How Can I Determine The Optimal Sample Size For A Two-sample T-test To Compare The Mean Response Times To A New Medication Versus A Placebo In A Clinical Trial, Given That The Population Standard Deviation Is Unknown, The Desired Power Is 0.9, And The Effect Size Is Expected To Be Small (Cohen's D = 0.2), While Also Accounting For The Potential Non-normality Of The Response Time Data Due To Outliers And Skewness?

To determine the optimal sample size for a two-sample t-test comparing the mean response times to a new medication versus a placebo, given a small effect size (Cohen's d = 0.2), a desired power of 0.9, and potential non-normality, follow these steps:

1. Effect Size and Desired Power: The effect size is small (d = 0.2), and the desired power is 0.9, which is standard for ensuring the study can detect the effect if it exists.

2. Significance Level: Assume a two-tailed test with α = 0.05.

3. Sample Size Calculation:

   • Using a z-test approximation for initial estimation:
     • Formula: ${ n = \left(\frac{Z_{\alpha/2} + Z_\beta}{d}\right)^2 }$
     • ${ Z_{\alpha/2} ≈ 1.96 }$ and ${ Z_\beta ≈ 1.28 }$ for power = 0.9.
     • Calculated ${ n ≈ 263 }$ per group.
   • Adjust for t-test: Since t-test is more conservative, increase the sample size slightly, approximating around 270-300 per group.

4. Consider Non-Normality: Due to potential outliers and skewness, increase the sample size to enhance robustness. Aim for a larger sample size, approximately 300 per group.

5. Final Recommendation: Considering all factors, including the need for robustness against non-normality, the optimal sample size is approximately 300 participants per group, totaling 600 participants.

Answer: The optimal sample size is approximately 300 participants per group, totaling 600 participants. This accounts for the desired power, small effect size, and potential non-normality of the data.