Good narratives for numbers justification:

• We used the JMP statistical package to perform power analysis to arrive at the proposed group size. Using parameters of alpha = 0.05; power = 80%; effect size = 50%; and standard deviation = 30% of the mean, we arrived at a group size of 7 for a comparison of two groups.  A group size of 10 was chosen because it provides an 80% chance of observing a difference of 50% in the histologic score at a level of significance of 0.05

• The formula for the sample size required at each time point is:
  where Tn-2,p denotes the (1-p) percentile of the t distribution with n-2 degrees of freedom with p equal to alpha/2 or beta, and q is the proportion of biological specimens allocated to each class; n is the sample size, alpha is the significance level (the probability of declaring that the expression level of a gene is different between classes/time points when in fact there is no difference, delta=0) and  beta is the false negative rate (the probability of declaring that the expression is not different when in fact the mean difference is not zero, delta≠0). The statistical power is 1-beta, the probability of obtaining statistical significance when the true difference in mean expression level between the two classes/time-points is delta. Since gene expression for 12,000 genes will be examined, the value of  should be set small.  is the standard deviation of expression level for the gene within each class. Common! choices for microarray experiments are alpha=0.001, beta= 0.05, sigma=0.5, and delta=1 (because gene expression levels utilize base 2 logarithms of the intensities, delta=1 corresponds to a 2-fold average difference between the two classes/time-points). An iterative computational procedure is used to solve the above equation, and gives approximately 15 as the sample size for each time point. Please note that this sample size calculation is very conservative in terms of minimizing the number of independent biological replicates (animals) per time-point, since we’re planning to analyze expression ratios for each of the 12,000 genes individually. This may require future increases in sample size based on the preliminary data from this protocol.