Chapter 13

Analysis of Variance ANOVA

A hypothesis-testing procedure that is used to evaluate mean differences between two or more treatments (or populations)

Scheffe test

A test that uses an F-ratio to evaluate the significance of the difference between any two treatment conditions. One of the safest of all possible post hoc tests.

What happens to the value of the F-ratio if differences between treatments are increased?

As differences between treatments increase, the F-ratio will increase.

interaction

Mean differences that cannot be explained by the main effects of the two factors. An interaction exits when the effects of one factor depend on the levels of the second factor.

Describe the distribution with large df values?

nearly all the F-ratios will be clustered very near to 1.00

Within-treatments variance

the difference that exist inside each treatment condition.

Two possible explanations for the difference or variance that exists between treatments.

1. Systematic Differences caused by treatments. 2. Random, unsystematic differences: individual differences and experimental error.

Eta squared (n^2)

A measure of effect size based on the percentage of variance accounted for by the sample mean differences.

two-factor design

A research study examining two factors (two independent or quasi-independent variables).

ANOVA summary table

A table that shows the source of variability (between treatments, within treatments, and total variability), SS, df, MS, and F.

Matrix

A two-dimensional table is a matrix and each box in the table is called a cell.

Experimentiwise alpha level

The risk of a Type 1 error that accumulates as you do more and more separate tests.

Tukey's HSD test

A test that allows you to compute a single value that determines the minimum difference between treatment means that is necessary for significance. A commonly used post hoc test.

Post Hoc Tests

A test that is conducted after an ANOVA with more than two treatment conditions where the null hypothesis was rejected. The purpose of post hoc tests is to determine exactly which treatment conditions are significantly different.

ANOVA is statistical procedure that compares two or more treatment conditions for differences in variance. (True/False)

False. Although ANOVA uses variance in the computations, the purpose of the test to evaluate differences in means between treatments.

What does the numerator of the F-ratio measure?

Measures between-treatment variance and always includes the same unsystematic variability of the error term in the denominator and any systematic differences caused by the treatment effect. The actual difference obtained from the sample data.

Testwise alpha level

Systematic differences that are caused by changing treatment conditions.

between-subjects variance

The differences that exist from one subject to another.

Main effect

The overall mean differences between the levels of one factor. When the data are organized in a matrix, the main effects are the mean differences among the rows (or the columns).

F-ratio

The test statistic for analysis of variance is call an F-ratio and compares the differences (variance) between treatments with the differences (variance) that are expected by chance.

In ANOVA , the total variability is partitioned into two parts. What are these two variability components called?

The two components are between-treatments variability and within-treatment variability.

In ANOVA what value is expected, on the average, for the F-ratio when the null Hypothesis is true?

When the null hypothesis is true, the expected value for the F-ratio is 1.00 because the top and bottom of the ratio are both measuring the same variance.

Describe the distribution of F-ratios

the distribution is cut off at zero (all positive values), piles up around 1.00, and then tapers off to the right.

individual differences

the naturally occurring differences from one individual to another that may cause the individuals to have different scores.

distribution of F-ratios

All of the possible F values when Ho is true.

What happens to the F-ratio if variability within treatments is increased?

As variability within treatments increases, the F-ratio will decrease.

In ANOVA , the total variability is partitioned into two parts. How are they used in the F-ratio?

Between-treatments variance is the numerator of the F-ratio, and within-treatments variance is the denominator.

Levels

In an experiment, the different values of the independent variable selected to create and define the treatment conditions. In other research studies, the different values of a factor.

mean square (MS)

In analysis of variance, a sample variance is called a mean square MS, indicating that variance measures the mean of squared deviations.

Factor

In analysis of variance, an independent variable (or quasi-independent variable) is called a factor.

What does the denominator of the F-ration measure?

Measures within-treatments variance and only random and unsystematic variability also called the error term. The difference that would be expected if there were no treatment effect.

What does the exact shape of the F distribution depend on?

The exact shape of the F distribution depends on the degrees of freedom for two variances in the F-ratio, the MS (variances) between the numerator and the denominator of the F-ratio.

Pairwise Comparisons

To go back through the data and compare the individual treatments two at a time.

error variance

Unexplained, unsystematic differences that are not caused by any known factor.

between-treatments variance

Values used to measure and describe the differences between treatments (mean differences).

cells

a two-dimensional table is a matrix and each box in the table is called a cell

error term

for ANOVA, the denominator of the F-ratio is called the error term. The error term provides a measure of the variance caused by random unsystematic differences. When treatment effect is zero (H0 is true), the error term measures the same sources of variance as the numerator of the F-ratio, so the value of the F-ratio is expected to be nearly equal to 1.00

Describe the distribution with smaller df values?

the F-distribution is more spread out.