This document summarizes model diagnostics and checks.

First, we set up by loading packages and generated R objects.

knitr::opts_chunk$set(echo = TRUE)
library(targets)
library(mgcv)
library(gt)
options(width = 250)
tar_load(c(
  multinomial_model,
  multinomial_model_alt,
  gam_posterior,
  posterior_stats,
  dat_prepped
))

Model Summary

summary(multinomial_model)
## 
## Family: multinom 
## Link function: 
## 
## Formula:
## outcome ~ s(sample_type, bs = "re", by = dummy_rectal) + s(day, 
##     bs = "tp", k = 5, by = sample_type, m = 2) + s(day, gender_age, 
##     bs = "fs", k = 5, xt = list(bs = "tp"), by = dummy_rectal, 
##     m = 1) + s(day_of_year, bs = "cc", by = sample_type, k = 5, 
##     m = 2) + s(day_of_year, gender_age, bs = "fs", xt = list(bs = "cc"), 
##     k = 5, by = dummy_rectal, m = 1) + s(fmi_normalized, k = 5, 
##     bs = "tp", by = dummy_rectal) + s(reproductive_condition, 
##     bs = "re", by = dummy_repro)
## <environment: 0x7f87dd1492a8>
## ~s(sample_type, bs = "re", by = dummy_rectal) + s(day, bs = "tp", 
##     k = 5, by = sample_type, m = 2) + s(day, gender_age, bs = "fs", 
##     k = 5, xt = list(bs = "tp"), by = dummy_rectal, m = 1) + 
##     s(day_of_year, bs = "cc", by = sample_type, k = 5, m = 2) + 
##     s(day_of_year, gender_age, bs = "fs", xt = list(bs = "cc"), 
##         k = 5, by = dummy_rectal, m = 1) + s(fmi_normalized, 
##     k = 5, bs = "tp", by = dummy_rectal) + s(reproductive_condition, 
##     bs = "re", by = dummy_repro)
## <environment: 0x7f87dd1492a8>
## ~s(sample_type, bs = "re", by = dummy_rectal) + s(day, bs = "tp", 
##     k = 5, by = sample_type, m = 2) + s(day_of_year, bs = "cc", 
##     by = sample_type, k = 5, m = 2)
## <environment: 0x7f87dd1492a8>
## 
## Parametric coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept)    -3.8526     0.2441 -15.784   <2e-16 ***
## (Intercept).1  -2.4051     0.1467 -16.391   <2e-16 ***
## (Intercept).2  -6.0192     0.6724  -8.953   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##                                                 edf Ref.df Chi.sq  p-value    
## s(sample_type):dummy_rectal1              1.807e-04  1.000  0.000 0.515589    
## s(day):sample_typeFecal                   2.574e+00  2.885  4.874 0.173261    
## s(day):sample_typeRectal                  1.000e+00  1.000  0.161 0.688685    
## s(day,gender_age):dummy_rectal1           1.219e-04 22.000  0.000 0.905557    
## s(day_of_year):sample_typeFecal           2.247e+00  3.000 24.721 7.52e-07 ***
## s(day_of_year):sample_typeRectal          1.351e+00  3.000  3.923 0.048421 *  
## s(day_of_year,gender_age):dummy_rectal1   1.902e-04 16.000  0.000 0.900047    
## s(fmi_normalized):dummy_rectal1           1.000e+00  1.000  3.184 0.074388 .  
## s(reproductive_condition):dummy_repro1    1.019e-04  3.000  0.000 0.911317    
## s.1(sample_type):dummy_rectal1            7.222e-05  1.000  0.000 0.458759    
## s.1(day):sample_typeFecal                 2.339e+00  2.689 13.628 0.001508 ** 
## s.1(day):sample_typeRectal                2.629e+00  2.873  6.911 0.055820 .  
## s.1(day,gender_age):dummy_rectal1         2.112e-04 20.000  0.000 0.866381    
## s.1(day_of_year):sample_typeFecal         2.412e+00  3.000 38.195  < 2e-16 ***
## s.1(day_of_year):sample_typeRectal        2.074e+00  3.000  9.862 0.000529 ***
## s.1(day_of_year,gender_age):dummy_rectal1 5.080e-04 16.000  0.000 0.802830    
## s.1(fmi_normalized):dummy_rectal1         1.000e+00  1.000  1.927 0.165134    
## s.1(reproductive_condition):dummy_repro1  1.833e+00  3.000  5.754 0.025429 *  
## s.2(sample_type):dummy_rectal1            1.475e-05  1.000  0.000 0.758139    
## s.2(day):sample_typeFecal                 1.000e+00  1.000  0.009 0.923636    
## s.2(day):sample_typeRectal                1.000e+00  1.000  0.344 0.557249    
## s.2(day_of_year):sample_typeFecal         2.073e+00  3.000 18.059 4.35e-05 ***
## s.2(day_of_year):sample_typeRectal        1.478e+00  3.000  4.624 0.036269 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Deviance explained = 14.6%
## -REML = -571.32  Scale est. = 1         n = 1430

Alterate Model Summary

This is an alternate model fomulation with simple singel random effects of age/sex category rather than time-varying ones. This is used to check if there age/sex effects are just insignificant because of the high complexity of the time-varying smooths.

summary(multinomial_model_alt)
## 
## Family: multinom 
## Link function: 
## 
## Formula:
## outcome ~ s(sample_type, bs = "re", by = dummy_rectal) + s(day, 
##     bs = "tp", k = 5, by = sample_type, m = 2) + s(day_of_year, 
##     bs = "cc", by = sample_type, k = 5, m = 2) + s(gender_age, 
##     bs = "re", by = dummy_rectal) + s(fmi_normalized, k = 5, 
##     bs = "tp", by = dummy_rectal) + s(reproductive_condition, 
##     bs = "re", by = dummy_repro)
## <environment: 0x7f88e3eec110>
## ~s(sample_type, bs = "re", by = dummy_rectal) + s(day, bs = "tp", 
##     k = 5, by = sample_type, m = 2) + s(day_of_year, bs = "cc", 
##     by = sample_type, k = 5, m = 2) + s(gender_age, bs = "re", 
##     by = dummy_rectal) + s(fmi_normalized, k = 5, bs = "tp", 
##     by = dummy_rectal) + s(reproductive_condition, bs = "re", 
##     by = dummy_repro)
## <environment: 0x7f88e3eec110>
## ~s(sample_type, bs = "re", by = dummy_rectal) + s(day, bs = "tp", 
##     k = 5, by = sample_type, m = 2) + s(day_of_year, bs = "cc", 
##     by = sample_type, k = 5, m = 2)
## <environment: 0x7f88e3eec110>
## 
## Parametric coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept)    -3.8528     0.2441 -15.782   <2e-16 ***
## (Intercept).1  -2.4051     0.1467 -16.391   <2e-16 ***
## (Intercept).2  -6.0192     0.6724  -8.953   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##                                                edf Ref.df Chi.sq  p-value    
## s(sample_type):dummy_rectal1             2.109e-04  2.000  0.000 0.516073    
## s(day):sample_typeFecal                  2.575e+00  2.886  4.878 0.172907    
## s(day):sample_typeRectal                 1.000e+00  1.000  0.160 0.688800    
## s(day_of_year):sample_typeFecal          2.247e+00  3.000 24.725 7.50e-07 ***
## s(day_of_year):sample_typeRectal         1.351e+00  3.000  3.923 0.048417 *  
## s(gender_age):dummy_rectal1              1.742e-04  4.000  0.000 0.976698    
## s(fmi_normalized):dummy_rectal1          1.000e+00  1.000  3.184 0.074371 .  
## s(reproductive_condition):dummy_repro1   9.877e-05  3.000  0.000 0.911349    
## s.1(sample_type):dummy_rectal1           2.896e-04  1.000  0.000 0.458759    
## s.1(day):sample_typeFecal                2.339e+00  2.689 13.628 0.001508 ** 
## s.1(day):sample_typeRectal               2.629e+00  2.873  6.909 0.055847 .  
## s.1(day_of_year):sample_typeFecal        2.412e+00  3.000 38.196  < 2e-16 ***
## s.1(day_of_year):sample_typeRectal       2.075e+00  3.000  9.863 0.000529 ***
## s.1(gender_age):dummy_rectal1            7.434e-04  4.000  0.000 0.659272    
## s.1(fmi_normalized):dummy_rectal1        1.000e+00  1.000  1.927 0.165131    
## s.1(reproductive_condition):dummy_repro1 1.833e+00  3.000  5.754 0.025432 *  
## s.2(sample_type):dummy_rectal1           1.858e-05  1.000  0.000 0.758138    
## s.2(day):sample_typeFecal                1.000e+00  1.000  0.009 0.923646    
## s.2(day):sample_typeRectal               1.000e+00  1.000  0.345 0.557247    
## s.2(day_of_year):sample_typeFecal        2.073e+00  3.000 18.059 4.35e-05 ***
## s.2(day_of_year):sample_typeRectal       1.478e+00  3.000  4.624 0.036270 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Deviance explained = 14.6%
## -REML = -571.32  Scale est. = 1         n = 1430

GAM Checks

gam.check() provides convergence diagnostics. Basis dimension (k) results should all be above 0.05 to indicate that there are not systematic patterns in the model residuals. Four diagnostic plot are provided. In the first (Q-Q) plot, we should see residuals and quantiles line up on the 1-1 line. Other plots are difficult to interpret in the multinomial case.

gam.check(multinomial_model)

## 
## Method: REML   Optimizer: efs
## $iter
## [1] 51
## 
## $score.hist
##  [1] -544.2658 -557.4188 -562.9083 -565.1258 -566.6785 -567.9323 -568.8746 -569.5572 -570.0543 -570.4158 -570.6714 -570.8419 -570.9473 -571.0080 -571.0413 -571.0592 -571.0687 -571.0738 -571.0768 -571.0786 -571.0799 -571.0811 -571.0822 -571.0835
## [25] -571.0850 -571.0868 -571.0888 -571.0913 -571.0942 -571.0977 -571.1019 -571.1070 -571.1131 -571.1206 -571.1296 -571.1405 -571.1536 -571.1693 -571.1874 -571.2077 -571.2293 -571.2507 -571.2699 -571.2855 -571.2971 -571.3049 -571.3098 -571.3127
## [49] -571.3144 -571.3152 -571.3160
## 
## $conv
## [1] "full convergence"
## 
## Model rank =  157 / 157 
## 
## Basis dimension (k) checking results. Low p-value (k-index<1) may
## indicate that k is too low, especially if edf is close to k'.
## 
##                                                 k'      edf k-index p-value
## s(sample_type):dummy_rectal1              2.00e+00 1.81e-04      NA      NA
## s(day):sample_typeFecal                   4.00e+00 2.57e+00    0.95    0.15
## s(day):sample_typeRectal                  4.00e+00 1.00e+00    0.95    0.25
## s(day,gender_age):dummy_rectal1           2.50e+01 1.22e-04    0.95    0.14
## s(day_of_year):sample_typeFecal           3.00e+00 2.25e+00    0.95    0.21
## s(day_of_year):sample_typeRectal          3.00e+00 1.35e+00    0.95    0.18
## s(day_of_year,gender_age):dummy_rectal1   2.00e+01 1.90e-04    0.95    0.20
## s(fmi_normalized):dummy_rectal1           4.00e+00 1.00e+00    0.97    0.39
## s(reproductive_condition):dummy_repro1    4.00e+00 1.02e-04      NA      NA
## s.1(sample_type):dummy_rectal1            2.00e+00 7.22e-05      NA      NA
## s.1(day):sample_typeFecal                 4.00e+00 2.34e+00    0.95    0.22
## s.1(day):sample_typeRectal                4.00e+00 2.63e+00    0.95    0.18
## s.1(day,gender_age):dummy_rectal1         2.50e+01 2.11e-04    0.95    0.21
## s.1(day_of_year):sample_typeFecal         3.00e+00 2.41e+00    0.95    0.18
## s.1(day_of_year):sample_typeRectal        3.00e+00 2.07e+00    0.95    0.21
## s.1(day_of_year,gender_age):dummy_rectal1 2.00e+01 5.08e-04    0.95    0.21
## s.1(fmi_normalized):dummy_rectal1         4.00e+00 1.00e+00    0.97    0.43
## s.1(reproductive_condition):dummy_repro1  4.00e+00 1.83e+00      NA      NA
## s.2(sample_type):dummy_rectal1            2.00e+00 1.48e-05      NA      NA
## s.2(day):sample_typeFecal                 4.00e+00 1.00e+00    0.95    0.16
## s.2(day):sample_typeRectal                4.00e+00 1.00e+00    0.95    0.15
## s.2(day_of_year):sample_typeFecal         3.00e+00 2.07e+00    0.95    0.24
## s.2(day_of_year):sample_typeRectal        3.00e+00 1.48e+00    0.95    0.20

Individual Smooth terms

These are plots of the indivudal smooths in the model on the linear scale. Inspect for strange behavior and check that their shape corresponds with significance terms in in the model summary.

for (i in seq_len(length(multinomial_model$smooth) + sum(multinomial_model$nsdf) - 3)) {
  plot(multinomial_model, scale = 0, select = i, all.terms = TRUE)
}

MCMC diagnostics

These are acceptance ratios from the 4 MCMC chains run.

From ?gam.mh:

The function reports the acceptance rate of the two types of step. If the random walk acceptance probability ($rw.accept) is higher than a quarter then rw.step should probably be increased. Similarly if the acceptance rate ($accept) is too low, it should be decreased. The random walk steps can be turned off altogether (see above), but it is important to check the chains for stuck sections if this is done.

attributes(gam_posterior)[c("accept", "rw.accept")]
## $accept
## [1] 0.18280 0.15795 0.15805 0.17980
## 
## $rw.accept
## [1] 0.16045 0.16250 0.15910 0.15660

Here is a summary of the dimensions of the MCMC chain output

tibble::tibble(
  dimension = names(dimnames(gam_posterior)),
  value = dim(gam_posterior)
)
## # A tibble: 3 × 2
##   dimension value
##   <chr>     <int>
## 1 Iteration   100
## 2 Chain         4
## 3 Parameter   157

Here are Stan-type per-parameter chain diagnostics. Rhat should be at or near 1 to indicate that the multiple MCMC chains are well-mixed. ess_bulk and ess_tail are the effective sample sizes for posterior samples from the parameter distributions and the tails of those distributions, respectively. From the Stan manual:

We recommend running at least four chains by default and only using the sample if R-hat is less than 1.05….

Both bulk-ESS and tail-ESS should be at least 100 (approximately) per Markov Chain in order to be reliable and indicate that estimates of respective posterior quantiles are reliable

Note that for coefficient values < 1e-6, mixing isn’t very relevant.

options(scipen=12)
posterior_stats |> 
  mutate(across(c("ess_bulk", "ess_tail"), round)) |> 
  mutate(across(2:7, \(x) signif(x, 2))) |> 
  mutate(rhat = signif(rhat, 3)) |> 
  mutate(across(2:8, \(x) gt:::align_to_char(as.character(x)))) |> 
  knitr::kable(format.args = list(scientific = FALSE))
variable mean median sd mad q5 q95 rhat ess_bulk ess_tail
(Intercept) -4         -4          0.27    0.25    -4.5    -3.6     1.01  447 376
s(sample_type):dummy_rectal1.1  0.00024    0.00034    0.0068  0.0062  -0.011   0.012   0.995 451 416
s(sample_type):dummy_rectal1.2  0.00014    0.00051    0.0068  0.0067  -0.011   0.012   1     415 340
s(day):sample_typeFecal.1  2.2        2.2        1.2     1.2      0.19    4.1     1     399 368
s(day):sample_typeFecal.2  0.067     -0.036      2.8     2.9     -4.6     4.8     1     493 402
s(day):sample_typeFecal.3 -1.4       -1.4        3.6     3.6     -7.6     4.4     1     536 416
s(day):sample_typeFecal.4 -2.6       -2.6        1.4     1.4     -4.6    -0.29    1     394 368
s(day):sample_typeRectal.1 -0.000075  -0.00009    0.0033  0.0035  -0.0053  0.0054  1     349 415
s(day):sample_typeRectal.2 -0.00014   -0.00021    0.0054  0.0057  -0.0096  0.008   0.997 337 340
s(day):sample_typeRectal.3  0.000016  -0.000033   0.01    0.0099  -0.018   0.016   0.999 319 395
s(day):sample_typeRectal.4 -0.09      -0.11       0.35    0.34    -0.64    0.49    0.999 436 365
s(day,gender_age):dummy_rectal1.1  0.000083   0.00015    0.0015  0.0015  -0.0024  0.0026  1.01  408 290
s(day,gender_age):dummy_rectal1.2 -0.000087  -0.00008    0.0015  0.0017  -0.0025  0.0025  0.994 415 370
s(day,gender_age):dummy_rectal1.3 -0.000014  -0.0000062  0.0015  0.0015  -0.0025  0.0024  1.02  492 391
s(day,gender_age):dummy_rectal1.4 -0.000051  -0.000023   0.0016  0.0015  -0.0028  0.0024  1     323 340
s(day,gender_age):dummy_rectal1.5 -0.00014   -0.000087   0.002   0.0021  -0.0033  0.003   1     339 324
s(day,gender_age):dummy_rectal1.6  0.000019  -0.000093   0.0016  0.0015  -0.0024  0.0026  1     402 398
s(day,gender_age):dummy_rectal1.7 -0.0000049  0.000046   0.0015  0.0015  -0.0025  0.0024  0.996 441 376
s(day,gender_age):dummy_rectal1.8 -0.000012   0.000054   0.0015  0.0015  -0.0025  0.0024  0.999 395 416
s(day,gender_age):dummy_rectal1.9 -0.000049  -0.000084   0.0016  0.0015  -0.0027  0.0026  0.996 422 397
s(day,gender_age):dummy_rectal1.10  0.00012    0.00016    0.0021  0.0023  -0.0034  0.0033  1.01  379 458
s(day,gender_age):dummy_rectal1.11 -0.00014   -0.00017    0.0015  0.0015  -0.0025  0.0023  1.02  345 407
s(day,gender_age):dummy_rectal1.12  0.0000061  0.0000027  0.0016  0.0017  -0.0025  0.0027  0.997 385 330
s(day,gender_age):dummy_rectal1.13  0.000045   0.00011    0.0016  0.0015  -0.0026  0.0026  1     381 374
s(day,gender_age):dummy_rectal1.14  0.000086   0.000047   0.0015  0.0014  -0.0022  0.0024  0.998 433 412
s(day,gender_age):dummy_rectal1.15 -0.00018   -0.00019    0.0021  0.0023  -0.0038  0.0032  1     411 415
s(day,gender_age):dummy_rectal1.16  0.000086   0.000043   0.0016  0.0016  -0.0025  0.0028  1     454 373
s(day,gender_age):dummy_rectal1.17 -0.00003    0.000076   0.0015  0.0015  -0.0025  0.0023  1.01  372 375
s(day,gender_age):dummy_rectal1.18  0.0000071  0.000021   0.0015  0.0016  -0.0024  0.0024  0.996 367 342
s(day,gender_age):dummy_rectal1.19 -0.000059  -0.000024   0.0015  0.0016  -0.0024  0.0023  1.01  397 365
s(day,gender_age):dummy_rectal1.20 -0.000048  -0.00014    0.0021  0.0021  -0.0033  0.0035  0.999 434 373
s(day,gender_age):dummy_rectal1.21  0.000082  -0.0000064  0.0015  0.0014  -0.0024  0.0026  1.02  334 389
s(day,gender_age):dummy_rectal1.22 -0.000021   0.0000095  0.0016  0.0015  -0.0027  0.0025  1.02  406 339
s(day,gender_age):dummy_rectal1.23  0.000033  -0.0000094  0.0015  0.0015  -0.0024  0.0026  1.01  337 347
s(day,gender_age):dummy_rectal1.24 -0.000079  -0.000065   0.0016  0.0016  -0.0026  0.0025  1     369 340
s(day,gender_age):dummy_rectal1.25  0.000066  -0.000047   0.002   0.002   -0.003   0.0034  0.996 476 410
s(day_of_year):sample_typeFecal.1  0.64       0.64       0.53    0.52    -0.2     1.5     1     384 269
s(day_of_year):sample_typeFecal.2 -2.4       -2.3        0.84    0.86    -3.8    -1.1     1     433 409
s(day_of_year):sample_typeFecal.3 -2         -2          0.53    0.55    -2.9    -1.2     1     485 362
s(day_of_year):sample_typeRectal.1  0.22       0.23       0.4     0.41    -0.42    0.83    1     513 346
s(day_of_year):sample_typeRectal.2 -0.44      -0.47       0.47    0.5     -1.2     0.3     1     479 360
s(day_of_year):sample_typeRectal.3 -0.71      -0.72       0.44    0.41    -1.4     0.0084  1     436 354
s(day_of_year,gender_age):dummy_rectal1.1 -0.000063   0.000079   0.0022  0.002   -0.0041  0.0037  0.998 398 402
s(day_of_year,gender_age):dummy_rectal1.2 -0.0000054  0.00003    0.0021  0.002   -0.0036  0.0034  1     401 308
s(day_of_year,gender_age):dummy_rectal1.3 -0.000022   0.000031   0.0021  0.002   -0.0038  0.0034  1     418 443
s(day_of_year,gender_age):dummy_rectal1.4 -0.000066  -0.00014    0.0023  0.0022  -0.0039  0.0038  0.995 422 366
s(day_of_year,gender_age):dummy_rectal1.5 -0.00011   -0.00022    0.002   0.002   -0.0035  0.0032  1     420 391
s(day_of_year,gender_age):dummy_rectal1.6 -0.00021   -0.00021    0.0022  0.0021  -0.0037  0.0033  1.01  439 308
s(day_of_year,gender_age):dummy_rectal1.7 -0.000012   0.000013   0.0021  0.0021  -0.0034  0.0035  0.998 556 367
s(day_of_year,gender_age):dummy_rectal1.8  0.00017    0.00028    0.0021  0.002   -0.0031  0.0037  0.999 479 373
s(day_of_year,gender_age):dummy_rectal1.9  0.000089   0.000036   0.0021  0.002   -0.0038  0.0036  1     460 401
s(day_of_year,gender_age):dummy_rectal1.10 -0.000039  -0.000064   0.0022  0.0021  -0.0033  0.0037  1.02  306 299
s(day_of_year,gender_age):dummy_rectal1.11  0.00012    0.000078   0.0021  0.0022  -0.0032  0.0037  0.997 424 346
s(day_of_year,gender_age):dummy_rectal1.12 -0.00016   -0.00011    0.0022  0.0023  -0.0038  0.0032  1     352 298
s(day_of_year,gender_age):dummy_rectal1.13  0.000089   0.000071   0.0022  0.0022  -0.0036  0.0037  0.998 433 357
s(day_of_year,gender_age):dummy_rectal1.14 -0.000055  -0.000075   0.0022  0.0023  -0.0034  0.0034  1     424 282
s(day_of_year,gender_age):dummy_rectal1.15 -0.0001    -0.00011    0.0022  0.0023  -0.0038  0.0034  1     446 396
s(day_of_year,gender_age):dummy_rectal1.16  0.00003   -0.000014   0.0021  0.002   -0.0035  0.0033  0.998 390 411
s(day_of_year,gender_age):dummy_rectal1.17  0.0000025 -0.000023   0.0022  0.0023  -0.0036  0.0035  0.999 448 466
s(day_of_year,gender_age):dummy_rectal1.18  0.00017    0.00023    0.0023  0.0024  -0.0035  0.0041  1.01  394 326
s(day_of_year,gender_age):dummy_rectal1.19 -0.000087   0.000038   0.0022  0.0022  -0.0036  0.0034  1.01  350 306
s(day_of_year,gender_age):dummy_rectal1.20 -0.000031   0.00000027 0.002   0.0022  -0.0033  0.0033  1     377 411
s(fmi_normalized):dummy_rectal1.1 -0.000055  -0.000011   0.0014  0.0014  -0.0022  0.0022  1.01  444 371
s(fmi_normalized):dummy_rectal1.2  0.0000093 -0.000025   0.00067 0.00061 -0.0011  0.0011  0.999 306 413
s(fmi_normalized):dummy_rectal1.3 -0.00051    0.000048   0.0081  0.0081  -0.014   0.012   0.999 464 401
s(fmi_normalized):dummy_rectal1.4 -0.29      -0.29       0.17    0.17    -0.56   -0.006   1     385 329
s(reproductive_condition):dummy_repro1.1 -0.00024   -0.000083   0.0067  0.0065  -0.012   0.011   1.01  337 263
s(reproductive_condition):dummy_repro1.2 -0.00025   -0.00037    0.0069  0.0069  -0.011   0.012   1.01  381 281
s(reproductive_condition):dummy_repro1.3 -0.000032  -0.00043    0.0069  0.0064  -0.012   0.011   0.998 407 412
s(reproductive_condition):dummy_repro1.4  0.00034    0.00043    0.0065  0.0059  -0.011   0.011   1     344 370
(Intercept).1 -2.5       -2.5        0.15    0.15    -2.7    -2.2     1     345 414
s.1(sample_type):dummy_rectal1.1  0.00023    0.00014    0.0028  0.0029  -0.0042  0.0048  0.999 389 374
s.1(sample_type):dummy_rectal1.2  0.0000096 -0.00011    0.0031  0.0033  -0.0046  0.0049  1.01  409 373
s.1(day):sample_typeFecal.1  0.33       0.34       0.55    0.53    -0.56    1.2     1.01  342 416
s.1(day):sample_typeFecal.2  0.27       0.27       1.1     1.2     -1.7     2.1     1     358 293
s.1(day):sample_typeFecal.3  1.6        1.6        1.5     1.6     -0.79    4       1.01  353 341
s.1(day):sample_typeFecal.4 -0.49      -0.48       0.58    0.57    -1.4     0.48    1     397 295
s.1(day):sample_typeRectal.1 -2.1       -2.1        1.3     1.3     -4.2     0.11    0.996 441 401
s.1(day):sample_typeRectal.2 -2.1       -2.1        3.4     3.3     -8       3.9     1.01  459 406
s.1(day):sample_typeRectal.3 -1         -1          4.8     4.5     -9       6.8     1.01  433 397
s.1(day):sample_typeRectal.4  2.1        2.2        1.7     1.6     -0.65    5       1     413 374
s.1(day,gender_age):dummy_rectal1.1  0.000057   0.0001     0.00093 0.00096 -0.0015  0.0015  0.997 420 409
s.1(day,gender_age):dummy_rectal1.2  0.000024  -0.000017   0.00095 0.00094 -0.0015  0.0016  1.01  507 416
s.1(day,gender_age):dummy_rectal1.3 -0.000052   0.0000044  0.00098 0.00098 -0.0017  0.0016  1     488 416
s.1(day,gender_age):dummy_rectal1.4 -0.000046  -0.00012    0.00092 0.00091 -0.0017  0.0015  1     484 454
s.1(day,gender_age):dummy_rectal1.5 -0.000097  -0.00018    0.0014  0.0013  -0.0022  0.0021  1     320 283
s.1(day,gender_age):dummy_rectal1.6  0.000064   0.000042   0.00097 0.00092 -0.0015  0.0016  1     415 466
s.1(day,gender_age):dummy_rectal1.7  0.000023   0.000044   0.001   0.00096 -0.0017  0.0017  1     458 413
s.1(day,gender_age):dummy_rectal1.8  0.000015   0.00005    0.00095 0.00087 -0.0016  0.0016  1.01  447 341
s.1(day,gender_age):dummy_rectal1.9 -0.000085  -0.000081   0.00093 0.00087 -0.0017  0.0014  1     431 340
s.1(day,gender_age):dummy_rectal1.10 -0.000024  -0.00003    0.0013  0.0013  -0.0021  0.0022  1     381 417
s.1(day,gender_age):dummy_rectal1.11 -0.000057  -0.000074   0.00095 0.0011  -0.0017  0.0016  1.01  370 432
s.1(day,gender_age):dummy_rectal1.12 -0.00002    0.0000004  0.00095 0.00094 -0.0016  0.0015  1.01  411 417
s.1(day,gender_age):dummy_rectal1.13  0.000068   0.0001     0.00093 0.00093 -0.0016  0.0016  1.02  347 367
s.1(day,gender_age):dummy_rectal1.14 -0.000025  -0.000015   0.00095 0.00098 -0.0016  0.0014  1     441 442
s.1(day,gender_age):dummy_rectal1.15  0.000073   0.000043   0.0014  0.0015  -0.0023  0.0022  1     394 374
s.1(day,gender_age):dummy_rectal1.16 -0.00003   -0.000018   0.00093 0.00087 -0.0016  0.0015  1.01  443 393
s.1(day,gender_age):dummy_rectal1.17  0.000059   0.0001     0.00098 0.00096 -0.0017  0.0016  1.01  437 409
s.1(day,gender_age):dummy_rectal1.18 -0.000037  -0.000011   0.00098 0.00099 -0.0016  0.0015  1.01  375 409
s.1(day,gender_age):dummy_rectal1.19  0.00002    0.000041   0.00099 0.0011  -0.0015  0.0015  0.996 429 357
s.1(day,gender_age):dummy_rectal1.20 -0.00021   -0.00018    0.0014  0.0014  -0.0023  0.0018  1.01  402 313
s.1(day,gender_age):dummy_rectal1.21  0.000083   0.00006    0.00091 0.0009  -0.0014  0.0016  0.995 426 417
s.1(day,gender_age):dummy_rectal1.22 -0.000014  -0.000047   0.00092 0.00098 -0.0015  0.0016  1     428 372
s.1(day,gender_age):dummy_rectal1.23 -0.000056  -0.000012   0.00096 0.00094 -0.0019  0.0014  0.995 428 429
s.1(day,gender_age):dummy_rectal1.24  0.000071   0.000089   0.00095 0.00093 -0.0015  0.0016  0.997 416 374
s.1(day,gender_age):dummy_rectal1.25 -0.000031   0.000032   0.0014  0.0013  -0.0024  0.0022  0.999 354 417
s.1(day_of_year):sample_typeFecal.1  1.6        1.6        0.29    0.29     1.1     2.1     1.01  394 359
s.1(day_of_year):sample_typeFecal.2  0.59       0.59       0.34    0.34     0.015   1.2     1     345 400
s.1(day_of_year):sample_typeFecal.3 -1.4       -1.4        0.39    0.38    -2.1    -0.78    1     407 373
s.1(day_of_year):sample_typeRectal.1  0.97       0.97       0.39    0.39     0.36    1.7     1     467 452
s.1(day_of_year):sample_typeRectal.2  0.0082     0.022      0.53    0.54    -0.9     0.89    0.999 479 461
s.1(day_of_year):sample_typeRectal.3 -0.47      -0.45       0.41    0.36    -1.2     0.18    0.999 455 414
s.1(day_of_year,gender_age):dummy_rectal1.1  0.00015    0.00018    0.0017  0.0017  -0.0026  0.003   0.996 462 342
s.1(day_of_year,gender_age):dummy_rectal1.2  0.000021   0.000023   0.0017  0.0018  -0.0027  0.0029  1     458 411
s.1(day_of_year,gender_age):dummy_rectal1.3  0.000053   0.000075   0.0016  0.0016  -0.0027  0.0027  0.999 436 409
s.1(day_of_year,gender_age):dummy_rectal1.4  0.0000004  0.000052   0.0022  0.0024  -0.0033  0.0036  1     366 414
s.1(day_of_year,gender_age):dummy_rectal1.5 -0.000094  -0.0001     0.0017  0.0016  -0.0029  0.0027  1     390 416
s.1(day_of_year,gender_age):dummy_rectal1.6 -0.00012   -0.0001     0.0016  0.0017  -0.0029  0.0024  0.995 595 365
s.1(day_of_year,gender_age):dummy_rectal1.7  0.000053  -0.000016   0.0016  0.0016  -0.0024  0.0028  0.996 500 450
s.1(day_of_year,gender_age):dummy_rectal1.8 -0.000038   0.000024   0.002   0.0021  -0.0034  0.0031  0.999 396 414
s.1(day_of_year,gender_age):dummy_rectal1.9 -0.000042  -0.000012   0.0017  0.0017  -0.0028  0.0028  1     375 330
s.1(day_of_year,gender_age):dummy_rectal1.10 -0.000035  -0.000031   0.0017  0.0017  -0.0029  0.0028  1     388 321
s.1(day_of_year,gender_age):dummy_rectal1.11 -0.000063  -0.000059   0.0017  0.0016  -0.0028  0.0029  0.995 450 359
s.1(day_of_year,gender_age):dummy_rectal1.12 -0.00011    0.000055   0.0021  0.002   -0.0034  0.0029  0.998 332 304
s.1(day_of_year,gender_age):dummy_rectal1.13  0.000023   0.00002    0.0017  0.0016  -0.0029  0.0028  1.01  446 417
s.1(day_of_year,gender_age):dummy_rectal1.14 -0.000023  -0.000079   0.0016  0.0016  -0.0027  0.0027  0.998 391 416
s.1(day_of_year,gender_age):dummy_rectal1.15 -0.00002   -0.000069   0.0017  0.0017  -0.0027  0.0027  0.998 396 410
s.1(day_of_year,gender_age):dummy_rectal1.16 -0.000083  -0.00011    0.0021  0.002   -0.0038  0.0036  0.995 456 410
s.1(day_of_year,gender_age):dummy_rectal1.17 -0.00022   -0.00022    0.0017  0.0018  -0.0031  0.0025  1.01  378 459
s.1(day_of_year,gender_age):dummy_rectal1.18 -0.000046  -0.000085   0.0017  0.0017  -0.0028  0.003   1     416 393
s.1(day_of_year,gender_age):dummy_rectal1.19  0.000043   0.000071   0.0017  0.0016  -0.0027  0.0026  1     390 354
s.1(day_of_year,gender_age):dummy_rectal1.20 -0.0000085  0.000049   0.0022  0.0022  -0.0038  0.0038  1     373 287
s.1(fmi_normalized):dummy_rectal1.1  0.0000063  0.000032   0.0013  0.0013  -0.0021  0.0021  1.01  388 401
s.1(fmi_normalized):dummy_rectal1.2 -0.000035  -0.000039   0.00064 0.00066 -0.0011  0.00097 1     348 414
s.1(fmi_normalized):dummy_rectal1.3  0.00032   -0.0002     0.0074  0.0075  -0.012   0.012   1     406 302
s.1(fmi_normalized):dummy_rectal1.4 -0.13      -0.13       0.092   0.097   -0.27    0.026   0.999 344 464
s.1(reproductive_condition):dummy_repro1.1 -0.037     -0.061      1.3     1.2     -2.1     2.3     1     380 305
s.1(reproductive_condition):dummy_repro1.2 -1.1       -1.1        0.97    1       -2.6     0.49    0.999 345 200
s.1(reproductive_condition):dummy_repro1.3 -0.7       -0.61       0.75    0.81    -1.9     0.47    1     352 337
s.1(reproductive_condition):dummy_repro1.4 -1.5       -1.4        0.67    0.63    -2.7    -0.46    0.997 337 331
(Intercept).2 -6.6       -6.6        0.75    0.75    -8      -5.4     1.01  440 413
s.2(sample_type):dummy_rectal1.1  0.000027   0.0003     0.0052  0.0053  -0.0094  0.0083  0.999 385 361
s.2(sample_type):dummy_rectal1.2  0.000031   0.00011    0.0053  0.0053  -0.0091  0.0086  1.01  363 376
s.2(day):sample_typeFecal.1  0.000038   0.00014    0.0031  0.0031  -0.0052  0.0049  1.01  411 390
s.2(day):sample_typeFecal.2  0.00007    0.00027    0.0052  0.0056  -0.0085  0.0086  1     434 372
s.2(day):sample_typeFecal.3 -0.00051   -0.00072    0.0094  0.0093  -0.017   0.014   0.999 395 372
s.2(day):sample_typeFecal.4  0.0075     0.022      0.35    0.35    -0.55    0.57    1     376 347
s.2(day):sample_typeRectal.1  0.000034   0.000063   0.0061  0.0061  -0.01    0.01    0.997 464 328
s.2(day):sample_typeRectal.2  0.0014     0.0013     0.01    0.0094  -0.015   0.018   1.01  341 411
s.2(day):sample_typeRectal.3  0.0024     0.0027     0.019   0.018   -0.031   0.034   1.01  402 372
s.2(day):sample_typeRectal.4 -0.82      -0.81       1.1     1.1     -2.8     0.87    1     399 373
s.2(day_of_year):sample_typeFecal.1 -1.4       -1.4        1.3     1.3     -3.6     0.88    1     345 409
s.2(day_of_year):sample_typeFecal.2  3.2        3.3        0.83    0.81     1.9     4.7     0.998 455 371
s.2(day_of_year):sample_typeFecal.3  3.6        3.6        0.99    1        2.1     5.2     0.999 494 435
s.2(day_of_year):sample_typeRectal.1 -0.31      -0.3        0.92    0.89    -1.8     1.1     0.997 393 360
s.2(day_of_year):sample_typeRectal.2  1.9        1.9        1.1     1.1      0.2     3.8     1     415 416
s.2(day_of_year):sample_typeRectal.3  1.7        1.8        1       1.1     -0.086   3.3     1     442 415