Dan Gillen Department of Statistics University of ...dgillen/STAT211/Handouts/Discussion... · Stat...

Lecture – Discussion

Stat 211 - D. Gillen

Confounding

Collapsibility

Confounding vs.Noncollapsibility

Adjustment forPrecision Variables

Simulation Study

Extra: TheProportional HazardsModelTwo conjectures

Simulation results

Summary

– Discussion.1


Confounding, Non-Collapsibility,Precision, and PowerStatistics 211 - Statistical Methods II

Presented February 27, 2018

Dan GillenDepartment of Statistics

University of California, Irvine



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.2

Confounding

Various definitions of confounding

1. A type of bias in estimating causal effects, resulting from amixing of effects of extraneous factors with the effect ofinterest. In this setting, a confounder is usually defined athird variable that causally effects both the predictor ofinterest and the outcome.

2. The phenomenon that occurs when stratum-specific andcrude measurements differ. The stratification variablewould then be termed a confounder.

3. Inseparablility of main effects and interactions under aparticular controlled design.



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.3

Confounding

Counterfactual approach to causation (Neyman, 1923)

I Suppose that N units are to be assigned one of Ktreatments x0, x1, ..., xK−1, with x0 the referent treatment.

I The outcome of interest for the i th unit is the value of theresponse variable Yi .

I Further, suppose that Yi will equal yik if unit i is assignedtreatment xk .

Then the causal effect of xk on Yi relative to x0 is defined to bea specified contrast of yik and yi0, say h(yik , yi0).

I For example, we may take h to be the difference yik − yi0.Of course, because only one of the potential outcomesyik (k ≥ 0) can be observed in any one unit, an individualeffect yik − yi0 cannot be observed.



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.4

Confounding

Counterfactual approach to causation (Neyman, 1923)

I Probabilistic Extension:

I Consider the joint distribution F (y0, ..., yK ) of yi0, ..., yiK in apopulation of units.

I Then consider population effects defined by differencesamong the marginal distributions F (y0), ...,F (yK ), or asummary measure of these marginal distributions, eg.µk − µ0 where µk represents the mean of the distributionF (yk ).



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.5

Confounding

Confounding based on the counterfactual model

I Suppose we wish to determine the effect of applying atreatment x1 on a parameter µ in population A, relative toapplying treatment x0.

I Suppose that µ will equal µA1 if x1 is administered topopulation A, and will equal µA0 if x0 is administered topopulation A.

I Of course, if treatment x1 is administered to the targetpopulation, A, then we will be able to observe µA1, but µA0will be unobserved. To obtain a comparison measure, wewill instead administer treatment x0 to a control population,B, allowing us to observe µB0.



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.6

Confounding


I The causal effect of x1 relative to x0 based on thecounterfactual model is defined as the change from µA0 toµA1, based on some specified contrast of the twomeasures. Since we cannot observe µA0 we must insteadbase our inference on the contrast between µB0 and µA1,eg. µA1 − µB0.

I Based on the counterfactual model, we say thatconfounding exists if

µA1 − µA0 6= µA1 − µB0

or equivalently if

µA0 6= µB0



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.7

Confounding


I Notice however that the counterfactual definition ofconfounding states no explicit differences betweenpopulations A and B with respect to covariates that mightaffect µ.

I Clearly, if µA0 and µB0 differ, then A and B must differ withrespect to covariates that effect µ, these covariates beingtermed confounders in the counterfactual context.

I This definition differs from that given by (1) and (2) above inthe sense that although drastic differences in covariatedistributions may occur between the comparisonpopulations, µA0 and µB0 may still be equal, resulting in noconfounding based on the counterfactual definition.



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.8

Confounding

(Unrealistic but possible) Example

I The effect of Statin use on mean total cholesterol

Potential confounder Effect on total cholesterol↑ age ↑↑ obesity ↑

I What if younger obese patients were more likely to berandomized to Statins?

I Possible Scenario: The adverse effect of the largeproportion of obese patients in the Statins group mayoffset the beneficial effect of the large proportion ofyounger patients, leaving

µControl,Young Obese = µControl,Older Non−obese



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.9

Collapsibility

Collapsibility (Greenland, et al, 1999)

I Consider

I a I × J × K contingency table representing the jointdistribution of three discrete variables X ,Y , and Z ,

I with the I × J marginal table representing the jointdistribution of X and Y ,

I and the set of K I × J subtables representing the jointdistribution of X and Y within levels of Z .

I Then a measure of association of X and Y is said to becollapsible across Z if it is constant across the strata of Zand this constant value equals the value obtained from themarginal table.



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.10

Collapsibility


Example:

Z=1 Z=0 MarginalX=1 X=0 X=1 X=0 X=1 X=0

Y =1 90 70 30 10 120 80Y =0 10 30 70 90 80 120

Risks (Pr[Y =1]) .90 .70 .30 .10 .60 .40

Risk Differences .20 .20 .20Risk Ratios 1.29 3.00 1.50Odds Ratio 3.86 3.86 2.25



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.11

Collapsibility


I In this case

1. The risk difference is strictly collapsible (stratum specificmeasures equal to the marginal measure)

2. The risk ratio is not collapsible (summary measure variesacross the strata of Z )

3. The odds ratio is not collapsible (stratum specific measuresnot equal to the marginal measure).



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.12

Collapsibility

Example: Noncollapsibility without confounding

I Objective: To investigate the effect of an experimentaltreatment (Tx) on the response probability for the outcomeY in a population A

I To investigate the effect of Tx a control sample B isenlisted

I Sample B is chosen so that the distribution of the potentialconfounder Z is the same as that in the sample frompopultion A



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.13

Collapsibility


Index Sample (A)Response probability if

Stratum Tx=1 Tx=0 Stratum SizeZ=1 0.9 0.7 1,000Z=0 0.3 0.1 1,000Unconditional on Z 0.6 0.4

Control Sample (B)Response probability if

Stratum Tx=1 Tx=0 Stratum SizeZ=1 – 0.7 1,000Z=0 – 0.1 1,000Unconditional on Z – 0.4



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.14

Collapsibility


I First note that no confounding exists (w/ respect to Z ) inthe ’covariate imbalance’ definition since this was fixed bydesign, nor in the counterfactual definition since:

True crude OR =µA1/(1− µA1)

µA0/(1− µA0)

=0.6/(1− 0.6)0.4/(1− 0.4)

= 2.25

=µA1/(1− µA1)

µB0/(1− µB0)= Observable crude OR



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.15

Collapsibility


I But within the levels of Z , we have

ORZ=1 =0.9/(1− 0.9)0.7/(1− 0.7)

= 3.86

ORZ=0 =0.3/(1− 0.3)0.1/(1− 0.1)

= 3.86

I Thus the stratum specific estimates of the OR are equal,yet different from the crude (marginal) OR, ie. the OR isnoncollapsible. Note that this phenomenon is not bias, butrequires careful interpretation of marginal andstratum-specific effects.



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.16

Collapsibility

Example: Confounding with collapsibility

Index Sample (A)Response probability if

Stratum Tx=1 Tx=0 Stratum SizeZ=1 0.9 0.7 1,000Z=0 0.3 0.1 1,000Unconditional on Z 0.6 0.4

Control Sample (B)Response probability if

Stratum Tx=1 Tx=0 Stratum SizeZ=1 – 0.7 350Z=0 – 0.1 1,650Unconditional on Z – 0.28



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.17

Collapsibility

Example: Confounding with collapsibility

I By changing the number of subjects with Z=0, we haveintroduced confounding. To see this, note that

Oberservable crude OR =µA1/(1− µA1)

µB0/(1− µB0)

=0.6/(1− 0.6)

0.28/(1− 0.28)= 3.86 6= True crude OR (2.25)

I On the other hand the crude OR of 3.86 does now equalthe stratum specific odds ratios computed previously.



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.18

Precision Variables

Extension to regression

I Consider a generalized linear model for the regression ofY on two covariates X and Z :

g[E(Y |X = x ,Z = z)] = β0 + β1x + β2z.

I Then the regression is said to be noncollapsible for β1 overZ if β1 6= β∗

1 in the regression omitting Z ,

g[E(Y |X = x)] = β∗0 + β∗

1 x .



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.19

Precision Variables


I Suppose that the full model is correct, then β1 isgauranteed to be collapsible over Z in the followingsituations:

1. β2 = 0 (ie. no association between Y and Z )

2. Neither β1 nor β2 is zero, X and Z are independent, AND gis the identity or log link (Gail, Wieand and Piantadosi,1984; Gail 1986).

I Also note that collapsibility for β1 over Z can occur even ifX and Z are associated.

I Thus we cannot equate collapsibility over Z withindependence of X .



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.20

Precision Variables


I In the case of situation (2), where we have independencebetween X and Z and noncollapsibility over Z , thedifference between β1 and β∗

1 is often interpreted as biasdue to confounding.

I However this is not generally true unless g is the identity orlog link.

I Instead, we must take extra precaution in interpretingstratum-specific and population-averaged (marginal)effects.

I That is, if X and Z are independent, it is possible for β1 tounbiasedly represent the effect of manipulating X withinlevels of Z , and at the same time, for β∗

1 to unbiasedlyrepresent the unconditional effect of manipulating X , eventhough β1 6= β∗

1 .



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.21

Precision Variables

Graphical representation of noncollapsibility in logisticregression

I X ∼ N (0,1)

I Z a 3-level categorical predictor (Zi representing anindicator for groups i=2,3), Z independent of X

I Full Model:

logit[E(Y |X = x ,Z = z)] = β0 + β1x + β2z2 + β3z3

I Reduced Model:

logit[E(Y |X = x)] = β∗0 + β∗

1 x



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.22

Precision Variables

Graphical representation of noncollapsibility in logisticregression

Precision Variables

Graphical representation of noncollapsibility inlogistic regression

X

Pro

b[Y

=1]

-10 -5 0 5 10

0.0

0.2

0.4

0.6

0.8

1.0

Stratum specific probabilities

Marginal probability

Page 27/41-1 D. Gillen/UCI Epi-2007/01.26.2007



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.23

Precision Variables

Effect of adjustment for precision variables on power

I Linear regression

I Consider the linear regression model:

Yi = β0 + β1Xi + β2Zi + εi

where εi ∼ N (0, σ2F ).

I Further consider the reduced model:

Yi = β∗0 + β∗

1 Xi + ε∗i

where ε∗i ∼ N (0, σ2R).



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.24

Precision Variables


I Linear regression (cont’d)

I Suppose X and Z are independent and β2 6= 0, then:

1. β1 = β∗1 (collapsible due to identity link)

2. Var(β̂1) < Var(β̂∗1 )

I Thus if Z is a predictor of Y and Z is independent of X , wecan gain power by adjusting for Z .



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.25

Precision Variables

Effect of adjustment for precision variables on power Precision Variables

Linear regression

0

( )

!^

1*

Reduced (unadjusted) model:

0

( )

!^

1

Full (adjusted) model:




Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.26

Precision Variables


I Logistic regression

I Let Yi ∼ B(1, πi) be a response such that

log(

πi

1− πi

)= β0 + β1Xi + β2Zi


log(

πi

1− πi

)= β∗

0 + β∗1 Xi .



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.27

Precision Variables


Linear regression

0

( )

!^

1*


0

( )

!^

1





Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.28

Precision Variables


I Logistic regression (cont’d)

I Suppose X and Z are independent and β2 6= 0, then:

1. |β∗1 | < |β1|

2. Var(β̂∗1 ) < Var(β̂1)

I Thus although adjustment for Z may increase variability ofthe estimate of β1, this can be offset by β∗

1 ’s position inrelation to the null hypothesis, resulting in relatively littleeffect on power.



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.29

Precision Variables


Logistic regression

0

( )

!^

1*


0

( )

!^

1





Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.30

Simulation results

Linear and logistic regression

Reduced Model Full ModelModel Mean β̂1 Power Mean β̂1 Power

Linear Regressionβ1 = 1, β2 = 0 0.994 0.996 0.997 0.996β1 = 1, β2 = 2 0.989 0.656 1.009 0.999β1 = 1, β2 = 4 0.960 0.235 0.996 0.998

Logistic Regressionβ1 = 1, β2 = 0 0.918 0.968 0.928 0.969β1 = 1, β2 = 1 0.834 0.935 0.961 0.968β1 = 1, β2 = 2 0.646 0.804 0.956 0.897



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.31

Precision Variables


I Proportional hazards regression

I The proportional hazards model specifies that the hazardfunction, λ(t |X ,Z ), is given by

λ(t |X ,Z ) = λ0(t)eβ1X+β2Z


λ(t |X ) = λ∗0(t)e

β∗1 X



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.32

Precision Variables


I Proportional hazards regression (cont’d)

I Conjecture 1:

I Suppose X and Z are independent, β2 6= 0, andproportional hazards holds in the adjusted model. Then:

1. |β∗1 | < |β1|

2. Var(β̂∗1 ) < Var(β̂1)



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.33

Precision Variables


I Proportional hazards regression (cont’d)

I Conjecture 2:

I Suppose X and Z are independent, β2 6= 0, proportionalhazards holds in the unadjusted model. Then for localdeviations from proportional hazards in the adjusted model:

1. |β∗1 | < |β1|

2. Var(β̂∗1 ) < Var(β̂1)



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.34

Simulation results

Poisson and Cox regression

Reduced Model Full ModelModel Mean β̂1 Power Mean β̂1 Power

Poisson Regressionβ1 = 1, β2 = 0 0.951 0.983 0.960 0.982β1 = 1, β2 = 1 0.951 0.965 0.993 0.980β1 = 1, β2 = 2 0.948 0.924 0.960 0.978

Cox Regression (case 1)β1 = 1, β2 = 0 1.038 0.992 1.054 0.989β1 = 1, β2 = 1 0.785 0.929 1.066 0.996β1 = 1, β2 = 2 0.529 0.684 1.070 0.987



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.35

Summary

Take-home messages

I Various definitions of confounding exist, many of which arenot distinguished.

I Noncollapsibility can occur in the absence of confounding,and confounding can occur in the absence ofnoncollapsibility.

I When noncollapsibility occurs in the absence ofconfounding, the phenomenon is not bias, as long as oneis careful to interpret estimates as either stratum-specificor population-averaged.

I In the context of generalized linear models confounding (inthe counterfactual sense) and noncollapsibility are notequivalent unless using an identity or log link.



Confounding

Collapsibility



Simulation Study


Simulation results

Summary

– Discussion.36

Summary

Take-home messages

I We conjecture that substantial gains in power can beobtained by modeling important predictors of outcomewhich are independent with the predictor of interest(precision variable) in the setting Cox regression, and theamount of gain depends on the strength of the effect ofprecision variable.

Dan Gillen Department of Statistics University of ...dgillen/STAT211/Handouts/Discussion... · Stat...

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