Bivariate Mixed Discrete and Continuous Responses

1
Bivariate Mixed Discrete and Continuous
Responses

• Majnu John
• BDMC, CHOP

2
Articles referenced

• R1) Fitzmaurice and Laird (JASA, 1995).
  Regression models for a bivariate discrete and
  continuous outcome with clustering
• R2) Fitzmaurice and Laird (Biometrics, 1997).
  Regression models for mixed discrete and
  continuous responses with potentially missing
  values
• Other papers (not referenced) Cox (1972)
  Catalano and Ryan (1992)

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National Toxicity Program (NTP) Study

• Ethylene glycol, EG, is a high volume industrial
  chemical.
• EG (at different doses) was applied to
  pregnant lab mice over the period of major
  organogenesis, beginning just after implantation
• Each live fetus found (after sacrifice) was
  examined for evidence of malformations (-- a
  discrete response)
• Fetal weight (-- a continuous response--) was
  also measured
• Primary question effects of dose on fetal weight
  and malformation

4
SimBaby Trial (A study at CHOP, Investigators
Aaron Donoghue and others)

• Randomized Trial
• 2 groups house staff performing mock
  resuscitation exercises on a patient simulator
  vs. standard manikin
• Hypothesis Patient simulator (SimBaby) group
  will have improved performance in test scenarios
  compared to the manikin group
• A list of tasks has to be performed in sequence
  by both the groups

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SimBaby Trial - Responses

• Dichotomous variables indicating whether each
  task was performed or not
• If a task was performed, then the time taken for
  that task is also measured
• Performance evaluated on both these variables

6
SCCOR (A study at CHOP, Investigators Elizabeth
Goldmuntz and others)

• Retrospective case-control study
• Main objective to test whether patients with
  22q11 deletion, a genetic mutation, has worse
  clinical outcomes than the non-deleted group
• A few of the outcomes are bivariate mixed
  discrete and continuous in nature

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SCCOR a few specific outcomes

• Cardiac Pulmonary Bypass (CPB) yes/no
• If yes, time taken for each run
• DHCA yes/no
• If yes, time taken for each run

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A brief note on the examples

• There is correlation between the bivariate
  responses (since they are both measured on the
  same subjects), which needs to be accounted for
• Last two examples are qualitatively a bit
  different from the first one The time variable
  gets switched on only if the dichotomous
  variable is yes so its more than just
  correlation

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Likelihood representation (as given in R1)

• Xi continuous response, Yi binary response
• Assume (1 x P vector) Zi predicts both Yi and Xi
• The marginal distribution of Yi is Bernoulli,
• f(yiZi) expyi?i log1 exp(?i),
  where
• ?i logµ1i/(1- µ1i) Ziß1 and
• µ1i E(Yi) Pr(Yi 1/Zi, ß1)

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Likelihood representation (as given in R1)

• The log-likelihood is
• where f Xi, Yi (xi, yi) fYi(yi)fXiYi(xiyi
  ) is the joint density
• We assume fXiYi(xiyi) (2ps2) -1/2
• ? is a parameter for the regression of Xi on Yi

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Likelihood representation (as given in R1)

• The continuous variable has a conditional mean
  that depends on the binary response.
• This dependency induces association or
  correlation between Yi and Xi.
• Also note that so
  that both ß1 and ß2 are regression parameter that
  have marginal interpretations

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Parameter estimates (as given in R1)

• The parameter estimates for
  may be obtained by solving the score
  equations
• The covariance of the parameter estimates can be
  approximated by the inverse of the Fisher
  information matrix
  
  
  
  
  

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Correlated Bivariate Model (as given in R1)

• Extensions of the previous model to allow for
  clustering
• The responses for the ith cluster consists of
  (Xi, Yi), where
• Xi (Xi1, Xi2, , Xini)', Yi (Yi1,
  Yi2, , Yini)'
• Let Zi (zi1, zi2, , zini)' represent the
  covariates for the ith cluster

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Correlated Bivariate Model (as given in R1)

• The model for the mean is assumed to be
• where
• ?1 ?2 association between binary and
  continuous responses made on the same unit within
  a cluster
• ?2 association between binary and continuous
  responses made on different units within a
  cluster

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Correlated Bivariate Model (as given in R1)

• Also assumed separate intracluster correlations,
  ?Y and ?X, respectively for the binary and
  continuous responses
• GEE methodology is used for the estimation of
  (ß1, ß2, ?1, ?2). Method of moments estimators
  for s2, ?Y and ?X.
• Maximum likelihood estimation is quite
  complicated in the clustered data setting

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A closer look at SimBaby and SCCOR examples

• The discrete variable (task performed yes/no)
  and the continuous variable (time taken for the
  task) is much more than correlated. The
  continuous variable gets switched on to a
  nonzero value only when the discrete variable is
  yes. For these examples, maybe the joint
  distribution for discrete and continuous
  variables should be reformulated to reflect this.
• If the continuous variable is e.g. time taken,
  its range is 0, 8). Maybe a gamma distribution
  assumption is better than a normal distribution
  assumption

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Thank you!