Bayesian Analysis of Vaccine Effectiveness against Pig Virus: Temperature Responses, Study Guides, Projects, Research of Statistics

Download Bayesian Analysis of Vaccine Effectiveness against Pig Virus: Temperature Responses and more Study Guides, Projects, Research Statistics in PDF only on Docsity! Stat 544 Final Project A Bayesian analysis of the effectiveness of different vaccines against a pig virus By Alireza Kabirian, Ming Li and Roger Zoh May 1st, 2008 1 1 Introduction A set of experiments at USDA National Animal Disease Center (located at Ames, Iowa) was conducted to study the effect of five vaccines on physiological responses of pigs infected by pseudo rabies virus around 1992. A total of 24 pigs were randomly selected and separated into six groups with four pigs in each group. Pigs at five of those groups were vaccinated with different vaccines twice, while one group was not vaccinated and served as the control group. Then all pigs were challenged four weeks after the second vaccine injection by around 105 pfu of virulent virus in a 2 ml volume, with ½ ml given in each nostril and 1 ml given orally. Body temperatures were measured daily from 3 days before the virus challenge to 11 days after the virus challenge (a total of 15 days). Body weights were also recorded twice weekly from 14 days before challenge to 21 days after challenge, which was not studied in this report. (Reference 1) The main goal of the original study was to determine the treatment effects of different vaccines through the comparison of pig’s temperature (as well as weight) response before and after the virus challenge across different groups. The non-linear and longitudinal temperature responses were not easy to be captured and analyzed with only four pigs in each group. Without an appropriate way to separate the random pig effects, there was no way to analysis the treatment effects. The extreme case of this situation was while there was only one pig in each treatment group and the control group researcher would have no chance to tell whether the temperature response difference across groups was from the vaccine effects or from the random variation of pigs (i.e. the pig effects). In this report, we apply a Bayesian method to the temperature response data and study the vaccine effects by separating the treatment effects and pig effects. Following the introduction, a section of preliminary exploration of the data suggests a suitable non- linear function family to describe the data. The model selection section proposes and compares two models. Then the analyses of the 2nd model are presented in the section of main results followed by a section of summary of conclusions. 2 3 Modeling There were maybe previous studies of similar problem or biological mechanism to roughly summarize some information about parameters in the mean response function, which can be used as priors for 1 2 3, ,β β β and 4β . Bayesian analysis may take advantage of that prior information to solve the problem of too few data in each group. But in this report, we will show even with non-informative priors; we can still get useful information with only four pigs in each group and estimate the vaccine treatment effects. Our starting model assigns each treatment group a set of parameter (1) (2) (3) (4, , ,i i i i )β β β β where i indexes for different groups (vaccine treatments), and with a common additive error term accounting for variation of pigs and all other variation factors as illustrated in Eq. (2) and direct graph of Figure 3. ( )(1) (2) (3) (4) 2, , , , , : ~ (0,ijk i i i i ijk ijk iY T k with N )β β β β ε ε σ= + (2) (1) iβ (2) iβ (3) iβ (4) iβ 1iY 2iY 3iY 4iY iσ … … Figure 3: DAG of model specified by Eq. (2) But this naive model is kind of assuming the variation of mean response due to the pig effects is small and can be absorbed into the common variation term 2(0, )iN σ which is what we want to estimate at the first place. By using this model, even if we conclude 5 from the Bayesian analysis that treatment of group 1 is different from treatment of group 2, we cannot conclude that difference is from different vaccines effects. As we mentioned at the introduction part, the random pig effects may be mixed with the treatment effects which makes the estimation of the treatment effects hard. Figure 4 conceptually illustrated the random pig effects to the estimation of treatment effects. For each treatment group, four pigs are randomly selected from the population of pigs with variance σ ; but due to the fact there are only four pigs in each group the variance in each of those groups ( 1, 2σ σ ) may different substantially. -2 0 2 4 6 8 10 38 39 40 41 42 Days Te m pe ra tu re individual pig overall estimation treatment estimation true response -2 0 2 4 6 8 10 38 39 40 41 42 Days Te m pe ra tu re individual pig overall estimation treatment estimation true response Population of Pigs Population of Pigs Sample four pigs Sample four pigs σ 1σ 2σ 38 39 40 41 42 Te m pe ra tu re 38 39 40 41 42 Te m pe ra tu re 38 39 40 41 42 Te m pe ra tu re 38 39 40 41 42 Te m pe ra tu re Figure 4: Conceptual illustration of the random pig effects to the estimation of treatment effects If we try to estimate the mean temperature response without taking into account the different variability, we will get the estimation of treatment effects as the red curves in Figure 4. But if the pig variability is considered, we may get the estimation of treatment 6 as the green curves which are closer to the true but unknown treatment responses shown as black curves in Figure 4. To account for the pig effects in each of the treatment groups, a revised version of the model is proposed. In the revised model, we put a perturbation terms to each of the four parameters to reflect the fact that each pig is indeed different in the responses of the same vaccine, i.e. the pig effects. This approach will work because we have 15 temperature responses for each pig and we have an overall 60 temperature responses for each treatment groups while there are 20 parameters to be estimated: four common parameters and four perturbation terms for each pig. More over it is reasonable to assume that the four parameters and four perturbation terms for different pigs follow the same prior distributions. The revised model is illustrated at Eq. (3) and Figure 5. ( )(1) (1) (2) (2) (3) (3) (4) (4) 2, , , , , : ~ (0,ijk i ij i ij i ij i ij ijk ijk iY T k with N )β β β β ε ε= + Δ + Δ + Δ + Δ + σ (3) (1) iβ (2) iβ (3) iβ (4) iβ 1iY 2iY 3iY 4iY iσ … … 2iΔ 3iΔ 4iΔ (1) (2) (3) (4) 1 1 1 1, , ,i i i iΔ Δ Δ Δ (1) (2) (3) (4), , ,i i i iτ τ τ τ Figure 5: DAG of model specified by Eq. (3) 7 In the next step, we generated posterior samples of the beta vector, denoted by (1) (2) (3) (4, , ,i i i i )β β β β ) ) ) ) ( . Then we plugged each posterior beta vector during simulation into Eq. (1) to get ( ,T k 1) (2) (3) (4), , , )i i i iβ β β β ) ) ) ) , which we call the posterior treatment curve. In fact, this curve is the nonlinear response function of our model evaluated at posterior samples of the parameters related to treatment effect (through the beta vector). By taking more samples of beta vector and computing the posterior treatment curves, we can get a (posterior sample) mean treatment curve for each treatment and 95% credible bands. Figure 7(a) shows the mean treatment curves for all 6 groups. Figures 7(b)-(f) compare the mean temperature response and credible bands of each vaccine group with the control group. Except vaccine 4, the other vaccines more or less have lower posterior mean temperatures. Since there is little to no overlap between credible bands of the control group and vaccine 5 around the peak temperature, our main statistically-valid conclusion is that vaccine 5 does have an effect against the virus especially in term of the highest temperature. While, vaccines 1, 2 and 3 seem to be effective on controlling the tail temperature behavior (i.e. the temperature decreases more rapidly after reach the highest temperature). Moreover, we can look at the posterior beta parameters for different treatments to get more insight about the effectiveness of the vaccines. Figure 8 illustrates the 95% credible intervals of the four parameters (group index 6 is the control treatment). The parameter (1) iβ in all the treatments ( ) are statistically the same which intuitively makes sense because this parameter is the baseline of the nonlinear function and it is measured before the challenge days of all groups and should be unaffected by the virus injected and vaccine effects. Parameter 1,...,6i = (2) iβ is the peak (relative height) of the response function and it is particularly interest. Figure 8 clearly implies that the temperature peak for vaccine 5 is statistically less than that of the control group. Another interesting observation is seen in the plot for (3)iβ which is the day in which the maximum temperature occurs. This plot implies that vaccine 1 “accelerates” the temperature change process in the pigs. The plot of parameter (4)iβ has no statistically significant difference in term of mean value, but the 10 variance of (4)iβ at vaccine groups (except vaccine 1) is smaller compared with the control group, which may also indicate some vaccine effects in term of controlling variation. Figure 7: Simulated results for posterior temperature response; (a) mean response across all groups; (b) – (f) comparison between treatment groups (V1-V5) and control group (C) with 95% credible band (CB) 11 38.0 38.5 39.0 39.5 1 2 3 4 5 6 Comparsion of estimated beta 1 Estimated beta 1 G ro up in de x 2 4 6 8 10 1 2 3 4 5 6 Comparsion of estimated beta 2 Estimated beta 2 G ro up in de x 3.0 3.5 4.0 4.5 5.0 1 2 3 4 5 6 Comparsion of estimated beta 3 Estimated beta 3 G ro up in de x 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 5 6 Comparsion of estimated beta 4 Estimated beta 4 G ro up in de x Figure 8: Comparison of estimated parameters and their 95% credible intervals across all groups 5 Conclusions In this report, we have summarized our Bayesian simulation method of how to separate the fixed treatment effects and random effects to make better inference of the fixed 12 (b) WinBugs Code with comment: (only the first groups code shows here, all other groups are the same except the data read in part. model{ ##### Winbugs code for treatment Group 5 ########### for(i in 1:4) { for(j in 1:15) { mu[i,j]<-b1[i]+b2[i]*exp(-(x[j]-b3[i])/b4[i]-exp(-(x[j]-b3[i])/b4[i])) y[i,j]~dnorm(mu[i,j],tau) ## Likelihood yp[i,j]~dnorm(mu[i,j],tau) ## simulating some new data set } b1[i] <- b10 + b11[i] b2[i] <- b20 + b21[i] ## defining the parameters of the expectation function b3[i] <- b30 + b31[i] b4[i] <- b40 + b41[i] b11[i]~dnorm(0,tn1) b21[i]~dnorm(0,tn2) # random pig effect b31[i]~dnorm(0,tn3) b41[i]~dnorm(0,tn4) } for(i in 1:15) { yn[i]<-b10+b20*exp(-(x[i]-b30)/b40-exp(-(x[i]-b30)/b40)) } tn1~dgamma(.001,.001) sig1<-sqrt(1/tn1) tn2~dgamma(.001,.001) sig2<-sqrt(1/tn2) ## defining the priors distributions tn3~dgamma(.001,.001) 15 sig3<-sqrt(1/tn3) tn4~dgamma(.001,.001) sig4<-sqrt(1/tn4) b10~dnorm(0,.001) b20~dnorm(0,.001) b30~dnorm(0,.0001) b40~dnorm(0,.0001) sigma <- 1/sqrt(tau) tau~dunif(0,1000) } list(x=c(-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11),# covariate for day y=structure(.Data=c( 38.89,…,38.33) ,.Dim=c(4,15)) ) # initialization of the data list(b10=10,b20=2,b30=- 2,b40=10,tn1=100,tn2=100,tn3=100,tn4=100,tau=100,b11=c(5,5,5,5),b21=c(6,6,6,6),b31 =c(6,6,6,6),b41=c(5,5,5,5)) ## initialization of a chain 16 (c) More model assessment plots for all other groups: 17