## Spearman's Rank-Order Correlation ..

### The Spearman correlation coefficient, ..

Interestingly, there is considerable debate, even among statisticians, regarding the appropriate use of one- versus two-tailed -tests. Some argue that because in reality no two population means are ever identical, that all tests should be one tailed, as one mean must in fact be larger (or smaller) than the other (). Put another way, the null hypothesis of a two-tailed test is always a false premise. Others encourage standard use of the two-tailed test largely on the basis of its being more conservative. Namely, the -value will always be higher, and therefore fewer false-positive results will be reported. In addition, two-tailed tests impose no preconceived bias as to the direction of the change, which in some cases could be arbitrary or based on a misconception. A universally held rule is that one should never make the choice of a one-tailed -test ^{} after determining which direction is suggested by your data In other words, if you are hoping to see a difference and your two-tailed -value is 0.06, don't then decide that you really intended to do a one-tailed test to reduce the -value to 0.03. Alternatively, if you were hoping for no significant difference, choosing the one-tailed test that happens to give you the highest -value is an equally unacceptable practice.

### How do you express the null hypothesis for this test?

Most importantly, the -value for this test will answer the question: If the null hypothesis is true, what is the probability that the following result could have occurred by chance sampling?

Of course, our experimental result that GFP^{wt} was greater than GFP^{mut } clearly fails to support this research hypothesis. In such cases, there would be no reason to proceed further with a -test, as the -value in such situations is guaranteed to be >0.5. Nevertheless, for the sake of completeness, we can write out the null hypothesis as

## Tests of Pearson's Correlation (1 of 6) - David Lane

One aspect of the -test that tends to agitate users is the obligation to choose either the one or two-tailed versions of the test. That the term “tails” is not particularly informative only exacerbates the matter. The key difference between the one- and two-tailed versions comes down to the formal statistical question being posed. Namely, the difference lies in the wording of the research question. To illustrate this point, we will start by applying a two-tailed -test to our example of embryonic GFP expression. In this situation, our typical goal as scientists would be to detect a difference between the two means. This aspiration can be more formally stated in the form of a or . Namely, that the average expression levels of ::GFP in wild type and in mutant are different. The must convey the opposite sentiment. For the two-tailed -test, the null hypothesis is simply that the expression of ::GFP in wild type and mutant backgrounds is the same. Alternatively, one could state that the difference in expression levels between wild type and mutant is zero.

## Tests of Pearson's Correlation ..

The Central Limit Theorem having come to our rescue, we can now set aside the caveat that the populations shown in are non-normal and proceed with our analysis. From we can see that the center of the theoretical distribution (black line) is 11.29, which is the actual difference we observed in our experiment. Furthermore, we can see that on either side of this center point, there is a decreasing likelihood that substantially higher or lower values will be observed. The vertical blue lines show the positions of one and two SDs from the apex of the curve, which in this case could also be referred to as SEDMs. As with other SDs, roughly 95% of the area under the curve is contained within two SDs. This means that in 95 out of 100 experiments, we would expect to obtain differences of means that were between “8.5” and “14.0” fluorescence units. In fact, this statement amounts to a 95% CI for the difference between the means, which is a useful measure and amenable to straightforward interpretation. Moreover, because the 95% CI of the difference in means does not include zero, this implies that the -value for the difference must be less than 0.05 (i.e., that the null hypothesis of no difference in means is not true). Conversely, had the 95% CI included zero, then we would already know that the -value will not support conclusions of a difference based on the conventional cutoff (assuming application of the two-tailed -test; see below).

## The sampling distribution of Pearson's r is normal only if ..

The rationale behind using the paired -test is that it takes meaningfully linked data into account when calculating the -value. The paired -test works by first calculating the difference between each individual pair. Then a mean and variance are calculated for all the differences among the pairs. Finally, a one-sample -test is carried out where the null hypothesis is that the mean of the differences is equal to zero. Furthermore, the paired -test can be one- or two-tailed, and arguments for either are similar to those for two independent means. Of course, standard programs will do all of this for you, so the inner workings are effectively invisible. Given the enhanced power of the paired -test to detect differences, it is often worth considering how the statistical analysis will be carried out at the stage when you are developing your experimental design. Then, if it's feasible, you can design the experiment to take advantage of the paired -test method.