable Y, if Y follows a generalized Gamma distribution with scale parameter μ and shape parameters σ and κ, then $Y \sim f_{GG} (y; μ, σ, κ)$ and

f_{GG} (y; μ, σ, κ) = \frac{γ^{γ}}{σ y \sqrt{γ} Γ (γ)} exp (z \sqrt{γ} - u)

(13)

in which

γ = | κ |^{- 2}, z = sign (κ) {log (y) - μ} / σ

, and

u = γ exp (| κ | z)

. If

Y \sim f_{GG} (y; μ, σ, κ)

, then

E (Y) = \frac{e^{μ} κ^{2 σ / κ} Γ (1 / κ^{2} + σ / κ)}{Γ (1 / κ^{2})}

(14)

The generalized Gamma distribution is very flexible. It includes Gamma, inverse Gamma, Weibull, exponetial, and log-normal distributions as its special cases (see Table 1 in Manning et al.⁶ for details). The Type I and Type II marginalized two-part random-effects models with the positive healthcare expenditures in Part (II) following a heteroscedastic generalized Gamma distribution (abbreviated as Type I and Type II generalized Gamma m2REMs) were introduced by Zhang et al.⁹ Part (I) of the Type I generalized Gamma m2REM is constructed by equations (1) and (2) as in the log-normal m2REMs in Section 2.1, but Part (II) of the model takes equation (3) and assumes $(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{GG} (y_{it}; μ_{it}, σ_{it}, κ)$ for equations (4) and (5). Part (I) of the Type II generalized Gamma m2REM is also constructed by equations (1) and (2), but Part (II) of the Type II generalized Gamma m2REM takes equation (8) and assumes $(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{GG} (y_{it}; μ_{it}, σ_{it}, κ)$ for equations (4) and (5). Model specification of the Type I and Type II generalized Gamma m2REMs is summarized in Table 1. For the Type I generalized Gamma m2REM, it can be derived from $(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{GG} (y_{it}; μ_{it}, σ_{it}, κ)$ and equations (4), (5), and (14) that

\begin{matrix} E (Y_{it} | Y_{it} > 0, X_{it}) = \int E (Y_{it} | Y_{it} > 0, X_{it}, b_{it}) f_{b_{it}} (b_{it}) d b_{it} \\ = \int exp [Λ_{it} + var (b_{it})^{1 / 2} s] \frac{κ^{2 exp (X_{it}' δ_{m}) / κ} Γ (1 / κ^{2} + exp (X_{it}' δ_{m}) / κ)}{Γ (1 / κ^{2})} φ (s) d s \\ = \frac{κ^{2 exp (X_{it}' δ_{m}) / κ} Γ (1 / κ^{2} + exp (X_{it}' δ_{m}) / κ)}{Γ (1 / κ^{2})} exp [Λ_{it} + \frac{var (b_{it})}{2}] \end{matrix}

(15)

This, combined with $E (Y_{i t} | Y_{i t} > 0, X_{i t}) = ℏ_{m_{2}}^{- 1} (X_{i t}' β_{m})$ in equation (3), implies that in the Type I generalized Gamma m2REM

Λ_{i t} = \log [\frac{ℏ_{m_{2}}^{- 1} (X_{i t}' β_{m}) Γ (1 / κ^{2})}{κ^{2 \exp (X_{i t}' δ_{m}) / κ} Γ (1 / κ^{2} + \exp (X_{i t}' δ_{m}) / κ)}] - \frac{var (b_{i t})}{2}

For the Type II generalized Gamma m2REM, it can be derived from $(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{GG} (y_{it}; μ_{it}, σ_{it}, κ)$ and equations (1) and (15) that

\begin{matrix} E (Y_{i t} | X_{i t}) = P (Y_{i t} > 0 | X_{i t}) E (Y_{i t} | Y_{i t} > 0, X_{i t}) \\ = ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m}) \frac{κ^{2 \exp (X_{i t}' δ_{m}) / κ} Γ (1 / κ^{2} + \exp (X_{i t}' δ_{m}) / κ)}{Γ (1 / κ^{2})} \exp [Λ_{i t} + \frac{var (b_{i t})}{2}] \end{matrix}

This, combined with $Λ_{i t} = \log [\frac{ℏ_{m_{3}}^{- 1} (X_{i t}' β_{m})}{ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m})}] + \log [1 - Φ^{2} (X'_{i t} δ)] - \frac{var (b_{i t})}{2}$ from equation (8), implies that in the Type II generalized Gamma m2REM

Λ_{i t} = \log [\frac{ℏ_{m_{3}}^{- 1} (X_{i t}' β_{m}) Γ (1 / κ^{2})}{ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m}) κ^{2 \exp (X_{i t}' δ_{m}) / κ} Γ (1 / κ^{2} + \exp (X_{i t}' δ_{m}) / κ)}] - \frac{var (b_{i t})}{2}

Weibull distribution is a special case of the generalized Gamma distribution when κ = 1. Therefore, by fixing κ = 1, the Type I and Type II generalized Gamma m2REMs degenerates to Type I and Type II Weibull m2REMs, respectively. Model specification of the Type I and Type II generalized Gamma and Weibull m2REMs is also summarized in Table 1.

2.3 Marginalized two-part random-effects (inverse) Gamma models

The generalized Gamma distribution (13) includes Gamma and inverse Gamma distributions as its special case. When $σ = κ$ , equation (13) degenerates to the Gamma distribution with a density of

f_{G} (y; μ, σ) = \frac{1}{σ^{2 / σ^{2}} y Γ (1 / σ^{2})} exp [\frac{log (y) - μ - y exp (- μ)}{σ^{2}}]

When $σ = - κ$ , equation (13) degenerates to the inverse Gamma distribution with a density of

f_{IG} (y; μ, σ) = \frac{1}{σ^{2 σ^{- 2}} y Γ (1 / σ^{2})} exp [\frac{μ - log (y) - exp (μ) / y}{σ^{2}}]

However, it should be noted that the Type I and Type II generalized Gamma m2REMs do not cover the Type I and Type II Gamma or inverse Gamma m2REMs, in which the positive healthcare expenditures in Part (II) follow a heteroscedastic Gamma or inverse Gamma distribution. The reason is that, in the Type I and Type II generalized Gamma m2REMs, the shape parameter κ or its negative number $- κ$ is a constant and therefore cannot be equal to $(σ_{it} | Y_{it} > 0, X_{it}) = exp (X_{it}' δ)$ in equation (5), unless the heteroscedasticity is given up. This motivates us to additionally construct the Type I and Type II Gamma and inverse Gamma m2REMs that can maintain the setup of heteroscedastic variance with the specification of Gamma or inverse Gamma distribution for the positive healthcare expenditures in Part (II).

Part (I) of the Type I Gamma m2REM is constructed by equations (1) and (2) as in the log-normal m2REMs in Section 2.1, but Part (II) of the model takes equation (3) and assumes $(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{G} (y_{it}; μ_{it}, σ_{it})$ for equations (4) and (5). It can be derived from $(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{G} (y_{it}; μ_{it}, σ_{it}), E (Y_{it} | Y_{it} > 0, X_{it}, b_{it}) = exp (μ_{it})$ , equations (4) and (5), that

\begin{matrix} E (Y_{it} | Y_{it} > 0, X_{it}) = \int E (Y_{it} | Y_{it} > 0, X_{it}, b_{it}) f_{b_{it}} (b_{it}) d b_{it} \\ = \int exp [Λ_{it} + var (b_{it})^{1 / 2} s] φ (s) d s \\ = exp [Λ_{it} + \frac{var (b_{it})}{2}] \end{matrix}

(16)

This, combined with $E (Y_{i t} | Y_{i t} > 0, X_{i t}) = ℏ_{m_{2}}^{- 1} (X_{i t}' β_{m})$ in equation (3), implies that in the Type I Gamma m2REM

Λ_{i t} = \log [ℏ_{m_{2}}^{- 1} (X_{i t}' β_{m})] - \frac{var (b_{i t})}{2}

Part (I) of the Type II Gamma m2REM is constructed by equations (1) and (2), but Part (II) of the model takes equation (8) and assumes $(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{G} (y_{it}; μ_{it}, σ_{it})$ for equations (4) and (5). It can be derived from $(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{G} (y_{it}; μ_{it}, σ_{it})$ , and equations (1), (4), (5), and (16) that

\begin{matrix} E (Y_{i t} | X_{i t}) = P (Y_{i t} > 0 | X_{i t}) E (Y_{i t} | Y_{i t} > 0, X_{i t}) \\ = ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m}) \exp [Λ_{i t} + \frac{var (b_{i t})}{2}] \end{matrix}

This, combined with $Λ_{i t} = \log [\frac{ℏ_{m_{3}}^{- 1} (X_{i t}' β_{m})}{ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m})}] + \log [1 - Φ^{2} (X'_{i t} δ)] - \frac{var (b_{i t})}{2}$ in equation (8), implies that in the Type II Gamma m2REM

Λ_{i t} = \log [\frac{ℏ_{m_{3}}^{- 1} (X_{i t}' β_{m})}{ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m})}] - \frac{var (b_{i t})}{2}

To formulate the Type I and Type II inverse Gamma m2REMs, some adjustment on equation (5) is required. Note that, if a non-negative response variable Y follows an inverse Gamma distribution with a density $f_{IG} (y; μ, σ)$ , then $E (Y) = e^{μ} / (1 - σ^{2})$ exists only when $0 < σ < 1$ . Therefore, we propose to parameterize the heteroscedastic variance of positive healthcare expenditures in the Type I and Type II inverse Gamma m2REMs by

Φ^{- 1} (σ_{it} | Y_{it} > 0, X_{it}) = X_{it}' δ

(17)

in which δ is the regression coefficient vector. Part (I) of the Type I inverse Gamma m2REM is still constructed by equations (1) and (2) as in the log-normal m2REMs in Section 2.1, but Part (II) of the model takes equation (3) and assumes

(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{IG} (y_{it}; μ_{it}, σ_{it})

for equations (4) and (17). It can be derived from

(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{IG} (y_{it}; μ_{it}, σ_{it}), E (Y_{it} | Y_{it} > 0, X_{it}, b_{it}) = exp (μ_{it}) / (1 - σ_{it}^{2})

, equations (4) and (17) that

\begin{matrix} E (Y_{it} | Y_{it} > 0, X_{it}) = \int E (Y_{it} | Y_{it} > 0, X_{it}, b_{it}) f_{b_{it}} (b_{it}) d b_{it} \\ = \int \frac{1}{1 - Φ^{2} (X'_{it} δ)} exp [Λ_{it} + var (b_{it})^{1 / 2} s] φ (s) d s \\ = \frac{1}{1 - Φ^{2} (X'_{it} δ)} exp [Λ_{it} + \frac{var (b_{it})}{2}] \end{matrix}

(18)

This, combined with $E (Y_{i t} | Y_{i t} > 0, X_{i t}) = ℏ_{m_{2}}^{- 1} (X_{i t}' β_{m})$ in equation (3), implies that in the Type I inverse Gamma m2REM

Λ_{i t} = \log [ℏ_{m_{2}}^{- 1} (X_{i t}' β_{m})] + \log [1 - Φ^{2} (X'_{i t} δ)] - \frac{var (b_{i t})}{2}

Part (I) of the Type II inverse Gamma m2REM is also constructed by equations (1) and (2), but Part (II) of the model takes equation (8) and assumes $(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{IG} (y_{it}; μ_{it}, σ_{it})$ for equations (4) and (17). It can be derived from $(Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f_{IG} (y_{it}; μ_{it}, σ_{it})$ , equations (4), (17), and (18) that

\begin{matrix} E (Y_{i t} | X_{i t}) = P (Y_{i t} > 0 | X_{i t}) E (Y_{i t} | Y_{i t} > 0, X_{i t}) \\ = \frac{ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m})}{1 - Φ^{2} (X'_{i t} δ)} \exp [Λ_{i t} + \frac{var (b_{i t})}{2}] \end{matrix}

Λ_{i t} = \log [\frac{ℏ_{m_{3}}^{- 1} (X_{i t}' β_{m})}{ℏ_{m_{1}}^{- 1} (X_{i t}' α_{m})}] + \log [1 - Φ^{2} (X'_{i t} δ)] - \frac{var (b_{i t})}{2}

Model specification of the Type I and Type II Gamma and inverse Gamma m2REMs is summarized in Table 1.

3 Maximum likelihood estimation and simulation studies

For all of the Type I and Type II m2REMs introduced in Section 2, let $ϑ = (α_{m}, β_{m}, δ, κ, θ_{A}, θ_{B}, θ_{AB})'$ denote the vector that contains all unknown parameters. The full likelihood of these models is

\begin{matrix} L (ϑ) = Π_{i = 1}^{N} \int_{a_{i}} \int_{b_{i}} {Π_{t = 1}^{T} P (y_{it} > 0 | x_{ij}, a_{it})^{I (y_{it} > 0)} P (y_{it} = 0 | x_{ij}, a_{it})^{1 - I (y_{it} > 0)} \\ f (y_{it} | y_{it} > 0, x_{ij}, b_{it})^{I (y_{it} > 0)}} f_{a_{i} b_{i}} (a_{i}, b_{i}) d a_{i} d b_{i} \end{matrix}

(19)

in which

P (y_{i t} > 0 | x_{i j}, a_{i t}) = ℏ_{c_{1}}^{- 1} (Δ_{i t} + a_{i t})

P (y_{i t} = 0 | x_{i j}, a_{i t}) = 1 - ℏ_{c_{1}}^{- 1} (Δ_{i t} + a_{i t})

, and

f_{a_{i} b_{i}} (a_{i}, b_{i})

is the joint distribution of a_i and b_i. In equation (19),

f (y_{it} | y_{it} > 0, x_{ij}, b_{it})

is equal to

f_{LN} (y_{it}; μ_{it} = Λ_{it} + b_{it}, σ_{it} = exp (X_{it}' δ))

for the Type I and Type II log-normal m2REMs, to

f_{LSN} (y_{it}; μ_{it} = Λ_{it} + b_{it}, σ_{it} = exp (X_{it}' δ), κ)

for the Type I and Type II log-skew-normal m2REMs, to

f_{GG} (y_{it}; μ_{it} = Λ_{it} + b_{it}, σ_{it} = exp (X_{it}' δ), κ)

for the Type I and Type II generalized Gamma m2REMs, to

f_{G} (y_{it}; μ_{it} = Λ_{it} + b_{it}, σ_{it} = exp (X_{it}' δ))

for the Type I and Type II Gamma m2REMs, to

f_{IG} (y_{it}; μ_{it} = Λ_{it} + b_{it}, σ_{it} = Φ (X_{it}' δ))

for the Type I and Type II inverse Gamma m2REMs. The Δ _it in equation (19) can be obtained as discussed in Section 2.1. The Λ _it in equation (19) can be found specifically for each of the Type I and Type II m2REMs introduced in Section 2. Maximum likelihood estimates

\hat{ϑ}

that maximizes (19) can be obtained using either the Newton-Raphson or quasi-Newton optimization algorithm. During this process, adaptive Gaussian quadrature can be applied to numerically evaluate the integrals within equation (19).

In the likelihood function (19), we assume the observed panel data are balanced (i.e. the panel data are collected on every participant in the sample at each time of the study). When the panel data are unbalanced (i.e. a participant’s response can be missing at one follow-up time and then may be measured at a later follow-up time), it is requires to slightly modify the likelihood function (19) to be precise. When the missing responses are missing completely at random (the probability that responses are missing does not depend on either the observed or the missing responses) or missing at random (the probability that responses are missing depends on the observed responses but is unrelated to any missing responses), the full likelihood of the m2REMs is

\begin{matrix} L (ϑ) = Π_{i = 1}^{N} \int_{a_{i}} \int_{b_{i}} {\underset{t \in T (i)}{Π} P (y_{it} > 0 | x_{ij}, a_{it})^{I (y_{it} > 0)} P (y_{it} = 0 | x_{ij}, a_{it})^{1 - I (y_{it} > 0)} f (y_{it} | y_{it} > 0, x_{ij}, b_{it})^{I (y_{it} > 0)}} f_{a_{i} b_{i}} (a_{i}, b_{i}) d a_{i} d b_{i} \end{matrix}

(20)

in which T(i) is a subset of

{1, 2, \dots, T}

representing the time points at which the ith participant has responses. When the Type I and Type II m2REMs are employed, maximum likelihood estimation based on equation (20) can yield valid statistical inference, if the missing responses in panel data are assumed to be missing completely at random or missing at random.¹⁹

A simulation study was conducted to examine the performance of maximum likelihood estimates of the parameters in the Type I and Type II m2REMs. Simulation data sets were generated from the Type I m2REMs

\begin{matrix} Part (I) : Φ^{- 1} [P (Y_{it} > 0 | X_{it})] = α_{m 0} + X_{it 1} α_{m 1}, \\ Φ^{- 1} [P (Y_{it} > 0 | X_{it}, a_{it})] = Δ_{it} (X_{it}' α_{m}, var (a_{it})) + a_{i 0}; \\ Part (II) : (Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f (y; μ_{it}, σ_{it}, κ) \\ log [E (Y_{it} | Y_{it} > 0, X_{it})] = β_{m 0} + X_{it 1} β_{m 1} + X_{it 2} β_{m 2} + X_{it 3} β_{m 3}, \\ (μ_{it} | Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} (X_{it}' β_{m}, X_{it}' δ_{m}, var (b_{it})) + b_{i 0}, \\ log (σ_{it} | Y_{it} > 0, X_{it}) = δ_{m 0} + X_{it 1} δ_{m 1} \end{matrix}

(21)

and from the Type II m2REMs

\begin{matrix} Part (I) : Φ^{- 1} [P (Y_{it} > 0 | X_{it})] = α_{m 0} + X_{it 1} α_{m 1}, \\ Φ^{- 1} [P (Y_{it} > 0 | X_{it}, a_{it})] = Δ_{it} (X_{it}' α_{m}, var (a_{it})) + a_{i 0}; \\ Part (II) : (Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f (y; μ_{it}, σ_{it}, κ) \\ log [E (Y_{it} | X_{it})] = β_{m 0} + X_{it 1} β_{m 1} + X_{it 2} β_{m 2} + X_{it 3} β_{m 3}, \\ (μ_{it} | Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} (X_{it}' α_{m}, X_{it}' β_{m}, X_{it}' δ_{m}, var (b_{it})) + b_{i 0}, \\ log (σ_{it} | Y_{it} > 0, X_{it}) = δ_{m 0} + X_{it 1} δ_{m 1} \end{matrix}

(22)

in which

X_{it 1} \sim Bernoulli (1 / 2), X_{it 2} \sim Uniform (0, 1), t = 1, 2, 3, 4, X_{it 3} = t / 4

, κ = 1,

α_{m} = (α_{m 0}, α_{m 1})' = (- 0.5, - 0.5)'

β_{m} = (β_{m 0}, β_{m 1}, β_{m 2}, β_{m 3})' = (2, 1, 1, 1)', δ_{m} = (δ_{m 0}, δ_{m 1})' = (- 0.2, - 0.4)'

, and

(a_{i 0}, b_{i 0})' \sim N_{2} (0, Σ)

with

var (a_{i 0}) = var (b_{i 0}) = 0.5

and

corr (a_{i 0}, b_{i 0}) = 0.5

. Zhang et al.⁹ had already carefully examined the performance of maximum likelihood estimation procedure for the Type I and Type II generalized Gamma m2REMs. Therefore, in this simulation study, the remaining five parametric distributions of the positive healthcare expenditures

f (y; μ_{it}, σ_{it}, κ)

were specified in equations (21) and (22): log-normal distribution

f (y; μ_{it}, σ_{it}, κ) = f_{LN} (y; μ_{it}, σ_{it})

, log-skew-normal distribution

f (y; μ_{it}, σ_{it}, κ) = f_{LSN} (y; μ_{it}, σ_{it}, κ)

, Weibull distribution

f (y; μ_{it}, σ_{it}, κ) = f_{GG} (y; μ_{it}, σ_{it}, κ = 1)

, Gamma distribution

f (y; μ_{it}, σ_{it}, κ) = f_{G} (y; μ_{it}, σ_{it})

, and inverse Gamma distribution

f (y; μ_{it}, σ_{it}, κ) = f_{IG} (y; μ_{it}, σ_{it})

. A total number of 100 simulation data sets were generated from each of the five parametric distributions of

f (y; μ_{it}, σ_{it}, κ)

, and then the corresponding Type I or Type II m2REM was fit using maximum likelihood estimation procedure described above for parameter estimation. Each of the simulation data sets consisted of N = 1000 individuals (

j = 1, 2, \dots, N

). For the data sets simulated from equations (21) and (22), 76.685% of the simulated healthcare expenditures were zero, and the average positive and overall healthcare expenditures were 51.271 (standard deviation or SD: 218.260) and 60.673 (SD: 629.328), respectively. Table 2 reports the true values of parameters, the means of parameter estimates, the standard deviations, and the means of standard errors for the Type I and Type II m2REMs. The simulation results in Table 2 demonstrate that the empirical biases of the parameter estimates are negligible. Furthermore, the standard deviations of the parameter estimates are small enough relative to the biases, so that we are confident to conclude that the estimates are empirically unbiased. The average standard errors of the estimates are generally close to the standard deviations.

Table 2.

True values, parameter estimates, standard deviations (SD), and standard errors (SE) obtained in the simulation study for the Type I and Type II m2REMs with various parametric distributions (log-normal, log-skew-normal, Weibull, Gamma, and inverse Gamma distributions) in the positive healthcare expenditures in Part (II).

		True	Log-normal			Log-skew-normal			Weibull			Gamma			Inverse Gamma
	Estimand	value	Estimate	SD	SE	Estimate	SD	SE	Estimate	SD	SE	Estimate	SD	SE	Estimate	SD	SE
Type I m2REMs
	$α_{m 0}$	−0.5	−0.500	0.041	0.037	−0.496	0.044	0.037	−0.501	0.041	0.037	−0.495	0.040	0.037	−0.497	0.041	0.037
	$α_{m 1}$	−0.5	−0.510	0.064	0.056	−0.510	0.060	0.056	−0.501	0.055	0.055	−0.512	0.062	0.056	−0.514	0.059	0.056
	$β_{m 0}$	2	1.995	0.120	0.110	2.016	0.194	0.206	1.992	0.108	0.114	2.002	0.107	0.111	1.987	0.078	0.081
	$β_{m 1}$	1	0.992	0.097	0.086	1.003	0.154	0.162	0.994	0.086	0.084	0.999	0.092	0.083	0.997	0.072	0.070
	$β_{m 2}$	1	1.007	0.122	0.130	1.020	0.165	0.182	0.995	0.131	0.138	0.998	0.120	0.134	1.011	0.117	0.109
	$β_{m 3}$	1	0.999	0.103	0.095	0.996	0.159	0.162	1.015	0.098	0.104	0.994	0.100	0.098	1.002	0.056	0.053
	$γ_{m 0}$	−0.2	−0.190	0.034	0.039	−0.212	0.141	0.132	−0.194	0.040	0.043	−0.204	0.033	0.033	−0.199	0.045	0.042
	$γ_{m 1}$	−0.4	−0.410	0.074	0.072	−0.483	0.395	0.266	−0.412	0.082	0.079	−0.409	0.065	0.065	−0.402	0.073	0.069
	$var (a_{i 0})$	0.5	0.492	0.065	0.065	0.493	0.080	0.066	0.492	0.077	0.065	0.500	0.067	0.066	0.505	0.072	0.066
	$var (b_{i 0})$	0.5	0.494	0.060	0.058	0.495	0.110	0.110	0.495	0.068	0.066	0.504	0.056	0.062	0.497	0.040	0.041
	$corr (a_{i 0}, b_{i 0})$	0.5	0.523	0.109	0.096	0.500	0.156	0.145	0.512	0.106	0.100	0.511	0.101	0.097	0.513	0.076	0.081
	κ	2				2.008	0.134	0.121
Type II m2REMs
	$α_{m 0}$	−0.5	−0.500	0.041	0.037	−0.497	0.039	0.037	−0.501	0.041	0.037	−0.495	0.040	0.037	−0.497	0.041	0.037
	$α_{m 1}$	−0.5	−0.511	0.064	0.056	−0.511	0.059	0.056	−0.502	0.055	0.055	−0.512	0.062	0.056	−0.514	0.059	0.056
	$β_{m 0}$	2	1.995	0.137	0.122	2.025	0.198	0.213	1.989	0.121	0.126	2.007	0.125	0.123	1.990	0.102	0.097
	$β_{m 1}$	1	0.976	0.145	0.126	0.986	0.167	0.187	0.990	0.123	0.125	0.983	0.145	0.124	0.977	0.111	0.116
	$β_{m 2}$	1	1.005	0.122	0.130	1.021	0.166	0.182	0.998	0.133	0.138	0.998	0.120	0.134	1.011	0.117	0.109
	$β_{m 3}$	1	0.998	0.103	0.095	0.996	0.160	0.162	1.014	0.097	0.104	0.994	0.100	0.098	1.002	0.056	0.053
	$γ_{m 0}$	−0.2	−0.190	0.034	0.039	−0.216	0.137	0.133	−0.194	0.040	0.043	−0.204	0.033	0.033	−0.199	0.045	0.042
	$γ_{m 1}$	−0.4	−0.409	0.076	0.072	−0.485	0.397	0.291	−0.412	0.082	0.079	−0.409	0.065	0.065	−0.402	0.073	0.069
	$var (a_{i 0})$	0.5	0.493	0.064	0.065	0.499	0.062	0.066	0.491	0.077	0.065	0.500	0.067	0.066	0.505	0.072	0.066
	$var (b_{i 0})$	0.5	0.493	0.061	0.058	0.492	0.109	0.110	0.496	0.068	0.066	0.504	0.056	0.062	0.497	0.040	0.041
	$corr (a_{i 0}, b_{i 0})$	0.5	0.524	0.108	0.096	0.499	0.155	0.146	0.513	0.105	0.100	0.511	0.101	0.097	0.513	0.076	0.081
	κ	2				2.007	0.127	0.121

4 Estimation of marginal and incremental effects

The quantities of interest in the empirical analysis of panel data on healthcare expenditures are often the marginal or increment effects of an independent variable on expected healthcare expenditure at a specific time point.^20,21 Zhang et al.⁹ derived specific formulas for estimating these effects in the context of Type I and Type II generalized Gamma m2REMs. These formulas can be directly applied to the Type I and Type II m2REMs introduced in Section 2 without modification, and therefore are summarized below.

Let y_it be the semicontinuous dependent variable that represents the amount of the healthcare expenditure of the ith individual at the time t. Let $x_{it} = (x_{it 1}, x_{it 2}, \dots, x_{itK})$ be a vector of K independent variables. When any of the Type II m2REMs is used, the marginal effect of the kth independent variable x_itk at the time t on the expected overall healthcare expenditure $E (y_{it} | x_{it})$ is

η_{k} (x_{i t}, ϑ) = \frac{\partial E (y_{i t} | x_{i t})}{\partial x_{i t k}} = \frac{\partial ℏ_{m_{3}}^{- 1} (x_{i t}' β_{m})}{\partial (x_{i t}' β_{m})} \cdot β_{m, k} = {\dot{ℏ}}_{m_{3}}^{- 1} (x_{i t}' β_{m}) \cdot β_{m, k}

in which

β_{m, k}

denotes the regression coefficient for x_itk, and the dot on

{\overset{\cdot}{ℎ}}_{m_{3}}^{- 1} (\cdot)

indicates that it is the derivative of

ℏ_{m_{3}}^{- 1} (\cdot)

(i.e.

{\dot{ℏ}}_{m_{3}}^{- 1} (x) = \frac{\partial ℏ_{m_{3}}^{- 1} (x)}{\partial x}

). When x_itk is a categorical variable representing multiple experimental groups, the quantity of interest is usually the incremental effect

\begin{matrix} π_{k} (x_{i t, - k}, ϑ) = E (y_{i t} | x_{i t k} = l_{2}, x_{i t, - k}) - E (y_{i t} | x_{i t k} = l_{1}, x_{i t, - k}), \\ = ℏ_{m_{3}}^{- 1} [(x_{i t 1}, \dots, x_{i t k} = l_{2}, \dots, x_{i t K}) β_{m}] - ℏ_{m_{3}}^{- 1} [(x_{i t 1}, \dots, x_{i t k} = l_{1}, \dots, x_{i t K}) β_{m}] \end{matrix}

in which l₁ and l₂ are two levels of x_itk. The marginal effect quantifies the marginal the change in expected overall healthcare expenditure with a small amount of increase in x_itk, while holding other variables

x_{it, - k} = (x_{it 1}, \dots, x_{it, k - 1}, x_{it, k + 1}, \dots, x_{itK})

constant. The incremental effect characterizes the difference in the expected overall healthcare expenditure

E (y_{it} | x_{it})

at the two levels of x_itk while holding other variables

x_{it, - k}

constant.

The above marginal effect $η_{k} (x_{it}, ϑ)$ and incremental effect $π_{k} (x_{it, - k}, ϑ)$ represent a specific portion of the entire population as they are evaluated at specified X_it values. In contrast, the average marginal effect quantifies the population-average rate of marginal change in the expected overall healthcare expenditure with respect to x_itk over time, and it is given by

{\bar{η}}_{k} (ϑ) = \frac{1}{T} \sum_{t = 1}^{T} E_{X_{it}} [η_{k} (X_{it}, ϑ)] = \frac{1}{T} \sum_{t = 1}^{T} \int η_{k} (x_{it}, ϑ) d F_{X_{it}} (x_{it})

where

F_{x_{it}} (\cdot)

is the joint distribution of random vector X_it at time t and the expectation

E_{X_{it}} [\cdot]

is taken with respect to independent variable X_it at time t. When x_itk is a categorical variable, the average incremental effect is given by

{\bar{π}}_{k} (ϑ) = \frac{1}{T} \sum_{t = 1}^{T} E_{X_{it, - k}} [π_{k} (X_{it, - k}, ϑ)]

The average incremental effect ${\bar{π}}_{k} (ϑ)$ quantifies the population-average contrast over time in the expected overall healthcare expenditure at the two levels of x_itk while adjusting for other independent variables. When any of the Type II m2REMs is used, the marginal and incremental effects are estimated by

{\hat{η}}_{k} (x_{it}, \hat{ϑ}) = {\overset{\cdot}{ℎ}}_{m_{3}}^{- 1} (x_{it}' {\hat{β}}_{m}) \cdot {\hat{β}}_{m, k}

(23)

and

{\hat{π}}_{k} (x_{i t, - k}, \hat{ϑ}) = ℏ_{m_{3}}^{- 1} [(x_{i t 1}, \dots, x_{i t k} = l_{2}, \dots, x_{i t K}) {\hat{β}}_{m}] - ℏ_{m_{3}}^{- 1} [(x_{i t 1}, \dots, x_{i t k} = l_{1}, \dots, x_{i t K}) {\hat{β}}_{m}]

(24)

Obtained by averaging the individual marginal effects are the estimators of average marginal and incremental effects

\bar{^} η_{k} (\hat{ϑ}) = \frac{1}{NT} \sum_{t = 1}^{T} \sum_{t = 1}^{T} {{\overset{\cdot}{ℎ}}_{m_{3}}^{- 1} (x_{it}' {\hat{β}}_{m}) \cdot {\hat{β}}_{m, k}}

and

\begin{matrix} {\hat{\bar{π}}}_{k} (\hat{ϑ}) = \frac{1}{N T} \sum_{i = 1}^{N} \sum_{t = 1}^{T} {ℏ_{m_{3}}^{- 1} [(x_{i t 1}, \dots, x_{i t k} = l_{2}, \dots, x_{i t K}) {\hat{β}}_{m}] \\ - ℏ_{m_{3}}^{- 1} [(x_{i t 1}, \dots, x_{i t k} = l_{1}, \dots, x_{i t K}) {\hat{β}}_{m}]} \end{matrix}

When any of the Type I m2REMs is used, it is convenient to estimate the marginal and incremental effects of an independent variable with respect to the expected positive healthcare expenditure $E (y_{it} | y_{it} > 0, x_{it})$ . The marginal effect of x_itk on the expected positive healthcare expenditure at time t is

η_{k}^{+} (x_{it}, ϑ) = \frac{\partial E (y_{it} | y_{it} > 0, x_{it})}{\partial x_{itk}} = {\overset{\cdot}{ℎ}}_{m_{2}}^{- 1} (x_{it}' β_{m}) \cdot β_{m, k}

It quantifies the marginal change in the expected positive healthcare expenditure with a small amount of increase in x_itk while holding other independent variables $x_{it, - k}$ constant. When x_itk is a categorical variable, the incremental effect on the expected positive healthcare expenditure $E (y_{it} | y_{it} > 0, x_{it})$ from level l₂ to level l₁ of x_itk is defined as

\begin{matrix} π_{k}^{+} (x_{i t, - k}, ϑ) = E (y_{i t} | y_{i t} > 0, x_{i t k} = l_{2}, x_{i t, - k}) - E (y_{i t} | y_{i t} > 0, x_{i t k} = l_{1}, x_{i t, - k}), \\ = ℏ_{m_{2}}^{- 1} [(x_{i t 1}, \dots, x_{i t k} = l_{2}, \dots, x_{i t K}) β_{m}] - ℏ_{m_{2}}^{- 1} [(x_{i t 1}, \dots, x_{i t k} = l_{1}, \dots, x_{i t K}) β_{m}] \end{matrix}

The average marginal and incremental effects on the expected positive healthcare expenditures are subsequently given by

{\bar{η}}_{k}^{+} (ϑ) = \frac{1}{T} \sum_{t = 1}^{T} E_{X_{it} | Y_{it} > 0} [η_{k}^{+} (x_{it}, ϑ)] and {\bar{π}}_{k}^{+} (ϑ) = \frac{1}{T} \sum_{t = 1}^{T} E_{X_{it, - k} | Y_{it} > 0} [π_{k}^{+} (x_{it, - k}, ϑ)]

in which

F_{x_{it}} (\cdot | y_{it} > 0)

is the conditional distribution of random vector X_it at time t given that the healthcare expenditure Y_it is greater than zero, and the expectation

E_{X_{it} | Y_{it} > 0} [\cdot]

is taken with respect to X_it conditioning on Y_it > 0 at time t. Estimators of

{\hat{η}}_{k}^{+} (x_{it}, \hat{ϑ})

and

{\hat{π}}_{k}^{+} (x_{it, - k}, \hat{ϑ})

for the Type I m2REMs can be obtained by replacing

ℎ_{m_{3}} (\cdot)

with

ℎ_{m_{2}} (\cdot)

in equations (23) and (24). The average marginal and incremental effects

{\bar{η}}_{k}^{+} (ϑ)

and

{\bar{π}}_{k}^{+} (ϑ)

for the Type I m2REMs are estimated by averaging the individual effects for which the healthcare expenditures are not zeros

\bar{^} η_{k}^{+} (\hat{ϑ}) = \frac{1}{T} \sum_{t = 1}^{T} \frac{1}{{\tilde{N}}_{t}} \sum_{t = 1}^{T} {I (y_{it} > 0) {\overset{\cdot}{ℎ}}_{m_{2}}^{- 1} (x_{it}' {\hat{β}}_{m}) {\hat{β}}_{m, k}}

and

\begin{matrix} {\hat{\bar{π}}}_{k}^{+} (\hat{ϑ}) = \frac{1}{T} \sum_{t = 1}^{T} \frac{1}{{\tilde{N}}_{t}} \sum_{i = 1}^{N} I (y_{i t} > 0) {ℏ_{m_{2}}^{- 1} [(x_{i t 1}, \dots, x_{i t k} = l_{2}, \dots, x_{i t K}) {\hat{β}}_{m}] \\ - ℏ_{m_{2}}^{- 1} [(x_{i t 1}, \dots, x_{i t k} = l_{1}, \dots, x_{i t K}) {\hat{β}}_{m}]} \end{matrix}

in which

I (\cdot)

is an indicator function and, for a fixed t,

{\tilde{N}}_{t} = \sum_{t = 1}^{T} I (y_{it} > 0)

Zhang et al.⁹ elaborated the procedure of variance estimation of the (average) marginal and incremental effects for conducting inference on these effects, when using either the Type I and Type II m2REMs. Variance estimators for these effects were derived using the delta methods or Taylor series approximations, and the details were reported in Zhang et al.⁹ In some empirical analysis of healthcare expenditures, (average) elasticity and semi-elasticity are also quantities of interest. Please refer to Zhang et al.⁹ for the specifics of estimation and variance estimation of these quantities.

5 Model comparison

In the m2REMs introduced in Section 2, the log-normal, log-skew-normal, generalized Gamma, Weibull, Gamma, and inverse Gamma distributions are employed to delineate the spectrum of the positive healthcare expenditures in Part (II) of the models. These marginalized models supply multiple choices for analyzing the healthcare expenditure panel data with excess zeros. For the studies in which the primary interest of investigation is to draw inference on the marginal or incremental effect of an independent variable with respect to the expected overall or positive expenditures, the empirical mean square error (MSE) criterion has been proven effective to choose among multiple candidate models.¹¹ While estimating the (average) marginal and incremental effects is the aim of investigation, the MSEs of the estimators of (average) marginal and incremental effects are the variance of the estimators plus the square of their bias

\begin{matrix} MSE [{\hat{η}}_{k} (x_{it,}, \hat{ϑ})] = var [{\hat{η}}_{k} (x_{it,}, \hat{ϑ})] + [{\hat{η}}_{k} (x_{it}, \hat{ϑ}) - η_{k} (x_{it}, ϑ)]^{2}, \\ MSE [{\hat{π}}_{k} (x_{it, - k}, \hat{ϑ})] = var [{\hat{π}}_{k} (x_{it, - k}, \hat{ϑ})] + [{\hat{π}}_{k} (x_{it, - k}, \hat{ϑ}) - π_{k} (x_{it, - k}, ϑ)]^{2} \end{matrix}

and

\begin{matrix} MSE [\bar{^} η_{k} (\hat{ϑ})] = var [\bar{^} η_{k} (\hat{ϑ})] + [\bar{^} η_{k} (\hat{ϑ}) - {\bar{η}}_{k} (ϑ)]^{2} \\ MSE [\bar{^} π_{k} (\hat{ϑ})] = var [\bar{^} π_{k} (\hat{ϑ})] + [\bar{^} π_{k} (\hat{ϑ}) - {\bar{π}}_{k} (ϑ)]^{2} \end{matrix}

The MSE criterion selects the model with the minimum MSE as the best model among the candidates. However, in practice, the true effects $η_{k} (x_{it}, ϑ), π_{k} (x_{it, - k}, ϑ), {\bar{η}}_{k} (ϑ)$ , and ${\bar{π}}_{k} (ϑ)$ are unknown. Therefore, the empirical MSEs

\begin{matrix} EMSE [{\hat{η}}_{k} (x_{it}, \hat{ϑ})] = var [{\hat{η}}_{k} (x_{it}, \hat{ϑ})] + [{\hat{η}}_{k} (x_{it}, \hat{ϑ}) - {\hat{η}}_{k}^{c} (x_{it}, {\hat{ϑ}}^{c})]^{2}, \\ EMSE [{\hat{π}}_{k} (x_{it, - k}, \hat{ϑ})] = var [{\hat{π}}_{k} (x_{it, - k}, \hat{ϑ})] + [{\hat{π}}_{k} (x_{it, - k}, \hat{ϑ}) - {\hat{π}}_{k}^{c} (x_{it, - k}, {\hat{ϑ}}^{c})]^{2} \end{matrix}

and

\begin{matrix} EMSE [\bar{^} η_{k} (\hat{ϑ})] = var [\bar{^} η_{k} (\hat{ϑ})] + [\bar{^} η_{k} (\hat{ϑ}) - \bar{^} η_{k}^{c} ({\hat{ϑ}}^{c})]^{2}, \\ EMSE [\bar{^} π_{k} (\hat{ϑ})] = var [\bar{^} π_{k} (\hat{ϑ})] + [\bar{^} π_{k} (\hat{ϑ}) - \bar{^} π_{k}^{c} ({\hat{ϑ}}^{c})]^{2} \end{matrix}

are adopted, in which the true effects are replaced by the estimated effects from a pre-specified model. The empirical MSE criterion selects among the candidate models the one with the smallest empirical MSE.

The information criteria, such as the Akaike information criterion (AIC) and the Bayesian information criterion, are also advocated here to conduct model selection among the m2REMs. These criteria are not effect-specific and thus may conclude differently with the empirical MSE criterion.

6 Empirical analysis: estimating incremental effects of the New Cooperative Medical Scheme on healthcare expenditures among older adults in China

6.1 Data and variables

A set of panel data on the healthcare expenditures among older (60 years old or older) rural residents in China were analyzed using the m2REMs introduced in Section 2. The primary objective of this empirical analysis is to evaluate the impact of the New Cooperative Medical Scheme (NCMS) program on healthcare expenditures among older adults in rural areas in China. The panel data were taken from the China Health and Nutrition Survey (CHNS), a collaborative project between the Carolina Population Center at the University of North Carolina at Chapel Hill and the National Institute for Nutrition and Health at the Chinese Center for Disease Control and Prevention. The CHNS is designed to investigate the effects of health and nutrition programs implemented by Chinese governments throughout the past 20 years, in order to determine how the social and economic transformation in China impacts the health and nutritional status of the population during China’s rapid economic ascendance. Nine waves of the survey were conducted in 1989, 1991, 1993, 1997, 2000, 2004, 2006, 2009, and 2011. The survey implemented a multistage, random cluster process of data collection and by now it has drawn a sample of approximately 7200 households with more than 30,000 the individuals. The household and individual surveys contain several modules. One of the modules is the health services section that collected information on respondent demographics, health, nutrition, and income. Nine provinces (Liaoning, Heilongjiang, Guangxi, Guizhou, Henan, Hubei, Hunan, Jiangsu, and Shandong) with substantial variation in geography, economic development, and public health facilities had participated in the CHNS by 2011. In this empirical analysis, the five most recent waves from 2000 to 2011 (i.e. 2000, 2004, 2006, 2009, and 2011 waves) were used. The initial sample included 74,619 observations from both urban and rural residents who were 18 years old or older. The criterion to distinguish whether a participant is an urban or a rural resident is based upon the resident registration information. The individuals with an agricultural resident registration were classified as rural residents. Because the NCMS is only available to the residents with an agricultural resident registration, we excluded the observations from urban residents, yielding a sample of 29,895 observations from rural residents who participated in the CHNS from 2000 to 2011. The empirical analysis concerns the healthcare expenditures among the older adults. Thus, 23,454 additional observations from the participants that were younger than 60 years old were excluded, which left 6441 observations for further consideration.

In this empirical analysis, the primary dependent variable is the out-of-pocket healthcare expenditures during the previous four weeks of the survey (denoted by Y_it for individual i at the tth survey year) reported by the CHNS older participants. The out-of-pocket healthcare expenditure is the summation of three portions: self-care expense on the illness, injury, chronic or acute disease (e.g. the expenditure from purchases of over-the-counter drugs), expense from utilization of formal healthcare facilities including both inpatient and outpatient services (e.g. the expenditure of registration, treatment, medicines, lab tests, surgeries, hospital beds, etc.), and expense from the utilization of preventive health services (e.g. spending on general physical examination, eye exam, blood tests, blood pressure screening, tumor screening, etc.). The expense from utilization of formal healthcare facilities or preventive health services was calculated by multiplying the reported expenditure before insurance coverage by one minus the percentage of the expenditure paid by insurance. The percentage of health insurance coverage was counted as zero if the participant reported having no health insurance coverage or unknown.

The scientific interest of this empirical analysis is to estimate the incremental effect of the NCMS coverage with respect to the overall and positive out-of-pocket healthcare expenditures among older adults in rural China. As such, the primary independent variable of the analysis is a time-varying three-level indicator of health insurance coverage status with 0 representing no health insurance coverage, 1 representing the NCMS coverage, and 2 representing the coverage from other types of health insurance. When the CHNS inquires about the type of health insurance of the participants, it adopted two slightly different categorization approaches. The 2009 and 2011 questionnaires asked the participants to choose one insurance type, if they were covered by health insurance, from the following options: the NCMS, the Urban Employee Basic Medical Insurance,¹⁵ the Urban Resident Basic Medical Insurance,¹⁵ any commercial medical insurance, government free medical insurance, and other types of health insurance. Thus, these questionnaires clearly distinguished the NCMS from other type of health insurance coverage. However, the 2006 questionnaire provided the following options to the participants who were covered by health insurance: the Cooperative Medical Scheme (CMS), the Urban Employee Basic Medical Insurance, the Urban Resident Basic Medical Insurance, any commercial medical insurance, government free medical insurance, health insurance program for women and children, and other types of health insurance; and the 2000 and 2004 questionnaires provided the following options: the CMS, any commercial medical insurance, government free medical insurance, health insurance program for women and children, worker’s compensation insurance, insurance coverage as family members, unified planning medical service, and other types of health insurance. The CMS referred to in the 2000, 2004, and 2006 questionnaires included both the NCMS and the original CMS¹⁴ (the original CMS, or the old CMS, was the major health insurance program for rural residents in China before early 1980s; due to China’s economic reform, it collapsed as a direct consequence of the lost of welfare funds from collective economy). Since the NCMS was launched in 2003, when the CMS was chosen in the 2004 and 2006 CHNS, it may represent either the old CMS or the NCMS. In order to distinguish between these two programs, we adopted the approach introduced by Lei and Lin¹² and use community level data to define the health insurance status for the individuals that chose the CMS. The community survey component asked local government officials whether the CMS had been implemented in their community. It is true that this survey question to the communities was too vague, because it did not specifically distinguish the original CMS or the NCMS. However, because the pilot implementation of the NCMS began in 2003, the communities must implement the NCMS, instead of the old CMS, in 2004 and 2006 if the local government officials reported absence of CMS in the 2000 CNHS survey or before, but notified the establishment of it in 2004 and 2006. Consequently, the individuals who lived in such communities that offered the NCMS and who chose the CMS in the 2004 and 2006 CHNS questionnaires were identified to be covered by the NCMS health insurance program.

In this empirical analysis, the independent variables on individual characteristics include age, gender, marital status, years of schooling, logarithm of one plus household income, and whether the individual reported at least one type of the chronic diseases listed in the survey questionnaire (diabetes, hypertension, myocardial infarction, or apoplexy). The independent variables also include four dummy variables for five survey waves and eight dummy variables for nine provinces. The out-of-pocket healthcare expenditure and household income, both in terms of Ren Min Bi or RMB (RMB is the official currency of China), were converted to the 2011 price level according to the annual customer price indices (CPI) published by the National Bureau of Statistics of China.²²

Based upon the information of missingness on the involved independent variables, additional 941 observations were excluded. Table 3 elaborates the details of the numbers of observations excluded due to the missingness of each variable. The final data set used in the empirical analysis includes a total number of 5500 observations from five survey waves. Summary statistics of overall and positive healthcare expenditures of these older adults are reported in Tables 10 –13 in Appendix 1. Tables 10 and 11 illustrate the sample means and standard deviations of overall and positive healthcare expenditures of the older adults with different insurance coverage types in each variable category. Tables 12 and 13 summarize the number of survey participants with different insurance coverage types in each category of variables and the proportion that the amount of participants with different insurance takes in each category. The categories of each variable in Tables 10 –13 were decided based upon the CHNS questionnaires.

Table 3.

Detailed description of the number of observations excluded from the original CHNS data.

	Number	Number
Data	excluded	remaining
CHNS 2000, 2004, 2006, 2009, 2011 waves		74,619
Exclude if with urban resident registration	44,724	29,895
Exclude if age <60	23,454	6441
Exclude if insurance coverage is missing	95	6346
Exclude if gender is missing	0	6346
Exclude if marital status is missing	158	6188
Exclude if years of schooling is missing	72	6116
Exclude if household income is negative	30	6086
Exclude if household income is missing	80	6006
Exclude if disease history is missing	181	5825
Exclude if total out-of-pocket healthcare expenditure is missing	325	5500

6.2 Empirical analysis strategy

The strategy of our empirical analysis is to employ the Type I and Type II m2REMs with various distributions for positive healthcare expenditures in Part (II) and estimate the incremental effect of the NCMS health insurance program on both overall and positive out-of-pocket healthcare expenditures. The Type I m2REMs that was used to analyze the CHNS data take the form of

\begin{matrix} Part (I) : \\ Φ^{- 1} [P (Y_{it} > 0 | X_{it})] = α_{m 0} + α_{m 1} {NCMS}_{it} + α_{m 2} {INSU}_{it} + \sum_{k_{1} = 3}^{8} X_{{itk}_{1}} α_{{mk}_{1}} + \sum_{k_{2} = 1}^{12} U_{{ik}_{2}} α_{m (k_{2} + 8)}, \\ Φ^{- 1} [P (Y_{it} > 0 | X_{it}, a_{it})] = Δ_{it} + a_{i 0}; \\ Part (II) : \\ (Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f (y; μ_{it}, σ_{it}, κ) \\ log [E (Y_{it} | Y_{it} > 0, X_{it})] = β_{m 0} + β_{m 1} {NCMS}_{it} + β_{m 2} {INSU}_{it} + \sum_{k_{1} = 3}^{8} X_{{itk}_{1}} β_{{mk}_{1}} + \sum_{k_{2} = 1}^{12} U_{{ik}_{2}} β_{m (k_{2} + 8)}, \\ (μ_{it} | Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} + b_{i 0}, \\ log (σ_{it} | Y_{it} > 0, X_{it}) = δ_{m 0} + δ_{m 1} {NCMS}_{it} + δ_{m 2} {INSU}_{it} + \sum_{k_{1} = 3}^{8} X_{{itk}_{1}} δ_{{mk}_{1}} + \sum_{k_{2} = 1}^{12} U_{{ik}_{2}} δ_{m (k_{2} + 8)} \end{matrix}

(25)

The Type II m2REMs that was used to analyze the CHNS data take the form of

\begin{matrix} Part (I) : \\ Φ^{- 1} [P (Y_{it} > 0 | X_{it})] = α_{m 0} + α_{m 1} {NCMS}_{it} + α_{m 2} {INSU}_{it} + \sum_{k_{1} = 3}^{8} X_{{itk}_{1}} α_{{mk}_{1}} + \sum_{k_{2} = 1}^{12} U_{{ik}_{2}} α_{m (k_{2} + 8)}, \\ Φ^{- 1} [P (Y_{it} > 0 | X_{it}, a_{it})] = Δ_{it} + a_{i 0}; \\ Part (II) : \\ (Y_{it} | Y_{it} > 0, X_{it}, b_{it}) \sim f (y; μ_{it}, σ_{it}, κ) \\ log [E (Y_{it} | X_{it})] = β_{m 0} + β_{m 1} {NCMS}_{it} + β_{m 2} {INSU}_{it} + \sum_{k_{1} = 3}^{8} X_{{itk}_{1}} β_{{mk}_{1}} + \sum_{k_{2} = 1}^{12} U_{{ik}_{2}} β_{m (k_{2} + 8)}, \\ (μ_{it} | Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} + b_{i 0}, \\ log (σ_{it} | Y_{it} > 0, X_{it}) = δ_{m 0} + δ_{m 1} {NCMS}_{it} + δ_{m 2} {INSU}_{it} + \sum_{k_{1} = 3}^{8} X_{{itk}_{1}} δ_{{mk}_{1}} + \sum_{k_{2} = 1}^{12} U_{{ik}_{2}} δ_{m (k_{2} + 8)} \end{matrix}

(26)

Type I m2REM (25) and Type II m2REM (26) are specific forms of the proposed Type I and Type II m2REMs in Section 2, where the link function specifications are $ℎ_{m_{1}} (\cdot) = ℎ_{c_{1}} (\cdot) = Φ^{- 1} (\cdot)$ and $ℎ_{m_{2}} (\cdot) = ℎ_{m_{3}} (\cdot) = log (\cdot)$ , and $f (y; μ_{it}, σ_{it}, κ)$ represents a general distributional form that is replaced by the parametric distributions of positive expenditures in Table 1: $f (y; μ_{it}, σ_{it}, κ) = f_{LN} (y; μ_{it}, σ_{it})$ , $f (y; μ_{it}, σ_{it}, κ) = f_{LSN} (y; μ_{it}, σ_{it}, κ)$ , $f (y; μ_{it}, σ_{it}, κ) = f_{GG} (y; μ_{it}, σ_{it}, κ)$ , $f (y; μ_{it}, σ_{it}, κ) = f_{G} (y; μ_{it}, σ_{it})$ , or $f (y; μ_{it}, σ_{it}, κ) = f_{IG} (y; μ_{it}, σ_{it})$ . The Weilbull distribution was not considered because it is nested in the generalize Gamma distribution. When $f (y; μ_{it}, σ_{it}, κ) = f_{IG} (y; μ_{it}, σ_{it})$ , the log link $log (\cdot)$ for σ_it needs to be replaced by $Φ^{- 1} (\cdot)$ . In equations (25) and (26), ${NCMS}_{it}$ is the indicator of the NCMS coverage ( ${NCMS}_{it} = 1$ for the NCMS; ${NCMS}_{it} = 0$ for no insurance), ${INSU}_{it}$ is the indicator of other types of health insurance coverage other than the NCMS ( ${INSU}_{it} = 1$ for other insurance; ${INSU}_{it} = 0$ for no insurance), $X_{it 3}$ represents age, $X_{it 4}$ is the indicator of gender ( $X_{it 4} = 1$ for males; $X_{it 4} = 0$ for females), $X_{it 5}$ is the indicator of marital status ( $X_{it 5} = 1$ for married; $X_{it 5} = 0$ otherwise), $X_{it 6}$ is years of schooling, $X_{it 7}$ is the logarithm of one plus household income, $X_{it 8}$ is the indicator of chronic disease status ( $X_{it 8} = 1$ for having disease; $X_{it 8} = 0$ for no disease or unknown status), $U_{i 1}, U_{i 2}, \dots, U_{i 8}$ are eight dummy variables that represent eight provinces ( $U_{i 1}$ for Heilongjiang; $U_{i 2}$ for Jiangsu; $U_{i 3}$ for Shandong; $U_{i 4}$ for Henan; $U_{i 5}$ for Hubei; $U_{i 6}$ for Hunan; $U_{i 7}$ for Guangxi; $U_{i 8}$ for Guizhou) with the province of Liaoning as reference, and $U_{i 9}, \dots, U_{i, 12}$ are four dummy variables that represent four survey waves ( $U_{i 9}$ for 2004; $U_{i, 10}$ for 2006; $U_{i, 11}$ for 2009; $U_{i, 12}$ for 2011) with the year of 2000 as reference. In equations (25) and (26), $α_{m} = (α_{m 0}, α_{m 1}, \dots, α_{m, 15})', β_{m} = (β_{m 0}, β_{m 1}, \dots, β_{m, 15})'$ , and $δ_{m} = (δ_{m 0}, δ_{m 1}, \dots, δ_{m, 15})'$ are the vectors of regression coefficients for the intercepts and independent variables, and $a_{i 0}$ and $b_{i 0}$ are correlated random intercepts that follow a bivariate normal distribution $N_{2} (0, Σ)$ with $var (a_{i 0}) = σ_{a}^{2}, var (b_{i 0}) = σ_{b}^{2}$ , and $corr (a_{i 0}, b_{i 0}) = ρ$ . The maximum likelihood estimates of the unknown parameters in equations (25) and (26) were numerically obtained by maximizing equation (20). The (average) incremental effects of the NCMS on the expected overall and positive healthcare expenditure were estimated as described in Section 4. Variance estimation and inference procedures were conducted following the variance estimation procedure presented in Zhang et al.⁹

6.3 Estimation of fixed effects, heteroscedasticity, and variance components

The empirical analysis results from fitting the Type I and Type II m2REMs to the CHNS healthcare expenditure data on older adults are summarized in Tables 4 –8 for the log-skew-normal, log-normal, generalized Gamma, Gamma, and inverse Gamma m2REMs, respectively. The tables report the estimates, estimated standard errors and p-values associated with Parts (I) fixed effects, Part (II) fixed effects, heteroscedasticity regression parameters, and variance components in the models. The tables include the empirical analysis results from the full models that include all covariates, as well as the results from corresponding reduced models. The reduced models exclude the covariates with an insignificant p-value (under the significant level of 0.05), but retain the covariates

{NCMS}_{it}

and

{INSU}_{it}

in Part (I) and Part (II) for the purpose of conducting statistical inference on their incremental effects.

Table 4.

Parameter estimates, estimated standard errors, and p-values from fitting the Type I and Type II log-skew-normal m2REMs to the CHNS data.

		Type I log-skew-normal m2REM						Type II log-skew-normal m2REM
		Full model			Reduced model			Full model			Reduced model
	Parameter	Estimate	SE	p-value	Estimate	SE	p-value	Estimate	SE	p-value	Estimate	SE	p-value
Part (I)	Intercept (α₀)	−2.265	0.321	<.0001	−2.335	0.216	<.0001	−2.123	0.318	<.0001	−2.077	0.196	<.0001
Fixed Effects	NCMS (α₁)	0.051	0.080	0.524	0.111	0.055	0.045	0.044	0.080	0.580	0.090	0.053	0.087
	Other Insurance (α₂)	0.141	0.090	0.116	0.214	0.083	0.010	0.064	0.091	0.483	0.114	0.084	0.175
	Age (α₃)	0.016	0.004	<.0001	0.016	0.003	<.0001	0.015	0.003	<.0001	0.012	0.003	<.0001
	Gender (α₄)	−0.141	0.052	0.007	−0.118	0.046	0.010	−0.127	0.051	0.013
	Marriage (α₅)	0.063	0.055	0.255				0.067	0.055	0.221
	Education (α₆)	0.003	0.008	0.696				0.002	0.008	0.765
	log(1+Income) (α₇)	−0.026	0.015	0.075				−0.031	0.015	0.034
	Disease (α₈)	0.498	0.049	<.0001	0.493	0.049	<.0001	0.519	0.049	<.0001	0.484	0.049	<.0001
	Heilongjiang (α₉)	−0.081	0.125	0.514				−0.097	0.124	0.433
	Jiangsu (α₁₀)	0.164	0.117	0.162				0.193	0.116	0.096
	Shandong (α₁₁)	−0.302	0.129	0.019	−0.308	0.090	0.001	−0.289	0.128	0.024	−0.248	0.075	0.001
	Henan (α₁₂)	0.019	0.114	0.865				0.044	0.113	0.699
	Hubei (α₁₃)	−0.090	0.120	0.456				−0.063	0.119	0.597
	Hunan (α₁₄)	−0.023	0.120	0.846				−0.006	0.119	0.962
	Guangxi(α₁₅)	0.285	0.109	0.009	0.275	0.060	<.0001	0.288	0.109	0.008	0.206	0.053	0.000
	Guizhou(α₁₆)	0.021	0.112	0.851				0.028	0.111	0.800
	Wave 2004(α₁₇)	0.262	0.084	0.002	0.191	0.062	0.002	0.244	0.083	0.003	0.200	0.056	0.000
	Wave 2006(α₁₈)	0.115	0.088	0.189				0.117	0.087	0.176
	Wave 2009(α₁₉)	0.394	0.107	0.000	0.243	0.049	<.0001	0.384	0.106	0.000	0.250	0.045	<.0001
	Wave 2011(α₂₀)	0.156	0.110	0.156				0.132	0.109	0.225
Part (II)	Intercept (β₀)	5.028	2.099	0.017	6.807	0.245	<.0001	4.860	3.015	0.107	5.034	0.310	<.0001
Fixed Effects	NCMS (β₁)	−0.638	0.347	0.066	−0.242	0.138	0.079	−0.479	0.374	0.201	−0.274	0.160	0.087
	Other Insurance (β₂)	−0.688	0.329	0.037	−0.196	0.238	0.412	−0.496	0.312	0.112	−0.373	0.282	0.186
	Age (β₃)	0.024	0.021	0.266				0.016	0.025	0.535
	Gender (β₄)	−0.082	0.294	0.781				−0.125	0.282	0.658
	Marriage (β₅)	0.752	0.280	0.007				0.657	0.257	0.011	0.464	0.150	0.002
	Education (β₆)	0.027	0.036	0.461				0.000	0.037	0.995
	log(1+Income) (β₇)	0.006	0.058	0.919				−0.091	0.063	0.151
	Disease (β₈)	1.031	0.314	0.001	0.888	0.140	<.0001	1.522	0.280	<.0001	1.594	0.162	<.0001
	Heilongjiang (β₉)	1.323	0.738	0.073				1.211	1.567	0.440
	Jiangsu (β₁₀)	−0.319	0.754	0.672				−0.226	1.614	0.889
	Shandong (β₁₁)	0.227	0.749	0.762				0.125	1.194	0.917
	Henan (β₁₂)	0.545	0.617	0.377				0.545	1.264	0.667
	Hubei (β₁₃)	0.170	0.744	0.819				0.064	1.537	0.967
	Hunan (β₁₄)	0.238	0.632	0.707				0.203	1.291	0.875
	Guangxi (β₁₅)	−0.157	0.624	0.801				0.048	1.467	0.974
	Guizhou (β₁₆)	−0.022	0.720	0.976				−0.092	1.423	0.948
	Wave 2004 (β₁₇)	−0.331	0.446	0.458				−0.344	0.437	0.432
	Wave 2006 (β₁₈)	−0.771	0.439	0.080				−0.930	0.429	0.030	−0.789	0.181	<.0001
	Wave 2009 (β₁₉)	−0.198	0.558	0.723				0.180	0.553	0.745
	Wave 2011 (β₂₀)	0.259	0.342	0.449				0.216	0.499	0.665
Heteroscedasticity	Intercept (δ₀)	−0.301	0.954	0.753	0.252	0.120	0.036	0.055	2.070	0.979	−0.029	0.161	0.856
	NCMS (δ₁)	−0.183	0.142	0.197				−0.203	0.215	0.346
	Other Insurance (δ₂)	−0.341	0.223	0.127				−0.514	0.298	0.085
	Age (δ₃)	0.002	0.010	0.821				−0.004	0.016	0.779
	Gender (δ₄)	−0.024	0.127	0.851				0.036	0.150	0.813
	Marriage (δ₅)	0.245	0.129	0.056				0.191	0.166	0.251
	Education (δ₆)	0.012	0.016	0.461				0.006	0.022	0.799
	log(1+Income) (δ₇)	0.015	0.030	0.606				0.025	0.042	0.553
	Disease (δ₈)	0.178	0.147	0.225				0.101	0.169	0.548
	Heilongjiang (δ₉)	0.206	0.299	0.492				0.317	0.916	0.729
	Jiangsu (δ₁₀)	−0.388	0.441	0.379				−0.583	1.203	0.628
	Shandong (δ₁₁)	0.122	0.381	0.749				0.351	0.750	0.640
	Henan (δ₁₂)	0.303	0.271	0.264				0.435	0.839	0.604
	Hubei (δ₁₃)	0.088	0.342	0.797				0.186	0.934	0.842
	Hunan (δ₁₄)	−0.111	0.337	0.742				−0.061	0.782	0.938
	Guangxi (δ₁₅)	−0.196	0.350	0.576				−0.418	0.969	0.666
	Guizhou (δ₁₆)	0.048	0.362	0.894				0.087	0.889	0.922
	Wave 2004 (δ₁₇)	−0.095	0.248	0.701				−0.238	0.290	0.411
	Wave 2006 (δ₁₈)	−0.136	0.259	0.601				−0.269	0.292	0.358
	Wave 2009 (δ₁₉)	−0.038	0.283	0.894				−0.022	0.274	0.935
	Wave 2011 (δ₂₀)	0.159	0.138	0.248				0.139	0.230	0.545
	κ	2.340	0.148	<.0001	2.182	0.214	<.0001	2.578	0.191	<.0001	2.485	0.168	<.0001
Variance	σ_a	0.533	0.055	<.0001	0.534	0.055	<.0001	0.536	0.054	<.0001	0.537	0.055	<.0001
Components	σ_b	0.372	0.355	0.296	0.539	0.273	0.048	0.410	0.126	0.001	0.904	0.145	0.039
	ρ	0.275	0.698	0.693	0.729	0.452	0.107	0.194	0.425	0.647	0.515	0.249	<.0001

The fixed effects in Part (II) of the Type II m2REMs in Tables 4 –8 are the marginal effects of the covariates with respect to the logarithm, not to the original scale, of expected overall healthcare expenditure. The results on the fixed effects demonstrate that whether or not the older participants in the CHNS were insured by either the NCMS or other health insurance plans is not a statistically significant factor on the logarithmic scale of their expected overall healthcare expenditure. Yet, the significant impact of disease status was revealed on the logarithm of the expected overall healthcare expenditure among older adults, and most of the Type II m2REMs confirm that age and marriage status have significant impact as well. Education level in the log-normal and inverse Gamma Type II m2REMs is significant, but not in other models. Only the log-skew-normal Type II m2REMs shows the older residents in the survey year of 2006 may be dissimilar from the other survey years in terms of the logarithm of expected overall healthcare expenditures. The fixed effects in Part (II) of the Type I m2REMs in Tables 4 –8 are the marginal effects on logarithm of the expected positive healthcare expenditures. The significant impact of the NCMS and other health insurance coverage on the logarithmic scale of the expected positive healthcare expenditure is discovered only by the Type I log-normal m2REM, but not by the other four Type I models. The results show that disease status is significant on the logarithm of the expected positive healthcare expenditure among older adults, whereas the five models disagree with each other on age, marriage status, and education level. The fixed effects in Part (I) of the Type I and Type II m2REMs in Tables 4 –8 are the marginal effects of covariates with respect to the probit-transformed probability of healthcare utilization. In most of the Type I and Type II m2REMs, the fixed effects of the NCMS and other health insurance coverage are significant or have a p-value less than 0.1. Most of the Type I and Type II m2REMs agree that disease status, gender, the province of Shandong, the province of Guangxi, the survey year of 2004 and 2009 are significant factors on the probit-transformed probability of healthcare utilization among older adults in the CNHS. Heteroscedasticity among the overall and positive healthcare expenditures is not verified in the variance models of the Type I and Type II m2REMs, except that the results from the log-norma m2REMs imply the heteroscedastic variance may exist due to marriage statue or other insurance coverage. The variance components σ_a and σ_b in the m2REMs all have a p-value less than 0.0001, suggesting that unexplained heterogeneity must exist among the overall and positive healthcare expenditures. The correlation coefficients are not always significantly positive, which impose uncertainties to the correlation between the healthcare expenditures and the healthcare utilizations among older adults in the CHNS.

Table 5.

Parameter estimates, estimated standard errors, and p-values from fitting the Type I and Type II log-normal m2REMs to the CHNS data.

		Type I log-normal m2REM						Type II log-normal m2REM
		Full model			Reduced model			Full model			Reduced model
	Parameter	Estimate	SE	p-value	Estimate	SE	p-value	Estimate	SE	p-value	Estimate	SE	p-value
Part (I)	Intercept (α₀)	−2.607	0.321	<.0001	−2.478	0.218	<.0001	−2.260	0.316	<.0001	−2.342	0.215	<.0001
Fixed Effects	NCMS (α₁)	0.067	0.080	0.402	0.108	0.055	0.050	0.006	0.079	0.941	0.095	0.053	0.073
	Other Insurance (α₂)	0.169	0.089	0.058	0.222	0.083	0.007	0.037	0.091	0.684	0.115	0.085	0.174
	Age (α₃)	0.019	0.004	<.0001	0.017	0.003	<.0001	0.017	0.003	<.0001	0.016	0.003	<.0001
	Gender (α₄)	−0.151	0.052	0.004	−0.117	0.046	0.011	−0.118	0.051	0.022	−0.087	0.041	0.036
	Marriage (α₅)	0.084	0.055	0.130				0.052	0.055	0.344
	Education (α₆)	0.005	0.008	0.490				0.004	0.008	0.631
	Log(1+Income) (α₇)	−0.020	0.015	0.183				−0.023	0.015	0.124
	Disease (α₈)	0.498	0.049	<.0001	0.501	0.049	<.0001	0.490	0.049	<.0001	0.509	0.049	<.0001
	Heilongjiang (α₉)	−0.014	0.126	0.909				−0.085	0.124	0.491
	Jiangsu (α₁₀)	0.216	0.118	0.068				0.181	0.116	0.118
	Shandong (α₁₁)	−0.238	0.130	0.067				−0.304	0.128	0.018	−0.272	0.078	0.001
	Henan (α₁₂)	0.091	0.115	0.431				−0.011	0.113	0.925
	Hubei (α₁₃)	−0.019	0.121	0.878				−0.071	0.119	0.553
	Hunan (α₁₄)	0.019	0.121	0.877				−0.015	0.118	0.897
	Guangxi (α₁₅)	0.347	0.110	0.002	0.303	0.060	<.0001	0.240	0.108	0.026	0.214	0.054	<.0001
	Guizhou (α₁₆)	0.084	0.113	0.456				0.017	0.111	0.880
	Wave 2004 (α₁₇)	0.286	0.084	0.001	0.192	0.062	0.002	0.248	0.083	0.003	0.156	0.056	0.006
	Wave 2006 (α₁₈)	0.126	0.088	0.153				0.116	0.086	0.181
	Wave 2009 (α₁₉)	0.377	0.107	0.001	0.244	0.049	<.0001	0.378	0.106	0.000	0.211	0.045	<.0001
	Wave 2011 (α₂₀)	0.143	0.110	0.194				0.146	0.109	0.179
Part (II)	Intercept (β₀)	1.943	1.635	0.235	2.088	0.841	0.013	0.539	0.055	<.0001	0.585	0.913	0.522
Fixed Effects	NCMS (β₁)	−0.275	0.363	0.448	−0.397	0.147	0.007	0.569	1.851	0.759	−0.209	0.164	0.204
	Other Insurance (β₂)	−0.511	0.344	0.137	−0.379	0.249	0.128	−0.863	0.448	0.054	−1.446	0.301	<.0001
	Age (β₃)	0.050	0.019	0.007	0.052	0.011	<.0001	−1.585	0.377	<.0001	0.053	0.012	<.0001
	Gender (β₄)	−0.464	0.247	0.061				0.059	0.020	0.003
	Marriage (β₅)	1.029	0.223	<.0001	0.926	0.226	<.0001	−0.306	0.270	0.258	0.331	0.159	0.038
	Education (β₆)	0.091	0.037	0.015	0.057	0.022	0.009	0.697	0.257	0.007	0.051	0.021	0.017
	log(1+Income) (β₇)	0.026	0.054	0.631				0.084	0.039	0.032
	Disease (β₈)	1.029	0.236	<.0001	0.957	0.145	<.0001	−0.045	0.062	0.469	1.662	0.161	<.0001
	Heilongjiang (β₉)	0.845	0.623	0.175				1.720	0.252	<.0001
	Jiangsu (β₁₀)	−0.691	0.546	0.205				0.232	0.698	0.740
	Shandong (β₁₁)	1.347	0.835	0.107				−0.098	0.665	0.883
	Henan (β₁₂)	0.894	0.643	0.165				0.218	0.879	0.804
	Hubei (β₁₃)	0.256	0.636	0.688				0.188	0.665	0.777
	Hunan (β₁₄)	0.012	0.579	0.984				0.074	0.744	0.920
	Guangxi (β₁₅)	−0.088	0.550	0.873				−0.048	0.690	0.945
	Guizhou (β₁₆)	−0.435	0.551	0.430				0.053	0.648	0.935
	Wave 2004 (β₁₇)	0.007	0.416	0.987				−0.415	0.663	0.531
	Wave 2006 (β₁₈)	−0.826	0.414	0.046	−0.447	0.177	0.011	0.056	0.507	0.912
	Wave 2009 (β₁₉)	−0.257	0.476	0.590				−0.632	0.504	0.210
	Wave 2011 (β₂₀)	0.088	0.501	0.860				0.184	0.559	0.742
Heteroscedasticity	Intercept (δ₀)	0.102	0.384	0.791	0.468	0.075	<.0001	0.343	0.590	0.561	0.627	0.048	<.0001
	NCMS (δ₁)	−0.006	0.093	0.949				−0.125	0.488	0.798
	Other Insurance (δ₂)	−0.116	0.102	0.255				−0.169	0.120	0.157	−0.317	0.100	0.002
	Age (δ₃)	0.005	0.004	0.218				−0.470	0.135	0.001
	Gender (δ₄)	−0.067	0.057	0.236				0.007	0.005	0.147
	Marriage (δ₅)	0.165	0.058	0.005	0.135	0.063	0.032	−0.024	0.067	0.719
	Education (δ₆)	0.016	0.008	0.068				0.102	0.068	0.136
	log(1+Income) (δ₇)	0.011	0.016	0.500				0.017	0.010	0.070
	Disease (δ₈)	0.064	0.054	0.236				0.020	0.020	0.314
	Heilongjiang (δ₉)	−0.063	0.141	0.656				0.077	0.064	0.232
	Jiangsu (δ₁₀)	−0.268	0.132	0.043				−0.059	0.168	0.727
	Shandong (δ₁₁)	0.221	0.143	0.123				−0.107	0.164	0.513
	Henan (δ₁₂)	0.158	0.125	0.207				0.225	0.179	0.208
	Hubei (δ₁₃)	0.024	0.136	0.861				0.159	0.148	0.282
	Hunan (δ₁₄)	−0.122	0.135	0.367				0.134	0.164	0.413
	Guangxi (δ₁₅)	−0.075	0.121	0.536				−0.086	0.173	0.618
	Guizhou (δ₁₆)	−0.143	0.128	0.263				−0.039	0.152	0.798
	Wave 2004 (δ₁₇)	−0.005	0.100	0.963				−0.065	0.159	0.681
	Wave 2006 (δ₁₈)	−0.114	0.109	0.293				−0.010	0.121	0.936
	Wave 2009 (δ₁₉)	−0.077	0.120	0.523				−0.069	0.128	0.591
	Wave 2011 (δ₂₀)	−0.011	0.125	0.929				−0.051	0.146	0.728
Variance	σ_a	0.479	0.280	0.087	0.547	0.055	<.0001	0.064	0.154	0.676	0.531	0.055	<.0001
Components	σ_b	0.829	0.507	0.102	0.839	0.193	<.0001	0.822	0.172	<.0001	0.714	0.215	0.001
	Ρ	0.540	0.054	<.0001	0.585	0.225	0.009	0.474	0.244	0.052	0.376	0.278	0.176

Table 6.

Parameter estimates, estimated standard errors, and p-values from fitting the Type I and Type II generalized Gamma m2REMs to the CHNS data.

		Type I generalized Gamma m2REM						Type II generalized Gamma m2REM
		Full model			Reduced model			Full model			Reduced model
	Parameter	Estimate	SE	p-value	Estimate	SE	p-value	Estimate	SE	p-value	Estimate	SE	p-value
Part (I)	Intercept (α₀)	−2.334	0.328	<.0001	−2.344	0.219	<.0001	−2.242	0.318	<.0001	−2.270	0.215	<.0001
Fixed Effects	NCMS (α₁)	0.049	0.080	0.543	0.061	0.055	0.265	0.018	0.080	0.820	0.065	0.053	0.216
	Other Insurance (α₂)	0.155	0.089	0.083	0.152	0.083	0.069	0.115	0.090	0.201	0.170	0.083	0.040
	Age (α₃)	0.017	0.004	<.0001	0.016	0.003	<.0001	0.016	0.003	<.0001	0.016	0.003	<.0001
	Gender (α₄)	−0.138	0.052	0.008	−0.126	0.046	0.006	−0.144	0.051	0.005	−0.129	0.042	0.002
	Married (α₅)	0.075	0.055	0.174				0.064	0.055	0.246
	Education (α₆)	0.003	0.008	0.706				0.003	0.008	0.683
	log(1+Income) (α₇)	−0.026	0.015	0.081				−0.024	0.015	0.111
	Disease (α₈)	0.494	0.050	<.0001	0.505	0.049	<.0001	0.491	0.050	<.0001	0.494	0.049	<.0001
	Heilongjiang (α₉)	−0.071	0.125	0.568				−0.094	0.125	0.452
	Jiangsu (α₁₀)	0.165	0.118	0.161				0.181	0.117	0.122
	Shandong (α₁₁)	−0.280	0.129	0.030	−0.311	0.090	0.001	−0.294	0.128	0.022	−0.285	0.081	0.000
	Henan (α₁₂)	0.028	0.115	0.807				0.014	0.114	0.900
	Hubei (α₁₃)	−0.076	0.121	0.527				−0.091	0.120	0.450
	Hunan (α₁₄)	−0.036	0.120	0.763				−0.037	0.120	0.756
	Guangxi (α₁₅)	0.302	0.110	0.006	0.258	0.061	<.0001	0.281	0.109	0.010	0.208	0.056	0.000
	Guizhou (α₁₆)	0.032	0.112	0.779				0.024	0.112	0.832
	Wave 2004 (α₁₇)	0.285	0.084	0.001	0.170	0.062	0.006	0.243	0.083	0.004	0.131	0.056	0.018
	Wave 2006 (α₁₈)	0.142	0.088	0.106				0.087	0.087	0.316
	Wave 2009 (α₁₉)	0.401	0.107	0.000	0.274	0.049	<.0001	0.385	0.107	0.000	0.228	0.045	<.0001
	Wave 2011 (α₂₀)	0.174	0.110	0.113				0.148	0.109	0.178
Part (II)	Intercept (β₀)	2.169	1.350	0.108	2.090	0.843	0.013	2.003	1.314	0.128	1.998	0.809	0.014
Fixed Effects	NCMS (β₁)	−0.171	0.320	0.593	−0.109	0.133	0.410	−0.150	0.293	0.609	0.000	0.150	0.998
	Other Insurance (β₂)	−0.478	0.290	0.099	−0.386	0.229	0.092	−0.277	0.326	0.396	−0.267	0.270	0.323
	Age (β₃)	0.053	0.015	0.000	0.055	0.011	<.0001	0.040	0.014	0.004	0.037	0.011	0.001
	Gender (β₄)	−0.162	0.214	0.450				−0.326	0.210	0.121
	Married (β₅)	0.671	0.233	0.004	0.606	0.174	0.001	0.586	0.203	0.004
	Education (β₆)	0.049	0.028	0.078				0.031	0.030	0.307
	log(1+Income)(β₇)	−0.011	0.053	0.837				−0.094	0.044	0.034
	Disease (β₈)	1.040	0.215	<.0001	1.152	0.152	<.0001	1.677	0.189	<.0001	1.689	0.165	<.0001
	Heilongjiang (β₉)	0.706	0.463	0.127				0.579	0.500	0.247
	Jiangsu (β₁₀)	−0.116	0.405	0.774				0.086	0.459	0.851
	Shandong (β₁₁)	0.287	0.530	0.589				0.124	0.552	0.822
	Henan (β₁₂)	0.215	0.412	0.602				0.207	0.464	0.656
	Hubei (β₁₃)	0.004	0.449	0.992				−0.148	0.494	0.765
	Hunan (β₁₄)	0.197	0.449	0.661				0.110	0.470	0.815
	Guangxi (β₁₅)	0.278	0.396	0.482				0.642	0.441	0.146
	Guizhou (β₁₆)	0.027	0.410	0.948				−0.021	0.456	0.963
	Wave 2004 (β₁₇)	0.034	0.458	0.941				0.037	0.455	0.935
	Wave 2006 (β₁₈)	−0.347	0.523	0.507				−0.604	0.487	0.215
	Wave 2009 (β₁₉)	−0.214	0.606	0.724				0.104	0.533	0.845
	Wave 2011 (β₂₀)	0.228	0.585	0.697				0.202	0.544	0.710
Heteroscedasticity	Intercept (δ₀)	−0.395	1.017	0.698	−0.464	0.093	<.0001	0.373	0.836	0.656	−0.461	0.083	<.0001
	NCMS (δ₁)	0.118	0.268	0.660				0.077	0.225	0.733
	Other Insurance (δ₂)	0.005	0.228	0.984				0.040	0.229	0.861
	Age (δ₃)	0.001	0.009	0.866				−0.008	0.009	0.336
	Gender (δ₄)	−0.001	0.132	0.991				−0.013	0.125	0.917
	Married (δ₅)	0.179	0.180	0.320				0.114	0.117	0.331
	Education (δ₆)	−0.012	0.019	0.544				−0.015	0.019	0.441
	log(1+Income)(δ₇)	0.001	0.043	0.991				−0.013	0.036	0.712
	Disease (δ₈)	0.202	0.144	0.163				0.116	0.126	0.358
	Heilongjiang (δ₉)	−0.118	0.299	0.694				−0.146	0.285	0.610
	Jiangsu (δ₁₀)	−0.280	0.290	0.335				−0.241	0.258	0.349
	Shandong (δ₁₁)	0.116	0.362	0.749				0.170	0.317	0.591
	Henan (δ₁₂)	0.075	0.253	0.768				0.088	0.267	0.742
	Hubei (δ₁₃)	0.058	0.262	0.826				0.052	0.272	0.849
	Hunan (δ₁₄)	−0.173	0.334	0.605				−0.202	0.269	0.454
	Guangxi (δ₁₅)	0.065	0.235	0.781				0.059	0.246	0.810
	Guizhou (δ₁₆)	−0.040	0.258	0.876				−0.076	0.255	0.765
	Wave 2004 (δ₁₇)	−0.148	0.394	0.707				−0.226	0.280	0.421
	Wave 2006 (δ₁₈)	−0.181	0.436	0.679				−0.274	0.316	0.387
	Wave 2009 (δ₁₉)	−0.311	0.577	0.589				−0.284	0.367	0.440
	Wave 2011 (δ₂₀)	−0.175	0.503	0.728				−0.192	0.357	0.590
	κ	1.913	1.142	0.094	2.315	0.284	<.0001	2.414	0.324	<.0001	2.372	0.242	<.0001
Variance	σ_A	0.543	0.055	<.0001	0.557	0.054	<.0001	0.545	0.054	<.0001	0.547	0.055	<.0001
Components	σ_B	1.632	0.071	<.0001	1.686	0.066	<.0001	1.605	0.060	<.0001	1.697	0.059	<.0001
	ρ_AB	0.195	0.313	0.533	0.226	0.115	0.050	0.144	0.112	0.199	0.113	0.090	0.209

Table 7.

Parameter estimates, estimated standard errors, and p-values from fitting the Type I and Type II Gamma m2REMs to the CHNS data.

		Type I Gamma m2REM						Type II Gamma m2REM
		Full model			Reduced model			Full model			Reduced model
	Parameter	Estimate	SE	p-value	Estimate	SE	p-value	Estimate	SE	p-value	Estimate	SE	p-value
Part (I)	Intercept (α₀)	−2.177	0.320	<.0001	−2.235	0.218	<.0001	−2.256	0.317	<.0001	−2.296	0.215	<.0001
Fixed Effects	NCMS (α₁)	0.119	0.080	0.136	0.107	0.055	0.052	0.044	0.080	0.581	0.080	0.053	0.127
	Other Insurance (α₂)	0.144	0.089	0.107	0.234	0.082	0.005	0.146	0.090	0.103	0.170	0.083	0.041
	Age (α₃)	0.016	0.004	<.0001	0.014	0.003	<.0001	0.016	0.003	<.0001	0.016	0.003	<.0001
	Gender (α₄)	−0.142	0.052	0.006	−0.112	0.046	0.014	−0.146	0.051	0.005	−0.110	0.041	0.008
	Marriage (α₅)	0.057	0.055	0.306				0.061	0.055	0.263
	Education (α₆)	0.004	0.008	0.575				0.004	0.008	0.653
	log(1+Income) (α₇)	−0.025	0.015	0.090				−0.026	0.015	0.086
	Disease (α₈)	0.490	0.049	<.0001	0.485	0.049	<.0001	0.499	0.049	<.0001	0.497	0.049	<.0001
	Heilongjiang (α₉)	−0.050	0.125	0.690				−0.084	0.124	0.498
	Jiangsu (α₁₀)	0.171	0.118	0.145				0.160	0.116	0.169
	Shandong (α₁₁)	−0.269	0.129	0.037	−0.313	0.089	0.001	−0.304	0.128	0.018	−0.291	0.080	0.000
	Henan (α₁₂)	0.037	0.115	0.744				0.015	0.113	0.896
	Hubei (α₁₃)	−0.077	0.121	0.523				−0.089	0.119	0.457
	Hunan (α₁₄)	0.018	0.120	0.880				−0.031	0.119	0.795
	Guangxi (α₁₅)	0.299	0.110	0.007	0.266	0.060	<.0001	0.286	0.108	0.008	0.198	0.055	0.000
	Guizhou (α₁₆)	0.023	0.112	0.837				0.018	0.111	0.871
	Wave 2004 (α₁₇)	0.156	0.081	0.055	0.189	0.062	0.002	0.268	0.084	0.001	0.132	0.056	0.018
	Wave 2006 (α₁₈)	−0.006	0.086	0.946				0.116	0.087	0.186
	Wave 2009 (α₁₉)	0.210	0.106	0.047	0.252	0.049	<.0001	0.385	0.107	0.000	0.226	0.045	<.0001
	Wave 2011 (α₂₀)	−0.008	0.108	0.940				0.151	0.110	0.168
Part (II)	Intercept (β₀)	1.234	1.278	0.334	1.288	0.874	0.141	0.658	1.365	0.630	0.651	0.914	0.477
Fixed Effects	NCMS (β₁)	−0.242	0.272	0.374	−0.224	0.143	0.117	−0.018	0.297	0.951	0.007	0.159	0.963
	Other Insurance (β₂)	−0.771	0.288	0.007	−0.728	0.243	0.003	−0.390	0.319	0.222	−0.368	0.275	0.181
	Age (β₃)	0.064	0.014	<.0001	0.062	0.012	<.0001	0.054	0.015	0.000	0.049	0.012	<.0001
	Gender (β₄)	−0.307	0.195	0.115				−0.441	0.212	0.037
	Marriage (β₅)	0.772	0.187	<.0001	0.651	0.172	0.000	0.712	0.208	0.001	0.431	0.173	0.013
	Education (β₆)	0.066	0.029	0.021	0.065	0.023	0.004	0.048	0.031	0.121
	log(1+Income) (β₇)	0.005	0.046	0.913				−0.086	0.050	0.081
	Disease (β₈)	1.091	0.177	<.0001	1.081	0.151	<.0001	1.736	0.188	<.0001	1.693	0.167	<.0001
	Heilongjiang (β₉)	0.673	0.462	0.146				0.467	0.504	0.355
	Jiangsu (β₁₀)	−0.186	0.422	0.660				0.072	0.459	0.876
	Shandong (β₁₁)	0.460	0.560	0.411				0.121	0.605	0.842
	Henan (β₁₂)	0.139	0.427	0.745				0.148	0.463	0.749
	Hubei (β₁₃)	0.008	0.463	0.986				−0.233	0.499	0.640
	Hunan (β₁₄)	0.014	0.431	0.974				−0.015	0.472	0.975
	Guangxi (β₁₅)	0.161	0.405	0.691				0.615	0.438	0.161
	Guizhou (β₁₆)	−0.214	0.414	0.606				−0.176	0.451	0.697
	Wave 2004 (β₁₇)	0.051	0.388	0.895				0.129	0.458	0.779
	Wave 2006 (β₁₈)	−0.514	0.382	0.179				−0.697	0.463	0.132
	Wave 2009 (β₁₉)	−0.211	0.421	0.616				0.037	0.507	0.941
	Wave 2011 (β₂₀)	0.059	0.441	0.894				0.105	0.524	0.842
Heteroscedasticity	Intercept (δ₀)	−0.649	0.527	0.219	0.082	0.032	0.011	−0.170	0.503	0.736	0.097	0.032	0.003
	NCMS (δ₁)	0.024	0.147	0.870				0.174	0.153	0.257
	Other Insurance (δ₂)	−0.172	0.163	0.292				−0.074	0.159	0.640
	Age (δ₃)	0.007	0.005	0.194				0.003	0.005	0.590
	Gender (δ₄)	−0.057	0.077	0.461				−0.010	0.076	0.898
	Marriage (δ₅)	0.143	0.076	0.060				0.137	0.076	0.073
	Education (δ₆)	0.003	0.012	0.782				−0.002	0.012	0.895
	log(1+Income) (δ₇)	0.018	0.026	0.492				0.008	0.024	0.732
	Disease (δ₈)	0.137	0.075	0.067				0.085	0.074	0.251
	Heilongjiang (δ₉)	−0.088	0.171	0.607				−0.109	0.172	0.525
	Jiangsu (δ₁₀)	−0.172	0.161	0.286				−0.171	0.160	0.287
	Shandong (δ₁₁)	0.147	0.192	0.443				0.154	0.190	0.419
	Henan (δ₁₂)	0.049	0.155	0.749				0.097	0.152	0.526
	Hubei (δ₁₃)	0.048	0.160	0.762				0.038	0.160	0.810
	Hunan (δ₁₄)	−0.213	0.181	0.241				−0.197	0.181	0.276
	Guangxi(δ₁₅)	0.024	0.142	0.866				0.038	0.141	0.787
	Guizhou(δ₁₆)	−0.103	0.150	0.491				−0.095	0.148	0.522
	Wave 2004 (δ₁₇)	0.021	0.186	0.910				−0.102	0.174	0.560
	Wave 2006 (δ₁₈)	−0.062	0.190	0.746				−0.219	0.184	0.235
	Wave 2009 (δ₁₉)	−0.040	0.220	0.854				−0.292	0.219	0.182
	Wave 2011 (δ₂₀)	−0.037	0.227	0.872				−0.230	0.222	0.300
Variance	σ_a	0.566	0.055	<.0001	0.530	0.055	<.0001	0.530	0.055	<.0001	0.532	0.055	<.0001
Components	σ_b	1.556	0.079	<.0001	1.619	0.071	<.0001	1.517	0.077	<.0001	1.580	0.069	<.0001
	ρ	0.424	0.136	0.002	0.451	0.129	0.001	0.385	0.144	0.008	0.263	0.151	0.082

Table 8.

Parameter estimates, estimated standard errors, and p-values from fitting the Type I and Type II inverse Gamma m2REMs to the CHNS data.

		Type I inverse Gamma m2REM						Type II inverse Gamma m2REM
		Full model			Reduced model			Full model			Reduced model
	Parameter	Estimate	SE	p-value	Estimate	SE	p-value	Estimate	SE	p-value	Estimate	SE	p-value
Part (I)	Intercept (α₀)	–2.248	0.319	<.0001	–2.482	0.220	<.0001	–2.245	0.315	<.0001	–2.290	0.216	<.0001
	NCMS (α₁)	0.044	0.080	0.578	0.097	0.055	0.080	0.041	0.080	0.606	0.098	0.052	0.060
	Other Insurance (α₂)	0.144	0.089	0.107	0.229	0.083	0.006	0.143	0.089	0.111	0.191	0.083	0.022
	Age (α₃)	0.016	0.003	<.0001	0.018	0.003	<.0001	0.016	0.003	<.0001	0.016	0.003	<.0001
	Gender (α₄)	–0.145	0.052	0.005	–0.101	0.046	0.028	–0.138	0.051	0.007	–0.117	0.046	0.010
	Married (α₅)	0.063	0.055	0.252				0.066	0.055	0.228
	Education (α₆)	0.003	0.008	0.683				0.001	0.008	0.874
	log(1+Income) (α₇)	–0.029	0.015	0.047				–0.027	0.015	0.067
	Disease (α₈)	0.496	0.049	<.0001	0.502	0.049	<.0001	0.487	0.049	<.0001	0.488	0.049	<.0001
	Heilongjiang (α₉)	–0.043	0.125	0.728				–0.044	0.124	0.724
	Jiangsu (α₁₀)	0.196	0.118	0.096				0.192	0.117	0.100
	Shandong (α₁₁)	–0.263	0.129	0.041	–0.299	0.090	0.001	–0.262	0.128	0.041	–0.252	0.078	0.001
	Henan (α₁₂)	0.054	0.114	0.635				0.054	0.114	0.633
	Hubei (α₁₃)	–0.053	0.121	0.662				–0.050	0.120	0.677
	Hunan (α₁₄)	–0.003	0.120	0.978				0.000	0.119	1.000
	Guangxi (α₁₅)	0.321	0.110	0.003	0.267	0.060	<.0001	0.313	0.109	0.004	0.209	0.055	0.000
	Guizhou (α₁₆)	0.055	0.112	0.622				0.059	0.112	0.600
	Wave 2004 (α₁₇)	0.278	0.084	0.001	0.188	0.062	0.003	0.278	0.083	0.001	0.126	0.055	0.021
	Wave 2006 (α₁₈)	0.129	0.088	0.142				0.130	0.087	0.137
	Wave 2009 (α₁₉)	0.403	0.107	0.000	0.260	0.049	<.0001	0.404	0.107	0.000	0.207	0.044	<.0001
	Wave 2011 (α₂₀)	0.171	0.110	0.118				0.167	0.110	0.128
Part (II)	Intercept (β₀)	2.152	0.949	0.023	2.777	0.812	0.001	0.816	1.076	0.448	0.825	0.990	0.405
	NCMS (β₁)	–0.564	0.219	0.010	–0.103	0.126	0.410	–0.158	0.250	0.528	0.064	0.145	0.657
	Other Insurance (β₂)	–0.491	0.229	0.032	–0.301	0.213	0.158	–0.110	0.264	0.678	–0.130	0.247	0.599
	Age (β₃)	0.072	0.010	<.0001	0.061	0.010	<.0001	0.072	0.012	<.0001	0.071	0.011	<.0001
	Gender (β₄)	–0.261	0.154	0.090				–0.482	0.176	0.006	–0.423	0.168	0.012
	Married (β₅)	0.547	0.156	0.001	0.592	0.153	0.000	0.565	0.181	0.002	0.429	0.156	0.006
	Education (β₆)	0.072	0.024	0.003				0.056	0.027	0.040	0.071	0.023	0.002
	log(1+Income) (β₇)	0.083	0.042	0.048				–0.033	0.047	0.485
	Disease (β₈)	0.854	0.138	<.0001	0.941	0.138	<.0001	1.534	0.155	<.0001	1.503	0.153	<.0001
	Heilongjiang (β₉)	1.185	0.376	0.002	1.011	0.257	<.0001	0.980	0.435	0.024	0.909	0.258	0.000
	Jiangsu (β₁₀)	0.041	0.341	0.904				0.474	0.394	0.230
	Shandong (β₁₁)	0.194	0.400	0.628				–0.115	0.463	0.804
	Henan (β₁₂)	0.132	0.337	0.696				–0.034	0.391	0.930
	Hubei (β₁₃)	0.102	0.363	0.779				–0.057	0.421	0.893
	Hunan (β₁₄)	0.362	0.354	0.307				0.374	0.412	0.365
	Guangxi (β₁₅)	0.084	0.318	0.791				0.567	0.369	0.124
	Guizhou (β₁₆)	0.049	0.332	0.882				0.102	0.386	0.792
	Wave 2004 (β₁₇)	–0.184	0.240	0.443				0.050	0.276	0.857
	Wave 2006 (β₁₈)	–0.472	0.241	0.051				–0.479	0.281	0.088
	Wave 2009 (β₁₉)	–0.147	0.293	0.615				0.177	0.338	0.601
	Wave 2011 (β₂₀)	0.016	0.305	0.959				0.105	0.352	0.766
Heteroscedasticity	Intercept (δ₀)	0.390	0.376	0.299	1.861	0.152	<.0001	1.318	0.370	0.000	2.101	0.204	<.0001
	NCMS (δ₁)	–0.140	0.089	0.116				0.012	0.089	0.896
	Other Insurance (δ₂)	–0.081	0.093	0.382				–0.007	0.092	0.937
	Age (δ₃)	0.019	0.004	<.0001				0.010	0.004	0.011
	Gender (δ₄)	–0.046	0.062	0.460				–0.056	0.062	0.365
	Married (δ₅)	0.088	0.064	0.166				0.049	0.063	0.437
	Education (δ₆)	0.014	0.009	0.136				0.008	0.009	0.418
	log(1+Income) (δ₇)	0.054	0.018	0.003				0.022	0.017	0.202
	Disease (δ₈)	–0.009	0.054	0.865				–0.005	0.054	0.927
	Heilongjiang (δ₉)	0.102	0.149	0.494				0.059	0.150	0.693
	Jiangsu (δ₁₀)	–0.064	0.137	0.642				0.017	0.136	0.899
	Shandong (δ₁₁)	0.040	0.158	0.800				0.077	0.156	0.624
	Henan (δ₁₂)	0.054	0.134	0.685				–0.035	0.134	0.797
	Hubei (δ₁₃)	0.063	0.145	0.662				0.044	0.145	0.762
	Hunan (δ₁₄)	–0.066	0.143	0.641				–0.047	0.142	0.739
	Guangxi (δ₁₅)	–0.039	0.127	0.757				–0.009	0.126	0.944
	Guizhou (δ₁₆)	–0.011	0.133	0.936				–0.012	0.132	0.929
	Wave 2004 (δ₁₇)	–0.055	0.095	0.566				–0.137	0.094	0.147
	Wave 2006 (δ₁₈)	–0.027	0.096	0.780				–0.122	0.095	0.200
	Wave 2009 (δ₁₉)	–0.044	0.118	0.711				–0.167	0.116	0.152
	Wave 2011 (δ₂₀)	–0.050	0.124	0.685				–0.136	0.122	0.264
Variance	σ_a	0.533	0.056	<.0001	0.544	0.055	<.0001	0.531	0.056	<.0001	0.525	0.055	<.0001
Components	σ_b	1.369	0.066	<.0001	1.410	0.069	<.0001	1.357	0.066	<.0001	1.370	0.064	<.0001
	ρ	0.361	0.150	0.016	0.348	0.144	0.016	0.361	0.150	0.016	0.132	0.163	0.418

6.4 Estimation and comparison of incremental effects of the NCMS

The primary objective of this empirical analysis is to examine the incremental effect and average incremental effect of the NCMS, as well as the coverage of other insurance, on both the positive and the overall healthcare expenditures among the older adults who participated in the CHNS. The estimation and variance estimation procedures summarized in Section 4 were applied to assess these effects and their standard errors (SEs). To remove any interference induced by the insignificant independent variables other than the NCMS and other insurance coverage, the incremental and average incremental effects were estimated and further tested under the reduced model in Tables 4 –8.

Table 9 reports the estimates, the estimated SEs, and the p-values of the average incremental effects of the NCMS

\bar{^} π_{NCMS} (\hat{ϑ})

and the average incremental effects of other insurance

\bar{^} π_{INSU} (\hat{ϑ})

on the expected overall healthcare expenditure, given by the five Type II m2REMs. Table 9 also reports the estimates, the estimated SEs, and the p-values of the average incremental effects of the NCMS and other insurance

\bar{^} π_{NCMS}^{+} (\hat{ϑ})

and

\bar{^} π_{INSU}^{+} (\hat{ϑ})

with respect to the expected positive healthcare expenditure, given the five Type I m2REMs. The average incremental effects of the NCMS estimated by the Type II m2REMs range from –80.89 (SE: 54.19, log-skew-normal) to 58.20 (SE: 132.71, inverse Gamma) on the expected overall healthcare expenditure. Large estimated SEs and p-values of these average incremental effects signify the ineffectiveness of the NCMS in reducing the average medical expenditure among the rural older adults in China. The average incremental effects of the NCMS on the expected positive healthcare expenditure obtained from the Type I m2REMs range from –336.86 (SE: 141.91, log-normal) to –108.94 (SE: 135.43, generalized Gamma). Four out of five Type I m2REMs confirm an insignificant average incremental effect of the NCMS on the positive healthcare expenditures with the log-normal m2REM as an exception. The average incremental effects of other insurance range from –137.32 (SE: 31.05, log-normal) to –42.59 (SE: 39.37, generalized Gamma) on the expected overall healthcare expenditure, and range from –788.59 (SE: 568.02, log-normal) to –212.33 (SE: 242.74, inverse Gamma) on the expected positive healthcare expenditure. Only the log-normal and Gamma m2REMs indicate that the average incremental effect of other insurance coverage may be significant on the overall healthcare expenditures.

Table 9.

Estimates, estimated standard errors, and p-values of the average incremental effects of the NCMS and other insurance on the expected overall and positive healthcare expenditure, as well as the AIC and the BIC values, given by the Type II and Type I m2REMs.

		Log-normal	Log-skew-normal	Generalized Gamma	Gamma	Inverse Gamma
Type I m2REM models
	AIC	17138	17096	17246	17223	17171
	BIC	17265	17206	17368	17344	17293
${\bar{π}}_{NCMS}^{+} (ϑ)$	Estimate	–336.86	–278.9	–108.94	–183.17	–302.99
	SE	141.91	174.95	135.43	124.45	389.33
	p-value	0.02	0.11	0.42	0.14	0.44
	EMSE	43759.17	39769.95	23852.39	15486.75	165938.67
${\bar{π}}_{INSU}^{+} (ϑ)$	Estimate	–283.39	–212.33	–333.75	–456.5	–788.59
	SE	169.27	242.74	180.40	138.47	568.02
	p-value	0.09	0.38	0.06	0.00	0.17
	EMSE	58622.32	118544.10	47611.52	19174.94	432925.91
Type II m2REM models
	AIC	17136	17124	17244	17224	17166
	BIC	17263	17239	17359	17340	17299
${\bar{π}}_{NCMS} (ϑ)$	Estimate	–34.52	–80.89	–0.05	1.08	58.20
	SE	28.98	54.19	26.78	22.93	132.71
	$p -$ value	0.23	0.14	1.00	0.96	0.66
	EMSE	2107.49	9654.81	718.36	525.91	20873.38
${\bar{π}}_{INSU} (ϑ)$	Estimate	–137.32	–95.24	–42.59	–45.61	–111.48
	SE	31.05	67.22	39.37	30.55	208.63
	p-value	0.00	0.16	0.28	0.14	0.59
	EMSE	9375.57	6982.03	1559.13	933.22	47866.53

By assuming the log-normal Type I and Type II m2REMs were consistent and consequently were used as the pre-specified model, the corresponding empirical MSEs of $\bar{^} π_{NCMS} (\hat{ϑ})$ , $\bar{^} π_{INSU} (\hat{ϑ})$ , $\bar{^} π_{NCMS}^{+} (\hat{ϑ})$ , and $\bar{^} π_{INSU}^{+} (\hat{ϑ})$ from the reduced m2REMs were calculated for the m2REMs and are reported in Table 4. Following the effect-specific model comparison criterion, the Type I and Type II Gamma m2REMs are selected as the best models for estimating the average incremental effects of both the NCMS and other insurance coverage. The AIC and the BIC values of the reduced m2REMs are reported in Table 4 as well. Both the AIC and the BIC select the log-skew-normal m2REM as the best model among the Type I m2REMs and the Type II m2REM.

Under the Type I log-skew-normal m2REM, the incremental effects (not the average incremental effects) of the NCMS and other insurance coverage on the expected positive healthcare expenditure were estimated as –472.45 (SE: 304.26, p-value: 0.12) and –390.49 (459.19, 0.40), respectively, for the diseased older adults and were estimated as –194.42 (122.57, 0.11) and –160.70 (186.08, 0.39), respectively, for the nondiseased older adults. Using the Type II log-skew-normal m2REM, the incremental effects of the NCMS on the expected overall healthcare expenditure were estimated under the six combinations of marriage status, disease status, and the survey year in 2006 versus other survey years. In 2006, the estimates, as well as the estimated SEs and the p-values are –131.08 (SE: 89.44, p-value: 0.14) for the married and diseased older adults, –26.62 (17.75, 0.13) for the married and nondiseased older adults, –82.39(56.33, 0.14) for the diseased older adults with other marriage status, and –16.73(11.20, 0.14) for the nondiseased older adults with other marriage status. In other survey years, the estimates, as well as the estimated SEs and the p-values are –288.42(199.67, 0.15) for the married and diseased older adults, –58.56 (39.63, 0.14) for the married and nondiseased older adults, –181.28 (125.84, 0.15) for the diseased older adults with other marriage status, and –36.80 (25.01, 0.14) for the nondiseased older adults with other marriage status. None of estimated incremental effects of the NCMS is significant on either the overall or the positive expected healthcare expenditures.

7 Concluding remarks

In this article, we explore a useful array of model specifications that may be used as alternatives to the generalized Gamma m2REMs, including log-normal, log-skew-normal, generalized Gamma, Weibull, Gamma, and inverse Gamma distributions in Part (II) of the m2REMs. These alternative models supply additional choices and needed flexibility for analyzing the healthcare expenditure panel data with excess zeros. We present approaches for deriving marginal inference, via marginal and incremental effects, for this set of models and provide guidance on choosing the best fitting model based upon the empirical MSE and information criteria. In the empirical analysis, using the proposed m2REMs, we examine the influence of the NCMS and other insurance coverage on individual healthcare expenditures among the older adults lived in rural China. Our conclusion is in alignment with the finding in Lei and Lin¹² who did not identify any significant impact of the NCMS on the average medical expenditures for residents in China.

A cautionary note for using the m2REMs should be raised when the panel data are unbalanced and the missingness responses are not missing at random (the probability that responses are missing depends on both the observed and the missing responses). With the assumption of not missing at random, we can develop either selection models or pattern-mixture models, combined with the m2REMs, to address such an issue of non-ignorable missingness.¹⁹ Yet, the developed methods must be valid with respect to the potential missing mechanism and sensitivity analysis may be required.

Footnotes

Acknowledgements

We sincerely thank two anonymous reviewers and Editors for their valuable comments, which had substantially improved this article. This research used data from China Health and Nutrition Survey (CHNS).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Dr Wei Liu’s research was partially supported by National Natural Science Foundation of China (Project 11601106). We thank the National Institute for Nutrition and Health, China Center for Disease Control and Prevention, Carolina Population Center (P2C HD050924, T32 HD007168), the University of North Carolina at Chapel Hill, the NIH (R01-HD30880, DK056350, R24 HD050924, and R01-HD38700) and the NIH Fogarty International Center (D43 TW009077, D43 TW007709) for financial support for the CHNS data collection and analysis files from 1989 to 2015 and future surveys, and the China–Japan Friendship Hospital, Ministry of Health for support for CHNS 2009, Chinese National Human Genome Center at Shanghai since 2009, and Beijing Municipal Center for Disease Prevention and Control since 2011.

Appendix 1. Summary statistical tables

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	Type I m2REM		Type II m2REM
Log-normal	Part (I):	$ℎ_{m_{1}} [P (Y_{it} > 0 \| X_{it})] = X_{it}' α_{m}$	Part (I):	$ℎ_{m_{1}} [P (Y_{it} > 0 \| X_{it})] = X_{it}' α_{m}$
		$ℎ_{c_{1}} [P (Y_{it} > 0 \| X_{it}, a_{it})] = Δ_{it} + a_{it}$		$ℎ_{c_{1}} [P (Y_{it} > 0 \| X_{it}, a_{it})] = Δ_{it} + a_{it}$
	Part (II):	$(Y_{it} \| Y_{it} > 0, X_{it}, b_{it}) \sim f_{LN} (y; μ_{it}, σ_{it})$	Part (II):	$(Y_{it} \| Y_{it} > 0, X_{it}, b_{it}) \sim f_{LN} (y; μ_{it}, σ_{it})$
		$ℎ_{m_{2}} [E (Y_{it} \| Y_{it} > 0, X_{it})] = X_{it}' β_{m}$		$ℎ_{m_{3}} [E (Y_{it} \| X_{it})] = X_{it}' β_{m}$
		$μ_{it} \| Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} + b_{it}$		$μ_{it} \| Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} + b_{it}$
		$log (σ_{it} \| Y_{it} > 0, X_{it}) = X_{it}' δ$		$log (σ_{it} \| Y_{it} > 0, X_{it}) = X_{it}' δ$
Log-skew-normal	Part (I):	$ℎ_{m_{1}} [P (Y_{it} > 0 \| X_{it})] = X_{it}' α_{m}$	Part (I):	$ℎ_{m_{1}} [P (Y_{it} > 0 \| X_{it})] = X_{it}' α_{m}$
		$ℎ_{c_{1}} [P (Y_{it} > 0 \| X_{it}, a_{it})] = Δ_{it} + a_{it}$		$ℎ_{c_{1}} [P (Y_{it} > 0 \| X_{it}, a_{it})] = Δ_{it} + a_{it}$
	Part (II):	$(Y_{it} \| Y_{it} > 0, X_{it}, b_{it}) \sim f_{LSN} (y; μ_{it}, σ_{it}, κ)$	Part (II):	$(Y_{it} \| Y_{it} > 0, X_{it}, b_{it}) \sim f_{LSN} (y; μ_{it}, σ_{it}, κ)$
		$ℎ_{m_{2}} [E (Y_{it} \| Y_{it} > 0, X_{it})] = X_{it}' β_{m}$		$ℎ_{m_{3}} [E (Y_{it} \| X_{it})] = X_{it}' β_{m}$
		$μ_{it} \| Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} + b_{it}$		$μ_{it} \| Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} + b_{it}$
		$log (σ_{it} \| Y_{it} > 0, X_{it}) = X_{it}' δ$		$log (σ_{it} \| Y_{it} > 0, X_{it}) = X_{it}' δ$
Generalized Gamma	Part (I):	$ℎ_{m_{1}} [P (Y_{it} > 0 \| X_{it})] = X_{it}' α_{m}$	Part (I):	$ℎ_{m_{1}} [P (Y_{it} > 0 \| X_{it})] = X_{it}' α_{m}$
(Weibull: κ = 1)		$ℎ_{c_{1}} [P (Y_{it} > 0 \| X_{it}, a_{it})] = Δ_{it} + a_{it}$		$ℎ_{c_{1}} [P (Y_{it} > 0 \| X_{it}, a_{it})] = Δ_{it} + a_{it}$
	Part (II):	$(Y_{it} \| Y_{it} > 0, X_{it}, b_{it}) \sim f_{GG} (y; μ_{it}, σ_{it}, κ)$	Part (II):	$(Y_{it} \| Y_{it} > 0, X_{it}, b_{it}) \sim f_{GG} (y; μ_{it}, σ_{it}, κ)$
		$ℎ_{m_{2}} [E (Y_{it} \| Y_{it} > 0, X_{it})] = X_{it}' β_{m}$		$ℎ_{m_{3}} [E (Y_{it} \| X_{it})] = X_{it}' β_{m}$
		$μ_{it} \| Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} + b_{it}$		$μ_{it} \| Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} + b_{it}$
		$log (σ_{it} \| Y_{it} > 0, X_{it}) = X_{it}' δ$		$log (σ_{it} \| Y_{it} > 0, X_{it}) = X_{it}' δ$
Gamma	Part (I):	$ℎ_{m_{1}} [P (Y_{it} > 0 \| X_{it})] = X_{it}' α_{m}$	Part (I):	$ℎ_{m_{1}} [P (Y_{it} > 0 \| X_{it})] = X_{it}' α_{m}$
		$ℎ_{c_{1}} [P (Y_{it} > 0 \| X_{it}, a_{it})] = Δ_{it} + a_{it}$		$ℎ_{c_{1}} [P (Y_{it} > 0 \| X_{it}, a_{it})] = Δ_{it} + a_{it}$
	Part (II):	$(Y_{it} \| Y_{it} > 0, X_{it}, b_{it}) \sim f_{G} (y; μ_{it}, σ_{it})$	Part (II):	$(Y_{it} \| Y_{it} > 0, X_{it}, b_{it}) \sim f_{G} (y; μ_{it}, σ_{it})$
		$ℎ_{m_{2}} [E (Y_{it} \| Y_{it} > 0, X_{it})] = X_{it}' β_{m}$		$ℎ_{m_{3}} [E (Y_{it} \| X_{it})] = X_{it}' β_{m}$
		$μ_{it} \| Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} + b_{it}$		$μ_{it} \| Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} + b_{it}$
		$log (σ_{it} \| Y_{it} > 0, X_{it}) = X_{it}' δ$		$log (σ_{it} \| Y_{it} > 0, X_{it}) = X_{it}' δ$
Inverse Gamma	Part (I):	$ℎ_{m_{1}} [P (Y_{it} > 0 \| X_{it})] = X_{it}' α_{m}$	Part (I):	$ℎ_{m_{1}} [P (Y_{it} > 0 \| X_{it})] = X_{it}' α_{m}$
		$ℎ_{c_{1}} [P (Y_{it} > 0 \| X_{it}, a_{it})] = Δ_{it} + a_{it}$		$ℎ_{c_{1}} [P (Y_{it} > 0 \| X_{it}, a_{it})] = Δ_{it} + a_{it}$
	Part (II):	$(Y_{it} \| Y_{it} > 0, X_{it}, b_{it}) \sim f_{IG} (y; μ_{it}, σ_{it})$	Part (II):	$(Y_{it} \| Y_{it} > 0, X_{it}, b_{it}) \sim f_{IG} (y; μ_{it}, σ_{it})$
		$ℎ_{m_{2}} [E (Y_{it} \| Y_{it} > 0, X_{it})] = X_{it}' β_{m}$		$ℎ_{m_{3}} [E (Y_{it} \| X_{it})] = X_{it}' β_{m}$
		$μ_{it} \| Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} + b_{it}$		$μ_{it} \| Y_{it} > 0, X_{it}, b_{it}) = Λ_{it} + b_{it}$
		$Φ^{- 1} (σ_{it} \| Y_{it} > 0, X_{it}) = X_{it}' δ$		$Φ^{- 1} (σ_{it} \| Y_{it} > 0, X_{it}) = X_{it}' δ$