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- SUMMARY
- 1. CAUSALITY AND DISPARITY MEASURES
- 2. STATISTICAL FRAMEWORKS
- 3. COMPARING CONDITIONAL AND MARGINAL DISPARITIES
- 4. FUTURE WORK
- REFERENCES

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Stat Med. Author manuscript; available in PMC 2009 September 10.

Published in final edited form as:

Stat Med. 2008 September 10; 27(20): 3941–3956.

doi: 10.1002/sim.3283PMCID: PMC2715701

NIHMSID: NIHMS66659

Naihua Duan,^{1,}^{2,}^{*} Xiao-Li Meng,^{3} Julia Y. Lin,^{4,}^{5} Chih-nan Chen,^{5} and Margarita Alegria^{4,}^{5}

See other articles in PMC that cite the published article.

Motivated by the need to meaningfully implement the Institute of Medicine’s (IOM’s) definition of health care disparity, this paper proposes statistical frameworks that lay out explicitly the needed causal assumptions for defining disparity measures. Our key emphasis is that a scientifically defensible disparity measure must take into account the direction of the causal relationship between *allowable covariates* that are not considered to be contributors to disparity and *non-allowable covariates* that are considered to be contributors to disparity, to avoid flawed disparity measures based on implausible populations that are not relevant for clinical or policy decisions. However, these causal relationships are usually unknown and undetectable from observed data. Consequently, we must make strong causal assumptions in order to proceed. Two frameworks are proposed in this paper, one is the *conditional disparity* framework under the assumption that allowable covariates impact non-allowable covariates but not vice versa. The other is the *marginal disparity* framework under the assumption that non-allowable covariates impact allowable ones but not vice versa. We establish theoretical conditions under which the two disparity measures are the same, and present a theoretical example showing that the difference between the two disparity measures can be arbitrarily large. Using data from the Collaborative Psychiatric Epidemiology Survey, we also provide an example where the conditional disparity is misled by Simpson’s paradox, while the marginal disparity approach handles it correctly.

The Institute of Medicine (IOM) [1] defines health care disparities as “racial or ethnic differences in the quality of health care that are not due to access-related factors or clinical needs, preference, and appropriateness of intervention.” This definition represents an important advance in disparity research, because it explicitly recognizes the role of causality in the determination of disparities through its reference to the causal expression “not due to”. However, it leaves open the interpretation of the causal model underlying this causal statement. In this paper we identify several causal models under which the IOM definition can be implemented meaningfully, and propose the corresponding frameworks for defining and comparing statistically justifiable disparity measures following these models. Our work can be viewed as a statistically oriented conceptualization of research in this area (e.g., [2, 3, 4, 5, 6]). Although our work was directly motivated by the IOM definition, the proposed general frameworks are equally applicable to other areas, such as in legal settings (e.g.,[7, 8, 9]).

The statistical frameworks proposed in this paper assume that the covariates of interest have been classified into *allowable* and *non-allowable* categories. Allowable covariates are considered to be justifiable to cause difference and hence should be adjusted before measuring disparity. The remaining covariates are classified as non-allowable.

It is important to note that the classification of allowable versus non-allowable covariates can, and should, vary from study to study, depending on the particular purpose for the study. For example, IOM’s classification of access-related factors as allowable is appropriate for studying disparity at the level of patient-clinician encounter, with the focus being the treatment delivered during the encounter, controlled for all historical factors that occurred prior to the encounter. However, when studying health care disparity at the level of service systems, it would be more appropriate to classify access-related factors as non-allowable, thus holding the service systems accountable for failure to engage disadvantaged patients into care. The statistical frameworks we establish in this paper apply to any of such classifications.

As a specific example for illustration, suppose that covariates that might be predictive of health care are classified as follows:

- Clinical needs and preference are considered allowable. Differences in health care due to these covariates are
*not*considered to be part of health care disparity. - All other covariates, such as knowledge about health, state of residency, insurance coverage, and education (to name a few), are considered non-allowable. Differences in health care due to these covariates are considered to be health-care disparity.

Given such a classification, our goal then is to measure the disparity that is “*not due to*” the allowable covariates.

A seemingly obvious, and hence very common, approach is to substitute the levels of allowable covariates of, for example, Afro-Caribbean with those of their non-Latino white counterparts, while leaving the levels of non-allowable covariates unchanged. This procedure is often used in Analysis of Covariance models that adjusts for allowable covariates across racial/ethnic groups. However, this approach is sensible in general only if the allowable covariates are statistically independent of the non-allowable covariates, a condition that is unlikely to hold in practice. Without this independence condition, this direct substitution may lead to an implausible population, such as a hypothetical population with high level of income (as a non-allowable covariate that remains unchanged) and a high level of chronic diseases (as an allowable covariate that was substituted with the levels from the reference population). As a result, the disparity estimates obtained from this procedure may not be relevant for clinical, policy or other purposes, because they are based on a postulated population that cannot be realized by policy changes or disparities interventions.

Not accounting properly for the causal relationships between allowable and non-allowable covariates is especially problematic when the two sets of covariates are highly correlated in the observed data, and both sets of variables are included in our outcome model. In such cases, the allowable covariates might appear to be very weak for predicting the outcome in the fitted model due to the well-known “collinearity” phenomenon. Consequently, replacing a minority group’s allowable covariates by their counterparts in the non-Latino white group in the fitted model may only produce trivial adjustment, even if in reality a substantial part of the observed racial/ethnic difference is indeed due to the difference in the allowable covariates. This could be either because of their direct impact on the outcome (which would not be detected by the fitted regression model because of the strong collinearity) or on the non-allowable covariates, or both. The frameworks proposed in this paper can help to substantially reduce such serious misestimation of disparity because they explicitly take into account the causal relationship between the allowable and non-allowable covariates. For example, our approaches permit an adjustment in allowable covariates to cause substantial adjustment in the non-allowable ones, which in turn may lead to substantial adjustment in the predicted outcome, even if the allowable predictor appears to be very weak in the fitted model for predicting the outcome.

In order to measure disparity meaningfully, such as to implement the IOM definition for health care disparity, one must be explicit about the underlying causal assumptions that are imbedded in any disparity measure. The fact that the exact causal mechanisms may not be known or may not even be knowable is not a reason to “sweep everything under the rug”. To the contrary, this is precisely the reason for us to be explicit about our assumptions so it is possible for policy makers and subsequent researchers to correctly interpret the disparity measures/estimates we obtain, as well as to determine the directions for correction or improvement when newer information becomes available for the underlying causal relationships.

The key reason that we need to make causal assumptions is that once an action is forced upon a particular variable (e.g., by changing a minority group’s distribution of clinical needs to match that of the non-Latino white population), it will have a ripple effect—in real life—on other variables (e.g., income level) that are impacted by the one adjusted. However, this ripple effect is not estimable without carrying out the actual (social) experiment, because the observed relationships in a natural population may or may not be preserved after an intervention. As an illustrative example, in a natural population, a person’s left-eye visual acuity (AV) may be highly correlated with the person’s right-eye AV. However, this correlation will be destroyed or at least reduced if we perform a vision correction laser surgery on the right eye only. The two AVs will become independent shortly after the surgery, but may become correlated again over time, though the cross-sectional data from a natural population would tell us little about how large this correlation could be or whether it would ever reach the same level as in the natural population.

Therefore, in order to measure the disparity “not due to” the allowable covariates, we must postulate causal directions, as well as how any relationships among relevant variables are preserved or altered with the change from a natural population to a hypothetical one. There are two extreme types of unidirectional causal relationships: (A) allowable covariates impact non-allowable covariates but not vice versa; and (B) non-allowable covariates impact allowable covariates but not vice versa. The more realistic relationships are likely to be either (C) allowable covariates and non-allowable covariates are inter-related and reciprocally impact each other, or (D), which is (C) plus the possibility that both allowable and non-allowable covariates are also impacted by the outcome itself (over time).

While (C) and (D) are most dynamic and realistic, they do not permit useful modeling without further specifications on how the variables involved impact each other. As these specifications are content dependent and can be extremely difficult to postulate, we will pursue them in future work. In this paper, we lay out the statistical frameworks for the simpler causal relationships (A) and (B). These two frameworks serve as building blocks for more complex causal specifications, and at the same time provide plausible specifications that might yield useful bounds on the true disparity when more complicated causal relationships are present. In some applications, such as the one presented in Section 3.2, such simplistic causal assumptions are actually reasonable, leading to sensible practical solutions.

Let **X*** _{N}* denote non-allowable covariates such as knowledge about health, and let

The goal of our modeling is to estimate the potential outcome if the group of interest has the same levels of allowable covariates as the reference group. The first step in setting up our proposed frameworks is to explicitly consider the joint distribution of (*Y*, **X*** _{A}*,

$${P}_{2}^{(H)}(Y,{\mathbf{X}}_{N},{\mathbf{X}}_{A})\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{P}_{2}^{(H)}\phantom{\rule{thinmathspace}{0ex}}(Y|{\mathbf{X}}_{N},{\mathbf{X}}_{A}){P}_{2}^{(H)}({\mathbf{X}}_{N},{\mathbf{X}}_{A}).$$

(1)

The importance of recognizing the dependence on *H* here is that only the natural population, *P*^{(N)} (*Y*, **X*** _{N}*,

$${P}_{2}^{\left(A\right)}(Y|{\mathbf{X}}_{N},{\mathbf{X}}_{A})={P}_{2}^{(N)}(Y|{\mathbf{X}}_{N},{\mathbf{X}}_{A}).$$

(2)

We will refer to (2) as the “predictively nature preserving” (PNP) assumption, meaning that the predictive nature of {**X*** _{N}*,

One can easily consider a scenario under which the PNP assumption is false, but without such an assumption, the estimation of the disparity is essentially impossible. For example, in our hypothetical eye vision example, two people may have identical AVs for both eyes (e.g., both are 20/20 in the right eye but 20/40 for in left eye), but they can have quite different probabilities of having automobile accidents if one of them was born with such vision, but the other achieved it via laser surgery to his right eye. Clearly, if this occurs, then it is impossible to estimate—using only the data collected from the natural population—the accident rate for the group of people with vision corrections done to their right eyes only.

To carry the decomposition (1) further, we can decompose the component ${P}_{2}^{(H)}({\mathbf{X}}_{N},{\mathbf{X}}_{A})$ in (1) into one conditional and one marginal distribution. This time, there are two possibilities:

$${P}_{2}^{(H)}\phantom{\rule{thinmathspace}{0ex}}({\mathbf{X}}_{N},{\mathbf{X}}_{A})\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{P}_{2}^{(H)}\phantom{\rule{thinmathspace}{0ex}}({\mathbf{X}}_{N}|{\mathbf{X}}_{A})\phantom{\rule{thinmathspace}{0ex}}{P}_{2}^{(H)}\phantom{\rule{thinmathspace}{0ex}}({\mathbf{X}}_{A});$$

(3)

and

$${P}_{2}^{(H)}\phantom{\rule{thinmathspace}{0ex}}({\mathbf{X}}_{N},{\mathbf{X}}_{A})\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{P}_{2}^{(H)}\phantom{\rule{thinmathspace}{0ex}}({\mathbf{X}}_{A}|{\mathbf{X}}_{N})\phantom{\rule{thinmathspace}{0ex}}{P}_{2}^{(H)}\phantom{\rule{thinmathspace}{0ex}}({\mathbf{X}}_{N}).$$

(4)

The first decomposition is the basis for our *conditional* framework, which assumes that non-allowable covariates **X*** _{N}* are causally dependent on allowable covariates

Under the conditional framework, we adjust the marginal distribution of the allowable covariates **X*** _{A}* from the natural population (such as Latinos) to the corresponding marginal distribution of the reference group (such as non-Latino whites), while preserving the conditional distribution for non-allowable covariates

$${P}_{2}^{(N)}(Y,{\mathbf{X}}_{N},{\mathbf{X}}_{A})={P}_{2}^{(N)}(Y|{\mathbf{X}}_{N},{\mathbf{X}}_{A}){P}_{2}^{(N)}({\mathbf{X}}_{N}|{\mathbf{X}}_{A}){P}_{2}^{(N)}({\mathbf{X}}_{A}),$$

(5)

by that of the reference population (e.g., non-Latino whites), and thereby creating the following hypothetical population distribution:

$${P}_{2}^{(C)}(Y,{\mathbf{X}}_{N},{\mathbf{X}}_{A})={P}_{2}^{(N)}(Y|{\mathbf{X}}_{N},{\mathbf{X}}_{A}){P}_{2}^{(N)}({\mathbf{X}}_{N}|{\mathbf{X}}_{A}){P}_{1}^{(N)}({\mathbf{X}}_{A}).$$

(6)

Although ${P}_{1}^{(N)}({\mathbf{X}}_{A})$ is taken from the natural population of the reference group, its insertion into (5) leads to a hypothetical population that retains the natural conditional distributions ${P}_{2}^{(N)}(Y|{\mathbf{X}}_{N},{\mathbf{X}}_{A})$ and ${P}_{2}^{(N)}({\mathbf{X}}_{N}|{\mathbf{X}}_{A})$, with the component ${P}_{2}^{(N)}({\mathbf{X}}_{A})$ “mutated” into ${P}_{1}^{(N)}({\mathbf{X}}_{A})$. We denote this adjustment rule under the *conditional* disparity framework as adjustment (*C*).

In order for (6) to be a meaningful hypothetical population, our assumptions are (i) the PNP assumption holds, and (ii) the adjustment action has no impact on the conditional distribution of **X*** _{N}* given

$${P}_{2}^{(C)}({\mathbf{X}}_{N}|{\mathbf{X}}_{A})={P}_{2}^{(N)}({\mathbf{X}}_{N}|{\mathbf{X}}_{A}),$$

(7)

which is plausible when the causal direction is from **X*** _{A}* to

The ratio between the adjusted joint density (6) and the natural joint density (5) is simply the ratio of the marginal densities

$${R}_{C}\phantom{\rule{thinmathspace}{0ex}}({\mathbf{X}}_{A})\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{{P}_{1}^{(N)}({\mathbf{X}}_{A})}{{P}_{2}^{(N)}({\mathbf{X}}_{A})}.$$

(8)

Following the principle of importance weighting, the expected outcome under the hypothetical population (6) can be expressed as the following *weighted* expectation of *Y* under the natural population (5), with the importance weight *R _{C}*(

$${E}_{2}^{(C)}[Y]\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{E}_{2}^{(N)}[Y\phantom{\rule{thinmathspace}{0ex}}{R}_{C}({\mathbf{X}}_{A})],$$

(9)

where ${E}_{2}^{(C)}$ denotes the expectation with respect to the hypothetical population in (6), and ${E}_{2}^{(N)}$ denotes the expectation with respect to the natural population in (5).

Expression (9) gives us a practical way to estimate ${E}_{2}^{(C)}$ [*Y*] because its right hand side only involves expectations with respect to the natural population (5), from which we can estimate from the sample data. Since the current paper focuses on setting up conceptual frameworks, the detailed estimation procedures, particularly for estimating *R** _{C}*(

Intuitively, the adjustment under our conditional framework amounts to weighting the level of health care (*Y*) among minorities by the density ratio *R** _{C}*(

$${D}_{C}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{E}_{2}^{(C)}[Y]-{E}_{1}^{(N)}[Y].$$

(10)

We term *D _{C}* of (10) as

Applying expression (9) to the definition (10), we have the following expression for conditional disparity that can be estimated using sample data:

$${D}_{C}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{E}_{2}^{(N)}[Y\phantom{\rule{thinmathspace}{0ex}}{R}_{C}({\mathbf{X}}_{A})]-{E}_{1}^{(N)}[Y].$$

(11)

Notice that this expression for conditional disparity does not involve the non-allowable covariates, *X _{N}*. This is possible because of the assumption that

In contrast to conditional disparity, which equates the two marginal distributions of **X*** _{A}*, the marginal disparity framework replaces the conditional distribution of

$${P}_{2}^{(N)}(Y,{\mathbf{X}}_{N},{\mathbf{X}}_{A})={P}_{2}^{(N)}(Y|{\mathbf{X}}_{N},{\mathbf{X}}_{A}){P}_{2}^{(N)}({\mathbf{X}}_{A}|{\mathbf{X}}_{N}){P}_{2}^{(N)}({\mathbf{X}}_{N}),$$

(12)

by that of the reference population to create the following hypothetical population

$${P}_{2}^{(M)}(Y,{\mathbf{X}}_{N},{\mathbf{X}}_{A})={P}_{2}^{(N)}(Y|{\mathbf{X}}_{N},{\mathbf{X}}_{A}){P}_{1}^{(N)}({\mathbf{X}}_{A}|{\mathbf{X}}_{N}){P}_{2}^{(N)}({\mathbf{X}}_{N}).$$

(13)

We denote this adjustment rule under the *marginal* disparity framework as adjustment (*M*).

Similar to the conditional disparity framework, in order for (13) to be a meaningful hypothetical population, we have assumed that (i) the PNP assumption holds, and (ii) the adjustment action has no impact on the marginal distribution of **X*** _{N}* either; that is,

$${P}_{2}^{(M)}({\mathbf{X}}_{N})={P}_{2}^{(N)}({\mathbf{X}}_{N}),$$

(14)

which is plausible when the causal direction is from **X*** _{N}* to

Similar to (8), the ratio between the joint densities (13) and (12) is given by the ratio between the two conditional densities

$${R}_{M}({\mathbf{X}}_{A};{\mathbf{X}}_{N})=\frac{{P}_{1}^{(N)}({\mathbf{X}}_{A}|{\mathbf{X}}_{N})}{{P}_{2}^{(N)}({\mathbf{X}}_{A}|{\mathbf{X}}_{N})}.$$

(15)

Again, the ratio (15) can be used as the importance weight to express:

$${E}_{2}^{(M)}[Y]={E}_{2}^{(N)}[Y{R}_{M}({\mathbf{X}}_{A};{\mathbf{X}}_{N})],$$

(16)

where ${E}_{2}^{(M)}$ denotes expectation under the hypothetical population (13), and ${E}_{2}^{(N)}$ denotes expectation under the natural population (12). Note that the right hand side of (16) can be estimated from sample data obtained in the natural population (12).

It is useful to visualize the adjustment under our marginal framework as first stratifying the minority population by the level of the non-allowable covariates (e.g., knowledge of health). We then apply the same weighting scheme as with the conditional disparity approach but now *within each stratum*, therefore the weights there, namely, the ratio of marginal densities *R** _{C}*(

The marginal disparity measure then is defined as the difference between the expected value of *Y* for the adjusted (hypothetical) population (13) and that of the reference population (12):

$${D}_{M}={E}_{2}^{(M)}[Y]-{E}_{1}^{(N)}[Y].$$

(17)

We term *D _{M}* as

$${D}_{M}={E}_{2}^{(N)}[Y\phantom{\rule{thinmathspace}{0ex}}{R}_{M}({\mathbf{X}}_{A};{\mathbf{X}}_{N})]-{E}_{1}^{(N)}[Y].$$

(18)

The estimation of *R** _{M}*(

With the two frameworks given above, a natural question is when do they give the same disparity estimates, or more profoundly, do they give different values that would matter in practice? The answer to the first part is a clean-cut theoretical result we present below. The answer to the second part is obviously “it depends” because it depends critically on the nature of the dependence structure between **X*** _{A}* and

The difference between *D _{C}* and

$$\Delta D\equiv {D}_{C}-{D}_{M}={E}_{2}\phantom{\rule{thinmathspace}{0ex}}[Y\xb7(\frac{{P}_{1}({\mathbf{X}}_{A})}{{P}_{2}({\mathbf{X}}_{A})}-\frac{{P}_{1}({\mathbf{X}}_{A}|{\mathbf{X}}_{N})}{{P}_{2}({\mathbf{X}}_{A}|{\mathbf{X}}_{N})}\left)\right]\phantom{\rule{thinmathspace}{0ex}}.$$

(19)

The two disparity measures will be identical, Δ*D* = 0, if

$$\frac{{P}_{1}({\mathbf{X}}_{A})}{{P}_{2}({\mathbf{X}}_{A})}=\frac{{P}_{1}({\mathbf{X}}_{A}|{\mathbf{X}}_{N})}{{P}_{2}({\mathbf{X}}_{A}|{\mathbf{X}}_{N})}.$$

(20)

This condition is equivalent to the condition that

$${G}_{1}({\mathbf{X}}_{N},{\mathbf{X}}_{A})\equiv \frac{{P}_{1}({\mathbf{X}}_{N},\phantom{\rule{thinmathspace}{0ex}}{\mathbf{X}}_{A})}{{P}_{1}({\mathbf{X}}_{N}){P}_{1}({\mathbf{X}}_{A})}=\frac{{P}_{2}({\mathbf{X}}_{N},\phantom{\rule{thinmathspace}{0ex}}{\mathbf{X}}_{A})}{{P}_{2}({\mathbf{X}}_{N}){P}_{2}({\mathbf{X}}_{A})}\equiv {G}_{2}({\mathbf{X}}_{N},\phantom{\rule{thinmathspace}{0ex}}{\mathbf{X}}_{A}).$$

(21)

Here the *G* function can be viewed as a measure of the dependence structure between **X*** _{N}* and

For continuous variables, the notion that *G* is a measure of dependence structure can also been examined through the *local dependence function* (LDF), as defined in [11] and studied in [12] and [13],

$$\gamma ({\mathbf{X}}_{N},{\mathbf{X}}_{A})=\frac{{\partial}^{2}\mathrm{log}P({\mathbf{X}}_{N},\phantom{\rule{thinmathspace}{0ex}}{\mathbf{X}}_{A})}{\partial {\mathbf{X}}_{N}\partial {\mathbf{X}}_{A}}.$$

(22)

Because

$$\frac{{\partial}^{2}\mathrm{log}G({\mathbf{X}}_{N},\phantom{\rule{thinmathspace}{0ex}}{\mathbf{X}}_{A})}{\partial {\mathbf{X}}_{N}\partial {\mathbf{X}}_{A}}=\frac{{\partial}^{2}\mathrm{log}P({\mathbf{X}}_{N},\phantom{\rule{thinmathspace}{0ex}}{\mathbf{X}}_{A})}{\partial {\mathbf{X}}_{N}\partial {\mathbf{X}}_{A}},$$

(23)

it is obvious that condition (21) implies that the LDF is independent of the group index, i.e., the LDF does not change with the racial/ethnic group. Note however that the reverse is not necessarily true; that is, we can have LDF invariant to group index, but the condition (21) does not hold. In this sense, the measure of dependence by the *G* function is more stringent than that by the LDF.

Finally we note that condition (21) is sufficient but not necessary for Δ*D* = 0. A simple example is that Δ*D* = 0 when the regression of *Y* on **X*** _{A}* and

We emphasize here that the statement we just made is true only when *both* **X*** _{N}* and

We start with a simple 2 × 2 × 2 contingency table example to both illustrate the basic calculations for *D _{C}* and

For simplicity, we focus on a dichotomous outcome, namely, *Y* = 1 means the respondent had at least one visit to any mental health service provider (either specialist or generalist) in the past year, and *Y* = 0 otherwise. The allowable covariate is also a binary variable indicating clinical need: *X _{A}* = 1 if there was a need, and

Table I provides the data for the non-Latino white population, from which we can easily calculate the service use rate for this population. In Table I, there are two numbers in each of the cells in the 2 × 2 layout. The top number is the percentage of individuals who fall into the (*i*, *j*)-cell defined by the values of (*X** _{N}* =

$$RD={E}_{2}[Y]-{E}_{1}[Y]=6.75\%-14.39\%=-7.64\%$$

(24)

This, however, is not necessarily the *disparity* in the sense of the IOM definition because it has not adjusted for the difference in clinical needs.

Comparing Table I and Table II, we observe an interesting phenomenon. The percentages of people in need are greater in the Afro-Caribbean population than in the non-Latino white population when *conditional on the nativity*—55.75% versus 41.62% for the US Born population and 33.90% versus 30.91% for the immigrant population. The pattern, however, is *reversed* for the *marginal rates*, that is, when we combine the US born and the immigrants together: 41.18% for Afro-Caribbean versus 41.28% for non-Latino whites. Although the difference between these two marginal rates is minimal (but there is no estimation error here since we are using the data as if they were the entire population), it is nevertheless an example of the well-known *Simpson’s paradox*[19]. The reason is the extreme imbalance of the nativity groups in the two populations: more than 95% of the non-Latino whites were US born, but only 1/3 of the Afro-Caribbean were US born.

The implication of this phenomenon for our disparity measure is clear. First, given that the difference in the marginal rates is so small, 41.18% verse 41.28%, one would expect that the *conditional disparity* which results from adjusting the Afro-Caribbean’s marginal rate from 41.18% to the non-Latino whites marginal rate of 41.28% will have a minimal impact on the value of *RD* of (24). Indeed, as shown below, the *conditional disparity* in this case is *D _{C}* = −7.62%, nearly identical to

Second, this adjustment in fact is in the wrong direction, because in this case the casual assumption underlying the conditional disparity, that is, the allowable covariate (clinical need) causes the non-allowable (nativity) is clearly a very implausible one. The *marginal disparity* approach is a much more sensible one, because it makes adjustment of clinical needs *within each nativity category*. Given the fact that the two nativity groups have very different levels of clinical needs, with the US Born having more needs, it is intuitive that we should make the adjustment after stratifying by nativity groups. Because the Afro-Caribbean population has more needs in each of the nativity groups, it is also intuitive that had their needs been the same as the non-Latino whites, the observed racial/ethnic difference would be even larger. Indeed, as shown below, the marginal disparity in this case is *D _{M}* = −8.84%. In contrast to

The calculations of *D _{C}* and

$${R}_{C}(0)=\frac{{P}_{1}({X}_{A}=0)}{{P}_{2}({X}_{A}=0)}=\frac{0.5872}{0.5882}=0.9983,\text{\hspace{1em}}{R}_{C}(1)=\frac{{P}_{1}({X}_{A}=1)}{{P}_{2}({X}_{A}=1)}=\frac{0.4128}{0.4118}=1.0024.$$

We can then multiply each of the three *un-bracketed* proportions in the “No (0)” column of Table II by *R _{C}*(0), and multiply each of the three

$${D}_{C}={E}_{2}^{(C)}[Y]-{E}_{1}[Y]=6.77\%-14.39\%=-7.62\%.$$

To calculate the marginal disparity, we need first to compute the *R _{M}* function of (15), which is determined by the right most columns labeled “

$$\begin{array}{l}{R}_{M}\phantom{\rule{thinmathspace}{0ex}}(0;\phantom{\rule{thinmathspace}{0ex}}0)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{{P}_{1}\phantom{\rule{thinmathspace}{0ex}}({X}_{A}\phantom{\rule{thinmathspace}{0ex}}=0|{X}_{N}=0)}{{P}_{2}\phantom{\rule{thinmathspace}{0ex}}({X}_{A}\phantom{\rule{thinmathspace}{0ex}}=0|{X}_{N}=0)}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{1-0.4162}{1-0.5575}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}1.3193;\text{\hspace{1em}}{R}_{M}\phantom{\rule{thinmathspace}{0ex}}(0;\phantom{\rule{thinmathspace}{0ex}}1)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{0.4162}{0.5575}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}0.7465;\\ {R}_{M}\phantom{\rule{thinmathspace}{0ex}}(1;\phantom{\rule{thinmathspace}{0ex}}0)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{{P}_{1}\phantom{\rule{thinmathspace}{0ex}}({X}_{A}\phantom{\rule{thinmathspace}{0ex}}=0|{X}_{N}=1)}{{P}_{2}\phantom{\rule{thinmathspace}{0ex}}({X}_{A}\phantom{\rule{thinmathspace}{0ex}}=0|{X}_{N}=1)}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{1-0.3091}{1-0.3390}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}1.0452;\text{\hspace{1em}}{R}_{M}\phantom{\rule{thinmathspace}{0ex}}(1;\phantom{\rule{thinmathspace}{0ex}}1)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{0.3091}{0.3390}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\mathrm{0.9118.}\end{array}$$

Table IV then is obtained by multiplying the (*i*, *j*)-cell proportion (the top un-bracketed percentage) in Table II with *R _{M}*(

To find the expectation of *Y* under this adjusted Afro-Caribbean population, we multiply the four cell percentages in Table IV respectively by the four μ* _{ij}* values in Table II and then sum them up. This yields ${E}_{2}^{(M)}[Y]=5.55\%$. Consequently, the marginal disparity, which in this example can be regarded as a sensible measure of disparity, is given by

$${D}_{M}={E}_{2}^{(M)}[Y]-{E}_{1}[Y]=5.55\%-14.39\%=-8.84\%.$$

This theoretical example establishes the mathematical fact that the difference in the conditional disparity and marginal disparity can be arbitrarily large. It also illustrates another form of the Simpson’s paradox, that is, even when there is no disparity in any strata defined by the non-allowable variables *X _{N}*, in the aggregated population one can still observe a disparity due to the correlation between

To see this, let us consider a simple linear regression case

$${E}_{k}[Y|{X}_{N},{X}_{A}]={\beta}^{(k)}+{\beta}_{N}^{(k)}{X}_{N}+{\beta}_{A}^{(k)}{X}_{A,}$$

(25)

where *k* = 1 indexes the non-Latino white population and *k* = 2 the minority population. To simplify algebra, suppose in the natural populations (*X _{N}, X_{A}*) is bivariate normal, with mean $({\mu}_{N}^{(k)},{\mu}_{A}^{(k)})$, unit variances and correlation ρ

$${\left(\begin{array}{c}{X}_{N}\\ {X}_{A}\end{array}\right)}_{k}\phantom{\rule{thinmathspace}{0ex}}~\phantom{\rule{thinmathspace}{0ex}}N\phantom{\rule{thinmathspace}{0ex}}\left[\left(\begin{array}{c}{\mu}_{N}^{(k)}\\ {\mu}_{A}^{(k)}\end{array}\right)\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}\left(\begin{array}{cc}1& {\rho}^{(k)}\\ {\rho}^{(k)}& 1\end{array}\right)\right]\phantom{\rule{thinmathspace}{0ex}},\text{\hspace{1em}}k=\phantom{\rule{thinmathspace}{0ex}}1,\phantom{\rule{thinmathspace}{0ex}}2.$$

(26)

Under this setting, for the conditional disparity, the hypothetical joint distribution ${P}_{2}^{(C)}({X}_{N},{X}_{A})={P}_{2}({X}_{N}|{X}_{A}){P}_{1}({X}_{A})$ is a bivariate normal with the following distribution:

$${\left(\begin{array}{c}{X}_{N}\\ {X}_{A}\end{array}\right)}_{2}^{(C)}\phantom{\rule{thinmathspace}{0ex}}~\phantom{\rule{thinmathspace}{0ex}}N\left[\left(\begin{array}{c}{\mu}_{N}^{(2)}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{\rho}^{(2)}\phantom{\rule{thinmathspace}{0ex}}({\mu}_{A}^{(1)}-{\mu}_{A}^{(2)})\\ {\mu}_{A}^{(1)}\end{array}\right)\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}\left(\begin{array}{cc}1& {\rho}^{(2)}\\ {\rho}^{(2)}& 1\end{array}\right)\right]\phantom{\rule{thinmathspace}{0ex}}.$$

(27)

In contrast, under the marginal disparity approach, the hypothetical joint distribution for (*X** _{N}*,

$${\left(\begin{array}{c}{X}_{N}\\ {X}_{A}\end{array}\right)}_{2}^{(M)}\phantom{\rule{thinmathspace}{0ex}}~\phantom{\rule{thinmathspace}{0ex}}N\left[\left(\begin{array}{c}{\mu}_{N}^{(2)}\\ {\mu}_{A}^{(1)}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{\rho}^{(1)}\phantom{\rule{thinmathspace}{0ex}}({\mu}_{N}^{(2)}-{\mu}_{N}^{(1)})\end{array}\right)\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}\left(\begin{array}{cc}1& {\rho}^{(1)}\\ {\rho}^{(1)}& 1\end{array}\right)\right]\phantom{\rule{thinmathspace}{0ex}}.$$

(28)

Simple algebra then yields that the difference between the two measures is

$$\Delta D={\rho}^{(2)}{\beta}_{N}^{(2)}({\mu}_{A}^{(1)}-{\mu}_{A}^{(2)})+{\rho}^{(1)}{\beta}_{A}^{(2)}({\mu}_{N}^{(1)}-{\mu}_{N}^{(2)}).$$

(29)

From (29), we have the following observations, two of which are special cases of what we have discussed in general in Section 3.1. Specifically, we see that Δ*D* = 0 whenever one of the following three condition holds:

- ρ
^{(1)}= ρ^{(2)}= 0; that is, when*X*and_{N}*X*are independent in_{A}*both*populations; - ${\beta}_{N}^{(2)}={\beta}_{A}^{(2)}=0$; that is, when the regression (25) does not depend on
*either**X*or_{N}*X*in the population of interest (not necessarily in the reference population);_{A} - ${\mu}_{N}^{(1)}={\mu}_{N}^{(2)}$
*and*${\mu}_{A}^{(1)}={\mu}_{A}^{(2)}$, that is, when the two populations have the same marginal distributions for*both**X*and_{N}*X*._{A}

Of course Δ*D* can be zero by many other (incidental) combinations of the parameter values, but the above three are most useful for theoretical insights. Note in particularly that conditions (a) and (b) are applicable in general, but condition (c) only works when the regression of *Y* is linear in both *X _{N}* and

We also remark a special case of interest, that is, when *E** _{k}*[

$${D}_{M}={\rho}^{(1)}{\beta}_{A}^{(2)}({\mu}_{N}^{(2)}-{\mu}_{N}^{(1)}).$$

(30)

This is zero only when (i) ρ^{(1)} = 0 and hence *X _{A}* and

Perhaps most important here is to notice the Simpson’s paradox again. Although in the aggregated population there is a marginal disparity for the case above, clearly there is no disparity in any subpopulation defined by a particular value of *X _{N}*, that is, when we condition on

The IOM definition of disparities takes an indirect approach of elimination, and defines health care disparity as the difference in health care that is *not due to* allowable covariates. While this approach is appropriate for capturing disparity in its entirety irrespective of source attribution, it leaves open the question of plausible causes for the disparity, and what can be done to eliminate or reduce the disparity.

An alternative direct, constructive approach, is to define health care disparity attributable to specific non-allowable covariates as the difference in health care that is *due to* these covariates. This alternative approach can be implemented using the similar statistical frameworks proposed above, but with the role of allowable and non-allowable covariates switched. This approach does not capture disparity in its entirety, because it only captures disparity attributable to the specific non-allowable covariates, and may miss the disparity attributable to other non-allowable covariates, including those that may not have been observed. However, this approach may have more direct policy implications, providing guidance on the potential to reduce or even eliminate health care disparities through specific policy implementations regarding the specific non-allowable covariates.

In practice, we believe both versions of the disparity are important. The elimination approach is useful for estimating the magnitude of the overall disparity, whereas the constructive approach is a tool for estimating how much disparity can be eliminated through specific policy interventions. A comparison between the two is also important in revealing how much of the overall disparity the policy intervention can eliminate. If a large portion remains, a new policy intervention needs to be identified. We plan to explore these issues in subsequent work, especially in the context of longitudinal data.

Another issue that we plan to investigate is the issue of variables that are not included in the model for predicting the outcome *Y* but may actually be important. Traditionally there is not much one can do about those variables other than trying one’s best to include as many variables as one can find and afford to measure. For the conditional disparity framework as we outlined, one may have noticed that the conditional disparity as defined by (11) does not involve the non-allowable variables. This provides an opportunity to realize the implicit assumption carried in the IOM definition, that is, the non-allowable category is the “catch all” category that includes all covariates that have not been named explicitly in the allowable category. Of course, without strong assumptions, nothing can be done for variables that are not even identified. Recall the fundamental assumption underlying our conditional disparity model is that the allowable variables, which clearly need to be identified and measured, are causes for non-allowable variables. Therefore, if in specific applications where such an assumption can be viewed as reasonable, even when the non-allowable variables form the “catch all” category, then the conditional disparity measure enjoys the property of being more general than we discussed in the current paper.

However, the “catch-all” formulation of the non-allowable variables would not produce anything meaningful under the marginal disparity model, because we simply cannot stratify on variables that are not measured, nor should it be as logically there is nothing can be done when the causes are not even identified. All these issues remind us again of the fundamental importance of explicitly formulating, identifying, and stating causal assumptions underlying any disparity measure.

We thank J. Gastwirth, X. Xie and A. Zaslavsky for helpful exchanges.

Contract/grant sponsor: NIH; contract/grant number: P50-MHO73469-03, U01-MH06220-06A2

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