Bayesian statistics in health economic evaluations

Quarto
R
Academia
health economics
Published

September 2, 2023

Hello folks, I hope you had some break time during summer as surely I did! After a whole year of stress and work it was nice to have some vacation period and to clear my mind for a while. Now that I am back to work I feel recharged and I am ready for a new year. One of my objective for this academic year is to find more time to dedicate to some new research projects as last year I only managed to do very little as most of my research energies went into the writing up of a research grant application. This year I hope to find more time to do something different, at least research wise.

So, with that spirit in mind, let’s start from today’s post where I follow-up from a past post introducing the concept of how to perform economic evaluations using standard statistical methods and power it up to what I normally do in this field, use Bayesian statistics! Perhaps some of you will not believe me but over time I am really sure that using Bayesian statistics made my life much easier when coming down to fit relatively complex models to health economics data. Since nowadays this seems to be very common in the literature, there is even more reason to go fully Bayesian when doing these analyses as the degree of flexibility it grants is so much more compared to what standard methods can typically achieve. Of course this is the opinion of someone totally biased! But before raising your finger, please try to come to the end of this post.

As I already mentioned in previous posts, the usual analysis task in economic evaluation based on individual-level data (e.g. QALYs and Total costs computed over a trial period) can be quite challenging due to the presence of a series of complexities that affect the data that need to be taken into account when choosing the statistical methods to use in the analysis; examples include: correlation between effects and costs, skewness of the outcome data, presence of structural values in the data. We saw before that different types of methods exist to deal with each of these problems but the general challenge comes from the fact that often these elements are present jointly in a single dataset and therefore the different methods used to handle each of them need to be combined in some way to perform the analysis. This, however, is easier said than done since, particularly under a frequentist framework, the complexity of fitting all these methods in combination with the need to quantify the impact of uncertainty on the results (e.g. via bootstrapping methods) can lead to extremely difficult-to-fit or expensive-to-implement models. This I believe the key reason why analysts often pretend to ignore some of these problems and prefer to implement easier methods in the hope that results will not be too much affected. Despite understanding their point, I feel that if they knew models that can account for all these problems together, then they would also like to fit them to improve the reliability of the results they obtain. Well, that is why today I talk about fitting the model under a Bayesian framework, which allows to achieve this task at the cost of learning a bit about Bayesian inference and how to interpret it.

Let’s start by simulating some non-Normal bivariate cost and QALY data from an hypothetical study for a total of \(300\) patients assigned to two competing intervention groups (\(t=0,1\)). When generating the data, we can try to mimic the typical skewness features of the outcome data by using alternative distributions such as Gamma for costs and Beta for QALYs. We also generate indicator variables that are used in order to determine which individuals should be assigned “structural values”, namely zero costs and one QALYs. The proportions of individuals assigned to these values is obtained by setting the probability of the Bernoulli distribution used to create the indicators.

set.seed(768)
n <- 300
id <- seq(1:n)
trt <- c(rep(0, n/2),rep(1, n/2))
mean_e1 <- c(0.5)
mean_e2 <- c(0.7)
sigma_e <- 0.15
tau1_e <- ((mean_e1*(1-mean_e1))/(sigma_e^2)-1)
tau2_e <- ((mean_e2*(1-mean_e2))/(sigma_e^2)-1)
alpha1_beta <- tau1_e*mean_e1
beta1_beta <- tau1_e*(1-mean_e1)
alpha2_beta <- tau2_e*mean_e2
beta2_beta <- tau2_e*(1-mean_e2)
e1 <- rbeta(n/2, alpha1_beta, beta1_beta)
e2 <- rbeta(n/2, alpha2_beta, beta2_beta)

mean_c1 <- 500
mean_c2 <- 1000
sigma_c <- 300
tau1_c <- mean_c1/(sigma_c^2)
tau2_c <- mean_c2/(sigma_c^2)
ln.mean_c1 <- log(500) + 5*(e1-mean(e1)) 
c1 <- rgamma(n/2, (exp(ln.mean_c1)/sigma_c)^2, exp(ln.mean_c1)/(sigma_c^2))
ln.mean_c2 <- log(1000) + 5*(e2-mean(e2)) + rgamma(n/2,0,tau2_c)
c2 <- rgamma(n/2, (exp(ln.mean_c2)/sigma_c)^2, exp(ln.mean_c2)/(sigma_c^2))

QALYs <- c(e1,e2)
Costs <- c(c1,c2)

p_zeros <- 0.25
d_zeros <- rbinom(n, 1, p_zeros)
p_ones <- 0.25
d_ones <- rbinom(n, 1, p_ones)

QALYs <- ifelse(d_ones==1,1,QALYs)
Costs <- ifelse(d_zeros==1,0,Costs)

data_sim_ec <- data.frame(id, trt, QALYs, Costs, d_zeros, d_ones)
data_sim_ec <- data_sim_ec[sample(1:nrow(data_sim_ec)), ]

In the code above, after simulating QALY and Cost data using Beta and Gamma distribution, indicator variables for the zero and one values were generated for each individual in the sample from a Bernoulli distribution. Whenever the indicator takes value 1, it denotes the presence of a structural value and the corresponding outcome value is then set equal to zero (Costs) or one (QALYs). We can now compute the correlation between variables and plot the two outcome variables against each other to show how the presence of these structural values affect their corresponding association pattern.

#empirical correlation between e and c (across groups)
cor(data_sim_ec$QALYs,data_sim_ec$Costs)
[1] 0.3582116
#scatterplot of e and c data by group
library(ggplot2)
data_sim_ec$trtf <- factor(data_sim_ec$trt)
levels(data_sim_ec$trtf) <- c("old","new")
ggplot(data_sim_ec, aes(x=QALYs, y=Costs)) +
  geom_point(size=2, shape=16) + theme_classic() +
  facet_wrap(~trtf)

In addition, we can also produce histograms of the distribution of the outcomes by treatment group to have a rough idea of the amount of structural values by type of outcome and treatment group in our sample.

data_sim_ec$trtf <- factor(data_sim_ec$trt)
levels(data_sim_ec$trtf) <- c("old","new")
QALY_hist <- ggplot(data_sim_ec, aes(x=QALYs))+
  geom_histogram(color="black", fill="grey")+
  facet_grid(trtf ~ .) + theme_classic()
Tcost_hist <- ggplot(data_sim_ec, aes(x=Costs))+
  geom_histogram(color="black", fill="grey")+
  facet_grid(trtf ~ .) + theme_classic()
gridExtra::grid.arrange(QALY_hist, Tcost_hist, nrow = 1, ncol = 2)

Step 1: fit a standard normal model

In order to explain the basics of how to fit a Bayesian model, let’s start by considering a (kind of) standard model based on Normal distributions for both QALYs and Total costs. However, I will slightly modify the model to allow for the correlation between the two outcomes, that is we fit a bivariate normal model \(p(e,c\mid \boldsymbol \theta)\), where \(e\) and \(c\) denote the QALYs and Total cost variables measured for each patient in the trial while \(\boldsymbol \theta\) denote the set of parameters indexing the model, including the key quantities of interest for the economic evaluations, i.e. the treatment-specific mean effect and cost \(\mu_{et}\) and \(\mu_{ct}\). To ease the task of modelling the data, we can re-express the joint distribution as:

\[ p(e,c\mid \boldsymbol \theta) = p(e\mid \boldsymbol \theta_e) p(c \mid e, \boldsymbol \theta_c), \]

where \(p(e\mid \boldsymbol \theta_e)\) is the marginal distribution of the effects and \(p(c \mid e, \boldsymbol \theta_c)\) is the conditional distribution of the cost given the effects, each indexed by corresponding set of parameters. The main reason for factoring the joint distribution into this product is the possibility to specify univariate distributions for \(e\) and \(c\), rather than a single bivariate distribution. This can be helpful when, for example, different covariates are considered for the two outcomes as it allows a higher degree of flexibility in specifying the model for each variable. But how are we going to fit the model? well, for that we can rely on freely-available Bayesian software which allows model fitting in a relatively simple way at the cost of learning how to code up the model in this new language. In this post I will consider the JAGS software although this is only one of the many that can be used. For the sake of making things clearer I will not focus here on the details of how the software works and which types of algorithms it uses to implement the model, but I will jump straight into the coding part. First, we need to write the code of the model into a txt file that will then be read by the program after providing the data as input. We can do all this in R.

model_bn <- "
model {

#model specification
for(i in 1:n){
QALYs[i] ~ dnorm(nu_e[i],tau_e)
nu_e[i] <- beta0 + beta1*trt[i]

Costs[i] ~ dnorm(nu_c[i],tau_c)
nu_c[i] <- gamma0 + gamma1*trt[i] + gamma2*QALYs[i]
}

#prior specification
tau_e <- 1/ss_e
ss_e <- s_e*s_e
tau_c <- 1/ss_c
ss_c <- s_c*s_c

s_c ~ dunif(0,1000)
s_e ~ dunif(0,1000)
beta0 ~ dnorm(0,0.000001)
beta1 ~ dnorm(0,0.000001)
gamma0 ~ dnorm(0,0.000001)
gamma1 ~ dnorm(0,0.000001)
gamma2 ~ dnorm(0,0.000001)

}
"
writeLines(model_bn, con = "model_bn.txt")

In the code above I first specify the model structure, i.e. assign normal distributions to QLAYs and Costs variable indexed by two parameters, the means \(\nu\) and precisions \(\tau\) (note that precisions correspond to inverse of the variance \(\tau=1/\sigma^2\)) since these are the default parameters used by JAGS to specify a normal distribution. For each outcome then I specify the mean structure, i.e. the mean of \(e\) depennds only on the treatment indicator while the mean of \(c\) depend both on treatment indicator and \(e\) (this is a conditional cost model!). Next, I specify the priors for non-deterministic parameters, namely using uniform distributions for standard deviations and normal distributions for regression coefficients. Finally, I save the model as a txt file in the current wd using the writeLines function. The model is now written and we can fit it by calling the JAGS software directly from R through dedicated functions. Before that, we need to convert the data as input for the software. Then, we load the package R2jags which allows to call the software from R through the function jags and after providing some technical parameters needed to run the model.

#save data input
n <- dim(data_sim_ec)[1]
QALYs <- data_sim_ec$QALYs
Costs <- data_sim_ec$Costs
trt <- data_sim_ec$trt

#load package and provide algorithm parameters
library(R2jags)
set.seed(2345) #set seed for reproducibility
datalist<-list("n","QALYs","Costs","trt") #pass data into a list
#set up initial values for algorithm
inits1 <- list(.RNG.name = "base::Wichmann-Hill", .RNG.seed = 1)
inits2 <- list(.RNG.name = "base::Wichmann-Hill", .RNG.seed = 2)
#set parameter easimates to save
params<-c("beta0","beta1","gamma0","gamma1","gamma2","s_c","s_e","nu_c","nu_e")
filein<-"model_bn.txt" #name of model file
n.iter<-20000 #n of iterations

We are now ready to fit the model, which we can do by typing

#fit model
jmodel_bn<-jags(data=datalist,inits=list(inits1,inits2),
                parameters.to.save=params,model.file=filein,
                n.chains=2,n.iter=n.iter,n.thin=1)
Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 600
   Unobserved stochastic nodes: 7
   Total graph size: 1345

Initializing model

After some time needed for the model to run, we end up with something like this

#posterior results
print(jmodel_bn)
Inference for Bugs model at "model_bn.txt", fit using jags,
 2 chains, each with 20000 iterations (first 10000 discarded)
 n.sims = 20000 iterations saved
           mu.vect sd.vect     2.5%      25%      50%      75%    97.5%  Rhat
beta0        0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
beta1        0.116   0.027    0.063    0.098    0.116    0.134    0.168 1.001
gamma0    -147.064 136.100 -414.191 -238.684 -146.655  -56.422  120.626 1.001
gamma1     443.594  89.617  268.643  383.247  443.401  504.381  618.307 1.001
gamma2    1003.566 185.780  642.338  878.610 1002.303 1128.245 1368.741 1.001
nu_c[1]    653.790  99.155  458.610  587.817  653.583  719.540  847.442 1.001
nu_c[2]   1019.968  62.799  897.609  977.709 1019.448 1062.243 1143.497 1.001
nu_c[3]    345.232  68.708  211.853  298.661  344.974  391.561  480.094 1.001
nu_c[4]   1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[5]    625.502  65.446  495.707  581.253  625.514  669.387  754.077 1.001
nu_c[6]   1194.620  66.124 1065.903 1150.333 1194.031 1238.974 1323.582 1.001
nu_c[7]    460.143  62.371  338.056  418.112  460.125  502.151  583.002 1.001
nu_c[8]    856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[9]   1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[10]   563.317  62.574  440.263  521.301  563.189  605.310  686.902 1.001
nu_c[11]  1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[12]  1191.867  65.950 1063.432 1147.673 1191.312 1235.984 1320.751 1.001
nu_c[13]   994.028  63.705  870.006  951.189  993.596 1036.803 1120.059 1.001
nu_c[14]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[15]  1165.410  64.460 1039.591 1122.082 1164.928 1208.722 1291.399 1.001
nu_c[16]   809.178  78.808  654.435  756.291  809.094  861.036  964.868 1.001
nu_c[17]   340.903  69.063  206.924  294.139  340.636  387.481  476.537 1.001
nu_c[18]   872.921  72.138  732.591  824.388  872.264  920.863 1015.917 1.001
nu_c[19]   500.837  61.750  379.933  459.165  500.833  542.352  622.304 1.001
nu_c[20]   973.213  64.682  847.399  929.984  972.492 1016.510 1100.975 1.001
nu_c[21]  1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[22]   974.684  64.606  848.977  931.523  973.961 1017.918 1102.260 1.001
nu_c[23]   837.468  75.693  690.179  786.761  837.409  887.411  987.207 1.001
nu_c[24]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[25]   806.961  79.061  651.919  753.944  806.819  859.092  963.011 1.001
nu_c[26]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[27]   361.790  67.419  230.804  316.064  361.775  407.072  494.146 1.001
nu_c[28]  1024.197  62.685  901.741  981.990 1023.696 1066.397 1147.427 1.001
nu_c[29]   383.746  65.890  255.443  338.978  383.877  428.377  513.959 1.001
nu_c[30]   227.424  80.696   70.418  173.207  227.225  282.027  385.480 1.001
nu_c[31]   929.086  67.436  797.581  883.929  928.507  974.166 1062.601 1.001
nu_c[32]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[33]   823.970  77.151  673.371  772.360  823.975  874.825  976.251 1.001
nu_c[34]  1132.331  63.085 1009.282 1089.397 1132.140 1174.378 1255.022 1.001
nu_c[35]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[36]  1118.608  62.681  996.209 1076.318 1118.460 1160.549 1240.764 1.001
nu_c[37]   359.446  67.594  228.237  313.609  359.371  404.935  492.136 1.001
nu_c[38]  1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[39]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[40]  1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[41]   588.930 108.778  374.370  516.785  588.994  661.198  802.627 1.001
nu_c[42]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[43]   455.381  62.503  332.926  413.332  455.433  497.558  578.525 1.001
nu_c[44]   740.735  87.187  570.340  682.575  740.532  798.336  912.542 1.001
nu_c[45]   430.231  63.398  306.488  387.377  430.247  473.506  555.669 1.001
nu_c[46]   441.056  62.972  317.871  398.451  441.096  483.796  565.390 1.001
nu_c[47]  1169.891  64.689 1043.692 1126.451 1169.458 1213.259 1296.198 1.001
nu_c[48]  1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[49]   288.014  73.972  144.226  238.023  287.678  338.282  432.055 1.001
nu_c[50]   928.271  67.496  796.571  883.110  927.761  973.420 1061.858 1.001
nu_c[51]  1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[52]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[53]  1072.145  62.071  950.668 1030.223 1071.914 1113.477 1193.582 1.001
nu_c[54]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[55]   988.362  63.950  863.777  945.424  987.777 1031.150 1114.910 1.001
nu_c[56]   128.235  93.581  -53.831   65.024  127.986  190.842  312.124 1.001
nu_c[57]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[58]   146.811  91.025  -29.725   85.618  146.778  208.016  325.036 1.001
nu_c[59]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[60]   607.099 106.033  398.304  536.753  606.724  677.340  815.537 1.001
nu_c[61]   152.831  90.209  -21.966   92.077  152.643  213.540  329.462 1.001
nu_c[62]   501.200  61.748  380.299  459.536  501.200  542.677  622.629 1.001
nu_c[63]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[64]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[65]   972.467  64.721  846.548  929.167  971.758 1015.811 1100.624 1.001
nu_c[66]   556.390  62.378  433.787  514.490  556.439  598.405  679.073 1.001
nu_c[67]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[68]   611.944 105.307  404.612  542.031  611.649  681.797  818.630 1.001
nu_c[69]  1135.728  63.200 1012.346 1092.687 1135.406 1177.868 1258.411 1.001
nu_c[70]  1055.716  62.140  934.313 1013.709 1055.420 1097.361 1177.248 1.001
nu_c[71]   831.932  76.284  683.435  780.881  831.983  882.173  982.980 1.001
nu_c[72]   778.404  82.440  617.205  723.291  778.203  832.481  940.706 1.001
nu_c[73]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[74]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[75]   865.762  72.822  723.928  816.706  865.138  914.183 1010.231 1.001
nu_c[76]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[77]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[78]  1084.918  62.121  963.388 1043.058 1084.629 1126.439 1206.235 1.001
nu_c[79]   200.283  84.019   37.557  143.579  200.108  257.303  364.512 1.001
nu_c[80]   331.705  69.843  196.626  284.343  331.304  378.963  469.184 1.001
nu_c[81]  1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[82]   300.374  72.737  159.373  251.200  300.150  349.635  442.519 1.001
nu_c[83]  1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[84]  1180.882  65.291 1053.657 1137.075 1180.348 1224.545 1308.583 1.001
nu_c[85]  1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[86]  1035.128  62.435  912.887  993.050 1034.908 1077.069 1157.367 1.001
nu_c[87]  1049.819  62.200  928.122 1007.790 1049.655 1091.573 1171.365 1.001
nu_c[88]   480.979  61.940  359.345  439.130  480.732  522.537  602.747 1.001
nu_c[89]  1129.820  63.003 1006.979 1087.061 1129.528 1171.899 1252.262 1.001
nu_c[90]   972.579  64.715  846.671  929.290  971.854 1015.907 1100.674 1.001
nu_c[91]   918.919  68.194  785.895  873.028  918.408  964.483 1053.651 1.001
nu_c[92]   431.336  63.351  307.756  388.563  431.312  474.563  556.622 1.001
nu_c[93]  1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[94]   449.700  62.676  326.853  407.428  449.703  492.178  573.204 1.001
nu_c[95]  1060.214  62.106  938.643 1018.300 1059.854 1101.722 1181.800 1.001
nu_c[96]   453.978  62.544  331.473  411.816  453.999  496.181  577.207 1.001
nu_c[97]   775.863  82.750  614.126  720.477  775.750  830.194  939.044 1.001
nu_c[98]   856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[99]  1075.338  62.075  953.857 1033.428 1075.046 1116.783 1197.246 1.001
nu_c[100] 1265.739  71.736 1126.516 1217.434 1265.248 1314.108 1407.430 1.001
nu_c[101] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[102]  567.378  62.701  444.136  525.268  567.360  609.454  691.153 1.001
nu_c[103]  471.741  62.102  350.168  430.014  471.659  513.439  593.879 1.001
nu_c[104]  390.904  65.439  263.424  346.660  391.057  435.257  520.019 1.001
nu_c[105] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[106]  545.692  62.125  423.762  503.696  545.709  587.546  667.842 1.001
nu_c[107] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[108] 1246.361  70.006 1110.454 1199.244 1245.694 1293.687 1384.062 1.001
nu_c[109]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[110]  453.798  62.550  331.319  411.643  453.823  496.027  577.068 1.001
nu_c[111]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[112]  829.187  76.581  679.831  777.979  829.247  879.665  980.609 1.001
nu_c[113] 1173.521  64.882 1047.074 1129.969 1173.028 1216.966 1300.152 1.001
nu_c[114]  579.849  63.146  455.542  537.246  579.870  622.206  704.546 1.001
nu_c[115]  878.268  71.639  739.032  830.161  877.622  926.056 1020.183 1.001
nu_c[116]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[117] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[118] 1087.045  62.138  965.397 1045.166 1086.754 1128.640 1208.452 1.001
nu_c[119]  269.480  75.916  121.941  218.248  269.373  320.905  417.606 1.001
nu_c[120]  217.486  81.893   58.804  162.411  217.173  272.976  377.719 1.001
nu_c[121] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[122]  411.898  64.255  286.333  368.495  411.972  455.562  538.746 1.001
nu_c[123]  340.754  69.076  206.779  293.997  340.491  387.344  476.440 1.001
nu_c[124]  393.508  65.281  266.302  349.330  393.631  437.807  522.578 1.001
nu_c[125]  993.860  63.712  869.841  951.039  993.414 1036.649 1119.920 1.001
nu_c[126]  918.847  68.200  785.814  872.950  918.333  964.407 1053.593 1.001
nu_c[127]  942.708  66.491  813.072  898.199  941.958  987.311 1074.899 1.001
nu_c[128]  600.639  64.066  474.121  557.393  600.442  643.701  726.447 1.001
nu_c[129]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[130]  238.751  79.362   84.821  185.310  238.637  292.281  394.532 1.001
nu_c[131]  291.722  73.596  148.600  241.944  291.345  341.674  435.143 1.001
nu_c[132]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[133]  272.240  75.620  125.369  221.126  272.136  323.435  419.896 1.001
nu_c[134]  862.012  73.188  719.604  812.768  861.429  910.596 1007.066 1.001
nu_c[135]  527.533  61.839  406.251  485.702  527.602  569.191  649.584 1.001
nu_c[136] 1109.593  62.471  987.488 1067.352 1109.397 1151.373 1231.314 1.001
nu_c[137]  357.166  67.767  225.565  311.261  357.093  402.750  490.291 1.001
nu_c[138] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[139] 1014.176  62.971  891.107  971.770 1013.619 1056.509 1138.054 1.001
nu_c[140] 1096.916  62.249  975.076 1054.895 1096.604 1138.436 1218.545 1.001
nu_c[141]  837.029  75.740  689.575  786.289  836.974  886.966  986.871 1.001
nu_c[142] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[143]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[144]  440.123  63.006  316.866  397.468  440.150  482.925  564.565 1.001
nu_c[145]  401.262  64.828  274.856  357.348  401.464  445.279  529.296 1.001
nu_c[146] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[147]  364.962  67.185  234.535  319.386  365.043  410.060  497.141 1.001
nu_c[148] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[149]  463.035  62.297  341.267  420.976  463.007  504.950  585.569 1.001
nu_c[150] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[151]  165.754  88.480   -5.480  106.133  165.719  225.351  339.038 1.001
nu_c[152]  377.056  66.333  247.901  331.963  377.180  421.751  507.572 1.001
nu_c[153]  507.359  61.735  386.562  465.664  507.384  548.947  628.816 1.001
nu_c[154]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[155] 1029.794  62.549  907.183  987.624 1029.433 1071.778 1152.676 1.001
nu_c[156]  676.339  95.935  487.461  612.554  676.129  739.855  863.658 1.001
nu_c[157]  452.035  62.603  329.448  409.841  452.074  494.344  575.421 1.001
nu_c[158] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[159] 1037.890  62.382  915.811  995.760 1037.673 1079.745 1159.808 1.001
nu_c[160] 1097.436  62.257  975.554 1055.434 1097.111 1138.986 1219.053 1.001
nu_c[161]  992.475  63.771  868.361  949.591  991.984 1035.225 1118.779 1.001
nu_c[162]  197.107  84.419   33.531  140.025  197.024  254.196  361.972 1.001
nu_c[163] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[164]  432.218  63.315  308.621  389.404  432.183  475.332  557.546 1.001
nu_c[165]  604.453  64.258  477.469  561.030  604.294  647.567  730.810 1.001
nu_c[166] 1108.516  62.448  986.372 1066.250 1108.262 1150.275 1230.153 1.001
nu_c[167]  490.674  61.820  369.481  449.052  490.515  532.279  612.184 1.001
nu_c[168]  297.238  73.045  155.364  247.821  296.911  346.776  439.597 1.001
nu_c[169]  757.527  85.033  591.281  700.792  757.319  813.510  924.869 1.001
nu_c[170] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[171]  769.315  83.557  606.126  713.484  769.219  824.229  933.923 1.001
nu_c[172]  503.620  61.741  382.632  461.915  503.588  545.131  624.894 1.001
nu_c[173]  130.676  93.242  -50.660   67.756  130.524  193.125  313.898 1.001
nu_c[174]  959.564  65.437  832.146  915.742  958.767 1003.510 1089.361 1.001
nu_c[175] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[176]  170.081  87.908   -0.093  110.791  169.977  229.324  342.003 1.001
nu_c[177] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[178]  305.910  72.200  166.109  257.031  305.625  354.740  447.209 1.001
nu_c[179]  481.343  61.934  359.708  439.493  481.098  522.917  603.097 1.001
nu_c[180]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[181]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[182]  534.462  61.927  412.766  492.529  534.620  576.220  656.674 1.001
nu_c[183]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[184] 1036.865  62.401  914.755  994.690 1036.621 1078.762 1158.836 1.001
nu_c[185]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[186] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[187]  228.208  80.602   71.421  174.065  227.969  282.654  386.077 1.001
nu_c[188]  453.709  62.552  331.241  411.562  453.718  495.945  576.989 1.001
nu_c[189] 1079.333  62.088  958.114 1037.493 1078.952 1120.800 1200.873 1.001
nu_c[190] 1072.557  62.071  951.115 1030.611 1072.321 1113.916 1193.961 1.001
nu_c[191] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[192]  436.323  63.151  313.002  393.679  436.323  479.210  561.095 1.001
nu_c[193] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[194]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[195] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[196] 1017.705  62.864  895.085  975.433 1017.095 1060.072 1141.432 1.001
nu_c[197]  162.162  88.957  -10.108  102.143  162.094  222.163  335.939 1.001
nu_c[198] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[199] 1067.667  62.075  945.754 1025.813 1067.496 1109.081 1188.903 1.001
nu_c[200] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[201]  323.111  70.601  186.414  275.174  322.813  371.022  462.002 1.001
nu_c[202] 1100.266  62.299  978.213 1058.213 1100.001 1141.787 1221.926 1.001
nu_c[203]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[204]  923.430  67.853  790.929  877.931  922.885  968.765 1057.648 1.001
nu_c[205]  808.787  78.852  653.982  755.927  808.727  860.683  964.561 1.001
nu_c[206]   73.696 101.388 -123.777    5.095   73.187  141.186  273.770 1.001
nu_c[207]  148.163  90.841  -27.996   87.123  148.121  209.274  325.924 1.001
nu_c[208] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[209]  780.940  82.132  620.262  726.041  780.662  834.767  942.495 1.001
nu_c[210]  772.066  83.217  609.404  716.391  771.970  826.645  936.222 1.001
nu_c[211]  566.636  62.677  443.372  524.561  566.575  608.673  690.331 1.001
nu_c[212] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[213] 1014.910  62.948  891.849  972.478 1014.298 1057.259 1138.751 1.001
nu_c[214]  712.363  90.953  533.536  651.553  712.275  772.596  890.455 1.001
nu_c[215]  992.563  63.767  868.445  949.670  992.060 1035.320 1118.845 1.001
nu_c[216]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[217]  259.008  77.060  109.562  207.075  258.796  310.995  409.677 1.001
nu_c[218]  910.403  68.862  776.619  864.265  909.869  956.347 1046.502 1.001
nu_c[219]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[220]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[221] 1135.591  63.195 1012.223 1092.577 1135.274 1177.724 1258.271 1.001
nu_c[222]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[223]  314.743  71.366  176.571  266.342  314.410  363.168  454.796 1.001
nu_c[224]  166.929  88.324   -4.136  107.394  166.891  226.387  339.874 1.001
nu_c[225] 1137.322  63.256 1013.909 1094.325 1137.065 1179.437 1260.155 1.001
nu_c[226]  902.146  69.538  766.438  855.432  901.529  948.629 1039.498 1.001
nu_c[227] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[228]  339.275  69.199  205.103  292.389  339.044  385.902  475.157 1.001
nu_c[229]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[230] 1183.029  65.416 1055.654 1139.089 1182.527 1226.752 1310.951 1.001
nu_c[231] 1048.167  62.221  926.479 1006.160 1047.966 1089.878 1169.762 1.001
nu_c[232]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[233]  471.388  62.109  349.843  429.703  471.298  513.122  593.505 1.001
nu_c[234]  756.718  85.136  590.368  699.890  756.510  812.821  924.330 1.001
nu_c[235] 1015.861  62.919  892.961  973.494 1015.225 1058.222 1139.715 1.001
nu_c[236]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[237]  433.514  63.262  310.113  390.746  433.533  476.546  558.628 1.001
nu_c[238]  354.821  67.947  222.710  308.802  354.675  400.591  488.379 1.001
nu_c[239] 1191.369  65.919 1062.956 1147.190 1190.825 1235.472 1320.154 1.001
nu_c[240] 1008.980  63.140  885.696  966.597 1008.513 1051.395 1133.476 1.001
nu_c[241]  710.029  91.269  530.568  649.105  709.985  770.399  888.743 1.001
nu_c[242] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[243]  379.830  66.147  251.073  334.858  380.032  424.445  510.398 1.001
nu_c[244]  472.134  62.094  350.590  430.446  472.051  513.847  594.245 1.001
nu_c[245]  568.522  62.739  445.109  526.389  568.511  610.636  692.440 1.001
nu_c[246]  525.065 118.679  290.495  445.865  525.356  604.315  757.512 1.001
nu_c[247]  355.505  67.895  223.552  309.538  355.419  401.219  488.949 1.001
nu_c[248] 1224.452  68.225 1091.572 1178.466 1223.865 1270.420 1358.569 1.001
nu_c[249]  307.717  72.027  168.248  259.022  307.392  356.456  448.801 1.001
nu_c[250] 1093.948  62.210  972.201 1052.079 1093.639 1135.454 1215.576 1.001
nu_c[251]  313.416  71.489  174.968  264.926  313.081  361.911  453.697 1.001
nu_c[252] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[253]  217.555  81.884   58.885  162.485  217.240  273.041  377.784 1.001
nu_c[254] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[255] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[256]  274.688  75.359  128.526  223.717  274.514  325.774  421.700 1.001
nu_c[257] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[258]  577.648  63.062  453.270  535.044  577.676  619.974  702.294 1.001
nu_c[259]  288.377  73.935  144.676  238.407  288.093  338.608  432.318 1.001
nu_c[260]  962.194  65.285  834.977  918.512  961.499 1005.948 1091.694 1.001
nu_c[261]  289.305  73.841  145.714  239.348  289.004  339.481  433.145 1.001
nu_c[262] 1081.546  62.099  960.130 1039.688 1081.260 1123.012 1202.804 1.001
nu_c[263]  188.313  85.538   22.210  130.477  188.244  246.045  355.528 1.001
nu_c[264]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[265]   56.582 103.919 -145.814  -14.036   56.083  125.730  261.937 1.001
nu_c[266]  793.987  80.571  636.301  739.769  793.778  847.068  952.291 1.001
nu_c[267] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[268] 1144.670  63.532 1020.502 1101.573 1144.416 1187.147 1268.349 1.001
nu_c[269]  231.478  80.215   75.627  177.611  231.251  285.529  388.696 1.001
nu_c[270]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[271]  327.033  70.252  191.121  279.389  326.818  374.628  465.338 1.001
nu_c[272]  946.117  66.268  817.019  901.739  945.340  990.589 1077.745 1.001
nu_c[273]  573.242  62.900  449.396  530.871  573.247  615.448  697.549 1.001
nu_c[274]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[275] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[276]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[277] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[278] 1232.160  68.830 1098.163 1185.729 1231.552 1278.396 1367.565 1.001
nu_c[279] 1009.228  63.132  885.931  966.839 1008.768 1051.619 1133.730 1.001
nu_c[280]  164.753  88.613   -6.774  105.008  164.741  224.484  338.222 1.001
nu_c[281] 1300.096  75.126 1155.050 1249.149 1299.742 1350.649 1449.069 1.001
nu_c[282]  626.000  65.477  496.128  581.724  626.001  669.931  754.663 1.001
nu_c[283]  257.657  77.210  107.917  205.546  257.496  309.786  408.557 1.001
nu_c[284]  372.097  66.675  242.342  326.834  372.134  416.931  503.345 1.001
nu_c[285]  715.856  90.481  537.992  655.407  715.729  775.835  893.483 1.001
nu_c[286]  403.727  64.691  277.568  359.971  403.867  447.772  531.564 1.001
nu_c[287] 1152.203  63.844 1027.452 1109.080 1151.926 1195.022 1276.850 1.001
nu_c[288] 1004.235  63.307  880.782  961.825 1003.827 1046.725 1129.082 1.001
nu_c[289]  426.669  63.551  302.407  383.738  426.792  469.977  552.243 1.001
nu_c[290]  782.928  81.891  622.579  728.116  782.675  836.602  944.040 1.001
nu_c[291]   95.061  98.280  -96.099   28.409   94.726  160.560  288.836 1.001
nu_c[292]  398.652  64.977  271.841  354.557  398.829  442.788  526.913 1.001
nu_c[293] 1142.963  63.466 1019.007 1099.823 1142.655 1185.382 1266.339 1.001
nu_c[294]  397.086  65.069  270.034  352.930  397.223  441.330  525.547 1.001
nu_c[295]  416.723  64.013  291.533  373.423  416.783  460.279  543.401 1.001
nu_c[296]  155.756  89.815  -18.145   95.239  155.576  216.315  331.500 1.001
nu_c[297]  988.642  63.937  864.104  945.693  988.055 1031.457 1115.224 1.001
nu_c[298]  139.710  91.995  -38.810   77.675  139.478  201.461  320.181 1.001
nu_c[299]  856.502  89.274  679.911  796.575  856.125  916.588 1031.153 1.001
nu_c[300] 1116.904  62.638  994.629 1074.631 1116.739 1158.830 1238.954 1.001
nu_e[1]      0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[2]      0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[3]      0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[4]      0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[5]      0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[6]      0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[7]      0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[8]      0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[9]      0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[10]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[11]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[12]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[13]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[14]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[15]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[16]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[17]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[18]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[19]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[20]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[21]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[22]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[23]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[24]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[25]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[26]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[27]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[28]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[29]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[30]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[31]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[32]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[33]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[34]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[35]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[36]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[37]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[38]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[39]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[40]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[41]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[42]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[43]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[44]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[45]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[46]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[47]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[48]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[49]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[50]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[51]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[52]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[53]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[54]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[55]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[56]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[57]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[58]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[59]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[60]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[61]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[62]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[63]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[64]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[65]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[66]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[67]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[68]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[69]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[70]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[71]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[72]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[73]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[74]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[75]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[76]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[77]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[78]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[79]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[80]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[81]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[82]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[83]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[84]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[85]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[86]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[87]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[88]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[89]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[90]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[91]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[92]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[93]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[94]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[95]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[96]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[97]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[98]     0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[99]     0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[100]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[101]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[102]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[103]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[104]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[105]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[106]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[107]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[108]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[109]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[110]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[111]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[112]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[113]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[114]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[115]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[116]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[117]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[118]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[119]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[120]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[121]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[122]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[123]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[124]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[125]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[126]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[127]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[128]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[129]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[130]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[131]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[132]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[133]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[134]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[135]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[136]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[137]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[138]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[139]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[140]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[141]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[142]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[143]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[144]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[145]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[146]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[147]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[148]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[149]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[150]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[151]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[152]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[153]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[154]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[155]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[156]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[157]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[158]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[159]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[160]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[161]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[162]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[163]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[164]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[165]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[166]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[167]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[168]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[169]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[170]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[171]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[172]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[173]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[174]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[175]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[176]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[177]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[178]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[179]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[180]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[181]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[182]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[183]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[184]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[185]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[186]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[187]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[188]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[189]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[190]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[191]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[192]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[193]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[194]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[195]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[196]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[197]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[198]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[199]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[200]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[201]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[202]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[203]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[204]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[205]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[206]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[207]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[208]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[209]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[210]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[211]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[212]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[213]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[214]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[215]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[216]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[217]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[218]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[219]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[220]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[221]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[222]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[223]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[224]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[225]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[226]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[227]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[228]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[229]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[230]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[231]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[232]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[233]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[234]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[235]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[236]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[237]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[238]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[239]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[240]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[241]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[242]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[243]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[244]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[245]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[246]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[247]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[248]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[249]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[250]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[251]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[252]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[253]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[254]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[255]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[256]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[257]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[258]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[259]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[260]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[261]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[262]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[263]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[264]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[265]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[266]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[267]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[268]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[269]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[270]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[271]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[272]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[273]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[274]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[275]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[276]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[277]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[278]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[279]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[280]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[281]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[282]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[283]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[284]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[285]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[286]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[287]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[288]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[289]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[290]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[291]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[292]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[293]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[294]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[295]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[296]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[297]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
nu_e[298]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[299]    0.656   0.019    0.619    0.643    0.656    0.669    0.693 1.001
nu_e[300]    0.772   0.019    0.735    0.759    0.772    0.785    0.809 1.001
s_c        759.603  31.405  701.198  737.789  758.687  780.114  824.233 1.001
s_e          0.231   0.009    0.214    0.225    0.231    0.237    0.251 1.001
deviance  4800.101   3.774 4794.804 4797.326 4799.423 4802.136 4809.117 1.001
          n.eff
beta0     20000
beta1     20000
gamma0    20000
gamma1    20000
gamma2    20000
nu_c[1]   13000
nu_c[2]    6900
nu_c[3]    9900
nu_c[4]   12000
nu_c[5]   12000
nu_c[6]    8800
nu_c[7]    9500
nu_c[8]   20000
nu_c[9]   12000
nu_c[10]  11000
nu_c[11]  12000
nu_c[12]   8800
nu_c[13]   7000
nu_c[14]  20000
nu_c[15]   8200
nu_c[16]   9300
nu_c[17]   9900
nu_c[18]   8100
nu_c[19]   9700
nu_c[20]   7100
nu_c[21]  12000
nu_c[22]   7100
nu_c[23]   8700
nu_c[24]  20000
nu_c[25]   9300
nu_c[26]  20000
nu_c[27]   9800
nu_c[28]   6900
nu_c[29]   9700
nu_c[30]  12000
nu_c[31]   7400
nu_c[32]  20000
nu_c[33]   9000
nu_c[34]   7600
nu_c[35]  20000
nu_c[36]   7500
nu_c[37]   9800
nu_c[38]  12000
nu_c[39]  20000
nu_c[40]  12000
nu_c[41]  14000
nu_c[42]  20000
nu_c[43]   9400
nu_c[44]  11000
nu_c[45]   9500
nu_c[46]   9400
nu_c[47]   8300
nu_c[48]  12000
nu_c[49]  11000
nu_c[50]   7400
nu_c[51]  12000
nu_c[52]  20000
nu_c[53]   7000
nu_c[54]  20000
nu_c[55]   7000
nu_c[56]  15000
nu_c[57]  20000
nu_c[58]  14000
nu_c[59]  20000
nu_c[60]  14000
nu_c[61]  14000
nu_c[62]   9700
nu_c[63]  20000
nu_c[64]  20000
nu_c[65]   7100
nu_c[66]  10000
nu_c[67]  20000
nu_c[68]  14000
nu_c[69]   7700
nu_c[70]   7000
nu_c[71]   8800
nu_c[72]   9900
nu_c[73]  20000
nu_c[74]  20000
nu_c[75]   8300
nu_c[76]  20000
nu_c[77]  20000
nu_c[78]   7100
nu_c[79]  13000
nu_c[80]   9900
nu_c[81]  12000
nu_c[82]  10000
nu_c[83]  12000
nu_c[84]   8500
nu_c[85]  12000
nu_c[86]   6900
nu_c[87]   6900
nu_c[88]   9500
nu_c[89]   7600
nu_c[90]   7100
nu_c[91]   7500
nu_c[92]   9500
nu_c[93]  12000
nu_c[94]   9400
nu_c[95]   7000
nu_c[96]   9400
nu_c[97]   9900
nu_c[98]  20000
nu_c[99]   7100
nu_c[100] 11000
nu_c[101] 12000
nu_c[102] 11000
nu_c[103]  9500
nu_c[104]  9600
nu_c[105] 12000
nu_c[106] 10000
nu_c[107] 12000
nu_c[108] 10000
nu_c[109] 20000
nu_c[110]  9400
nu_c[111] 20000
nu_c[112]  8900
nu_c[113]  8400
nu_c[114] 11000
nu_c[115]  8100
nu_c[116] 20000
nu_c[117] 12000
nu_c[118]  7100
nu_c[119] 11000
nu_c[120] 12000
nu_c[121] 12000
nu_c[122]  9500
nu_c[123]  9900
nu_c[124]  9600
nu_c[125]  7000
nu_c[126]  7500
nu_c[127]  7300
nu_c[128] 12000
nu_c[129] 20000
nu_c[130] 12000
nu_c[131] 11000
nu_c[132] 20000
nu_c[133] 11000
nu_c[134]  8300
nu_c[135] 10000
nu_c[136]  7400
nu_c[137]  9800
nu_c[138] 12000
nu_c[139]  6900
nu_c[140]  7200
nu_c[141]  8700
nu_c[142] 12000
nu_c[143] 20000
nu_c[144]  9400
nu_c[145]  9600
nu_c[146] 12000
nu_c[147]  9800
nu_c[148] 12000
nu_c[149]  9500
nu_c[150] 12000
nu_c[151] 13000
nu_c[152]  9700
nu_c[153]  9700
nu_c[154] 20000
nu_c[155]  6900
nu_c[156] 12000
nu_c[157]  9400
nu_c[158] 12000
nu_c[159]  6900
nu_c[160]  7200
nu_c[161]  7000
nu_c[162] 13000
nu_c[163] 12000
nu_c[164]  9400
nu_c[165] 12000
nu_c[166]  7300
nu_c[167]  9600
nu_c[168] 10000
nu_c[169] 10000
nu_c[170] 12000
nu_c[171] 10000
nu_c[172]  9700
nu_c[173] 14000
nu_c[174]  7200
nu_c[175] 12000
nu_c[176] 13000
nu_c[177] 12000
nu_c[178]  9300
nu_c[179]  9500
nu_c[180] 20000
nu_c[181] 20000
nu_c[182] 10000
nu_c[183] 20000
nu_c[184]  6900
nu_c[185] 20000
nu_c[186] 12000
nu_c[187] 12000
nu_c[188]  9400
nu_c[189]  7100
nu_c[190]  7000
nu_c[191] 12000
nu_c[192]  9400
nu_c[193] 12000
nu_c[194] 20000
nu_c[195] 12000
nu_c[196]  6900
nu_c[197] 14000
nu_c[198] 12000
nu_c[199]  7000
nu_c[200] 12000
nu_c[201]  9800
nu_c[202]  7300
nu_c[203] 20000
nu_c[204]  7500
nu_c[205]  9300
nu_c[206] 16000
nu_c[207] 14000
nu_c[208] 12000
nu_c[209]  9800
nu_c[210] 10000
nu_c[211] 11000
nu_c[212] 12000
nu_c[213]  6900
nu_c[214] 11000
nu_c[215]  7000
nu_c[216] 20000
nu_c[217] 11000
nu_c[218]  7600
nu_c[219] 20000
nu_c[220] 20000
nu_c[221]  7700
nu_c[222] 20000
nu_c[223]  9700
nu_c[224] 13000
nu_c[225]  7700
nu_c[226]  7700
nu_c[227] 12000
nu_c[228]  9900
nu_c[229] 20000
nu_c[230]  8600
nu_c[231]  6900
nu_c[232] 20000
nu_c[233]  9500
nu_c[234] 10000
nu_c[235]  6900
nu_c[236] 20000
nu_c[237]  9400
nu_c[238]  9800
nu_c[239]  8800
nu_c[240]  6900
nu_c[241] 11000
nu_c[242] 12000
nu_c[243]  9700
nu_c[244]  9500
nu_c[245] 11000
nu_c[246] 15000
nu_c[247]  9800
nu_c[248]  9600
nu_c[249]  9400
nu_c[250]  7200
nu_c[251]  9600
nu_c[252] 12000
nu_c[253] 12000
nu_c[254] 12000
nu_c[255] 12000
nu_c[256] 11000
nu_c[257] 12000
nu_c[258] 11000
nu_c[259] 11000
nu_c[260]  7100
nu_c[261] 11000
nu_c[262]  7100
nu_c[263] 13000
nu_c[264] 20000
nu_c[265] 17000
nu_c[266]  9600
nu_c[267] 12000
nu_c[268]  7800
nu_c[269] 12000
nu_c[270] 20000
nu_c[271]  9800
nu_c[272]  7300
nu_c[273] 11000
nu_c[274] 20000
nu_c[275] 12000
nu_c[276] 20000
nu_c[277] 12000
nu_c[278]  9900
nu_c[279]  6900
nu_c[280] 14000
nu_c[281] 12000
nu_c[282] 12000
nu_c[283] 11000
nu_c[284]  9700
nu_c[285] 11000
nu_c[286]  9500
nu_c[287]  7900
nu_c[288]  6900
nu_c[289]  9500
nu_c[290]  9800
nu_c[291] 16000
nu_c[292]  9600
nu_c[293]  7800
nu_c[294]  9600
nu_c[295]  9500
nu_c[296] 14000
nu_c[297]  7000
nu_c[298] 14000
nu_c[299] 20000
nu_c[300]  7400
nu_e[1]   20000
nu_e[2]   20000
nu_e[3]   20000
nu_e[4]   20000
nu_e[5]   20000
nu_e[6]   20000
nu_e[7]   20000
nu_e[8]   20000
nu_e[9]   20000
nu_e[10]  20000
nu_e[11]  20000
nu_e[12]  20000
nu_e[13]  20000
nu_e[14]  20000
nu_e[15]  20000
nu_e[16]  20000
nu_e[17]  20000
nu_e[18]  20000
nu_e[19]  20000
nu_e[20]  20000
nu_e[21]  20000
nu_e[22]  20000
nu_e[23]  20000
nu_e[24]  20000
nu_e[25]  20000
nu_e[26]  20000
nu_e[27]  20000
nu_e[28]  20000
nu_e[29]  20000
nu_e[30]  20000
nu_e[31]  20000
nu_e[32]  20000
nu_e[33]  20000
nu_e[34]  20000
nu_e[35]  20000
nu_e[36]  20000
nu_e[37]  20000
nu_e[38]  20000
nu_e[39]  20000
nu_e[40]  20000
nu_e[41]  20000
nu_e[42]  20000
nu_e[43]  20000
nu_e[44]  20000
nu_e[45]  20000
nu_e[46]  20000
nu_e[47]  20000
nu_e[48]  20000
nu_e[49]  20000
nu_e[50]  20000
nu_e[51]  20000
nu_e[52]  20000
nu_e[53]  20000
nu_e[54]  20000
nu_e[55]  20000
nu_e[56]  20000
nu_e[57]  20000
nu_e[58]  20000
nu_e[59]  20000
nu_e[60]  20000
nu_e[61]  20000
nu_e[62]  20000
nu_e[63]  20000
nu_e[64]  20000
nu_e[65]  20000
nu_e[66]  20000
nu_e[67]  20000
nu_e[68]  20000
nu_e[69]  20000
nu_e[70]  20000
nu_e[71]  20000
nu_e[72]  20000
nu_e[73]  20000
nu_e[74]  20000
nu_e[75]  20000
nu_e[76]  20000
nu_e[77]  20000
nu_e[78]  20000
nu_e[79]  20000
nu_e[80]  20000
nu_e[81]  20000
nu_e[82]  20000
nu_e[83]  20000
nu_e[84]  20000
nu_e[85]  20000
nu_e[86]  20000
nu_e[87]  20000
nu_e[88]  20000
nu_e[89]  20000
nu_e[90]  20000
nu_e[91]  20000
nu_e[92]  20000
nu_e[93]  20000
nu_e[94]  20000
nu_e[95]  20000
nu_e[96]  20000
nu_e[97]  20000
nu_e[98]  20000
nu_e[99]  20000
nu_e[100] 20000
nu_e[101] 20000
nu_e[102] 20000
nu_e[103] 20000
nu_e[104] 20000
nu_e[105] 20000
nu_e[106] 20000
nu_e[107] 20000
nu_e[108] 20000
nu_e[109] 20000
nu_e[110] 20000
nu_e[111] 20000
nu_e[112] 20000
nu_e[113] 20000
nu_e[114] 20000
nu_e[115] 20000
nu_e[116] 20000
nu_e[117] 20000
nu_e[118] 20000
nu_e[119] 20000
nu_e[120] 20000
nu_e[121] 20000
nu_e[122] 20000
nu_e[123] 20000
nu_e[124] 20000
nu_e[125] 20000
nu_e[126] 20000
nu_e[127] 20000
nu_e[128] 20000
nu_e[129] 20000
nu_e[130] 20000
nu_e[131] 20000
nu_e[132] 20000
nu_e[133] 20000
nu_e[134] 20000
nu_e[135] 20000
nu_e[136] 20000
nu_e[137] 20000
nu_e[138] 20000
nu_e[139] 20000
nu_e[140] 20000
nu_e[141] 20000
nu_e[142] 20000
nu_e[143] 20000
nu_e[144] 20000
nu_e[145] 20000
nu_e[146] 20000
nu_e[147] 20000
nu_e[148] 20000
nu_e[149] 20000
nu_e[150] 20000
nu_e[151] 20000
nu_e[152] 20000
nu_e[153] 20000
nu_e[154] 20000
nu_e[155] 20000
nu_e[156] 20000
nu_e[157] 20000
nu_e[158] 20000
nu_e[159] 20000
nu_e[160] 20000
nu_e[161] 20000
nu_e[162] 20000
nu_e[163] 20000
nu_e[164] 20000
nu_e[165] 20000
nu_e[166] 20000
nu_e[167] 20000
nu_e[168] 20000
nu_e[169] 20000
nu_e[170] 20000
nu_e[171] 20000
nu_e[172] 20000
nu_e[173] 20000
nu_e[174] 20000
nu_e[175] 20000
nu_e[176] 20000
nu_e[177] 20000
nu_e[178] 20000
nu_e[179] 20000
nu_e[180] 20000
nu_e[181] 20000
nu_e[182] 20000
nu_e[183] 20000
nu_e[184] 20000
nu_e[185] 20000
nu_e[186] 20000
nu_e[187] 20000
nu_e[188] 20000
nu_e[189] 20000
nu_e[190] 20000
nu_e[191] 20000
nu_e[192] 20000
nu_e[193] 20000
nu_e[194] 20000
nu_e[195] 20000
nu_e[196] 20000
nu_e[197] 20000
nu_e[198] 20000
nu_e[199] 20000
nu_e[200] 20000
nu_e[201] 20000
nu_e[202] 20000
nu_e[203] 20000
nu_e[204] 20000
nu_e[205] 20000
nu_e[206] 20000
nu_e[207] 20000
nu_e[208] 20000
nu_e[209] 20000
nu_e[210] 20000
nu_e[211] 20000
nu_e[212] 20000
nu_e[213] 20000
nu_e[214] 20000
nu_e[215] 20000
nu_e[216] 20000
nu_e[217] 20000
nu_e[218] 20000
nu_e[219] 20000
nu_e[220] 20000
nu_e[221] 20000
nu_e[222] 20000
nu_e[223] 20000
nu_e[224] 20000
nu_e[225] 20000
nu_e[226] 20000
nu_e[227] 20000
nu_e[228] 20000
nu_e[229] 20000
nu_e[230] 20000
nu_e[231] 20000
nu_e[232] 20000
nu_e[233] 20000
nu_e[234] 20000
nu_e[235] 20000
nu_e[236] 20000
nu_e[237] 20000
nu_e[238] 20000
nu_e[239] 20000
nu_e[240] 20000
nu_e[241] 20000
nu_e[242] 20000
nu_e[243] 20000
nu_e[244] 20000
nu_e[245] 20000
nu_e[246] 20000
nu_e[247] 20000
nu_e[248] 20000
nu_e[249] 20000
nu_e[250] 20000
nu_e[251] 20000
nu_e[252] 20000
nu_e[253] 20000
nu_e[254] 20000
nu_e[255] 20000
nu_e[256] 20000
nu_e[257] 20000
nu_e[258] 20000
nu_e[259] 20000
nu_e[260] 20000
nu_e[261] 20000
nu_e[262] 20000
nu_e[263] 20000
nu_e[264] 20000
nu_e[265] 20000
nu_e[266] 20000
nu_e[267] 20000
nu_e[268] 20000
nu_e[269] 20000
nu_e[270] 20000
nu_e[271] 20000
nu_e[272] 20000
nu_e[273] 20000
nu_e[274] 20000
nu_e[275] 20000
nu_e[276] 20000
nu_e[277] 20000
nu_e[278] 20000
nu_e[279] 20000
nu_e[280] 20000
nu_e[281] 20000
nu_e[282] 20000
nu_e[283] 20000
nu_e[284] 20000
nu_e[285] 20000
nu_e[286] 20000
nu_e[287] 20000
nu_e[288] 20000
nu_e[289] 20000
nu_e[290] 20000
nu_e[291] 20000
nu_e[292] 20000
nu_e[293] 20000
nu_e[294] 20000
nu_e[295] 20000
nu_e[296] 20000
nu_e[297] 20000
nu_e[298] 20000
nu_e[299] 20000
nu_e[300] 20000
s_c       20000
s_e       15000
deviance  20000

For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).

DIC info (using the rule, pD = var(deviance)/2)
pD = 7.1 and DIC = 4807.2
DIC is an estimate of expected predictive error (lower deviance is better).

The print function allows to see key posterior summaries for all parameters saved from the model, including values for posterior mean estimates, different quantiles of the posterior distribution for each parameter and diagnostic statistics such as potential scale-reduction factor or Rhat and the number of effective sample size or n.eff. Here I will not go into details about these quantities but it is enough to say that they can be used to check whether some problems occurred in the algorithm. From a first look everything seems ok. Additional checks should also be done to ensure the model behaves somewhat reasonably (e.g. no incorrect prior specification), such as posterior predictive checks, but given the simplicity of the setting I will not go into that now.

At this point however you should be asking, but what about the quantities I want to estimate, i.e. the mean QALYs and Total costs per treatment arm? how can I obtain these? Well a possible way to retrieve these is to post-process the results of the model. In particular, we can use our estimates for the conditional means \(\nu_e, \nu_c\) and standard deviations \(s_e,s_c\) in order to generate, through simulation methods, estimates for the marginal means \(\mu_e,\mu_c\) we are looking for. Although this process may seem quite complicated it is relatively simple to implement and, most importantly, can be done for most of the models we will fit, even the most complicated ones. So, in R, we do this by typing.

#obtain estimates of means by arm

#extract estimates for each mean parameter by trt group
nu_e0 <- jmodel_bn$BUGSoutput$sims.list$nu_e[,trt==0]
nu_e1 <- jmodel_bn$BUGSoutput$sims.list$nu_e[,trt==1]
nu_c0 <- jmodel_bn$BUGSoutput$sims.list$nu_c[,trt==0]
nu_c1 <- jmodel_bn$BUGSoutput$sims.list$nu_c[,trt==1]
#extract estimates for std
s_e <- jmodel_bn$BUGSoutput$sims.list$s_e
s_c <- jmodel_bn$BUGSoutput$sims.list$s_c

#create empty vectors to contain results for means by trt group
mu_e0 <- mu_c0 <- c()
mu_e1 <- mu_c1 <- c()

#set number of replications
L <- 5000

set.seed(2345) #set seed for reproducibility
#generate replications and take mean at each iteration of the posterior
for(i in 1:n.iter){
 mu_e0[i] <- mean(rnorm(L,nu_e0[i,],s_e[i])) 
 mu_e1[i] <- mean(rnorm(L,nu_e1[i,],s_e[i]))
 mu_c0[i] <- mean(rnorm(L,nu_c0[i,],s_c[i])) 
 mu_c1[i] <- mean(rnorm(L,nu_c1[i,],s_c[i])) 
}

At this point we obtained the final posterior estimates for our desired quantities, namely \(\mu_{ct}\) and \(\mu_{et}\), which can be summarised as usual, or we can even compute the incremental quantities \(\Delta_e=\mu_{e1}-\mu_{e0}\) and \(\Delta_c=\mu_{c1}-\mu_{c0}\) to see the distribution of the differences between mean outcomes by trt group (i.e. we look at the usual Cost-Effectivenss Plane).

#compute differences by arm
Delta_e <- mu_e1 - mu_e0
Delta_c <- mu_c1 - mu_c0

#plot the differences against each other
data_delta_ec <- data.frame(Delta_e,Delta_c)
ggplot(data_delta_ec, aes(x=Delta_e, y=Delta_c)) +
  geom_point(size=2, shape=16) + theme_classic()

We can then produce all standard CEA output, e.g. CEAC or CE Plane, by post-processing these posterior distributions. If you want to skip the fun, we can take advantage of the R package BCEA which is dedicated to post-processing the results from a Bayesian CEA model.

#load package and provide means e and c by group as input 
library(BCEA)
mu_e <- cbind(mu_e0,mu_e1)
mu_c <- cbind(mu_c0,mu_c1)
#produce CEA output
cea_res <- bcea(eff = mu_e, cost = mu_c, ref = 2)

#CE Plane (set wtp value)
ceplane.plot(cea_res, graph = "ggplot2", wtp = 10000)

#CEAC 
ceac.plot(cea_res, graph = "ggplot2")

#other output
summary(cea_res)

Cost-effectiveness analysis summary 

Reference intervention:  intervention 2
Comparator intervention: intervention 1

Optimal decision: choose intervention 1 for k < 4900 and intervention 2 for k >= 4900


Analysis for willingness to pay parameter k = 25000

               Expected net benefit
intervention 1                15889
intervention 2                18223

                                    EIB   CEAC   ICER
intervention 2 vs intervention 1 2334.1 0.9996 4827.2

Optimal intervention (max expected net benefit) for k = 25000: intervention 2
             
EVPI 0.088877

So, what you think? pretty cool…. Today we only scratch the surface of fitting Bayesian models for CEA with a very simple example based on normal distributions. In next posts I will show how these models can be tailored in a way to handle all problems of CEA data without the need to become crazy to figure out a way to fit the model or how to quantify the impact of uncertainty on the CEA results.I hope that I was able to catch your attention!