Statistical Methods for Infectious Diseases Household Based Studies - PowerPoint PPT Presentation

Outline Background Pneumococcal Studies Statistical Methods for Infectious Diseases Household Based Studies II Lecture 11A M. Elizabeth Halloran Fred Hutchinson Cancer Research Center and University of Washington Seattle, WA, USA February 9, 2009

Outline Background Pneumococcal Studies Background Pneumococcal Studies Data structure Models and analysis Bayesian latent variable model Hidden Markov model

Outline Background Pneumococcal Studies Background Pneumococcal Studies Data structure Models and analysis Bayesian latent variable model Hidden Markov model

Outline Background Pneumococcal Studies Studies conditioning on exposure to infection: VE S (VEcol), VE I , VE T ❼ Households, partnerships, or other small transmission units ❼ − → Households assuming independence of households ❼ − → Households assumed within communities ❼ Data structure: ❼ − → final-value data ❼ − → time-to-event data ❼ − → longitudinal panel data over time (pneumococcal carriage)

Outline Background Pneumococcal Studies Table: Parameters used for measuring various effects of vaccination ∗ Comparison groups and effect Level Parameter Susceptibility Infectiousness Combined change in choice susceptibility and infectiousness Conditional on exposure: VE S , p † = 1 − p · 1 VE I , p = 1 − p 1 · VE T , p = 1 − p 11 I Transmission p · 0 p 0 · p 00 probability Study design I IIA IIB III direct indirect total overall Unconditional: VE S , IR = 1 − IR A 1 VE IIA , IR = 1 − IR A 0 VE IIB , IR = 1 − IR A 1 VE III , IR = 1 − IR A · II Incidence IR A 0 IR B 0 IR B 0 IR B · or hazard VE S ,λ = 1 − λ A 1 VE IIA ,λ = 1 − λ A 0 VE IIB ,λ = 1 − λ A 1 VE III ,λ = 1 − λ A · rate, IR, λ λ A 0 λ B 0 λ B 0 λ B · VE S , PH = 1 − e β 1 III Proport. NA NA NA hazards, PH VE S , CI = 1 − CI A 1 VE IIA , CI = 1 − CI A 0 VE IIB , CI = 1 − CI A 1 VE III , CI = 1 − CI A · IV Cumulative CI A 0 CI B 0 CI B 0 CI B · incidence ∗ From Halloran, Struchiner, Longini, Am. J. Epidemiol 1997; 146;789–803.

Outline Background Pneumococcal Studies Background Pneumococcal Studies Data structure Models and analysis Bayesian latent variable model Hidden Markov model

Outline Background Pneumococcal Studies Background Pneumococcal Studies Data structure Models and analysis Bayesian latent variable model Hidden Markov model

Outline Background Pneumococcal Studies Longitudinal (panel) carriage data ❼ Collection of transmission units, such as households or schools ❼ Carriage data at certain time intervals ❼ Relevant covariates, such as age or vaccine status ❼ Choice of analyses

Outline Background Pneumococcal Studies Cauchemez et al (2006)

Outline Background Pneumococcal Studies Background Pneumococcal Studies Data structure Models and analysis Bayesian latent variable model Hidden Markov model

Outline Background Pneumococcal Studies Statistical Models ❼ Models of transmission within units such as households or schools assumed within communities. ❼ Two types of parameters: − → rate of acquisition from the community (unconditional) − → rate of acquisition from infective within transmission unit (conditional). ❼ Covariate (vaccine) effects are modeled affecting these parameters. ❼ Vaccine effect on duration of carriage (clearance rate) can be modeled directly.

Outline Background Pneumococcal Studies Longitudinal data on pneumococcal carriage ❼ Problem: neither the acquisition or clearance times are observed. ❼ Different approaches to statistical models can deal with this problem making a variety of assumptions. ❼ Other problems: combining multiple serotype data, missing data, competition, errors in diagnosis.

Outline Background Pneumococcal Studies Background Pneumococcal Studies Data structure Models and analysis Bayesian latent variable model Hidden Markov model

Outline Background Pneumococcal Studies ❼ Auranen, Arjas, Leino, Takala (2000) � n � λ ( s ) C ( s ) ˜ � = α ( t − T i ) + β ( t − T i ) k ( t ) i k =1 ×{ 1 − C i ( t ) } µ i ( t ) ˜ = µ C i ( t ) ❼ α is rate to acquire carriage of serotype s from community ❼ β is rate of transmission from infective in family to susceptible ❼ µ is the clearance rate (no serotype specific rates) ❼ n is size of family, C ( s ) ( t ) is 0,1 indicator if individual is i carrier of serotype s at time t , C i ( t ) is indicator of any of the three serotypes ❼ Problem solved: data augmented by unobserved event times of acquiring and clearing carriage (latent variables) using MCMC methods.

Outline Background Pneumococcal Studies Cauchemez et al (2006)

Outline Background Pneumococcal Studies ❼ Cauchemez, Temime, Valleron, Varon, Thomas, Guillemot, Bo¨ elle (2006) ❼ similar approach to Auranen et al (2000), school data rather than household ❼ included serotype data, clustered by community acquisition of infection into two groups ❼ used same model for influenza household studies

Outline Background Pneumococcal Studies Background Pneumococcal Studies Data structure Models and analysis Bayesian latent variable model Hidden Markov model

Outline Background Pneumococcal Studies ❼ Auranen et al (1996) (Hib), Melegaro, et al (2004), Melegaro, et al (2007) � α i + β 1 i I 1 ( t ) + β 2 i I 2 ( t ) � Pr i ( S → C ) δ t = · δ t ( z − 1) w Pr i ( C → S ) δ t = µ i · δ t ❼ I 1 ( t ) and I 2 ( t ) are number of infected children ( < 5 yrs) and adults in household, i is age group. ❼ Problem solved: use a Markov model with 1 day intervals to analyze 28-day interval data assuming only one person can change in household per day. ❼ Vaccine parameters for susceptibility and infectiousness, also vaccine-dependent clearance rates can be added.

Outline Background Pneumococcal Studies Intensity matrix Q 000 001 010 011 100 101 110 111 α (3) α (2) α (1) 000 q 11 0 0 0 0 α (2) + β (2) α (1) + β (1) µ (3) 001 0 0 0 0 q 22 α (3) + β (3) α (1) + β (1) µ (2) 010 0 q 33 0 0 0 µ (2) µ (3) α (1) + 2 β (1) 011 0 q 44 0 0 0 α (3) + 2 β (3) α (2) + β (2) µ (1) 100 0 0 0 0 q 55 α (2) + 2 β (2) µ (1) µ (3) 101 0 0 0 q 66 0 µ (1) µ (2) α (3) + 2 β (3) 110 0 0 0 0 q 77 µ (1) µ (2) µ (3) 111 0 0 0 0 q 88 ❼ The elements on the diagonals represent the intensity of staying in the same state. ❼ The q ii = 1 − � j � = i q ij (Karlin and Taylor 1975). ❼ The element (4,8) represents the transition from state 011 to state 111.

Outline Background Pneumococcal Studies Unconditional acquisition rates ❼ Smith et al (1993) ❼ Data did not include transmission units ❼ Statistical model included unconditional acquisition rates and and clearance rates

Outline Background Pneumococcal Studies Summary ❼ Pneumococcal carriage studies fit well within the context of a general framework of vaccine effects. ❼ Transmission models with parameters conditional on exposure are important for understanding a range of effects. ❼ Not all effects rely on transmission models. ❼ The relation to pivotal trials for licensure is open.

Outline Background Pneumococcal Studies Thank You!