L5 ANOVA Derivations

What do degrees of freedom (df) represent in statistical calculations?

The number of independent values that can vary after constraints have been imposed.

In the paint mix example, if Red = 4ml and Blue = 3ml, how much Green is needed to make 10ml?

Green must be 3ml, indicating df = 2.

How is the within-group degrees of freedom (df_R) calculated?

df_R = ∑(s_i - 1) for each group, where s_i is the number of subjects in each group.

What is the formula for between-group degrees of freedom (df_A)?

df_A = a - 1, where a is the number of groups.

How is total degrees of freedom (df_T) calculated?

df_T = a * s - 1, where a is the number of groups and s is the number of subjects per group.

What does the F-ratio in ANOVA represent?

The ratio of variance between groups to variance within groups.

What indicates a treatment effect in the context of the F-ratio?

If F-ratio > 1, it suggests a treatment effect; if F-ratio = 1, no treatment effect is indicated.

What are the key properties of the F-distribution?

Always positive, shape depends on numerator and denominator degrees of freedom, more skewed with smaller df.

What is the critical F-value used for in hypothesis testing?

It is compared to the calculated F-ratio to determine whether to reject or fail to reject the null hypothesis (H0).

What are the three assumptions that must be met for ANOVA?

Independence of observations, normality of data within groups, and homogeneity of variance across groups.

How can homogeneity of variance be tested?

Using Bartlett's test or Levene's test.

What are common tests for normality in ANOVA?

Skewness, Kolmogorov-Smirnov test, and Shapiro-Wilk test.

What does a skewness value indicate?

It indicates the asymmetry of the distribution; a value of zero indicates a normal distribution.

How is skewness tested for significance?

Using a Z-test where z = γ / SE_γ, with SE_γ calculated as √(6/N).

What is the purpose of data transformations in ANOVA?

To meet assumptions of normality or homogeneity of variance while maintaining the rank order of data.

What transformation is recommended for positive skewness?

Square root, logarithm, or reciprocal transformations depending on the severity of skewness.

What is the effect of data transformation on Type II error?

It increases statistical power, reducing the chance of missing a true effect.

What is the significance of the ANOVA summary table?

It summarizes the sources of variation, sum of squares, degrees of freedom, mean square error, F-ratio, and significance level.

What does it mean if the significance value (p-value) is less than 0.05 in ANOVA?

It indicates that the results are statistically significant, leading to rejection of the null hypothesis.

What is the null hypothesis (H0) in the context of ANOVA?

H0 states that there are no differences between group means.

What is the formula for calculating the F-ratio?

F-ratio = (Variance between groups) / (Variance within groups).

What does a significant main effect in ANOVA indicate?

It suggests that at least one group mean is different from the others.

What is the relationship between the F-ratio and critical F-value in hypothesis testing?

If the F-ratio exceeds the critical F-value, we reject the null hypothesis.

What does a large F-ratio suggest about the treatment effect?

It suggests that the treatment effect is likely to exist and is not due to random chance.