Baysian Weibull Model: A Problem with Parameter Estimates

2014-05-22
2014-05-22
  • Daniel Eacker
    Daniel Eacker
    2014-05-22

    Dear JAGS users,

    I have an inquiry about the parameter estimates from a Bayesian Proportional Hazards Weibull Model that I am fitting for an elk calf survival dataset that has right-censoring. Specifically, the Bayesian posterior estimates for the shape and scale parameters are different from the frequentist parameter estimates.

    However, if I compare the two approaches using randomly drawn numbers from a Weibull distribution without right-censoring, the estimates are nearly identical from the two methods.

    I am wondering if I have a problem with the priors that I have specified, or perhaps with the coding of the dinterval() to deal with the censoring?

    Here is the data set:

     weib.data
     $y
      [1]  NA  NA  NA  NA  NA  NA  68   6  NA  NA 234  NA  NA  NA  NA  NA  NA  NA
     [19]  NA  62  NA  54  NA  NA  NA 104  18 283  35 152 215  NA  NA  NA 258  NA
     [37]  NA  NA 218  NA 237  NA  NA  NA 292  NA  83 225   6  14  NA  NA  15  88
     [55]  71  NA  NA  NA  NA  NA  NA  NA  61  NA  NA  NA  19  15 244  NA  97  NA
     [73]  14 115  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA 262  NA  NA
     [91]  NA  NA  NA  NA  NA  NA  NA  NA  NA   8  12  NA  NA  NA  NA 142  NA  NA
    [109]  NA 335 121  NA  NA  NA  25  NA  NA 161  NA  24  NA  NA 145  NA 100 301
    [127]  NA  NA 173  83  NA  NA  NA   3  22  NA  NA  NA  NA  83  21  86  27  22
    [145]  NA  16  14  NA  29  NA  25  97  15  NA  25  NA  NA  40  NA  67  NA  NA
    [163] 247  NA 224  NA 195 316 257 320 193  NA 203  NA  NA  NA 192  NA  11  NA
    [181] 156  15  NA  NA  NA  NA  NA 211 100  88  NA  21  NA  NA  NA  NA  34  NA
    [199]  NA  NA  84  NA  NA  94  NA  28  20  61  NA  NA  NA 144  NA  14  13 136
    [217]  NA  NA  NA  NA  NA  NA  NA  NA  NA 273  NA  NA  NA 266  NA  13  NA   4
    [235]  NA 283   9  13  NA  12  NA  NA  NA  NA  NA  37  19 173 290  71  NA  NA
    [253]  NA  NA 154 302  NA  48  24 110  NA 157  NA 274 233  NA  NA 143  NA 271
    [271] 153  78  NA  NA  NA 128  NA
    
    $y.cens
      [1] 136 126  45  91 199  67   0   0 143  91   0 172 143 152  77 163 172 199
     [19] 199   0 143   0 219 102 193   0   0   0   0   0   0 363 363 363   0 363
     [37] 363 363   0 363   0 363 363 363   0  70   0   0   0   0 198 232   0   0
     [55]   0 218 360 168  77  42  57 163   0 113 346 330   0   0   0  14   0  51
     [73]   0   0 337 296  11  42 360 360 360 360 360 360 360 360 360   0 360 360
     [91] 360 360 190 300  76 362 168 362 362   0   0 362 362 102 362   0  76 158
    [109] 362   0   0 362  67 362   0  52 362   0  65   0 362 362   0 173   0   0
    [127] 362 362   0   0  61  88   3   0   0 132 219 363 112   0   0   0   0   0
    [145]  84   0   0 258   0  81   0   0   0 221   0  63  95   0 258   0 143 132
    [163]   0 363   0 363   0   0   0   0   0 363   0 363 248 363   0 363   0  47
    [181]   0   0 232 118 300  37 220   0   0   0  38   0 132  48  54  39   0 360
    [199] 146 198   0  53 118   0 302   0   0   0 289  27  20   0  78   0   0   0
    [217] 294 190 360 360 360 360 360 360 360   0 356 360 360   0 362   0 362   0
    [235] 122   0   0   0 114   0 362 362 362 362 362   0   0   0   0   0 362 108
    [253] 173 362   0   0 120   0   0   0 362   0 362   0   0  93 362   0 362   0
    [271]   0   0 362 362 362   0 362
    
    $nCalves
    [1] 277
    

    And here is the JAGS code:

    ##Jags code for Weibull Model
    
    data {
    for(i in 1:nCalves){
      one[i] <- 1
    }
    }
    
    model {
    for(i in 1:nCalves){ 
      one[i]~dinterval(y[i], y.cens[i])
      y[i]~dweib(shape,lambda[i])
      log(scale[i])<-alpha
      lambda[i]<-pow(scale[i],-shape)
    
    }
    # Priors
      shape ~ dunif(0,5)
      alpha ~ dnorm(0.0, 0.0001)
    
    }
    

    Here are the Bayesian estimates from the elk calf survival data:

    1. Empirical mean and standard deviation for each variable,
    plus standard error of the mean:
    
        Mean      SD Naive SE Time-series SE
    alpha 6.3239 0.14072 0.000995       0.007822
    shape 0.8025 0.06704 0.000474       0.003765
    

    Here are the frequentist estimates:

    Covariate          W.mean      Coef Exp(Coef)  se(Coef)    Wald p
    log(scale)                    5.944   381.564     0.172     0.000 
    log(shape)                   -0.595     0.551     0.143     0.000
    

    I have tried varying the number of posterior samples and nothing seems to provide a more agreeable estimate. Any advice or ideas on this topic would be greatly appreciated.

    Thank you.

    Sincerely,

    Daniel Eacker
    Wildlife Biology Program
    University of Montana

     
    Last edit: Daniel Eacker 2014-05-22