grant_hw_13
Dani Grant
4/23/2022
13M1. Revisit the Reed frog survival data, data(reedfrogs), and add the predation and size treatment variables to the varying intercepts model. Consider models with either main effect alone, both main effects, as well as a model including both and their interaction.
library(rstan)## Loading required package: StanHeaders## Loading required package: ggplot2## rstan (Version 2.21.3, GitRev: 2e1f913d3ca3)## For execution on a local, multicore CPU with excess RAM we recommend calling
## options(mc.cores = parallel::detectCores()).
## To avoid recompilation of unchanged Stan programs, we recommend calling
## rstan_options(auto_write = TRUE)## Do not specify '-march=native' in 'LOCAL_CPPFLAGS' or a Makevars filelibrary(rethinking)## Loading required package: cmdstanr## This is cmdstanr version 0.4.0## - Online documentation and vignettes at mc-stan.org/cmdstanr## - CmdStan path set to: C:/Users/Dani Grant/Documents/.cmdstanr/cmdstan-2.29.2## - Use set_cmdstan_path() to change the path## Loading required package: parallel## rethinking (Version 2.21)##
## Attaching package: 'rethinking'## The following object is masked from 'package:rstan':
##
## stan## The following object is masked from 'package:stats':
##
## rstudentdata(reedfrogs)
d <- reedfrogs
d_list <- list(
survive = d$surv, #number surviving
density = d$density, #initial tadpole density
predation = ifelse(d$pred == "no", 1, 2), #factor: predators present or absent
size = ifelse(d$size == "small", 1, 2), #factor: big or small tadpoles
tank = 1:nrow(d)
)#model for predation
m.pred <- ulam(
alist(
survive ~ dbinom(density, p),
logit(p) <- alpha[tank] + beta*predation,
alpha[tank] ~ dnorm(a_bar, sigma_a),
a_bar ~ dnorm(0, 1.5),
sigma_a ~ dexp(1),
beta ~ dnorm(0, .5)
), data = d_list, chains = 4, iter = 4000, log_lik = TRUE)## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c7bdb4149.stan', line 2, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c7bdb4149.stan', line 3, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c7bdb4149.stan', line 4, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c7bdb4149.stan', line 5, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c7bdb4149.stan', line 6, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Running MCMC with 4 sequential chains, with 1 thread(s) per chain...
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## Total execution time: 10.4 seconds.#model for size
m.size <- ulam(
alist(
survive ~ dbinom(density, p),
logit(p) <- alpha[tank] + beta*size,
alpha[tank] ~ dnorm(a_bar, sigma_a),
a_bar ~ dnorm(0, 1.5),
sigma_a ~ dexp(1),
beta ~ dnorm(0, .5)
), data = d_list, chains = 4, iter = 4000, log_lik = TRUE)## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c17c37679.stan', line 2, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c17c37679.stan', line 3, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c17c37679.stan', line 4, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c17c37679.stan', line 5, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c17c37679.stan', line 6, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Running MCMC with 4 sequential chains, with 1 thread(s) per chain...
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## Total execution time: 10.1 seconds.#model for predation & size
m.cont <- ulam(
alist(
survive ~ dbinom(density, p),
logit(p) <- alpha[tank] + beta1*predation + beta2*size,
alpha[tank] ~ dnorm(a_bar, sigma_a),
a_bar ~ dnorm(0, 1.5),
sigma_a ~ dexp(1),
beta1 ~ dnorm(0, .5),
beta2 ~ dnorm(0, .5)
), data = d_list, chains = 4, iter = 4000, log_lik = TRUE)## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c26ac2454.stan', line 2, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c26ac2454.stan', line 3, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c26ac2454.stan', line 4, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c26ac2454.stan', line 5, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c26ac2454.stan', line 6, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Running MCMC with 4 sequential chains, with 1 thread(s) per chain...
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## Mean chain execution time: 3.4 seconds.
## Total execution time: 14.4 seconds.#model for interaction
m.int <- ulam(
alist(
survive ~ dbinom(density, p),
logit(p) <- alpha[tank] + beta1*predation + beta2*size + beta3*predation*size,
alpha[tank] ~ dnorm(a_bar, sigma_a),
a_bar ~ dnorm(0, 1.5),
sigma_a ~ dexp(1),
beta1 ~ dnorm(0, .5),
beta2 ~ dnorm(0, .5),
beta3 ~ dnorm(0, .5)
), data = d_list, chains = 4, iter = 4000, log_lik = TRUE)## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c36c731f5.stan', line 2, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c36c731f5.stan', line 3, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c36c731f5.stan', line 4, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c36c731f5.stan', line 5, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Warning in 'C:/Users/DANIGR~1/AppData/Local/Temp/RtmpWcWQhl/model-e6c36c731f5.stan', line 6, column 4: Declaration## of arrays by placing brackets after a variable name is deprecated and## will be removed in Stan 2.32.0. Instead use the array keyword before the## type. This can be changed automatically using the auto-format flag to## stanc## Running MCMC with 4 sequential chains, with 1 thread(s) per chain...
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## Mean chain execution time: 4.9 seconds.
## Total execution time: 20.8 seconds.Instead of focusing on inferences about these two predictor variables, focus on the inferred variation across tanks. Explain why it changes as it does across models.
precis(m.pred, depth = 2)## mean sd 5.5% 94.5% n_eff Rhat4
## alpha[1] 3.8614444 0.7877880 2.6142868 5.137268 1176.3632 1.002638
## alpha[2] 4.4192820 0.8473987 3.1140385 5.817503 1739.6769 1.001626
## alpha[3] 3.0219706 0.7439505 1.8457776 4.209076 859.2952 1.002816
## alpha[4] 4.4123856 0.8669714 3.0917088 5.840798 1700.8527 1.001912
## alpha[5] 3.8648885 0.7796313 2.6674294 5.106660 1260.0488 1.002798
## alpha[6] 3.8662894 0.7955597 2.6293893 5.162456 1212.8584 1.001995
## alpha[7] 4.4045468 0.8514005 3.1022639 5.793951 1642.3484 1.001795
## alpha[8] 3.8681018 0.7854253 2.6055647 5.141178 1134.7298 1.001727
## alpha[9] 3.1372044 0.8153234 1.7977141 4.397956 521.4561 1.005476
## alpha[10] 4.7097058 0.7940307 3.4652684 5.983163 788.3030 1.003888
## alpha[11] 4.0189326 0.7675327 2.7885062 5.204260 580.4882 1.005191
## alpha[12] 3.7095295 0.7727444 2.4429765 4.905111 571.9615 1.005572
## alpha[13] 4.0172344 0.7839900 2.7719834 5.245011 616.9760 1.005158
## alpha[14] 3.4269844 0.7882217 2.1411701 4.643138 517.6671 1.006385
## alpha[15] 4.6965864 0.7933193 3.4540286 5.970312 768.5663 1.003854
## alpha[16] 4.6977009 0.7931827 3.4519262 5.964923 787.4008 1.003846
## alpha[17] 4.3875593 0.6985702 3.3067869 5.526095 1500.2243 1.002196
## alpha[18] 4.0384914 0.6692958 2.9967840 5.128770 1001.6254 1.003474
## alpha[19] 3.7383003 0.6329981 2.7518445 4.781536 856.2592 1.003668
## alpha[20] 4.8166807 0.7565965 3.6752052 6.080957 1805.4195 1.003696
## alpha[21] 4.0325592 0.6628160 3.0170921 5.118752 1031.1507 1.003316
## alpha[22] 4.0394726 0.6672503 3.0115339 5.115173 1090.3173 1.002672
## alpha[23] 4.0333196 0.6705748 3.0064962 5.109583 933.1504 1.003246
## alpha[24] 3.4829071 0.6127597 2.5147417 4.468733 758.1243 1.004517
## alpha[25] 2.4095496 0.7470839 1.1725335 3.567725 426.8587 1.007087
## alpha[26] 3.4219923 0.7132247 2.2614524 4.510342 418.1725 1.008169
## alpha[27] 2.0662424 0.7801108 0.7965408 3.288081 429.4623 1.007030
## alpha[28] 2.8674562 0.7169119 1.6946824 3.965116 408.3586 1.007435
## alpha[29] 3.4246204 0.7091818 2.2819216 4.539141 414.0886 1.008339
## alpha[30] 4.5046620 0.6994811 3.3793605 5.612874 483.4896 1.006646
## alpha[31] 2.7230494 0.7259116 1.5347550 3.847142 414.4234 1.006923
## alpha[32] 3.0036245 0.7126807 1.8553129 4.107994 410.6536 1.006605
## alpha[33] 4.5992093 0.6814038 3.5624993 5.735456 1308.8388 1.001543
## alpha[34] 4.2816995 0.6371800 3.3010748 5.332022 1094.6030 1.002344
## alpha[35] 4.2826257 0.6364354 3.3055797 5.326725 1058.0897 1.002892
## alpha[36] 3.7583516 0.5809995 2.8403756 4.712901 784.0299 1.003124
## alpha[37] 3.7639227 0.5768381 2.8409414 4.686879 769.8942 1.004223
## alpha[38] 4.9954787 0.7532019 3.8382726 6.243362 1971.8722 1.002437
## alpha[39] 4.2826245 0.6511242 3.2899578 5.349342 1167.7707 1.002303
## alpha[40] 3.9949221 0.5977325 3.0536329 4.961593 866.1936 1.003638
## alpha[41] 1.7202536 0.7693261 0.4585765 2.913043 406.2187 1.007072
## alpha[42] 2.7621609 0.6974765 1.6323889 3.866659 387.2709 1.008999
## alpha[43] 2.8672438 0.6935427 1.7343814 3.940114 391.8765 1.007588
## alpha[44] 2.9731969 0.6848161 1.8763588 4.021553 376.2419 1.007677
## alpha[45] 3.8082552 0.6734465 2.7232373 4.867294 390.5952 1.007873
## alpha[46] 2.7561993 0.6999378 1.6409380 3.839145 384.9466 1.008563
## alpha[47] 5.0202578 0.6833927 3.9214089 6.102240 491.1593 1.006462
## alpha[48] 3.2844055 0.6830467 2.1738236 4.374022 382.1225 1.008868
## a_bar 3.7200870 0.4880815 2.9579665 4.492757 330.3164 1.009709
## sigma_a 0.9518595 0.1782999 0.6997280 1.258817 847.7148 1.003925
## beta -1.6437978 0.3053246 -2.1246922 -1.160826 307.9413 1.010417precis(m.size, depth = 2)## mean sd 5.5% 94.5% n_eff Rhat4
## alpha[1] 2.35200616 1.1120394 0.67179210 4.1812103 884.0872 1.003531
## alpha[2] 3.27780922 1.2721460 1.36023970 5.4008824 1287.0538 1.001723
## alpha[3] 1.21255926 0.9588194 -0.28561999 2.7843946 647.6443 1.005239
## alpha[4] 3.27719032 1.2680912 1.38985625 5.4129597 1235.0458 1.001849
## alpha[5] 2.26978637 0.9666576 0.83318295 3.8831133 1799.8419 1.001321
## alpha[6] 2.27796255 0.9510336 0.83108705 3.8466787 1795.1956 1.001296
## alpha[7] 3.22710273 1.1960738 1.50714965 5.2717809 2147.9860 1.001401
## alpha[8] 2.28210371 0.9781528 0.82574183 3.9165743 1645.4202 1.001525
## alpha[9] 0.03979528 0.9308054 -1.44875190 1.5319397 614.8201 1.006737
## alpha[10] 2.34787053 1.0958805 0.66306920 4.1540416 870.1693 1.004734
## alpha[11] 1.22464022 0.9573764 -0.27832838 2.7865864 687.1256 1.004998
## alpha[12] 0.80284625 0.9302018 -0.68026117 2.3055298 628.4446 1.004958
## alpha[13] 1.13239704 0.7774992 -0.07506026 2.3977004 1296.5172 1.002420
## alpha[14] 0.32167335 0.7275710 -0.84823184 1.4763259 1197.7715 1.002101
## alpha[15] 2.27857638 0.9612185 0.85241537 3.9061576 1743.2807 1.001401
## alpha[16] 2.27941958 0.9701164 0.83837765 3.9165648 1628.3850 1.000851
## alpha[17] 3.13574664 1.0376193 1.54698210 4.8312333 817.4266 1.004110
## alpha[18] 2.62058395 0.9600378 1.10774910 4.1958302 673.9254 1.004567
## alpha[19] 2.23015076 0.9010498 0.78457510 3.6897953 585.5621 1.006361
## alpha[20] 3.87428230 1.1670422 2.10667790 5.8241593 1073.2863 1.003786
## alpha[21] 2.51843431 0.7790900 1.32386460 3.8016303 1379.9281 1.001627
## alpha[22] 2.51964843 0.7673882 1.34677955 3.8029838 1209.4140 1.002880
## alpha[23] 2.52679453 0.7686595 1.38768900 3.8415582 1337.1108 1.002492
## alpha[24] 1.82756669 0.6492575 0.82453535 2.8826822 970.0484 1.003473
## alpha[25] -0.77400117 0.8342830 -2.13100585 0.5598437 499.3063 1.006860
## alpha[26] 0.38697804 0.8189778 -0.94733164 1.7425236 482.6942 1.007098
## alpha[27] -1.20303790 0.8671779 -2.60110565 0.1720714 527.1182 1.006174
## alpha[28] -0.24418287 0.8119211 -1.53215730 1.0567976 483.9669 1.006601
## alpha[29] 0.27706361 0.5437581 -0.60107566 1.1374048 743.8013 1.003952
## alpha[30] 1.56581521 0.6298649 0.60330980 2.6193771 898.7367 1.005058
## alpha[31] -0.51879236 0.5549134 -1.40935825 0.3511165 774.3110 1.003789
## alpha[32] -0.19227308 0.5487483 -1.06688280 0.6906228 734.7594 1.004349
## alpha[33] 3.42449068 1.0209610 1.86780525 5.0783137 786.7297 1.004829
## alpha[34] 2.93398274 0.9433993 1.48896880 4.5040951 640.7874 1.006211
## alpha[35] 2.92295331 0.9333313 1.45811020 4.4391346 651.1884 1.005821
## alpha[36] 2.28285688 0.8629598 0.92372716 3.7018886 540.4975 1.005697
## alpha[37] 2.17856054 0.6279462 1.18150365 3.2032598 934.7629 1.002456
## alpha[38] 4.02847705 1.0708183 2.46889925 5.8301088 2219.1386 1.001885
## alpha[39] 2.84375672 0.7597061 1.69468080 4.0942598 1319.4402 1.002119
## alpha[40] 2.47331822 0.6931354 1.41293470 3.6029556 1046.7297 1.002791
## alpha[41] -1.58582243 0.8595965 -3.01498180 -0.2530945 514.6133 1.007210
## alpha[42] -0.34266647 0.7944442 -1.59500355 0.9439870 444.5724 1.007378
## alpha[43] -0.22413161 0.7911386 -1.51716585 1.0480772 440.0113 1.008250
## alpha[44] -0.10818407 0.7868420 -1.34672280 1.1719705 439.9004 1.007587
## alpha[45] 0.69614313 0.5057148 -0.10774510 1.5052986 664.8347 1.004532
## alpha[46] -0.45347935 0.5072334 -1.26914055 0.3577754 660.9345 1.004475
## alpha[47] 2.17714938 0.6328360 1.19570745 3.2024493 859.5124 1.003083
## alpha[48] 0.12133971 0.5021313 -0.68272059 0.9297588 654.6093 1.005197
## a_bar 1.51731187 0.5835884 0.58753892 2.4667370 433.6792 1.008356
## sigma_a 1.62151229 0.2182334 1.30620505 1.9957296 4515.9180 1.000454
## beta -0.11393194 0.3586493 -0.69518817 0.4605004 369.6313 1.009785precis(m.cont, depth = 2)## mean sd 5.5% 94.5% n_eff Rhat4
## alpha[1] 4.0906247 0.9197688 2.6102697 5.5466515 615.2932 1.008312
## alpha[2] 4.6157620 0.9248516 3.1542863 6.0952670 766.7069 1.008525
## alpha[3] 3.2523898 0.9033756 1.7861857 4.7009157 477.1650 1.011197
## alpha[4] 4.6054816 0.9496783 3.0925179 6.1495925 820.4214 1.008055
## alpha[5] 4.0194506 0.8616286 2.5948171 5.3703195 732.6129 1.007294
## alpha[6] 4.0402064 0.8490588 2.6944503 5.4215630 693.6187 1.009453
## alpha[7] 4.5817064 0.9074357 3.1819370 6.0695328 922.9692 1.006712
## alpha[8] 4.0232854 0.8658585 2.6525675 5.4009893 671.6540 1.008293
## alpha[9] 3.3755495 0.9514786 1.8044067 4.8381054 355.7788 1.014700
## alpha[10] 4.9116664 0.9002913 3.4477542 6.3315562 443.9795 1.014196
## alpha[11] 4.2444479 0.9173577 2.7477139 5.6844759 388.8351 1.015157
## alpha[12] 3.9386851 0.9167302 2.4403689 5.3833217 366.4775 1.013973
## alpha[13] 4.1628574 0.8429586 2.7832581 5.4790473 438.4062 1.013376
## alpha[14] 3.5664659 0.8590492 2.1620473 4.9050715 388.0587 1.013955
## alpha[15] 4.8464105 0.8475907 3.4913954 6.2186427 602.2156 1.011169
## alpha[16] 4.8536964 0.8426199 3.5018874 6.1845001 579.3837 1.010840
## alpha[17] 4.6153251 0.8415473 3.2826256 5.9968777 593.5862 1.010128
## alpha[18] 4.2663288 0.8108745 2.9616210 5.5578458 497.2987 1.011523
## alpha[19] 3.9664914 0.7960146 2.6844179 5.2357832 451.9110 1.013485
## alpha[20] 5.0275850 0.8744062 3.6677062 6.4323112 714.2517 1.009999
## alpha[21] 4.1902353 0.7414003 3.0370159 5.3950604 591.1584 1.011107
## alpha[22] 4.1884735 0.7414346 3.0211575 5.3963446 593.8999 1.010732
## alpha[23] 4.1924220 0.7428804 3.0093817 5.3724185 598.4107 1.008689
## alpha[24] 3.6251266 0.6897434 2.5111190 4.7260363 502.6334 1.009739
## alpha[25] 2.6489218 0.9328392 1.1236269 4.0940218 313.9107 1.016985
## alpha[26] 3.6627314 0.8818903 2.2016386 5.0234436 305.1006 1.016685
## alpha[27] 2.3141422 0.9569383 0.7382753 3.7818376 313.6659 1.016943
## alpha[28] 3.1083575 0.9016141 1.6176641 4.4932952 295.2886 1.019066
## alpha[29] 3.5606233 0.7694941 2.2927478 4.7422819 316.9749 1.017857
## alpha[30] 4.6236874 0.7602422 3.3839929 5.8158943 365.4609 1.013567
## alpha[31] 2.8571970 0.7971687 1.5297479 4.0849994 341.2623 1.014929
## alpha[32] 3.1445445 0.7927822 1.8534578 4.3659967 316.7180 1.016996
## alpha[33] 4.8317397 0.8273963 3.5239552 6.1707285 609.9445 1.009922
## alpha[34] 4.5105623 0.7925973 3.2320431 5.7708512 507.5985 1.012396
## alpha[35] 4.5058667 0.7892215 3.2637669 5.7861883 500.4449 1.011334
## alpha[36] 4.0018913 0.7605207 2.7736102 5.2025628 402.6280 1.014990
## alpha[37] 3.9114107 0.6573181 2.8654585 4.9812992 493.7918 1.010989
## alpha[38] 5.1469436 0.7951759 3.8855362 6.4169719 984.1037 1.007434
## alpha[39] 4.4311430 0.7060047 3.3279534 5.5722808 578.8054 1.008670
## alpha[40] 4.1491375 0.6771740 3.0760118 5.2271164 537.6282 1.010625
## alpha[41] 1.9701035 0.9563947 0.3896573 3.4523127 301.8810 1.018002
## alpha[42] 3.0021231 0.8821087 1.5742785 4.3470826 284.5445 1.019201
## alpha[43] 3.1137527 0.8803245 1.6639020 4.4704166 283.4359 1.020064
## alpha[44] 3.2177156 0.8726557 1.7713947 4.5684339 294.2595 1.019163
## alpha[45] 3.9413110 0.7406427 2.7078218 5.0811939 319.4139 1.017332
## alpha[46] 2.8898618 0.7666928 1.6129868 4.0879506 322.3617 1.016213
## alpha[47] 5.1401487 0.7472879 3.9065395 6.3215475 370.1947 1.014779
## alpha[48] 3.4141820 0.7518950 2.1956697 4.5704027 308.7008 1.017801
## a_bar 3.9106286 0.6298914 2.8857332 4.8828388 257.6151 1.022170
## sigma_a 0.9461209 0.1812201 0.6813081 1.2554322 782.6301 1.003105
## beta1 -1.6350700 0.3070999 -2.0995311 -1.1165786 377.5889 1.011993
## beta2 -0.1361554 0.2666650 -0.5678786 0.2857243 561.5505 1.011213precis(m.int, depth = 2)## mean sd 5.5% 94.5% n_eff Rhat4
## alpha[1] 3.2882213 0.8302016 1.9680444 4.6357161 438.4838 1.013940
## alpha[2] 3.7046233 0.8804660 2.3031967 5.1284713 463.4400 1.011040
## alpha[3] 2.6032405 0.8281743 1.2715354 3.8984925 411.1151 1.013818
## alpha[4] 3.6896047 0.8734053 2.2943598 5.1044059 456.8042 1.010354
## alpha[5] 3.2761025 0.8067009 1.9975730 4.5712580 463.4999 1.013373
## alpha[6] 3.2650298 0.7992805 1.9954867 4.5677086 457.9681 1.012859
## alpha[7] 3.6702971 0.8390378 2.3476284 5.0407458 523.2340 1.009117
## alpha[8] 3.2570251 0.7951581 1.9993619 4.5244570 544.5930 1.010594
## alpha[9] 3.1982803 0.7900583 1.9265227 4.4449958 339.3615 1.017130
## alpha[10] 4.4360595 0.8186387 3.1472505 5.7677134 382.7846 1.014581
## alpha[11] 3.9225799 0.7860689 2.6879740 5.1688701 354.2710 1.015016
## alpha[12] 3.6829447 0.7885312 2.4216684 4.9481756 319.3850 1.016694
## alpha[13] 3.4007973 0.7623992 2.1796830 4.6102893 370.0489 1.015076
## alpha[14] 2.9172030 0.7647229 1.6562528 4.0994332 351.1364 1.016686
## alpha[15] 3.9513298 0.7853267 2.7085489 5.2029974 423.5267 1.012144
## alpha[16] 3.9474622 0.7852345 2.6938707 5.1917983 374.0278 1.014925
## alpha[17] 3.6878858 0.7954759 2.4540698 4.9605565 398.3011 1.015765
## alpha[18] 3.3992925 0.7721189 2.1608378 4.6483560 369.5034 1.015595
## alpha[19] 3.1423373 0.7531334 1.9225923 4.3468312 353.5132 1.016482
## alpha[20] 4.0118912 0.8338009 2.7078409 5.3401114 393.3021 1.015981
## alpha[21] 3.3670032 0.7115368 2.2495846 4.5192482 444.0299 1.011787
## alpha[22] 3.3664269 0.7091958 2.2422643 4.5133808 401.4803 1.014244
## alpha[23] 3.3718484 0.7045209 2.2483177 4.4943455 477.5016 1.012747
## alpha[24] 2.8923778 0.6677432 1.8438971 3.9610947 443.7928 1.013232
## alpha[25] 2.6380741 0.7558213 1.3999524 3.7919447 295.0847 1.018325
## alpha[26] 3.5644097 0.7220115 2.3927502 4.6950853 282.6939 1.019229
## alpha[27] 2.3351172 0.7659644 1.0863417 3.5232038 304.0763 1.020077
## alpha[28] 3.0524331 0.7378577 1.8365741 4.2003620 281.1940 1.020055
## alpha[29] 2.8347994 0.6898514 1.7216156 3.8977983 294.2773 1.022065
## alpha[30] 3.8142001 0.7031716 2.6957896 4.9389702 298.0502 1.018359
## alpha[31] 2.2064846 0.7006448 1.0628601 3.2817685 300.5300 1.018560
## alpha[32] 2.4577257 0.6939495 1.3516994 3.5125228 287.5509 1.021638
## alpha[33] 3.8482272 0.7775901 2.6453481 5.0865560 391.6402 1.015551
## alpha[34] 3.5855183 0.7563039 2.3926468 4.7803980 342.0505 1.017142
## alpha[35] 3.5927057 0.7663892 2.3876423 4.8075849 405.3562 1.016825
## alpha[36] 3.1519128 0.7271678 1.9839373 4.3091387 322.3834 1.019472
## alpha[37] 3.1047321 0.6472586 2.0497041 4.1219506 396.8352 1.014165
## alpha[38] 4.1397874 0.7989306 2.9262104 5.4679133 575.7207 1.009628
## alpha[39] 3.5626595 0.6988341 2.4566842 4.6893282 422.7992 1.014383
## alpha[40] 3.3205599 0.6623979 2.2545315 4.3641381 395.5568 1.013235
## alpha[41] 2.0225485 0.7720652 0.7400945 3.2028879 289.6117 1.021464
## alpha[42] 2.9839181 0.7204370 1.8134685 4.1041676 269.1476 1.021163
## alpha[43] 3.0827947 0.7247199 1.9130751 4.2014392 264.5290 1.020626
## alpha[44] 3.1846426 0.7159481 2.0333212 4.3127648 267.5172 1.021601
## alpha[45] 3.1787992 0.6621656 2.1052339 4.1950162 271.8292 1.021097
## alpha[46] 2.2058758 0.6790100 1.1129619 3.2717579 281.4314 1.021142
## alpha[47] 4.2695748 0.6928798 3.1521923 5.3654765 292.1844 1.019623
## alpha[48] 2.6895897 0.6695310 1.6075675 3.7031164 269.8182 1.023223
## a_bar 3.2792181 0.5679393 2.3406424 4.1556466 212.2464 1.027593
## sigma_a 0.7541380 0.1496939 0.5347812 1.0019432 1667.0839 1.002098
## beta1 -0.8319445 0.3317046 -1.3523975 -0.3075241 356.5598 1.017944
## beta2 0.8391718 0.3458422 0.2770660 1.3980133 405.2542 1.017581
## beta3 -0.8984637 0.2126665 -1.2334199 -0.5494532 987.6224 1.007733plot(precis(m.pred, depth = 2))
plot(precis(m.size, depth = 2))
plot(precis(m.cont, depth = 2))
plot(precis(m.int, depth = 2))
plot(precis(m.pred, depth = 1))## 48 vector or matrix parameters hidden. Use depth=2 to show them.
plot(precis(m.size, depth = 1))## 48 vector or matrix parameters hidden. Use depth=2 to show them.
plot(precis(m.cont, depth = 1))## 48 vector or matrix parameters hidden. Use depth=2 to show them.
plot(precis(m.int, depth = 1))## 48 vector or matrix parameters hidden. Use depth=2 to show them.
There is a lot of variation among alphas in the model when the sole predictor is size, whereas. This suggests two things: (1) that whether or not a predator is present as a better predictor of fog survival in each tank, and (2) the tank effects are in-part due to presence of predators rather than something else specific to the tanks. Additionally, based on the variation in betas for the interaction model compared to the control model, I suspect the interaction is accounting for more variation in tank effects better than the control model.
13M2. Compare the models you fit just above, using WAIC. Can you reconcile the differences in WAIC with the posterior distributions of the models?
compare(m.pred, m.size, m.cont, m.int)## WAIC SE dWAIC dSE pWAIC weight
## m.pred 199.1586 7.857826 0.000000 NA 19.54749 0.3780954
## m.int 200.3103 8.911003 1.151636 3.7064892 19.32751 0.2125822
## m.size 200.3858 7.253405 1.227177 3.6702721 21.00322 0.2047026
## m.cont 200.3866 7.770532 1.227986 0.8007993 20.05174 0.2046198The WAIC values are not very different from one another, dWAIC is smallest for the predation and interaction models with a weight around the same. Size of frog is the worst model along with controlling model based on the higher WAIC, dWAIC, and lower weights. I think the best model to use here is the interaction model–similarly to my conclusion for the comparison of plots.