20.2 Nonparametric ANOVA

20.2.1 Kruskal-Wallis

Generalization of independent samples Wilcoxon Rank sum test for 2 independent samples (like F-test of one-way ANOVA is a generalization to several independent maples of the two sample t-test)

Consider the one-way case:

We have

  • \(a\ge2\) treatments
  • \(n_i\) is the sample size for the ith treatment
  • \(Y_{ij}\) is the j-th observation from the ith treatment.
  • we make no assumption of normality
  • We only assume that observations on the ith treatment are a random sample from the continuous CDF \(F_i\), i = 1,..,n, and are mutually independent.

\[ H_0: F_1 = F_2 = ... = F_a \\ H_a: F_i < F_j \text{ for some } i \neq j \]

or if distribution is from the location-scale family, \(H_0: \theta_1 = \theta_2 = ... = \theta_a\))

Procedure

  • Rank all \(N = \sum_{i=1}^a n_i\) observations in ascending order. Let \(r_{ij} = rank(Y_{ij})\), note \(\sum_i \sum_j r_{ij} = 1 + 2 .. + N = \frac{N(N+1)}{2}\)
  • Calculate the rank sums and averages:
    \[ r_{i.} = \sum_{j=1}^{n_i} r_{ij} \] and \[ \bar{r}_{i.} = \frac{r_{i.}}{n_i}, i = 1,..,a \]
  • Calculate the test statistic on the ranks: \[ \chi_{KW}^2 = \frac{SSTR}{\frac{SSTO}{N-1}} \] where \(SSTR = \sum n_i (\bar{r}_{i.}- \bar{r}_{..})^2\) and \(SSTO = \sum \sum (\bar{r}_{ij}- \bar{r}_{..})^2\)
  • FOr large \(n_i\) (\(\ge 5\) observations) the Kruskal-Wallis statistic is approximated by a \(\chi^2_{a-1}\) distribution when all the treatment means are equal. Hence, reject \(H_0\) if \(\chi^2_{KW} > \chi^2_{(1-\alpha;a-1)}\).
  • If sample sizes are small, one can exhaustively work out all possible distinct ways of assigning N ranks to the observations from a treatments and calculate the value of the KW statistic in each case (\(\frac{N!}{n_1!..n_a!}\) possible combinations). Under \(H_0\) all of these assignments are equally likely.

20.2.2 Friedman Test

When the responses \(Y_{ij} = 1,..,n, j = 1,..,r\) in a randomized complete block design are not normally distributed (or do not have constant variance), a nonparametric test is more helpful.

A distribution-free rank-based test for comparing the treatments in this setting is the Friedman test. Let \(F_{ij}\) be the CDF of random \(Y_{ij}\), corresponding to the observed value \(y_{ij}\)

Under the null hypothesis, \(F_{ij}\) are identical for all treatments j separately for each block i.

\[ H_0: F_{i1} = F_{i2} = ... = F_{ir} \text{ for all i} \\ H_a: F_{ij} < F_{ij'} \text{ for some } j \neq j' \text{ for all } i \]

For location parameter distributions, treatment effects can be tested:

\[ H_0: \tau_1 = \tau_2 = ... = \tau_r \\ H_a: \tau_j > \tau_{j'} \text{ for some } j \neq j' \]

Procedure

  • Rank observations from the r treatments separately within each block (in ascending order; if ties, each tied observation is given the mean of ranks involved). Let the ranks be called \(r_{ij}\)
  • Calculate the Friedman test statistic
    \[ \chi^2_F = \frac{SSTR}{\frac{SSTR + SSE}{n(r-1)}} \] where \[ SSTR = n \sum (\bar{r}_{.j}-\bar{r}_{..})^2 \\ SSE = \sum \sum (r_{ij} - \bar{r}_{.j})^2 \\ \bar{r}_{.j} = \frac{\sum_i r_{ij}}{n}\\ \bar{r}_{..} = \frac{r+1}{2} \]

If there is no ties, it can be rewritten as

\[ \chi^2_{F} = [\frac{12}{nr(n+1)}\sum_j r_{.j}^2] - 3n(r+1) \]

with large number of blocks, \(\chi^2_F\) is approximately \(\chi^2_{r-1}\) under \(H_0\). Hence, we reject \(H_0\) if \(\chi^2_F > \chi^2_{(1-\alpha;r-1)}\)
The exact null distribution for \(\chi^2_F\) can be derived since there are r! possible ways of assigning ranks 1,2,…,r to the r observations within each block. There are n blocks and thus \((r!)^n\) possible assignments to the ranks, which are equally likely when \(H_0\) is true.