Subgroup analysis using regression modeling multiple regression. Aeilko H Zwinderman

Transcription

1 Subgroup analysis using regression modeling multiple regression Aeilko H Zwinderman

2 who has unusual large response? Is such occurrence associated with subgroups of patients? such question is hypothesis-generating: to refine patient- or dose-selection subgroup-analyses are -by nature- almost surely underpowered: => regression model regression modeling may increase efficiency correct for confounding investigfate interaction / synergism be used for prediction

3 regression models: many possibilities quantitative data: linear/nonlinear regression models discrete data: (probit) logistic regression censored data: Cox regression

4 general form: E[Y i X i ] = g -1 ( X 1i + 2 X 2i k X ki ) Var[Y i X i ] = e 2 Y is the dependent variable (primary efficacy variable) X is a covariate, predictor or independent variable g is the link-function is a regression parameter (which must be estimated from the data)

5 linear model: Y = quantitative variable X = quantitative or discrete variable Y i X X i 2 2i k X ki e i is a direct effect: difference in mean of Y if X changes 1 unit assumptions: a. linarity of the relation between Y and X b. normality: Y is normally distributed for any given value of X c. homogeneity: Y has the same variance for any given value of X

6 logistic model: Y = binary variable (i.e. 1 or 0) X = quantitative or discrete variable P( Y i 1 X i ) exp( 1 exp( X1 i... X i k X ki ) X k ki ) is a log odds-ratio: change in the log(odds) that Y=1 if X changes 1 unit assumptions: a. linarity of the relation between log odds(of Y=1) and X b. the link-function g -1 has the logistic form

7 Cox proportional hazards regression model: Y(t) = binary status variable (i.e. 1 or 0) occurring at time t X = quantitative or discrete variable S exp( 1X1i... k Xki ) ( ti Xi) S0( t) is a log-relative risk: change in the log(hazard(t)) if X changes 1 unit assumptions: a. linarity of the relation between the log hazard(t) and X b. the relative risk is constant with time

8 Cum hazard Survival log(hazard) hazard X=0 X=20 X=80 X=0 X=20 X= time time X=0 X=20 X=80 X=0 X=20 X= time time

9 survival hazard X=0 X=80 X= X=40 X= X=80 time time time time

10 regression modeling to increase precision * placebo (n=434) or pravastatin (n=438) * two years treatment * average LDL-decrease: # pravastatin: 1.23 (SD 0.68, se = 0.68/438) # placebo: (SD 0.59, se = 0.59/434) * efficacy: = standard error = 0.043

11 LDL-reduction: Y i = X 1i + e i X 1 = 1 if a patient receives pravastatin and zero if he/she placebo => 1 is efficacy: 1.27 (SE = is a function of e 2 ) Suppose there is a covariate X 2 which is related to Y, but not to X 1: Y i = X 1i + 2 X 2i + e i 1 remains the same but e 2 will be (much) smaller => SE( 1 ) will be smaller => increased precision

12 An example of a variable that might be related to Y but not to treatment is baseline LDL * is not related to treatment (randomized trial) placebo: 4.32 (SD 0.78) pravast: 4.29 (SD 0.78) p=0.60 * is (almost surely) related to LDL-decrease 2 = 0.41 (SE 0.024, p<0.0001) => efficacy: 1 = 1.27 (SE 0.037, was 0.043: 15% gain in efficiency)

13 LDL decrease baseline LDL

14 usually there are many many many candidates to consider: specify which ones will be used in the protocol in non-linear regression models 1 always changes by including covariates, thus its interpretation changes (often not much, but it can be greatly inflated)

15 regression modeling to correct for confounding a confounder is a covariate Z that is associated with both Y and X 1 distorts the interpretation of the efficacy estimate 1 what is thought to be efficacy may just reflect the unbalance of Z between treatment groups

16 * will not happen often in randomized trials * will happen almost always in non-randomized research * when it happens, adjustment of 1 is required Y i = 0 + * 1 X 1i + 2 Z i + e i if r xz >0 and r yz >0 then * 1 < 1 if r xz >0 and r yz <0 then * 1 > 1 if r xz <0 and r yz >0 then * 1 > 1 if r xz <0 and r yz <0 then * 1 < 1

17 X Y X Y direct effect of treatment X on outcome Y: no need for regression modeling Z effect of treatment X on outcome Y is confounded bij Z: regression model may correct for this X Z Y effect of treatment X on outcome Y is partly through Z: Z is an intermediate not a confounder. Do not use regression modeling: in the regression model the effect of X is split between a direct and an indirect effect.

18 check only the necessary (known) confounders beware of multiple testing

19 interaction/synergism looking for subgroups with different efficacy Y i = X 1i + 2 X 2i + 3 X 1i.X 2i + e i Suppose X 2 =0 or 1: X 2 =1: Y i = ( )+ ( ) X 1i + e i X2=0: Y i = X 1i + e i

20 Primary question: H 0 : 3 = 0 Example: is there interaction between statins and CCBs? Y = change of diameter of coronary vessels during statin/placebo treatment placebo no CCB (0.20) CCB (0.22) statin no CCB (0.19) CCB (0.19)

21 Diameter decrease placebo statin 0 no CCB CCB

22 Efficacy: no CCB: 1 = = CCB: = = = = 0.084, p=0.011 thus, statins are significantly more effective in patients who also were prescribed CCBs.

23 Fellstrom et al. Rosuvastatin and cardiovascular events in patients undergoing hemodialysis. NEJM, 2009.

24 be careful investigating interactions: multiple testing problem do not enter too many covariates in a regression model: (k<n/10)

25 good models check assumptions use selection algorithms sparsely. (Instead use penalized methods, shrink regression weights) caution against optimistic results: (cross-) validation