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J Am Stat Assoc. Author manuscript; available in PMC 2011 April 4.

Published in final edited form as:

J Am Stat Assoc. 2010 December; 105(492): 1310–1322.

doi: 10.1198/jasa.2010.ap09581PMCID: PMC3070363

NIHMSID: NIHMS188741

Yang Yang,^{1} M. Elizabeth Halloran,^{1,}^{2} Michael J. Daniels,^{3} Ira M. Longini, Jr.,^{1,}^{2} Donald S. Burke,^{4} and Derek A. T. Cummings^{5}

See other articles in PMC that cite the published article.

In seasonal influenza epidemics, pathogens such as respiratory syncytial virus (RSV) often co-circulate with influenza and cause influenza-like illness (ILI) in human hosts. However, it is often impractical to test for each potential pathogen or to collect specimens for each observed ILI episode, making inference about influenza transmission difficult. In the setting of infectious diseases, missing outcomes impose a particular challenge because of the dependence among individuals. We propose a Bayesian competing-risk model for multiple co-circulating pathogens for inference on transmissibility and intervention efficacies under the assumption that missingness in the biological confirmation of the pathogen is ignorable. Simulation studies indicate a reasonable performance of the proposed model even if the number of potential pathogens is misspecified. They also show that a moderate amount of missing laboratory test results has only a small impact on inference about key parameters in the setting of close contact groups. Using the proposed model, we found that a non-pharmaceutical intervention is marginally protective against transmission of influenza A in a study conducted in elementary schools.

In seasonal influenza epidemics, pathogens such as respiratory syncytial virus (RSV) often co-circulate with influenza and cause influenza-like illness (ILI) in human hosts (Fleming *et al*., 2003). Consequently, ILI alone is not a reliable indicator for the infection status of influenza. Ideally, ILI-triggered specimen collection and laboratory tests are used to ascertain the responsible pathogen. However, test results may be incomplete because of either budgetary limits or administrative difficulty. Inference may be biased if the missing infection status is not appropriately imputed or integrated out (Rubin, 1976; Little and Rubin, 2002). In situations with small close contact groups such as households, the contact groups of individuals with missing infection status may be excluded from the analysis without invalidating the inference if there is no between-group transmission and assuming the data are missing at random (MAR). By MAR we mean that whether a subject with ILI is tested or not is independent of his/her infection status given the observed data. However, with larger contact groups such as schools, it is impractical to discard a contact group because a few cases are missing laboratory test results.

The issue of incomplete laboratory test results has been addressed in the setting of influenza vaccine studies with validation sets. In these studies, surveillance cultures were obtained from a small proportion of study subjects. Under the assumption of MAR, the mean score approach can be used to estimate the vaccine efficacy (VE) (Halloran *et al*., 2003). In the situation of nonignorable missingness, e.g., the vaccine reduces symptom severity but cases with severe symptoms are likely to be over-sampled, an informative prior distribution can be imposed on the selection bias of the validation set to estimate the VE (Scharfstein *et al*., 2006). These two methods are unconditional on individual-level exposure to the risk of infection and are suitable for cumulative incidence or time-to-event (Halloran *et al*., 2007) data with categorical covariates. When individual-level exposure to infection is considered, missing data impose a greater challenge because of both the dependence among individuals and the continuous nature of the individual-level exposure history.

To account for individual exposure to infection and missingness in infection status, we propose a Bayesian competing-risk model for multiple co-circulating pathogens under certain pathogenicity and interference assumptions. The use of MCMC for posterior computation in the setting of transmission of infectious diseases can be traced back to the late 1990s (Gibson, 1997; Gibson and Renshaw, 1998; O’Neill *et al*., 2000; Auranen *et al*., 2000). Recent applications to influenza include analyses of household data with illness onset dates (Cauchemez *et al*., 2004) and asymptomatic infections identified by laboratory tests (Yang *et al*., 2008). However, none of this methodology has explicitly addressed missingness in infection status. For an ILI episode with unknown infection status, the evaluation of *p*(infection with influenza | ILI is untested) requires knowledge about *p*(ILI is untested | infection with a non-influenza pathogen). Therefore, it seems natural to simultaneously model the spreading of both influenza and non-influenza pathogens in the framework of competing risks. The performance of the proposed model and the effect of missing data on the inference about key parameters, such as the transmissibility of pathogens and the efficacies of an intervention, are evaluated in simulation studies. The model is used to analyze a recent study of non-pharmaceutical intervention (NPI) against influenza illness conducted in elementary schools in Pittsburgh, Pennsylvania, in the United States.

Consider a prospective clinical study to evaluate the efficacy of an intervention against influenza that is implemented in a collection of close contact groups. These groups could be households in a community or schools in a school district. The intervention could be a vaccine, an antiviral agent or a non-pharmaceutical program. Let *N* denote the total number of individuals under study. Starting from day 1, *M* different pathogens including influenza are co-circulating in the community, each imposing a hazard of community-to-person infection on susceptibles. Individuals infected with any of the pathogens can initiate within-group transmission of the same pathogen. Let day *T* be the stopping time of the study.

During the study period, we observe sequential ILI onset times in infected subjects. The observation is made on a daily basis with an integer day *t* corresponding to the continuous time scale (*t* − 1, *t*]. Each individual may have no episodes, a single episode, or multiple episodes of ILI. Let **_{i}** = (

While transmissibility of the pathogens, primarily influenza, and intervention efficacies are the major quantities of inferential interest, our parameters for transmissibility are closely related to how we define the infectious period, part of the natural history of disease. For a clear presentation of the model, we first introduce assumptions about the natural history of disease that has the following elements: the incubation period (time from infection to ILI onset), the latent period (time from infection to infectiousness onset), the infectious period for each pathogen, and the interference, or competing risk, mechanism among pathogens. The parameterization of the natural history is chosen to reduce the need for strong empirical information that may not be available for newly emerging pathogens, for example, the novel influenza A(H1N1) virus. Figure 1 presents a diagram for the type of natural history under consideration.

A possible scenario of the natural history of two pathogens that infect the same individual. (a) two ILI onsets are observed at _{i1} and _{i2} for individual i. The lab-test result v_{i1} (v_{i2}) indicates that the responsible **...**

Without loss of generality, we assume the actual infection and symptom onset occur at the end of the days * _{ik}* and

While the data are generally observed on a discrete-time basis, it is more convenient to consider the infectious period in continuous-time. Yang *et al*. (2009) use a beta density, *f _{beta}*(·|

Within a host, the infection by one pathogen may interfere with the infection by another pathogen via two immunological mechanisms: (1) a certain level of specific immunity to a spectrum of pathogens that share similar antigenic structures with the invaded pathogen, and (2) non-specific immunity activated by the innate and/or the complement immune system to a much broader range of pathogens. Specific immunity lasts relatively long, e.g., two influenza strains of the same type rarely infect the same person in the same season, whereas non-specific immunity often lasts for days. Epidemiological effects of biological interference among pathogens have been previously studied in simulations (Ackerman *et al*., 1990), but its implications for statistical inference have, to our knowledge, never been addressed.

We assume infection with pathogen *m* provides immunity against pathogen *n* from the time of infection up to and including the ILI onset day and for an additional period of *d _{mn}* days beyond. After that, the immunity against pathogen

Let *ω _{m}* be the constant hazard of community-to-person transmission for pathogen

$${\lambda}_{j\to i}(m,t)={\delta}_{i}(m,t)exp\{{\mathit{\beta}}_{S,m}^{\prime}{\mathit{x}}_{i}(t)+{\mathit{\beta}}_{I,m}^{\prime}{\mathit{x}}_{i}(t)\}{\gamma}_{m}{f}_{\mathit{beta}}(\frac{t-{\widehat{t}}_{jk}}{{\mathrm{\Delta}}_{m}}{a}_{m},{b}_{jk}),$$

(1)

where *β** _{S.m}* and

$${\lambda}_{i}(m,t)={\delta}_{i}(m,t)exp\{{\mathit{\beta}}_{S,m}^{\prime}{\mathit{x}}_{i}(t)\}{\omega}_{m}+\sum _{\begin{array}{c}j{s}_{i}& j\ne i\end{array}}$$

(2)

The probability of escaping infection with pathogen *m* during day *t* is given by exp {− Λ* _{i}*(

$${P}_{i}(m,t)={\int}_{t-1}^{t}\{{\lambda}_{i}(m,\tau )\underset{exp\{-{\int}_{t-1}^{\tau}{\lambda}_{i}(m,\epsilon )d\epsilon \}}{\overset{}{l=0M}}\}d\tau .$$

(3)

The evaluation of (3) requires numerical integration and hence substantially increases the computational burden. If *λ _{i}*(

$${P}_{i}(m,t)=\frac{{\mathrm{\Lambda}}_{i}(m,t)}{{\sum}_{l=1}^{M}{\mathrm{\Lambda}}_{i}(l,t)}(1-\underset{exp\{-{\mathrm{\Lambda}}_{i}(l,t)\}}{\overset{}{l=1M}}).$$

(4)

The above expression implies an infection process that first selects infection from any pathogen via the probability
$1-{l=1M}_{exp\{-{\mathrm{\Lambda}}_{i}(l,t)\}}^{}$ and then chooses pathogen *m* via the probability
${\mathrm{\Lambda}}_{i}(m,t)/{\sum}_{l=1}^{M}{\mathrm{\Lambda}}_{i}(l,t)$. This mechanism of competition is computationally affordable. Consequently, although the true hazards are not constant over time, we assume (4), not (3), is the true mechanism of competition.

Let ** ψ** = {

$$p({\widehat{\mathit{t}}}_{i},{\mathit{v}}_{i}\mathit{\psi},{R}_{i})=(\underset{(\underset{{P}_{i}({v}_{ik},{\widehat{t}}_{ik})}{\overset{}{k=1{K}_{i}}}),}{t{\widehat{\mathit{t}}}_{i}\underset{exp\{-{\mathrm{\Lambda}}_{i}(m,t)\}}{\overset{}{m=1M}})}$$

(5)

the individual-level probability for the incubation period is

$$p({\stackrel{~}{\mathit{t}}}_{i}{\widehat{\mathit{t}}}_{i},{\mathit{v}}_{i},\mathit{q})=\underset{{q}_{{v}_{ik}({\stackrel{~}{t}}_{ik}-{\widehat{t}}_{ik})}}{\overset{.}{k=1{K}_{i}}}$$

(6)

The conditioning on *R _{i}*, the exposure history, in (5) is for notational convenience only, because the infection status of person

The following flat priors are assumed:

$$\begin{array}{ll}{\omega}_{m}~\text{Uniform}({10}^{-6},10),\hfill & {\gamma}_{m}~\text{Uniform}({10}^{-6},10),\hfill \\ {a}_{m}~\text{Uniform}(2,10),\hfill & {\mathit{q}}_{m}~\text{Dirichlet}({\mathbf{1}}_{{H}_{m}\times 1}),\hfill \end{array}$$

(7)

*m* = 1, …, *M*. A flat prior with a wide range is adopted for most parameters to allow for assessment of the posterior with minimal prior information. For transmission parameters, the boundaries are chosen to be wide and contain historic estimates for *ω _{m}* and

We assume that the missingness of the laboratory test results is ignorable (MAR), i.e., the probability of *u** _{i}* does not depend on

Define ** = {*** _{i}* :

$$p(\stackrel{~}{\mathit{t}},\mathit{v},\stackrel{~}{\mathit{t}},\mathit{\psi},\mathit{q})=(\pi (\mathit{\psi})\underset{\{p({\widehat{\mathit{t}}}_{i},{\mathit{v}}_{i}\mathit{\psi},{R}_{i})\}}{\overset{)}{i=1N}}(\pi (\mathit{q})\underset{\{p({\stackrel{~}{\mathit{t}}}_{i}{\widehat{\mathit{t}}}_{i},{\mathit{v}}_{i},\mathit{q})\}}{\overset{)}{i=1N}}$$

(8)

The factorization in (8) suggests that, conditioning on ** and **** v**,

$$p(\stackrel{~}{\mathit{t}},\mathit{v},\widehat{\mathit{t}},\mathit{\psi})=\int p(\stackrel{~}{\mathit{t}},\mathit{v},\widehat{\mathit{t}},\mathit{\psi},\mathit{q})d\mathit{q}=(\underset{B({\mathit{C}}_{m}+\mathbf{1})}{m})(\pi (\mathit{\psi})\underset{\{p({\widehat{\mathit{t}}}_{i},{\mathit{v}}_{i}\mathit{\psi},{R}_{i})\}}{\overset{)}{i=1N}},$$

(9)

where
$B({\mathit{C}}_{m}+\mathbf{1})={h=1{H}_{m}}_{\mathrm{\Gamma}({C}_{mh}+1)/\mathrm{\Gamma}\left({\sum}_{h=1}^{{H}_{m}}({C}_{mh}+1)\right)}^{}$ is the multinomial beta function with respect to the vector *C** _{m}* +

$$p(\mathit{\psi}\stackrel{~}{\mathit{t}},\mathit{v},\widehat{\mathit{t}})=p(\mathit{\psi}\mathit{v},\widehat{\mathit{t}})p(\mathit{v},\widehat{\mathit{t}},\mathit{\psi})=\pi (\mathit{\psi})\underset{\{p({\widehat{\mathit{t}}}_{i},{\mathit{v}}_{i}\mathit{\psi},{R}_{i})\}.}{\overset{}{i=1N}}$$

(10)

The posterior distribution of ** ψ** given the observed data,

All parameters in ** ψ** are sampled using a random walk Metropolis-Hastings algorithm. We describe the sampling procedure for

The data augmentation step involves sampling all infection times and missing pathogen types, if any, per individual simultaneously. Let {(
${\mathit{v}}_{i}^{(l)},{\widehat{\mathit{t}}}_{i}^{(l)}$) : *l* = 1, …, *L _{i}*} be the collection of candidate values of (

$$\begin{array}{l}p({\mathit{v}}_{i}={\mathit{v}}_{i}^{(l)},{\widehat{\mathit{t}}}_{i}={\widehat{\mathit{t}}}_{i}^{(l)}{\stackrel{~}{\mathit{t}}}_{i},\mathit{\psi},\{{\mathit{v}}_{j},{\widehat{\mathit{t}}}_{j},{\stackrel{~}{\mathit{t}}}_{j}:j{s}_{i},j\ne i\})=\frac{\{{m}_{B({\mathit{C}}_{m}^{(l)}+\mathbf{1})}\}p({\mathit{v}}_{i}^{(l)},{\widehat{\mathit{t}}}_{i}^{(l)}\mathit{\psi},{R}_{i})\{{\begin{array}{c}j{s}_{i}& j\ne i\end{array}p({\mathit{v}}_{j},{\widehat{\mathit{t}}}_{j}\mathit{\psi},{R}_{j}^{(l)})\}}_{}{\sum}_{k=1}^{{L}_{i}}\{{m}_{B({\mathit{C}}_{m}^{(k)}+\mathbf{1})}\}p({\mathit{v}}_{i}^{(k)},{\widehat{\mathit{t}}}_{i}^{(k)}\mathit{\psi},{R}_{i})\{{\begin{array}{c}j{s}_{i}& j\ne i\end{array}p({\mathit{v}}_{j},{\widehat{\mathit{t}}}_{j}\mathit{\psi},{R}_{j}^{(k)})\}}_{},l=1,\dots ,{L}_{i}.}{}\end{array}$$

(11)

Note that because *R _{j}* for

We conduct simulations to evaluate the performance of the model and the impact of missing laboratory test results under several different settings. We simulate epidemics in a population composed of 100 households, each of size 5. Prior to the epidemics, the individuals are randomized to either vaccine or placebo, and vaccine status is the only covariate for simplicity. Let *z _{i}* denote the intervention arm, 1 for vaccine and 0 for control. Then, in equations (1) and (2), we have
${\mathit{\beta}}_{S,m}^{\prime}{\mathit{x}}_{i}(t)={z}_{i}log({\theta}_{m})$ and
${\mathit{\beta}}_{I,m}^{\prime}{\mathit{x}}_{j}(t)={z}_{j}log({m}_{)}$. The parameters

$${\theta}_{m}~\text{Uniform}({10}^{-2},{10}^{2}),\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}{m}_{~}$$

We start by evaluating the performance of the proposed model when the number of co-circulating non-primary pathogens is either correctly specified or mis-specified. We simulate epidemics of three pathogens (*M*=3) for this purpose, and index the primary pathogen by 1 and the non-primary pathogens by 2 and 3. The primary pathogen has transmission parameters similar to that of interpandemic influenza, with about 0.26 for the community probability of infection (CPI) during the 100-day epidemic and 0.19 for the secondary attack rate (SAR) (Yang *et al*., 2006). To ensure a sufficient number of cases, the two non-primary pathogens have parameters not much different from those of the primary pathogen. When simulating the epidemics, we set *d _{mn}* = 7 for

We generate missing data under an assumption of MAR. In particular, we let the proportion of missing laboratory test results depend on the ILI onset day * _{ik}* : 50% for
${\stackrel{~}{t}}_{ik}<{\scriptstyle \frac{T}{2}}$ and 10% otherwise. Misspecification is introduced by analyzing the simulated data as if there were only two pathogens (not three), the primary one and a hypothetical non-primary one indexed by 1 and 2 respectively. We have to assume in the analysis that the non-primary pathogen does not induce specific immunity to itself, so that the misspecified model is compatible with possible situations of three ILI episodes per individual in the simulated data. For illustration, we set

The true parameter settings and simulation results for the primary pathogen are shown in Table 1. The estimates are very similar under the correctly specified and the misspecified scenarios, suggesting that the model is tolerant to this type of model misspecification at least when *M* is small. The results also imply that the transmission hazards, *ω*_{1} and *γ*_{1}, and the vaccine effect in reducing susceptibility, *θ*_{1}, can be estimated with little or no bias even for a moderate sample size. The decreased information for _{1} as compared to *θ*_{1} in the household data has been reported elsewhere (Yang *et al*., 2006). The last two columns of Table 1 show that increasing the sample size generally leads to improved inference for all parameters, particularly for _{1} and *α*_{1}. In addition, given an equal total sample size, having larger contact groups is preferable to having more contact groups in terms of both reducing bias and narrowing the credible sets (CS).

Average posterior summaries over 100 simulated epidemics of 3 co-circulating pathogens with laboratory test results missing for 30% of the ILI episodes. Estimates for the primary pathogen (m=1) are presented. Comparison is made between the scenarios that **...**

It is also of interest to investigate how the proportion of missing laboratory test results would alter inference about the primary pathogen, and whether such an effect is related to the number co-circulating pathogens. We first simulate pathogens with identical transmission characteristics and natural history. This is because any difference in parameters can help to differentiate the pathogens from each other and will thus limit the impact of missing laboratory results. In addition, we assume specific immunity within and between pathogens for simplicity. The assumption of within-pathogen specific immunity is also employed for similarity of the primary pathogen to influenza. In this simulation study, we let *ω _{m}* = 0.002,

Comparison of posterior estimates for parameters by the proportion of missing laboratory test results and the number of co-circulating pathogens. Bars with circles (triangles) denote average 95% credible sets, the middle point representing the average **...**

Overall, when two pathogens are co-circulating, a missingness proportion of 70% or less does not increase the uncertainty about the parameters by much. Dramatic increases in the length of 95% CS are observed for the efficacy parameters *θ*_{1} and _{1} when the proportion of missingness reaches 90%. Even more dramatic changes in the two efficacy parameters are seen at 95% missingness, but only moderate increases are observed for the transmission rates. The more co-circulating pathogens there are, the more sensitive the parameters are to the proportion of missingness. When three pathogens are co-circulating, a substantial rise in uncertainty is observed for _{1} at the missingness proportion of 70% and for both *θ*_{1} and _{1} at 90% and higher. In addition, there is a large increase in the length of the 95% CS for *γ*_{1} at 95% missingness. The reduced information about _{1} in the simulated epidemics compared to other parameters accounts for its extreme level of sensitivity. The pattern of larger bias and higher uncertainty with increasing missingness is also obvious for all the parameters in Figure 2.

Figure 3 compares the variability in parameter estimates in response to increasing the proportion of missing laboratory test results between two co-circulating pathogens with very different transmission rates. The weaker (stronger) pathogen has *ω*_{1} = 0.0015 (0.0025) and *γ*_{1} = 0.02 (0.04), corresponding to a CPI of 0.14 (0.22) and a SAR of 0.13 (0.24). We index the weaker (stronger) pathogen by 1 (2) in the subscripts. The estimates for all parameters are presented on the log scale. As expected, credible sets are shorter for the stronger pathogen that generates more cases. The parameter estimates associated with the weaker pathogen are much more sensitive to the proportion of missing laboratory results compared to those of the stronger pathogen. However, for all parameters except for _{1}, uncertainty does not increase dramatically until the missingness reaches 90% even for the weaker pathogen. Similar to Figure 2, a dramatic increase in uncertainty and bias may occur for the transmission rate *γ*_{1} when the information is decreased by an extremely high missingness rate. Among the key parameters, *ω*_{1} and _{1} are the parameters least and most affected by missingness respectively. Figure 3 also suggests that a fairly transmissible pathogen (with a SAR of 0.24) can provide nearly unbiased estimates for _{1}, and thus the VE* _{I}*, even at high missingness proportions.

The Pittsburgh Influenza Prevention Project (PIPP) is a prospective randomized study of the effectiveness of non-pharmaceutical interventions (NPI) conducted in ten public elementary schools in the city of Pittsburgh. The five schools in the intervention arm were provided with NPI training program including hand hygiene, hand etiquette, and cover your cough behaviors, as well as supplies of alcohol-based hand sanitizer for all classrooms. In addition, parents received educational materials about NPIs. Prior to and during the influenza season, absences were monitored and absentees were screened by phone for the presence of ILI. During the season, home visits were made to collect specimens for laboratory tests (rapid test and PCR). ILI is defined as fever (≥ 100°F) plus cough or sore throat.

The 2007–2008 influenza season started near the end of December and ended around the end of April. The epidemic curves for overall and pathogen-specific ILI episodes are shown in Figure 4. Most ILI episodes that occurred prior to the influenza epidemic were not lab-tested by design. We restrict all analyses to the period of 12/31/2007 – 4/18/2008, during which a total of 380 ILI episodes were observed in 352 out of 3959 students. Further breakdown of these numbers by pathogen type and laboratory test is given in Table 2. The influenza B epidemic occurred slightly later than influenza A, and only a single person was infected by both influenza types. We assume that ILI episodes with negative laboratory results for both influenza types were caused by a single non-influenza pathogen. The distribution of pathogen types among infected students does not differ significantly across schools (not shown).

Epidemic curves of ILI onsets in the PIPP study during the 2007–2008 influenza season. (a) All ILI onsets; (b) Influenza A; (c) Influenza B; (d) ILI episodes tested negative for influenza.

Summary of outcomes during 12/31/2007–4/18/2008 by intervention arm in the PIPP study of NPI effectiveness in the 2007–2008 winter season. Five schools were in each intervention arm.

We denote the non-influenza pathogen, influenza A and influenza B as pathogens 1, 2 and 3. For a study of large contact groups, it is reasonable to assume that the community-to-person infection rate is time dependent and proportional to the number of cases in the community which may be represented by the observed number of ILI onsets in the schools. Let *T*_{0}=12/31/2007 and *T*_{1}=4/18/2008 be the starting and stopping dates. We assume each student in the study population was susceptible to the three pathogens at *T*_{0}, unless infection with either influenza A or influenza B was lab-confirmed before *T*_{0} in the same season. If vaccination and infection history before the epidemic season were known at the individual level, prior immunity could be built into the model. Let *C _{m}*(

The observed numbers of influenza A and B cases in this study are relatively low, on average about 5 cases per school. To avoid non-identifiability of parameters, we assume the two types of influenza share the same natural history parameters, i.e., *a*_{2} = *a*_{3} and *q*_{2}* _{l}* =

$$\mathit{D}=\left\{\begin{array}{ccc}3& 3& 3\\ 3& \infty & 7\\ 3& 7& \infty \end{array}\right\},$$

(12)

which reflects that infection with influenza A (pathogen 2) or influenza B (pathogen 3) generates specific immunity to the same kind of infection. For infections with the non-influenza pathogen, the short-term nonspecific immunity against all pathogens is the only choice compatible with the data. The exact duration of non-specific immunity is arbitrarily chosen and subject to sensitivity analysis.

In an initial analysis assuming that all schools shared the same *ω _{m}* for all

Geographic location of schools in the PIPP study downloaded from google.com. Schools labeled red (blue) are in the active intervention (control) arm. The four schools in the box, three in the intervention arm and one in the control arm, have the highest **...**

As the intervention was implemented at the school level, only the product *ξ _{m}* =

- Model I :
*ρ*= 1, and*η*= 1. - Model II :
*ρ*is unknown, but*η*= 1. - Model III : both
*ρ*and*η*are unknown.

A convenient criterion for model selection in the Bayesian paradigm is the deviance information criterion (DIC) (Spiegelhalter *et al*., 2002). However, minimal exploration has been done with the DIC in missing data problems (Celeux *et al*., 2006; Daniels and Hogan, 2008). Two proposals for the DIC in models with missing data are the DIC based on the observed data likelihood and the DIC based on the full data likelihood. The former is very difficult to compute for the models proposed here as the observed data likelihood cannot be expressed in closed form; the latter is computationally more tractable but has not been well studied. Alternatively, a Reversible-Jump MCMC algorithm (Green, 1995) could be used to account for uncertainty about the underlying model. Here we compare models based on posterior credible sets due to the nesting structure of the three models under consideration.

As there are 395 students for whom grade is unknown, an extra data augmentation step is necessary in Model III to incorporate these students into the analysis. Note that there are 10 schools, each with 6 grades, in the PIPP study. We assume the grade of individual *i* in school *s* follows a multinomial distribution with probability vector *ξ** _{s}* = (

$$p({G}_{i}={G}_{i}^{(l)}\mathit{\psi},\{{\mathit{v}}_{j},{\widehat{\mathit{t}}}_{j},{\stackrel{~}{\mathit{t}}}_{j}:j{s}_{i}\})=\frac{p({\mathit{y}}_{{s}_{i}}^{(l)}{\mathit{\xi}}_{{s}_{i}})\{{j{s}_{i}p({\mathit{v}}_{j},{\widehat{\mathit{t}}}_{j}\mathit{\psi},{R}_{j}^{(l)})\}}_{}{\sum}_{k=1}^{6}p({\mathit{y}}_{{s}_{i}}^{(l)}{\mathit{\xi}}_{{s}_{i}})\{{j{s}_{i}p({\mathit{v}}_{j},{\widehat{\mathit{t}}}_{j}\mathit{\psi},{R}_{j}^{(k)})\}}_{},\phantom{\rule{0.16667em}{0ex}}l=1,\dots ,6,}{}$$

(13)

if individual *i* had no ILI; otherwise, sample *G _{i}* together with (

$$\begin{array}{l}p({\mathit{v}}_{i}={\mathit{v}}_{i}^{(l)},{\widehat{\mathit{t}}}_{i}={\widehat{\mathit{t}}}_{i}^{(l)},{G}_{i}={G}_{i}^{(l)}{\stackrel{~}{\mathit{t}}}_{i},\mathit{\psi},\{{\mathit{v}}_{j},{\widehat{\mathit{t}}}_{j},{\stackrel{~}{\mathit{t}}}_{j}:j{s}_{i},j\ne i\})=\frac{\{{m}_{B({\mathit{C}}_{m}^{(l)}+\mathbf{1})}\}p({\mathit{v}}_{i}^{(l)},{\widehat{\mathit{t}}}_{i}^{(l)}\mathit{\psi},{R}_{i}^{(l)})p({\mathit{y}}_{{s}_{i}}^{(l)}{\mathit{\xi}}_{{s}_{i}})\{{\begin{array}{c}j{s}_{i}& j\ne i\end{array}p({\mathit{v}}_{j},{\widehat{\mathit{t}}}_{j}\mathit{\psi},{R}_{j}^{(l)})\}}_{}{\sum}_{k=1}^{{L}_{i}}\{{m}_{B({\mathit{C}}_{m}^{(k)}+\mathbf{1})}\}p({\mathit{v}}_{i}^{(k)},{\widehat{\mathit{t}}}_{i}^{(k)}\mathit{\psi},{R}_{i}^{(k)})p({\mathit{y}}_{{s}_{i}}^{(k)}{\mathit{\xi}}_{{s}_{i}})\{{\begin{array}{c}j{s}_{i}& j\ne i\end{array}p({\mathit{v}}_{j},{\widehat{\mathit{t}}}_{j}\mathit{\psi},{R}_{j}^{(k)})\}}_{},l=1,\dots ,{L}_{i},}{}\end{array}$$

(14)

where *L _{i}* is the number of all possible values of the triple (

The estimates for *η*, 1.91 (95% CS:0.18, 9.4), in model III do not provide strong evidence for *η >* 1, whereas those for *ρ*, 2.05 (95% CS:1.31, 3.01) in model II and 2.02 (95% CS:1.36, 2.91) in model III, do for *ρ >* 1. Therefore, we chose model II for the subsequent analyses. Table 3 presents results in the left and the right panels assuming that the RIC mode is located at the end of day * _{ik}* and of day

Analysis of the PIPP Study with influenza A and influenza B epidemics in the 2007–2008 winter season. Posterior medians and 95% credible sets are presented.

Influenza A tended to be more transmissible person-to-person than influenza B in this epidemic, the posterior median of *γ*_{2}/*γ*_{3} being 4.73 (95% CS: 0.77, 112.33). While the two pathogens generated about the same number of cases, the average length between successive ILI onsets within schools in the control arm is 4.4 days for influenza A and 9.7 days for influenza B. Consequently, influenza B cases are far less likely to be explained by secondary infections. The estimates for the SAR in Table 3 do not take into account the possibility of absence due to the disease or holidays. To obtain an estimate for the effective SAR adjusted for absence, we fixed the duration of the incubation period for influenza A and influenza B at two days, the average duration estimated for seasonal influenza. As a result, the infectious period is now fixed relative to the symptom onset day and can be represented by days 1–7 with day 2 being the symptom onset day. Let *e _{k}* be the probability of absence for day

The community-to-person infection hazard during the holidays is more than 2 times that during school days. The four schools in the south experienced twice the community-to-person hazard of influenza B compared to the six schools in the north, but the 95% CS of the difference covers zero, possibly due to the small number of cases per school.

The estimates were not sensitive to the location of the mode of the RIC, as indicated by the similarity between the two panels in Table 3. Sensitivity to the natural history parameter setting was assessed assuming (1) a longer incubation period with *H _{m}* = 4; (2) a longer maximum infectious period with Δ

We have demonstrated via a simulation study the usefulness of the Bayesian approach in estimating transmissibility of co-circulating pathogens and pathogen-specific intervention efficacies with incomplete laboratory test results. Under the assumption of ignorability (MAR), a moderate amount of missing laboratory results for ILI episodes had a very limited effect on the inference for most key parameters. The reason for such insensitivity is intuitive: uncertainty about the responsible pathogen is restricted to two or three candidates, providing far more information than not observing the ILI episode at all. This finding justifies the efforts to obtain a small validation set of laboratory test results to improve statistical inference. However, the simulation study also warns that the missing results may be a concern if the number of co-circulating pathogens is high or the number of cases of the pathogen of interest is very low.

Randomized studies of large close contact groups provide unique opportunities for estimating characteristics of the natural history of infectious pathogens. Yet both our simulation study and the data analysis showed that reasonable accuracy is difficult to obtain without a sufficiently large number of secondary cases. To minimize the required amount of information for such estimation, we have assumed that the mode of the RIC is strongly associated with symptom onset. This assumption of a known location of the RIC mode can be slightly relaxed. An additional simulation study (results not shown) suggests that allowing the mode to vary uniformly over a short period, such as (* _{ik}* − 1,

We have assumed the number of co-circulating pathogens is known in the data analysis. As shown in the simulation study, the model is robust to misspecification of the number of co-circulating pathogens, *M*, when there are few. When *M* is large and unknown, a Reversible-Jump MCMC approach could be used. However, when laboratory results are not available for two or more pathogens, one or more of the following conditions must hold for identifiability of parameters for the non-tested pathogens: (1) these pathogens differ substantially in transmissibility; (2) sufficiently many clusters are infected by each pathogen; and (3) an informative prior is assumed for *M*. In addition, the specification of the immunity matrix and/or other natural history characteristices may be difficult for the unknown pathogens.

Our analysis also demonstrates the need for careful randomization of groups in studies like PIPP. Balance in geographical distributions between study arms is often critical to minimize the effects of unmeasured risk factors that are geographically heterogeneous. In the PIPP study, it was not possible to predict before hand that influenza B would circulate more in one area of the study. Accordingly, it is also important to examine and adjust for such potential imbalance to obtain interpretable results.

In the best scenario, the ILI cases that are confirmed biologically would be a random sample, possibly stratified by covariate values. We have assumed MAR for the missing laboratory test results, which may not necessarily hold in some studies, as discussed in Scharfstein *et al*. (2006). The biased selection of severe cases for surveillance culture mentioned in the introduction is such an example. Another possible scenario is that one pathogen is highly pathogenic and more likely to cause immediate hospitalization and thus prevent the collection of the specimen. A viable approach to allow for nonignorable missingness is a pattern mixture model (Rotnitzky *et al*., 2001; Daniels and Hogan, 2008) that assumes different transmission and competing probability structures for the two data-missing patterns: laboratory test result is available vs. missing. An example is an exponential tilt relationship: *P _{i}*(

Unobserved pathogen types are updated individual by individual in our algorithm. In a cluster whose laboratory test results are completely missing, the MCMC algorithm can get stuck with a single pathogen. This situation is likely to occur if the cluster sizes are small, for example, in household transmission studies. Updating the infection status for a whole cluster at the same iteration could circumvent this problem. However, it would be necessary to specify a candidate distribution for proposing the set of infection status for all individuals in the cluster since enumerating all possible infection status, as we did for individual-level updating, is not computationally feasible even with a moderate cluster size. A possible approach to improve the mixing of the MCMC is the multiset sampler proposed in Leman *et al*. (2009), in which multisets of missing pathogen types and infection times for each cluster or the whole study population may be constructed and assigned appropriate probability densities to facilitate the move of the MCMC across potential pathogen types. The feasibility and operating characteristics of this approach in the presence of high-dimensional missing data has yet to be explored.

This work was supported by the National Institute of Allergy and Infectious Diseases grant R01-AI32042 and the National Institute of General Medical Sciences MIDAS grant U01-GM070749. We thank Donald S. Burke and Derek A. T. Cummings, the primary investigators of the PIPP study, for sharing the data with us.

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