One-Way ANOVA Tests

The NPAR1WAY Procedure

One-Way ANOVA Tests

PROC NPAR1WAY computes a one-way ANOVA test for each score type that you specify. Under the null hypothesis of no difference among class levels (or samples), this test statistic has an asymptotic chi-square distribution with r-1 degrees of freedom, where r is the number of class levels. For Wilcoxon scores, this test is known as the Kruskal-Wallis test.

Exact one-way ANOVA tests are also available for multisample data (where the data are classified into more than two levels). For two-sample data, exact simple linear rank tests are available. PROC NPAR1WAY computes exact tests for each score type that you specify in the EXACT statement. See the "Exact Tests" section for details on exact tests.

PROC NPAR1WAY computes the one-way ANOVA test statistic as

$C = ( \sum_{i=1}^r (T_i - E_0(T_i))^2 / n_i ) / S^2$

where T_i is the total of scores for the class level i, E₀(T_i) is the expected total for level i under the null hypothesis of no difference among levels, n_i is the number of observations in level i, and S² is the sample variance of the scores.

$T_i = \sum_{j=1}^n c_{ij} a(R_j)$

where a(R_j) is the score for observation j, and c_ij indicates whether observation j is in level i.

$E_0(T_i) = n_i \bar{a}$

where $\bar{a}$ is the average score,

$\bar{a} = \sum_{j=1}^n a(R_j) / n$

$S^2 = \sum_{j=1}^n (a(R_j) - \bar{a})^2 / (n-1)$

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