Part VI: NonParametric Analogues
Overview of Non-Parametric Tests
| Why NonParametric Tests? | NonParametric Analogues |
 
 
Up until now, we have been primarily discussing parametric tests; that is, tests that hold certain assumptions about when they are valid. Notice that t-tests and ANOVA both had assumptions regarding the shape of the distribution (normal) and about the necessity of having similar groups (homogeneity of variance). These are part of a whole set of parametric assumptions.

Under these parametric assumptions, it was possible to devise fairly simple sample statistics (such as the mean and the standard deviation) that could accurately estimate population parameters. Furthermore, by holding these assumptions we could also designate a set of common, standard sampling distributions. This, in turn, allowed us to obtain probabilities regarding our null hypotheses quickly.

However, it is not always possible to uphold these assumptions and obtain accurate probabilities regarding our hypotheses. In these situations, it is necessary to turn to tests that do not have such stringent assumptions--non-parametric or "distribution-free" tests. Specifically, there are three cases which necessitate the use of non-parametric tests:

  • The data for the dependent variable is not interval. Usually, non-interval data exists in the form of ranks or dichotomized scores. Interval level data, as you may recall, is an absolute necessity for the use of means and standard deviations, and therefore, parametric tests.
  • The distribution of the data for the dependent variable is highly skewed. Recall that a relatively normal distribution is assumed for parametric tests.
  • There exists severely unequal variances between groups. This is obviously a violation of the homogeneity of variance assumption for parametric tests.

In the last two cases, we have interval level data, but it violates our parametric assumptions. Therefore, we no longer treat this data as interval, but as ordinal. In a sense, we demote it because it fails to meet specific assumptions.
 
 
Most parametric tests have their non-parametric analogues. In other words, non-parametric tests exist for most situations you commonly test: 2 independent groups, 2 matched groups, and multiple groups. The primary difference is that the data is no longer interval; instead it is ordinal (or is treated as ordinal). Below is a chart comparing parametric tests with their analogues:

Parametric Test

Non-Parametric Tests
Independent t-Test (df = N-2)

(X=0,1; Y=interval & normal)

 Mann-Whitney (X=0,1; Y=ordinal)

Median (X=0,1; Y=0,1)
Matched Pairs t (df = n-1)

(X=0,1 & matched; Y=interval & normal)

 Wilcoxon (X=0,1 & matched;Y=ordinal)

Sign Test (X=0,1 & matched; Y=0,1)
One-Way ANOVA

(X=1,2,3,...k; Y=interval & normal)

 Kruskall-Wallis(X=1,2,3,..n;Y=ordinal)

While non-parametric tests make fewer assumptions regarding the nature of distributions, they are usually less powerful than their parametric counterparts. However, in cases where assumptions are violated and interval data is treated as ordinal, not only are non-parametric tests more proper, they can also be more powerful.