We have a variable A which denotes a percentage between 0 and 1. The variable is not normally distributed but has a very left skewed distribution. About 5% of the observations are zero, most observations are below 0.02, the others are below 0.04.
There is a second nominally scaled variable B with four factors.
We want to know if A and B are independent.
The t-Test (test if the mean of A within a subgroup of B differs from the overall mean) or ANOVA seem to be inappropriated because A is not normally distributed.
Moreover the $\chi^2$-Goodness-Of-Fit-Test (test if the means of A over each subgroup of B fit a equally distributed variable) should not be used because the expected values are below 5 (actually they are between 0.02 and 0.03).
The next idea is to transform the variable A into an ordinal variable (e.g. [0.0,0.01) [0.01,0.02) ...) and to apply the $\chi^2$-Test of independency. However, by this way some information is lost.
Is there any other way to test if A and B are correlated or if the means and the variances of the subgroups differ significantly? Is there any correlation measure between a categorial and a interval scaled variable when the later is not normally distributed?
EDIT with respect to the first answer:
- The overall goal is to analyse if A depends on B. In a first step it would be good to show that the within-subgroup-means are (not) significantly different. Alternatively one or more correlation measure(s) which reflect(s) the interrelation between the variables would help. We are thankful for any hint on appropriate methods.
- Normal distribution of variable A: A is neither normally distributed at all nor within the subgroups of B.
- $\chi^2$: A denotes a percentage of hours. If A is 0.1 then 10 hours out of 100 are dropped out. One can assume that "10 hours" is a count. If you have 0.1, 0.11, 0.09 in the sub groups B1, B2, and B3, then one could test the observed counts 10, 11, and 9 against the expected ones 10, 10, and 10. (i) Is this a valid view point? And if yes, (ii) is it still valid if A is 0.02? And if (i) is answered "yes" and (ii) is answered "no" what is if A is transformed as "x hours out of 1000" [0.02 get 20 counts]?
- Correlation: We want to have some sort of "index" which measures the interrelation between A and B, a measure comparable to Pearson product-moment correlation coefficient or the point-biserial correlation coefficient. Unfortunately we have not found such a coefficient for the special case where one variable is nominal scaled and the other interval scaled. The answer to the question Correlation of a nominally scaled variable and another of interval scale type points to ANOVA. Under Assumptions it is stated that ANOVA assumes that "the distributions of the residuals are normal." We wonder if the assumption is violated by A?