Lecture Discussion. Confounding, Non-Collapsibility, Precision, and Power Statistics Statistical Methods II. Presented February 27, 2018

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1 , Non-, Precision, and Power Statistics Statistical Methods II Presented February 27, 2018 Dan Gillen Department of Statistics University of California, Irvine Discussion.1

2 Various definitions of confounding 1. A type of bias in estimating causal effects, resulting from a mixing of effects of extraneous factors with the effect of interest. In this setting, a confounder is usually defined a third variable that causally effects both the predictor of interest and the outcome. 2. The phenomenon that occurs when stratum-specific and crude measurements differ. The stratification variable would then be termed a confounder. 3. Inseparablility of main effects and interactions under a particular controlled design. Discussion.2

3 Counterfactual approach to causation (Neyman, 1923) Suppose that N units are to be assigned one of K treatments x 0, x 1,..., x K 1, with x 0 the referent treatment. The outcome of interest for the ith unit is the value of the response variable Y i. Further, suppose that Y i will equal y ik if unit i is assigned treatment x k. Then the causal effect of x k on Y i relative to x 0 is defined to be a specified contrast of y ik and y i0, say h(y ik, y i0 ). For example, we may take h to be the difference y ik y i0. Of course, because only one of the potential outcomes y ik (k 0) can be observed in any one unit, an individual effect y ik y i0 cannot be observed. Discussion.3

4 Counterfactual approach to causation (Neyman, 1923) Probabilistic Extension: Consider the joint distribution F(y0,..., y K ) of y i0,..., y ik in a population of units. Then consider population effects defined by differences among the marginal distributions F(y 0 ),..., F (y K ), or a summary measure of these marginal distributions, eg. µ k µ 0 where µ k represents the mean of the distribution F(y k ). Discussion.4

5 based on the counterfactual model Suppose we wish to determine the effect of applying a treatment x 1 on a parameter µ in population A, relative to applying treatment x 0. Suppose that µ will equal µ A1 if x 1 is administered to population A, and will equal µ A0 if x 0 is administered to population A. Of course, if treatment x 1 is administered to the target population, A, then we will be able to observe µ A1, but µ A0 will be unobserved. To obtain a comparison measure, we will instead administer treatment x 0 to a control population, B, allowing us to observe µ B0. Discussion.5

6 based on the counterfactual model The causal effect of x 1 relative to x 0 based on the counterfactual model is defined as the change from µ A0 to µ A1, based on some specified contrast of the two measures. Since we cannot observe µ A0 we must instead base our inference on the contrast between µ B0 and µ A1, eg. µ A1 µ B0. Based on the counterfactual model, we say that confounding exists if µ A1 µ A0 µ A1 µ B0 or equivalently if µ A0 µ B0 Discussion.6

7 based on the counterfactual model Notice however that the counterfactual definition of confounding states no explicit differences between populations A and B with respect to covariates that might affect µ. Clearly, if µ A0 and µ B0 differ, then A and B must differ with respect to covariates that effect µ, these covariates being termed confounders in the counterfactual context. This definition differs from that given by (1) and (2) above in the sense that although drastic differences in covariate distributions may occur between the comparison populations, µ A0 and µ B0 may still be equal, resulting in no confounding based on the counterfactual definition. Discussion.7

8 (Unrealistic but possible) Example The effect of Statin use on mean total cholesterol Potential confounder age obesity Effect on total cholesterol What if younger obese patients were more likely to be randomized to Statins? Possible Scenario: The adverse effect of the large proportion of obese patients in the Statins group may offset the beneficial effect of the large proportion of younger patients, leaving µ Control,Young Obese = µ Control,Older Non obese Discussion.8

9 (Greenland, et al, 1999) Consider a I J K contingency table representing the joint distribution of three discrete variables X, Y, and Z, with the I J marginal table representing the joint distribution of X and Y, and the set of K I J subtables representing the joint distribution of X and Y within levels of Z. Then a measure of association of X and Y is said to be collapsible across Z if it is constant across the strata of Z and this constant value equals the value obtained from the marginal table. Discussion.9

10 (Greenland, et al, 1999) Example: Z =1 Z =0 Marginal X=1 X=0 X=1 X=0 X=1 X=0 Y = Y = Risks (Pr[Y =1]) Risk Differences Risk Ratios Odds Ratio Discussion.10

11 (Greenland, et al, 1999) In this case 1. The risk difference is strictly collapsible (stratum specific measures equal to the marginal measure) 2. The risk ratio is not collapsible (summary measure varies across the strata of Z ) 3. The odds ratio is not collapsible (stratum specific measures not equal to the marginal measure). Discussion.11

12 Example: without confounding Objective: To investigate the effect of an experimental treatment (Tx) on the response probability for the outcome Y in a population A To investigate the effect of Tx a control sample B is enlisted Sample B is chosen so that the distribution of the potential confounder Z is the same as that in the sample from popultion A Discussion.12

13 Example: without confounding Index Sample (A) Response probability if Stratum Tx=1 Tx=0 Stratum Size Z= ,000 Z= ,000 Unconditional on Z Control Sample (B) Response probability if Stratum Tx=1 Tx=0 Stratum Size Z= ,000 Z= ,000 Unconditional on Z 0.4 Discussion.13

14 Example: without confounding First note that no confounding exists (w/ respect to Z ) in the covariate imbalance definition since this was fixed by design, nor in the counterfactual definition since: True crude OR = µ A1/(1 µ A1 ) µ A0 /(1 µ A0 ) 0.6/(1 0.6) = 0.4/(1 0.4) = 2.25 = µ A1/(1 µ A1 ) µ B0 /(1 µ B0 ) = Observable crude OR Discussion.14

15 Example: without confounding But within the levels of Z, we have OR Z =1 = OR Z =0 = 0.9/(1 0.9) 0.7/(1 0.7) = /(1 0.3) 0.1/(1 0.1) = 3.86 Thus the stratum specific estimates of the OR are equal, yet different from the crude (marginal) OR, ie. the OR is noncollapsible. Note that this phenomenon is not bias, but requires careful interpretation of marginal and stratum-specific effects. Discussion.15

16 Example: with collapsibility Index Sample (A) Response probability if Stratum Tx=1 Tx=0 Stratum Size Z= ,000 Z= ,000 Unconditional on Z Control Sample (B) Response probability if Stratum Tx=1 Tx=0 Stratum Size Z= Z= ,650 Unconditional on Z 0.28 Discussion.16

17 Example: with collapsibility By changing the number of subjects with Z =0, we have introduced confounding. To see this, note that Oberservable crude OR = µ A1/(1 µ A1 ) µ B0 /(1 µ B0 ) 0.6/(1 0.6) = 0.28/(1 0.28) = 3.86 True crude OR (2.25) On the other hand the crude OR of 3.86 does now equal the stratum specific odds ratios computed previously. Discussion.17

18 Extension to regression Consider a generalized linear model for the regression of Y on two covariates X and Z : g[e(y X = x, Z = z)] = β 0 + β 1 x + β 2 z. Then the regression is said to be noncollapsible for β 1 over Z if β 1 β1 in the regression omitting Z, g[e(y X = x)] = β 0 + β 1x. Discussion.18

19 Extension to regression Suppose that the full model is correct, then β 1 is gauranteed to be collapsible over Z in the following situations: 1. β 2 = 0 (ie. no association between Y and Z ) 2. Neither β 1 nor β 2 is zero, X and Z are independent, AND g is the identity or log link (Gail, Wieand and Piantadosi, 1984; Gail 1986). Also note that collapsibility for β 1 over Z can occur even if X and Z are associated. Thus we cannot equate collapsibility over Z with independence of X. Discussion.19

20 Extension to regression In the case of situation (2), where we have independence between X and Z and noncollapsibility over Z, the difference between β 1 and β1 is often interpreted as bias due to confounding. However this is not generally true unless g is the identity or log link. Instead, we must take extra precaution in interpreting stratum-specific and population-averaged (marginal) effects. That is, if X and Z are independent, it is possible for β 1 to unbiasedly represent the effect of manipulating X within levels of Z, and at the same time, for β1 to unbiasedly represent the unconditional effect of manipulating X, even though β 1 β1. Discussion.20

21 Graphical representation of noncollapsibility in logistic regression X N (0, 1) Z a 3-level categorical predictor (Z i representing an indicator for groups i=2,3), Z independent of X Full : logit[e(y X = x, Z = z)] = β 0 + β 1 x + β 2 z 2 + β 3 z 3 Reduced : logit[e(y X = x)] = β 0 + β 1 x Discussion.21

22 Graphical representation of noncollapsibility in logistic regression Graphical representation of noncollapsibility in logistic regression Prob[Y=1] Stratum specific probabilities Marginal probability X Page 27/41-1 Discussion.22 D. Gillen/UCI Epi-2007/

23 Effect of adjustment for precision variables on power Linear regression Consider the linear regression model: Y i = β 0 + β 1 X i + β 2 Z i + ɛ i where ɛ i N (0, σf 2 ). Further consider the reduced model: Y i = β 0 + β 1 X i + ɛ i where ɛ i N (0, σ 2 R). Discussion.23

24 Effect of adjustment for precision variables on power Linear regression (cont d) Suppose X and Z are independent and β2 0, then: 1. β 1 = β1 (collapsible due to identity link) 2. Var( ˆβ 1 ) < Var( ˆβ 1 ) Thus if Z is a predictor of Y and Z is independent of X, we can gain power by adjusting for Z. Discussion.24

25 Effect of adjustment for precision variables on power inear regression Reduced (unadjusted) model: 0 ( )!^1 * Full (adjusted) model: 0 ( )!^1 Discussion.25

26 Effect of adjustment for precision variables on power Logistic regression Let Yi B(1, π i ) be a response such that log ( πi 1 π i ) = β 0 + β 1 X i + β 2 Z i Further consider the reduced model: log ( πi 1 π i ) = β0 + β1 X i. Discussion.26

27 Effect of adjustment for precision variables on power inear regression Reduced (unadjusted) model: 0 ( )!^1 * Full (adjusted) model: 0 ( )!^1 Discussion.27

28 Effect of adjustment for precision variables on power Logistic regression (cont d) Suppose X and Z are independent and β2 0, then: 1. β 1 < β 1 2. Var( ˆβ 1 ) < Var( ˆβ 1 ) Thus although adjustment for Z may increase variability of the estimate of β 1, this can be offset by β1 s position in relation to the null hypothesis, resulting in relatively little effect on power. Discussion.28

29 Effect of adjustment for precision variables on power gistic regression Reduced (unadjusted) model: 0 ( )!^1 * Full (adjusted) model: 0 ( )!^1 Discussion.29

30 Linear and logistic regression Reduced Full Mean ˆβ 1 Power Mean ˆβ 1 Power Linear Regression β 1 = 1, β 2 = β 1 = 1, β 2 = β 1 = 1, β 2 = Logistic Regression β 1 = 1, β 2 = β 1 = 1, β 2 = β 1 = 1, β 2 = Discussion.30

31 Effect of adjustment for precision variables on power Proportional hazards regression The proportional hazards model specifies that the hazard function, λ(t X, Z ), is given by λ(t X, Z ) = λ 0 (t)e β 1X+β 2 Z Further consider the reduced model: λ(t X) = λ 0(t)e β 1 X Discussion.31

32 Effect of adjustment for precision variables on power Proportional hazards regression (cont d) Conjecture 1: Suppose X and Z are independent, β2 0, and proportional hazards holds in the adjusted model. Then: 1. β 1 < β 1 2. Var( ˆβ 1 ) < Var( ˆβ 1 ) Discussion.32

33 Effect of adjustment for precision variables on power Proportional hazards regression (cont d) Conjecture 2: Suppose X and Z are independent, β2 0, proportional hazards holds in the unadjusted model. Then for local deviations from proportional hazards in the adjusted model: 1. β 1 < β 1 2. Var( ˆβ 1 ) < Var( ˆβ 1 ) Discussion.33

34 Poisson and Cox regression Reduced Full Mean ˆβ 1 Power Mean ˆβ 1 Power Poisson Regression β 1 = 1, β 2 = β 1 = 1, β 2 = β 1 = 1, β 2 = Cox Regression (case 1) β 1 = 1, β 2 = β 1 = 1, β 2 = β 1 = 1, β 2 = Discussion.34

35 Take-home messages Various definitions of confounding exist, many of which are not distinguished. can occur in the absence of confounding, and confounding can occur in the absence of noncollapsibility. When noncollapsibility occurs in the absence of confounding, the phenomenon is not bias, as long as one is careful to interpret estimates as either stratum-specific or population-averaged. In the context of generalized linear models confounding (in the counterfactual sense) and noncollapsibility are not equivalent unless using an identity or log link. Discussion.35

36 Take-home messages We conjecture that substantial gains in power can be obtained by modeling important predictors of outcome which are independent with the predictor of interest (precision variable) in the setting Cox regression, and the amount of gain depends on the strength of the effect of precision variable. Discussion.36