What is effect size and how do you calculate it in your experiment?

Effect size is a measure of the strength of the relationship between two variables in an experiment. It is used to quantify the size of the difference or the strength of the association between two groups or conditions. In general, the formula for effect size calculation involves dividing the difference between the means of the two groups by the standard deviation of the entire sample. However, the specific calculation method may vary based on the statistical test being used and the nature of the data.

I typically try to pick an effect measure that is: interpretable, relevant, and has good numerical properties. In conversion experiments, "risk difference" (i.e. the difference in conversion rates between groups) is an effect measure I think is all three. The risk difference is simple, speaks directly to what we care about, and can rarely mislead us. Compare this to "lift" (i.e. difference in conversion divided by conversion in control). When the control conversion is very small, the lift can seem enormous, even though the change in conversion between groups was very small. In general, lift is not comparable between experiments, unless the control conversion rate is identical.

Effect size can be defined from a perspective as the "sensitivity" you want to measure with a statistical test. For instance, you would like to check whether layout A or B is a better option to attract customers for your website. If you consider notion of being better as even a difference of %0.1 than it is evident you need to collect much more sample to validate a difference of 1%.

This information can be really useful for newcomers into statistics, however it is highly important to have a clear vision of what is the expected outcome of the analysis in order to pick the most suitable effect measure.

Measures of effect size, the magnitude of the difference or association between variables, are usually standardised. For example, Cohen's d is a difference between means divided by (i.e. standardised by) the pooled standard deviation of the dataset; the Pearson's correlation coefficient r is the covariance between two variables divided by (i.e. standardised by) the product of their respective standard deviations. And so forth. The advantage of representing differences or associations in a standardised form is that this transforms the results of one study into numbers that are comparable across studies. Conceptually, this transformation "frees" one's results from the "constraints" of one's specific study.

Formulas for Cohen's d and Pearson's r. Cohen's d = (Mean_A - Mean_B) / StDev_pooled(A,B) Pearson's r = Covariance(A,B) / (StDev_A * StDev_B)

It is important to understand derivation of test effect size calculation rather than just relying on them. The distibution of test metrics for various hypotheses and related assumptions should be checked. And effect magnitude should be observed visually on these distributions.

El tama?o de un efecto No se puede calcular como valor exacto, solo se puede estimar (aproximar) estadísticamente con cierto Nivel de Confiabilidad (generalmente 95%), existen varios métodos Post-Anova para estimar el tama?o de dicho efecto tales como: Tukey, Fisher, Bonferroni y otras El tama?o del efecto se estimará mediante un Intervalo de Confianza. Si el tama?o del efecto es significativo ó No lo decidirá el Investigador ó Experimentador según el problema que enfrenta.

Effect size can be categorized into weak, moderate and strong according to a measure that is used to quantify it like, Cohen's d, Pearson's Correlation etc. after ensuring enough statistical power and minimum sample size for the experiment. The following table can be used a reference: Effect Cohen's d Pearson's r Weak <0.5 0.1-0.3 or (-0.3)-(-0.1) Mod <0.8 0.3 - 0.6 or (-0.6)-(-0.3) Strong >0.8 0.6 - 1.0 or (-1.0)-(-0.6) Pearson's measure additionally infers the direction of the relationship. For example, a +ve value 0.56 indicates both variables inc. or dec. together. a -ve value -0.56 indicates as one variable dec. the other one inc. & vice-versa.

First of all, effect size cannot be generalized - it's not an inference. It's just another method to describe the *current* data. Moreover, it has no connection with any practical relevance, unless the outcomes are standardized (like many topics in psychology, expressed in z-scores). It simply ignores *any context*, like the MCID (minimal clinically important difference). People much overstate its usefulness in deciding whether the outcome is, say, clinically meaningful. One may obtain a small effect of high precision (small variance), which will yield both p<0.0001 in some testing and "very large" effect size, and still it can be totally of no importance form the domain's perspective. Don't confuse effect size with practical significance.

It is evident that effect size is a very subjective matter and task dependent. If your client or collaborators have no strict specifications on them, best option is to present various effect sizes and their impact on experimentation settings.

I would suggest refraining from interpreting effects solely from their size, e.g. 0.2 is small. Rather, put the effect in context and consider uncertainty in the estimate as well.

Be careful about the following things: - The scale of the effect size. Sometimes, effect size can be represented in various scales. For example, for a Ratio of Means or a Ratio of Proportions test, it is a common practice to call either the ratio or the log(ratio) as effect size depending on its usage. One solution can be to check the symbol being used. For the example above, typically Rho represents the ratio scale and Delta represents the log(Ratio) scale. - Synonyms of Effect Size. Sometimes, in some papers and industries, some other names are commonly used for effect size. For instance, in clinical trial industries, Treatment Effect is often used as a common synonym for effect size.

A template I like to use is "We estimate the <effect estimate name> was <estimate> (confidence interval, additional required information)". For example, if I ran a t test to compute the risk difference, I might say "We estimate the risk difference between treatment and control to be 2.0 (95% CI: 1.2 -- 2.8, df=99)". The inclusion of the degrees of freedom and confidence interval + confidence level should provide readers everything they need to deduce any other information (e.g. the p value).

I haven't seen many people reporting the CI for effect size yet. Still, SPSS now calculates it automatically with most basic analyses, and there's no reason not to start including this information by default.

Due to the nature of frequentist statistics, you have to do your analysis in one go if you want to robustly prove your hypothesis. One analysis plan, one sample, one analysis moment. All the extra statistical tests you do lead to chance capitalization, which requires costly p-value corrections that only decrease the robustness of your analysis. This is why it's crucial to have an idea of how large the effect size is of whatever it is you're studying. Because you need an estimate of the effect size to understand how big your sample needs to be in order to prove your hypothesis. If you go in blind, you'll likely not be able to prove the effect you think is there, even if you're correct.

One thing I have found helpful is to calculate the minimum sample size to ensure enough statistical power with an expected effect size even before I begin the experiment. Maximizing power translates to maximizing the likelihood of a hypothesis test detecting a true effect if there is one. The test is then more likely to reject a false negative (Type II error) By performing this power analysis, in turn I can use a pre-determined effect size(inferred from the domain understanding/context of the experiment) and significance level to determine the sample size needed for a certain power level.

Let's have some fun There are many effect sizes, yup yup for example: Cohen's d: This is a widely used effect size Hedges' g: Similar to Cohen's dbut it includes a correction for small sample sizes, making it more accurate in these scenarios. Glass's Delta:unlike Cohen's d, it uses only the standard deviation of the control group or the group not exposed to the treatment/intervention. Eta Squared (η2): This is used in the context of ANOVA tests Omega Squared (ω2): Similar to Eta Squared, but provides a less biased estimate Cohen's f: This is used primarily in the context of ANOVA and multiple regression analyses.

While researchers often report just a single effect size, Cohen's D, there are lots of other measures. Even the Cohen's D has multiple versions, especially in terms of model-based testing (e.g. longitudinal studies). They are like statistical tests: both parametric (meaningful under normal distribution, like the probability of overlap, superiority) and non-parametric, like rank-biserial (links with Mann-Whitney-Wilcoxon, Sommer's D, Kendall tau, stochastic superiority). Therefore it's not surprising to have a discrepancy between two effect-size methods, e.g. small Cohen's D (or Hedges' G for small samples) and large rank-biserial. This highlights the importance of *always* starting with a research hypothesis rather than "juggling" methods.

Statistical testing can be highly intimidating for new comers. And it really is as you delve more. One should always start from basic visuals to understand notions of hypotheses etc before being drawned in equations and rule of thumbs which does not make sense to him/her.