Suppose that scores given by judges to competitors in the...

Suppose that scores given by judges to competitors in the ski-jumping events of the Winter Olympics were analyzed. For the men's ski-jumping competition, suppose there were 22 contestants and 9 judges. Each judge in seven subevents assessed each contestant. The scores given can, thus, be treated in the framework of a two-way analysis of variance with 198 contestant-judge cells, seven observations per cell. The sums of squares are given in the following table: $$ \begin{array}{lc} \hline \text { Source of Variation } & \text { Sum of Squares } \\ \hline \text { Between contestants } & 364.50 \\ \text { Between judges } & 0.81 \\ \text { Interaction } & 4.94 \\ \text { Error } & 1,069.94 \\ \hline \end{array} $$ a. Complete the analysis of variance table. b. Carry out the associated $F$ tests and interpret your findings.

Suppose that scores given by judges to competitors in the ski-jumping events of the Winter Olympics were analyzed. For the men's ski-jumping competition, suppose there were 22 contestants and 9 judges. Each judge in seven subevents assessed each contestant. The scores given can, thus, be treated in the framework of a two-way analysis of variance with 198 contestant-judge cells, seven observations per cell. The sums of squares are given in the following table:
$$
\begin{array}{lc}
\hline \text { Source of Variation } & \text { Sum of Squares } \\
\hline \text { Between contestants } & 364.50 \\
\text { Between judges } & 0.81 \\
\text { Interaction } & 4.94 \\
\text { Error } & 1,069.94 \\
\hline
\end{array}
$$
a. Complete the analysis of variance table.
b. Carry out the associated $F$ tests and interpret your findings.

Key Concepts

F-Tests

F-tests in the two?way ANOVA context are used to determine whether the variances explained by the main effects (and the interaction) are significantly larger than what would be expected by chance, relative to the error variance. By comparing the mean square for a given effect with the mean square error using the F-distribution, one can assess the statistical significance of that effect, testing hypotheses such as the equality of group means across the levels of a factor.

Interaction Effects

The interaction effect in two-way ANOVA indicates whether the effect of one factor depends on the level of the other factor. A significant interaction suggests that the influence of one independent variable is not consistent across the levels of the other variable, which can complicate the interpretation of the main effects and often provides deeper insights into the underlying patterns in the data.

Mean Squares Calculation

Mean squares are derived by dividing each sum of squares by its corresponding degrees of freedom. They provide an estimate of the variance attributable to each source of variation, making them a critical step in the ANOVA table. The ratio of these mean squares, for the factors or interaction relative to the error mean square, forms the basis of the F-tests.

Degrees of Freedom

Degrees of freedom (df) describe the number of independent values that can vary in an analysis without breaking any constraints. In the context of two-way ANOVA, separate degrees of freedom are calculated for each main effect (for each factor, df = number of levels minus one), for the interaction (the product of the individual dfs), and for the error (calculated typically as the total number of observations minus the number of parameters estimated). This partitioning is essential both for computing mean squares and performing F-tests.

Two-Way Analysis of Variance

This is a statistical method used to assess the effect of two categorical independent variables (factors) on a continuous outcome variable. In such an analysis, the total variability in the data is partitioned among the main effects of the two factors, their interaction effect, and random error. It allows researchers to examine not only the individual impact of each factor but also whether the effect of one factor depends on the level of the other.

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