Decomposition Techniques for Social Epidemiology

Today:

• Inequality decomposition: Concentration Index
• Decomposing two-group differences: Kitagawa-Blinder-Oaxaca

Not covered here:

• Life table decomposition
• Effect decomposition (i.e., mediation)
• Decomposition of population rates
• Inequality decomposition for nominal social groups

• Ultimately, we want to know why health inequalities are changing over time—what changed?

  • Risk factors?
  • Demographic composition?
  • Social conditions?

• Unpacking the ‘components’ of health inequality is an opportunity to better integrate the monitoring of health inequalities with the etiology of health inequalities.

• These techniques often involve various kinds of ‘counterfactual’ scenarios

Relative Concentration Curve

Recall that we can write the CI as:

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

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:

• The basic idea here is to develop a model for predicting \(y\) using several determinants, then plug that model back into the equation for the \(RCI\)
• This allows us to estimate how much of the overall inequality in \(y\) is due to the association between income and other factors that predict health, and how much is ‘unexplained’ by these factors.

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

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

One part \((\beta_{k}\bar{x}_{k}/\mu)RCI_{k}\) that is due to the association between:

• determinants that predict health (i.e., \(\beta_{k}\bar{x}_{k}/\mu\)); and
• the systematic variation in these determinants across income groups (i.e., \(RCI_{k}\))

The other part \((gRCI_{e}/\mu)\) is ‘unexplained’, i.e., inequality that cannot be explained by systematic variation across income groups in the determinants of health.

The influence of determinants depends on two things:

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. Estimate a regression equation predicting \(y\) (‘health’) from its determinants \((\beta_{k}x_{k})\):

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

2. Calculate the mean of \(y\) \((\mu)\) and of each of the \(x_{k}\) determinants (e.g., education, age)

3. Calculate the overall Concentration Index for the health variable (\(CI\)) and for each determinant in the equation predicting health \((CI_{k})\). This means using each determinant \(x_{k}\) as the “outcome” and estimate a \(CI\) for age, \(CI\) for education, etc.

4. Calculate the absolute contribution of each determinant by multiplying its ‘elasticity’ by its concentration index \((CI_{k})\):

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

5. Calculate the percentage contribution of each determinant:

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

Concentration curve for smoking

Overall CI = -0.20

The poorest 50% of the population accounts for around 70% of current smokers.

Note that income decile (the ranking variable) is excluded from the regression.

This is deliberate, since including it can be misleading:

“the residual component will be zero, or close to zero, suggesting that we have explained all or most of the variation in the Concentration Index”

Excluding it means the residual will:

• Capture unexplained income-related variation
• Allow for a direct income → smoking pathway
• Not artificially inflate the “explained” share

The appropriate interpretation is that the covariates in the model account for part of the income gradient in smoking, and the rest remains either directly attributable to income or unexplained by the observed determinants.

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

• Estimated \(\beta\) coeff on low education (logit scale): 0.359 (OR = 1.43)
• Marginal effect on probability scale: 0.024 (2.4 pct points)
• Mean low education: 0.452
• Mean smoking rate: 7.3%

With these parameters, the elasticity of smoking with respect to 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!).

What about the RCI for low education?

Concentration curve for low education

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

Recall the decomposition formula:

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

So the elasticity of smoking (from the previous slide) with respect to low education is (0.024 * 0.452 / .073) = 0.148

Now we have the RCI for low education = -0.265

So now we can calculate the contribution of low education as:

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

Thus low education accounts for -.039/ -0.203 = 19% of the overall \(RCI\)

Decomposition of Income-Related Inequality in Smoking: Sweden

Overall RCI = -0.203

Contributions of each factor can be positive or negative

• Binge drinking contribution is negative since it increases smoking but is more common among higher income.
• Married contribution is positive since it decreases smoking but is more common among higher income.

Each contribution is a product of two estimated quantities:

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

Both carry sampling variability — so do the contributions.

Bootstrap resamples the full dataset and re-runs the decomposition \(B\) times to get empirical CIs.

Decomposition results will be sensitive to the choice of determinants included (i.e., how well-specified the model is for predicting \(y\)).

The regression equations are predictive and not causal models.

Main utility is not in estimating the potential impact on \(y\) of changing the distribution of socioeconomic position, but in indicating the potential role that other factors may play in generating socioeconomic inequalities in health.

The core idea is to explain the distribution of the outcome variable in question by a set of factors that vary systematically with exposure status.

Thus, we want to know, on average, why the mean level of health or disease differs between exposed and unexposed groups.

Since, for most health outcomes there are multiple determinants, we may want to know which of these determinants plays more or less important roles in explaining the difference in average outcomes.

“Unpacking” or “decomposing” difference.

Evelyn Kitagawa (1955) was a sociologist and demographer who devised a non-parametric method for decomposing differences between rates, refined by Prithwis das Gupta in 1978.

• Focused on understanding group contributions to rate differences.

Studies by 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

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.

If data allow identification of causal effects, then the factors that generate the inequality are identified.

Then one can (potentially) draw conclusions about how policies would impact on inequality.

Two potential sources of mean differences in outcomes

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)

The overall gap between exposed and unexposed can be written as a function of differences the respective beta coefficients, evaluated at the mean for each group:

First method

• Coefficients of unexposed

• Means of exposed

\(y^{exp} - y^{unexp} = \Delta\bar{x}\beta^{unexp}-\Delta\beta x^{exp}\)

Second method

• Coefficients of exposed

• Means of unexposed

\(y^{exp} - y^{unexp} = \Delta\bar{x}\beta^{exp}-\Delta\beta x^{unexp}\)

In the first, the differences in the \(X\)s are weighted by the coefficients of the unexposed group and the differences in the coefficients are weighted by the \(X\)s of the exposed group: \[y^{exp} - y^{unexp} = \Delta\bar{x}\beta^{unexp}-\Delta\beta x^{exp}\]

whereas, in the second, the differences in the \(X\)s are weighted by the coefficients of the exposed group and the differences in the coefficients are weighted by the \(X\)s of the unexposed group: \[y^{exp} - y^{unexp} = \Delta\bar{x}\beta^{exp}-\Delta\beta x^{unexp}\]

See Oaxaca (1973), Blinder (1973) and Cotton (1988) for details.

Basic question

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

How much of this difference is due to the fact that determinants of BMI (age, gender, smoking, drinking, income) differ between education groups?

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

• Low-educated Italians differ from high-educated on several covariates

• These differences could explain part of the BMI gap

• Each covariate may have a different association with BMI by education

• Differences in returns form the unexplained component

Interpretation:

• 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 — closing 38% of the gap.

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

Interpretation:

• If the high-educated group’s covariates had the same relationship with BMI as they do in the low-educated group (i.e., if low-ed coefficients applied to high-ed covariate means), their BMI would be 1.030 higher, accounting for 62% of the gap.

• Note 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

Methods frontier

Attempting to reconcile the non-causal framework of KBO with mediation methods, new estimators.

Various decomposition techniques exist that may be useful for analyzing social determinants of health:

• Regression-based decomposition of Concentration Index
• Oaxaca decomposition of mean health between groups

All of these techniques make assumptions that need to be evaluated in the course of analysis.

When used properly, decomposition techniques can help to provide key evidence on why health inequalities exist and change over time.