David Vaux argues that experimental biologists should be better versed in classical statistics (Nature 492, 180–181; 2012). We suggest that they might also join the shift to Bayesian statistics that is already under way in many other areas of science.
He defines the 95% confidence interval (CI) as “with 95% confidence, the population mean will lie in this interval”, adding that it is commonly used “to infer where the population mean lies, and to compare two populations”.
However, a 95% CI merely tells us that if we were to sample from the population many times and calculate a 95% CI for each sample, 95% of the calculated CIs would, on average, contain the true population mean. Because classical statistics concern conditional probabilities of data based on assumed true parameter values (namely, the plausibility of the observed or more extreme data, given our assumptions), the 95% CI does not allow a probabilistic inference about “where the population mean lies”.
Bayesian statistics, by contrast, provide conditional probabilities of parameter values — the plausibility of different parameter values — given the data. Bayesian statistics therefore allow for probabilistic inferences about the true population mean and other parameters (J. K. Kruschke J. Exp. Psychol. Gen. http://doi.org/kdb; 2012).
Researchers often confuse probabilities derived from classical statistics (P values, for example) with Bayesian posterior probabilities (G. Gigerenzer J. Socio-Econ. 33, 587–606; 2004). This is because the latter represent what scientists are ultimately interested in: the conditional probability of parameter values or hypotheses, given the data.
David Vaux replies: I agree that it is often preferable to use Bayesian rather than classical statistics, and that I did not give the full and precise definition of 95% confidence intervals in my Comment. However, the distinction is unimportant for experiments in which N is 3 or less (where N is the number of independent samples).
Cell and molecular biologists could learn from physicist Ernest Rutherford, who said: “If your experiment needs statistics, you ought to have done a better experiment.” Where N is small, they would do well to plot all data points, rather than showing any statistics, classical or Bayesian.
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