01/20151 EPI 5344: Survival Analysis in Epidemiology Confounding and Effect Modification March 24, 2015 Dr. N. Birkett, School of Epidemiology, Public Health & Preventive Medicine, University of Ottawa

01/20152 Objectives Review confounding and effect modification (EM) Graphical view of confounding and EM Linear regression model for confounding/EM How to detect/adjust for confounding in Cox modeling Effect modification and Cox modeling.

01/20153 I used many of the early slides for my second session in EPI5340 Will review this material

01/20154 Introduction (1) In etiological study, interest is on effect of ‘X’ on ‘Y’ Suppose we have a third variable ‘C’ which explains some of the effect of ‘X’: C X Y Confounding.

01/20155 Introduction (2) A common definition is based on ‘change in estimate of coefficient’ –Estimate the crude OR/RR/etc. (RR crude ) –Estimate OR/RR after adjusting for ‘C’ (RR adj ) –If RR adj differs from RR crude by ‘enough to matter’, we have confounding. Usually look for 10%+ in RR (or ln(RR)):

01/20156 Introduction (3) I am suggesting using: –Does adjusting change the crude RR by more than 10% Some books use this criterion: –Does ‘not adjusting’ change the adjusted RR by more than 10%

01/20157 Introduction (4) As long as confounding shifts RR towards null, we have My criterion will smaller than other one Hence, if mine is > 0.10, the other one will be too If confounding is in other direction, the reverse is true. Areas with disagreement are marginal effects

01/20158 Introduction (5) There is controversy over this definition of confounding. Pearl and others have shown that, in some situations, an unadjusted estimate for RR can be more valid than an adjusted one. –Argues against using the ‘change in estimate’ method. This course isn’t the place to explore the debate. We will use the ‘change in estimate’ approach.

01/20159 Introduction (6) Effect Modification –Our main interest is relationship of ‘X’ to ‘Y’ –What if a third variable ‘modifies’ that relationship so that it is very different in two or more groups?

01/ Introduction (7) Effect Modification –Example RR crude = 1.7 ( ) RR male =15.6 ( ) **** RR female = 0.4 ( ) **** –What is the effect of ‘X’ on ‘Y’?

01/ Introduction (8) IT DEPENDS! –In men: risk is much higher –In women:risk is much lower Effect Modification occurs when –a third variable defines groups where the relationship of ‘X’ and ‘Y’ differs ‘enough to matter’

01/ Introduction (9) For OR/RR/etc., usually want Can also look for statistically significant differences in the RR estimates for the groups –Tests have low power

01/ Introduction (10) How do we know if we have either of confounding or effect modification? –There are rules-of-thumb –These are not ‘truth’. Answer depends on the impact of the results Does the modification of the OR/RR by a third factor introduce enough difference ‘to matter’? –If so, then confounding/EM –If not, then no confounding/EM

01/ Introduction (11) This is an empirical/practical approach Requires knowledge of the scientific field under study Impact on clinical/scientific/policy decisions –if the HR is 10.0, then an adjustment to 9.0 or 11.0 won’t matter Some other views –Confounding applies in the ‘population’, not your study. Always matters –Causal modeling –Impact on understanding causal paths as opposed to making practical decisions.

01/ Confounding (1) Consider this research situation OutcomeSystolic Blood Pressure (SBP) QuestionDoes a new R x  a decrease in SBP? AssumptionsSBP and age are linearly related Issues1.Age is related to SBP levels 2.Drug is given mainly to older people since it has fewer side effects. OR 3.Drug is given mainly to younger people because it interacts with meds given commonly for arthritis.

01/ Age vs. SBP SBP age All subjects combined Now, look at the Rx and control groups separately

01/ Age vs. SBP SBP age Control RxRx Δ true Treatment effect is the distance between the lines It is the same at every age

01/ Assume Rx group is older Control RxRx Δ true Δ observed

01/ Assume control group is older Control RxRx Δ true Δ observed

01/ Assume groups are same age Control RxRx Δ true Δ observed

01/ Control RxRx Δ true Δ observed Assume groups are same age

01/ Confounding (2) We can adjust for confounding by making the age in the two groups ‘the same’ –Design Matching RCT Restriction –Analysis Graphs are hard to work with, so let’s switch to regression models. –I’ll use linear regression for now

01/ Confounding (3) Cont: y i = B 0,cont + B 1 (age) i R x :y i = B 0,Rx + B 1 (age) i Note: the slopes are assumed to be the same True effect confounding effect

01/ Confounding (4) True effect confounding effect Confounding = 0 iff Β 1 =0 OR mean ages are the same

01/ Confounding (5) How to estimate the Beta’s? Could do stratified regression (in each treatment group) –Limits your options Instead, let’s create one regression model which includes both stratified models as sub-models.

01/ Control RxRx 0

01/ Confounding (6) Define z i = 0 if in control group = 1 if in Rx group which can be written as:

01/ Confounding (7) β 2 = adjusted effect of R x Can use this model to test hypotheses about adjusted effect, get 95% CI’s, etc.

01/ Confounding (8) To check for confounding, fit this model with and without ‘age’. –Compare the change in β 2 from model without age to the model with age –Look for 10% change –Note that this is equivalent to previous ‘rule’ using the logs of the RR/OR/etc.

01/ age Control RxRx Now, consider this situation SBP The treatment effect DEPENDS on the age of the subject Small for low age Large for high age EFFECT MODIFICATION

01/ Effect Modification (1) Can also put this into regression models Key is to note that the slopes are not the same. As before, define: z i = 0 if in control group = 1 if in Rx group

01/ Effect Modification (2)

01/ Effect Modification (2) which can be written as: β 3 ≠0  Effect Modification

01/ Confounding/EM in Cox (1) Linear regression model is called: –Parallel line analysis Extends easily to Cox models. –Replace ‘y’ by ‘ln(HR)’ –Remember, no intercept for the Cox model (contained in h 0 (t)) Our combined regression model with Cox is:

01/ Confounding/EM in Cox (2) To test for EM, look at the β 3 term –Statistical significance –magnitude of exp(β 3 ) ‘large values’ suggest important EM

01/ Confounding/EM in Cox (3) To test for confounding, compare to model without the ‘x’: Look at difference in the Betas If there is no EM, we get: Effect of ‘z’ adjusted for ‘x’

01/ SAS code (1) TO CHECK FOR EM: proc phreg data=njb1; class z/param=ref ref=first; model time*cens(0)=x z x*z; run; Look at the ‘Beta’ for ‘x*z’ to check for EM.

01/ SAS code (2) If there is no EM, run two more models to check for confounding. ADJUSTED MODEL proc phreg data=njb1; class z/param=ref ref=first; model time*cens(0)=x z; run; CRUDE MODEL proc phreg data=njb1; class z/param=ref ref=first; model time*cens(0)=z; run; Compare the ‘Beta’ for ‘z’ in these two models

01/ SAS code (3) What if ‘z’ has more than 2 levels? Have more than one indicator variable for ‘z’ Can use the same code Effect Modification –Use the likelihood ratio test (-2Δlog(L)) Available directly in SAS output –Can have EM for one level of ‘z’ but not for another one.

01/ SAS code (4) Confounding –Need to consider possible confounding for each level of ‘z’ –Overall decision to adjust for ‘x’ depends on presence of any confounding.

Our Cox model is: What is the effect of ‘z’ on the HR? If there were no interaction, can just use exp(β 2 ) But, this is not appropriate with interaction: 01/ R x effect when there is EM (1)

01/ R x effect when there is EM (2) There is no single treatment effect. –Misleading to present a single RR Depends on value of EM variable Need to present effect estimates in a graph or table (or formula)

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