Esra Kurum

Transcript

1. Joint Modeling of Longitudinal Binary and Continuous Responses Esra K¨ ur¨ um,1 Runze Li,2 Saul Shiffman,3 and Weixin Yao1 1 Department of Statistics, University of California, Riverside 2Department of Statistics and The Methodology Center, The Pennsylvania State University 3 Department of Psychology, University of Pittsburgh
2. Smoking Cessation Study (Shiffman et al., 1996) Cigarette smoking is one of the leading preventable causes of coronary heart disease (CHD) and cancer (U.S. Department of Health and Human Services, 1982, 1983, and 2004) Source: The Methodology Center, Penn State University.
3. Smoking Cessation Study (Shiffman et al., 1996) Cigarette smoking is one of the leading preventable causes of coronary heart disease (CHD) and cancer (U.S. Department of Health and Human Services, 1982, 1983, and 2004) 304 participants Smoking for at least 2 years with 10 cigarettes/day Source: The Methodology Center, Penn State University.
4. Data Collection – Ecological Momentary Assessment Data Current activities and setting, such as coffee consumption, alcohol use, and presence of other smokers Current mood and urge to smoke
5. Data Collection – Ecological Momentary Assessment Data Current activities and setting, such as coffee consumption, alcohol use, and presence of other smokers Current mood and urge to smoke Source: http://mhealth.jmir.org/2014/1/e4/
6. Data Collection in 1996 Source: http://www.bpsmedicine.com/content/2/1/13 At each random prompt, each subject answers a series of questions about his/her current activities and setting
7. Smoking Cessation Study
8. Literature on Smoking Smoking is likely to occur in certain situations or contexts, such as alcohol consumption Source: The Methodology Center, Penn State University.
9. Literature on Smoking Smoking is likely to occur in certain situations or contexts, such as alcohol consumption Alcohol consumption (Shiffman and Balabanis, 1995; Shiffman et al., 2002) increases the odds of smoking is associated with increased risk of lapsing back to smoking Source: The Methodology Center, Penn State University.
10. The Goal Primary interest: Investigate the relationship between alcohol use and urge to smoke controlling for confounders such as mood variables
11. Relationship between Urge to smoke and Alcohol Use Regression? Urge to smoke → Alcohol use OR Alcohol use → Urge to smoke
12. Relationship between Urge to smoke and Alcohol Use Regression? Urge to smoke → Alcohol use OR Alcohol use → Urge to smoke Association: Urge to smoke ←→ Alcohol use
13. Relationship between Urge to smoke and Alcohol Use Regression? Urge to smoke → Alcohol use OR Alcohol use → Urge to smoke Association: Urge to smoke ←→ Alcohol use Time variation: Before and after the quit day
14. The Goal Primary interest: Investigate the relationship between alcohol use and urge to smoke controlling for confounders such as mood variables
15. The Goal Primary interest: Investigate the relationship between alcohol use and urge to smoke controlling for confounders such as mood variables Primary interest: Estimate the time-varying association between alcohol use and urge to smoke controlling for the temporal patterns of mood variables
16. The Goal Primary interest: Investigate the relationship between alcohol use and urge to smoke controlling for confounders such as mood variables Primary interest: Estimate the time-varying association between alcohol use and urge to smoke controlling for the temporal patterns of mood variables Approach: Joint modeling of longitudinal binary and continuous responses 1 1K¨ ur¨ um, E., Li, R., Shiffman, S., and Yao, W. (2016). Time-varying coefficient models for joint modeling binary and continuous outcomes in longitudinal data. Statistica Sinica
17. Joint Models for Binary and Continuous Responses Challenge: Lack of a natural multivariate distribution
18. Joint Models for Binary and Continuous Responses Challenge: Lack of a natural multivariate distribution Solution: Latent variable approach Introduce a continuous latent outcome underlying the binary outcome Assume that the latent variable and the continuous response follow a joint normal distribution
19. Joint Models for Binary and Continuous Responses Challenge: Lack of a natural multivariate distribution Solution: Latent variable approach Introduce a continuous latent outcome underlying the binary outcome Assume that the latent variable and the continuous response follow a joint normal distribution Factorize the joint model: a marginal model for the continuous variable and a conditional model for the binary variable given the continuous variable
20. Associations W(t
    ) W(t) 2(t) 2(t
    ) ⌧(t
    ) ⌧(t) ⇢12 (t ,t  ) ⇢1 (t
    , t) ⇢2 (t
    , t) t t C o n t i n u o u s o u t c o m e B i n a r y O u t c o m e
21. The Model Consider the following bivariate model: W (t) = XT(t)β(t) + ε1 (t) (1) Y (t) = XT(t)α(t) + ε2 (t) where ε1 (t) ∼ N 0, σ2 1 (t) with corr {ε1 (tj ), ε1 (tk )} = ρ1 (tj , tk ) for j = k ε2 (t) ∼ N 0, σ2 2 (t) with corr {ε2 (tj ), ε2 (tk )} = ρ2 (tj , tk ) for j = k corr {ε1 (t), ε2 (t)} = τ(t) corr {ε1 (tj ), ε2 (tk )} = ρ12 (tj , tk ) for j = k
22. The relation between the latent variable and the binary variable: Q(t) = 1 if Y (t) > 0 and Q(t) = 0 if Y (t) ≤ 0 Factorize the joint model—a marginal model for the continuous variable W (t) and a conditional model for Q(t) given W (t) f {q(t), w(t)} = fW {w(t)} f {q(t)|w(t)}
23. Conditional Model First, obtain the conditional model Y (t)|W (t) Applying the standard normal theory Y (t)|W (t) ∼ N µ(t), σ2 2 (t) 1 − τ2(t) where µ(t) = XT(t)α(t) + σ2 (t) σ1 (t) τ(t)ε1 (t) and ε1 (t) = W (t) − XT(t)β(t) (2)
24. Conditional Model (Cont’d) Q(t) = 1 if Y (t) > 0 and Q(t) = 0 if Y (t) ≤ 0 Y (t)|W (t) ∼ N µ(t), σ2 2 (t) 1 − τ2(t)
25. Conditional Model (Cont’d) Q(t) = 1 if Y (t) > 0 and Q(t) = 0 if Y (t) ≤ 0 Y (t)|W (t) ∼ N µ(t), σ2 2 (t) 1 − τ2(t) The conditional model for Q(t) given W (t) is P {Q(t) = 1|W (t)} = Φ µ(t) σ2 2 (t) {1 − τ2(t)}
26. Conditional Model (Cont’d) Q(t) = 1 if Y (t) > 0 and Q(t) = 0 if Y (t) ≤ 0 Y (t)|W (t) ∼ N µ(t), σ2 2 (t) 1 − τ2(t) The conditional model for Q(t) given W (t) is P {Q(t) = 1|W (t)} = Φ µ(t) σ2 2 (t) {1 − τ2(t)} Reparameterize P {Q(t) = 1|W (t)} = Φ XT(t)α∗(t) + α∗ p+1 (t)ε1 (t) (3) where α∗(t) = α∗ 1 (t), . . . , α∗ p (t) T
27. Time-Varying Association P {Q(t) = 1|W (t)} = Φ XT(t)α∗(t) + α∗ p+1 (t)ε1 (t) (4) It can be seen that α∗ p+1 (t) = 1 σ1 (t) · τ(t) 1 − τ2(t) Let b(t) = α∗ p+1 (t)σ1 (t). Then τ(t) = b(t) 1 + b2(t) (5)
28. Two-Stage Estimation Procedure First stage Marginal model W (t) = XT(t)β(t) + ε1 (t) Second stage Conditional model P {Q(t) = 1|W (t)} = Φ XT(t)α∗(t) + α∗ p+1 (t)e(t) Time-varying Association b(t) = α∗ p+1 (t)σ1 (t) τ(t) = b(t) √ 1+b2(t)
29. Application: Smoking Cessation Study Time-varying association between Urge to smoke (Continuous response) and Alcohol use (Binary response) 149 subjects lapsed Time period: Two weeks before and after the quit day
30. Alcohol Use versus Urge to Smoke – First Stage Continuous outcome: Urge to smoke (Recorded on a scale ranging from 0 to 11) Predictors: Mood variables (Negative affect factor, arousal factor, attention disturbance factor) (Shiffman et al., 2002)
31. Alcohol Use versus Urge to Smoke – First Stage Continuous outcome: Urge to smoke (Recorded on a scale ranging from 0 to 11) Predictors: Mood variables (Negative affect factor, arousal factor, attention disturbance factor) (Shiffman et al., 2002) The Model: W (t) = β0 (t) + β1 (t)X1 (t) + β2 (t)X2 (t) + β3 (t)X3 (t) + ε1 (t) (6) where W (t) : The score of urge to smoke of the ith subject at time t X1 (t) : The centered score of negative affect of the ith subject at time t X2 (t) : The centered score of arousal of the ith subject at time t X3 (t) : The centered score of attention disturbance of the ith subject at time t
32. Controlling for the Time-Varying Confounders: Negative Affect -10 -5 0 5 10 15 0.0 0.5 1.0 1.5 2.0 Days since quit smoking β1 (T)
33. Second Stage After obtaining the residuals from the marginal model, we fit the following generalized time-varying coefficient model: P {Q(t) = 1 | W (t)} = Φ α∗ 0 (t) + α∗ 1 (t)X1 (t) + α∗ 2 (t)X2 (t) + α∗ 3 (t)X3 (t) + α∗ 4 (t)e(t) where Q(t) : the alcohol usage of the ith subject at time t X1 (t) : the centered score of negative affect of the ith subject at time t X2 (t) : the centered score of arousal of the ith subject at time t X3 (t) : the centered score of attention disturbance of the ith subject at time t
34. Two-Stage Estimation Procedure First stage Marginal model W (t) = XT(t)β(t) + ε1 (t) Second stage Conditional model P {Q(t) = 1|W (t)} = Φ XT(t)α∗(t) + α∗ 4 (t)e(t)} Time-varying Association b(t) = α∗ 4 (t)σ1 (t) τ(t) = b(t) √ 1+b2(t)
35. Time-Varying Association between Alcohol Use and Urge to Smoke -10 -5 0 5 10 15 -0.4 -0.2 0.0 0.2 0.4 Days since quit smoking τ(T) Before quit day, the relationship is positive, i.e, increased drinking is associated with increased urge in smoking
36. Our Contributions A joint modeling method that can estimate the time-varying association between longitudinal binary and continuous responses allow all parameters to be time-varying: response–predictor relationships, all associations, and variances be applied to longitudinal data sets with irregular time points
37. Generalization Association between binary and continuous response
38. Generalization Association between binary and continuous response Copula approach: any dimension and any type of outcome 2 3K¨ ur¨ um, E., Hughes, J., Li, R., and Shiffman, S. (2017). Time-varying copula models for longitudinal mixed data, Statistics and Its Interface
39. Acknowledgements The project described was supported by National Institute on Drug Abuse/National Institute of Health grants P50-DA10075 and P50 DA039838, National Cancer Institute grant R01 CA168676, and National Science Foundation grants DMS-1512422 and DMS-1461677 The authors acknowledge the Research Computing and Cyberinfrastructure unit of Information Technology Services at The Pennsylvania State University for providing advanced computing resources and services that have contributed to the research. URL: http://rcc.its.psu.edu