Inference procedures.

I will outline the procedure that we can use for constructing confidence intervals for the mutation rate using the parameterized distribution that I described in the previous section.

Assume that we start with a datum x, representing the proportion of mutants in the culture, and the we know parameter r of the cell cycle time distribution. Knowing r, we can first calculate the growth rate of the culture, using Eq. determine, with T_B = r/(r+1) and $\lambda = r+1$. We then calculate c, the correction for the mutation rate, by the formula . We retrieve the value of the parameter b from Table . We can calculate the value of the scaled variable as a function of the mutation rate

$$\xi(\mu) = \frac{x}{c \mu} - \frac{1}{b \mu N}.$$ (6.38)

Assume that we want to find the 1 - a confidence interval for the mean. All we need to do is to find the values of the mutation rate for which the given datum $\xi(\mu)$ corresponds to the a/2 and 1-a/2 quantiles, respectively, of the distribution specified by the formula . As the quantile to which x corresponds is a monotonic function of $\mu$, a simple search algorithm on $\mu$ would give us these values.

This procedure can be easily automated. The interface to it would be simple, the query being specified by only two variables: the ratio r of the division time to the mean waiting time (or, even simpler, the coefficient of variation of the cell cycle time), and the observed proportion of mutants in the culture. The program would construct confidence intervals for the mutation rate. I believe that this approach would provide a very useful tool in the study of mutational processes.