I've some data that is divided into a series of groups and am testing whether the mean is different from 1. The data is highly skewed, with skewness ranging from 1.10 to 26!
I did a one-sample t-test to see were the means different from one, in spite of the skewness. I also did a one-sample median test (Wilcoxon Signed Rank) and have produced p-values for both.
Am I at risk of a TYPEII error when looking at the one sample t-test as the data is highly skewed?
Would there be any benefit to using a skewness-adjusted t-statistic (as in Johnson (1978) bagusco.staff.ipb.ac.id/files/2014/01/modified-t-test.pdf)?
If I'm showing a table of this, should I produce both the one sample mean and the one sample median statistics?
Any other suggestions?
EDIT: A few clarifications based on the answer below.
The underlying data is a series of sequential bets. Each bet has a certain stake size and is a win or a loss. Some of the stake sizes are very small, some are quite large, so I standardised the first bet to 1. I'm interested in the path-dependent changes in stake size, so the ratio is ok, I think. The groups (which are nodes in the betting/result sequence) go from 'L' and 'W' for the first bet, to 'WWWW', 'WLLW' etc for the fourth bet. If stake sizes are independent of the result/path, the stake size ratio should still be 1 after 4 bets. If it's not, there's path-dependency. I want to test whether the stake sizes in each group are changing and whether they are statistically significantly different from 1.
- Are you testing each group individually for mean=1? (my subsequent discussion assumes so)
Yes. There's 16 groups. If the mean standardised stake is still 1, the stake size changes are independent of the previous results. I expect there to be a difference from one for those that had 3/4 winning bets and those that had 3/4 losing bets.
- mean 1 sounds a little unusual. Are we dealing with ratios in any way?
Yes. As explained above.
- What sample sizes are these?
There's at least a few hundred observations in most groups, while some only have 60/70 observations.
- Are the data discrete or continuous?
The data are continuous, but as mentioned above, it's stake/bet sizes so highly skewed. Some agents bet small, some bet very big.
- Is it definitely the population mean you're interested in?
I'm not too sure. Maybe not.
Could I use the Sign Test since it doesn't assume symmetry? If the median in each group is changing, that might answer the question of whether there's anything happening along the result path of these groups.