to interpret the results of statistical tests and about our programs
GraphPad InStat and GraphPad Prism.
are described in this book and are performed by InStat, except for tests
marked with asterisks. Tests labeled with a single asterisk are briefly
mentioned in this book, and tests labeled with two asterisks are not
mentioned at all.
REVIEW OF NONPARAMETRIC TESTS
Choosing the right test to compare measurements is a bit tricky, as you must choose between two families of tests: parametric and nonparametric. Many -statistical test are based upon the assumption that the data are sampled from a Gaussian distribution. These tests are referred to as parametric tests. Commonly used parametric tests are listed in the first column of the table and include the t test and analysis of variance.
Tests that do not make assumptions about the population distribution are referred to as nonparametric- tests. You've already learned a bit about nonparametric tests in previous chapters. All commonly used nonparametric tests rank the outcome variable from low to high and then analyze the ranks. These tests are listed in the second column of the table and include the Wilcoxon, Mann-Whitney test, and Kruskal-Wallis tests. These tests are also called distribution-free tests.
CHOOSING BETWEEN PARAMETRIC AND NONPARAMETRIC TESTS: THE EASY CASES
Choosing between parametric and nonparametric tests is sometimes easy. You should definitely choose a parametric test if you are sure that your data are sampled from a population that follows a Gaussian distribution (at least approximately). You should definitely select a nonparametric test in three situations:
• The outcome
is a rank or a score and the population is clearly not Gaussian. Examples
include class ranking of students, the Apgar score for the health of
newborn babies (measured on a scale of 0 to IO and where all scores
are integers), the visual analogue score for pain (measured on a continuous
scale where 0 is no pain and 10 is unbearable pain), and the star scale
commonly used by movie and restaurant critics (* is OK, ***** is fantastic).
CHOOSING BETWEEN PARAMETRIC AND NONPARAMETRIC TESTS: THE HARD CASES
It is not always easy to decide whether a sample comes from a Gaussian population. Consider these points:
• If you
collect many data points (over a hundred or so), you can look at the
distribution of data and it will be fairly obvious whether the distribution
is approximately bell shaped. A formal statistical test (Kolmogorov-Smirnoff
test, not explained in this book) can be used to test whether the distribution
of the data differs significantly from a Gaussian distribution. With
few data points, it is difficult to tell whether the data are Gaussian
by inspection, and the formal test has little power to discriminate
between Gaussian and non-Gaussian distributions.
CHOOSING BETWEEN PARAMETRIC AND NONPARAMETRIC TESTS: DOES IT MATTER?
Does it matter whether you choose a parametric or nonparametric test? The answer depends on sample size. There are four cases to think about:
• Large sample.
What happens when you use a parametric test with data from a nongaussian
population? The central limit theorem (discussed in Chapter 5) ensures
that parametric tests work well with large samples even if the population
is non-Gaussian. In other words, parametric tests are robust to deviations
from Gaussian distributions, so long as the samples are large. The snag
is that it is impossible to say how large is large enough, as it depends
on the nature of the particular non-Gaussian distribution. Unless the
population distribution is really weird, you are probably safe choosing
a parametric test when there are at least two dozen data points in each
Thus, large data sets present no problems. It is usually easy to tell if the data come from a Gaussian population, but it doesn't really matter because the nonparametric tests are so powerful and the parametric tests are so robust. Small data sets present a dilemma. It is difficult to tell if the data come from a Gaussian population, but it matters a lot. The nonparametric tests are not powerful and the parametric tests are not robust.
ONE- OR TWO-SIDED P VALUE?
With many tests, you must choose whether you wish to calculate a one- or two-sided P value (same as one- or two-tailed P value). The difference between one- and two-sided P values was discussed in Chapter 10. Let's review the difference in the context of a t test. The P value is calculated for the null hypothesis that the two population means are equal, and any discrepancy between the two sample means is due to chance. If this null hypothesis is true, the one-sided P value is the probability that two sample means would differ as much as was observed (or further) in the direction specified by the hypothesis just by chance, even though the means of the overall populations are actually equal. The two-sided P value also includes the probability that the sample means would differ that much in the opposite direction (i.e., the other group has the larger mean). The two-sided P value is twice the one-sided P value.
A one-sided P value is appropriate when you can state with certainty (and before collecting any data) that there either will be no difference between the means or that the difference will go in a direction you can specify in advance (i.e., you have specified which group will have the larger mean). If you cannot specify the direction of any difference before collecting data, then a two-sided P value is more appropriate. If in doubt, select a two-sided P value.
If you select a one-sided test, you should do so before collecting any data and you need to state the direction of your experimental hypothesis. If the data go the other way, you must be willing to attribute that difference (or association or correlation) to chance, no matter how striking the data. If you would be intrigued, even a little, by data that goes in the "wrong" direction, then you should use a two-sided P value. For reasons discussed in Chapter 10, I recommend that you always calculate a two-sided P value.
PAIRED OR UNPAIRED TEST?
When comparing two groups, you need to decide whether to use a paired test. When comparing three or more groups, the term paired is not apt and the term repeated measures is used instead.
Use an unpaired test to compare groups when the individual values are not paired or matched with one another. Select a paired or repeated-measures test when values represent repeated measurements on one subject (before and after an intervention) or measurements on matched subjects. The paired or repeated-measures tests are also appropriate for repeated laboratory experiments run at different times, each with its own control.
You should select a paired test when values in one group are more closely correlated with a specific value in the other group than with random values in the other group. It is only appropriate to select a paired test when the subjects were matched or paired before the data were collected. You cannot base the pairing on the data you are analyzing.
FISHER'S TEST OR THE CHI-SQUARE TEST?
When analyzing contingency tables with two rows and two columns, you can use either Fisher's exact test or the chi-square test. The Fisher's test is the best choice as it always gives the exact P value. The chi-square test is simpler to calculate but yields only an approximate P value. If a computer is doing the calculations, you should choose Fisher's test unless you prefer the familiarity of the chi-square test. You should definitely avoid the chi-square test when the numbers in the contingency table are very small (any number less than about six). When the numbers are larger, the P values reported by the chi-square and Fisher's test will he very similar.
The chi-square test calculates approximate P values, and the Yates' continuity correction is designed to make the approximation better. Without the Yates' correction, the P values are too low. However, the correction goes too far, and the resulting P value is too high. Statisticians give different recommendations regarding Yates' correction. With large sample sizes, the Yates' correction makes little difference. If you select Fisher's test, the P value is exact and Yates' correction is not needed and is not available.
REGRESSION OR CORRELATION?
Linear regression and correlation are similar and easily confused. In some situations it makes sense to perform both calculations. Calculate linear correlation if you measured both X and Y in each subject and wish to quantity how well they are associated. Select the Pearson (parametric) correlation coefficient if you can assume that both X and Y are sampled from Gaussian populations. Otherwise choose the Spearman nonparametric correlation coefficient. Don't calculate the correlation coefficient (or its confidence interval) if you manipulated the X variable.
Calculate linear regressions only if one of the variables (X) is likely to precede or cause the other variable (Y). Definitely choose linear regression if you manipulated the X variable. It makes a big difference which variable is called X and which is called Y, as linear regression calculations are not symmetrical with respect to X and Y. If you swap the two variables, you will obtain a different regression line. In contrast, linear correlation calculations are symmetrical with respect to X and Y. If you swap the labels X and Y, you will still get the same correlation coefficient.