Explain GLMM practical.

README.md

@@ -24,8 +24,6 @@ Text and figures are licensed under Creative Commons Attribution [CC BY 4.0](htt
 
 + Replace crazy function using cosinus by prediction w/ rainfall/temperature (beware standardisation); Or do both
 
-+ Debug GLMM practical
-
 + Complete example on survival, w/ hypothesis testing, illustrate with Bayes factors? See https://rstudio-pubs-static.s3.amazonaws.com/358672_09291d0b37ce43f08cf001cfd25c16c2.html, https://stackoverflow.com/questions/60278806/bayes-factor-in-r-with-jaggs, https://www.martinmodrak.cz/2021/03/28/three-ways-to-compute-a-bayes-factor/, http://yourdomain.com/statistics,/modeling/2017/07/07/BF_computation.html ou encore https://michael-franke.github.io/statistics,/modeling/2017/07/07/BF_computation.html
 
 + Properly introduce GLMs; Illustrate how to find p from logit(p)
@@ -44,8 +42,6 @@ Text and figures are licensed under Creative Commons Attribution [CC BY 4.0](htt
 
 + Finish up writing that book
 
-+ ~~Add a plot with several lines from posterior distribution of regression parameters to a plot of mean response function of a covariate; then get the credible interval on the prediction~~ 
-
 + Prior predictive check for logistic storks and lmm plants
 sample_mu <- rnorm( 1e4 , 178 , 20 )
 sample_sigma <- runif( 1e4 , 0 , 50 )
@@ -65,6 +61,9 @@ dens( prior_h )
 + Add something on equivalence w/ MLE: say binomial lik $Bin(n,k)$ and beta prior $Beta(a,b)$ then posterior is beta $Beta(a+k, b+n-k)$; posterior mean is $(a+k)/(a+b+n)$ which can be written $(1-w)(a/a+b) + w k/n$. Posterior mean is weighted average of prior mean and MLE. When sample size is big, $n$ tends to infinity and posterior mean tends to MLE, whatever the prior. Same reasoning with variance shows that Bayes gives reasonable results, even w/ small sample size
 
 
++ ~~Debug GLMM practical~~
+
++ ~~Add a plot with several lines from posterior distribution of regression parameters to a plot of mean response function of a covariate; then get the credible interval on the prediction~~ 
 
 + ~~Make code/outputs fit in slides~~ 
 

docs/practicals/practical9.Rmd

@@ -196,6 +196,47 @@ round(fit$BUGSoutput$summary[, -c(4,6)], 3)
 
 ---
 
+Convert regression coefficients from scaled to non-scaled and compare to values used to generate data (from <https://stats.stackexchange.com/questions/361995/how-to-convert-coefficients-from-quadratic-function-from-scaled-to-not-scaled-co>)
+
+```{r}
+sbzero <- fit$BUGSoutput$sims.matrix[,'mu.a']
+sbun <- fit$BUGSoutput$sims.matrix[,'b[1]']
+sbdeux <- fit$BUGSoutput$sims.matrix[,'b[2]']
+
+mu <- mean(boo)
+sg <- sd(boo)
+```
+
+---
+
+```{r}
+bzero <- sbzero - sbun*mu/sg + sbdeux*mu^2/(sg^2)
+hist(bzero)
+abline(v = -14, col = "red", lwd = 2)
+mean(bzero)
+```
+
+---
+
+```{r}
+bun <- sbdeux/sg - 2 * sbdeux * mu / (sg^2)
+hist(bun)
+abline(v = 1.8, col = "red", lwd = 2)
+mean(bun)
+```
+
+---
+
+```{r}
+bdeux <- sbdeux/(sg^2)
+hist(bdeux)
+abline(v = - 0.045, col = "red", lwd = 2)
+mean(bdeux)
+```
+
+---
+## Frequentist approach
+
 .tiny-font[
 ```{r}
 library(lme4)
@@ -213,3 +254,4 @@ Parameter estimates obtained with both approaches are very similar.
 visreg::visreg(fit_lme4, xvar = 'Temp')
 ```
 ]
+

docs/practicals/practical9.html

@@ -19,7 +19,7 @@
 ### Olivier Gimenez
 ]
 .date[
-### last updated: 2023-11-11
+### last updated: 2023-11-12
 ]
 
 ---
@@ -142,13 +142,13 @@
 ```
 
 ```
-##   Transect Temperature Anemones      Temp
-## 1        1    20.33973       53 0.1338975
-## 2        1    20.47855       48 0.2125261
-## 3        1    20.61737       40 0.2911548
-## 4        1    20.75619       41 0.3697834
-## 5        1    20.89500       32 0.4484121
-## 6        1    21.03382       35 0.5270407
+##   Transect Temperature Anemones       Temp
+## 1        1    18.31928       48 -0.9742340
+## 2        1    18.31673       56 -0.9757048
+## 3        1    18.31417       51 -0.9771756
+## 4        1    18.31162       51 -0.9786464
+## 5        1    18.30906       48 -0.9801172
+## 6        1    18.30651       39 -0.9815880
 ```
 ]
 
@@ -237,26 +237,98 @@
 
 ```
 ##              mean    sd     2.5%      50%    97.5%  Rhat n.eff
-## a[1]        3.748 0.046    3.659    3.750    3.838 1.001  3000
-## a[2]        3.612 0.054    3.508    3.612    3.716 1.001  3000
-## a[3]        4.096 0.035    4.027    4.097    4.163 1.001  3000
-## a[4]        4.486 0.050    4.386    4.487    4.581 1.001  3000
-## a[5]        4.057 0.042    3.978    4.056    4.138 1.001  3000
-## a[6]        3.401 0.053    3.295    3.400    3.507 1.002  1700
-## a[7]        3.621 0.068    3.484    3.621    3.754 1.003  1100
-## a[8]        3.562 0.038    3.486    3.563    3.636 1.002   910
-## a[9]        3.866 0.033    3.798    3.867    3.930 1.001  1800
-## a[10]       3.853 0.041    3.774    3.854    3.934 1.001  3000
-## b[1]       -0.040 0.028   -0.094   -0.039    0.014 1.001  3000
-## b[2]       -0.129 0.017   -0.163   -0.129   -0.095 1.001  3000
-## deviance 1333.022 4.910 1325.304 1332.328 1344.842 1.001  3000
-## mu.a        3.828 0.131    3.570    3.829    4.068 1.001  3000
-## sigma.a     0.377 0.121    0.229    0.351    0.664 1.006  3000
+## a[1]        4.142 0.050    4.045    4.142    4.240 1.001  3000
+## a[2]        4.290 0.059    4.177    4.289    4.405 1.002  1100
+## a[3]        4.372 0.049    4.276    4.371    4.468 1.001  3000
+## a[4]        3.579 0.098    3.391    3.578    3.767 1.001  1900
+## a[5]        4.286 0.028    4.228    4.286    4.341 1.001  3000
+## a[6]        4.840 0.030    4.782    4.840    4.898 1.001  3000
+## a[7]        3.366 0.046    3.276    3.366    3.453 1.001  3000
+## a[8]        4.175 0.032    4.113    4.175    4.237 1.001  2600
+## a[9]        4.630 0.023    4.585    4.631    4.677 1.001  3000
+## a[10]       3.563 0.043    3.476    3.563    3.648 1.001  3000
+## b[1]       -0.023 0.029   -0.079   -0.023    0.035 1.001  2800
+## b[2]       -0.153 0.023   -0.196   -0.153   -0.109 1.001  3000
+## deviance 1401.831 4.854 1394.333 1401.129 1413.014 1.002  1100
+## mu.a        4.125 0.191    3.746    4.127    4.500 1.001  3000
+## sigma.a     0.568 0.167    0.341    0.535    0.980 1.007   580
 ```
 ]
 
 ---
 
+Convert regression coefficients from scaled to non-scaled and compare to values used to generate data (from <https://stats.stackexchange.com/questions/361995/how-to-convert-coefficients-from-quadratic-function-from-scaled-to-not-scaled-co>)
+
+
+```r
+sbzero <- fit$BUGSoutput$sims.matrix[,'mu.a']
+sbun <- fit$BUGSoutput$sims.matrix[,'b[1]']
+sbdeux <- fit$BUGSoutput$sims.matrix[,'b[2]']
+
+mu <- mean(boo)
+sg <- sd(boo)
+```
+
+---
+
+
+```r
+bzero <- sbzero - sbun*mu/sg + sbdeux*mu^2/(sg^2)
+hist(bzero)
+abline(v = -14, col = "red", lwd = 2)
+```
+
+![](practical9_files/figure-html/unnamed-chunk-11-1.png)<!-- -->
+
+```r
+mean(bzero)
+```
+
+```
+## [1] -15.89737
+```
+
+---
+
+
+```r
+bun <- sbdeux/sg - 2 * sbdeux * mu / (sg^2)
+hist(bun)
+abline(v = 1.8, col = "red", lwd = 2)
+```
+
+![](practical9_files/figure-html/unnamed-chunk-12-1.png)<!-- -->
+
+```r
+mean(bun)
+```
+
+```
+## [1] 1.939693
+```
+
+---
+
+
+```r
+bdeux <- sbdeux/(sg^2)
+hist(bdeux)
+abline(v = - 0.045, col = "red", lwd = 2)
+```
+
+![](practical9_files/figure-html/unnamed-chunk-13-1.png)<!-- -->
+
+```r
+mean(bdeux)
+```
+
+```
+## [1] -0.05066283
+```
+
+---
+## Frequentist approach
+
 .tiny-font[
 
 ```r
@@ -271,15 +343,15 @@
 ##  Family: poisson  ( log )
 ## Formula: Anemones ~ Temp + I(Temp^2) + (1 | Transect)
 ##    Data: data
-##       AIC       BIC    logLik  deviance  df.resid 
-## 1382.2781 1395.4714 -687.1391 1374.2781       196 
+##      AIC      BIC   logLik deviance df.resid 
+## 1461.916 1475.109 -726.958 1453.916      196 
 ## Random effects:
 ##  Groups   Name        Std.Dev.
-##  Transect (Intercept) 0.3026  
+##  Transect (Intercept) 0.4552  
 ## Number of obs: 200, groups:  Transect, 10
 ## Fixed Effects:
 ## (Intercept)         Temp    I(Temp^2)  
-##     3.83058     -0.04063     -0.12902
+##     4.12492     -0.02394     -0.15259
 ```
 ]
 
@@ -293,8 +365,9 @@
 visreg::visreg(fit_lme4, xvar = 'Temp')
 ```
 
-![](practical9_files/figure-html/unnamed-chunk-11-1.svg)<!-- -->
+![](practical9_files/figure-html/unnamed-chunk-15-1.svg)<!-- -->
 ]
+
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