The Shape of Code

The statistical design and analysis of experiments is a relatively recent invention (around 150 years old; verifying scientific hypotheses using experiments was first proposed over 1,000 years ago).

Once an experiment has been run, and performance measurements collected, what techniques are available to analyse the data?

Before electronic computers were invented, the practical statistical techniques were those that could be performed manually (perhaps with some help from a mechanical calculator).

Once computers became available, these manual use techniques were widely implemented in statistical applications and libraries. New statistical techniques have been invented that are only of practical when a computer is available, e.g., the bootstrap.

Most researchers in software engineering remain stuck in the pre-computer world of statistical analysis, i.e., failing to use more powerful techniques that make use of computers. Software engineering is not unique in being stuck using pre-computer statistical techniques, there are many other fields that have failed to move on to the more flexible and powerful techniques now available.

Performance comparison by benchmarking (i.e., running an experiment) is a common activity in scientific and engineering fields. Once the measurements have been made, what is the difference between pre-computer and post-manual statistical comparison techniques?

Performance comparison invariably involves comparing the mean/average of each set of measurements (one set for each product benchmarked). Random variations will have some impact on the measured values, and it’s possible that any difference in mean values is the result of these random variations. There are statistical tests for estimating the likelihood of the difference being due to random variation.

Statistical tests invariably come with preconditions on the characteristics of their input data, i.e., the test only works as advertised if the preconditions are met.

The t.test is a popular pre-computer method of estimating the likelihood that the difference between the means of two sets of measurements occurred by chance. Two of its preconditions are that the two sets of input data have a Normal/Gaussian distribution, and have the same standard deviation (if the standard deviations are not equal, Welch’s t-test should be used).

The Mann–Whitney U test (also known as the Wilcoxon rank-sum test) is a pre-computer method with a more relaxed precondition on the distribution of the inputs, requiring only that they are drawn from the same distribution (which need not be a Normal distribution; it is a non-parametric test). This test returns the likelihood of one group being less/greater than the other, and says nothing about the magnitude of the difference.

Removing the constraint that it must be practical to manually perform the calculations, led, in 1979, to a paper proposing the Bootstrap (major issues around the theory underpinning the bootstrap were answered by the early 1990s).

The Bootstrap, a post-manual method, does not have any preconditions on the distribution or standard deviation of the input data. An important precondition is that it is reasonable to assume that the elements of the two sets of measurements are exchangeable (more on this below).

The Bootstrap is a general technique that is not limited to only comparing mean values, it can be used to compare any quantity of interest (assuming the appropriate data is available).

The bootstrap algorithm starts by assuming that, if the two samples are combined into an aggregate sample, all possible permutations of values in this aggregate into two samples are equally likely (this is the exchangeability assumption). If this assumption is true, then any compared difference in the measured two samples will be roughly the same as the compared difference in a pair of random samples drawn from the aggregate.

The computational intensive component of a bootstrap test is generating many pairs of random samples (5,000 is a common choice; sampling with replacement, and saving the result of the comparison).

The comparison results of the paired random samples and the two measured samples are sorted. The position of the measured comparison in this sorted list is the test statistic, i.e., how extreme is the result from the measured samples, compared to the random samples?

News of the flexibility and power of the bootstrap are slowly permeating through the research community. There is an existing way of doing things, and there is little incentive to change. As Plank famously observed: “Science advances one funeral at a time.”