The question asks to identify the component representing random error variance in Analysis of Variance (ANOVA).

ANOVA Error Component Identification

In ANOVA, the primary goal is to partition the total variation in the data into different sources. The comparison of variances helps determine if group means are significantly different. The 'error' component specifically refers to the variability within the groups, which is assumed to be random.

Understanding Mean Squares in ANOVA

• Between Mean Square (MSB): This measures the variance between the different groups. It reflects differences between sample means. It is used in the numerator of the F-statistic.
• Within Mean Square (MSW): This measures the variance within each individual group (also known as error mean square or MSE). It represents the average variance attributable to random error after accounting for group differences. It is used in the denominator of the F-statistic.
• Interaction Mean Square (MSI): Used in two-way ANOVA and beyond, this measures the variance due to the interaction effect between two or more independent variables.
• Total Mean Square (MST): Represents the total variance in the data, combining both between-group and within-group variances.

Identifying the Error Component

The variability that cannot be explained by the independent variable(s) or their interactions is considered random error. In the context of ANOVA, this unexplained or random variation is captured by the variance calculated from the data points *within* each treatment group.

Therefore, the Within Mean Square (MSW), often referred to as the Mean Squared Error (MSE), represents the error component in ANOVA. It estimates the error variance (${\sigma^2}$) assuming the null hypothesis is true.

Conclusion: The Within Mean Square is the error component in ANOVA.