Causal difference-in-differences estimation for evaluating the impact of semi-continuous medical home scores on health care for children

Abstract

Difference-in-differences (DID) is a popular approach in observational and quasi-experimental studies to estimate the effects of a treatment with discrete statuses. In many studies, however, the treatment can have a range of dosages or exposure levels. In our paper, “medical homeness” is a semi-continuous score ranging from 0 to 100 to indicate the extent to which a patient-centered medical home model is achieved. We developed a causal DID approach to estimating the effects of a treatment with semi-continuous dosages. The proposed approach allows for mixed-type designs as well as different propensity models. We applied the proposed approach to evaluate the dosage effect of medical homeness scores on the utilization and quality of children’s health care. We found that there was a roughly linear effect of medical homeness scores on the annual number of visits to doctor offices when medical homeness scores were below 60 points. The number of office visits did not further increase when medical homeness scores were above 60. A similar relationship was found between medical homeness scores and ratings for health care quality.

Appendix

Properties 1 to 3 are simply by the definition of ${\mathcal {G}}$, $r^\delta$, and R.

Proof of Property 4

For any $\zeta \ge 0$ and by iterative expectations,

$$\begin{aligned} \begin{aligned}&P(d \le \zeta |Y_1(\delta ) -Y_0, {\mathcal {G}}) = E\Big \{ E \big [ I\{ d \le \zeta \}|Y_1(\delta ) -Y_0, {\mathbf {X}} \big ] \Big | Y_1(\delta ) -Y_0, {\mathcal {G}} \Big \}. \\ \end{aligned} \end{aligned}$$

By the ignorability assumption, the last expression is equal to

$$\begin{aligned} \begin{aligned} E\Big \{ E\big [ I\{ d \le \zeta \}|{\mathbf {X}} \big ] | Y_1(\delta ) -Y_0, {\mathcal {G}} \Big \} = E \Big \{ \int _0^\zeta r^a \text {d}a \big | Y_1(\delta ) -Y_0, {\mathcal {G}} \Big \} , \text { by Property 1.} \end{aligned} \end{aligned}$$

Since $r^a$ is fully determined by ${\mathcal {G}}$, the last expression is equal to $\int _0^\zeta r^a \text {d}a= P(d \le \zeta | {\mathbf {X}}).$□

Proof of Property 5

By Property 2,

$$\begin{aligned} E[ Y_1(d) - Y_0 | d=\delta , R=s] = E[ Y_1(\delta ) - Y_0 | d=\delta , R=s]= E[ Y_1(\delta ) - Y_0 | d=\delta , r^\delta =s]. \end{aligned}$$

(17)

Then following essentially the same technique in Theorem 2 in Hirano and Imbens (2002), we first show $f(Y_1(\delta ) - Y_0 | d=\delta , r^\delta =s) = f(Y_1(\delta ) - Y_0 | r^\delta =s).$

Given $s \ne 0$,

$$\begin{aligned} \begin{aligned} f(Y_1(\delta ) - Y_0 | d=\delta , r^\delta =s)&=\frac{f(d=\delta | Y_1(\delta ) - Y_0, r^\delta =s)f(Y_1(\delta ) - Y_0| r^\delta =s)}{f(d=\delta |r^\delta =s)}\\&=s^{-1}f(d=\delta | Y_1(\delta ) - Y_0, r^\delta =s)f(Y_1(\delta ) - Y_0| r^\delta =s). \end{aligned} \end{aligned}$$

In addition,

$$\begin{aligned} \begin{aligned}&f(d=\delta | Y_1(\delta ) - Y_0, r^\delta =s) \\&\quad = \int f(d=\delta , {\mathbf {X}} | Y_1(\delta ) - Y_0, r^\delta =s) \text {d}{\mathbf {X}} \\&\quad =\int f(d=\delta | {\mathbf {X}} , Y_1(\delta ) - Y_0, r^\delta =s) f ({\mathbf {X}} | Y_1(\delta ) - Y_0, r^\delta =s) \text {d} {\mathbf {X}} \\&\quad =\int f(d=\delta | {\mathbf {X}} , r^\delta =s) f ({\mathbf {X}} | Y_1(\delta ) - Y_0, r^\delta =s) \text {d} {\mathbf {X}},\quad \text {by ignorability,}\\&\quad = \int s f ({\mathbf {X}} | Y_1(\delta ) - Y_0, r^\delta =s) \text {d} {\mathbf {X}}= s. \end{aligned} \end{aligned}$$

Therefore, $f(Y_1(\delta ) - Y_0 | d=\delta , r^\delta =s) = f(Y_1(\delta ) - Y_0| r^\delta =s)$.

Revisit the last expression in (17),

$$\begin{aligned} \begin{aligned}&E[ Y_1(\delta ) - Y_0 | d=\delta , r^\delta =s]\\&\quad = \int (Y_1(\delta ) - Y_0 ) f(Y_1(\delta ) - Y_0 | d=\delta , r^\delta =s) \text {d}(Y_1(\delta ) - Y_0)\\&\quad =\int (Y_1(\delta ) - Y_0 ) f(Y_1(\delta ) - Y_0| r^\delta =s) \text {d}(Y_1(\delta ) - Y_0)\\&\quad = E[ Y_1(\delta ) - Y_0 | r^\delta =s]. \end{aligned} \end{aligned}$$

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