Adjusting for Time-Varying Confounding in

Survival Analysis

Natalya Verbitsky

Joint Work with Jennifer BarberSusan MurphyUniversity of Michigan

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Outline

Introduction Problem Possible Solutions

Evaluation Data Simulation Statistical Models

Results “Path Analysis”

Rules Simulation Results

Conclusions Future Work

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Introduction We are concerned with cause-and-

effect relationship Q: If educational opportunities for

children in poor countries are increased, would couples limit their total family size via contraception?

Q: Does having a teenage birth curtail educational attainment?

Problem:

How do we assess the effect of an exposure on a response in the presence of confounders in a time-varying setting?Def: Confounders are variables that affect both response and predictors either directly or indirectly.

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Example:Q: If educational opportunities for

children in poor countries are increased, would couples limit their total family size via contraception?

Exposure: schooling children Response: limiting family size via contraception Some Possible Confounders: political power,

parents’ education, number of children in the family, proximity of school nearby

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Possible Solutions: Some Traditional Methods

Naïve approach: regular logistic regression of response on the exposure

Standard approach: regular logistic regression of response on exposure including confounders as covariates

New Method: Weighted logistic regression

)|expPr(

exp)Pr(

confConfExpos

ExposWeight

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Data Simulation Diagram

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Conf10 Conf11

Expos0 Resp0 Expos1 Resp1

Conf20 Conf21

Time 0 Time 1

UnmeasUnmeasured Confounder

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Variables: Unmeas=unmeasured confounder

Ex: political power Unmeas=2, high Unmeas=1, medium Unmeas=0, low

Conf1j = binary measured confounder 1 at time j, j=0 or 1 Ex: presence of school nearby

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Variables (Con’d)

Conf2j = binary measured confounder 2 at time j, j=0 or 1 Ex: having a small family

Exposj = binary exposure at time j, j=0 or 1 Ex: any child in the family has attended school

by time j

Respj = binary response at time j, j=0 or 1 initiating permanent contraception at time j

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Statistical Models: Model 1: Regular Logistic Regression (non-

parametric) Logit Pr(Resp0 = 1)=a0 + a2 Expos0 Logit Pr(Resp1 = 1)=a0 + a1 + a3 Expos0 + a4 Expos1

Model 2: Logistic Regression with Confounders Logit Pr(Resp0 = 1)=a0 + a2 Expos0 + a5 Conf10 + a6 Conf20

Logit Pr(Resp1 = 1)=a0 + a1 + a3 Expos0 + a4 Expos1 + a5 Conf11 + a6 Conf21

Model 3 and 4: Regular Logistic Regression with Weights (do not include confounders as covariates) Model 3 weights: both confounders included Model 4 weights: only Conf1 is included

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“Path Analysis” Rules Find all possible path between your

predictor and response To calculate the effect of a particular

path: Multiply the effect within the path If you conditioned on a variable and the

arrows of the path meet there, multiply the arrow effect by -1

Add the effect of all possible paths to get the total effect

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Path Analysis forExpos0 on Resp1 in Naïve Model:

Logit Pr(Resp0 = 1)=a0 + a2 Expos0

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Conf1 0 Conf1 1

Expos0 Resp 0 Expos 1 Resp 1

Conf2 0 Conf2 1

Time 0 Time 1

UnmeasUnmeasured Confounder

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Path Analysis for Expos0 on Resp1 in Naïve Model:Logit Pr(Resp1 = 1)=a0 + a1 + a3 Expos0 + a4 Expos1

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Conf1 0 Conf1 1

Expos0 Resp 0 Expos 1 Resp 1

Conf2 0 Conf2 1

Time 0 Time 1

UnmeasUnmeasured Confounder

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Results (general) The values inside the cells represent

1st line: average estimate and average std. error

2nd line: the proportion of the time you would reach the wrong conclusion

All results are based on 1,000 samples (unless otherwise specified) of 1,000 observations each

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Results (Table 1)Naive Standard Weighted

(Conf1, Conf2)Prt. Wtd(Conf1)

Effect of Expos0 on Resp0

0.31 (.15)0.56

-0.03 (.16)0.07

-.002 (.14)0.04

0.13 (.14)0.13

Effect of Expos0 on Resp1

0.22 (.20)0.24

-0.43 (.23)0.48

-.003 (.20)0.05

0.09 (.20)0.09

Effect of Expos1 on Resp1

0.33 (.30)0.22

-0.14 (.39)0.07

0.02 (.30)0.05

0.15 (.30)0.08

Note: intercepts are 0.0; alphas=etas=1.5; gammas=0.5

• biased estimates in regular logistic regression (Model 1)• biased estimates in Model 2 in the estimates of past values of predictors; no bias in the estimates of present values of predictors• According to “path analysis” rules expect positive bias in 1st and 3rd cells of Model 1; and a negative bias in the middle cell of Model 2

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Path Analysis forExpos0 on Resp1 in Standard Model:Logit Pr(Resp1 = 1)=a0 + a1 + a3 Expos0 + a4 Expos1 + a5 Conf11 + a6 Conf21

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Conf1 0 Conf1 1

Expos0 Resp 0 Expos 1 Resp 1

Conf2 0 Conf2 1

Time 0 Time 1

UnmeasUnmeasured Confounder

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Path Analysis for Weighted Model

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Conf10 Conf11

Expos0 Resp0 Expos1 Resp1

Conf20 Conf21

Time 0 Time 1

UnmeasUnmeasured Confounder

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Results (Table 2)

Naïve Standard Weighted Prt. Wtd

Effect of Expos0 on Resp0

-0.29 (.13)0.58

-.001 (.14)0.04

-0.01 (.12)0.02

-0.13 (.13)0.14

Effect of Expos0 on Resp1

-0.18 (.18)0.14

-1.17 (.24)0.99

-.0004 (.18)0.03

-0.08 (.18)0.06

Effect of Expos1 on Resp1

-0.31 (.25)0.24

0.10 (.31)0.06

0.002 (.25)0.04

-.13 (.25)0.08

Note: intercepts are 0.0; alphas=etas=-1.5; gammas=0.5

• According to “path analysis” rules of thumb, expect negative bias in the 1st and 3rd estimates of Model 1 and in the middle row of Model 2; on the other hand, expect unbiased estimates in Model 3

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Results (Table 3)

Naïve Standard Weighted Prt. Wtd.

Effect of Expos0 on Resp0

0.17 (.14)0.21

0.01 (.14)0.05

-0.01 (.14)0.05

0.08 (.14)0.09

Effect of Expos0 on Resp1

0.07 (.19)0.06

-.004 (.20)0.05

-0.01 (.20)0.05

0.03 (.19)0.06

Effect of Expos1 on Resp1

0.10 (.27)0.07

-.04 (.28)0.05

-.004 (.28)0.04

0.04 (.28)0.04

Note: intercepts are 0.0; alphas=etas=0.5; gammas=0.5

• Compare with Table 1, degree of bias depends on the values of alphas and etas• Including weights does not hurt your analysis

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Results (Table 4)

Naive (parsim.)

Standard (parsim.)

Weighted (parsim.)

Prt. Wtd. (parsim.)

Effect of Expos0 on Resp0

0.31 (.15)0.56

-0.09 (.16)0.10

-.002 (.14)0.04

0.13 (.14)0.13

Effect of Expos1 on Resp1

0.49 (.25)0.51

-0.45 (.34)0.30

0.02 (.25)0.04

0.22 (.26)0.14

Note: intercepts=0.0; alphas=etas=1.5; gammas=0.5

• Compare with Table 1, in parsimonious models 1 and 2 bias in the estimates of effect of Expos1 on Resp1 has increased• Model 3 has unbiased estimates

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Results (Table 5)Naïve Standard Weighted Prt. Wtd.

(Conf1)Prt. Wtd.(Conf2)

Effect of Expos0 on Resp0

.27 (.14)0.47

-.03 (.15)0.04

0.003 (.14)0.03

.09 (.14)0.08

.15 (.14)0.15

Effect of Expos0 on Resp1

.17 (.20)0.16

-.51 (.23)0.62

-0.03 (.20)0.06

.01 (.20)0.06

.11 (.20)0.10

Effect of Expos1 on Resp1

.30 (.29)0.18

-.07 (.36)0.05

-.002 (.29)0.06

.06 (.29)0.06

.20 (.30)0.10

Note: intercepts=0.0; alpha1j=2.25, alpha2j=0.75 j=0 or 1; etas=1.5Gammas=0.5

• Model 3 is weighted logistic regression, weights use Conf1 and Conf2• Model 4 is weighted logistic regression, weights use only Conf1 (3/4)• Model 5 is weighted logistic regression, weights use only Conf2 (1/4)

•Weighting Model is robust to missing confounders

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Conclusions Regular logistic regression (Model 1)—biased

estimates Regular logistic regression with confounders

(Model 2)—bias due to confounders affected by past exposure

The sign of the bias can be found by using “path analysis” type rules of thumb

Degree of the bias depends on the strength of correlations between Unmeas and Conf, Conf and Expos, and past Expos and Conf

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Conclusions (Con’d):

Parsimonious Models have biased estimates of effect of Expos on Resp

The use of weights incorporating all confounders eliminates this bias

Weighting method is robust to missing confounders

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Future Work

Diagnostics for situations with “bad” weights