Supplementary Online Materials for "Dynamic Latent Trait Models with
Mixed Hidden Markov Structure for Mixed Longitudinal Outcomes"
Yue Zhanga,b* and Kiros Berhanec
a
b
c
Department of Internal Medicine, University of Utah, Salt Lake City, UT
Department of Family and Preventive Medicine, University of Utah, Salt Lake City, UT
Department of Preventive Medicine, University of Southern California, Los Angeles, CA
Correspondence to: Yue Zhang, Division of Epidemiology, Department of Internal Medicine,
University of Utah, 295 Chipeta Way, Salt Lake City, UT, 84018
E-mail: zhang.yue@hsc.utah.edu
Tel: 801-213-3735
Fax: 801-581-3623
1
Appendix A: Likelihood
Based on the assumption of independence among outcomes, given latent variables and random
effects, the observed-data likelihood for this model is
𝐿(𝒀𝟏 , ⋯ , 𝒀𝒑 , 𝒀𝑜 | ∙)
𝑝
𝑛
=
∬ ∏ {{∑ 𝑃(𝒀𝑜𝑖 |𝒀𝑟𝑖 , 𝑼1 , 𝑼2 , 𝑽, 𝑳)𝑃( 𝒀𝑟𝑖 |𝑼1 , 𝑽, 𝑳)} ∏ 𝑓(𝒀𝒊𝒎 |𝑳, 𝑽, 𝑢𝑖𝑚 , 𝜺)} 𝑓(𝑳|𝑽)
𝑖=1
𝑚=1
𝒀𝑟𝑖
∙ 𝑓(𝑽)𝑓(𝑼1 )𝑓(𝑼2 )𝑓(𝜺)𝑓(𝑢𝑖𝑚 )𝑑𝑢𝑖𝑚 𝑑𝜺𝑑𝑼1 𝑑𝑼2 𝑑𝑽𝑑𝑳
𝑇𝑖
𝑛
=
𝑟
∫ ∬ ∬ ∏ {{∑ 𝑃(𝑌𝑖1𝑜 |𝑌𝑖1𝑟 , 𝑼1 , 𝑼2 )𝑃( 𝑌𝑖1𝑟 |𝑼1 ) ∏ 𝑃(𝑌𝑖𝑗𝑜 |𝑌𝑖𝑗𝑟 , 𝑼1 , 𝑼2 )𝑃(𝑌𝑖𝑗𝑟 |𝑌𝑖𝑗−1
, 𝑼1 )}
𝑖=1
𝑗=2
𝒀𝑟𝑖
𝑇𝑖
𝑝
∙ ∏ ∏ 𝑓(𝑌𝒊𝒋𝒎 |𝑳, 𝑽, 𝑢𝑖𝑚 , 𝜺)} 𝑓(𝑳|𝑽)𝑓(𝑽) 𝑓(𝑼1 )𝑓(𝑼2 )𝑓(𝜺)𝑓(𝑢𝑖𝑚 )𝑑𝑢𝑖𝑚 𝑑𝜺𝑑𝑼1 𝑑𝑼2 𝑑𝑽𝑑𝑳 .
𝑚=1 𝑗=1
Appendix B: Details of MCMC Algorithm
The joint posterior distribution for the parameters and latent variables in Section 3 is
𝑓(𝚯𝟏 , 𝚯𝟐 , 𝝃, 𝝍𝟏 , 𝝍𝟐 , 𝑼𝟏 , 𝑼𝟐 , 𝒖, 𝑽, 𝜺, 𝒀𝒓 , 𝑳|𝒀𝟏 , ⋯ , 𝒀𝒑 , 𝒀𝑜 )
∝ 𝑓(𝒀𝑜 |𝚯𝟏 , 𝝍𝟏 , 𝝍𝟐 , 𝑼𝟏 , 𝑼𝟐 , 𝑽, 𝒀𝒓 )𝑓(𝒀𝑟 |𝑼𝟏 )𝑓(𝑼𝟏 )𝑓(𝑼𝟐 )𝑓(𝚯𝟏 )𝑓(𝝍𝟏 )𝑓(𝝍𝟐 )
× 𝑓(𝒀𝟏 , ⋯ , 𝒀𝒑 |𝚯𝟐 , 𝝃, 𝒖, 𝑽, 𝜺, 𝑳)𝑓(𝑳|𝑽, 𝜺)𝑓(𝒖)𝑓(𝚯𝟐 )𝑓(𝑽)𝑓(𝜺)𝑓(𝝃)
𝑁
∝
𝑇𝑖
𝑟
∏ {𝑃(𝑌𝑖1𝑜 |𝑌𝑖1𝑟 , 𝑼1 , 𝑼2 )𝑃(𝑌𝑖1𝑟 |𝑼1 , 𝑽) ∏ 𝑃(𝑌𝑖𝑗𝑜 |𝑌𝑖𝑗𝑟 , 𝑼1 , 𝑼2 )𝑃(𝑌𝑖𝑗𝑟 |𝑌𝑖𝑗−1
, 𝑼1 )
𝑖=1
𝑗=2
2
𝑇𝑖
𝑝
𝑞
2
1
(3)
𝑇
∏ ∏ 𝑓(𝑌𝑖𝑗𝑚 |𝚯𝟐 , 𝝃, 𝒖, 𝑽, 𝜺, 𝑳) exp (− ∑ (𝐿𝑖1𝑚 − 𝑣𝑚
∙ 𝑍𝑖1 − 𝑐𝑚 ∙ 𝑉𝑖1 )
2
𝑗=1 𝑙=1
𝑚=1
2
𝑗−1
−
1
(3)
(3)
𝑟
𝑇
𝑇
∑ (𝐿𝑖𝑗𝑚 −𝑣𝑚
∙ 𝑍𝑖𝑗 − ∑(𝜙1𝐾𝑚
∙ (𝐿𝑖𝐾 − 𝑣 𝑇 ∙ 𝑍𝑖𝐾 ) + 𝜙2𝐾𝑚 ∙ 𝑦𝑖𝐾𝑝+1
)) )
2
𝑗=2,𝑚
𝑝
×∏
𝑙=1
𝐾=0
1
1
2 ′
−1
exp
(−
𝑢
𝜉
𝑢
)
}
exp
(−
𝚯 𝚯 )
𝑖𝑙
𝑙
𝑖𝑙
|𝝃𝒍 |0.5
2
9 𝟏 𝟏
2
×∏
𝑔=1
2
×∏
𝑘=1
|𝚺𝐠 |
𝑚𝒈 /2
|𝝍𝒈 |
−(𝑚𝒈 +𝑝𝒈 +1)/2 −𝑡𝑟𝑎𝑐𝑒(𝚺 𝝍 −2 )/2
𝐠 𝒈
𝑒
2𝑚𝒈 𝑝𝒈/2 Γ𝑝𝒈 (𝑚𝒈 ⁄2)
1
1 ′
1
−𝟏
exp
(−
𝑼
𝝍
𝑼
)
exp
(−
∙ 𝚯′ 𝚯 )
𝒌
𝒌
𝒌
|𝝍𝒌 |0.5
2
200 𝟐 𝟐
Each parameter or parameter vector was updated by conditioning on all other parameters via
Gibbs sampling. For simplicity of development, we denote the regression parameters 𝚯𝟏 in
prevalence, transition and misclassification probability models for observed categorical outcome
with
misclassification
by 𝚯𝟏𝟏 = (𝒂𝟏 , 𝜶𝟏 , 𝑐𝑞+1 ) , 𝚯𝟏𝟐 = (𝒂𝟐 , 𝒂𝟑 , 𝜶𝟐 , 𝜷, 𝝓𝟑 ) and 𝚯𝟏𝟑 =
(𝒃𝟏 , 𝒃𝟐 , 𝜸, 𝜹). We also denote the regression parameters 𝚯𝟐 in modeling outcomes from the
exponential family by 𝚯𝟐𝟏 = (𝝀, 𝜻), 𝚯𝟐𝟐 = (𝒗) and 𝚯𝟐𝟑 = (𝒄, 𝝓𝟏 , 𝝓𝟐 ). The latent true health
states for subject i are sampled directly from their full conditional distributions:
𝑝(𝑌𝑖1𝑟 = 𝑘|𝚯𝟏𝟏 , 𝚯𝟏𝟐 , 𝚯𝟏𝟑 , 𝑼𝟏 , 𝑼𝟐 , 𝒀𝑜 ) =
𝑟
𝑝(𝑌𝑖1
= 𝑘|𝚯𝟏𝟏 , 𝑼𝟏 )∙𝑝(𝑌𝑖2𝑟|𝑌𝑖1𝑟 = 𝑘, 𝚯𝟏𝟐 , 𝑼𝟏 )∙𝑝(𝑌𝑖1𝑜 |𝑌𝑖1𝑟 =𝑘,𝚯𝟏𝟑 ,𝑼𝟐 )
,
𝑟
𝑟 𝑟
𝑜 |𝑌 𝑟 =ℎ,𝚯 ,𝑼 )
∑ℎ=1⋯𝑆1 𝑝(𝑌𝑖1 = ℎ|𝚯𝟏𝟏 , 𝑼𝟏 )∙𝑝(𝑌𝑖2 |𝑌𝑖1 = ℎ, 𝚯𝟏𝟐 , 𝑼𝟏 )∙𝑝(𝑌𝑖1
𝟏𝟑 𝟐
𝑖1
(A1)
3
𝑝(𝑌𝑖𝑗𝑟 = 𝑘|𝚯𝟏𝟐 , 𝚯𝟏𝟑 , 𝑼𝟏 , 𝑼𝟐 , 𝒀𝑜 ) =
𝑟
𝑝(𝑌𝑖𝑗
𝑟
𝑟
𝑟
= 𝑘|𝑌𝑖𝑗−1
, 𝚯𝟏𝟐 , 𝑼𝟏 )∙𝑝(𝑌𝑖𝑗+1
|𝑌𝑖𝑗 = 𝑘, 𝚯𝟏𝟐 , 𝑼𝟏 )∙𝑝(𝑌𝑖𝑗𝑜 |𝑌𝑖𝑗𝑟 =𝑘,𝚯𝟏𝟑 ,𝑼𝟐 )
, for
𝑟
𝑟
𝑟
𝑟
∑ℎ=1⋯𝑆1 𝑝(𝑌𝑖𝑗 = ℎ|𝑌𝑖𝑗−1 , 𝚯𝟏𝟐 , 𝑼𝟏 )∙𝑝(𝑌𝑖𝑗+1 |𝑌𝑖𝑗 = ℎ, 𝚯𝟏𝟐 , 𝑼𝟏 )∙𝑝(𝑌𝑖𝑗𝑜 |𝑌𝑖𝑗𝑟 =ℎ,𝚯𝟏𝟑 ,𝑼𝟐 )
𝑗 = 1, ⋯ , 𝑇𝑖 .
(A2)
The latent variable 𝑳𝒊𝒋 for subject i at time j is sampled from full conditional distributions:
𝑝
𝑝(𝑳𝒊𝒋 |𝒀𝟏 , ⋯ , 𝒀𝒑 , 𝒀𝑟 , 𝑳𝒊𝟏 , ⋯ , 𝑳𝒊𝒋−𝟏 , 𝚯𝟐 , 𝝃, 𝒖, 𝜺) ∝ exp {∑𝑙=1 log𝑓𝑙 (𝑦𝑖𝑗𝑙 |𝚯𝟐𝟏 , 𝑳𝒊𝒋 , 𝝃, 𝒖) −
∑𝑞𝑚=1
(𝐿𝑖𝑗𝑚 −𝐄(𝐿𝑖𝑗𝑚 ))2
2
},
(A3)
The subject-level random intercept 𝑢𝑖𝑚 is sample from full conditional distributions:
T
i
𝑝(𝑢𝑖𝑚 |𝒀𝟏 , ⋯ , 𝒀𝒑 , 𝒀𝑟 , 𝑳𝒊𝟏 , ⋯ , 𝑳𝒊𝒋 , 𝚯𝟐 , 𝝃, 𝒖, 𝜺) ∝ exp {∑𝑗=1
log𝑓𝑚 (𝑦𝑖𝑗𝑚 |𝚯𝟐𝟏 , 𝑳𝒊𝒋 , 𝝃, 𝑢𝑖𝑚 ) −
(𝑢𝑖𝑚 )2
2𝜉𝑚
}.
(A4)
The full conditional distributions of scale parameters 𝝍𝒊 and 𝜉𝑚 are sampled from
𝑝(𝝍𝒊 |𝑼𝒊 )~𝐈𝐧𝐯 − 𝐖𝐢𝐬𝐡𝐚𝐫𝐭(𝚺𝐢 + 𝑼′𝒊 𝑼𝒊 , 𝑚𝒊 + 𝑛),
(A5)
and
𝜉𝑚 ~𝑖𝑛𝑣 − 𝑔𝑎𝑚𝑚𝑎(em +
∑n
i=1 Ti
2
1
T
i
, 𝑓𝑚 + 2 ∑𝑛𝑖=1 ∑𝑗=1
(𝑢𝑖𝑚 )2 ), 𝑚 = 1,2, ⋯ , 𝑝.
(A6)
The full conditional distribution of 𝚯𝟏𝟏 is
2
𝑝(𝚯𝟏𝟏 |𝑼𝟏 , 𝒀𝑟 ) ∝ exp {∑𝒊 𝐥𝐨𝐠𝒑( 𝑌𝑖1𝑟 |𝚯𝟏𝟏 , 𝑼𝟏 ) − 9 𝚯′𝟏𝟏 𝚯𝟏𝟏 },
(A7)
The full conditional distribution of 𝚯𝟏𝟐 is
2
𝑟
𝑝(𝚯𝟏𝟐 |𝚯𝟏𝟏 , 𝑼𝟏 , 𝒀𝑟 ) ∝ exp {∑𝒊,𝒋=𝟐 𝐥𝐨𝐠𝒑( 𝑌𝑖𝑗𝑟 |𝚯𝟏𝟑 , 𝑼𝟏 , 𝑌𝑖𝑗−1
) − 9 𝚯′𝟏𝟐 𝚯𝟏𝟐 },
(A8)
The full conditional distribution of 𝚯𝟏𝟑 is
2
𝑝(𝚯𝟏𝟑 |𝑼𝟐 , 𝒀𝑟 , 𝒀𝑜 ) ∝ exp {∑𝒊,𝒋 𝐥𝐨𝐠𝒑( 𝑌𝑖𝑗𝑜 |𝚯𝟏𝟑 , 𝑼𝟐 , 𝒀𝑟 ) − 9 𝚯′𝟏𝟑 𝚯𝟏𝟑 }.
4
(A9)
The full conditional distribution of 𝚯𝟐𝟏 is
1
𝑝(𝚯𝟐𝟏 |𝑼𝟏 , 𝒀𝑟 ) ∝ exp {∑𝑖,𝑗,𝑚 log𝑓𝑚 (𝑦𝑖𝑗𝑚 |𝚯𝟐𝟏 , 𝑳𝒊𝒋 , 𝝃, 𝑢𝑖𝑚 ) − 200 𝚯′𝟐𝟏 𝚯𝟐𝟏 }.
(A10)
if each of 𝑓𝑚 is normal, then this posterior distribution is normal; otherwise, a Metropolis step
can be used (Hastings 1970).
The full conditional distribution of 𝚯𝟐𝟐 is
′
𝑝(𝚯𝟐𝟐 |𝒁(3) , 𝑳) = 𝑵((𝒁(3) 𝒁(3) + 1/100𝑰dim(𝚯𝟐𝟐 ) )−1 𝒁(3) 𝑰𝑞 ∑ni=1 Ti 𝑳 , (𝒁(3) 𝒁(3)′ +
1/100𝑰dim(𝚯𝟐𝟐 ) )−1 ),
(A11)
(𝟑) 𝑻
where 𝒁(3) is a matrix with row vectors {𝒛𝑖𝑗𝑚 ; 𝑖 = 1, … , 𝑛; 𝑗 = 1, … , 𝑇𝑖 ; 𝑚 = 1, … , 𝑞}, where
(𝟑) 𝑻
𝒛𝑖𝑗𝑚 represents the covariate vector corresponding to 𝚯𝟐𝟐 .
The full conditional distribution of 𝚯𝟐𝟑 is
𝑝(𝚯𝟐𝟑 |𝒁𝚯𝟐𝟑 , 𝑳) = 𝑵((𝒁𝚯𝟐𝟑 𝒁′𝚯𝟐𝟑 + 1/100𝑰dim(𝚯𝟐𝟑 ) )−1 𝒁𝚯𝟐𝟑 𝑰𝑞 ∑ni=1 Ti 𝑳 , (𝒁𝚯𝟐𝟑 𝒁′𝚯𝟐𝟑 +
1/100𝑰dim(𝚯𝟐𝟑 ) )−1 ),
(A12)
(𝚯
)𝑻
where 𝒁𝚯𝟐𝟑 is a matrix with row vectors {𝒛𝑖𝑗𝑚𝟐𝟑 ; 𝑖 = 1, … , 𝑛; 𝑗 = 1, … , 𝑇𝑖 ; 𝑚 = 1, … , 𝑞}, where
(𝚯
𝒛𝑖𝑗𝑚𝟐𝟑
)𝑻
represents the covariate vector corresponding to 𝚯𝟐𝟑 .
Appendix C: Validation of MCMC Algorithm
A simple joint model for continuous and binary outcomes was simulated to test the ability of the
MCMC based estimation algorithm to lead to valid parameter estimation in terms of bias and
coverage probability. This joint modeling simulation study was conducted using a process in
which 100 data sets were generated with a structure that closely followed the CHS data. In each
data set, there were 400 subjects with 4 yearly follow-up. In the simulated data, three continuous
5
outcomes and one categorical outcome were generated. The categorical outcome was defined as
a binary, subject to misclassification. Even though an absorbing state is defined in some
applications as a special real state in which the latent process will never leave it once it enters
(e.g., death), no absorbing state was assumed in this simulation in order to allow for more
general settings. In the MCMC sampling process, the first 10000 iterations were discarded as a
burn-in, then the next 10000 iterations were used to calculate the posterior summaries of
parameters of interest with thinning rate equal to 10. The Gelman-Rubin statistic was used to
check for convergence (Gelman, et al. 1992). The simulation scenario could be described as
follows:
Measurement model:
For three observed continuous outcomes:
𝑌𝑖𝑗𝑘 , = 𝑎0𝑘 , + 𝛾𝑘 , ∙ 𝐿𝑖𝑗 + 𝑏𝑖𝑘 , + 𝑒𝑖𝑗𝑘 , , 𝑘 , = 1,2,3.
For an observed categorical outcome subject to misclassification:
𝑃(𝑌𝑖𝑗𝑜 = 1|𝑌𝑖𝑗𝑟 ) =
𝑟
𝑟
exp(𝑏1 +𝑏2 ∙𝑦𝑖𝑗
+𝛼1𝑇 ∙𝑋1𝑖𝑗 +𝛽1𝑇 ∙𝑦𝑖𝑗
∙𝑋1𝑖𝑗 )
𝑟 +𝛼𝑇 ∙𝑋1 +𝛽𝑇 ∙𝑦 𝑟 ∙𝑋1 )
1+exp(𝑏1 +𝑏2 ∙𝑦𝑖𝑗
𝑖𝑗
𝑖𝑗
1
1 𝑖𝑗
,
where 𝑏𝑖𝑘 , ~𝑁(0, 𝜉𝑘2, ) and 𝑒𝑖𝑗𝑘 , ~𝑁(0, 𝜏𝑘2, ).
Structural model at Baseline:
For the latent variable at baseline, 𝐿𝑖1 , the model is given as follows:
𝐿𝑖1 = 𝛼 ∙ 𝑋2𝑖1 + 𝑐0 ∙ 𝑉𝑖1 + 𝜀𝑖1 .
𝑟
Similarly, the model for the real latent binary variable at baseline, 𝑦𝑖14
, is
𝑃(𝑌𝑖1𝑟
(𝑐)
= 1) =
(1)
exp(𝑎1 +𝛼1 ∙𝑋3𝑖1 +𝑐1 ∙𝑉𝑖1 +𝑈𝑖
(𝑐)
)
(1)
1+exp(𝑎1 +𝛼1 ∙𝑋3𝑖1 +𝑐1 ∙𝑉𝑖1 +𝑈𝑖
(1)
)
,
where 𝜀𝑖11 ~𝑁(0,1), 𝑉𝑖1 ~𝑁(0,1) for identifiability and 𝑈𝑖 ~𝑁(0, 𝜎12 ). We further assume 𝑐0 =
1 to ensure identifiability of the model.
6
Structural model at follow-up:
For the latent variable at follow-up, 𝐿𝑖𝑗 ,
(𝑐)
𝑟
𝐿𝑖𝑗 = 𝛼 ∙ 𝑋2𝑖𝑗 +𝑐3 ∙ 𝐿𝑖𝑗−1 + 𝑐4 ∙ 𝑌𝑖𝑗−1
+ 𝜀𝑖𝑗 .
For real latent binary variable at follow-up, 𝑌𝑖𝑗𝑟 ,
𝑟
𝑃(𝑌𝑖𝑗𝑟 = 1|𝑌𝑖𝑗−1
)
(𝑐)
=
(𝑐)
(𝑐)
(1)
𝑟
𝑟
exp(𝑎2 + 𝑎3 ∙ 𝑌𝑖𝑗−1
+ 𝛼2 ∙ 𝑋3𝑖𝑗 + 𝛼3 ∙ 𝑋3𝑖𝑗 ∙ 𝑌𝑖𝑗−1
+ 𝑐2 ∙ 𝐿𝑖𝑗−1 + 𝑈𝑖 )
(𝑐)
(𝑐)
(𝑐)
(1)
𝑟
𝑟
1 + exp(𝑎2 + 𝑎3 ∙ 𝑌𝑖𝑗−1
+ 𝛼2 ∙ 𝑋3𝑖𝑗 + 𝛼3 ∙ 𝑋3𝑖𝑗 ∙ 𝑌𝑖𝑗−1
+ 𝑐2 ∙ 𝐿𝑖𝑗−1 + 𝑈𝑖 )
,
(1)
where 𝜀𝑖𝑗1 ~𝑁(0,1) for identifiability reasons and 𝑈𝑖 ~𝑁(0, 𝜎12 ). The superscript (𝑐) indicates
that the variable is centered by its mean.
Results from the simulation study are given in Table S.1. Average mean, median, standard
deviation, nominal 95% coverage rate and bias were reported. The results showed that estimated
posterior means for parameters of interest were close to the true values on average (Bias
range=(0,0.44)). About half of the median estimates were above the true parameter values. In
addition, coverage rates of the estimated 95% credible intervals containing the true parameter
values were at least 95/100.
7
8
Appendix D: Full results from DLTM-MHMM in Southern California Children Health
Study (extended from Table 4) with Information of Convergence Diagnosis, DIC and
Number of Effective Sample Size.
Table S.2: Full Results of DLTM-MHMM in Southern California Children Health Study
mean
sd
2.50%
97.50%
PSRF*
N.effect
intercept
-3.97
0.45
-4.91
-3.14
1
4000
hispanic
-0.44
0.57
-1.57
0.66
1
4000
black
0.25
0.95
-1.72
1.97
1
4000
asian
-0.96
1.05
-3.13
0.96
1
4000
others+mixed
-0.41
0.92
-2.2
1.31
1
4000
age
0.29
0.54
-0.78
1.34
1
4000
gender
-0.05
0.49
-1.01
0.9
1
4000
allergy
0.88
0.48
-0.06
1.83
1
4000
Prevalence Probability
severe wheezing
1.8
0.6
0.61
2.99
1
4000
medication use
5.34
0.68
4.11
6.8
1
2200
Baseline health status effect
-0.48
0.34
-1.15
0.16
1
4000
intercept
-3.43
0.54
-4.52
-2.39
1
4000
hispanic
0.21
0.47
-0.78
1.07
1
4000
black
-0.03
0.96
-2.2
1.59
1
4000
asian
0.34
0.69
-1.11
1.6
1
3400
others+mixed
-0.87
1.1
-3.23
0.97
1
4000
age
-0.9
0.39
-1.75
-0.23
1
540
gender
0.27
0.39
-0.5
1.04
1
750
allergy
1.07
0.4
0.28
1.86
1
750
Current wheezing
0.84
0.46
-0.12
1.69
1
4000
Family asthma history
0.98
0.42
0.19
1.81
1
4000
Ozone
-0.03
0.02
-0.06
0.00
1
4000
Number of Sports
-0.62
0.78
-2.35
0.69
1
2000
Ozone*Number of Sports
Latent lung function transition effect on latent
asthma
Misclassification Probability
0.77
1.21
-1.99
2.82
1
4000
-0.17
0.21
-0.59
0.25
1
2300
intercept
-4.37
0.42
-5.21
-3.59
1
340
Latent asthma
5.4
0.68
4.12
6.73
1
1300
age
-0.02
0.22
-0.53
0.36
1
520
Current wheezing
0.96
0.46
0.07
1.9
1
2200
-0.08
0.43
-0.9
0.8
1.01
230
Transition Probability
Above HS
When Latent True Asthma=0
9
1.33
0.6
0.17
2.5
1.01
960
0.15
0.31
-0.43
0.8
1
4000
age
1.55
0.04
1.47
1.64
1
2300
height
28.08
1.23
25.7
30.47
1
3800
gender
0.13
0.12
-0.1
0.37
1.01
4000
NO2
-0.01
0
-0.02
0
1
4000
Latent lung function transition effect on itself
Latent asthma transition effect on latent lung
function
Lung Function Measurements Modeling
0.87
0.02
0.83
0.92
1
4000
-0.24
0.08
-0.4
-0.08
1
2900
intercept of lnfev
7.62
0.01
7.6
7.63
1
1000
intercept of lnfef
7.21
0.01
7.19
7.23
1
4000
intercept of lnmef
7.72
0.01
7.71
7.74
1
2300
Latent lung function effect on lnfev
0.07
0
0.07
0.07
1
1400
Latent lung function effect on lnfef
0.08
0
0.08
0.09
1
2900
Latent lung function effect on lnmef
0.07
0
0.07
0.08
1
1000
Subject-level random effect in lnfev
0.06
0
0.05
0.07
1.01
4000
Subject-level random effect in lnfef
0.22
0.01
0.2
0.23
1
1900
Subject-level random effect in lnmef
0.15
0.01
0.14
0.16
1
4000
Town-level random effect in latent asthma
0.29
0.1
0.15
0.54
1
4000
Residual in lnfev
0.03
0
0.03
0.03
1
1100
Residual in lnfef
0.14
0
0.13
0.14
1
3600
Residual in lnmef
0.09
0
0.09
0.09
1
4000
When Latent True Asthma=1
interaction b/w age and True Asthma
Latent Lung Function Modeling
Standard Deviation (S.D.)
DIC
-4402.3
Brooks-Gelman PSRF
1.0
*: PSRF: potential scale reduction factor.
10
Appendix E: Prior Distribution Sensitivity Analysis
Table S.3: Results of DLTM-MHMM in Southern California Children Health Study using alternative
uniform prior distribution instead of normal prior distribution.
Effect Estimate (95% CI)
Prevalence Probability
Age
Allergy
Severe wheezing
Transition Probability
Age
Allergy
Family Asthma History
-0.89(-1.92,-0.21)
1.00(0.14,1.79)
1.04(0.18,2.00)
Latent lung function transition effect on latent asthma
-0.11(-0.54,0.38)
0.24(-0.81,1.29)
0.91(-0.01,1.87)
1.7(0.49,2.91)
Misclassification Probability
Current wheezing
Above HS
When Latent True Asthma=0
When Latent True Asthma=1
Latent Lung Function Modeling
Age
Height
NO2
1.21(0.25,1.53)
-0.09(-0.97,0.83)
1.28(0.11,2.48)
1.56(1.48,1.64)
28.51(26.15,30.85)
Latent lung function transition effect on itself
Latent asthma transition effect on latent lung function
Lung Function Measurements Modeling
Latent lung function effect on FEV1
Latent lung function effect on FEF75
-0.01(-0.02,0)
0.87(0.83,0.92)
-0.25(-0.4,-0.09)
0.07(0.07,0.07)
0.08(0.08,0.09)
0.07(0.07,0.08)
Latent lung function effect on MMEF
11
Appendix F: Posterior Predictive Checks
Time 1
8.0
6.8
7.2
7.6
Observed lnFEV
7.6
7.2
Observed lnFEV
8.0
Time 2
7.2
7.4
7.6
7.8
8.0
7.2
7.6
7.8
Time 3
Time 4
Observed lnFEV
8.0
7.8
7.6
7.6
7.8
8.0
8.2
7.4
Predicted lnFEV
8.0
8.2
8.2
8.4
7.4 7.6 7.8 8.0 8.2 8.4
Predicted lnFEV
7.4
Observed lnFEV
7.4
7.4
Predicted lnFEV
8.2
7.0
7.6
7.8
8.0
Predicted lnFEV
Figure S.1: QQ-plots comparing posterior predicted values to observed values for the logtransformed lung function FEV at different times of visit.
12
Observed lnFEF
7.5
7.0
6.5
6.0
6.5
7.0
7.5
8.0
6.0
7.0
7.5
8.0
Predicted lnFEF
Time 3
Time 4
7.0
7.5
Observed lnFEF
8.0
Predicted lnFEF
6.5
Observed lnFEF
6.5
6.5
7.0
7.5
8.0
6.0 6.5 7.0 7.5 8.0 8.5
Observed lnFEF
6.0
6.0 6.5 7.0 7.5 8.0 8.5
Time 2
8.0
Time 1
6.5
Predicted lnFEF
7.0
7.5
8.0
8.5
Predicted lnFEF
Figure S.2: QQ-plots comparing posterior predicted values to observed values for the logtransformed lung function FEF at different times of visit.
13
Time 2
7.5
7.5
8.0
7.0
7.5
8.0
Predicted lnMEF
Time 3
Time 4
8.0
8.5
8.0
7.5
7.0
Observed lnMEF
8.0
7.5
7.5
7.0
Predicted lnMEF
8.5
8.5
Predicted lnMEF
7.0
Observed lnMEF
7.0
8.0
8.5
7.0
8.5
6.5
7.0
Observed lnMEF
8.0
7.5
7.0
6.5
Observed lnMEF
8.5
Time 1
7.5
8.0
8.5
Predicted lnMEF
Figure S.3: QQ-plots comparing posterior predicted values to observed values for the logtransformed lung function MMEF at different times of visit.
Table S.4: Comparison of predicted and observed asthma at different times of visit.
Observed Asthma vs. Predicted Asthma
Equal
Not Equal
Accuracy
Time 1
637
6
99%
Time 2
637
6
99%
Time 3
631
12
98%
Time 4
622
21
97%
14