“Assessing the Total Effect of Time-Varying Predictors in
Prevention Research”
Bethany Cara Bray
Department of Human Development and Family Studies
The Methodology Center
Pennsylvania State University
University Park, PA 16803
bcbray@psu.edu
Phone: 814.865.1225
Rick S. Zimmerman
Department of Communications
University of Kentucky
Donald Lynam
Department of Psychology
University of Kentucky
Susan Murphy
Department of Statistics
The Institute for Social Research
University of Michigan
Preparation of this article and presentation was supported by Grant # P50 DA10075 from the
National Institute on Drug Abuse to the Methodology Center at Pennsylvania State University
and by the National Institute on Drug Abuse award #1 K02 DA15674-01.
For more information or copies of the manuscript please contact Bethany Cara Bray.
Assessing the Total Effect of Time-Varying Predictors in Prevention Research
4.7.03
Definitions

Time-varying: A variable (i.e. peer pressure resistance) that has different values through
out time.

Non-time-varying: A variable (i.e. sex) that does not have different values through out
time.

Confounder: A variable that is correlated with both the predictor and the response.
Confounders are alternate explanations of the observed relationship between the predictor
and response. For instance, if the response is marijuana initiation and the predictor is
conduct disorder initiation, peer pressure resistance is one confounder.

Compositional Differences: The unequal distribution of the confounder (i.e. peer pressure
resistance) between the types of participants that initiate the predictor (i.e. conduct
disorder) and those who do not. For instance, of the participants who initiate conduct
disorder, there is a higher percentage of participants who have low peer pressure
resistance and of the participants who do not initiate conduct disorder, there is a higher
percentage of participants who have high peer pressure resistance.

Spurious Correlation: A false, accidental correlation. A spurious correlation is an
accidental correlation between the predictor and response, created by including
confounders in the response regression model. Spurious correlations make the
relationship between the predictor and response appear different than what is actually
true.

Total Effect: The entire effect of the predictor on the response through all direct and
indirect influences. For example, in Figure 1 the total effect of the predictor (i.e. conduct
disorder, Cd) on the response (i.e. marijuana initiation, Mj) is represented by all paths
following the direction of the arrows from Cd1 and Cd2 to Mj2 and Mj3.
 Response Regression Model: Regression model of the response (i.e. marijuana initiation)
on the predictor (i.e. conduct disorder) and possibly other covariates (i.e. sex and race).
The goal of this model is to estimate the total effect of the predictor on the response.
Bethany Cara Bray
bcbray@psu.edu
Assessing the Total Effect of Time-Varying Predictors in Prevention Research
4.7.03
Themes

Problems with confounders: Since confounders are correlated with both the predictor and
response they offer alternate explanations of the observed relationship between the
predictor and response. When confounding is not controlled, the unequal distribution of
levels of the confounders among levels of the predictor (called compositional differences)
causes bias in the estimated total effect of the predictor on the response. When
confounding is not controlled, the estimated coefficient of the effect of the predictor on
the response reflects the difference between the predictor groups, in addition to the causal
effect of the predictor on the response. In other words, it is unclear whether the estimated
effect of the predictor on the response represents the consequence that delayed predictor
initiation has on the initiation of the response, or whether the estimated effect merely
reflects compositional differences in the confounder, or if the estimated effect reflects a
combination of the two.

Standard Model: The standard model, which includes confounders as covariates in the
response regression model, attempts to do two things simultaneously. The first is to
control for confounding. The second is to estimate direct effects. We should worry when
we are using one model to do two different things. Here we are going to focus on the
problems with using the standard model to control for confounding while estimating the
total effect of a predictor on a response when confounders are affected by the predictor,
as often happens when the predictor and confounders are time-varying.

Spurious correlations: One example is when confounders are included as covariates in the
response regression model. A pathway opens between the predictor at time 1 and the
response at time 3. This pathway is a spurious correlation that makes the relationship
between the predictor and response (the estimated effect) appear different than what is
actually true. When these spurious correlations cause the estimated effect to be different
than what it actually is (a biased estimate), false conclusions regarding the consequences
that the timing of the predictor has on the timing of the response may be made, leading to
inaccurate conclusions, treatment, and intervention decisions.

Why weighting works: Weighting attempts to do what randomization does – equalize the
compositional differences in the confounders among levels of the predictor. This makes
the groups of people in the different predictor levels comparable. By equalizing the
compositional differences between the predictor levels, the confounders are controlled
and the correlation between the confounder and predictor is eliminated. Thus, the
confounder does not need to be controlled by including it as a covariate in the response
regression model, which eliminates the spurious correlation. In other words, weighting
eliminates the path of the spurious correlation by not conditioning on the confounder in
the final response regression model while controlling for confounders by equalizing the
compositional differences between initiators and non-initiators of the predictor. Hence,
the estimates from the final response regression model are unbiased.
Bethany Cara Bray
bcbray@psu.edu
Assessing the Total Effect of Time-Varying Predictors in Prevention Research
4.7.03
Notation

Predictor: Conduct Disorder Initiation, Cd

Response: Marijuana Initiation, Mj

Confounder: Peer Pressure Resistance, Ppress

Unmeasured Confounder: Parent-child Relationship Quality, U
Selected References

Barber, J. S., Murphy, S. A., & Verbitsky, N. (2002). Adjusting for time-varying
confounding in survival analysis. Manuscript submitted for publication.

Clayton, R. R., Cattarello, A. M., & Johnstone, B. M. (1996). The effectiveness of drug
abuse resistance education (project DARE): 5-year follow-up results. Preventive
Medicine, 25, 307-318.

Hernán, M., Brumback, B., & Robins, J. M. (2000). Marginal structural models to
estimate the causal effect of zidovudine on the survival of HIV-positive men.
Epidemiology, 11, 5, 561-570.

Pearl, J. (1998). Graphs, causality, and structural equation models. Sociological
Methods and Research, 27, 226-284.

Robins, J. M. (1986). A new approach to causal inference in mortality studies with
sustained exposure periods – application to control of the healthy worker survivor
effect. Mathematical Modeling, 7, 1393-1512.

Robins, J. M. (1989). The analysis of randomized and nonrandomized AIDS treatment
trials using a new approach to causal inference in longitudinal studies. In L.
Sechrest, H. Freeman, & A. Mulley (Eds.), Health Service Research
Methodology: A Focus on AIDS (pp. 113-159). Washington, DC: NCHSR, U.S.
Public Health Service.

Robins, J. M. (1998). Marginal structural models. 1997 proceedings of the American
Statistical Association, section on Bayesian statistical science (pp. 1-10).
Retrieved from: http://www.biostat.harvard.edu/~robins/research.html.

Robins, J. M. & Greenland, S. (1994). Adjusting for differential rates of PCP
prophylaxis in high- versus low dose AZT treatment arms in an AIDS randomized
trial. Journal of the American Statistical Association, 89, 737-749.

Robins, J. M., Hernán, M. & Brumback, B. (2000). Marginal structural models and
causal inference in epidemiology. Epidemiology, 11, 5, 550-560.
Bethany Cara Bray
bcbray@psu.edu
Assessing the Total Effect of Time-Varying Predictors in Prevention Research
4.7.03
Figure 1. Illustration of a spurious correlation between predictors and response in the sprinkler example
RAINING
(time 1)
RAINING
(time 2)
U1
U2
c
c
c
c
a
Conf1
Predict = Predictor
Conf = Confounder
Resp = Response
U = Unmeasured Predictor
a
Predict1
Resp2
Conf2
Predict2
Resp3
Front
Yard
Grass
(time 2)
Front
Yard
Sprinkler
(time 2)
Back
Yard
Grass
(time 3)
b
Front
Yard
Grass
(time 1)
Front
Yard
Sprinkler
(time 1)
Back
Yard
Grass
(time 2)
Sprinkler example follows examples often used by Pearl.
Bethany Cara Bray
bcbray@psu.edu
Assessing the Total Effect of Time-Varying Predictors in Prevention Research
4.7.03
Figure 2. Some relationships among conduct disorder, peer pressure resistance, and marijuana
Parent-Child
Relationship Quality
(time 1)
Parent-Child
Relationship Quality
(time 2)
U1
U2
a
a
Ppress1
c
c
c
c
Cd = Predictor
Ppress = Confounder
Mj = Response
U = Unmeasured Predictor
Cd1
Mj2
Ppress2
Cd2
Mj3
Peer
Pressure
Resistance
(time 2)
Conduct Disorder
Initiation
Marijuana
Initiation
(time 2)
(time 3)
b
Peer
Pressure
Resistance
(time 1)
Conduct Disorder
Initiation
Marijuana
Initiation
(time 1)
(time 2)
Notes:
 The arrows represent causal paths.
 Many arrows that would naturally be in this figure are omitted for simplicity.
 Time progresses from left to right.
 This is not an SEM diagram.
Bethany Cara Bray
bcbray@psu.edu