My approach to comparing pre-molt and post-molt crab sizes using a Monte Carlo permutation test is a sound way to address the potential non-normality of your data. Here’s a step-by-step summary of proposed method:
Data Collection: You have collected data on pre-molt and post-molt crab sizes.
Kurtosis Assessment: You’ve noted that the kurtosis values for both groups are relatively high, which suggests that the data distributions have heavier tails than a normal distribution.
Hypothesis Testing Challenge: Given the non-normality of the data, using a traditional t-test may not be appropriate, as it assumes normality. Hence, you’re considering an alternative approach.
Monte Carlo Test:
Data Pooling: You combine the data from both pre-molt and post-molt groups into one dataset.
Random Sampling: You randomly split this combined dataset into two groups of equal size many times (10 million times in your case).
Calculate Mean Differences: For each split, you calculate the mean difference between the two groups.
Distribution of Mean Differences: After all iterations, you have a distribution of mean differences, which represents what you might expect under the null hypothesis (i.e., no real difference between pre-molt and post-molt crab sizes).
Compare Observed Difference: You compare the observed mean difference in your actual data to the distribution of permuted mean differences.
Calculate p-value: The p-value is the proportion of permuted mean differences that are as extreme as or more extreme than the observed mean difference. A low p-value suggests that the observed difference is unlikely to have occurred by chance, supporting the rejection of the null hypothesis.
This Monte Carlo permutation test approach allows you to assess the significance of the observed mean difference while accounting for the non-normality of your data. It’s a robust method for hypothesis testing when the assumptions of traditional parametric tests like the t-test are not met. If your calculated p-value is below your chosen significance level (e.g., 0.05), you can conclude that there is a significant difference between pre-molt and post-molt crab sizes.