BIOS 312: MODERN REGRESSION ANALYSISbiostat.mc.vanderbilt.edu/wiki/pub/Main/CourseBios312/effmod.pdf · 4.3.3 Diagnosing Confounding Confounding typically produces a difference between

BIOS 312: MODERN REGRESSION ANALYSIS

James C (Chris) SlaughterDepartment of Biostatistics

Vanderbilt University School of [email protected]

biostat.mc.vanderbilt.edu/CourseBios312

Copyright 2009-2012 JC Slaughter All Rights ReservedUpdated January 5, 2012

Contents

4 Effect Modification and Confounding 5

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4.2 Effect Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.2.1 Examples of Effect Modification . . . . . . . . . . . . . . . . 7

4.2.2 Analysis of Effect Modification . . . . . . . . . . . . . . . . 10

4.3 Confounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3.1 Simpson’s Paradox . . . . . . . . . . . . . . . . . . . . . . . 11

4.3.2 Definition of Confounding . . . . . . . . . . . . . . . . . . . 12

4.3.3 Diagnosing Confounding . . . . . . . . . . . . . . . . . . . 15

4.4 Precision Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.4.2 Adjusting for Precision Variables . . . . . . . . . . . . . . . 18

4.5 Diagnosing Confounding . . . . . . . . . . . . . . . . . . . . . . . . 21

4.5.1 Adjustment for Covariates . . . . . . . . . . . . . . . . . . . 21

4.5.2 Confounder Detection . . . . . . . . . . . . . . . . . . . . . 22

3

CONTENTS 4

4.6 Graphical Methods for Visualizing Effect Modification, Confound-ing, and Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.6.1 Effect Modifiers . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.6.2 Confounders . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.6.3 Precision Variables . . . . . . . . . . . . . . . . . . . . . . . 28

Chapter 4

Effect Modification and Confounding

4.1 Overview

· Scientific questions

– Most often scientific questions are translated into comparing the distribu-tion of some response variable across groups of interest

– Groups are defined by the predictor of interest (POI)∗ Categorical predictors of interest: Treatment or control, knockout or

wild type, ethnic group

∗ Continuous predictors of interest: Age, BMI, cholesterol, blood pres-sure

· Often we need to consider additional variables other than POI because...

– We want to make comparisons in different strata∗ e.g if we stratify by gender, we may get different answers to our scien-

tific question in men and women

5

CHAPTER 4. EFFECT MODIFICATION AND CONFOUNDING 6

– Groups being compared differ in other ways∗ Confounding: A variable that is related to both the outcome and pre-

dictor of interest

– Less variability in the response if we control for other variables∗ Precision: If we restrict to looking within certain strata, may get smallerσ2

· Covariates other than the Predictor of Interest are included in the model as...

– Effect modifiers

– Confounders

– Precision variables

– Not necessarily mutually exclusive

4.2 Effect Modification

· The association between the Response and the Predictor of Interest differsin strata defined by the effect modifier

– Statistical term: “Interaction” between the effect modifier and the POI

– Effect modification depends on the measure of effect that you choose∗ Choice of summary measure: mean, median, geometric mean, odds,

hazard

∗ Choice of comparisons across groups: differences, ratios


4.2.1 Examples of Effect Modification

· Example 1: Is serum LDL by gender modified by smoking?

Mean MedianSmoke Non-smoke Smoke Non-smoke

Men 120 122 120 115Women 133 122 133 124

Diff -13 0 -13 -9Ratio 0.90 1.00 0.90 0.93

– Yes for mean, not really for median

– Difference or ratio

· Example 2: Creatinine by stroke (modified by gender?)

Mean MedianWomen Men Women Men

No Stroke 0.72 1.08 0.7 1.1Stroke 1.01 1.51 1.0 1.5

Diff -0.29 -0.43 -0.3 -0.4Ratio 0.71 0.72 0.70 0.73

– Yes for difference, not really for ratio

– Mean or median

· Example 3: Stroke by smoking (modified by gender?)


Proportion OddsWomen Men Women Men

No Smoke 0.10 0.16 0.03 0.19Smoke 0.16 0.26 0.19 0.35

Diff -0.06 -0.10 -0.16 -0.16Ratio 0.62 0.62 0.16 0.54

– Proportion: No for ratio, more for difference

– Odds: No for difference, yes for ratio

· Example 4: Stroke by smoking (modified by CVD?)

Proportion OddsNone CVD None CVD

No Smoke 0.02 0.33 0.02 0.50Smoke 0.04 0.50 0.04 1.00

Diff -0.02 -0.17 -0.02 -0.50Ratio 0.50 0.67 0.50 0.50

– Proportion: Yes for ratio, yes for difference

– Odds: Yes for difference, no for ratio

· Example 5: CHD by current smoking (modified by gender?)


No Smoke 0.18 0.26 0.22 0.35Smoke 0.05 0.24 0.05 0.32

Diff 0.13 0.02 0.17 0.03Ratio 3.60 1.08 4.17 1.11


– Proportion: Yes for ratio, yes for difference

– Odds: Yes for difference, yes for ratio

· Example 6: CHD by ever smoke (modified by gender?)


Never 0.16 0.25 0.19 0.33Ever 0.16 0.26 0.19 0.35

Diff 0.00 -0.01 0.00 -0.02Ratio 1.00 0.96 1.00 0.95

– Proportion: No for ratio, no for difference

– Odds: No for difference, no for ratio

· Summary comments on examples

– If there is an effect, will see effect modification on at least one of thedifference and ratio scale

– If there is no effect (example 6), will see no effect modification on bothdifference and ratio scale

– In real world, will usually see effect modification on both scales. The realquestion is the effect modification scientifically meaningful.∗ If we find there is important effect modification, science will go forward

estimating effects separately

∗ Models with interaction terms are useful for testing if effect modifica-tion is present (statistically)

· Aside: Be careful with ratios


– How close are two ratios?∗ 0.20 and 0.25 VERSUS 5.0 and 4.0?

∗ 0.10 and 0.15 VERSUS 10.0 and 6.7?

– We might consider ratios to be more different when both ratios are > 1than when both are < 1. But, that would be wrong.

4.2.2 Analysis of Effect Modification

· When the scientific question involves effect modification

– Conduct analysis within each stratum separately

– If we want to estimate the degree of effect modification or test its exis-tence, use a regression model including∗ Predictor of interest (main effect)

∗ Effect modifying variable (main effect)

∗ A covariate modeling the interaction (usually a product)

· Ignoring effect modification

– By design or mistake, we sometimes do not model effect modification

– Might perform∗ Unadjusted analysis: POI only

∗ Adjusted analysis: POI and third variable, but no interaction term

· If effect modification exists, an unadjusted analysis will give different resultsaccording to the association between the POI and effect modifier in the sam-ple


– If the POI and the effect modifier are not associated∗ Unadjusted analysis tends toward an (approximate) weighted average

of the stratum specific effects

∗ With means, exactly a weighted average

∗ With odds and hazards, an approximate weighted average (becausethey are non-linear functions of the mean)

– If the POI and the effect modifier are associated in the sample∗ The “average” effect is confounded and thus unreliable

∗ (variables can be both effect modifiers and confounders)

· If effect modification exists, an analysis adjusting only for the third variable(but no interaction) will tend toward a weighted average of the stratum spe-cific effects

– Hence, an association in one stratum and not the other will make anadjusted analysis look like an association (provide the sample size islarge enough)

4.3 Confounding

4.3.1 Simpson’s Paradox

· Confounding has its roots in Simpson’s Paradox

· Given binary variables Y (response), X (POI), and Z (strata) it is possible tohave ...

– Pr(Y = 1|X = 1, Z = 1) > Pr(Y = 1|X = 0, Z = 1)


– Pr(Y = 1|X = 1, Z = 0) > Pr(Y = 1|X = 0, Z = 0)

· ... but to have ...

– Pr(Y = 1|X = 1) < Pr(Y = 1|X = 0)

4.3.2 Definition of Confounding

· The association between a predictor of interest and the response is con-founded by a third variable if

– The third variable is associated with the predictor of interest in the sam-ple, AND

– The third variable is associated with the response∗ Causally (in truth)

∗ In groups that are homogeneous with respect to the predictor of inter-est

∗ Not in the causal pathway of interest

· We must consider our belief about the causal relationships among the mea-sured variables

– There is no statistical test for causality

– Inference about causation comes only from the study design

– BUT, consideration of the causal relationships helps us to decide whichstatistical questions to answer

· Classic confounder


– A clear case of confounding occurs when some third variable is a “cause”of both the POI and response

– We generally adjust for such a confounder

– Example: Ice cream (POI), murder rate (outcome), and temperature(confounder) in New York City during the summer

– Directed Acyclyic Graph:

–

–

· Causal pathways

– A variable in the causal pathway of interest is not a confounder∗ We would not adjust for such a variable

∗ If we did adjust, we would lose ability to detect associations betweenthe POI and the outcome

∗ Example: Second hand smoke (POI), stunted growth (confounder),FEV1 (outcome)· Scientific question is about the impact of smoking on lung function

· Stunted growth addresses lung anatomy, not lung function, whichwe don’t care about it

∗ Directed Acyclyic Graph:

∗


∗

– We want to adjust for a variable in a causal pathway not of interest∗ Example: Work stress causing ulcers by hormonal effects versus al-

coholism

∗ Directed Acyclyic Graph:

∗

∗

· Surrogate for response

– Adjustment for a surrogate is a bad idea

– As the name implies, surrogates are a substitute for the response vari-able


–

–

· Complicated causal pathway (M shape)

– There is no confounding in this diagram

– However, adjustment for the observable surrogate would produce a spu-rious association

– Example: Smoking marijuana (POI), heart attack (Response), some lab-oratory measurement (surrogate)


∗ Latent variable A: Some (genetic, environmental?) characteristic thatmakes you more likely to smoke marijuana and increases the lab value

∗ Latent variable B: Having cardiovascular disease may elevate both thelab measurement and risk for heart attack


–

–

–

–

· Greenland, Pearl, and Robins. Causal Diagrams for Epidemiologic Re-search. Epidemiology. (1999)

– http://www.jstor.org/stable/3702180

4.3.3 Diagnosing Confounding

· Confounding typically produces a difference between unadjusted and ad-justed analyses

– This symptom is not proof of confounding∗ Such a difference can occur when there is no confounding

∗ Symptom is more indicative of confounding when modeling means(linear regression) than when modeling odds (logistic regression) orhazards (Cox, proportional hazards regression)


· Estimates of association from unadjusted analysis are markedly differentfrom estimates of association from adjusted analysis

– Association within each stratum is similar to each other, but different fromthe association in the combined data

· In linear regression, differences between adjusted and unadjusted analysesare diagnostic of confounding

– Precision variables tend to change standard errors, but not slope esti-mates

– Effect modification would show differences between adjusted analysisand unadjusted analysis, but would also show different associations inthe strata

· More difficult to diagnosis confounding with non-linear functions of the mean

– Common non-linear functions: Odds (odds ratios), hazards (hazard ra-tios)

– May show the symptoms of confounding when confounding is not present

– Adjusting for precision variables can appear to be confounding

– In logistic and PH regression, difference between adjusted and unad-justed analyses are more difficult to judge∗ Comparison in more homogeneous groups (i.e. after adjustment for a

precision variable) will drive slope estimates away from the null

· Example: Suppose you have a sample where 50% of the subjects die

– What is the variability?

– We can reduce this variability by changing p, the probability of death


– Estimate p in different stratum. One stratum may have a higher p, anothera lower p.

– By making the estimate more precise, we have also impacted the mean

4.4 Precision Variables

4.4.1 Overview

· Sometimes the scientific question to be answered is chosen based on whichquestions can be answered most precisely

– In general, questions can be answered more precisely when the withingroup distribution is less variable

– Comparing groups that are similar with respect to other important riskfactors decreases variability

· The precision variability is independent of the cause of the response

– If we adjust for such a variable, we tend to gain precision


–

–

· Standard errors are the key to precision

· Greater precision is achieved with smaller standard errors

· Standard errors are decreased by either increasing V or decreasing n


– Typically: se(θ̂) =√Vn

– Width of CI: 2× (crit value)× se(θ̂)

– Test statistic: Z = θ̂−θ0se(θ̂)

· Options for increasing precision

– Increase sample size

– Decrease V

– (Decrease confidence level)

4.4.2 Adjusting for Precision Variables

Precision for Difference of Independent Means

· Independent observations where group 1 has a different mean and variancethan group 2

– ind Yij ∼ (µj, σ2j ), j = 1, 2; i = 1, . . . , nj

– n = n1 + n2; r = n1/n2

– θ = µ1 − µ2, θ̂ = Y 1 − Y 2

– V = (r + 1)(σ21

r + σ22)

– se(θ̂) =√Vn =

√σ21

n1+ σ2

2

n2


· Might control for some variable in order to decrease the within group vari-ability

– Restrict population sampled

– Standardize ancillary treatments

– Standardize measurement procedure

Precision for Linear Regression

· Independent continuous outcome associated with covariate (X)

– ind Yi|Xi ∼ (β0 + β1Xi, σ2Y |X), i = 1, . . . , n

– θ = β1, θ̂ = β̂1 from LS regression

– V =σ2Y |X

Var(X)

– se(θ̂) =

√√√√ σ̂2Y |X

nˆVar(X)

· Adjusting for covariates (W ) decreases the within group standard deviation

– Var(Y |X) versus Var(Y |X,W )

· Independent continuous outcome associated with covariate (X) and preci-sion variable (W )

– ind Yi|Xi,Wi ∼ (β0 + β1Xi + β2Wi, σ2Y |X,W ), i = 1, . . . , n

– θ = β1, θ̂ = β̂1 from LS regression

– V =σ2Y |X,W

Var(X)(1−r2X,W )


– se(θ̂) =

√√√√ σ̂2Y |X

nˆVar(X)(1−r2X,W )

– σ2Y |X,W = σ2Y |X − β22Var(W |X)

Precision for Difference of Proportions

· When analyzing proportions (means), the mean variance relationship is crit-ical

– Precision is greatest when proportion is close to 0 or 1

– Greater homogeneity of groups makes results more deterministic (this isthe goal, at least)

· Independent binary outcomes

– ind Yij ∼ B(1, pj), i = 1, . . . , nj; j = 1, 2

– n = n1 + n2; r = n1/n2

– θ = p1 − p2, θ̂ = p̂1 − p̂2 = Y 1 − Y 2

– σ2j = pj(1− pj)

– V = (r + 1)(σ21

r + σ22)

– se(θ̂) =√Vn =

√σ21

n1+ σ2

2

n2

Precision for Odds

· When analyzing odds (a nonlinear function of the mean), adjusting for preci-sion variables results in more extreme estimates


· Odds = p1−p

· Odds using average of stratum specific p is not the average of stratum spe-cific odds

· Example: Stroke by smoking (in CVD strata)

– No association between smoking and CVD in the sample: 10% smokersin each group

– CVD is not a confounder, but is clearly a precision variable

– Note that the unadjusted odds ratio is attenuated toward the null com-pared to the adjusted odds ratios

No CVD CVD CombinedN p odds N p odds N p odds

Smoke 1000 0.04 0.04 100 0.50 1.00 1100 0.082 0.089Nonsmk 10000 0.02 0.02 1000 0.33 0.50 11000 0.048 0.051

Ratio 2.00 2.00 1.75

4.5 Diagnosing Confounding

4.5.1 Adjustment for Covariates

· We include predictors in an analysis for a number of reasons. In order ofimportance...

1. Scientific question– Predictor of Interest

– Effect Modifiers


2. Adjust for confounding3. Gain precision

· Adjustment for covariates changes the question being answered by the sta-tistical analysis

– Adjustments can be made to isolate associations that are of particularinterest

– When consulting with a scientist, it is often difficult to decide whether theinterest in an additional covariate is due to confounding, effect modifica-tion, or precision∗ The distinction is important because I tend to treat these variable dif-

ferently in the analysis

– Often the scientific question dictates inclusion of particular predictors∗ Predictor of interest: The scientific parameter of interest can be mod-

eled by multiple predictors (e.g. dummy variables, polynomials, splines)

∗ Effect Modifiers: The scientific question relates to the detection of ef-fect modification

∗ Confounders: The scientific question may be state in terms of adjust-ing for known (or suspected) confounders

4.5.2 Confounder Detection

· Unanticipated confounding

– Some times we must explore our data to assess whether our results wereconfounded by some variable

– Goal is to assess the “independent effect” of the predictor of interest onthe outcome


· Confounders

– Variables (causally) predictive of the outcome, but not in the causal path-way∗ Best method: Think about the scientific problem beforehand (perhaps

draw DAG)

∗ Using data, often assessed in the control group

– Variables associated with the predictor of interest in the sample∗ Note that statistical significance is not relevant because this tells us

about associations in the population

– Detection of confounding ultimately must rely on our best knowledgeabout the possible scientific mechanisms

· Effect of confounding: A confounder can make the association between thepredictor of interest and the response variable look...

– Stronger than the true association

– Weaker than the true association

– The complete reverse of the true association (“qualitative confounding”)

4.6 Graphical Methods for Visualizing Effect Modification, Confounding,and Precision

· Conduct stratified analysis to distinguish between

– Effect modifiers

– Confounders


– Precision variables

4.6.1 Effect Modifiers

· Estimates of treatment effect differ among strata

– When analyzing difference of means of continuous data, stratified smoothcurves of the data are non-parallel

– Graphical techniques difficult in other settings


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4.6.2 Confounders

· Estimates of treatment effect the same across strata, AND

– Confounder is causally associated with the response, AND

– Confounder associated with the POI in the sample

· When analyzing difference of means of continuous data

– Stratified smooth curve of data are parallel

– Distribution of POI differs across strata

– Unadjusted and adjusted analyses give different estimates


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4.6.3 Precision Variables

· Estimates of treatment effect the same across strata, AND

– Variable is causally associated with the response, AND

– Variable is not associated with the POI in the sample

· When analyzing difference of means of continuous data

– Stratified smooth curve of data are parallel

– Distribution of POI same across strata

– Unadjusted and adjusted analyses give similar estimates but with smallerstandard errors


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Documents

BIOS 312: MODERN REGRESSION ANALYSISbiostat.mc.vanderbilt.edu/wiki/pub/Main/CourseBios312/effmod.pdf · 4.3.3 Diagnosing Confounding Confounding typically produces a difference between