Interpreting Cohen's d Effect Size

An Interactive Visualization

The Cohen’s d effect size is immensely popular in psychology. However, its interpretation is not straightforward and researchers often use general guidelines, such as small (0.2), medium (0.5) and large (0.8) when interpreting an effect. Moreover, in many cases it is questionable whether the standardized mean difference is more interpretable than the unstandardized mean difference.

In order to aid the interpretation of Cohen’s d, this visualization offers these different representations of Cohen’s d: visual overlap, Cohen’s U₃, the probability of superiority, percentage of overlap, and the number needed to treat. It also lets you change the standard deviation and displays the unstandardized difference.

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Cohen's U₃

% Overlap

Probability of Superiority

Number Needed to Treat

A Common Language Explanation

With a Cohen's d of 0.80, 78.8% of the "treatment" group will be above the mean of the "control" group (Cohen's U₃), 68.9% of the two groups will overlap, and there is a 71.4% chance that a person picked at random from the "treatment" group will have a higher score than a person picked at random from the "control" group (probability of superiority). Moreover, in order to have one more favorable outcome in the "treatment" group compared to the "control" group, we need to treat 3.5 people on average. This means that if there are 100 people in each group, and we assume that 20 people have favorable outcomes in the "control" group, then 20 + 28.3 people in the "treatment" group will have favorable outcomes.¹

¹The values are averages, and it is assumed that 20 (CER) of the "control" group have "favorable outcomes," i.e., their outcomes are below some cut-off. Change this by pressing the settings symbol to the right of the slider. Go to the formula section for more information.

Colors

Dist1

Overlap

Dist2

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Parameters

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Written by Kristoffer Magnusson, a researcher in clinical psychology. You should follow him on Bluesky or on Twitter.

FAQ

How do I use this visualization?

Change Cohen’s d

Use the slider to change Cohen’s d, or open the settings drawer and change the parameters. The inputs can also be controlled using the keyboard arrows.

Settings

You can change the following settings by clicking on the settings icon to the right of the slider.

Parameters
- Mean 1
- Mean 2
- SD
- Control group event rate (CER)
Labels
- X axis
- Distribution 1
- Distribution 2
Slider settings
- Slider Max
- Slider Step: Controls the step size of the slider

Save settings

The settings can be saved in your browser’s localStorage and will thus persist across visits.

Pan and rescale

You can pan the x axis by clicking and dragging the visualization. Double-click the visualization to center and rescale it.

Offline use

This site is cached using a service worker and will work even when you are offline.

What are the formulas?

Cohen’s d

Cohen’s d is simply the standardized mean difference,

$δ = \frac{μ _{2} - μ _{1}}{σ}$ ,

where $δ$ is the population parameter of Cohen’s d. Where it is assumed that $σ_{1} = σ_{2} = σ$ , i.e., homogeneous population variances. And $μ_{i}$ is the mean of the respective population.

Cohen’s U₃

Cohen (1977) defined U₃ as a measure of non-overlap, where “we take the percentage of the A population which the upper half of the cases of the Β population exceeds”. Cohen’s d can be converted to Cohen’s U₃ using the following formula

$U_{3} = Φ (δ)$

where $Φ$ is the cumulative distribution function of the standard normal distribution, and $δ$ the population Cohen’s d.

Overlap

Generally called the overlapping coefficient (OVL). Cohen’s d can be converted to OVL using the following formula (Reiser and Faraggi, 1999)

$OVL = 2 Φ (- ∣ δ ∣ / 2)$

where $Φ$ is the cumulative distribution function of the standard normal distribution, and $δ$ the population Cohen’s d.

Probability of superiority

This is effect size with many names: common language effect size (CL), Area under the receiver operating characteristics (AUC) or just A for its non-parametric version (Ruscio & Mullen, 2012). It is meant to be more intuitive for persons without any training in statistics. The effect size gives the probability that a person picked at random from the treatment group will have a higher score than a person picked at random from the control group. Cohen’s d can be converted CL using the following formula (Ruscio, 2008)

$CL = Φ (\frac{δ}{2})$

where $Φ$ is the cumulative distribution function of the standard normal distribution, and $δ$ the population Cohen’s d.

Number Needed to Treat

NNT is the number of patients we would need to treat with the intervention to achieve one more favorable outcome compared to the control group. Furukawa and Leucht (2011) gives the following formula for converting Cohen’s d into NNT

$NNT = \frac{1}{Φ ( δ + Ψ ( C E R ) ) - C E R}$

where $Φ$ is the cumulative distribution function of the standard normal distribution and $Ψ$ its inverse, CER is the control group’s event rate and $δ$ the population Cohen’s d. N.B. CER is set to 20 % in the visualization above. You can change this be pressing the settings symbol to the right of the slider. The definition of an “event” or a “response” is arbitrary and could be defined as the proportion of patients who are in remission, e.g. bellow some cut-off on a standardized questionnaire. It is possible to convert Cohen’s d into a version of NNT that is invariant to the event rate of the control group. The interested reader should look at Furukawa and Leucht (2011) where a convincing argument is given to why this complicates the interpretation of NNT.

R code to calculate NNT from Cohen’s d

Since many have asked about R code for the formula above, here it is

References

Baguley, T. (2009). Standardized or simple effect size: what should be reported? British journal of psychology, 100(Pt 3), 603–17.
Cohen, J. (1977). Statistical power analysis for the behavioral sciencies. Routledge.
Furukawa, T. A., & Leucht, S. (2011). How to obtain NNT from Cohen’s d: comparison of two methods. PloS one, 6(4).
Reiser, B., & Faraggi, D. (1999). Confidence intervals for the overlapping coefficient: the normal equal variance case. Journal of the Royal Statistical Society, 48(3), 413-418.
Ruscio, J. (2008). A probability-based measure of effect size: robustness to base rates and other factors. Psychological methods, 13(1), 19–30.
Ruscio, J., & Mullen, T. (2012). Confidence Intervals for the Probability of Superiority Effect Size Measure and the Area Under a Receiver Operating Characteristic Curve. Multivariate Behavioral Research, 47(2), 201–223.

How do I cite this page?

Cite this page according to your favorite style guide. The references below are automatically generated and contain the correct information.

APA 7

Magnusson, K. (2023). A Causal Inference Perspective on Therapist Effects. PsyArXiv. https://DOI

BibTex

I fund a bug/error/typo or want to make an suggestion!

Please report errors or suggestions by opening an issue on GitHub, if you want to ask a question use GitHub discussions

I'm gonna ask a large number of students to visit this site. Will it crash your server?

No, it will be fine. The app runs in your browser so the server only needs to serve the files.

The overlap statistic differs from Cohen's calculations

This is intentional, you can read more about my reasons in this blog post: Where Cohen went wrong – the proportion of overlap between two normal distributions

Can I include this visualization in my book/article/etc?

Yes, go ahead! I did not invent plotting two overlapping Gaussian distributions. This visualization is dedicated to the public domain, which means “you can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission” (see Creative common’s CC0-license). Although, attribution is not required it is always appreciated!

The source code for this page is licensed using MIT, and the text on the page is CC-BY 4.0.

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