Chapter 25: Analysis of Variance Flashcards
Define ‘Mean square treatment (or between) (MSt)’.
Another estimate of the error variance -under the assumption that the treatment means are all equal. If the (null) assumption is not true, the MSt will tend to be larger than the error variance.
Define ‘F-distribution’.
The sampling distribution of the F-statistic when the null hypothesis that the treatment means are equal is true. It has two degrees of freedom parameters, one for the numerator, k-1, and one for the denominator, N-k, where N is the total number of observations and k is the number of groups.
Define ‘ANOVA’.
An analysis method for testing equality of means across treatment groups.
Define ‘Mean square error (or within) (MSe)’.
The estimate of the error variance obtained by pooling the variances of each treatment group. The square root of the MSe is the estimate of the error standard deviation, Sp.
Define ‘ANOVA table’.
A table that shows the degrees of freedom, the mean square treatment, the mean square error, their ratio, the F-stat, and its P-value. Other quantities of lesser interest are usually included as well.
Define ‘F-stat’.
The ratio of MSt/MSe. When the F-stat is sufficiently large, we reject the null hypothesis that the group means are equal.
Define ‘ANOVA model’.
The model for a 1-way (one response, one factor) ANOVA is
Yij = μj + ϵij
Estimating above with Yij = Y-bar j + eij gives predicted values Y-hat ij = Y-bar j and residuals eij = Yij - Y-bar j.
Define ‘Residual standard dev’.
Gives an idea of the underlying variability of the response values:
Sp = sqrt(MSe) = sqrt(sum (e^2) / (N-k)).
Define ‘Assumptions for ANOVA (and conditions to check)’.
- Equal Variance Assumption (Similar Variance Condition). Look at side-by side boxplots to check for similar spreads, or look at residuals verses predicted to see if the plot thickens.
- Normal Population Assumption (Nearly Normal Condition)/ Check a histogram or Normal prob. plot of the residuals.
- Independence Assumption.
Define ‘F-test’.
Tests the null hypothesis that all the group means are equal against the alternative that they are not all equal. We reject the hypothesis of equal means if the F-stat exceeds the critical value from the F-distribution corresponding to the specified significance level and degrees of freedom.
Define ‘Balance’.
An experiment’s design is balanced if each treatment level has the same number of experimental units. Balanced designs make calculations simpler and are generally more powerful.
Define ‘Multiple comparisons’.
If we reject the null hypothesis of equal means, we often then want to investigate further and compare pairs of treatment group means to see if they differ. If we want to test several such pairs, we must adjust for performing several tests to keep the overall risk of a Type I error from growing too large. Such adjustments are called methods for multiple comparisons.
Define ‘Least sign. difference (LSD)’.
The standard margin of error in the confidence interval for the difference of two means. It has the correct Type I error rate for a single test, but not when performing more than one comparison.
How are the F-test and the t-test related? (Similar to chi-square and z-test)
For two groups, F-test stat is squared t-test stat?
