Group rate: \(R_{i} = y_{i}/n_{i}\)

Group share: \(P_{i} = n_{i}/\sum_{i} n_{i}\)

Crude rate: \(\frac{\sum_{i=1}^{I} y_{i}}{\sum_{i=1}^{I} n_{i}} = \sum_{i=1}^{I} P_{i}R_{i}\)

Standardized rate: \(\frac{\sum_{i=1}^{I} y_{i}}{\sum_{i=1}^{I} s_{i}} = \sum_{i=1}^{I} S_{i}R_{i}\)

where \(S_{i}\) is the share of group i in a standard population (\(\sum_{i} S_{i} = 1\))

Data for crude rates

…if the difference between two standardized rates is subtracted from the corresponding difference in crude rates…the result is a weighted average of differences between the composition of the two groups.

Crude difference = \(\sum P_{Ai}R_{Ai} - \sum P_{Bi}R_{Bi}\)

→ Reflects differences in both rates and composition

\(A\)-Standardized difference = \(\sum P_{Ai}R_{Ai} - \sum P_{Ai}R_{Bi}\)

→ Reflects only differences in rates (\(P_{Ai}\) is constant)

What accounts for differences in neonatal mortality rates across European countries?

GA-specific rates or distribution?

Implications for interventions?

1. Means

Differences in the prevalence of determinants of outcome

2. Coefficients

Differences in the coefficient of a given determinant on the outcome (i.e., effect measure modification)

Our findings show that unexplained factors associated to immigrant status determine to a great extent disparities in the probability of using hospital, specialist and emergency services of immigrants relative to Spaniards, while individual characteristics, in particular self-reported health and chronic conditions, are much more important in explaining the differences in the probability of using general practitioner services between immigrants and Spaniards

Studies by R. Oaxaca (1973) and Blinder (1973) applied regression-based decomposition methods to analyze the wage gap between men and women and between whites and blacks in the USA.

Focused on how much of wage gap was ‘explained’ by differences in observable characteristics

Recent attempts to reconcile O-B with Kitagawa (equivalent for binary outcome and categorical covariates)

Decomposition methods are based on regression analyses, and thus all of the usual caveats about good specification apply.

If regressions are purely descriptive, they reveal the associations that characterize the health inequality. Then inequality is explained in a statistical sense but implications for policies to reduce inequality are limited.

Why does this work?

OLS regression line always passes through \((\bar{x}, \bar{y})\) because residuals sum to zero.

Why does this work?

The gap decomposes into endowments (different \(\bar{x}\), same coefficients) and coefficients (same \(\bar{x}\), different slopes/intercepts).

Equally valid to use the exposed group’s slope

Same total gap, but different split

What is the average difference in body mass index (BMI) between those with low vs. high education?

How much is due to differing determinants of BMI (age, gender, smoking, drinking, income)?

Any residual difference is due to education differences in the associations between those risk factors and BMI — i.e., the coefficients differ.

Differences in age, smoking, etc. could explain part of the BMI gap

Differences in returns (i.e., coefficients) form the ‘unexplained’ component

If the low-educated had the same covariate means as the high-educated (keeping the low-educated group’s own coefficients), their BMI would be 0.608 \(kg/m^2\) lower (38% of the gap).

Most of this is due to older age among the low-educated, which predicts higher BMI.

If the high-educated group’s covariates had the same relationship with BMI as they do in the low-educated group, their BMI would be 1.030 higher, accounting for 62% of the gap.

Smoking is negative, since smoking predicts higher BMI among the high-educated and is more common among high educated.

• Variables where groups differ and that predict BMI contribute most

• Income and age are the biggest drivers

Relative Concentration Curve

Explicitly accounts for changing composition

\[RCI= \frac{2}{n\mu} \sum_{i=1}^{n}y_{i}R_{i}-1\]

where

• \(\mu\) is the mean of \(y_{i}\) (e.g., smoking);
• \(R_{i}\) is the fractional rank of the ith person in the socioeconomic (i.e., income) distribution.

Decomposition:

Develop a model for predicting \(y\) using several determinants, then plug back into the \(RCI\) equation

Estimates how much of the overall inequality in \(y\) is due to the association between income and other factors that predict health

RCI Decomposition

\(RCI\) is a function of health \((y_{i})\) and socioeconomic rank \((R_{i})\), i.e. \[RCI= \frac{2}{n\mu} \sum_{i=1}^{n}{\color{red}{y_{i}}}R_{i}-1\]

Then write a regression expressing health \((y_{i})\) as a function of several \(k_{i}\) determinants (e.g., age, gender, urban/rural status): \[\color{red}{y_{i}}=\alpha + \sum{\beta_{x}x_{k_{i}}}+\epsilon_{i}\]

RCI Decomposition

Now re-express \(RCI\) as:

\[RCI=\sum{(\beta_{k}\bar{x}_{k}/\mu)RCI_{k}}+gRCI_{e}/\mu\]

Where

• \(\mu\) is the mean of \(y\),
• \(\bar{x}_{k}\) is the mean of determinant \(x_{k}\),
• \(\beta_{k}\) is the regression coefficient for \(x_{k}\), and
• \(RCI_{k}\) is the relative concentration index for \(x_{k}\).

Determinant impact depends on:

1. the strength of the relationship between each factor and income \(\color{blue}{RCI_{k}}\)

2. the strength of the relationship between each factor and health, and its prevalence in the population (elasticity) \(\color{red}{\beta_{k}\bar{x}_{k}/\mu}\)

1. Regress \(y\) (‘health’) on its determinants \((\beta_{k}x_{k})\):

\[y_{i}=\alpha + \sum{\beta_{x}x_{k_{i}}}+\epsilon_{i}\]

2. Calculate means of \(y\) \((\mu)\) and all \(x_{k}\) determinants

3. Calculate (\(RCI\)) for health and for each determinant \((RCI_{k})\), i.e., use each \(x_{k}\) as the “outcome” and estimate a \(RCI\) for age, education, etc.

4. Absolute contribution: for each \(x\) multiply ‘elasticity’ by its concentration index \((RCI_{k})\):

\[(\beta_{k}\bar{x}_{k}/\mu)RCI_{k}\]

5. Calculate the % contribution of each determinant:

\[[(\beta_{k}\bar{x}_{k}/\mu)RCI_{k}]/RCI\]

What are we explaining?

Overall CI = -0.20

Smoking more concentrated among the poor

• \(\beta_{\text{low education}}\) = 0.36 (OR = 1.4)
• Marginal effect: 0.024
• Mean low education: 0.452
• Mean smoking rate: 7.3%

Elasticity for low education is: (0.024 * 0.452 / .073) = 0.148

Interpretation: a 1% increase in low education increases smoking by 14.8% (not percentage points!).

Step 3: \(RCI_{k}\)

Note the y-axis is cumulative share of low education

The poorest 50% account for roughly 60% of the share of low educated.

Estimation for a specific factor: Low education

\[RCI=\sum{ \underbrace{(\beta_{k}\bar{x}_{k}/\mu)}_{\text{elasticity}} RCI_{k}}+gRCI_{e}/\mu\]

The elasticity of smoking (prior slide) = 0.148

Now we have \(RCI_{k}\) for low education = -0.265

The contribution of low education:

\[\text{Elasticity}\times RCI_{ed} = 0.148 * -0.265 = -.039\]

RCI = -0.203

(-) Binge drinking: ↑ smoking but more common at higher incomes

(+) Married: ↓ smoking but more common at higher incomes

Each contribution has sampling variability:

\[\underbrace{\frac{\hat{\beta}_k \bar{x}_k}{\hat{\mu}}}_{\text{elasticity}} \times \underbrace{\widehat{RCI}_k}_{\text{inequality in }x_k}\]

Can bootstrap the decomp \(B\) times to get empirical CIs.

Prior decompositions (e.g., Kitagawa, O-B) ‘descriptive’

Causal mediation challenges

Challenges with non-manipulable exposures

How much would gaps change if we intervened on a target?

Steps for pursuing interventional decomposition

IDIE-exposed is the expected reduction in 1-year mortality among low-income heart failure patients in a hypothetical world where they initiated medication with the same probability as observed among similar high-income patients, rather than their observed probability.

Various decomposition techniques exist that may be useful for analyzing social inequalities in health

Moves beyond measuring to explanations and interventions

Used responsibly, they can help to provide key evidence on why health inequalities exist and change over time.