fsml_anova_1way Interface

public interface fsml_anova_1way

The one-way ANOVA (Analysis of Variance) tests whether three or more population means $\mu_1, \mu_2, \dots, \mu_k$ are equal.

Hypotheses:

The null hypothesis $H_0$ and alternative hypothesis $H_1$ are defined as: $H_0$: $\mu_1 = \mu_2 = \cdots = \mu_k$, and $H_1$: At least one $\mu_j$ differs from the others.

Procedure:

The data is passed to the procedure as a rank-2 array x, where each column is a group of observations. The procedure partitions then the total variability in the data ( $SS_{total}$ ) into variability between groups ( $SS_{between}$; variability explained by groups ), and variability within groups ( $SS_{within}$; unexplained or residual variability ), so that $$SS_{total} = SS_{between} + SS_{within}$$

The F-statistic (f) is the ratio of the mean sum of squares between groups to the mean sum of squares within groups: $$F = \frac{SS_{between} / (k - 1)}{SS_{within} / (n - k)}$$ where $k$ is the number of groups, $n$ is the total number of observations, $SS_{between}$ is the sum of squares between groups, and $SS_{within}$ is the sum of squares within groups.

The degrees of freedom are $\nu_1 = k - 1$ between groups (df_b) and $\nu_2 = n - k$ within groups (df_w).

The resulting p-value (p) is computed from the F-distribution: $$p = P(F_{\nu_1, \nu_2} > F_{observed}) = 1 - \text{CDF}(F_{observed})$$

It is computed with the elemental procedure f_dst_f_cdf_core.

The ANOVA makes the assumptions that a) the groups are independent, b) the observations within each group are normally distributed, and c) The variances within groups are equal.

Module Procedures