# A Priori (Power)Sample Size Analysis with cohen's d

How can I do an a priori power analysis to calculate the sample size for a study with DeclareDesign based on cohen’s d? By now most social science journals request this.
For example, if I have the effect size as cohen’s d (e.g. =.4) from a meta-analysis. And one wants to run a two arm experiment with a covariate (e.g. a covariate that has not been tested with the treatment before). How can I transfer cohen’s d into the parameters needed by DeclareDesign (e.g. Mean scores and ATE)?

Cohen’s `d` is unitless, so multiply it by the standard deviation to get the treatment effect (with units). Just remember to be careful with units.

but usually I don’t know/have an SD, neither the mean. If I would run several simulations with a range of SDs and means I would end up with a power distribution. In the end journals want to see one value for N that needs to be sampled max given a certain effect size and power of .8 without assuming any SD or means.

It sounds like you just want a classical power analysis - all you need is a t-table, there’s no need to use DeclareDesign to numerically approximate it.

I think my view differs somewhat from Neals.

Imagine that your potential outcomes function were something like this

`declare_potential_outcomes(Y ~ tau*Z + rnorm(N, 0, 1))`

Here the control group’s outcome is normal distributed with mean zero and variance 1 (which is close to what happens when you standardize a variable. The treatment group’s outcome is also distributed standard normal, but with a higher mean.

If you use declare design, then you can ALSO build in other design features like blocking, clustering, or covariate adjustment. Classical power analysis doesn’t let you do that.

This example is solved perfectly well by `power.t.test()`, though. Do you have a case of actually doing a covariate adjustment (w/o means and SDs assumed, as @colonus above asked)? I suppose you would have a d for both the treatment variable and the covariate, or maybe a correlation if the covariate were continuous? Not sure.

thank you both for taking this issue on. Yes, I would assume that with the covariate I would get a higher power and would need a smaller sample size and therefore power.t.test() does not suffice. I do have a d-score for the treatment effect and the correlation of the covariate with the outcome.

To the best of my understanding of your question, I think you could use the two arm with covariate design from the design library. I’ve adapted it a little to match your parameters, d and r:

``````library(DeclareDesign)

d = .8
r = .2
N = 20

a_design <- declare_population(N=N, X=rnorm(N)) +
declare_estimand(d=d) +
declare_potential_outcomes(Y_Z_0 = rnorm(n=length(X), mean=r*X, sd=sqrt(1-r^2)), Y_Z_1=Y_Z_0 + d) +
declare_assignment( ) +
declare_reveal() +
declare_estimator(Y~Z , model=lm, label="no covariate", estimand='d') +
declare_estimator(Y~Z+X, model=lm, label="with covariate", estimand='d')

diagnose_design(a_design)

power <- diagnose_design(redesign(a_design, N=as.list(1:10*10)))
``````

However, we can compare your results to the t table:

``````# Must divide N by two, it expects *per group*
t_power <- transform(data.frame(N=10:100),
no_cov = power.t.test(n=N/2, delta = d, sd = 1, type='two.sample')\$power,
with_cov = power.t.test(n=N/2, delta = d, sd = sqrt(1 - r^2), type='two.sample')\$power)

ggplot() +
geom_line(aes(x=N, y=power, col=estimator_label), data=power\$diagnosands_df) +
geom_line(aes(x=N, y=no_cov), data=t_power, linetype="dashed", color='pink') +
geom_line(aes(x=N, y=with_cov), data=t_power, linetype="dashed", color='cyan')
``````

Which yields this: EDIT:

• I would tend to trust the t-table in this case, at least to the extent that the assumptions of Gaussianity are reasonable.
• You might need to add 1 to the N for the t-table above, it doesn’t know you are fitting a three parameter model.
• On the other hand, the `lm` model as above is using pooled variance, and the t-test can use unequal variance, so maybe the t-table is more conservative.
• If getting samples is easy / cheap, just get 100 and there’ll be plenty of power.
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Final thought: as @Alex_Coppock said above, if you wanted to mess around with the assignment (eg not do a 50/50 split) DeclareDesign will make that easier to diagnose - on the other hand, I would expect that design to perform worse / be less data-efficient than a 50/50 split. DeclareDesign also removes the burden of counting degrees of freedom.

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thank you so much! This helps a lot even though I am surprised that adding the covariate does not noticably increase power