Comparing EC50s between two or more GLMM models

In this post I am going to describe how to compare EC50s from two or more GLMM curves. To begin, run the code in my recent post

Automated three factor dose-response with binomial GLMMs and ECx’s

In the post above we calculated the 95% CI around the ECx using an approximation by multiplying the standard error by 1.96 and adding or subtracting it from the mean ECx. But this makes it difficult to compare the ECx’s except to say that the 95% CI do or not overlap. In this example, we bootstrap the data which creates a distributions around the ECx. From there we can visually present the distributions or compare the distributions statistically.

Let’s first check out our old EC50 of ‘factor a’. The EC50 = 46.12 (95% CI 36.85 – 55.39)

> ec.df1
ecx1   lower1    upper1
(Intercept) 46.11674   36.84638      55.3871

Now let’s bootstrap the EC50. Remember in the previous code I had “ecx1 <- (eta1 – betas1[1])/betas1[2] “, which interpolates the 50% response onto the curve? I will run this 200 times, but resampling the data each time. This will give 200 EC50 values that we can make a distribution out of. I will use ‘lme4’ again, ignoring the initial convergence issue. The 200 EC50s will be stored in ‘bb1$t’. We can also order then and get the bottom 2.5% and 97.5% i.e the 95% CI

#Bootstrapped confidence intervals on ECx values
library(lme4)
md1 =glmer(cbind(suc,(tot-suc))~raw.x*factor+(1|obs) ,data1,family=’binomial’ )
sims = 200 #number of simulations.
lowerCI = 0.025*sims
median.CI = 0.5*sims
upperCI = 0.975*sims
# ecx <- (logit(0.5) – coef(md1)[1])/coef(md1)[2]
predFun2<-function(.)((eta1 – fixef(.)[1])/fixef(.)[2]) #define the function
bb1<-bootMer(md1,FUN=predFun2,nsim=200, parallel = “multicore”, .progress = ‘txt’ ) #get bootstrapping, 200 sims
bb_se1<-apply(bb1$t,2,function(X) X[order(X)]) #apply function
median<-bb_se1[median.CI,] #find the 50%
low<-bb_se1[lowerCI,] #find the bottom 2.5%
up<-bb_se1[upperCI,] #find the top 2.5%

ec.df1.boot = data.frame(median, low, up)

> ec.df1.boot
median   low   up
(Intercept) 45.22589   36.04048    55.48538

You can see they are similar to the approximated confidence intervals above but slightly different. Generally when random effects are involved, bootstrapped confidence intervals are the way to go. Now let’s see the distribution using a density plot

plot(density(bb1$t))

3

So here is the EC50 distribution for ‘factor a’ that we want to compare factors b and c against. We will need to again relevel the model for the other factors and repeat the code.

#Bootstrapped confidence intervals for factor b and c
library(lme4)
data1$factor <- relevel(data1$factor, ref = “b”) #set reference levels for GLMs
md2 <- glmer(cbind(suc,(tot-suc))~raw.x*factor+(1|obs) ,data1,family=’binomial’ )
bb2<-bootMer(md2,FUN=predFun2,nsim=200, parallel = “multicore”, .progress = ‘txt’ ) #get bootstrapping, 200 sims
data1$factor <- relevel(data1$factor, ref = “c”) #set reference levels for GLMs
md3 <- glmer(cbind(suc,(tot-suc))~raw.x*factor+(1|obs) ,data1,family=’binomial’ )
bb3<-bootMer(md3,FUN=predFun2,nsim=200, parallel = “multicore”, .progress = ‘txt’ ) #get bootstrapping, 200 sims
all.dist = data.frame(‘a’ = as.vector(bb1$t), ‘b’ = as.vector(bb2$t), ‘c’ = as.vector(bb3$t))
all.dist.long <- gather(all.dist, factor, ecx, a:c, factor_key=TRUE)

#Plotting
library(ggplot2)
library(tidybayes)
p1 = ggplot(all.dist.long, aes(x=ecx))+geom_density(aes(group=factor, color =factor , fill=factor), alpha=0.3)+
stat_pointintervalh(aes(y = 0.00, x = ecx, group=factor),.width = c(.66, .95))+#+facet_wrap(~contrast+time, nrow = 3, ncol = 2)+
theme_light()
p1 = p1+scale_fill_manual( values = c(“steelblue4”, “orange”, ‘red’))+
scale_color_manual( values = c(“steelblue4″,”grey”, “steelblue1″,”steelblue4”, “grey”,”grey”, “grey”,”grey”))+theme(legend.position=”none”)#nice
p1 = p1 + scale_y_continuous(name =”Density”)
p1 = p1 + coord_cartesian(xlim = c(0.0, 50))
p1 = p1 + coord_cartesian(ylim = c(0.0, 0.1))
p1

4

Starting to look pretty. The blue is ‘factor ‘a’, yellow ‘factor b’, and red ‘factor c’. By looking at the 95% CI on the bottom, the red looks statistically different than the factor a EC50.

Now let’s compare the distributions against ‘factor a’ by subtracting the two distributions.

#Differences posterior
df4.s = data.frame(atob= all.dist$a – all.dist$b, atoc = all.dist$a – all.dist$c)
df4.s.long <- gather(df4.s, factor, diff, atob:atoc, factor_key=TRUE)
plot(density(df4.s$atob))
p2 = ggplot(df4.s.long, aes(x=diff))+geom_density(aes(group=factor, color =factor , fill=factor), alpha=0.3)+
stat_pointintervalh(aes(y = 0.00, x = diff, group=factor),.width = c(.66, .95))+#+facet_wrap(~contrast+time, nrow = 3, ncol = 2)+
theme_light()
p2 = p2+scale_fill_manual( values = c(“steelblue4”, “orange”, ‘red’))+
scale_color_manual( values = c(“steelblue4″,”grey”, “steelblue1″,”steelblue4”, “grey”,”grey”, “grey”,”grey”))+theme(legend.position=”none”)#nice
p2 = p2 + scale_y_continuous(name =”Density”)
p2 = p2 +geom_vline(xintercept = 0, color = “red”, lty = 2)+ theme_light()
p2 = p2 + coord_cartesian(xlim = c(0.0, 50))
p2 = p2 + coord_cartesian(ylim = c(0.0, 0.1))
# p2 = p2 + scale_x_continuous(name =”Standardized effect size”)
p2 = p2+facet_wrap(~factor)
p2

5

I’ve added a line through zero, which represents ‘no difference’ i.e if both distribution had the same ECx, the black dot for the median ECx would be on zero. Values less than zero mean that factor b or c are greater than factor a, and vice versa for positive numbers. So the interpretation is that if the 95% interval see by the thin black line overlaps with the red line, the two distributions are not different. Here ‘a’ and ‘b’ are similar, and ‘factor a’ and ‘factor c’ are different.

You can of course get the values for these 95% CI.

atob.diff = sort(df4.s$atob)
nrow(df4.s)
atob.diff[median.CI]
median.ab<-atob.diff[median.CI] #find the 50%
low.ab<-atob.diff[lowerCI] #find the bottom 2.5%
up.ab<-atob.diff[upperCI] #find the top 2.5%
ab.diff = data.frame(median.ab, low.ab, up.ab)
ab.diff

> ab.diff
median.ab    low.ab       up.ab
3.940983   -8.379134   16.5668

You may also be interested in making a probability type statement. This probably should be reserved to Bayesian analysis (and yes you can do this exact analysis above using Bayesian model parameter outputs). But in a frequentist sense, we could say, ‘if this experiment was repeated 100 times, the EC50 of factor b would be greater than the EC50 of factor a, X many times‘.

length(which (all.dist$a < all.dist$b))/sims*100

length(which (all.dist$a < all.dist$b))/sims*100
[1] 28

length(which (all.dist$a < all.dist$c))/sims*100

length(which (all.dist$a < all.dist$c))/sims*100
[1]

0.5

‘if this experiment was repeated 100 times, the EC50 of factor b would be greater than the EC50 of ‘factor a’ 28 of the times’.

‘if this experiment was repeated 100 times, the EC50 of factor c would be greater than the EC50 of ‘factor a’ less than 1 time’.

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