Generalized Additive Models — or GAMs in short — have been somewhat of a mystery to me. I’ve known about them, but didn’t know exactly what they did, or when they’re useful. That came to an end when I found out about this tutorial by Noam Ross.
In this beautiful, online, interactive course, Noam allows you to program several GAMs yourself (in R) and to progressively learn about the different functions and features. I am currently halfway through, but already very much enjoy it.
If you’re already familiar with linear models and want to learn something new, I strongly recommend this course!
Created by Andrew Stewart, and tweeted by John Holbein, the visuals show samples taken from a normal distributed variable with a mean of 10 and a standard deviation of 2. In the left section, Andrew took several samples of 20. In the right section, the sample size was increased to 500.
Just look at how much the distribution and the estimated mean change for small samples!
Andrew shared his code via Github, so I was able to download and tweak it a bit to make my own version.
Andrew’s version seems to be concerned with potential Type 1 errors when small samples are taken. A type 1 error occurs when you reject your null hypothesis (you reject “there is no effect”) while you should not have (“there is actually no effect”).
You can see this in the distributions Andrew sampled from in the tweet above. The data for conditions A (red) and B (blue) are sampled from the same distribution, with mean 10 and standard deviation 2. While there should thus be no difference between the groups, small samples may cause researchers to erroneously conclude that there is a difference between conditions A and B due to the observed data.
We could use Andrew’s basic code and tweak it a bit to simulate a setting in which Type 2 errors could occur. A type 2 error occurs when you do not reject your null hypothesis (you maintain “there is no effect”) whereas there is actually an effect, which you thus missed.
To illustrate this, I adapted Andrew’s code: I sampled data for condition B using a normal distribution with a slightly higher mean value of 11, as opposed to the mean of 10 for condition A. The standard deviation remained the same in both conditions (2).
Next, I drew 10 data samples from both conditions, for various sample sizes: 10, 20, 50, 100, 250, 500, and even 1000. After drawing these samples for both conditions, I ran a simple t-test to compare their means, and estimate whether any observed difference could be considered significant (at the alpha = 0.05 level [95%]).
In the end, I visualized the results in a similar fashion as Andrew did. Below are the results.
As you can see, only in 1 of our 10 samples with size 10 were we able to conclude that there was a difference in means. This means that we are 90% incorrect.
After increasing the sample size to 100, we strongly decrease our risk of Type 2 errors. Now we are down to 20% incorrect conclusions.
At this point though, I decided to rework Andrew’s code even more, to clarify the message.
I was not so much interested in the estimated distribution, which currently only distracts. Similarly, the points and axes can be toned down a bit. Moreover, I’d like to be able to see when my condition samples have significant different means, so let’s add a 95% confidence interval, and some text. Finally, let’s increase the number of drawn samples per sample size to, say, 100, to reduce the influence that chance may have on our Type 2 error rate estimations.
Let’s rerun the code and generate some GIFs!
The below demonstrates that small samples of only 10 observations per condition have only about a 11% probability of detecting the difference in means when the true difference is 1 (or half the standard deviation [i.e., 2]). In other words, there is a 89% chance of a Type 2 error occuring, where we fail to reject the null hypothesis due to sampling error.
Doubling the sample size to 20, more than doubles our detection rate. We now correctly identify the difference 28% of the time.
With 50 observations the Type 2 error rate drops to 34%.
Finally, with sample sizes of 100+ our results become somewhat reliable. We are now able to correctly identify the true difference over 95% of the times.
With a true difference of half the standard deviation, further increases in the sample size start to lose their added value. For instance, a sample size of 250 already uncovers the effect in all 100 samples, so doubling to 500 would not make sense.
I hope you liked the visuals. If you are interested in these kind of analysis, or want to estimate how large of a sample you need in your own study, have a look at power analysis. These analysis can help you determine the best setup for your own research initiatives.
If you’d like to reproduce or change the graphics above, here is the R code. Note that it is strongly inspired by Andrew’s original code.
# setup -------------------------------------------------------------------
# The new version of gganimate by Thomas Lin Pedersen - @thomasp85 may not yet be on CRAN so use devtools
# devtools::install_github('thomasp85/gganimate')
library(ggplot2)
library(dplyr)
library(glue)
library(magrittr)
library(gganimate)
# main function to create and save the animation --------------------------
save_created_animation = function(sample_size,
samples = 100,
colors = c("red", "blue"),
Amean = 10, Asd = 2,
Bmean = 11, Bsd = 2,
seed = 1){
### generate the data
# set the seed
set.seed(seed)
# set the names of our variables
cnames <- c("Score", "Condition", "Sample")
# create an empty data frame to store our simulated samples
df <- data.frame(matrix(rep(NA_character_, samples * sample_size * 2 * length(cnames)), ncol = length(cnames), dimnames = list(NULL, cnames)), stringsAsFactors = FALSE)
# create an empty vector to store whether t.test identifies significant difference in means
result <- rep(NA_real_, samples)
# run a for loop to iteratively simulate the samples
for (i in seq_len(samples)) {
# draw random samples for both conditions
a <- rnorm(sample_size, mean = Amean, sd = Asd)
b <- rnorm(sample_size, mean = Bmean, sd = Bsd)
# test whether there the difference in the means of samples is significant
result[i] = t.test(a, b)$p.value < 0.05
# add the identifiers for both conditions, and for the sample iteration
a <- cbind(a, rep(glue("A\n(μ={Amean}; σ={Asd})"), sample_size), rep(i, sample_size))
b <- cbind(b, rep(glue("B\n(μ={Bmean}; σ={Bsd})"), sample_size), rep(i, sample_size))
# bind the two sampled conditions together in a single matrix and set its names
ab <- rbind(a, b)
colnames(ab) <- cnames
# push the matrix into its reserved spot in the reserved dataframe
df[((i - 1) * sample_size * 2 + 1):((i * (sample_size * 2))), ] <- ab
}
### prepare the data
# create a custom function to calculate the standard error
se <- function(x) sd(x) / sqrt(length(x))
df %>%
# switch data types for condition and score
mutate(Condition = factor(Condition)) %>%
mutate(Score = as.numeric(Score)) %>%
# calculate the mean and standard error to be used in the error bar
group_by(Condition, Sample) %>%
mutate(Score_Mean = mean(Score)) %>%
mutate(Score_SE = se(Score)) ->
df
# create a new dataframe storing the result per sample
df_result <- data.frame(Sample = unique(df$Sample), Result = result, stringsAsFactors = FALSE)
# and add this result to the dataframe
df <- left_join(df, df_result, by = "Sample")
# identify whether not all but also not zero samples identified the difference in means
# if so, store the string "only ", later to be added into the subtitle
result_mention_adj <- ifelse(sum(result) != 0 & sum(result) < length(result), "only ", "")
### create a custom theme
textsize <- 16
my_theme <- theme(
text = element_text(size = textsize),
axis.title.x = element_text(size = textsize),
axis.title.y = element_text(size = textsize),
axis.text.y = element_text(hjust = 0.5, vjust = 0.75),
axis.text = element_text(size = textsize),
legend.title = element_text(size = textsize),
legend.text = element_text(size = textsize),
legend.position = "right",
plot.title = element_text(lineheight = .8, face = "bold", size = textsize),
panel.border = element_blank(),
panel.grid.minor = element_blank(),
panel.grid.major = element_blank(),
axis.line = element_line(color = "grey", size = 0.5, linetype = "solid"),
axis.ticks = element_line(color = "grey")
)
# store the chosen colors in a named vector for use as palette,
# and add the colors for (in)significant results
COLORS = c(colors, "black", "darkgrey")
names(COLORS) = c(levels(df$Condition), "1", "0")
### create the animated plot
df %>%
ggplot(aes(y = Score, x = Condition, fill = Condition, color = Condition)) +
geom_point(aes(y = Score), position = position_jitter(width = 0.25), alpha = 0.20, stroke = NA, size = 1) +
geom_errorbar(aes(ymin = Score_Mean - 1.96 * Score_SE, ymax = Score_Mean + 1.96 * Score_SE), width = 0.10, size = 1.5) +
geom_text(data = . %>% filter(as.numeric(Condition) == 1),
aes(x = levels(df$Condition)[1], y = Result * 10 + 5,
label = ifelse(Result == 1, "Significant!", "Insignificant!"),
col = as.character(Result)), position = position_nudge(x = -0.5), size = 5) +
transition_states(Sample, transition_length = 1, state_length = 2) +
guides(fill = FALSE) +
guides(color = FALSE) +
scale_x_discrete(limits = rev(levels(df$Condition)), breaks = rev(levels(df$Condition))) +
scale_y_continuous(limits = c(0, 20), breaks = seq(0, 20, 5)) +
scale_color_manual(values = COLORS) +
scale_fill_manual(values = COLORS) +
coord_flip() +
theme_minimal() +
my_theme +
labs(x = "Condition") +
labs(y = "Dependent variable") +
labs(title = glue("When drawing {samples} samples of {sample_size} observations per condition")) +
labs(subtitle = glue("The difference in means is identified in {result_mention_adj}{sum(result)} of {length(result)} samples")) +
labs(caption = "paulvanderlaken.com | adapted from github.com/ajstewartlang") ->
ani
### save the animated plot
anim_save(paste0(paste("sampling_error", sample_size, sep = "_"), ".gif"),
animate(ani, nframes = samples * 10, duration = samples, width = 600, height = 400))
}
# call animation function for different sample sizes ----------------------
# !!! !!! !!!
# the number of samples is set to 100 by default
# if left at 100, each function call will take a long time!
# add argument `samples = 10` to get quicker results, like so:
# save_created_animation(10, samples = 10)
# !!! !!! !!!
save_created_animation(10)
save_created_animation(20)
save_created_animation(50)
save_created_animation(100)
save_created_animation(250)
save_created_animation(500)
Data visualizations that make smart use of icons have a way of conveying information that sticks. Dataviz professionals like Moritz Stefaner know this and use the practice in their daily work.
A recent #tidytuesday entry by Georgios Karamanis demonstrates how easy it is to integrate visual icons in your data figures when you write code in R. You can simply store the URL location of an icon as a data column, and map it to an aesthetic using the ggplot2::geom_image function.
Do have a closer look at Georgios’ github repository for week 21 of tidytuesday. You will probably have to alter the code a bit to get it to work. though!
For those who haven’t moved away from base R plotting functions yet, here’s a good StackOverflow item showing how to use icons in both base R and tidyverse.
Hugo Toscano wrote a great blog providing an overview of all the helpful functionalities of the R forcats package. The package includes functions that help handling categorical data, by setting a random or fixed order, and by recategorizing or anonymizing. These functions are specifically helpful when visualizing data with R’s ggplot2.
A comprehensive overview is provided in the form of the RStudio forcats cheat sheet, but, on his blog, Hugo demonstrates some of its functionalities using a dataset on suicides and people’s ages:
Other functions can be used to automatically surpress infrequent categories, to reverse the order of categories, to shuffle or shift categories, to quickly relabel or anonimize categories, and many more…
Yan Holtz recently created a neat little overview of handy R Markdown tips and tricks that improve the appearance of output documents. He dubbed this overview Pimp my RMD. Have a look, it’s worth it!
Propensity score matching (wiki) is a statistical matching technique that attempts to estimate the effect of a treatment (e.g., intervention) by accounting for the factors that predict whether an individual would be eligble for receiving the treatment. The wikipedia page provides a good example setting:
Say we are interested in the effects of smoking on health. Here, smoking would be considered the treatment, and the ‘treated’ are simply those who smoke. In order to find a cause-effect relationship, we would need to run an experiment and randomly assign people to smoking and non-smoking conditions. Of course such experiments would be unfeasible and/or unethical, as we can’t ask/force people to smoke when we suspect it may do harm. We will need to work with observational data instead. Here, we estimate the treatment effect by simply comparing health outcomes (e.g., rate of cancer) between those who smoked and did not smoke. However, this estimation would be biased by any factors that predict smoking (e.g., social economic status). Propensity score matching attempts to control for these differences (i.e., biases) by making the comparison groups (i.e., smoking and non-smoking) more comparable.
Lucy D’Agostino McGowan is a post-doc at Johns Hopkins Bloomberg School of Public Health and co-founder of R-Ladies Nashville. She wrote a very nice blog explaining what propensity score matching is and showing how to apply it to your dataset in R. Lucy demonstrates how you can use propensity scores to weight your observations in such a way that accounts for the factors that correlate with receiving a treatment. Moreover, her explainations are strenghtened by nice visuals that intuitively demonstrate what the weighting does to the “pseudo-populations” used to estimate the treatment effect.
paulvanderlaken.com 