6.14. Odds-ratio’s

There are a multitude of statistical measures that are presented as odds-ratios (hereafter OR). They are often seen in terms of evaluating clinical studies. For instance, the odds-ratio of statin medication increasing life-span. We will be concerned with using the odds-ratio to distinguish between competing hypotheses.

Consider an odds ratio that contrast the probability of a sequence under an alternate hypothesis (\(p_{alt}\)) against the probability of the sequence under a null hypothesis (\(p_{null}\)). In this instance, an OR equal to 1 if \(p_{alt} = p_{null}\), i.e. the sequence is equally likely under both models. OR > 1 indicates the sequence is more likely under the alternate, while a value less than 1 implies the converse.

[1]

Recalling that \(\log(a/b)=\log(a) - \log(b)\), and \(\log(a \times b) = \log(a) + \log(b)\).

6.14.1. Log-odds – the logarithm variant

A log-odds ratio is simply the log-transformation of the odds ratio [1]. For example, for OR=1, the log-odds \(\log_{10}(OR)=0\) [2].

[2]

If you find a log-odds value hard to interpret, convert it back into a natural number by raising the logarithm base to the power of the log-odds. For example, if the log-odds value you have is \(LOR\), and it was obtained as \(LOR = \log_2(OR)\), then the reverse operation is \(OR = 2^{LOR}\).