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Assessing the Total Effect of Time-Varying Predictors in Prevention Research. Bethany Bray April 7, 2003 University of Michigan, Dearborn. OUTLINE. Introduction to the problem Standard model Problems with the standard model Suggested solution Data example Future directions. GOAL.

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Assessing the total effect of time varying predictors in prevention research

Assessing the Total Effect of Time-Varying Predictors in Prevention Research

Bethany Bray

April 7, 2003

University of Michigan, Dearborn


OUTLINE Prevention Research

  • Introduction to the problem

  • Standard model

  • Problems with the standard model

  • Suggested solution

  • Data example

  • Future directions


GOAL Prevention Research

Assess the total effect that delaying the timing of a predictor has on the timing of a response.


OBJECTIVE Prevention Research

To estimate total effect of conduct disorder initiation on marijuana initiation.

OUR QUESTION

“Does delaying conduct disorder initiation lead to a delay in the initiation of marijuana?”


CONFOUNDERS Prevention Research

Common correlates of the predictor and the response.

Alternate explanations for the observed relationship between the predictor and response.

Must be controlled for when estimating the total effect.


COMPOSITIONAL DIFFERENCES Prevention Research

The unequal distribution of levels of the confounder between the types of children that initiate the predictor and those who do not.


WHY WORRY? Prevention Research

The coefficient of the predictor is a biased estimate of the total effect.


COEFFICIENT ESTIMATES Prevention Research

Estimated coefficient reflects the difference between the predictor groups, in addition tothe causal effect.


WHY ONLY IN OBSERVATIONAL STUDIES? Prevention Research

Compositional differences are minimized by randomization.

Observational studies require statistical methods and scientific assumptions to adjust for compositional differences.


WHAT DO WE NORMALLY DO? Prevention Research

The standard model.


THE STANDARD MODEL Prevention Research

Includes confounders as covariates in the response regression model.



PROBLEM Prevention Research

The confounder is affected by the predictor.

If the confounder is included as a covariate, a spurious correlationis created.


SPRINKLER EXAMPLE Prevention Research

Consider a simple example involving sprinklers.



RESULT Prevention Research

Spurious correlations are dangerous.


DANGER OF SPURIOUS CORRELATIONS Prevention Research

Degree of bias related to the strength of the correlations.

In simulations, false conclusions reached in up to 80% of the data sets.


WEIGHTING? Prevention Research

Weighting attempts to make people with different predictor levels comparable in all other respects.

WHAT DO WE DO NOW?

Use sample weights to statistically control for time-varying confounders*.

*Hernán, Brumback, and Robins, 2000


HOW DOES IT WORK? Prevention Research

Equalizes the compositional differences of the confounder among the predictor levels.


Original frequencies – conduct disorder initiation by peer press. resistance

Conduct Disorder Initiation Status

Non-Initiator Initiator Total

High Peer Pressure Resistance 40 10 50

Low Peer Pressure Resistance 30 30 60

Total 70 40 110

Ideal frequencies – conduct disorder initiation by peer press. resistance

Conduct Disorder Initiation Status

Non-Initiator Initiator Total

High Peer Pressure Resistance 25 25 50

Low Peer Pressure Resistance 30 30 60

Total 55 55 110

Weighted frequencies – conduct disorder initiation by peer press. resistance

Conduct Disorder Initiation Status

Non-Initiator Initiator Total

High Peer Pressure Resistance 50 50 100

Low Peer Pressure Resistance 60 60 120

Total 110 110 220


HOW DO WE GET THE WEIGHTS? press. resistance

Inverse of the conditional probability of predictor status given confounder status.

10 Initiators w/ high peer pressure resistance:

Weight of (10/50)-1 = 5

40 Non-initiators w/ high peer pressure resistance:

Weight of (40/50)-1 = 5/4

60 Children w/ low peer pressure resistance:

Weight of (30/60)-1 = 2


IN PRACTICE press. resistance

Eliminate the elevation of the total sample size.

EQUATION 1


WHY DOES THIS WORK? press. resistance

  • Eliminates the problematic spurious correlation.

  • Controls for confounders by equalizing compositional differences.


HOW DO WE DO IT? press. resistance

1. Ratio of two predicted probabilities

a. Denominator: predicted probability of observed conduct disorder initiation given confounders and baseline variables.

b. Numerator: predicted probability of observed conduct disorder initiation given baseline variables.

2. Weight at time t, Wt: product of these ratios up to time t.


EQUATION 2 press. resistance*

*The “over-bars” above Alci-1 and Mji-1 signal that the probability is conditional on the complete past predictor and response patterns.


NOW WHAT? press. resistance

Weighted logistic regression of the response on the predictor.


DATA EXAMPLE press. resistance

  • Naïve Model

  • Standard Model

  • Weighted Model


NOTES: press. resistance

+Coefficients for intercepts and baseline variables are omitted.

†These models do not include confounders by definition.

One tailed tests:

*p<0.05

**p<0.01

***p<0.001

RESPONSE REGRESSION MODELS WITH CONDUCT DISORDER AS THE PREDICTOR+

Naïve† Standard Weighted†

Predictor:

Conduct Disorder 1.2544*** 0.3628 0.6565**

Odds 3.51 1.44 2.06

(<0.0001) (0.1203) (0.0054)

Time-Varying Confounders:

Cigarettes 0.4085

Alcohol 0.8238**

Other Drug Use 1.2848**

Peer Pressure Res. -0.0470***

Non-Time-Varying Confounders:

Heart Rate -0.0118

Verbal IQ -0.0265**

Performance IQ -0.0117

Ave. Sen. Seeking 0.0191


SUMMARY press. resistance

  • Worry about confounders in observational studies.

  • Standard method of controlling for confounders results in biased estimates from spurious correlation issues.

  • The weighting method is one way to reduce bias.


ASSUMPTIONS press. resistance

  • Sequential Ignorability

  • Past confounder patterns do not exclude particular levels of exposure


FUTURE DIRECTIONS press. resistance

  • Generalization of method to multilevel data structures

  • Procedures to detect assumption violations

  • Robustness to assumption violations


ROBUSTNESS TO ASSUMPTION VIOLATIONS press. resistance

Assumption 1: Adjusting for more and more confounders leads to decreased bias using the weighted model.

Assumption 2: Biased estimators from the weighted model.


EXTRA INFO press. resistance


PATH ANALYSIS press. resistance

  • A Few Rules:

  • Paths with no converging arrows and variables not in model do contribute to correlation

  • Paths with converging arrows and variable not in model do not contribute to correlation

  • Paths with no converging arrows and a variable in model do not contribute to correlation, path is blocked

  • Paths with converging arrows and a variable in model do contribute to correlation, multiply path’s sign by -1


OUR DATA press. resistance

  • Lexington Longitudinal Study

  • 121 Female, 41 non-white

  • Multiple confounders

  • Time measured ever 1/3 of a school year


WEIGHT CALCULATIONS press. resistance

  • Numerator Regression Model:

  • Denominator Regression Model:

  • Weight (conduct disorder initiation at time t):


  • Intercept Term:

  • Standard Model:

  • Confounders:


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