Supporting Information S8
for
Vuong, Chiu, Smouse, Fonseca, Brisson, Morin, and Ostfeld (2016). PLOS ONE.
library(coda)
library(runjags)
head(tick2)## d m n lyme hostHindexAll emptyVariable log_hostPELEprop
## 601 360 4 22 8 1.819 -1.1301030
## 602 360 13 14 6 1.890 -0.7550226
## 603 240 44 44 15 2.568 -1.1488535
## 604 360 17 13 4 1.582 -0.7918632
## 605 360 40 16 7 2.017 -0.4019712
## 606 330 0 22 11 0.555 -0.1996712
## log_hostTASTprop log_hostBLBRprop
## 601 -5.298317 -1.316768
## 602 -1.987774 -1.241329
## 603 -1.067114 -2.830218
## 604 -2.180367 -1.619488
## 605 -5.298317 -2.465104
## 606 -5.298317 -1.931022Preliminary JAGS model
Realistic model reflecting RLB success/failure
\[ \begin{aligned} \text{[2.2a]} && \left. z_{ij} \right| p_i^B \sim \text{(ind) Bernoulli}(p_i^B) \\ \text{[2.3]} && \left. \text{logit } p_i^B \right| \alpha_0, \boldsymbol{\alpha}, \tau^2, \boldsymbol{x}_i \sim \text{(ind) } N(\alpha_0 + \boldsymbol{x}_i ' \boldsymbol{\alpha}, \tau^2) \\ \text{[2.6]} && \left. v_{ij} \right| \{z_{ij}=1\}, p_i^S \sim \text{(ind) Bernoulli}(p_i^S) \\ \text{[2.4c]} && \left. t_{ij} \right| \{z_{ij}=1\}, \{v_{ij}=1\}, p_i^{SH} \sim \text{(ind) Bernoulli}(p_i^{SH}) \\ \text{[2.4c1]} && \left. t_{ij} \right| \{z_{ij}=1\}, \{v_{ij}=0\}, p_i^{FH} \sim \text{(ind) Bernoulli}(p_i^{FH}) \\ \text{[2.7]} && p_i^S = \frac{1-p_i^c/p_i^{FH}}{1-p_i^{SH}/p_i^{FH}} \\ && p_i^{SH} < p_i^c < p_i^{FH} \\ \text{[2.5]} && \text{logit } p_i^c \left| \gamma_0, \boldsymbol{\gamma}, \omega^2, \boldsymbol{x}_i \right. \sim \text{(ind) } N(\gamma_0 + \boldsymbol{x}_i ' \boldsymbol{\gamma}, \omega^2) \\ \end{aligned} \]
print(dat.c) # covariates have been centered## $S
## [1] 30
##
## $x
## hostHindexAll emptyVariable log_hostPELEprop log_hostTASTprop
## 601 -0.15313333 -0.386414919 -2.77860025
## 602 -0.08213333 -0.011334547 0.53194277
## 603 0.59586667 -0.405165468 1.45260350
## 604 -0.39013333 -0.048175117 0.33934966
## 605 0.04486667 0.341716818 -2.77860025
## 606 -1.41713333 0.544016842 -2.77860025
## 607 0.73686667 -0.258705394 1.39270536
## 608 0.01186667 0.062469427 -2.77860025
## 609 -0.13613333 -0.059274010 2.01055678
## 610 0.67486667 0.163869542 0.47949629
## 611 -0.21713333 0.459997986 -0.86167763
## 612 -0.08413333 -0.017737984 1.14929611
## 613 -0.16713333 -0.272423030 1.54155098
## 614 -0.20913333 0.387013093 1.67574705
## 615 0.06086667 0.195506627 -0.04423274
## 616 0.36086667 0.268872851 0.78244584
## 617 0.66686667 0.247751026 -0.21365089
## 618 0.30586667 0.311365475 0.94949992
## 619 0.33586667 0.188562154 0.03080245
## 620 0.56186667 0.127501898 0.24669083
## 621 -0.35113333 -0.518620344 1.19921050
## 622 0.69686667 -0.275189284 0.46399211
## 623 -0.31313333 -0.323425585 1.12941474
## 624 0.38586667 -0.075022367 0.32149204
## 625 0.21386667 0.278472924 -2.77860025
## 626 -0.18013333 0.458669082 -2.77860025
## 627 -0.37313333 0.249391715 1.56520518
## 628 -0.91313333 -2.036932857 1.85806861
## 629 -0.36913333 0.408215301 -2.77860025
## 630 -0.29713333 -0.004971854 1.44969229
## log_hostBLBRprop
## 601 0.83156557
## 602 0.90700528
## 603 -0.68188397
## 604 0.52884562
## 605 -0.31677015
## 606 0.21731233
## 607 0.48760266
## 608 0.16055952
## 609 0.73364003
## 610 0.45551435
## 611 -0.28208460
## 612 -0.71637014
## 613 0.72951632
## 614 -0.46896197
## 615 0.21731233
## 616 0.02807033
## 617 0.39387018
## 618 1.09565051
## 619 -3.14998350
## 620 0.52378232
## 621 -0.27078504
## 622 -0.19507322
## 623 1.29501794
## 624 0.45551435
## 625 0.76603153
## 626 -3.14998350
## 627 0.20342322
## 628 -3.14998350
## 629 0.73774682
## 630 1.61389838
##
## $n_noLyme
## 601 602 603 604 605
## 14 8 29 9 9
## 606 607 608 609 610
## 11 22 28 4 17
## 611 612 613 614 615
## 9 1 7 11 27
## 616 617 618 619 620
## 20 21 28 19 22
## 621 622 623 624 625
## 26 13 6 18 24
## 626 627 628 629 630
## 24 21 26 8 13
##
## $z_noLyme
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## [1,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [2,] 0 0 0 0 0 0 0 0 -999 -999 -999 -999 -999
## [3,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [4,] 0 0 0 0 0 0 0 0 0 -999 -999 -999 -999
## [5,] 0 0 0 0 0 0 0 0 0 -999 -999 -999 -999
## [6,] 0 0 0 0 0 0 0 0 0 0 0 -999 -999
## [7,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [8,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [9,] 0 0 0 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [10,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [11,] 0 0 0 0 0 0 0 0 0 -999 -999 -999 -999
## [12,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] 0 0 0 0 0 0 0 -999 -999 -999 -999 -999 -999
## [14,] 0 0 0 0 0 0 0 0 0 0 0 -999 -999
## [15,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [16,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [17,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [18,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [19,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [20,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [21,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [22,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [23,] 0 0 0 0 0 0 -999 -999 -999 -999 -999 -999 -999
## [24,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [25,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [26,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [27,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [28,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [29,] 0 0 0 0 0 0 0 0 -999 -999 -999 -999 -999
## [30,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24]
## [1,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [2,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [3,] 0 0 0 0 0 0 0 0 0 0 0
## [4,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [5,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [6,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [7,] 0 0 0 0 0 0 0 0 0 -999 -999
## [8,] 0 0 0 0 0 0 0 0 0 0 0
## [9,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [10,] 0 0 0 0 -999 -999 -999 -999 -999 -999 -999
## [11,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [12,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [14,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [15,] 0 0 0 0 0 0 0 0 0 0 0
## [16,] 0 0 0 0 0 0 0 -999 -999 -999 -999
## [17,] 0 0 0 0 0 0 0 0 -999 -999 -999
## [18,] 0 0 0 0 0 0 0 0 0 0 0
## [19,] 0 0 0 0 0 0 -999 -999 -999 -999 -999
## [20,] 0 0 0 0 0 0 0 0 0 -999 -999
## [21,] 0 0 0 0 0 0 0 0 0 0 0
## [22,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [23,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [24,] 0 0 0 0 0 -999 -999 -999 -999 -999 -999
## [25,] 0 0 0 0 0 0 0 0 0 0 0
## [26,] 0 0 0 0 0 0 0 0 0 0 0
## [27,] 0 0 0 0 0 0 0 0 -999 -999 -999
## [28,] 0 0 0 0 0 0 0 0 0 0 0
## [29,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [30,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [,25] [,26] [,27] [,28] [,29]
## [1,] -999 -999 -999 -999 -999
## [2,] -999 -999 -999 -999 -999
## [3,] 0 0 0 0 0
## [4,] -999 -999 -999 -999 -999
## [5,] -999 -999 -999 -999 -999
## [6,] -999 -999 -999 -999 -999
## [7,] -999 -999 -999 -999 -999
## [8,] 0 0 0 0 -999
## [9,] -999 -999 -999 -999 -999
## [10,] -999 -999 -999 -999 -999
## [11,] -999 -999 -999 -999 -999
## [12,] -999 -999 -999 -999 -999
## [13,] -999 -999 -999 -999 -999
## [14,] -999 -999 -999 -999 -999
## [15,] 0 0 0 -999 -999
## [16,] -999 -999 -999 -999 -999
## [17,] -999 -999 -999 -999 -999
## [18,] 0 0 0 0 -999
## [19,] -999 -999 -999 -999 -999
## [20,] -999 -999 -999 -999 -999
## [21,] 0 0 -999 -999 -999
## [22,] -999 -999 -999 -999 -999
## [23,] -999 -999 -999 -999 -999
## [24,] -999 -999 -999 -999 -999
## [25,] -999 -999 -999 -999 -999
## [26,] -999 -999 -999 -999 -999
## [27,] -999 -999 -999 -999 -999
## [28,] 0 0 -999 -999 -999
## [29,] -999 -999 -999 -999 -999
## [30,] -999 -999 -999 -999 -999
##
## $t_noLyme
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## [1,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [2,] 0 0 0 0 0 0 0 0 -999 -999 -999 -999 -999
## [3,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [4,] 0 0 0 0 0 0 0 0 0 -999 -999 -999 -999
## [5,] 0 0 0 0 0 0 0 0 0 -999 -999 -999 -999
## [6,] 0 0 0 0 0 0 0 0 0 0 0 -999 -999
## [7,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [8,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [9,] 0 0 0 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [10,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [11,] 0 0 0 0 0 0 0 0 0 -999 -999 -999 -999
## [12,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] 0 0 0 0 0 0 0 -999 -999 -999 -999 -999 -999
## [14,] 0 0 0 0 0 0 0 0 0 0 0 -999 -999
## [15,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [16,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [17,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [18,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [19,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [20,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [21,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [22,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [23,] 0 0 0 0 0 0 -999 -999 -999 -999 -999 -999 -999
## [24,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [25,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [26,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [27,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [28,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [29,] 0 0 0 0 0 0 0 0 -999 -999 -999 -999 -999
## [30,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24]
## [1,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [2,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [3,] 0 0 0 0 0 0 0 0 0 0 0
## [4,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [5,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [6,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [7,] 0 0 0 0 0 0 0 0 0 -999 -999
## [8,] 0 0 0 0 0 0 0 0 0 0 0
## [9,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [10,] 0 0 0 0 -999 -999 -999 -999 -999 -999 -999
## [11,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [12,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [14,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [15,] 0 0 0 0 0 0 0 0 0 0 0
## [16,] 0 0 0 0 0 0 0 -999 -999 -999 -999
## [17,] 0 0 0 0 0 0 0 0 -999 -999 -999
## [18,] 0 0 0 0 0 0 0 0 0 0 0
## [19,] 0 0 0 0 0 0 -999 -999 -999 -999 -999
## [20,] 0 0 0 0 0 0 0 0 0 -999 -999
## [21,] 0 0 0 0 0 0 0 0 0 0 0
## [22,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [23,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [24,] 0 0 0 0 0 -999 -999 -999 -999 -999 -999
## [25,] 0 0 0 0 0 0 0 0 0 0 0
## [26,] 0 0 0 0 0 0 0 0 0 0 0
## [27,] 0 0 0 0 0 0 0 0 -999 -999 -999
## [28,] 0 0 0 0 0 0 0 0 0 0 0
## [29,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [30,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [,25] [,26] [,27] [,28] [,29]
## [1,] -999 -999 -999 -999 -999
## [2,] -999 -999 -999 -999 -999
## [3,] 0 0 0 0 0
## [4,] -999 -999 -999 -999 -999
## [5,] -999 -999 -999 -999 -999
## [6,] -999 -999 -999 -999 -999
## [7,] -999 -999 -999 -999 -999
## [8,] 0 0 0 0 -999
## [9,] -999 -999 -999 -999 -999
## [10,] -999 -999 -999 -999 -999
## [11,] -999 -999 -999 -999 -999
## [12,] -999 -999 -999 -999 -999
## [13,] -999 -999 -999 -999 -999
## [14,] -999 -999 -999 -999 -999
## [15,] 0 0 0 -999 -999
## [16,] -999 -999 -999 -999 -999
## [17,] -999 -999 -999 -999 -999
## [18,] 0 0 0 0 -999
## [19,] -999 -999 -999 -999 -999
## [20,] -999 -999 -999 -999 -999
## [21,] 0 0 -999 -999 -999
## [22,] -999 -999 -999 -999 -999
## [23,] -999 -999 -999 -999 -999
## [24,] -999 -999 -999 -999 -999
## [25,] -999 -999 -999 -999 -999
## [26,] -999 -999 -999 -999 -999
## [27,] -999 -999 -999 -999 -999
## [28,] 0 0 -999 -999 -999
## [29,] -999 -999 -999 -999 -999
## [30,] -999 -999 -999 -999 -999
##
## $n_RLB
## 601 602 603 604 605
## 4 2 12 1 3
## 606 607 608 609 610
## 7 11 3 3 14
## 611 612 613 614 615
## 7 0 4 6 4
## 616 617 618 619 620
## 14 10 13 0 7
## 621 622 623 624 625
## 2 3 3 13 1
## 626 627 628 629 630
## 8 6 2 2 2
##
## $z_RLB
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## [1,] 1 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [2,] 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [3,] 1 1 1 1 1 1 1 1 1 1 1 1 -999
## [4,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [5,] 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [6,] 1 1 1 1 1 1 1 -999 -999 -999 -999 -999 -999
## [7,] 1 1 1 1 1 1 1 1 1 1 1 -999 -999
## [8,] 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [9,] 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [10,] 1 1 1 1 1 1 1 1 1 1 1 1 1
## [11,] 1 1 1 1 1 1 1 -999 -999 -999 -999 -999 -999
## [12,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] 1 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [14,] 1 1 1 1 1 1 -999 -999 -999 -999 -999 -999 -999
## [15,] 1 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [16,] 1 1 1 1 1 1 1 1 1 1 1 1 1
## [17,] 1 1 1 1 1 1 1 1 1 1 -999 -999 -999
## [18,] 1 1 1 1 1 1 1 1 1 1 1 1 1
## [19,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [20,] 1 1 1 1 1 1 1 -999 -999 -999 -999 -999 -999
## [21,] 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [22,] 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [23,] 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [24,] 1 1 1 1 1 1 1 1 1 1 1 1 1
## [25,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [26,] 1 1 1 1 1 1 1 1 -999 -999 -999 -999 -999
## [27,] 1 1 1 1 1 1 -999 -999 -999 -999 -999 -999 -999
## [28,] 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [29,] 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [30,] 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [,14]
## [1,] -999
## [2,] -999
## [3,] -999
## [4,] -999
## [5,] -999
## [6,] -999
## [7,] -999
## [8,] -999
## [9,] -999
## [10,] 1
## [11,] -999
## [12,] -999
## [13,] -999
## [14,] -999
## [15,] -999
## [16,] 1
## [17,] -999
## [18,] -999
## [19,] -999
## [20,] -999
## [21,] -999
## [22,] -999
## [23,] -999
## [24,] -999
## [25,] -999
## [26,] -999
## [27,] -999
## [28,] -999
## [29,] -999
## [30,] -999
##
## $v_RLB
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## [1,] 1 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [2,] 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [3,] 1 1 1 1 1 1 1 1 1 1 1 1 -999
## [4,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [5,] 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [6,] 1 1 1 1 1 1 1 -999 -999 -999 -999 -999 -999
## [7,] 1 1 1 1 1 1 1 1 1 1 1 -999 -999
## [8,] 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [9,] 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [10,] 1 1 1 1 1 1 1 1 1 1 1 1 1
## [11,] 1 1 1 1 1 1 1 -999 -999 -999 -999 -999 -999
## [12,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] 1 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [14,] 1 1 1 1 1 1 -999 -999 -999 -999 -999 -999 -999
## [15,] 1 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [16,] 1 1 1 1 1 1 1 1 1 1 1 1 1
## [17,] 1 1 1 1 1 1 1 1 1 1 -999 -999 -999
## [18,] 1 1 1 1 1 1 1 1 1 1 1 1 1
## [19,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [20,] 1 1 1 1 1 1 1 -999 -999 -999 -999 -999 -999
## [21,] 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [22,] 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [23,] 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [24,] 1 1 1 1 1 1 1 1 1 1 1 1 1
## [25,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [26,] 1 1 1 1 1 1 1 1 -999 -999 -999 -999 -999
## [27,] 1 1 1 1 1 1 -999 -999 -999 -999 -999 -999 -999
## [28,] 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [29,] 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [30,] 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [,14]
## [1,] -999
## [2,] -999
## [3,] -999
## [4,] -999
## [5,] -999
## [6,] -999
## [7,] -999
## [8,] -999
## [9,] -999
## [10,] 1
## [11,] -999
## [12,] -999
## [13,] -999
## [14,] -999
## [15,] -999
## [16,] 1
## [17,] -999
## [18,] -999
## [19,] -999
## [20,] -999
## [21,] -999
## [22,] -999
## [23,] -999
## [24,] -999
## [25,] -999
## [26,] -999
## [27,] -999
## [28,] -999
## [29,] -999
## [30,] -999
##
## $t_RLB
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## [1,] 1 1 0 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [2,] 0 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [3,] 1 0 0 1 1 1 1 1 1 1 0 0 -999
## [4,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [5,] 0 1 0 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [6,] 0 0 1 0 1 1 0 -999 -999 -999 -999 -999 -999
## [7,] 0 1 1 1 1 1 1 0 0 1 1 -999 -999
## [8,] 1 0 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [9,] 0 1 0 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [10,] 1 1 0 0 0 1 1 1 0 1 1 1 1
## [11,] 0 1 0 0 0 0 1 -999 -999 -999 -999 -999 -999
## [12,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] 0 1 1 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [14,] 0 0 0 0 0 0 -999 -999 -999 -999 -999 -999 -999
## [15,] 1 1 0 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [16,] 0 1 1 1 0 1 0 1 1 1 0 1 1
## [17,] 0 1 1 1 1 0 1 0 0 1 -999 -999 -999
## [18,] 1 1 1 0 1 0 0 0 0 1 1 0 1
## [19,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [20,] 0 0 0 0 0 1 1 -999 -999 -999 -999 -999 -999
## [21,] 0 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [22,] 0 1 0 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [23,] 1 0 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [24,] 1 1 1 0 0 1 0 1 0 1 0 1 1
## [25,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [26,] 0 0 0 0 0 0 1 0 -999 -999 -999 -999 -999
## [27,] 0 1 1 1 1 1 -999 -999 -999 -999 -999 -999 -999
## [28,] 0 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [29,] 0 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [30,] 0 0 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [,14]
## [1,] -999
## [2,] -999
## [3,] -999
## [4,] -999
## [5,] -999
## [6,] -999
## [7,] -999
## [8,] -999
## [9,] -999
## [10,] 1
## [11,] -999
## [12,] -999
## [13,] -999
## [14,] -999
## [15,] -999
## [16,] 1
## [17,] -999
## [18,] -999
## [19,] -999
## [20,] -999
## [21,] -999
## [22,] -999
## [23,] -999
## [24,] -999
## [25,] -999
## [26,] -999
## [27,] -999
## [28,] -999
## [29,] -999
## [30,] -999
##
## $n_noRLB
## 601 602 603 604 605
## 4 4 3 3 4
## 606 607 608 609 610
## 4 3 1 1 1
## 611 612 613 614 615
## 1 1 0 5 0
## 616 617 618 619 620
## 5 3 5 1 2
## 621 622 623 624 625
## 5 0 1 10 3
## 626 627 628 629 630
## 3 1 2 1 1
##
## $z_noRLB
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 1 1 1 1 -999 -999 -999 -999 -999 -999
## [2,] 1 1 1 1 -999 -999 -999 -999 -999 -999
## [3,] 1 1 1 -999 -999 -999 -999 -999 -999 -999
## [4,] 1 1 1 -999 -999 -999 -999 -999 -999 -999
## [5,] 1 1 1 1 -999 -999 -999 -999 -999 -999
## [6,] 1 1 1 1 -999 -999 -999 -999 -999 -999
## [7,] 1 1 1 -999 -999 -999 -999 -999 -999 -999
## [8,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [9,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [10,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [11,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [12,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [14,] 1 1 1 1 1 -999 -999 -999 -999 -999
## [15,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [16,] 1 1 1 1 1 -999 -999 -999 -999 -999
## [17,] 1 1 1 -999 -999 -999 -999 -999 -999 -999
## [18,] 1 1 1 1 1 -999 -999 -999 -999 -999
## [19,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [20,] 1 1 -999 -999 -999 -999 -999 -999 -999 -999
## [21,] 1 1 1 1 1 -999 -999 -999 -999 -999
## [22,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [23,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [24,] 1 1 1 1 1 1 1 1 1 1
## [25,] 1 1 1 -999 -999 -999 -999 -999 -999 -999
## [26,] 1 1 1 -999 -999 -999 -999 -999 -999 -999
## [27,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [28,] 1 1 -999 -999 -999 -999 -999 -999 -999 -999
## [29,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [30,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999
##
## $v_noRLB
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0 0 0 0 -999 -999 -999 -999 -999 -999
## [2,] 0 0 0 0 -999 -999 -999 -999 -999 -999
## [3,] 0 0 0 -999 -999 -999 -999 -999 -999 -999
## [4,] 0 0 0 -999 -999 -999 -999 -999 -999 -999
## [5,] 0 0 0 0 -999 -999 -999 -999 -999 -999
## [6,] 0 0 0 0 -999 -999 -999 -999 -999 -999
## [7,] 0 0 0 -999 -999 -999 -999 -999 -999 -999
## [8,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [9,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [10,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [11,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [12,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [14,] 0 0 0 0 0 -999 -999 -999 -999 -999
## [15,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [16,] 0 0 0 0 0 -999 -999 -999 -999 -999
## [17,] 0 0 0 -999 -999 -999 -999 -999 -999 -999
## [18,] 0 0 0 0 0 -999 -999 -999 -999 -999
## [19,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [20,] 0 0 -999 -999 -999 -999 -999 -999 -999 -999
## [21,] 0 0 0 0 0 -999 -999 -999 -999 -999
## [22,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [23,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [24,] 0 0 0 0 0 0 0 0 0 0
## [25,] 0 0 0 -999 -999 -999 -999 -999 -999 -999
## [26,] 0 0 0 -999 -999 -999 -999 -999 -999 -999
## [27,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [28,] 0 0 -999 -999 -999 -999 -999 -999 -999 -999
## [29,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [30,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
##
## $t_noRLB
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] NA NA NA NA -999 -999 -999 -999 -999 -999
## [2,] NA NA NA NA -999 -999 -999 -999 -999 -999
## [3,] NA NA NA -999 -999 -999 -999 -999 -999 -999
## [4,] NA NA NA -999 -999 -999 -999 -999 -999 -999
## [5,] NA NA NA NA -999 -999 -999 -999 -999 -999
## [6,] NA NA NA NA -999 -999 -999 -999 -999 -999
## [7,] NA NA NA -999 -999 -999 -999 -999 -999 -999
## [8,] NA -999 -999 -999 -999 -999 -999 -999 -999 -999
## [9,] NA -999 -999 -999 -999 -999 -999 -999 -999 -999
## [10,] NA -999 -999 -999 -999 -999 -999 -999 -999 -999
## [11,] NA -999 -999 -999 -999 -999 -999 -999 -999 -999
## [12,] NA -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [14,] NA NA NA NA NA -999 -999 -999 -999 -999
## [15,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [16,] NA NA NA NA NA -999 -999 -999 -999 -999
## [17,] NA NA NA -999 -999 -999 -999 -999 -999 -999
## [18,] NA NA NA NA NA -999 -999 -999 -999 -999
## [19,] NA -999 -999 -999 -999 -999 -999 -999 -999 -999
## [20,] NA NA -999 -999 -999 -999 -999 -999 -999 -999
## [21,] NA NA NA NA NA -999 -999 -999 -999 -999
## [22,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [23,] NA -999 -999 -999 -999 -999 -999 -999 -999 -999
## [24,] NA NA NA NA NA NA NA NA NA NA
## [25,] NA NA NA -999 -999 -999 -999 -999 -999 -999
## [26,] NA NA NA -999 -999 -999 -999 -999 -999 -999
## [27,] NA -999 -999 -999 -999 -999 -999 -999 -999 -999
## [28,] NA NA -999 -999 -999 -999 -999 -999 -999 -999
## [29,] NA -999 -999 -999 -999 -999 -999 -999 -999 -999
## [30,] NA -999 -999 -999 -999 -999 -999 -999 -999 -999jags.script.c <- "
model{ # realistic, RLB, alpha_2=gamma_2=0 (ignore emptyVariable)
########################### NOTE ##########################
##
## To reflect conditional binomials, partition n_i for
## each i into
##
## n_i = n_i_noLyme + n_i_RLB + n_i_noRLB
##
## and similarly for the associated data vectors z_i, t_i, v_i.
## Thus, if the partition is irrelevant to the model statement
## (e.g. Eq. [2.2a]), then must use same distribution (with
## same parameters) over all loop partitions under likelihood.
##
##############################################################
# ---------- definitions
tau <- 1/sqrt(tausq.inv) # SD in eq 2.3
omega <- 1/sqrt(omsq.inv) # SD in eq 2.5
alph[2] <- 0
gam[2] <- 0
for(i in 1:S){
pB[i] <- ilogit(logitpB[i])
pC[i] <- ilogit(logitpC[i])
pS[i] <- (1-pC[i]/pFH[i])/(1-pSH[i]/pFH[i]) # eq 2.7
logitpS[i] <- logit(pS[i])
logitpSH[i] <- logit(pSH[i])
logitpFH[i] <- logit(pFH[i])
nuL[i] <- alph0 + alph[1] * x[i,1] + inprod(alph[3:5], x[i,3:5]) # mean in eq 2.3
nuH[i] <- gam0 + gam[1] * x[i,1] + inprod(gam[3:5], x[i,3:5]) # mean in eq 2.5
}
# ---------- likelihood
for(i in 1:S){
logitpB[i] ~ dnorm(nuL[i], tausq.inv) # eq 2.3
logitpC[i] ~ dnorm(nuH[i], omsq.inv) # eq 2.5
for(j in 1:n_noLyme[i]){ # z=0
z_noLyme[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR-
}
}
for(i in 1:12){ # z=1, v=0
for(j in 1:n_noRLB[i]){
z_noRLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 0
v_noRLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 0
t_noRLB[i,j] ~ dbern(pFH[i]) # eq 2.4c1
}
}
for(i in 14:14){ # z=1, v=0
for(j in 1:n_noRLB[i]){
z_noRLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 0
v_noRLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 0
t_noRLB[i,j] ~ dbern(pFH[i]) # eq 2.4c1
}
}
for(i in 16:21){ # z=1, v=0
for(j in 1:n_noRLB[i]){
z_noRLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 0
v_noRLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 0
t_noRLB[i,j] ~ dbern(pFH[i]) # eq 2.4c1
}
}
for(i in 23:S){ # z=1, v=0
for(j in 1:n_noRLB[i]){
z_noRLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 0
v_noRLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 0
t_noRLB[i,j] ~ dbern(pFH[i]) # eq 2.4c1
}
}
for(i in 1:11){ # z=v=1
for(j in 1:n_RLB[i]){
z_RLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 1
v_RLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 1
t_RLB[i,j] ~ dbern(pSH[i]) # eq 2.4c
}
}
for(i in 13:18){ # z=v=1
for(j in 1:n_RLB[i]){
z_RLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 1
v_RLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 1
t_RLB[i,j] ~ dbern(pSH[i]) # eq 2.4c
}
}
for(i in 20:S){ # z=v=1
for(j in 1:n_RLB[i]){
z_RLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 1
v_RLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 1
t_RLB[i,j] ~ dbern(pSH[i]) # eq 2.4c
}
}
# ---------- priors
tausq.inv ~ dgamma(1, .01)
omsq.inv ~ dgamma(1, .01)
alph0 ~ dnorm(0, .001)
gam0 ~ dnorm(0, .001)
alph[1] ~ dnorm(0, .001)
gam[1] ~ dnorm(0, .001)
for(k in 3:5){
alph[k] ~ dnorm(0, .001)
gam[k] ~ dnorm(0, .001)
}
for(i in 1:S){
pSH[i] ~ dunif(0, pC[i])
pFH[i] ~ dunif(pC[i], 1)
}
}
"fit.c <- run.jags(jags.script.c, data=dat.c, n.chains=2,
inits=list(tausq.inv=1, omsq.inv=1),
adapt=10000, burnin=5000, sample=5000, thin=10,
monitor=c(
"t_noRLB[1,1]","t_noRLB[1,2]","t_noRLB[1,3]","t_noRLB[1,4]",
"t_noRLB[2,1]","t_noRLB[2,2]","t_noRLB[2,3]","t_noRLB[2,4]",
"t_noRLB[3,1]","t_noRLB[3,2]","t_noRLB[3,3]",
"t_noRLB[4,1]","t_noRLB[4,2]","t_noRLB[4,3]",
"t_noRLB[5,1]","t_noRLB[5,2]","t_noRLB[5,3]","t_noRLB[5,4]",
"t_noRLB[6,1]","t_noRLB[6,2]","t_noRLB[6,3]","t_noRLB[6,4]",
"t_noRLB[7,1]","t_noRLB[7,2]","t_noRLB[7,3]",
"t_noRLB[8,1]",
"t_noRLB[9,1]",
"t_noRLB[10,1]",
"t_noRLB[11,1]",
"t_noRLB[12,1]",
"t_noRLB[14,1]","t_noRLB[14,2]","t_noRLB[14,3]","t_noRLB[14,4]","t_noRLB[14,5]",
"t_noRLB[16,1]","t_noRLB[16,2]","t_noRLB[16,3]","t_noRLB[16,4]","t_noRLB[16,5]",
"t_noRLB[17,1]","t_noRLB[17,2]","t_noRLB[17,3]",
"t_noRLB[18,1]","t_noRLB[18,2]","t_noRLB[18,3]","t_noRLB[18,4]","t_noRLB[18,5]",
"t_noRLB[19,1]",
"t_noRLB[20,1]","t_noRLB[20,2]",
"t_noRLB[21,1]","t_noRLB[21,2]","t_noRLB[21,3]","t_noRLB[21,4]","t_noRLB[21,5]",
"t_noRLB[23,1]",
"t_noRLB[24,1]","t_noRLB[24,2]","t_noRLB[24,3]","t_noRLB[24,4]","t_noRLB[24,5]",
"t_noRLB[24,6]","t_noRLB[24,7]","t_noRLB[24,8]","t_noRLB[24,9]","t_noRLB[24,10]",
"t_noRLB[25,1]","t_noRLB[25,2]","t_noRLB[25,3]",
"t_noRLB[26,1]","t_noRLB[26,2]","t_noRLB[26,3]",
"t_noRLB[27,1]",
"t_noRLB[28,1]","t_noRLB[28,2]",
"t_noRLB[29,1]",
"t_noRLB[30,1]",
"logitpB","logitpC","logitpS","logitpSH","logitpFH",
"alph0","alph","gam0","gam","tau","omega","deviance",
"pd","dic"))## module dic loaded
## Compiling rjags model...
## Calling the simulation using the rjags method...
## Adapting the model for 10000 iterations...
## |++++++++++++++++++++++++++++++++++++++++++++++++++| 100%
## Burning in the model for 5000 iterations...
## |**************************************************| 100%
## Running the model for 50000 iterations...
## |**************************************************| 100%
## Extending 50000 iterations for pD/DIC estimates...
## |**************************************************| 100%
## Simulation complete
## Calculating summary statistics...
## Note: The monitored variables 'alph[2]' and 'gam[2]' appear to be non-stochastic; they will not be included in the
## convergence diagnostic
## Calculating the Gelman-Rubin statistic for 243 variables....
## Finished running the simulation
## Warning messages:
## 1: The length of the initial values argument supplied found does not correspond to the number of chains specified. Some initial values were recycled or ignored.
## 2: In rjags::jags.model(model, data = dataenv, inits = inits, n.chains = length(runjags.object$end.state), :
## Unused variable "t_noLyme" in data
## 3: In rjags::jags.samples(rjags, variable.names = monitor, n.iter = extra.options$sample, :
## Failed to set trace monitor for pD
## pD is infinite because at least one observed node does not have fixed supportSee this for discussion on infinite pD.
scans.c <- as.mcmc.list(fit.c)
scans.c.pooled <- rbind(scans.c[[1]], scans.c[[2]])fit.c$runjags.version## [1] "2.0.2-8" "R version 3.2.2 (2015-08-14)"
## [3] "unix" "RStudio"
## [5] "mac.binary.mavericks" "2015-09-20 17:07:51"fit.c$psrf$mpsrf # Gelman-Rubin convergence check## [1] 1.03492fit.c.dic.alt <- mean(scans.c.pooled[,"deviance"]) + var(scans.c.pooled[,"deviance"])/2
fit.c.dic.alt## [1] 1511.262pval <- apply(scans.c.pooled[,c(230,232:234,236,238:240)], 2, ecdf) # placeholder
pval <- sapply(pval,do.call,args=list(0))
pval <- data.frame(lefttail=pval,righttail=1-pval,
median=apply(scans.cNoS.pooled[,c(230,232:234,236,238:240)], 2, median))
tmp <- subset(pval,lefttail<.5)
pval <- rbind(tmp, subset(pval,righttail<.5))
print(pval) # posterior probs of < 0 (lefttail) and > 0 (righttail) -- tails sum to 1## lefttail righttail median
## alph[3] 0.0370 0.9630 0.52465413
## alph[4] 0.1062 0.8938 0.09940828
## alph[5] 0.0993 0.9007 0.12994497
## gam[1] 0.2579 0.7421 0.20047632
## gam[5] 0.0461 0.9539 0.25656229
## alph[1] 0.5721 0.4279 -0.04478082
## gam[3] 0.9614 0.0386 -0.72719620
## gam[4] 0.6976 0.3024 -0.06117828par(mfrow=c(4,3))
traceplot(scans.c)
Note with dgamma(1, .001) then larger DIC and poorer mixing (i.e., not our model of choice):
## > fit.c.dic.alt # with dgamma(1, .001)
## [1] 1558.504
Reduced realistic model (as if HIS missingness is unrelated to any mechanism, i.e. failure data ignored)
This parametrization allows easy JAGS coding without the need to partition {\((i,j)\)} into {\((i,j)\): \(t_{i,j}\) observed} and {\((i,j)\): \(t_{i,j}\) unobserved}.
\[ \begin{aligned} \text{[2.2a]} && \left. z_{ij} \right| p_i \sim \text{(ind) Bernoulli}(p_i^B) \\ \text{[2.3]} && \left. \text{logit } p_i^B \right| \alpha_0, \boldsymbol{\alpha}, \tau^2, \boldsymbol{x}_i \sim \text{(ind) } N(\alpha_0 + \boldsymbol{x}_i ' \boldsymbol{\alpha}, \tau^2) \\ \text{[3.1aa]} && \left. t_{ij} \right| z_{ij}, p_i^c \sim \text{(ind) Bernoulli}(z_{ij} p_i^c) \\ \text{[2.5]} && \text{logit } p_i^c \left| \gamma_0, \boldsymbol{\gamma}, \omega^2, \boldsymbol{x}_i \right. \sim \text{(ind) } N(\gamma_0 + \boldsymbol{x}_i ' \boldsymbol{\gamma}, \omega^2) \\ \end{aligned} \]
print(dat.a)## $S
## [1] 30
##
## $n
## [1] 22 14 44 13 16 22 36 32 8 32 17 2 11 22 31 39 34 46 20 31 33 16 10
## [24] 41 28 35 28 30 11 16
##
## $z
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## [1,] 1 0 0 1 0 1 0 0 1 0 0 0 0
## [2,] 1 0 0 0 0 1 0 1 1 0 0 0 1
## [3,] 0 1 0 0 0 0 0 1 0 1 0 1 1
## [4,] 1 1 0 0 0 0 0 1 0 0 0 0 1
## [5,] 0 1 1 0 1 0 0 1 1 0 0 1 0
## [6,] 0 0 1 1 0 1 1 1 1 0 1 1 0
## [7,] 0 0 1 0 0 0 0 0 0 1 1 0 0
## [8,] 0 0 0 0 0 1 1 0 0 1 0 0 0
## [9,] 1 1 0 1 0 0 1 0 -999 -999 -999 -999 -999
## [10,] 1 0 0 0 1 1 0 1 1 0 0 0 1
## [11,] 1 0 1 1 0 1 0 0 1 0 0 1 1
## [12,] 0 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] 1 1 1 0 0 0 1 0 0 0 0 -999 -999
## [14,] 0 0 0 0 0 1 0 1 1 1 1 0 0
## [15,] 0 0 0 1 0 0 0 0 0 1 0 0 0
## [16,] 0 0 1 0 1 0 0 1 1 1 1 0 0
## [17,] 0 1 0 1 0 0 1 0 1 0 1 1 0
## [18,] 0 0 1 0 0 0 0 0 0 0 1 1 0
## [19,] 0 0 1 0 0 0 0 0 0 0 0 0 0
## [20,] 0 1 0 0 0 0 0 1 0 0 1 0 0
## [21,] 0 1 0 0 0 0 0 0 0 0 0 0 1
## [22,] 0 0 0 0 0 0 0 0 1 1 0 0 1
## [23,] 0 0 0 0 0 1 1 1 0 1 -999 -999 -999
## [24,] 0 0 1 0 1 1 1 1 0 1 1 0 1
## [25,] 0 0 0 1 0 0 0 0 0 0 0 1 0
## [26,] 0 0 1 0 0 0 0 0 0 0 0 0 0
## [27,] 0 0 0 0 0 0 0 0 0 0 0 1 0
## [28,] 0 0 1 0 0 0 0 0 0 0 0 0 0
## [29,] 1 0 0 0 0 0 1 1 0 0 0 -999 -999
## [30,] 0 0 0 0 0 0 1 0 0 1 0 1 0
## [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24]
## [1,] 0 0 1 1 0 0 0 1 1 -999 -999
## [2,] 1 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [3,] 0 0 0 1 0 0 0 1 0 0 0
## [4,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [5,] 0 0 1 -999 -999 -999 -999 -999 -999 -999 -999
## [6,] 0 1 1 0 1 0 0 0 0 -999 -999
## [7,] 1 1 1 0 0 1 1 0 0 0 1
## [8,] 0 0 0 0 1 0 0 0 0 0 0
## [9,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [10,] 1 0 1 1 0 1 0 1 0 0 1
## [11,] 1 0 0 0 -999 -999 -999 -999 -999 -999 -999
## [12,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [14,] 0 1 1 1 1 0 0 1 1 -999 -999
## [15,] 0 1 0 0 0 0 0 0 0 0 0
## [16,] 0 0 0 0 1 1 0 0 1 0 1
## [17,] 0 0 0 1 0 0 0 1 0 1 1
## [18,] 0 0 0 0 1 1 0 1 1 0 1
## [19,] 0 0 0 0 0 0 0 -999 -999 -999 -999
## [20,] 0 0 0 1 1 0 0 0 0 1 0
## [21,] 0 0 0 0 1 0 0 1 0 0 0
## [22,] 0 0 0 -999 -999 -999 -999 -999 -999 -999 -999
## [23,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [24,] 1 1 0 1 1 1 1 0 0 0 0
## [25,] 0 0 0 0 0 1 0 0 0 0 0
## [26,] 1 0 0 0 0 1 1 0 0 0 1
## [27,] 0 1 0 0 1 1 0 0 0 0 1
## [28,] 0 0 0 0 0 0 1 0 1 0 0
## [29,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [30,] 0 0 0 -999 -999 -999 -999 -999 -999 -999 -999
## [,25] [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35]
## [1,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [2,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [3,] 1 0 0 1 0 0 1 0 1 0 1
## [4,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [5,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [6,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [7,] 1 1 0 0 0 1 0 1 0 1 0
## [8,] 0 0 0 0 0 0 0 0 -999 -999 -999
## [9,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [10,] 1 0 0 1 0 0 0 1 -999 -999 -999
## [11,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [12,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [14,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [15,] 0 0 0 0 1 0 0 -999 -999 -999 -999
## [16,] 1 0 0 0 1 1 1 1 0 0 1
## [17,] 1 0 1 0 0 0 0 0 1 0 -999
## [18,] 1 1 1 1 1 0 0 0 1 0 0
## [19,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [20,] 0 0 0 1 0 1 1 -999 -999 -999 -999
## [21,] 0 1 0 0 0 1 0 1 0 -999 -999
## [22,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [23,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [24,] 1 0 1 1 0 1 0 0 0 1 1
## [25,] 1 0 0 0 -999 -999 -999 -999 -999 -999 -999
## [26,] 1 0 0 1 0 1 1 0 0 1 1
## [27,] 1 0 1 0 -999 -999 -999 -999 -999 -999 -999
## [28,] 1 0 0 0 0 0 -999 -999 -999 -999 -999
## [29,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [30,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [,36] [,37] [,38] [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46]
## [1,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [2,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [3,] 0 0 0 0 0 0 1 1 1 -999 -999
## [4,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [5,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [6,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [7,] 0 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [8,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [9,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [10,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [11,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [12,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [13,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [14,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [15,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [16,] 1 0 1 1 -999 -999 -999 -999 -999 -999 -999
## [17,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [18,] 0 1 1 0 0 1 0 0 0 0 1
## [19,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [20,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [21,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [22,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [23,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [24,] 0 1 0 1 1 0 -999 -999 -999 -999 -999
## [25,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [26,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [27,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [28,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [29,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [30,] -999 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
##
## $t
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## 601 1 0 0 1 0 NA 0 0 NA 0 0
## 602 NA 0 0 0 0 0 0 NA NA 0 0
## 603 0 1 0 0 0 0 0 0 0 NA 0
## 604 NA 1 0 0 0 0 0 NA 0 0 0
## 605 0 0 1 0 NA 0 0 NA NA 0 0
## 606 0 0 0 NA 0 0 1 NA 0 0 1
## 607 0 0 NA 0 0 0 0 0 0 NA 0
## 608 0 0 0 0 0 NA 1 0 0 0 0
## 609 0 1 0 NA 0 0 0 0 -999 -999 -999
## 610 1 0 0 0 1 0 0 0 0 0 0
## 611 0 0 1 0 0 0 0 0 0 0 0
## 612 0 NA -999 -999 -999 -999 -999 -999 -999 -999 -999
## 613 0 1 1 0 0 0 1 0 0 0 0
## 614 0 0 0 0 0 NA 0 0 0 0 NA
## 615 0 0 0 1 0 0 0 0 0 1 0
## 616 0 0 0 0 1 0 0 1 NA 1 0
## 617 0 0 0 NA 0 0 1 0 1 0 1
## 618 0 0 1 0 0 0 0 0 0 0 NA
## 619 0 0 NA 0 0 0 0 0 0 0 0
## 620 0 0 0 0 0 0 0 0 0 0 NA
## 621 0 0 0 0 0 0 0 0 0 0 0
## 622 0 0 0 0 0 0 0 0 0 1 0
## 623 0 0 0 0 0 1 NA 0 0 1 -999
## 624 0 0 NA 0 NA NA 1 1 0 1 NA
## 625 0 0 0 1 0 0 0 0 0 0 0
## 626 0 0 0 0 0 0 0 0 0 0 0
## 627 0 0 0 0 0 0 0 0 0 0 0
## 628 0 0 NA 0 0 0 0 0 0 0 0
## 629 NA 0 0 0 0 0 0 1 0 0 0
## 630 0 0 0 0 0 0 0 0 0 NA 0
## [,12] [,13] [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21]
## 601 0 0 0 0 0 NA 0 0 0 1
## 602 0 NA 1 -999 -999 -999 -999 -999 -999 -999
## 603 NA 0 0 0 0 1 0 0 0 NA
## 604 0 NA -999 -999 -999 -999 -999 -999 -999 -999
## 605 0 0 0 0 NA -999 -999 -999 -999 -999
## 606 NA 0 0 1 NA 0 0 0 0 0
## 607 0 0 1 NA 1 0 0 1 1 0
## 608 0 0 0 0 0 0 1 0 0 0
## 609 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 610 0 1 1 0 1 0 0 1 0 1
## 611 0 1 NA 0 0 0 -999 -999 -999 -999
## 612 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 613 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 614 0 0 0 NA 0 NA NA 0 0 0
## 615 0 0 0 0 0 0 0 0 0 0
## 616 0 0 0 0 0 0 NA 1 0 0
## 617 NA 0 0 0 0 1 0 0 0 0
## 618 1 0 0 0 0 0 1 0 0 NA
## 619 0 0 0 0 0 0 0 0 0 -999
## 620 0 0 0 0 0 0 0 0 0 0
## 621 0 1 0 0 0 0 NA 0 0 NA
## 622 0 0 0 0 0 -999 -999 -999 -999 -999
## 623 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 624 0 0 0 NA 0 1 0 1 0 0
## 625 NA 0 0 0 0 0 0 NA 0 0
## 626 0 0 0 0 0 0 0 NA 0 0
## 627 0 0 0 1 0 0 1 1 0 0
## 628 0 0 0 0 0 0 0 0 NA 0
## 629 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 630 0 0 0 0 0 -999 -999 -999 -999 -999
## [,22] [,23] [,24] [,25] [,26] [,27] [,28] [,29] [,30] [,31]
## 601 NA -999 -999 -999 -999 -999 -999 -999 -999 -999
## 602 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 603 0 0 0 1 0 0 1 0 0 1
## 604 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 605 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 606 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 607 0 0 1 1 0 0 0 0 0 0
## 608 0 0 0 0 0 0 0 0 0 0
## 609 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 610 0 0 1 1 0 0 1 0 0 0
## 611 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 612 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 613 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 614 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 615 0 0 0 0 0 0 0 1 0 0
## 616 NA 0 0 1 0 0 0 1 NA 1
## 617 0 1 NA 0 0 0 0 0 0 0
## 618 1 0 0 0 NA 0 NA NA 0 0
## 619 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 620 0 NA 0 0 0 0 0 0 1 1
## 621 0 0 0 0 NA 0 0 0 NA 0
## 622 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 623 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 624 0 0 0 1 0 NA 0 0 1 0
## 625 0 0 0 NA 0 0 0 -999 -999 -999
## 626 0 0 0 0 0 0 0 0 NA NA
## 627 0 0 1 1 0 NA 0 -999 -999 -999
## 628 0 0 0 1 0 0 0 0 0 -999
## 629 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 630 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [,32] [,33] [,34] [,35] [,36] [,37] [,38] [,39] [,40] [,41]
## 601 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 602 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 603 0 1 0 1 0 0 0 0 0 0
## 604 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 605 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 606 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 607 1 0 1 0 0 -999 -999 -999 -999 -999
## 608 0 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 609 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 610 NA -999 -999 -999 -999 -999 -999 -999 -999 -999
## 611 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 612 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 613 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 614 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 615 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 616 0 0 0 1 NA 0 1 1 -999 -999
## 617 0 1 0 -999 -999 -999 -999 -999 -999 -999
## 618 0 0 0 0 0 1 1 0 0 0
## 619 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 620 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 621 NA 0 -999 -999 -999 -999 -999 -999 -999 -999
## 622 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 623 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 624 0 0 1 NA 0 NA 0 NA NA 0
## 625 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 626 0 0 1 0 -999 -999 -999 -999 -999 -999
## 627 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 628 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 629 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## 630 -999 -999 -999 -999 -999 -999 -999 -999 -999 -999
## [,42] [,43] [,44] [,45] [,46]
## 601 -999 -999 -999 -999 -999
## 602 -999 -999 -999 -999 -999
## 603 1 0 0 -999 -999
## 604 -999 -999 -999 -999 -999
## 605 -999 -999 -999 -999 -999
## 606 -999 -999 -999 -999 -999
## 607 -999 -999 -999 -999 -999
## 608 -999 -999 -999 -999 -999
## 609 -999 -999 -999 -999 -999
## 610 -999 -999 -999 -999 -999
## 611 -999 -999 -999 -999 -999
## 612 -999 -999 -999 -999 -999
## 613 -999 -999 -999 -999 -999
## 614 -999 -999 -999 -999 -999
## 615 -999 -999 -999 -999 -999
## 616 -999 -999 -999 -999 -999
## 617 -999 -999 -999 -999 -999
## 618 0 0 0 0 1
## 619 -999 -999 -999 -999 -999
## 620 -999 -999 -999 -999 -999
## 621 -999 -999 -999 -999 -999
## 622 -999 -999 -999 -999 -999
## 623 -999 -999 -999 -999 -999
## 624 -999 -999 -999 -999 -999
## 625 -999 -999 -999 -999 -999
## 626 -999 -999 -999 -999 -999
## 627 -999 -999 -999 -999 -999
## 628 -999 -999 -999 -999 -999
## 629 -999 -999 -999 -999 -999
## 630 -999 -999 -999 -999 -999
##
## $x
## hostHindexAll emptyVar log_hostPELEprop log_hostTASTprop
## 601 -0.15313333 -0.386414919 -2.77860025
## 602 -0.08213333 -0.011334547 0.53194277
## 603 0.59586667 -0.405165468 1.45260350
## 604 -0.39013333 -0.048175117 0.33934966
## 605 0.04486667 0.341716818 -2.77860025
## 606 -1.41713333 0.544016842 -2.77860025
## 607 0.73686667 -0.258705394 1.39270536
## 608 0.01186667 0.062469427 -2.77860025
## 609 -0.13613333 -0.059274010 2.01055678
## 610 0.67486667 0.163869542 0.47949629
## 611 -0.21713333 0.459997986 -0.86167763
## 612 -0.08413333 -0.017737984 1.14929611
## 613 -0.16713333 -0.272423030 1.54155098
## 614 -0.20913333 0.387013093 1.67574705
## 615 0.06086667 0.195506627 -0.04423274
## 616 0.36086667 0.268872851 0.78244584
## 617 0.66686667 0.247751026 -0.21365089
## 618 0.30586667 0.311365475 0.94949992
## 619 0.33586667 0.188562154 0.03080245
## 620 0.56186667 0.127501898 0.24669083
## 621 -0.35113333 -0.518620344 1.19921050
## 622 0.69686667 -0.275189284 0.46399211
## 623 -0.31313333 -0.323425585 1.12941474
## 624 0.38586667 -0.075022367 0.32149204
## 625 0.21386667 0.278472924 -2.77860025
## 626 -0.18013333 0.458669082 -2.77860025
## 627 -0.37313333 0.249391715 1.56520518
## 628 -0.91313333 -2.036932857 1.85806861
## 629 -0.36913333 0.408215301 -2.77860025
## 630 -0.29713333 -0.004971854 1.44969229
## log_hostBLBRprop
## 601 0.83156557
## 602 0.90700528
## 603 -0.68188397
## 604 0.52884562
## 605 -0.31677015
## 606 0.21731233
## 607 0.48760266
## 608 0.16055952
## 609 0.73364003
## 610 0.45551435
## 611 -0.28208460
## 612 -0.71637014
## 613 0.72951632
## 614 -0.46896197
## 615 0.21731233
## 616 0.02807033
## 617 0.39387018
## 618 1.09565051
## 619 -3.14998350
## 620 0.52378232
## 621 -0.27078504
## 622 -0.19507322
## 623 1.29501794
## 624 0.45551435
## 625 0.76603153
## 626 -3.14998350
## 627 0.20342322
## 628 -3.14998350
## 629 0.73774682
## 630 1.61389838jags.script.aa <- "
model{ # same as 'c' except ignore RLB failure, alpha_2=gamma_2=0 (ignore emptyVar)
# ---------- definitions
tau <- 1/sqrt(tausq.inv) # SD in eq 2.3
omega <- 1/sqrt(omsq.inv) # SD in eq 2.5
alph[2] <- 0
gam[2] <- 0
for(i in 1:S){
pB[i] <- ilogit(logitpB[i])
pC[i] <- ilogit(logitpC[i])
nuL[i] <- alph0 + alph[1] * x[i,1] + inprod(alph[3:5], x[i,3:5]) # mean in eq 2.3
nuH[i] <- gam0 + gam[1] * x[i,1] + inprod(gam[3:5], x[i,3:5]) # mean in eq 2.5
for(j in 1:n[i]){
pH[i,j] <- z[i,j] * pC[i] # inside eq 3.1aa
}
}
# ---------- likelihood
for(i in 1:S){
logitpB[i] ~ dnorm(nuL[i], tausq.inv) # eq 2.3
logitpC[i] ~ dnorm(nuH[i], omsq.inv) # eq 2.5
for(j in 1:n[i]){
z[i,j] ~ dbern(pB[i]) # eq 2.2a
t[i,j] ~ dbern(pH[i,j]) # eq 3.1aa
}
}
# ---------- priors
tausq.inv ~ dgamma(1, .01)
omsq.inv ~ dgamma(1, .01)
alph0 ~ dnorm(0, .001)
gam0 ~ dnorm(0, .001)
alph[1] ~ dnorm(0, .001)
gam[1] ~ dnorm(0, .001)
for(k in 3:5){
alph[k] ~ dnorm(0, .001)
gam[k] ~ dnorm(0, .001)
}
}
"fit.aa <- run.jags(jags.script.aa, data=dat.a, n.chains=2,
inits=list(tausq.inv=1, omsq.inv=1),
adapt=1000, burnin=10000, sample=5000, thin=10,
monitor=c(
"t[1,6]","t[1,9]","t[1,17]","t[1,22]","t[2,1]",
"t[2,8]","t[2,9]","t[2,13]","t[3,10]","t[3,12]",
"t[3,21]","t[4,1]","t[4,8]","t[4,13]","t[5,5]",
"t[5,8]","t[5,9]","t[5,16]","t[6,4]","t[6,8]",
"t[6,12]","t[6,16]","t[7,3]","t[7,10]","t[7,15]",
"t[8,6]","t[9,4]","t[10,32]","t[11,14]","t[12,2]",
"t[14,6]","t[14,11]","t[14,15]","t[14,17]","t[14,18]",
"t[16,9]","t[16,18]","t[16,22]","t[16,30]","t[16,36]",
"t[17,4]","t[17,12]","t[17,24]","t[18,11]","t[18,21]",
"t[18,26]","t[18,28]","t[18,29]","t[19,3]","t[20,11]",
"t[20,23]","t[21,18]","t[21,21]","t[21,26]","t[21,30]",
"t[21,32]","t[23,7]","t[24,3]","t[24,5]","t[24,6]",
"t[24,11]","t[24,15]","t[24,27]","t[24,35]","t[24,37]",
"t[24,39]","t[24,40]","t[25,12]","t[25,19]","t[25,25]",
"t[26,19]","t[26,30]","t[26,31]","t[27,27]","t[28,3]",
"t[28,20]","t[29,1]","t[30,10]",
"logitpB","logitpC","alph0","gam0","alph","gam","tau","omega","deviance","pd","dic"))## module dic loaded
## Compiling rjags model...
## Calling the simulation using the rjags method...
## Adapting the model for 1000 iterations...
## |++++++++++++++++++++++++++++++++++++++++++++++++++| 100%
## Burning in the model for 10000 iterations...
## |**************************************************| 100%
## Running the model for 50000 iterations...
## |**************************************************| 100%
## Extending 50000 iterations for pD/DIC estimates...
## |**************************************************| 100%
## Simulation complete
## Calculating summary statistics...
## Note: The monitored variables 'alph[2]' and 'gam[2]' appear to be non-stochastic;
## they will not be included in the convergence diagnostic
## Calculating the Gelman-Rubin statistic for 153 variables....
## Finished running the simulation
## Warning message:
## The length of the initial values argument supplied found does not correspond to the number of chains specified. Some initial values were recycled or ignored. scans.aa <- as.mcmc.list(fit.aa)
scans.aa.pooled <- rbind(scans.aa[[1]], scans.aa[[2]])fit.aa$psrf$mpsrf # Gelman-Rubin convergence check## [1] 1.031637fit.aa.dic.alt <- mean(scans.aa.pooled[,"deviance"]) + var(scans.aa.pooled[,"deviance"])/2
fit.aa.dic.alt## [1] 1190.508par(mfrow=c(4,3))
traceplot(scans.aa)
No apparent mixing issues, even for logit(\(p_i^c\)) for \(i\) = 12 (ID=612), 14 (ID=614), 19 (ID=619), 30 (ID=630) where RLB tests gave either all NAs or most nonNAs that were 0.
Hypothetical model as if RLB test had been administered to entire sample
\[ \begin{aligned} \text{[2.2a]} && \left. z_{ij} \right| p_i^B \sim \text{(ind) Bernoulli}(p_i^B) \\ \text{[2.3]} && \left. \text{logit } p_i^B \right| \alpha_0, \boldsymbol{\alpha}, \tau^2, \boldsymbol{x}_i \sim \text{(ind) } N(\alpha_0 + \boldsymbol{x}_i ' \boldsymbol{\alpha}, \tau^2) \\ \text{[2.6]} && \left. v_{ij} \right| \{z_{ij}=1\}, p_i^S \sim \text{(ind) Bernoulli}(p_i^S) \\ \text{[3.1a]} && \left. t_{ij} \right| p_i^B, p_i^c \sim \text{(ind) Bernoulli}(p_i^B p_i^c) \\ \text{[2.7a]} && p_i^S = \frac{1-p_i^c/p_i^{FH}}{1-p_i^{SH}/p_i^{FH}} \\ && p_i^{SH} < p_i^c < p_i^{FH} \\ \text{[2.5]} && \text{logit } p_i^c \left| \gamma_0, \boldsymbol{\gamma}, \omega^2, \boldsymbol{x}_i \right. \sim \text{(ind) } N(\gamma_0 + \boldsymbol{x}_i ' \boldsymbol{\gamma}, \omega^2) \\ \end{aligned} \]
jags.script.cc <- "
model{ # hypothetical, RLB, alpha_2=gamma_2=0 (ignore emptyVariable)
########################### NOTE ##########################
##
## To reflect conditional binomials, partition n_i for
## each i into
##
## n_i = n_i_noLyme + n_i_RLB + n_i_noRLB
##
## and similarly for the associated data vectors z_i, t_i, v_i.
## Thus, if the partition is irrelevant to the model statement
## (e.g. Eq. [3.1a]), then must use same distribution (with
## same parameters) over all loop partitions under likelihood.
##
##############################################################
# ---------- definitions
tau <- 1/sqrt(tausq.inv) # SD in eq 2.3
omega <- 1/sqrt(omsq.inv) # SD in eq 2.5
alph[2] <- 0
gam[2] <- 0
for(i in 1:S){
pB[i] <- ilogit(logitpB[i])
pC[i] <- ilogit(logitpC[i])
pS[i] <- (1-pC[i]/pFH[i])/(1-pSH[i]/pFH[i]) # eq 2.7
logitpS[i] <- logit(pS[i])
pH[i] <- pB[i] * pC[i] # inside eq. 3.1a
logitpSH[i] <- logit(pSH[i])
logitpFH[i] <- logit(pFH[i])
nuL[i] <- alph0 + alph[1] * x[i,1] + inprod(alph[3:5], x[i,3:5]) # mean in eq 2.3
nuH[i] <- gam0 + alph[1] * x[i,1] + inprod(gam[3:5], x[i,3:5]) # mean in eq 2.5
}
# ---------- likelihood
for(i in 1:S){
logitpB[i] ~ dnorm(nuL[i], tausq.inv) # eq 2.3
logitpC[i] ~ dnorm(nuH[i], omsq.inv) # eq 2.5
for(j in 1:n_noLyme[i]){ # z=0
z_noLyme[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR-
t_noLyme[i,j] ~ dbern(pH[i]) # eq 3.1a for PCR-
}
}
for(i in 1:12){ # z=1, v=0
for(j in 1:n_noRLB[i]){
z_noRLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 0
v_noRLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 0
t_noRLB[i,j] ~ dbern(pH[i]) # eq 3.1a for PCR+, v[i,j] = 0
}
}
for(i in 14:14){ # z=1, v=0
for(j in 1:n_noRLB[i]){
z_noRLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 0
v_noRLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 0
t_noRLB[i,j] ~ dbern(pH[i]) # eq 3.1a for PCR+, v[i,j] = 0
}
}
for(i in 16:21){ # z=1, v=0
for(j in 1:n_noRLB[i]){
z_noRLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 0
v_noRLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 0
t_noRLB[i,j] ~ dbern(pH[i]) # eq 3.1a for PCR+, v[i,j] = 0
}
}
for(i in 23:S){ # z=1, v=0
for(j in 1:n_noRLB[i]){
z_noRLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 0
v_noRLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 0
t_noRLB[i,j] ~ dbern(pFH[i]) # eq 3.1a for PCR+, v[i,j] = 0
}
}
for(i in 1:11){ # z=v=1
for(j in 1:n_RLB[i]){
z_RLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 1
v_RLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 1
t_RLB[i,j] ~ dbern(pSH[i]) # eq 3.1a for PCR+, v[i,j] = 1
}
}
for(i in 13:18){ # z=v=1
for(j in 1:n_RLB[i]){
z_RLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 1
v_RLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 1
t_RLB[i,j] ~ dbern(pSH[i]) # eq 3.1a for PCR+, v[i,j] = 1
}
}
for(i in 20:S){ # z=v=1
for(j in 1:n_RLB[i]){
z_RLB[i,j] ~ dbern(pB[i]) # eq 2.2 for PCR+, v[i,j] = 1
v_RLB[i,j] ~ dbern(pS[i]) # eq 2.6 for PCR+, v[i,j] = 1
t_RLB[i,j] ~ dbern(pSH[i]) # eq 3.1a for PCR+, v[i,j] = 1
}
}
# ---------- priors
tausq.inv ~ dgamma(1, .01)
omsq.inv ~ dgamma(1, .01)
alph0 ~ dnorm(0, .001)
gam0 ~ dnorm(0, .001)
alph[1] ~ dnorm(0, .001)
gam[1] ~ dnorm(0, .001)
for(k in 3:5){
alph[k] ~ dnorm(0, .001)
gam[k] ~ dnorm(0, .001)
}
for(i in 1:S){
pSH[i] ~ dunif(0, pC[i])
pFH[i] ~ dunif(pC[i], 1)
}
}
"fit.cc <- run.jags(jags.script.cc, data=dat.c, n.chains=2,
inits=list(tausq.inv=1, omsq.inv=1),
adapt=10000, burnin=10000, sample=5000, thin=10,
monitor=c(
"t_noRLB[1,1]","t_noRLB[1,2]","t_noRLB[1,3]","t_noRLB[1,4]",
"t_noRLB[2,1]","t_noRLB[2,2]","t_noRLB[2,3]","t_noRLB[2,4]",
"t_noRLB[3,1]","t_noRLB[3,2]","t_noRLB[3,3]",
"t_noRLB[4,1]","t_noRLB[4,2]","t_noRLB[4,3]",
"t_noRLB[5,1]","t_noRLB[5,2]","t_noRLB[5,3]","t_noRLB[5,4]",
"t_noRLB[6,1]","t_noRLB[6,2]","t_noRLB[6,3]","t_noRLB[6,4]",
"t_noRLB[7,1]","t_noRLB[7,2]","t_noRLB[7,3]",
"t_noRLB[8,1]",
"t_noRLB[9,1]",
"t_noRLB[10,1]",
"t_noRLB[11,1]",
"t_noRLB[12,1]",
"t_noRLB[14,1]","t_noRLB[14,2]","t_noRLB[14,3]","t_noRLB[14,4]","t_noRLB[14,5]",
"t_noRLB[16,1]","t_noRLB[16,2]","t_noRLB[16,3]","t_noRLB[16,4]","t_noRLB[16,5]",
"t_noRLB[17,1]","t_noRLB[17,2]","t_noRLB[17,3]",
"t_noRLB[18,1]","t_noRLB[18,2]","t_noRLB[18,3]","t_noRLB[18,4]","t_noRLB[18,5]",
"t_noRLB[19,1]",
"t_noRLB[20,1]","t_noRLB[20,2]",
"t_noRLB[21,1]","t_noRLB[21,2]","t_noRLB[21,3]","t_noRLB[21,4]","t_noRLB[21,5]",
"t_noRLB[23,1]",
"t_noRLB[24,1]","t_noRLB[24,2]","t_noRLB[24,3]","t_noRLB[24,4]","t_noRLB[24,5]",
"t_noRLB[24,6]","t_noRLB[24,7]","t_noRLB[24,8]","t_noRLB[24,9]","t_noRLB[24,10]",
"t_noRLB[25,1]","t_noRLB[25,2]","t_noRLB[25,3]",
"t_noRLB[26,1]","t_noRLB[26,2]","t_noRLB[26,3]",
"t_noRLB[27,1]",
"t_noRLB[28,1]","t_noRLB[28,2]",
"t_noRLB[29,1]",
"t_noRLB[30,1]",
"logitpB","logitpC","logitpS","logitpSH","logitpFH",
"alph0","alph","gam0","gam","tau","omega","deviance","pd","dic"))## module dic loaded
## Compiling rjags model...
## Calling the simulation using the rjags method...
## Adapting the model for 10000 iterations...
## |++++++++++++++++++++++++++++++++++++++++++++++++++| 100%
## Burning in the model for 10000 iterations...
## |**************************************************| 100%
## Running the model for 50000 iterations...
## |**************************************************| 100%
## Extending 50000 iterations for pD/DIC estimates...
## |**************************************************| 100%
## Simulation complete
## Calculating summary statistics...
## Note: The monitored variables 'alph[2]' and 'gam[2]' appear to be non-stochastic; they will not be
## included in the convergence diagnostic
## Calculating the Gelman-Rubin statistic for 243 variables....
## Finished running the simulation
## Warning messages:
## 1: The length of the initial values argument supplied found does not correspond to the number of chains specified. Some initial values were recycled or ignored.
## 2: In rjags::jags.samples(rjags, variable.names = monitor, n.iter = extra.options$sample, :
## Failed to set trace monitor for pD
## pD is infinite because at least one observed node does not have fixed supportscans.cc <- as.mcmc.list(fit.cc)
scans.cc.pooled <- rbind(scans.cc[[1]], scans.cc[[2]])fit.cc$psrf$mpsrf # Gelman-Rubin convergence check## [1] 1.032797fit.cc.dic.alt <- mean(scans.cc.pooled[,"deviance"]) + var(scans.cc.pooled[,"deviance"])/2
fit.cc.dic.alt## [1] 1660.277par(mfrow=c(4,3))
traceplot(scans.cc)
Diagnosis for realistic model with RLB
Now we diagnose the “residuals” \(\eta\) and \(\xi\) (note that normality of posterior of each \(\eta_i\) and \(\xi_i\) is irrelevant):
eta <- scans.c.pooled[,79:108] # just logit(pB)
eta <- eta - scans.c.pooled[,"alph0"] # logit(pB) - alph0
eta <- eta - scans.c.pooled[,c(230,232:234)] %*% t(dat$x[,c(1,3:5)]) # actual eta
xi <- scans.c.pooled[,109:138] # just logit(pC)
xi <- xi - scans.c.pooled[,"gam0"] # logit(pC) - gam0
xi <- xi - scans.c.pooled[,c(236,238:240)] %*% t(dat$x[,c(1,3:5)]) # actual xipar(mfrow=c(4,1))
par(cex.lab=1.5)
par(cex.main=2)
zoomlim <- 2.5
densfac <- 4
covar <- colnames(dat$x)[c(1,3:5)]
dens <- apply(eta, 2, density)
dens.x <- NULL
dens.y <- NULL
for(i in 1:nrow(dat$x)){
dens.x <- cbind(dens.x, dens[[i]]$x)
dens.y <- cbind(dens.y, dens[[i]]$y)
}
for(k in covar){
plot(1:2, type="n", xlim=range(dat$x[,k]), ylim=range(dens.x),
xlab=paste("2006",k), ylab="eta posterior") # dummy plot
abline(h=0)
for(i in 1:nrow(dat$x))
points(rep(dat$x[i,k],length(dens.x[,i])), dens.x[,i], cex=dens.y[,i]*densfac)
}
tmp <- rbind( c(-.4,.8),
c(-.6,.6),
c(-1,2.1),
c(-.8,1.8)
)
rownames(tmp) <- covar
for(k in covar){
plot(1:2, type="n", ylim=c(-zoomlim,zoomlim),
xlim=tmp[k,], main="omit influential resid, zoomed in",
xlab=paste("2006",k), ylab="eta posterior") # dummy plot
abline(h=0)
for(i in 1:nrow(dat$x))
points(rep(dat$x[i,k],length(dens.x[,i])), dens.x[,i], cex=dens.y[,i]*densfac)
}
plot(1:nrow(dat$x), type="n", ylim=range(dens.x),
xlab="2006 site i", ylab="eta posterior") # dummy plot
abline(h=0)
for(i in 1:nrow(dat$x))
points(rep(i,length(dens.x[,i])), dens.x[,i], cex=dens.y[,i]*3)
plot(1:nrow(dat$x), type="n", ylim=c(-zoomlim,zoomlim),
xlab="2006 site i", ylab="eta posterior", main="zoomed in") # dummy plot
abline(h=0)
for(i in 1:nrow(dat$x))
points(rep(i,length(dens.x[,i])), dens.x[,i], cex=dens.y[,i]*3)
par(mfrow=c(4,1))
par(cex.lab=1.5)
par(cex.main=2)
zoomlim <- .6
densfac <- 1.5
covar <- colnames(dat$x)[c(1,3:5)]
dens <- apply(xi, 2, density)
dens.x <- NULL
dens.y <- NULL
for(i in 1:nrow(dat$x)){
dens.x <- cbind(dens.x, dens[[i]]$x)
dens.y <- cbind(dens.y, dens[[i]]$y)
}
tmp <- rbind( c(-.4,.8),
c(-.6,.6),
c(-1,2.1),
c(-.8,1.8)
)
rownames(tmp) <- covar
for(k in covar){
plot(1:2, type="n", xlim=range(dat$x[,k]), ylim=range(dens.x),
xlab=paste("2006",k), ylab="xi posterior") # dummy plot
abline(h=0)
for(i in 1:nrow(dat$x))
points(rep(dat$x[i,k],length(dens.x[,i])), dens.x[,i], cex=dens.y[,i]*densfac)
}
for(k in covar){
plot(1:2, type="n", ylim=c(-zoomlim,zoomlim),
xlim=tmp[k,], main="omit influential resid, zoomed in",
xlab=paste("2006",k), ylab="xi posterior") # dummy plot
abline(h=0)
for(i in 1:nrow(dat$x))
points(rep(dat$x[i,k],length(dens.x[,i])), dens.x[,i], cex=dens.y[,i]*densfac)
}
plot(1:nrow(dat$x), type="n", ylim=range(dens.x),
xlab="2006 site i", ylab="xi posterior") # dummy plot
abline(h=0)
for(i in 1:nrow(dat$x))
points(rep(i,length(dens.x[,i])), dens.x[,i], cex=dens.y[,i]*densfac)
plot(1:nrow(dat$x), type="n", ylim=c(-zoomlim,zoomlim),
xlab="2006 site i", ylab="xi posterior", main="zoomed in") # dummy plot
abline(h=0)
for(i in 1:nrow(dat$x))
points(rep(i,length(dens.x[,i])), dens.x[,i], cex=dens.y[,i]*densfac)
The full model fit.c shows no mixing issues nor undesirable residual patterns => can use these results as final inference.
Compare fit (posterior SD) and goodness-of-fit (default DIC)
| model | z | t | RLB | realistic? | reduced? | DIC |
|---|---|---|---|---|---|---|
| aa | Y | Y | N | Y (except for ignoring RLB failure) | Y | 1191 |
| c | Y | Y | Y | Y | N | 1466 |
| cc | Y | Y | Y | N | N | 1623 |
NOTE: DIC is based on deviance => can only compare DICs among models that consider the same set of responses.
# names(fit.aa$summary$stat[,"SD"])
# rbind(names(fit.c$summary$stat[,"SD"]),names(fit.cc$summary$stat[,"SD"]))
pBpCcolname <- names(fit.aa$summary$stat[,"SD"])[79:138]
coef_prec_colname <- names(fit.aa$summary$stat[,"SD"])[c(139:141,143:146,148:153)]
paramcolname <- names(fit.aa$summary$stat[,"SD"])[c(139:141,143:146,148:153,79:138)]
library(boot)
merge(
t(data.frame(c(mod="aa",inv.logit(fit.aa$summary$quant[pBpCcolname,"50%"])))),
merge(t(data.frame(c(mod="c",inv.logit(fit.c$summary$quant[pBpCcolname,"50%"])))),
t(data.frame(c(mod="cc",inv.logit(fit.cc$summary$quant[pBpCcolname,"50%"])))),all=T),
all=T) # posterior medians on original scale ## mod logitpB[1] logitpB[2] logitpB[3]
## 1 aa 0.286275680341725 0.377200798547525 0.312362346164428
## 2 c 0.287744327216969 0.378251170025369 0.314126431482953
## 3 cc 0.224213947625426 0.307636152454255 0.256245009761382
## logitpB[4] logitpB[5] logitpB[6] logitpB[7]
## 1 0.333320129802471 0.333546224376753 0.404328899350201 0.358182213551589
## 2 0.33273271444382 0.335304338494897 0.405307694333071 0.357910001271798
## 3 0.252931498295195 0.292291375968954 0.345234553420738 0.290652223313192
## logitpB[8] logitpB[9] logitpB[10] logitpB[11]
## 1 0.212150507330821 0.404272020611921 0.409252881052269 0.390894096861515
## 2 0.209849124469809 0.402433816764527 0.410004416304936 0.389385091694235
## 3 0.149586526014099 0.331262650409587 0.35675183027141 0.340923617350502
## logitpB[12] logitpB[13] logitpB[14] logitpB[15]
## 1 0.335313072059939 0.348905889180321 0.434472747369546 0.251674227818513
## 2 0.334363696729813 0.348753517137964 0.436121365539287 0.249206735297587
## 3 0.275679026033247 0.270449305788599 0.387066324241151 0.178041036487967
## logitpB[16] logitpB[17] logitpB[18] logitpB[19]
## 1 0.42955600748724 0.366330141310985 0.400648030661053 0.191891032235729
## 2 0.42953950755372 0.367430551944616 0.400119611856332 0.189647923512377
## 3 0.378756111704248 0.306012892909537 0.326235417062445 0.153553361498798
## logitpB[20] logitpB[21] logitpB[22] logitpB[23]
## 1 0.326363546450276 0.252764360230601 0.261232120759884 0.358721044777837
## 2 0.325046572032911 0.252004501866955 0.260978001564976 0.359017621768765
## 3 0.255709291238385 0.182133079083367 0.198205735555762 0.273760014552085
## logitpB[24] logitpB[25] logitpB[26] logitpB[27]
## 1 0.451536382056986 0.243381260659997 0.270201618433772 0.332412097240632
## 2 0.454896083121185 0.241001728322721 0.269694471557962 0.330568945665612
## 3 0.404354554762651 0.175073322027557 0.245871971801113 0.251673468560199
## logitpB[28] logitpB[29] logitpB[30] logitpC[1]
## 1 0.123071300629036 0.318363385503664 0.335355588162847 0.745288558734118
## 2 0.122653217680628 0.31785890240741 0.333356098851662 0.74388082083711
## 3 0.0830672822253752 0.245608286904683 0.234302833709962 0.559651674957678
## logitpC[2] logitpC[3] logitpC[4] logitpC[5]
## 1 0.626628558440849 0.597238496585605 0.57131403341174 0.472648042910999
## 2 0.652178303522756 0.642987368466495 0.625826888457087 0.570161824386182
## 3 0.494951590809613 0.453976445580962 0.472258280285624 0.374390546012562
## logitpC[6] logitpC[7] logitpC[8] logitpC[9]
## 1 0.351022157092187 0.701857163421253 0.595649608340639 0.584528800489353
## 2 0.49331412832672 0.692932518824053 0.643607335243804 0.620779773105128
## 3 0.364065428879117 0.515663992989087 0.44031440266289 0.479608686889698
## logitpC[10] logitpC[11] logitpC[12] logitpC[13]
## 1 0.619981884519604 0.387082348967783 0.43203406833043 0.645979711554642
## 2 0.634799850487406 0.50034685668626 0.542793211748421 0.664176002906938
## 3 0.455033795838808 0.345152614333685 0.38642975154125 0.515231754948423
## logitpC[14] logitpC[15] logitpC[16] logitpC[17]
## 1 0.331262483904011 0.532384373076934 0.518854031589226 0.597483589768923
## 2 0.465570988022531 0.588687159748592 0.577234469934066 0.622682898118569
## 3 0.33732555449222 0.419110254564096 0.412985899846002 0.43555757236538
## logitpC[18] logitpC[19] logitpC[20] logitpC[21]
## 1 0.60319303785903 0.200119506928761 0.609532090156252 0.576609262170153
## 2 0.618993706294134 0.393220113339315 0.628411363018045 0.646718539820754
## 3 0.461571985056062 0.226340328894845 0.447520960831188 0.482882534579423
## logitpC[22] logitpC[23] logitpC[24] logitpC[25]
## 1 0.637273827204701 0.704482774696421 0.64104203738729 0.634928209434662
## 2 0.658876245770762 0.701785964702165 0.662655482862873 0.658035822970187
## 3 0.462680358109986 0.55936643975727 0.485360682165771 0.456002959681775
## logitpC[26] logitpC[27] logitpC[28] logitpC[29]
## 1 0.164209456005324 0.453401538519058 0.534496449198115 0.541704043509629
## 2 0.355983607150089 0.540992886488328 0.683607311790795 0.60067451775838
## 3 0.199624866560396 0.407026081407832 0.496699927115124 0.427557274251797
## logitpC[30]
## 1 0.658603558781841
## 2 0.6631499792292
## 3 0.527277910527855merge(
t(data.frame(c(mod="aa",fit.aaNoS$summary$quant[coef_prec_colname,"50%"]))),
merge(t(data.frame(c(mod="c",fit.c$summary$quant[coef_prec_colname,"50%"]))),
t(data.frame(c(mod="cc",fit.cc$summary$quant[coef_prec_colname,"50%"]))),all=T),
all=T) # posterior medians## mod alph0 alph[1] alph[3]
## 1 aa -0.744991260848557 -0.037064029359063 0.526552167166945
## 2 c -0.746272681636666 -0.0447808198563683 0.524654133413428
## 3 cc -1.06436348483086 0.056222679861029 0.612238438073267
## alph[4] alph[5] gam0
## 1 0.0990121670573621 0.127944746116117 0.159750354003106
## 2 0.0994082840930224 0.129944965409824 0.427962474216662
## 3 0.0972637241323568 0.0909995174312724 -0.263709832052086
## gam[1] gam[3] gam[4]
## 1 0.40271445400255 -0.964487762106847 -0.075542216120264
## 2 0.200476322719819 -0.727196202093208 -0.06117827688635
## 3 -0.126804139800104 -0.596788498112673 -0.0114243798173063
## gam[5] tau omega deviance
## 1 0.476582194793155 0.380267456196503 0.120862543981439 1128.28973675357
## 2 0.256562292312631 0.38582967125466 0.111443679616285 1419.30066145254
## 3 0.275272719667932 0.444427261166902 0.112951969251933 1575.88401583129merge(
t(data.frame(c(mod="aa",fit.aaNoS$summary$stat[paramcolname,"SD"]))),
merge(t(data.frame(c(mod="c",fit.c$summary$stat[paramcolname,"SD"]))),
t(data.frame(c(mod="cc",fit.cc$summary$stat[paramcolname,"SD"]))),all=T),
all=T) # posterior SDs (logit scale for p)## mod alph0 alph[1] alph[3]
## 1 aa 0.116453901998254 0.244497922882195 0.292844147177524
## 2 c 0.116234075166985 0.247367349146739 0.299950929706252
## 3 cc 0.12612288369119 0.180670977643597 0.320528626585234
## alph[4] alph[5] gam0 gam[1]
## 1 0.0809370352946895 0.105976347573409 0.187544594176602 0.357759912569506
## 2 0.0823956227297001 0.10598949954466 0.152483952758831 0.310660053295692
## 3 0.0855679511750812 0.1153994651087 0.158217678502502 31.7869838247839
## gam[3] gam[4] gam[5] tau
## 1 0.543777405365543 0.144392324232291 0.21224718903663 0.154169351522706
## 2 0.420954839735748 0.118711063852355 0.156644397871067 0.153569925700505
## 3 0.430344819566998 0.117445416698138 0.164123373657617 0.142929070984102
## omega deviance logitpB[1] logitpB[2]
## 1 0.115232543966504 11.0503189324571 0.36808507794432 0.330575156724792
## 2 0.0795333626354588 13.4941555969172 0.374554139418522 0.33039448070943
## 3 0.0785776236636045 12.9210558854353 0.378324097879355 0.353445382380385
## logitpB[3] logitpB[4] logitpB[5] logitpB[6]
## 1 0.256970715815465 0.340148497841952 0.347524194992219 0.385138520035015
## 2 0.25635580013168 0.345754568303797 0.356625171203561 0.386074950180639
## 3 0.268091262086179 0.362519201759414 0.373006655266224 0.36791872930912
## logitpB[7] logitpB[8] logitpB[9] logitpB[10]
## 1 0.26779456435986 0.36405441589406 0.382725347054261 0.278744229790735
## 2 0.269921003649936 0.365808055108282 0.381791845827488 0.280702650581445
## 3 0.27653220675284 0.378743689170874 0.412093591241131 0.289335660022653
## logitpB[11] logitpB[12] logitpB[13] logitpB[14]
## 1 0.32810386565998 0.421507625290027 0.364966834101772 0.331562473142562
## 2 0.335996233153091 0.425613414082882 0.361599609696398 0.327252531923079
## 3 0.353704085215468 0.472024328314693 0.381715583170823 0.341323400458301
## logitpB[15] logitpB[16] logitpB[17] logitpB[18]
## 1 0.365112755140872 0.261514128392857 0.268227851056739 0.247752388273688
## 2 0.367125280541311 0.261254561431267 0.270708208583391 0.243397566333423
## 3 0.37257259938427 0.268633047657384 0.279394059100425 0.252300246372432
## logitpB[19] logitpB[20] logitpB[21] logitpB[22]
## 1 0.478385544590157 0.287757608347857 0.303472381427287 0.361929729820525
## 2 0.495000252307315 0.288818114726033 0.308889287305192 0.370991578715777
## 3 0.496705227799186 0.296412686414443 0.323209468904882 0.390507431316728
## logitpB[23] logitpB[24] logitpB[25] logitpB[26]
## 1 0.375520511981175 0.305375533751416 0.372589726571813 0.334870839638384
## 2 0.384299240968608 0.303384279710179 0.374632246712722 0.331759022978076
## 3 0.40734517039777 0.290919004909208 0.38665726537675 0.347122441790222
## logitpB[27] logitpB[28] logitpB[29] logitpB[30]
## 1 0.326059674728195 0.510438702973802 0.382488800297149 0.386179255525152
## 2 0.329575159442413 0.514251361870474 0.38342310784036 0.388579425123633
## 3 0.333244309064732 0.527682586757859 0.414233137813887 0.404113344620195
## logitpC[1] logitpC[2] logitpC[3] logitpC[4]
## 1 0.65718772396178 0.319481171724085 0.363891573678911 0.349091954389417
## 2 0.509771983403403 0.257627393401854 0.306107880675853 0.284672599232602
## 3 0.537904822600131 0.25580059921283 0.298916414586777 0.254900518085732
## logitpC[5] logitpC[6] logitpC[7] logitpC[8]
## 1 0.437459647852321 0.592349749243151 0.322520390645636 0.479528853242403
## 2 0.371514591861682 0.479221083279775 0.279690395052763 0.391229757671555
## 3 0.359832619973532 0.420877360250204 0.257948142837326 0.393867622480815
## logitpC[9] logitpC[10] logitpC[11] logitpC[12]
## 1 0.39199819707069 0.281501647392307 0.331906596410157 0.380823587579195
## 2 0.331201230928907 0.243507740181076 0.265493356341518 0.288193988646584
## 3 0.307430388751561 0.218073470999664 0.264784335866168 0.289007304268889
## logitpC[13] logitpC[14] logitpC[15] logitpC[16]
## 1 0.378452739759891 0.510901099136485 0.255151536444294 0.28597753782729
## 2 0.317834397386474 0.376367627545367 0.211954261059861 0.2363951486285
## 3 0.295927607369386 0.376989093886644 0.212600634245666 0.241710186300164
## logitpC[17] logitpC[18] logitpC[19] logitpC[20]
## 1 0.306316390968879 0.308997360640142 0.769368219658805 0.298262539199432
## 2 0.270766595207796 0.25610339327358 0.572462522295954 0.246742752175197
## 3 0.228942140094307 0.267565882313455 0.590913957442594 0.231629700286442
## logitpC[21] logitpC[22] logitpC[23] logitpC[24]
## 1 0.43874009995545 0.353826454496628 0.456091014347591 0.256636873053592
## 2 0.351421886786474 0.308140633446226 0.377791457732199 0.221833981760168
## 3 0.327152649754851 0.271588976331774 0.346740220122096 0.216316922335039
## logitpC[25] logitpC[26] logitpC[27] logitpC[28]
## 1 0.508223038794655 0.746674255890884 0.421884219640203 1.21616410728608
## 2 0.420807453238395 0.57298422080111 0.347338963174625 0.946725155311658
## 3 0.404313406457059 0.589574894313576 0.311446286722094 0.924409310399316
## logitpC[29] logitpC[30]
## 1 0.470943016358126 0.469062695761407
## 2 0.380520353772254 0.3891048479601
## 3 0.38986686966711 0.35670145197181
- Posterior median (original scale) and SD (logit scale) of each \(p^B\) are mostly comparable among models, similarly for median and SD of each \(\alpha\) and \(\tau\)
- to be expected because \(\boldsymbol{z}\) fully observed and modeled identically in all cases.
- RLB models (
candcc)- have noticeably smaller posterior SD for each logit(\(p^c\)), relevant \(\gamma\), and \(\omega\) => modeling \(\boldsymbol{v}\) improves HIS-related inference
- additionally gain inference for \(p^S, p^{SH}, p^{FH}\)
- have noticeably smaller posterior SD for each logit(\(p^c\)), relevant \(\gamma\), and \(\omega\) => modeling \(\boldsymbol{v}\) improves HIS-related inference
- Unrealistic model (
cc)- doesn’t reflect actual lab procedure (RLB tested only when Lyme detected)
- poorer predictive performance (larger DICs)
- posterior median for \(\gamma_0\) and some \(p^c\) differ noticeably
- posterior SD for some \(\gamma\) and some logit(\(p^c\)) differ noticeably
- doesn’t reflect actual lab procedure (RLB tested only when Lyme detected)
Some useful info for rest of document:
names(site)## [1] "siteName" "emptyVar" "siteFragHa"
## [4] "siteFragCat" "siteDenDragLen" "siteDenNymphsDragged"
## [7] "siteDON" "siteLabNymphs" "siteLabNIP"
## [10] "siteDON.NIP" "siteOspCdiv" "siteHISProp"
## [13] "hostHindexAll" "siteRich" "hostPELEprop"
## [16] "hostTASTprop" "hostBLBRprop" "hostPropPTBtot"
## [19] "sitePELEden" "siteTASTden" "siteBLBRden"
## [22] "Level"For each site,
- \(p_B\) = true rate of “Lyme+” (model parameter)
- \(\lambda\) = true spatial density of ticks (parameter, unmodeled)
- \(\lambda p_B\) = true rate of “Lyme+” per unit space (parameter, partially modeled)
- \(n\) =
siteLabNymphs= # test ticks (by design)
- \(y\) = # observed “Lyme+” test ticks (random)
- \(\hat{p}_B = y/n\) =
siteLabNIP= proportion of observed Lyme+ (random)
- \(a\) =
siteDenDragLen= drag area (by design)
- \(m\) =
siteDenNymphsDragged= # ticks observed in \(a\) (random)
- \(\hat{\lambda} = m/a\) =
siteDON= observed spatial density of ticks (random)
- \(\hat{\lambda}\hat{p}_B\) = site’s DIN = naive estimate of “Lyme+” rate per unit space (random)
=siteDON.NIP=siteDON\(\times\)siteLabNIP
Naive NIP.All estimates and their modeled counterparts
tmp <- data.frame( site[,c("siteLabNIP","siteLabNymphs")] )
pB.naive.ci95 <- 1.96*sqrt(tmp[,1]*(1-tmp[,1])/tmp[,2]) # half width only
pB.naive.ci95 <- data.frame(lo=tmp[,1]-pB.naive.ci95, hi=tmp[,1]+pB.naive.ci95)
pB.c.ci95 <- apply(pB.c, 2, quantile, prob=c(.025,.5,.975))
tmp <- data.frame( tmp, t(pB.c.ci95), pB.naive.ci95 )
tmp <- tmp[ order(tmp[,1], decreasing = TRUE), ]
tmp1 <- rownames(tmp)
par(las=2) # make label text perpendicular to axis
par(mar=c(9,5,5,2)) # increase x-axis margin
par(mgp=c(3,.4,0))
tmp2 <- barplot( tmp[,1],
names.arg = paste(tmp1, "(",
format(tmp[,2], justify="right"),
")"),
col="gray95", ylab=expression(paste(p[B], " = NIP")),
ylim=c(-0.1,1) )
title("GRAY: naive NIP[All] estimate in desc. order and\nnaive 95% CI\nBLACK: modeled posterior median ('o') and\n95% credible interval")
text(tmp2[15], -.08, "site name (n)")
points(tmp2-.3, tmp[,6], pch="-", font=2, col="gray60")
points(tmp2-.3, tmp[,7], pch="-", font=2, col="gray60")
segments(tmp2-.3, tmp[,6], tmp2-.3, tmp[,7], lwd=2, col="gray60")
points(tmp2+.3, tmp[,4], pch="o", font=2)
points(tmp2+.3, tmp[,3], pch="-", font=2)
points(tmp2+.3, tmp[,5], pch="-", font=2)
segments(tmp2+.3, tmp[,3], tmp2+.3, tmp[,5], lwd=2)
Naive NIP[HIS] estimates and their modeled counterparts
library(boot)
pC.c <- inv.logit(scans.c.pooled[,109:138])
pC.c.ci95 <- apply(pC.c, 2, quantile, prob=c(.025,.5,.975))
colnames(pC.c.ci95) <- site[,"siteName"]
pC.naive.ci95 <- sapply(site[,"siteName"],
function(x){
sum(subset(tick1.lyme, siteName==x)[,"his"])
}) # HIS+ only
pC.naive.ci95 <- pC.naive.ci95 / tick2[,"lyme"] # naive pC est only
tmp <- data.frame(est=pC.naive.ci95, nh=tick2[,"lyme", drop=FALSE])
tmp## est lyme
## 601 NA 8
## 602 NA 6
## 603 NA 15
## 604 NA 4
## 605 NA 7
## 606 NA 11
## 607 NA 14
## 608 NA 4
## 609 NA 4
## 610 NA 15
## 611 NA 8
## 612 NA 1
## 613 0.7500000 4
## 614 NA 11
## 615 0.7500000 4
## 616 NA 19
## 617 NA 13
## 618 NA 18
## 619 NA 1
## 620 NA 9
## 621 NA 7
## 622 0.3333333 3
## 623 NA 4
## 624 NA 23
## 625 NA 4
## 626 NA 11
## 627 NA 7
## 628 NA 4
## 629 NA 3
## 630 NA 3pC.naive.ci95 <- 1.96*sqrt(tmp[,1]*(1-tmp[,1])/tmp[,2]) # half width only
pC.naive.ci95 <- data.frame(est=tmp[,1], lo=tmp[,1]-pC.naive.ci95, hi=tmp[,1]+pC.naive.ci95)
rownames(pC.naive.ci95) <- rownames(tmp)
tmp <- data.frame( tmp, t(pC.c.ci95), pC.naive.ci95[,2:3] )
tmp <- tmp[ tmp1, ]
par(las=2) # make label text perpendicular to axis
par(mar=c(9,5,5,2)) # increase x-axis margin
par(mgp=c(3,.4,0))
tmp2 <- barplot( tmp[,1],
names.arg = paste(tmp1, "(",
format(tmp[,2], justify="right"),
")"),
col="gray95", ylab=expression(paste(p[C], " = NIP[HIS]")),
ylim=c(-0.1,1) )
title("GRAY: naive NIP[HIS] estimate\nBLACK: modeled posterior median ('o') and\n95% credible interval")
text(tmp2[15], -.04, "site name (y)")
points(tmp2, tmp[,4], pch="o", font=2)
points(tmp2, tmp[,3], pch="-", font=2)
points(tmp2, tmp[,5], pch="-", font=2)
segments(tmp2, tmp[,3], tmp2, tmp[,5], lwd=2)
Naive DIN estimates and their modeled (posterior predictive) counterparts
For each site:- CAUTION:
- posterior of \(\hat{\lambda}p_B\) isn’t a posterior predictive!
- pointless to model \(m, \lambda\) because
- just one data point for \(m\),
- no spatial info about sites so can’t model \(\lambda\) spatially
- just one data point for \(m\),
- posterior of \(\hat{\lambda}p_B\) isn’t a posterior predictive!
- let \(y^\ast\) = # dragged ticks that would’ve been “Lyme+” (random, unobserved)
- \(E(y^\ast \mid m) = mp_B\)
- get conditional posterior predictive for \(y^\ast\):
- let \(p_B^{(s)}\) = \(s\)th MCMC scan of \(p_B\)
- simulate \(y^\ast\mid m\sim\) Bin(\(m,p_B^{(s)}\))
- let \(p_B^{(s)}\) = \(s\)th MCMC scan of \(p_B\)
- conditional posterior predictive for
siteDON.NIPis that for \(y^\ast/a\)
- can’t use for cross validation because don’t have an observed version of \(y^\ast\)
- \(E(y^\ast \mid m) = mp_B\)
library(boot)
pB.c <- inv.logit(scans.c.pooled[,79:108])
pB.c.ls <- as.list( as.data.frame(pB.c) )
library(parallel)
ystar.c <- mcmapply( rbinom,
site$siteDenNymphsDragged,
pB.c.ls,
MoreArgs = list(n=length(pB.c.ls)),
mc.cores=6)
din.c <- t( t(ystar.c)/site$siteDenDragLen )
colnames(din.c) <- site$siteName
din.c.ci95 <- apply(din.c, 2, quantile, prob=c(.025,.5,.975))tmp <- site[,c("siteDON.NIP","siteDenNymphsDragged")]
tmp <- tmp[ tmp1, ]
par(las=2) # make label text perpendicular to axis
par(mar=c(9,5,4,2)) # increase x-axis margin
par(mgp=c(3,.4,0))
tmp2 <- barplot( tmp[,1],
names.arg = paste(tmp1, "(",
format(tmp[,2], justify="right"),
")"),
col="gray95", ylab=expression(paste(p[B]*m/a, " = DIN")),
ylim=c(-0.015,.15) )
title("GRAY: naive DIN estimate\nBLACK: modeled posterior median ('o') and\n95% 'predictive' interval")
text(tmp2[15], -.008, "site name (m)")
points(tmp2, din.c.ci95[2,tmp1], pch="o", font=2)
points(tmp2, din.c.ci95[1,tmp1], pch="-", font=2)
points(tmp2, din.c.ci95[3,tmp1], pch="-", font=2)
segments(tmp2, din.c.ci95[1,tmp1], tmp2, din.c.ci95[3,tmp1], lwd=2)
Cross-validation via posterior prediction of \(\hat{p}^B_{\text{med}}\) (an example in ecology)
Can do this for \(\hat{p}^B_{\text{med}}\) = median{ \(y_i/n_i\) } (but not for median{ \(y^\ast_i/n_i\) }) and for \(i_{(15)} = \{i: \hat{p}^B_i=\hat{p}^B_{(15)}\}\) and \(i_{(16)}\).
Due to the discreteness of \(y_i\), the true 50th percentile among \(S\)=30 sites could be a value shared identically among more than 2 sites. Here, for pure cross-validation (i.e. not necessarily the true 50th percentile technically defined for a distribution), we take
\[ \hat{p}^B_{\text{med}} = \frac{\hat{p}^B_{(15)}+\hat{p}^B_{(16)}}{2} \]
where the subscript “\((i)\)” denotes the \(i\)th sorted value, and sorting is done by a combination of the value of \(\hat{p}^B_i\) and the corresponding site label, irrespective of whether either \(\hat{p}^B_{(15)}\) or \(\hat{p}^B_{(16)}\) is equal to another site’s value. For example, suppose
\[ \hat{p}^B_{(14)} = \hat{p}^B_{(15)} < \hat{p}^B_{(16)} < \hat{p}^B_{(17)} \]
with “(14), (15), (16)” corresponding to “614, 615, 616” respectively in alphabetical order. Then, even though “614” would’ve been another contributor to a technically defined 50th percentile, we report
our median (for cross-validation purposes only) is the mean of \(\hat{p}^B_{(15)}\) and \(\hat{p}^B_{(16)}\) corresponding to “614, 615.”
pBhat <- site[order(site$siteLabNIP),"siteLabNIP",drop=FALSE]
medpBhat <- mean(pBhat[15:16,])
medpBhat.site <- rownames(pBhat)[15:16]
medpBhat # observed median## [1] 0.364medpBhat.site # observed median site(s)## [1] " 601" " 613"library(parallel)
y.c <- mcmapply( rbinom,
site$siteLabNymphs,
pB.c.ls,
MoreArgs = list(n=nrow(pB.c)),
mc.cores=6)
phat.c <- t( t(y.c)/site$siteLabNymphs )
find.med <- function(vect, rname, loc){
ord <- order(vect)
mid <- vect[ord][loc]
md <- mean( mid )
mdname <- rname[ord][loc]
md <- data.frame( c(mid,md),
check.names=FALSE,
row.names=c(mdname,"median"))
colnames(md) <- "pBhat"
return(md)
}
phat.med.c <- apply(phat.c, 1,
find.med,
rname=as.character(site$siteName),
loc=c(15:16))tmp <- sapply(
phat.med.c,
function(df){
return(df[3,1])
})
hist( tmp,
main=paste("posterior predictive distr'n for median{naive NIP estimates}\nred = observed median (",
medpBhat.site[1], ",", medpBhat.site[2], ")"),
cex.main=1, font.main=1,
xlab=expression(paste("median ", hat(p)[B]))
)
abline(v=medpBhat, col="red")
sum( tmp > medpBhat ) / length(tmp) # 1-sided posterior predictive p## [1] 0.1336quantile(tmp, prob=c(.05,.95)) # 90% predictive interval## 5% 95%
## 0.2727273 0.3798077phat.med.c.site <- t( sapply(phat.med.c,
function(x){ rownames(x)[1:2] },
simplify="array" ))
head(phat.med.c.site)## [,1] [,2]
## [1,] " 619" " 617"
## [2,] " 630" " 606"
## [3,] " 601" " 614"
## [4,] " 630" " 625"
## [5,] " 601" " 611"
## [6,] " 610" " 624"phat.med.c.site.tab <- apply(phat.med.c.site, 2, table)
phat.med.c.site15 <- sort(phat.med.c.site.tab[,1], decreasing=TRUE)
phat.med.c.site16 <- sort(phat.med.c.site.tab[,2], decreasing=TRUE)tmp <- names(phat.med.c.site15)==medpBhat.site[1]
tmp1 <- rep("grey",S)
tmp1[tmp] <- "red"
par(mfrow=c(2,1))
par(las=2) # make label text perpendicular to axis
par(mgp=c(3,1,0)) # default
par(mar=c(10,5,4,2)) # increase x-axis margin
barplot( phat.med.c.site15, col=tmp1 )
title("posterior predictive dist'n for site \nwith '15th largest' naive NIP estimate\nred = observed")
tmp <- names(phat.med.c.site16)==medpBhat.site[2]
tmp1 <- rep("grey",S)
tmp1[tmp] <- "red"
barplot( phat.med.c.site16, col=tmp1 )
title("posterior predictive dist'n for site \nwith '16th largest' naive NIP estimate\nred = observed")