L6: Family Wise Error & Comparison

What does the F-value indicate in statistical analysis?

The F-value indicates if there is a difference somewhere among the means, but does not specify which levels differ.

What is required after obtaining a significant omnibus F?

Further analysis is required to determine which specific groups differ.

What are the two types of analytical comparisons?

Planned (a priori) comparisons and Post-hoc (a posteriori) comparisons.

What is a planned comparison?

A comparison made based on direct predictions made before the experiment, such as expecting one teaching style to be more effective.

What is a post-hoc comparison?

A comparison made without prior expectations, conducted after the experiment to explore unexpected differences.

What does the Greek letter Ψ (psi) represent in contrasts?

Ψ represents the contrast defined as the difference between group means, such as Ψ=μ_1−μ_2.

What is the design matrix in statistical analysis?

The design matrix consists of coefficients that help combine group means for testing expected differences.

What is the significance of coefficients in the design matrix?

Coefficients indicate how to combine group means, with positive coefficients predicting higher outcomes and negative coefficients predicting lower outcomes.

How is the F-ratio calculated in a planned comparison?

The F-ratio is calculated using the mean squared error from the planned comparison divided by the within-group mean squared error.

What is the family-wise error rate (FWER)?

FWER is the probability of making at least one Type I error across multiple comparisons.

What is a Type I error?

A Type I error occurs when the null hypothesis is rejected when it is actually true, resulting in a false positive.

What is the Bonferroni correction?

A method to control the family-wise error rate by adjusting the significance level to α/m, where m is the number of comparisons.

What is the Scheffé Test used for?

The Scheffé Test adjusts the critical F-value based on the number of treatment levels, allowing for less conservative comparisons than Bonferroni.

What does the Tukey HSD Test compare?

The Tukey HSD Test compares the differences between group means and determines if they are significant based on a critical value from a Q table.

What is the impact of post-hoc tests on Type I and Type II errors?

Post-hoc tests reduce Type I errors but increase the chance of Type II errors, making it harder to detect real differences.

What does the term 'per-comparison error' refer to?

Per-comparison error refers to the risk of Type I error associated with individual statistical tests.

What is the purpose of adjusting error rates in post-hoc comparisons?

Adjusting error rates in post-hoc comparisons accounts for the increased risk of Type I errors when multiple tests are conducted.

What does the notation SS_(A1,A2) represent?

SS_(A1,A2) represents the sum of squares for a planned comparison between groups A1 and A2.

How is mean squared error (MSE) calculated for a planned comparison?

MSE is calculated by dividing the sum of squares for the comparison by the degrees of freedom, which is always 1 for planned comparisons.

What is the significance of the degrees of freedom in planned comparisons?

The degrees of freedom for a planned comparison is always 1, meaning that the mean squared error is equal to the sum of squares.

What is the formula for calculating the F-ratio in a planned comparison?

F_(A1,A2) = MS_(A1,A2) / MS_R, where MS_R is the within-group mean squared error.

What does the notation F(α=0.05) (1,12) represent?

It represents the critical F-value at a significance level of 0.05 with 1 and 12 degrees of freedom.

What is the result of a planned comparison that shows significant differences?

The result indicates that one group performed significantly better than another, supported by statistical evidence.

What is the role of statistical software in post-hoc tests?

Statistical software packages facilitate the execution of various post-hoc tests, providing options for different error rate adjustments.