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

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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

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

OBJECTIVE

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

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

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

WHY WORRY?

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

COEFFICIENT ESTIMATES

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

WHY ONLY IN OBSERVATIONAL STUDIES?

Compositional differences are minimized by randomization.

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

WHAT DO WE NORMALLY DO?

The standard model.

THE STANDARD MODEL

Includes confounders as covariates in the response regression model.

Sprinkler example follows examples often used by Pearl.

PROBLEM

The confounder is affected by the predictor.

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

SPRINKLER EXAMPLE

Consider a simple example involving sprinklers.

Sprinkler example follows examples often used by Pearl.

RESULT

Spurious correlations are dangerous.

DANGER OF SPURIOUS CORRELATIONS

Degree of bias related to the strength of the correlations.

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

WEIGHTING?

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?

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-InitiatorInitiatorTotal

High Peer Pressure Resistance 40 1050

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-InitiatorInitiatorTotal

High Peer Pressure Resistance 25 2550

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-InitiatorInitiatorTotal

High Peer Pressure Resistance 50 50 100

Low Peer Pressure Resistance 60 60 120

Total 110 110 220

HOW DO WE GET THE WEIGHTS?

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

Eliminate the elevation of the total sample size.

EQUATION 1

WHY DOES THIS WORK?

- Eliminates the problematic spurious correlation.
- Controls for confounders by equalizing compositional differences.

HOW DO WE DO IT?

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*

*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?

Weighted logistic regression of the response on the predictor.

DATA EXAMPLE

- Naïve Model
- Standard Model
- Weighted Model

NOTES:

+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**

Odds3.511.442.06

(<0.0001)(0.1203)(0.0054)

Time-Varying Confounders:

Cigarettes0.4085

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. Seeking0.0191

SUMMARY

- 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

- Sequential Ignorability
- Past confounder patterns do not exclude particular levels of exposure

FUTURE DIRECTIONS

- Generalization of method to multilevel data structures
- Procedures to detect assumption violations
- Robustness to assumption violations

ROBUSTNESS TO ASSUMPTION VIOLATIONS

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

PATH ANALYSIS

- 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

- Lexington Longitudinal Study
- 121 Female, 41 non-white
- Multiple confounders
- Time measured ever 1/3 of a school year

WEIGHT CALCULATIONS

- Numerator Regression Model:

- Denominator Regression Model:

- Weight (conduct disorder initiation at time t):

- Naïve Model and Weighted Model:

- Intercept Term:

- Standard Model:

- Confounders: