Tweaking -khb- to control for post-treatment confounders in mediation analysis

Kristian Bernt Karlson

Department of SociologyUniversity of Copenhagen

Email: kbk@soc.ku.dk

Slide 1 :: Aarhus University, September 5, 2014

Mediation analysis

Decomposition:

Total effect:β+θγDirect effect: βIndirect effect: θγ

Slide 2 :: Aarhus University, September 5, 2014

X Y

M

β

θ γ

Post-treatment confounders

Even if X is a randomized treatment, M may still be endogenous.

Potential strategy: Control for proxies for U to obtain unbiased estimate of γ*.

Slide 3 :: Aarhus University, September 5, 2014

X Y

M

*β

θ γ*

U

δλ

Post-treatment confounders

However: If (some element in) U depends on X, then everything breaks down…

Direct effect of on Y other than through M is not identified, even if proxies for U are included…

… damned if you do (control), damned if you don’t (control)!Slide 4 :: Aarhus University, September 5, 2014

X Y

M

*β

θ* γ*

U

δλπ

Solutions

Inverse probability weighting or g-computation (James Robins and colleagues).

Logic: Reweigh data in such a way that the X-U edge disappears, i.e., X and (proxies for) U become statistically independent.

In this talk: Residualize (observed proxies for) U

for O by regression...

Slide 5 :: Aarhus University, September 5, 2014

Solutions

Let Z be observed elements of U and assume linearity.

Direct effect:

Indirect effect:

Slide 6 :: Aarhus University, September 5, 2014

X Y

M

*β

θ* γ*

Z

δλπ

*

* *

Solutions

Estimate the direct effect other than through M via

Where is the residual from the auxiliary regression

That is Slide 7 :: Aarhus University, September 5, 2014

X Y

M

*β

θ* γ*

Z

δλπ

E Z X

*

*E Y X M Z %

Z%

- and then back out indirect effect.

The logistic case and -khb-

Breen, Karlson, and Holm (2013, Sociological Methods and Research) suggest a way of decomposing total effects into direct and indirect effects in logistic regression models (that accounts for the attenuation bias implied in these models).

Method implemented in -khb-.

In this presentation: Use “residualization trick” to construct Z that are uncorrelated with X, and then apply -khb- to the transformed variables…Slide 8 :: Aarhus University, September 5, 2014

Example

Education (E) is viewed as the key mediator of the association between class origins (O) and class destinations (D) in sociological stratification research.

Yet the impact of education (E) on class destinations (D), i.e., the “status returns” to education, may be biased by omitted ability (A) – or so the economists say…

But ability (A) also depends strongly on class origins (O); that is, ability (A) is a post-treatment confounder.Slide 9 :: Aarhus University, September 5, 2014

Example

Data from GB cohorts born 1958 and 1970Class origins and destinations measured by binary distinction between service class and working class. Education is a ordinal variable with 6 levels. Ability is a standardized cognitive test score (measured at around 11).

The conclusion – percent mediated by education:

Conventional estimate: 54 %“Ability-corrected” estimate: 44 %

Slide 10 :: Aarhus University, September 5, 2014

Example (conventional approach)

Slide 11 :: Aarhus University, September 5, 2014

OSC 2,1822874 54,18 1,2868005 Variable Conf_ratio Conf_Pct Resc_Fact

Summary of confounding

Diff 1,117957 ,0667154 16,76 0,000 ,9871974 1,248717 Full ,9455884 ,0892704 10,59 0,000 ,7706216 1,120555 Reduced 2,063546 ,0963444 21,42 0,000 1,874714 2,252377OSC DSC Coef. Std. Err. z P>|z| [95% Conf. Interval] Concomitant: male surveyZ-variable(s): educ2 educ3 educ4 educ5 educ6Variables of Interest: OSC Pseudo R2 = 0,32Model-Type: logit Number of obs = 3791

Decomposition using the KHB-Method

. khb logit DSC OSC || educ2-educ6, c(male survey) s

Example (ability-corrected)

Slide 12 :: Aarhus University, September 5, 2014

OSC 1,7785452 43,77 1,1690219 Variable Conf_ratio Conf_Pct Resc_Fact

Summary of confounding

Diff ,9320424 ,0632029 14,75 0,000 ,8081671 1,055918 Full 1,197159 ,094891 12,62 0,000 1,011176 1,383142 Reduced 2,129201 ,09886 21,54 0,000 1,935439 2,322963OSC DSC Coef. Std. Err. z P>|z| [95% Conf. Interval] Concomitant: male survey abilresZ-variable(s): educ2 educ3 educ4 educ5 educ6Variables of Interest: OSC Pseudo R2 = 0,35Model-Type: logit Number of obs = 3791

Decomposition using the KHB-Method

. khb logit DSC OSC || educ2-educ6, c(male survey abilres) s

. predict abilres, res

. qui reg abil OSC male survey

Conclusion

Example: We overestimate the role of education in social mobility in conventional studies (but perhaps not much?).

Generally, difficult to know the bias, but even without ability observed, we could have figured this out, given the evidence we have on the causes and effects of ability.

Convenient “tweak” of -khb-: Under linearity assumption, one can quite easily control for post-treatment confounders.

To do: Standard errors in ability-corrected analysis are wrong. Bootstrap the easy solution (delta method another).

Slide 14 :: Aarhus University, September 5, 2014