A one-sample ratio t test compares the geometric mean of a single column of numbers against a hypothetical geometric mean that you provide.

The P value answers this question:

If the data were sampled from a population following a lognormal distribution that has a geometric mean equal to the hypothetical value you entered, what is the chance of randomly selecting N data points and obtaining a geometric mean as extreme from the hypothetical value as observed here? The sample geometric mean and the hypothetical geometric mean are compared by taking their ratio. This is equivalent to comparing the difference between the mean of the log-transformed data and the log-transformed hypothetical value.

If the P value is large, the data do not give you any reason to conclude that the population geometric mean differs from the hypothetical value you entered. This is not the same as concluding that the true geometric mean must equal the hypothetical value. You just don't have evidence to reject the null hypothesis that they're the same.

If the P value is small (usually defined to mean less than 0.05), then it is unlikely that the discrepancy you observed between sample geometric mean and hypothetical geometric mean is due to a coincidence arising from random sampling. You can reject the hypothesis that the size of the ratio between these two values is a coincidence, and conclude instead that the population has a geometric mean different than the hypothetical value you entered. The ratio is statistically significant. But is the difference scientifically important? The confidence interval helps you decide.

Prism also reports the 95% confidence interval for the ratio of the sample and hypothetical geometric means. If the experiment were conducted a large number of times, 95% of the intervals constructed this way would contain the true ratio between the geometric mean of the population from which the data were sampled and the hypothetical geometric mean you provided.

Assumptions

The one sample ratio t test assumes that you have sampled your data from a population that follows a lognormal distribution. While this assumption is not too important with large samples, it is important with small sample sizes, especially when N is less than 10. If your data do not come from a lognormal distribution, you have three options. Your best option is to transform the values to make the distribution more Gaussian (for example by transforming all values to their reciprocals) then performing a one-sample t test. Another choice is to use the Wilcoxon signed rank nonparametric test instead of the t test.

The one sample t test also assumes that the “errors” are independent. The term “error” here refers to the ratio between each value and the group geometric mean. The results of a one-sample ratio t test only make sense when this multiplicative error is random – that whatever factor caused a value to be too high or too low affects only that one value. Prism does test this assumption.

How the one-sample ratio t test works

Prism starts by log-transforming the entered data and the entered hypothetical geometric mean. The geometric mean of the entered data is equivalent to the mean of the log-transformed data. Prism then calculates the difference between the mean of the log-transformed data and the log-transformed hypothetical value. By raising 10 to the power of this difference, you obtain the geometric mean of ratios reported by Prism. The standard deviation of the log-transformed data is then calculated. By raising 10 to the power of this value, you obtain the geometric standard deviation of ratios reported by Prism. Dividing the calculated standard deviation by the square root of the number of values provides the standard error of the log-transformed data. By raising 10 to the power of this value, you obtain the geometric standard error of ratios reported by Prism.

With the calculated difference and standard deviation, the t ratio can be calculated. This is simply the ratio of the difference divided by the standard error. A P value is computed from the t ratio and the numbers of degrees of freedom (which equals sample size minus 1).