# Permutation test for equality between the distribution of $g$ population

I have a data matrix of 42000 observation and 12 variables

I suppose to observe 12 samples of size $n_j$ from 12 indipendent random variables $Y_j,j=1,...,g$ I want do a permutation test for $$H_0: Y_1\stackrel{d}=Y_2\stackrel{d}=...\stackrel{d}=Y_{12}\ vs\ H_1: \exists j_1,j_2 \in \{1...g\} \ s.t \ Y_{j1}\neq{Y_{j2}}$$

As test statistics i use Fisher's $$T_F=\frac{\sum\limits_{j=1}^g (\overline{Y_j}-\overline{Y})^2/(g-1)}{\sum\limits_{j=1}^g\sum\limits_{i=1}^{n_j}({Y_{ij}}-\overline{Y})^2/(n_j-g)}$$ where ${\overline{Y}=\sum\limits_{j=1}^g\sum\limits_{i=1}^{n_j}({Y_{ij}})/(n_1+...+n_g)}$ is the common sample mean and $\overline{Y_j}$ is the common sample mean of group $j$.

My main problem is to understand how many permutation i have to do.

I'm not good on permutation test on this amount of data,are there another non-parameetric tests that handle so many observations?

• (1) Your expression for $T_F$ often divides by zero and so is not defined. (The denominators in the bottom also look incorrect, regardless of any other errors that might have crept into this expression.) (2) Your $H_0$ is not a valid hypothesis. A hypothesis refers to a definite state of affairs, not to the values of random variables. To the extent these are just typographical errors, please edit your post to fix them. If they reflect misunderstandings, then you ought to review the theory of Analysis of Variance and permutation tests first, just to fix the basic ideas in your mind. – whuber May 12 '16 at 16:03
• @whuber yes sorry i edited,it was typographical errors. – Federico May 12 '16 at 16:11
• Please check again -- should there be a squared term in the numerator of your test statistic? – Glen_b -Reinstate Monica May 13 '16 at 1:24
• @Glen_b yes sorry,i edited,but the main topic is not the $T_f$ but if i can do permutation test like this in my case. – Federico May 13 '16 at 13:50