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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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posted:8/18/2011
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