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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

I.   Introduction to the problem
II. Standard model
III. Problems with the standard model
IV. Suggested solution
V. Data example
VI. Future directions

2
GOAL
Assess the total effect that delaying the timing of a
predictor has on the timing of a response.

3
OUR QUESTION
“Does delaying conduct disorder initiation lead to a
delay in the initiation of marijuana?”

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

4
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.

5
COMPOSITIONAL
DIFFERENCES
The unequal distribution of levels of the confounder
between the types of children that initiate the predictor
and those who do not.

6
WHY WORRY?
The coefficient of the predictor is a biased estimate of the
total effect.

7
COEFFICIENT ESTIMATES
Estimated coefficient reflects the difference between
the predictor groups, in addition to the causal effect.

8
WHY ONLY IN
OBSERVATIONAL STUDIES?
Compositional differences are minimized by
randomization.

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

9
WHAT DO WE NORMALLY
DO?
The standard model.

10
THE STANDARD MODEL
Includes confounders as covariates in the response
regression model.

11
Figure 1. Illustration of a spurious correlation between predictors and response in the sprinkler example

RAINING                                                         RAINING
(time 1)                                                        (time 2)                                    Predict = Predictor
Conf = Confounder
U1                                                               U2                                      Resp = Response
U = Unmeasured Predictor

c               c                                                  c                     c

a                                                                a
Conf1              Predict1              Resp2                    Conf2                 Predict2     Resp3
b

Front              Front                 Back                    Front                  Front        Back
Yard               Yard                  Yard                    Yard                   Yard         Yard
Grass             Sprinkler              Grass                   Grass                 Sprinkler     Grass
(time 1)           (time 1)              (time 2)                (time 2)               (time 2)     (time 3)

Sprinkler example follows examples often used by Pearl.
12
PROBLEM
The confounder is affected by the predictor.
If the confounder is included as a covariate, a spurious
correlation is created.

13
SPRINKLER EXAMPLE
Consider a simple example involving sprinklers.

14
Figure 1. Illustration of a spurious correlation between predictors and response in the sprinkler example

RAINING                                                         RAINING
(time 1)                                                        (time 2)                                    Predict = Predictor
Conf = Confounder
U1                                                               U2                                      Resp = Response
U = Unmeasured Predictor

c               c                                                  c                     c

a                                                                a
Conf1              Predict1              Resp2                    Conf2                 Predict2     Resp3
b

Front              Front                 Back                    Front                  Front        Back
Yard               Yard                  Yard                    Yard                   Yard         Yard
Grass             Sprinkler              Grass                   Grass                 Sprinkler     Grass
(time 1)           (time 1)              (time 2)                (time 2)               (time 2)     (time 3)

Sprinkler example follows examples often used by Pearl.
15
Figure 2. Some relationships among conduct disorder, peer pressure resistance, and marijuana

Parent-Child                                                     Parent-Child
Relationship Quality                                             Relationship Quality
(time 1)                                                         (time 2)                                    Cd = Predictor
Ppress = Confounder
U1                                                               U2                                        Mj = Response
U = Unmeasured Predictor

c                c                                                 c                c

a                                                          a
Ppress1                  Cd1             Mj2                   Ppress2                  Cd2             Mj3
b

Peer              Conduct Disorder   Marijuana                Peer               Conduct Disorder   Marijuana
Pressure               Initiation      Initiation             Pressure                Initiation      Initiation
Resistance                                                    Resistance
(time 1)                (time 1)        (time 2)              (time 2)                 (time 2)        (time 3)

16
RESULT
Spurious correlations are dangerous.

17
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.

18
WHAT DO WE DO NOW?
Use sample weights to statistically control for time-
varying confounders*.

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

*Hernán, Brumback, and Robins, 2000                          19
HOW DOES IT WORK?
Equalizes the compositional differences of the
confounder among the predictor levels.

20
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
21
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                 22
IN PRACTICE
Eliminate the elevation of the total sample size.

EQUATION 1
P[Cd i ]
W
P[Cd i | Conf i ]

23
WHY DOES THIS WORK?
•Eliminates the problematic spurious correlation.
•Controls for confounders by equalizing compositional
differences.

24
Figure 3. Elimination of relationship among conduct disorder and peer pressure resistance by using sample weights

Parent-Child                                                     Parent-Child
Relationship Quality                                             Relationship Quality
(time 1)                                                         (time 2)                                        Cd = Predictor
Ppress = Confounder
U1                                                               U2                                            Mj = Response
U = Unmeasured Predictor

c                c                                                 c                    c

a                                                             a
Ppress1                  Cd1             Mj2                   Ppress2                       Cd2            Mj3
b

Peer              Conduct Disorder   Marijuana                Peer                   Conduct Disorder   Marijuana
Pressure               Initiation      Initiation             Pressure                    Initiation      Initiation
Resistance                                                    Resistance
(time 1)                (time 1)        (time 2)              (time 2)                      (time 2)       (time 3)

25
Figure 4. Some Relationships in a weighted sample when peer pressure resistance is omitted

Parent-Child                                                    Parent-Child
Relationship Quality                                            Relationship Quality
(time 1)                                                        (time 2)                            Cd = Predictor
Mj = Response
U1                                                              U2                                U = Unmeasured Predictor

Cd1                  Mj2                                   Cd2             Mj3

Conduct Disorder        Marijuana                           Conduct Disorder   Marijuana
Initiation           Initiation                             Initiation      Initiation

(time 1)            (time 2)                               (time 2)        (time 3)

26
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.                                               27
EQUATION 2*
t
P[Cdi | Cd i-1 , Sex, Race, Mji-1 ]
Wt  
i 1 P[Cdi | Cd i -1 , Conf i , Sex, Race, Mji -1 ]

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

28
NOW WHAT?
Weighted logistic regression of the response on the
predictor.

29
DATA EXAMPLE
•Naïve Model
•Standard Model
•Weighted Model

30
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             NOTES:
+Coefficients for
Alcohol                                      0.8238**          intercepts and
baseline
Other Drug Use                               1.2848**          variables are
omitted.
Peer Pressure Res.                          -0.0470***       †These models do
not include
Non-Time-Varying Confounders:                                  confounders by
Heart Rate                                  -0.0118            definition.

Verbal IQ                                   -0.0265**         One tailed tests:
*p<0.05
**p<0.01
Performance IQ                              -0.0117            ***p<0.001

Ave. Sen. Seeking                            0.0191                      31
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.

32
ASSUMPTIONS
1. Sequential Ignorability
2. Past confounder patterns do not exclude
particular levels of exposure

33
FUTURE DIRECTIONS
•Generalization of method to multilevel data structures
•Procedures to detect assumption violations
•Robustness to assumption violations

34
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.

35
EXTRA INFO

36
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
37
OUR DATA
•Lexington Longitudinal Study
•121 Female, 41 non-white
•Multiple confounders
•Time measured ever 1/3 of a school year

38
WEIGHT CALCULATIONS
•Numerator Regression Model:
numprti
log(             )   t Schyr  1 * Sexi   2 * Racei
1  numprti

•Denominator Regression Model:
denprti
log(             )   t Schyr   1 * Sexi   2 * Racei   * Confti
1  denprti

•Weight (conduct disorder initiation at time t):
 numpri     1  numpri          1  numpri   
Wt  
 denpr    
  1  denpr     
       1  denpr    

      i   t          i    t 1           i   1   39
•Naïve Model and Weighted Model:
pt
log(        )   t Schyr  1 * Sex   2 * Race   3Cd t
1  pt
•Intercept Term:
 t Schyr   1 * Schyr1   2 * Schyr2     t * Schyrt

•Standard Model:
pt
log(        )   t Schyr  1 * Sex   2 * Race   3Cd t   tConft
1  pt
•Confounders:
Conft  [ Ppress t , Odgat , Hr , Piq,Viq, Asss , Cig t , Alct ]
40

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