How to analyse a contingency table with specific expected distribution?

Question

I have a 2x2 table of single nucleotide substitutions in DNA. The data rows represent the nucleotides in one genome (A) and columns are the nucleotides in the same positions of a different genome (B). The numbers are the times such substitution occurred.

        A or T   C or G
A or T  3        12
C or G  12       3

I want to analyse whether these mutations are random. The problem is that since any "substitutions" to the same base (eg A->A) are not recorded you would expect to have twice as many substitutions from A/T to C/G than from A/T to A/T etc. The expected values for n=30:

        A or T   C or G
A or T  5        10
C or G  10       5

Tests like chi-squared or Fisher's exact test determine if there are nonrandom associations between rows and columns. But I know that there is an association anyway and want to test whether the data is significantly different from it.

What statistical tests would be appropriate?

When performing chi-squared for independence is it possible to use specific expected values instead of using row and column totals?

Would it be correct to transform the data by dividing the A/T->C/G and C/G->A/T cells by 2, to get the mean number of substitutions to the other base category? The n would then change.

Answer 1

In the formula of chi-square, for each observed frequency you:

-- 1) take the difference between the observed frequency and its expected one;

-- 2) square the above difference;

-- 3) divide the above squared difference by the expected frequency of point 1;

-- 4) sum all the results (of points 1-3 applied to every cell in your contingency table) to obtain the chi-square statistic.

Therefore, in the formula, you can set a different expected frequency for each cell in your contingency table and test the hypothesis you need.