Statistical analyses of growth curves data
At this point, we’ve now summarized our growth curves data into some metrics. How can we best go about drawing statistical conclusions from these data?
When should we average replicates?
Before we dig into what we to do next, I want to emphasize something we did not do in this workflow: averaging different wells together before summarization. In my opinion, averaging should only occur after summarization, not before. Why? Even wells that have the same contents (i.e. are technical replicates) can still differ in their growth due to biological variation (e.g. stochastic growth dynamics). If we average our density values at the beginning, we may introduce bias and we will not have the ability to visualize or assess the biological variation present in our data.
Let’s look at a simple example to demonstrate this point. I’m going to simulate bacterial growth using the Baranyi-Roberts mathematical model of growth, which is logistic growth but with a period of acclimation at the beginning:
\[\frac{dN}{dt} = \frac{q_{0}}{q_{0} + e^{-mt}} * r N\left(1-\frac{N}{k}\right)\]
Where \(N\) is the population size, \(q_{0}\) is a parameter controlling the initial acclimatization state of the population, \(m\) is the rate of acclimation, \(r\) is the rate of growth, and \(k\) is the carrying capacity of the population.
In the code below, I’ve simulated the growth of 96 different wells of bacteria. All the bacteria are identical, except that they differ in the rate at which they acclimate.
Now, let’s calculate the growth rate of each well and plot the growth:
sim_dat_tdy <- mutate(group_by(sim_dat_tdy, Well),
percap_deriv = calc_deriv(y = Measurements, x = time,
percapita = TRUE, blank = 0))
# Plot the growth in our wells
ggplot(data = filter(sim_dat_tdy, Well != "averaged"),
aes(x = time, y = Measurements, group = Well)) +
geom_line(alpha = 0.1) +
geom_line(data = filter(sim_dat_tdy, Well == "averaged"), color = "red") +
scale_y_continuous(trans = "log10")
Here we’ve plotted each individual well in black, with the “average well” plotted in red. We can clearly see that our different wells are varying in how quickly they’re acclimating. Our average well appears to reflect the data pretty well, does it give a good measure for our average maximum per-capita growth rate?
# Summarize our data
sim_dat_sum <- summarize(group_by(sim_dat_tdy, Well),
max_growth_rate = max(percap_deriv, na.rm = TRUE))
# Plot the maximum per-capita growth rates of each well
# Add a red line for the max growth rate of the "average well"
# Add a dashed line for the average growth rate of all the wells
ggplot(data = filter(sim_dat_sum, Well != "averaged"),
aes(x = max_growth_rate)) +
geom_histogram() +
geom_vline(data = filter(sim_dat_sum, Well == "averaged"),
aes(xintercept = max_growth_rate), color = "red") +
geom_vline(xintercept =
mean(filter(sim_dat_sum, Well != "averaged")$max_growth_rate),
lty = 2)
#> `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Here we can see that the maximum growth rate of the “average well” (red line) is not the same as the true average maximum growth rate of all the wells (dashed line). While this bias might seem small, it is in fact very consistent: if you ran this simulation many more times, the “average well” growth rate would nearly always be lower than the true average growth rate. Moreover, this bias extends to many other statistics as well.
Thus, I recommend against averaging wells together before summarizing, since it often introduces bias and eliminates our ability to visualize variation between replicates.
Carrying out statistical testing
How do you go about running statistics on analyzed growth curve data? Typically, growth curves experiments will have a highly nested structure. You probably have multiple wells with the same contents (i.e. technical replicates) in each plate. You may also have multiple plates from different runs (creating the possibility of batch effects).
In order to pull apart these effects and test for differences between your treatments, you’ll likely need to do mixed-effects modeling. Unfortunately, it’s beyond the scope of this vignette to provide a sufficient explanation of how to do mixed-effects statistics. However, I can provide some guidance:
For frequentist statistics, the R package
lme4 is one of the most-popular implementations of
mixed-effects modeling.
For Bayesian statistics, the R packages
brms or rstanarm are popular implementations
that can incorporate mixed-effects modeling.
Regardless of your approach, you should:
- use your your summarized statistics (e.g.
auc,max_growth_rate,lag_time, etc.) as your response variable - use your design elements (e.g.
Bacteria_strain,Phage) as your explanatory variables (i.e. fixed effects) - incorporate random effects for any technical replicates you have
- incorporate random effects for any potential batch effects in-play
There are a number of excellent resources out there to learn how to do this sort of mixed-effects modeling, including what I think is a good introductory guide to the process by Michael Clark.
