Question

I need to test the effects of different treatments on plants and I would like to know the optimal sample size (number of plants that I need to sample). The variable I need to measure is the average number of insects present in a single plant. In the experiment, I have 3 main treatments and, in each treatment, I have 2 sub-treatments. The information I can provide are the following: Average number of larvae per plant: ranging between 0.03 and 0.1Standard deviation: ranging between 0.09 and 0.28The desiredZZ-score is 1.96I have no idea about the minimum detectable effect size but I can guess it could be 10% Looking online, I have found different formulas that return different results. The first formula was: n1=(σ21+σ22/K)(z1−α/2+z1−β)2Δ2,n1=(σ12+σ22/K)(z1−α/2+z1−β)2Δ2,where: Δ=|μ2−μ1|Δ=|μ2−μ1|is the absolute difference between two means,σ1σ1,σ2σ2are variances of mean #1 and #2,n1n1,n2n2are sample sizes for group #1 and #2,ααis the probability of type I error (usually 0.05),ββis the probability of type II error (usually 0.2),zzis the criticalZZvalue for a givenααorββ, andkkis the ratio of sample size for group #2 to group #1. With this formula, comparing these two means (0.03 and 0.09) and using a standard deviation of 0.09, I am gettingn=35n=35for each treatment. A different formula I found is: n=2σ2Z2d2,n=2σ2Z2d2,where nnis the required sample size per sub-treatment group,σ2σ2is the estimated variance,ZZis theZZscore, andddis the minimum detectable effect size. With this formula, using a mean of 0.03 and a standard deviation of 0.09, I am gettingn>6000n>6000for each treatment. Could you suggest me a simple and efficient formula for calculating the sample size?

Answer 1

The optimal sample size for your experiment can be calculated using different formulas or a power analysis, which takes into account the power of the test, the significance level, effect size, and standard deviation.

Step-by-step explanation:

Calculating the optimal sample size for an experiment can be complex, involving a range of statistical considerations. Given the provided average number of larvae per plant, standard deviation, desired Z-score, and an estimated minimum detectable effect size of 10%, there are multiple formulas at our disposal. However, commonly used formulas for determining sample sizes in experiments include the following:

• The first formula you provided seems to be based on comparing two means and takes into account the desired level of type I and type II error rates (commonly α = 0.05 and β = 0.2). This is typically used for calculating the sample size for each group in a two-sample t-test scenario.
• The second formula seems to be for a setting where you want to detect a minimum effect size with a given certainty. This simpler formula does not directly take the sub-treatment groups into account and could result in an overestimation of the required sample size, especially if the effect size is small.

Considering the experimental design with main and sub-treatments, I might suggest a power analysis to determine your sample size, which is a more comprehensive approach and can help to estimate the sample size needed to detect an effect of a given size with a certain degree of confidence.

Power analysis typically involves specifying the desired power of the test (β, with common practice using 0.8), the significance level (α), the effect size (difference you wish to detect), and the standard deviation. It adjusts for multiple groups and can be done using specialized statistical software.