Drummond:Akashi's Test

Introduction
In 1994, Hiroshi Akashi developed an elegant test for translational accuracy selection on coding sequences Akashi-Genetics-1994. Some codons, particularly those corresponding to abundant tRNAs, are translated more accurately than others. Under selection for translational accuracy, usage of those more-accurate synonymous codons will be favored at important (e.g., evolutionarily conserved) amino-acid sites, where translation errors could disrupt protein folding or function. At less-important (e.g., evolutionarily variable) amino-acid sites, errors are presumably more tolerable, and therefore more-accurate codons are less likely to be favored. Akashi's test asks:
 * 1) How strong is the association between preferred codons and conserved amino acids, controlling for differences between amino acids and between genes?
 * 2) How likely is that association to have occurred by chance?

Akashi's test on a single gene
Akashi's test is technically very simple to carry out. The hard part is just tabulating data. Suppose you have two aligned codon sequences (a target sequence and an orthologous sequence) and a list of preferred codons. From the aligned codon sequences, build a 2x2 contingency table with entries $$a$$, $$b$$, $$c$$, and $$d$$ like this:

for each amino acid. You'll usually have 18 tables; W and M have no synonymous codon alternatives and therefore don't contribute to Akashi's test.


 * $$a$$ = the number of codons in your target sequence that encode amino acid AA, are PREFERRED, and encode an AA which is unchanged (CONSERVED) in the orthologous sequence
 * $$b$$ = the number of codons in your target sequence that encode amino acid AA, are PREFERRED and encode an AAwhich is different (VARIABLE) in the orthologous sequence
 * $$c$$ = the number of codons in your target sequence that encode amino acid AA, are UNPREFERRED and encode an AA which is unchanged (CONSERVED) in the orthologous sequence
 * $$d$$ = the number of codons in your target sequence that encode amino acid AA, are UNPREFERRED and encode an AA which is different (VARIABLE) in the orthologous sequence

Now the statistics. Assuming no association -- that is, assuming that the probability of a codon being preferred (which we designate $$p$$) is independent of the probability that it encodes a conserved amino acid (which we designate $$q$$) -- we can write down estimates for the expected value and variance of $$a$$, $$E(a)$$ and $$V(a)$$:


 * $$n = a + b + c + d\!$$


 * $$\hat{p} = \frac{a + b}{n}$$
 * $$\hat{q} = \frac{a + c}{n}$$


 * $$\hat{E}(a) = n \hat{p}\hat{q} = \frac{(a+b)(a+c)}{n}$$
 * $$\hat{V}(a) = \frac{n}{n-1} n\hat{p}(1-\hat{p}) \hat{q}(1-\hat{q}) = \frac{(a+b)(a+c)(b+d)(c+d)}{n^2(n-1)}$$

With the mean and variance, we can write down a $Z$-score for one table:
 * $$\hat{Z} = \frac{a - \hat{E}(a)}{\sqrt{\hat{V}(a)}}$$

And because a $$Z$$-score only gives us a measure of statistical significance (question #2 above), we also want an effect size -- the magnitude of the association between preferred codons and conserved sites -- which we can compute as an odds ratio, the odds of finding a preferred/conserved association divided by the odds of finding a nonpreferred/variable association. The odds ratio answers question #1 above. An unbiased estimate of the odds ratio $$\psi$$ is given by:


 * $$\hat{\psi} = \frac{ad}{bc}$$.

Akashi's test on multiple genes
Estimating $$Z$$ and $$\psi$$ for a single amino acid in a single gene is perhaps of limited interest. How do we combine tables so that we can ask questions like, "What is the overall association between preferred codons and conserved sites across the genome?" or, "How statistically significant is the preferred/conserved association for alanine compared to glycine?"

To combine tables, we use the Mantel-Haenszel procedure. The basic principle is that tables are independent. Expectations add, variances add, and observed values add. That is, indexing tables by $$i$$, with table $$i$$ given by and $$\hat{E}(a_i)$$ and $$\hat{V}(a_i)$$ computed as before for each table, we have the combined $$Z$$-score


 * $$\hat{Z} = \frac{\sum_i{a_i} - \sum_i{\hat{E}(a_i)}}{\sqrt{\sum_i{\hat{V}(a_i)}}}$$

and the Mantel-Haenszel estimator for the common odds ratio (i.e., the single odds ratio $$\psi$$ assumed to underlie all tables being analyzed),


 * $$\hat{\psi} = \frac{\sum_i{\frac{a_i d_i}{n_i}}}{\sum_i{\frac{b_i c_i}{n_i}}}$$

With enough tables, we assume that $$\hat{Z}$$ follows the standard normal distribution $$\Phi$$, so that a $$P$$-value can be computed as $$\Phi(\hat{Z})$$.

Akashi's test equates statistical significance of a preferred-codon/conserved-site association with the influence of translational accuracy selection. One simply asks how statistically significant $$\Phi(\hat{Z})$$ is.

Implementation
The open-source statistical package R includes an implementation of the Mantel-Haenszel test (mantelhaen.test), which is sufficient to carry out Akashi's test.

Examples
Coming...