Checked 02/05/2023

Similar to my previous blog on deriving ‘Effect Concentrations’ from GLMMs, this blog details a way to code bootstrapped 95% CI from GAMMs and their bootstrapped ECx. I know of no method to extract ECx from GAMMs analytically like we can for GLMMs, so here I interpolate from the predictions. This isn’t super accurate but the error will be quite small – make sure to keep the prediction length >100 though.

This dataset doesn’t really require a GAMM, I’ve just used it for demonstration purposes. I’ve added an observational-level random effect, which is typically used for tank experiments.

Lets load in some data.

```
# Load packages (if not installed Tools->Install Packages)
library(mgcv)
library(lattice)
library(MASS)
library(MuMIn)
data1 <- read.table(file="https://raw.githubusercontent.com/gerard-ricardo/data/master/2015_uppersurface", header= TRUE, dec=",")
data1
# 2 Labeling and wrangling -----------------------------------------------
data1$sus.sed <- as.numeric(as.character(data1$sus.sed))
data1$log.x <- log10(data1$sus.sed) #log10 trans
data1$fert <- as.numeric(as.character(data1$fert))
data1$suc <- data1$fert
data1$tot <- as.numeric(as.character(data1$tot))
data1$obs <- factor(formatC(1:nrow(data1), flag="0", width = 3))# unique tank ID for later on
data1$log.x <- ifelse(data1$log.x < 0.1, 0, data1$log.x) #Replace neg with zero
```

Next let’s fit a regular GAMM for demonstration sake. Here we are fitting a GAMM, and predicting along raw.x in the data.frame df1. We convert the predicted SE to approximate 95% CI by multiplying them by 1.96.

```
m2.gamm<-gamm(cbind(suc,(tot - suc))~s(log.x, fx = F, k = 3),
random = list(obs=~1),
family=binomial,
data=data1,
method = "REML") #
df1 <- data.frame(log.x = seq(min(data1$log.x), max(data1$log.x), length = 30)) #setting up new data frame (df) defining log.x values to run
pred.md1 <- predict(m2.gamm, newdata=df1, type = "response", se.fit=T, lpmatrix=TRUE)
df1$pred = pred.md1$fit
plot(data1$log.x,(data1$suc / data1$tot),main="GAM") #second plot
lines(df1$log.x, df1$pred, type = "l", lwd = 2, col = 2, xaxt = "n", las = 1) #plot model mean line
df1$up.ci = pred.md1$fit + 1.96 * pred.md1$se.fit #predicting the data1 95% CI
df1$lo.ci = pred.md1$fit - 1.96 * pred.md1$se.fit #predicting the lower 95% CI
lines(df1$log.x, df1$up.ci, type = "l", lwd = 2, xaxt = "n", las = 1) #plot data1 95% CI
lines(df1$log.x, df1$lo.ci, type = "l", lwd = 2, xaxt = "n", las = 1) #plot lower 95% CI
```

The code used to interpolate the ECx is below. This function calculates the ECx from the maximum of the curve. It simply works by finding the nearest value in the prediction data.frame to X% of the maximum response, and then the corresponding value in the predictor column. If you have a unimodal ‘hump-shaped’ trend, it will only calculate one ECx (I think).

```
ecx.interp <- function(ecx, df1) {
inhibx <- max(df1$pred) * ecx
nearest_idx <- which.min(abs(df1$pred - inhibx))
df2 <- data.frame(
ecx = df1$log.x[nearest_idx],
lower = df1$log.x[which.min(abs(df1$lo.ci - inhibx))],
data = df1$log.x[which.min(abs(df1$up.ci - inhibx))]
)
return(df2)
}
ecx.interp(0.5, df1) #this is the ECx calculate between the top and 0.
```

Ok, now for a GAMM with bootstrapped 95% CI. Make sure to keep sims = as multiple of 200, otherwise the CIs can’t calculate properly. Ideally your finally attempt will want around 1000 otherwise the CIs will look wobbly, like below.

Here we have the code above wrapped into a function called m2.gamm, which outputs the prediction. In the for-loop, we resample the raw data.frame and then apply this function. The prediction is stored into a list and then a data.frame called df3. This data.frame is then ordered and next subset by the 0.025 (2.5%) or 0.975 (97.5%) columns. Because columns are integers, make sure your sims multiplied by these proportions result in an integer e.g. 200.

```
set.seed(123)
lst = list() #initial list
fit_gam <- function(resampled) {
m2.gamm <- gamm(cbind(suc, (tot - suc)) ~ s(log.x, fx = F, k = 3),
random = list(obs = ~1),
family = binomial,
data = resampled,
method = "REML")
pred.md1 <- predict(m2.gamm, newdata=df1, type = "response",se.fit=T)
out = unname(pred.md1$fit)
# lst[[1]] = dd
# return(lst)
return(out)
} #gam function returns prediction
#Now resample the residuals x times
resid_gam <- residuals(m2.gamm$gam, type = "response")
sims = 200
for (i in 1:sims) {
resampled_resid <- resid_gam[sample(length(resid_gam), replace = TRUE)]
new_response <- m2.gamm$gam$fitted.values + resampled_resid
resampled_data <- data1
resampled_data$suc <- round(pmin(pmax(new_response * resampled_data$tot, 0), resampled_data$tot)) #pmax prevents values less than 0
out <- fit_gam(resampled_data)
lst[[i]] <- out
}
lst #returns all predictions in list
df3 <- do.call(rbind, lst) #add this to 'loop into a list'
df3 #so all the predicted fits from the x simulations are stored as rows in this list
```

Now lets see how each resampled curve looks

```
#plot output of predictions
df4 <- as.data.frame(lst)
df4$log.x = df1$log.x
df4$log.x
library(tidyverse)
data1.long = df4 %>% tidyr::pivot_longer(-log.x, names_to = "factors" ,values_to = "meas") %>% data.frame() #keep vec.x, add all other columns to factors , add all their values to meas)
ggplot(data1.long, aes(x = log.x, y = meas, color = 'steelblue1', group = factors)) +
geom_line() +
#geom_point(aes(y = meas)) +
labs(x = "Predictor", y = "Response", title = "Multiple Model Fits") +
theme_minimal() + theme(legend.position="none")
```

Let’s take the 2.5% and the 97.5% lines to see the bootstrapped CIs

```
#ordering and take the bottom 2.5% and top 97.5% lines
eta = 0.5*sims
lowerCI = 0.025*sims
dataCI = 0.975*sims
bb_se1<-apply(df3,2,function(X) X[order(X)]) #orders the predictions from lowest to highest
df1$boot.pred = bb_se1[eta,] #find the bottom 2.5%
df1$boot_lo = bb_se1[lowerCI,] #find the bottom 2.5%
df1$boot_up = bb_se1[dataCI,] #find the top 2.5%
#plotting
plot(data1$log.x,(data1$suc / data1$tot),main="GAM") #second plot
lines(df1$log.x, df1$boot.pred, type = "l", lwd = 2, col = 2, xaxt = "n", las = 1) #plot model mean line
lines(df1$log.x, df1$boot_lo, type = "l", lwd = 2, xaxt = "n", las = 1) #plot data2 95% CI
lines(df1$log.x, df1$boot_up, type = "l", lwd = 2, xaxt = "n", las = 1) #plot lower 95% CI
```

**With that in mind, we can now bootstrap the ECx.**

Just to recap:

1) we fit a GAMM to the data

2) we extracted the residuals and resampled them

3) made new predictions from the resampled residuals

4) repeated this 200 times

Ok. Let resample just once to see how it look, but keeping everything in a function, which we will need below.

```
set.seed(123)
lst1 = list() #initial list
fit_gam2 <- function(resampled, ecx) {
m2.gamm <- gamm(cbind(suc, (tot - suc)) ~ s(log.x, fx = F, k = 3),
random = list(obs = ~1),
family = binomial,
data = resampled,
method = "REML")
pred<- predict(m2.gamm, newdata = df1, type = "response",se.fit=T)
df2 = data.frame(log.x = df1$log.x, pred = pred$fit)
ecx.interp2 <- function(ecx, df2) {
inhibx <- max(df2$pred) * ecx
nearest_idx <- which.min(abs(df2$pred - inhibx))
ecx = df2$log.x[nearest_idx]
return(ecx)
}
out1 = ecx.interp2(ecx, df2)
out = unname(out1)
# lst[[1]] = dd
# return(lst)
return(out)
} #gam function returns prediction
fit_gam2(resampled_data, 0.5)
```

Now let’s run it in a loop for each resample

```
#Now run in loop for sims
ecx = 0.5
sims = 200
for(i in 1:sims) {
# resampled = data2[sample(nrow(data2), replace = TRUE), ]
# out = fit_gam2(resampled, ecx)
# lst[[i]] = out
resampled_resid <- resid_gam[sample(length(resid_gam), replace = TRUE)]
new_response <- m2.gamm$gam$fitted.values + resampled_resid
resampled_data <- data1
resampled_data$suc <- round(pmin(pmax(new_response * resampled_data$tot, 0), resampled_data$tot)) #pmax prevents values less than 0
out <- fit_gam2(resampled_data, ecx)
lst[[i]] <- out
}
lst #returns all predictions in list
df3 = unlist(lst)
df3
hist(df3)
plot(density(df3)) #density plot
df4 = df3[order(df3)]
df1$boot.pred = df4[eta] #find the bottom 2.5%
df1$boot_lo = df4[lowerCI] #find the bottom 2.5%
df1$boot_up = df4[dataCI] #find the top 2.5%
df1
```

# Plotting

```
library(ggplot2)
library(tidybayes)
df5 = data.frame(ecx = df3)
p1 = ggplot(df5, aes(x=ecx))+geom_density(aes(fill = 'steelblue4'), alpha=0.3)+
stat_pointinterval(aes(y = 0.00, x = ecx),.width = c(.66, .95))
theme_light()
p1 = p1+scale_fill_manual( values = c("steelblue4"))+
scale_color_manual( values = c("steelblue4"))+theme(legend.position="none")#nice
p1 = p1 + scale_y_continuous(name ="Density")
p1 = p1 + coord_cartesian(ylim = c(0.0, 3.5))
p1
```

And if you want everything wrapped into a function

```
gamm_ecx <- function(data1, ecx) {
library(mgcv)
library(lattice)
library(MASS)
library(MuMIn)
data1$suc <- as.numeric(as.character(data1$suc))
data1$tot <- as.numeric(as.character(data1$tot))
data1$obs <- factor(formatC(1:nrow(data1), flag="0", width = 3))# unique tank ID for later on
lst1 = list() #initial list
data1$log.x <- ifelse(data1$log.x < 0.1, 0, data1$log.x) #Replace neg with zero
set.seed(123)
m2.gamm<-gamm(cbind(suc,(tot - suc))~s(log.x, fx = F, k = 3),
random = list(obs=~1),
family=binomial,
data=data1,
method = "REML") #
resid_gam <- residuals(m2.gamm$gam, type = "response")
df1 <- data.frame(log.x = seq(min(data1$log.x), max(data1$log.x), length = 30)) #setting up new data frame (df) defining log.x values to run
fit_gam2 <- function(resampled, ecx) {
m2.gamm <- gamm(cbind(suc, (tot - suc)) ~ s(log.x, fx = F, k = 3),
random = list(obs = ~1),
family = binomial,
data = resampled,
method = "REML")
pred<- predict(m2.gamm, newdata = df1, type = "response",se.fit=T)
df2 = data.frame(log.x = df1$log.x, pred = pred$fit)
ecx.interp2 <- function(ecx, df2) {
inhibx <- max(df2$pred) * ecx
nearest_idx <- which.min(abs(df2$pred - inhibx))
ecx = df2$log.x[nearest_idx]
return(ecx)
}
out1 = ecx.interp2(ecx, df2)
out = unname(out1)
return(out)
} #gam function returns prediction
#fit_gam2(resampled_data, 0.5) #gives the same asnwer as above but in a function
sims = 200
lst = list() #initial list
for(i in 1:sims) {
resampled_resid <- resid_gam[sample(length(resid_gam), replace = TRUE)]
new_response <- m2.gamm$gam$fitted.values + resampled_resid
resampled_data <- data1
resampled_data$suc <- round(pmin(pmax(new_response * resampled_data$tot, 0), resampled_data$tot)) #pmax prevents values less than 0
out <- fit_gam2(resampled_data, ecx)
lst[[i]] <- out
}
lst #returns all predictions in list
df3 = unlist(lst)
df3
hist(df3)
plot(density(df3)) #density plot
df4 = df3[order(df3)]
eta = 0.5*sims
lowerCI = 0.025*sims
dataCI = 0.975*sims
boot.pred = df4[eta] #find the bottom 2.5%
boot_lo = df4[lowerCI] #find the bottom 2.5%
boot_up = df4[dataCI] #find the top 2.5%
df5 = data.frame(boot.pred, boot_lo, boot_up)
df5
output_list = list(df4, df5)
return(output_list)
}
boot_ecx = gamm_ecx(data1, 0.5)
boot_ecx[[1]]
boot_ecx[[2]]
```

If you fit multiple curves, you can then compare their ECx distributions together to assess for evidence of differences.

Happy bootstrapping!