Efficiency of Thai provincial public hospitals during the introduction of universal health coverage using capitation

Abstract

We investigate the impact of implementing capitated-based Universal Health Coverage (UC) in Thailand on technical efficiency in larger public hospitals during the policy transition period. We measure efficiency before and during the transition period of UC using a two-stage analysis with Data Envelopment Analysis, bootstrap DEA, and truncated regressions. Our analysis indicates that during the transition period efficiency in larger public hospitals across the country increased. The findings differed by region, and hospitals in provinces with more wealth not only started with greater efficiency, but improved their relative position during the transitional phases of the UC system.

Appendices

Appendices

1.1 Appendix A: bootstrap DEA estimators algorithm

This appendix borrows heavily from Badin and Simar [3].

Step 1. Find the original efficiency estimates from the sample of n producers $\chi _n = \left\{ {\left( {x_i ,y_i } \right),i = 1, \ldots ,n} \right\}$, where x _i  ∈ R ^p is the input vector and y _i  ∈ R ^q is the output vector of producer i. For each observed producer $\left( {x_i ,y_i } \right) \in \chi _n$, compute the DEA estimator of the efficiency score $\hat \theta _i = \hat \theta _{{\text{DEA}}} \left( {x_i ,y_i } \right),\,i = 1, \ldots ,\,n$.

Step 2. If $\left( {x_0 ,y_0 } \right) \notin \chi _n$ repeat step 1 for $\left( {x_0 ,y_0 } \right)$ to obtain $\hat \theta _i = \hat \theta _{{\text{DEA}}} \left( {x_0 ,y_0 } \right)$.

Step 3. Define $S_m = \left\{ {\hat \theta _i , \ldots ,\,\hat \theta _m } \right\}$ where $m = \# \left\{ {\hat \theta _i < 1} \right\}_{1 \leqslant i \leqslant n}$, i.e. the number of inefficient producers.

Step 4. Giving n is the number of decision-making units (DMUs), estimate the distribution $f\left( {\hat \theta } \right)$ from the remaining $\hat \theta$ and generate B samples of the boundary condition $\hat \theta < 1$ (the size n − 1), which is $\left\{ {\hat \theta _1^{*b} , \ldots ,\,\hat \theta _{n - 1}^{*b} .} \right\}_{b = 1}^B$. The steps are as follows:

4.1. Given a random sample $\theta _1 , \ldots ,\theta _n$ with a continuous, univariate density f, the kernel density estimator is defined by:¹⁸

$$\hat f\left( \theta \right) = \frac{1}{{nh}}\sum\limits_{i = 1}^n {K\left( {\frac{{\theta - \theta _i }}{h}} \right)} $$

(5)

where K( ) is the kernel function and h is the bandwidth parameter. Under mild conditions (h must decrease with increasing n) the kernel estimate converges in probability to the true density. Performance of kernel is measured by Mean Integrated Squared Error (MISE).

Bandwidth selection is a crucial issue in the application of the smoothing procedure. Refer to Silverman [44] for a completed review of several approaches of bandwidth selection. In this paper, the bandwidth function rule for univariate data recommended by is

$$h = 0.9\left( {\min \left\{ \begin{aligned}& \hat \sigma _{\hat \theta } \\& {{R_{13} } \mathord{\left/{\vphantom {{R_{13} } {1.34}}} \right.\kern-\nulldelimiterspace} {1.34}} \\ \end{aligned} \right.} \right)n^{ - {1 \mathord{\left/{\vphantom {1 5}} \right.\kern-\nulldelimiterspace} 5}} $$

where R ₁₃ denotes the inter-quartile range of the sample $\left\{ {\hat \theta _i } \right\}$ and denotes the standard deviation estimate of the efficiency estimates $\left\{ {\hat \theta _i } \right\}$, respectively.¹⁹

4.2. Using the reflection method [44] we estimate $f\left( {\hat \theta } \right)$ under the boundary condition $\hat \theta < 1$. Suppose we have m inefficient producers, denoting $S_m = \left\{ {\hat \theta _i , \ldots ,\,\hat \theta _m } \right\}$. In order to find a consistent estimator of $f\left( {\hat \theta } \right)$, let$\left\{ {\beta _1^ * , \ldots ,\beta _{n - 1}^ * } \right\}$ be a bootstrap sample, obtained by sampling with replacement from S _m and $\left\{ {\varepsilon _1^ * , \ldots ,\varepsilon _{n - 1}^ * } \right\}$ a random variable of standard normal deviates. By the convolution formula, we have:

$$\tilde \theta _i^* = \beta _i^* + h\varepsilon _i^* \sim \frac{1}{m}\sum\limits_{j = 1}^m {\frac{1}{h}\phi \left( {\frac{{z - \hat \theta _{\left( j \right)} }}{h}} \right)} $$

for i = 1,…, n  − 1. Define now for i = 1,…, n − 1 the bootstrap data:

$$\theta _i^* = \left\{ \begin{aligned} &\tilde \theta _i^* \,\quad \quad \;\;{\text{if}}\,\tilde \theta _i^* < 1 \\&2 - \tilde \theta _i^* \,\quad {\text{otherwise}}{\text{.}} \\ \end{aligned} \right.$$

(6)

where $\theta _i^ *$ defined in Eq. 6 is proved to be random variables distributed according to $\hat f_h \left( z \right)$. The final smoothed resample efficiencies are obtained by rescaling the bootstrap data making the variance is approximately the sample variance of $\hat \theta _i$. We employ the following transform: $\hat \theta _i^* = \bar \hat \theta + \frac{1}{{\sqrt {\left( {1 + {{h^2 } \mathord{\left/ {\vphantom {{h^2 } {\bar \hat \sigma ^2 }}} \right. \kern-\nulldelimiterspace} {\bar \hat \sigma ^2 }}} \right)} }}\left( {\theta _i^* - \bar \hat \theta } \right)$,

where $\bar \hat \theta = \frac{1}{n}\sum\limits_{j = 1}^n {\hat \theta _j }$ and $\hat \sigma ^2 = \frac{1}{n}\sum\limits_{j = 1}^n {\left( {\hat \theta _j - \bar \hat \theta } \right)^2 }$.

Step 5. Then, draw n − 1 bootstrap values $\hat \theta _i^* ,\,i = 1, \ldots ,\,n - 1$ from the kernel density estimate of $f\left( {\hat \theta } \right)$ and sort in ascending order: $\hat \theta _{\left( 1 \right)}^* \leqslant \ldots \leqslant \hat \theta _{\left( {n - 1} \right)}^* .$

Step 6. Repeat step 5 (drawing n − 1 bootstrap values $\hat \theta _i^*$) B times (in this study, 1,000 times), to obtain a set of B bootstrap estimates $\left\{ {\hat \theta _{\left( {n - j} \right)}^{*b} } \right\}_{b = 1}^B$, for some $1 \leqslant j \leqslant n - 1.$ ²⁰

Step 7. Finally, approximate $\tilde \theta _{\left( {n - j} \right)}$ for some j $\left( {1 \leqslant j \leqslant n - 1} \right)$ by the average of $\hat \theta _{\left( {n - j} \right)}^{*b}$ over the B simulations:

$$\tilde \theta _{\left( {n - j} \right)} = \frac{1}{B}\sum\limits_{b = 1}^B {\hat \theta _{\left( {n - j} \right)}^{*b} } $$

(7)

Step 8. The confidence intervals for θ ₀ are:

$$\tau _1 = \left( {\hat \theta _0 - \frac{{\left( {1 - \alpha } \right)^{{1 \mathord{\left/{\vphantom {1 \delta }} \right.\kern-\nulldelimiterspace} \delta }} }}{{\left( {1 - } \right.\left( {1 - \alpha } \right)^{{1 \mathord{\left/{\vphantom {1 \delta }} \right.\kern-\nulldelimiterspace} \delta }} }}\left( {{{\hat \theta _0 } \mathord{\left/{\vphantom {{\hat \theta _0 } {\tilde \theta _{\left( {n - j} \right)} }}} \right.\kern-\nulldelimiterspace} {\tilde \theta _{\left( {n - j} \right)} }} - \hat \theta _0 } \right),\hat \theta _0 } \right)$$

$$\tau _2 = \left( {2\hat \theta _0 - {{\hat \theta _0 } \mathord{\left/ {\vphantom {{\hat \theta _0 } {\tilde \theta _{\left( {n - j} \right)} }}} \right. \kern-\nulldelimiterspace} {\tilde \theta _{\left( {n - j} \right)} }},\hat \theta _0 } \right)$$

1.2 Appendix B: the adjustment factor

Without some measure of case mix it is possible that what shows up as inefficiency may in fact be due to case mix variation. The Ministry of Public Health (MOPH) does not gather direct information about patient severity so no common case mix measure is available. However, in the output mix, MOPH differentiates between large surgeries and small surgeries. “Large surgeries” are more complicated, require more staff time and equipment, and usually indicate a more severely ill patient. MOPH classifies large surgeries as those that apply epidural/spinal blocks or general anesthesia. Small surgeries are uncomplicated minor surgeries that do not need general anesthesia. Under the assumption that a hospital that does a greater share of large surgeries is more likely to attract more severely ill patients in all areas of care provision, we estimate relative severity for each hospital compared to severity of all hospitals in the sample by first computing the surgery mix in each hospital. Thus for hospital i:

mix _i  = (number of large surgeries in hospital i/number of small surgeries in hospital i).

Relative severity (or our case mix adjustment factor) for hospital i is then computed by the formula:

$${\text{adjustment}}_i = {{{\text{mix}}_i } \mathord{\left/{\vphantom {{{\text{mix}}_i } {\left( {\max \, \times \,{\text{mix}}} \right)}}} \right.\kern-\nulldelimiterspace} {\left( {\max \, \times \,{\text{mix}}} \right)}}$$

where (max × mix) is the largest value of mix in the sample over all i. Thus, INSUR which is the adjusted inpatient visits in acute surgical—general surgery and orthopedic surgery multiplies the number of inpatient visits within this category by the adjustment factor.

Table 17 shows simple statistics for the adjustment factor by year. Overall, there was little change in the adjustment factor over time. In fact, a hospital-by-hospital analysis of changes in the adjustment factor year to year showed that the average change in the adjustment factor by hospital from 2000 to 2001 was 0.009 with a standard deviation of 0.062 indicating that the change was not statistically different from zero with any reasonable level of confidence. For the change from 2001 to 2002 the mean was 0.009 with a standard deviation of 0.067, again not statistically significant. It is interesting to note that on average the net effect from 2000 to 2002 was zero. A more detailed analysis looking at the number of hospitals that increased or decreased year to year split almost evenly each year, again with no statistically significant pattern to these changes.

Table 17 Severity adjustment factor by year

While there seems to be no change in the adjustment factor year to year, there are clear differences by hospital type, as shown in Table 18. Both the mean and median of the adjustment factor for regional and large general hospitals exceeds that of small general hospitals, indicating, as expected, that the former two types of hospitals handle more severely ill patients. Means for both regional and large general hospitals are significantly different from the mean for small general hospitals with over 95% confidence.

Table 18 Severity adjustment factor by hospital type

Finding differences in the adjustment factor by hospital type, but not over years, supports using this ratio as a severity adjustor for outputs, especially as we look for changes in efficiency over time. It indicates that individual hospital severity remained relatively constant. Moreover, our focus on changes in relatively efficiency of individual hospitals over time means the adjustment factor is less important than if we were most concerned with a comparing efficiency among hospitals. There is sometimes difficulty discerning differences in efficiency from differences in quality or case mix, with higher quality hospitals or those with more severe case mixes appearing inefficient. Since we concentrate on the change in efficiency, if we have measurement errors due to quality or case mix differences those should carryover year-to-year (recall that the adjustment factor showed little year-to-year changes but significant cross-section changes) and the responses we see should therefore be robust nonetheless.

1.3 Appendix C

Table 19 Bootstrap DEA estimations results