Short R script to plot effect sizes (Cohen's d) and shade overlapping area
Update: I have created an interactive effect size visualization here
Introduction to effect sizes
Many times you read in a study that “x and y were significantly different, p < .05”, which is another way of saying that “assuming that the null hypothesis is true, the probability of getting the observed value simply by chance alone is less than 0.05” But that’s not really that interesting, though is it? Say you are reading an intervention study that are comparing a treatment group to a control group, I bet you are more interested in finding out the amount of difference between the groups, rather than the chances of the differences popping up under the null hypothesis. Luckily it’s getting more and more common to also report effect sizes in addition to p-values. Effect sizes, in this case, are metrics that represent the amount of differences between two sample means. One of the most common effect size measure in psychology is Cohen’s d or the standardized mean difference. As you can see by the name it’s a measure of the standardized difference between two means. Commonly Cohen’s d is categorized in 3 broad categories: 0.2–0.3 represents a small effect, ~0.5 a medium effect and over 0.8 to infinity represents a large effect. What that means is that with two samples with a standard deviation of 1, the mean of group 1 is 0.8 sd away from the other group’s mean if Cohen’s d = 0.8. That might sound very intuitive to some, but I find it’s more explanatory to present different d values visually. Which is really easy to do in R statistical software.
Some quick R code to visualize Cohen’s d
The thing I actually wanted to try out here was to shade the overlapping area of the two distributions. It turned to be pretty easy to do in R.
require("ggplot2") # Standardized Mean Difference (Cohen's d)ES <- 0.8# get mean2 depending on value of ES from d = (u1 - u2)/sdmean1 <- ES*1 + 1# create x sequencex <- seq(1 - 3*1, mean1 + 3*1, .01)# generate normal dist #1y1 <- dnorm(x, 1, 1)# put in data framedf1 <- data.frame("x" = x, "y" = y1)# generate normal dist #2y2 <- dnorm(x, mean1, 1)# put in data framedf2 <- data.frame("x" = x, "y" = y2)# get y values under overlapy.poly <- pmin(y1,y2)# put in data framepoly <- data.frame("x" = x, "y" = y.poly) # Cohen's U3, proportion of control > 50th perc. treatmentu3 <- 1 - pnorm(1, mean1,1)u3 <- round(u3,3) # plot with ggplot2ggplot(df1, aes(x,y, color="treatment")) + # add line for treatment group geom_line(size=1) + # add line for control group geom_line(data=df2, aes(color="control"),size=1) + # shade overlap geom_polygon(aes(color=NULL), data=poly, fill="red", alpha=I(4/10), show_guide=F) + # add vlines for group means geom_vline(xintercept = 1, linetype="dotted") + geom_vline(xintercept = mean1, linetype="dotted") + # add plot title opts(title=paste("Visualizing Effect Sizes (Cohen's d = ",ES,"; U3 = ",u3,")", sep="")) + # change colors and legend annotation scale_color_manual("Group", values= c("treatment" = "black","control" = "red")) + # remove axis labels ylab(NULL) + xlab(NULL)And some plots of the different effect size values
A “large” effect size really look insignificant compared to the ridiculously large effect size reported by Clark et al. (2006) in their study Cognitive Therapy Versus Exposure and Applied Relaxation in Social Phobia: A Randomized Controlled Trial
Written by Kristoffer Magnusson, a researcher in clinical psychology. You should follow him on Bluesky or on Twitter.
Published April 23, 2012 (View on GitHub)
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I've been looking for applets that show this for YEARS, for demonstrations for classes. Thank you so much! Students do not need to tolarate my whiteboard scrawl now. I'm sure they'd appreciate you, too.l
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Thank you for putting this together! I am using these visuals and this information to teach my Advanced Quant class.
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You finally helped me understand correlation! Many, many thanks... 😄
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Good stuff! It's been so helpful for teaching a Psych Stats class. Cheers!
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I wish I could learn more from you about stats and math -- you use language in places that I do not understand. Cohen's D visualizations opened my understanding. Thank you
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Thank you, Kristoffer
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Hi Kristoffer, Some time ago I've come up with a similar illustration about CIs as you have produced, and I'm now also referring to your work:<br>https://hugoquene.github.io/QMS-EN/ch-testing.html#sec:t-confidenceinterval-mean<br>With kind regards, Hugo Quené<br>(Utrecht University, Netherlands)
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I've been learning about power analysis and effect sizes (trying to decide on effect sizes for my planned study to calculate sample size) and your Cohen's d interactive tool is incredibly useful for understanding the implications of different effect sizes!
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Really wonderful visuals, and such a fantastic and effective teaching tool. So many thanks!
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Thanks for the assistance for RSCH 8210.
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Archived Comments (5)
Graphs are yours or from another author?
These are very useful visuals. Thank you for sharing them!
Outstandind!
Thank you! I have wrapped this script in a function and I show it on my R course for undergrads students.
Hi Kristoffer,
I am Sithara from India, just completed my master of public health from Kerala, India. I am very much passionate about new things in R. The details in your blog regarding effect size gives and understanding about bothe topic and the functions in R.
Thank you so much!
Keep up the good spirit
Expect new posts.
Bookmarking! As an I/O practitioner I usually run into trouble when presenting this kind of information to clients. This looks like it could help out!
Thanks and keep up the good work!