1. Mixed latent Markov models for longitudinal multiple diagnostics data with an application to salmonella in Malawi
Marc Henrion, Angeziwa Chirambo, Tonney S. Nyirenda, Melita Gordon
Malawi – Liverpool - Wellcome Trust Clinical Research Programme
Liverpool School of Tropical Medicine
JSM Vancouver, Canada
30 July 2018
Don’t talk too much here!

2. Latent Markov Models
Latent process, observed outcome variables at each time point
Conditional on latent state, outcome variables assumed independent
Salmonella data from Malawi:
longitudinal data on multiple (binary) diagnostic tests
no gold standard D1 D2
y1,1
y2,1
yk,1 … DT
y1,2
y2,2
yk,2
y1,T
y2,T
yk,T
TP1
TP2
TPT-1
CRP IP
MEASUREMENT MODEL
STRUCTURAL MODEL
LMM = latent variable model
Extends LCA to longitudinal data
Structural model + measurement model
Completely specified by IP, TP and CRP
To keep maths tractable, typically assume independence of outcomes conditional on latent state
Application to Salmonella data:
5 tests, 4 molecular tests, reference stool culture
Conditional independence unlikely to hold: 4 tests use same PCR technology and 2-by-2 use the same primers

3. Relaxing the conditional independence assumption
𝑃 𝒀=𝒚;ϑ = 𝑖=1 𝑛 𝑑 𝑖 𝑃( 𝐷 𝑖 (1) = 𝑑 𝑖 (1) ) 𝑡=2 𝑇 1− 𝜏 01,𝑡 (1− 𝑑 𝑖 (𝑡) )(1− 𝑑 𝑖 (𝑡−1) ) 𝜏 01,𝑡 𝑑 𝑖 𝑡 (1− 𝑑 𝑖 (𝑡−1) ) 1− 𝜏 11,𝑡 (1− 𝑑 𝑖 (𝑡) ) 𝑑 𝑖 (𝑡−1) 𝜏 11,𝑡 𝑑 𝑖 (𝑡) 𝑑 𝑖 (𝑡−1)
𝑚=1 𝑘 𝜑 10,𝑘 𝑡 𝑦 𝑖𝑘 (𝑡) 1− 𝑑 𝑖 (𝑡) 1− 𝜑 10,𝑘 𝑡 1− 𝑦 𝑖𝑘 (𝑡) 1− 𝑑 𝑖 (𝑡) 𝜑 11,𝑘 𝑡 𝑦 𝑖𝑘 (𝑡) 𝑑 𝑖 𝑡 1− 𝜑 11,𝑘 𝑡 1− 𝑦 𝑖𝑘 (𝑡) 𝑑 𝑖 (𝑡)
Initial state probabilities
Conditional response probabilities
𝜑 𝑦𝑑,𝑚 =𝑃 𝑌 𝑚 =𝑦 | 𝑠𝑡𝑎𝑡𝑒=𝑑
Transition probabilities
Sum over all possible latent state combinations
Product over all patients
Basic LMM
Now add random effect to CRPs: 𝜑 𝑦𝑑,𝑖,𝑚 = 𝑒𝑥𝑝 𝛼 𝑑,𝑚 + 𝛽 𝑚 𝑍 𝑖 1+𝑒𝑥𝑝 𝛼 𝑑,𝑚 + 𝛽 𝑚 𝑍 𝑖 , 𝑍 𝑖 ~𝑁 0, 𝜎 2
Bayesian approach
Frequentist much harder.
Principled way of dealing with missing values.
Other authors have considered mixed LMMs, but typically consider random effects in the latent process / structural model.
Basic LMM likelihood: product over sum of latent states combinations of product of IP, TP and CRPs
CRP term further factorises into product over outcome variables – this is what keeps the maths tractable
Add subject specific random effects using a logistic function– trivial to add covariates into the CRPs
Now you get large integrals to compute: Bayesian approach much easier, also better for missing data
Could do similar extension to the transmission probas – but this has been done before

4. Simulations Assess convergence, identifiability issues, …
Converges (up to label switching)
Compare performance of basic and mixed LMMs: DIC, …
basic LMM mixed LMM
Simulations to check:
Convergence, identifiability issues
Compare performance: DIC, bias of MAP estimates, importance of prior, …
Results:
Yes, identifiable and converges – up to label switching typical for Bayesian mixture models; easily fixed after running MCMC
Less biased MAP (or other) estimates

5. Application to Malawi salmonella data
4 new molecular PCR tests + stool culture
60 patients observed monthly for 12 months
Results
TTR primer PCR test achieves best sensitivity/specificity trade-off
Data too sparse for time heterogeneous and mixed LMMs despite being biologically more plausible
One specific PCR test is best: TTR
Figure shows MAP estimates with 95% credible, highest posterior density intervals
Unfortunately [embarrassingly?], even though this data motivated the model development, the more complex models: mixed, time heterogeneous do not converge [further work?]
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