STATS test 4 review

Interaction

An interaction between two factors occurs whenever the mean differences between individual treatment conditions, or cells, are different from what would be predicted from the overall main effects of the factors.

ANOVA

Analysis of Variance is a statistical technique that is used to test the significance of mean differences among two or more treatment conditions.

Factor

In ANOVA, the variable (independent or quasi-independent) that designates the groups being compared.

ANOVA summary tables

It is useful to organize the results of the analysis in one table called the ANOVA summary table. The table shows the source of variability (between treatments, within treatments, and total variability), SS, df, MS, and F.

Level

The individual conditions or values that make up a factor are called the levels of the factor.

Testwise alpha level

The risk of Type I error, or alpha level, for an individual hypothesis test.

Mean square (MS)

The variance between treatments and the variance within treatments.

Within-Treatments Variance

The within-treatments variance provides a measure of the variability inside each treatment condition.

F-ratio

Using variance to measure sample mean differences when there are two or more samples. F = variance (differences) between sample means/variance (differences) expected with no treatment effect F = MS(between)/MS(within)

When you reject the null and the experiment contains more than two treatment conditions

it is necessary to continue the analysis with a post hoc test such as Tukey's HSD test or the Scheffe test. The purpose of these tests is to determine exactly which treatments are significantly different and which are not.

Between-subjects variance

measure the size of individual differences

The two-factor ANOVA

produces three F-ratios: one for factor A, one for factor B, and one for the A x B interaction. Each F-ratio has the same basic structure: F(A) = MS(A)/MS(within) F(B) = MS(B)/MS(within) F(AxB) = MS(AxB)/MS(within)

When there is no treatment effect (H0 is true)

the numerator and the denominator of the F-ratio are measuring the same variance, and the obtained ratio should be near 1.00. If there is a significant treatment effect, then the numerator of the ratio should be larger than the denominator, and the obtained F value should be much greater than 1.00.

Matrix

two factors are used to create a matrix with one factor defining the rows and the other factor defining the columns

Post hoc tests

Additional hypothesis tests that are done after an ANOVA to determine exactly which mean differences are significant and which are not.

Distribution of F-ratios

All of the possible F values when H0 is true.

Tukey's HSD Test

Allows you to compute a single value that determines the minimum difference between treatment means that is necessary for significance.

Independent-measures ANOVA

F = MS(between)/MS(within)

The F-ratio has the following structure:

F = treatment effect + random unsystematic differences/ random unsystematic differences

The test statistic for ANOVA

F-ratio

Error term

For ANOVA, this is the denominator of the F-ratio. The error term provides a measure of variance caused by random, unsystematic differences. When the 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.

Scheffe Test

Has the smallest possible risk of Type I error. Uses the F-ratio to evaluate the significance of the difference between any two treatment conditions.

Analysis of Variance (ANOVA) text book definition

Hypothesis testing procedure that is used to evaluate mean difference between two or more treatments (or populations).

Individual differences

Participant characteristics such as age, personality and gender that vary from one person to another and may influence the measures you obtain for each person. These are present in the independent-measure F-ratio but are eliminated in the repeated-measures F-ratio.

Treatment Effects

The differences between treatments not caused by sampling error.

Main effect

The mean differences among the levels of one factor are referred to as the main effect of that factor. When design of the research study is represented as a matrix with one factor determining the columns and the other determining the rows, then the mean differences among the rows describe the main effect of one factor, and the mean differences among the columns describe the main effect of the other factor.

Eta squared (n^2)

The percentage of variance accounted for by the treatment effect. The method for measuring treatment effect. n^2 = SS(betweent)/SS(total)

Error variance

The remaining variance in the denominator after the between subjects variance is subtracted out.

Pairwise Comparisons

Using post hoc tests to go back through the data and compare the individual treatments two at a time.

Between-Treatments Variance

We calculate the variance between treatments to provide a measure of the overall differences between treatment conditions. The variance between treatments is really measuring the difference between sample means.

Experimentwise alpha level

When an experiment involves several different hypothesis tests, the experimentwise alpha level is the total probability of a Type I error that is accumulated from all the individual tests in the experiment. Typically the experimentwise alpha level is substantially greater than the value of alphas used for any one of the individual tests.

Two-factor design

a research design that has two independent variables

Cell

a single box within the matrix of four boxes

The second stage of the repeated-measures analysis

individual differences are computed and removed from the denominator of the F-ratio. To remove the individual differences, you first compute the variability between subjects (SS and df) and then subtract these values from the corresponding within-in treatment values. The residual provides a measure of error excluding individual differences, which is the appropriate denominator for the repeated-measures ANOVA: n^2 = SS(between treatments)/SS(total)-SS(between subjects) = SS(between treatments)/SS(between treatments)+SS(error) Because part of the variability (the SS caused by individual differences) is removed before computing n^2, this measure of effect size is often called a partial eta squared.

A research study with two independent variables

is called a two-factor design. Such a design can be diagrammed as a matrix with the levels of one factor defining the rows and the levels of the other factor defining the columns. Each cell in the matrix corresponds to a specific combination of the two factors.

The first stage of the repeated-measures ANOVA

is identical to the independent-measures ANOVA and separates the total variability into two components: between treatments and within treatments. Because a repeated-measures design uses the same subjects in every treatment condition, the difference between treatments cannot be caused by individual differences. Thus individual differences are automatically eliminated from the between treatments variance in the numerator of the F-ratio.

Effect size for the independent-measures ANOVA

is measured by computing eta squared, the n^2 = SS(between)/SS(between) + SS(within) = SS(between)/SS(total)

The purpose of the ANOVA

is to determine whether there are any significant mean differences among the treatments or cells in the experimental matrix. These treatment effects are classified as follows: a. The A-effect: Overall mean differences among the levels of factor A. b. The B-effect: Overall mean differences among the levels of factor B. c. The A x B interaction: Extra mean differences that are not accounted for by the main effects.

The repeated-measures ANOVA

is used to evaluate the mean differences obtained in a research study comparing two or more treatment conditions using the same sample of individuals in each condition. The test statistic is an F-ratio, in which the numerator measures the variance (differences) between treatment and the denominator measures the variance (differences) that is expected without any treatment effects or individual differences. F = MS(between treatments)/MS(error)

MS between

measures differences between the treatments by computing the variance for the treatment means or totals These differences are assumed to be produced by: a. Treatment Effects (if they exist) b. Random, unsystematic differences (chance)

MS within

measures variance inside each of the treatment conditions. Because individuals inside a treatment condition are all treated exactly the same, any differences within treatments cannot be caused by treatment effects. Thus, the within-treatments MS is produced only by random, unsystematic differences.

Null hypothesis for ANOVA

there are no mean differences among the treatments in the general population (u1 = u2 = u3 = etc.)

Alternative hypothesis for ANOVA

there is at least one mean different from another (not H0)

What the F-ratio has

two values for degrees of freedom, one associated with the MS in the numerator and one associated with the MS in the denominator. These df values are used to find the critical value for the F-ratio in the F distribution table.