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.

<[requests for clarification removed - it's now dealt with in the question]>

Any additional information you can give about what you're trying to find out may help.

2) The Signed Rank Test

The Wilcoxon signed rank test assumes symmetry under the null, and its really only a test for the median when that symmetry holds (it's actually a test for the median of pairwise averages, but with symmetry, the population quantity that converges to - the pseudo-median - will coincide with the population median). You don't have symmetry.

A sign test actually is a test for medians and doesn't assume symmetry. But if you want your inference to be about means that doesn't help you.

3) Skewness and the t-test

Your risk isn't just a type II error. High skewness affects the type I error rate, and can produce a biased test (that is, a test with power lower than $\alpha$ when $H_0$ is false). So you aren't doing the test at the significance level you think you are, for starters.

4) Other suggestions

(a) Your data are skew, so the obvious thing to do is use a model for skewed data. This is easily accomplished by a simple GLM of the form $E(y)=\mu$ (e.g. in R, a model fitted with the formula y~1). GLMs give a variety of possible skewed models, and are explicitly models for the mean. The obvious first choice is gamma-family, but there are others if that's not suitable (inverse Gaussian is more skew, for example)

Here's an example, done in R.

The data:

 0.8347 0.9837 0.8740 0.8471 0.8251 1.0461 0.7796 0.8302 0.9559 0.7557 0.8850
0.8695 0.9451 0.8436 0.8216 0.8431 0.7182 0.8753 0.7793 0.7483


First, here's what the data look like (this is a kernel density estimate with a rugplot): Now for the hypothesis test that the mean is 1, done via GLM (some output omitted):

> summary(glm(y~1,family=Gamma(link="identity")))

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.85305    0.01835   46.49   <2e-16 ***
---
(Dispersion parameter for Gamma family taken to be 0.009251426)

Null deviance: 0.17154  on 19  degrees of freedom
Residual deviance: 0.17154  on 19  degrees of freedom


Then the t-statistic we want would be (0.85305-1)/0.01835 = -8.008

(the values used in that calculation come from the line labelled "(Intercept)", and the null hypothesis)

You can even do it all directly in GLM through use of an offset:

> mu0=rep(1,length(y))

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.14695    0.01835   -8.01 1.65e-07 ***

(Dispersion parameter for Gamma family taken to be 0.009251426)

Null deviance: 0.17154  on 19  degrees of freedom
Residual deviance: 0.17154  on 19  degrees of freedom


And now the t-value and the p-value for the test are right there in the output.

(in this case, the data are not actually gamma distributed, but it's pretty close, and the test should give excellent results.)

(b) Perhaps a resampling-based test, such as seeing whether a bootstrap confidence interval for the mean includes 1.

bootmean=replicate(10000,mean(sample(y,20,replace=TRUE)))  # bootstrap

plot(density(bootmean),xlim=c(0.76,1)) # draw a smoothed density, include 1 on axis
rug(bootmean,col=8)                    # mark bootstrapped mean values in gray
rug(quantile(bootmean,p=c(.025,.975)),col=2,lwd=2) # mark on 95% CI limits in red
abline(v=1,lty=2,col="green3")                     # draw H0 value


Here's the plot: Clearly, the null is rejected here as well.

• I've edited the question to provide more clarity there. – user2146441 Feb 8 '14 at 10:39

Here is a simple approach: Transform all your values (ratios) to logarithms, and then analyze those logarithms. My guess is that the logs won't be skewed so a regular one-sample t test can be used to compare the mean of the logarithms with the hypothetical mean of 0.0 (the log of 1.).