ANOVA
Analysis of Variance is a statistical technique that is used to test the significance of mean differences among two or more treatment conditions.
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)
The test statistic for ANOVA
F-ratio
Independent-measures ANOVA
F = MS(between)/MS(within)
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 produc
The F-ratio has the following structure:
F = treatment effect + random unsystematic differences/ random unsystematic differences
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 obtain
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.
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)
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.
Analysis of Variance (ANOVA) text book definition
Hypothesis testing procedure that is used to evaluate mean difference between two or more treatments (or populations).
Factor
In ANOVA, the variable (independent or quasi-independent) that designates the groups being compared.
Level
The individual conditions or values that make up a factor are called the levels of the factor.
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)
Testwise alpha level
The risk of Type I error, or alpha level, for an individual hypothesis test.
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 subs
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.
Within-Treatments Variance
The within-treatments variance provides a measure of the variability inside each treatment condition.
Treatment Effects
The differences between treatments not caused by sampling error.
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 numera
Mean square (MS)
The variance between treatments and the variance within treatments.
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.
Distribution of F-ratios
All of the possible F values when H0 is true.
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)
Post hoc tests
Additional hypothesis tests that are done after an ANOVA to determine exactly which mean differences are significant and which are not.
Pairwise Comparisons
Using post hoc tests to go back through the data and compare the individual treatments two at a time.
Tukey's HSD Test
Allows you to compute a single value that determines the minimum difference between treatment means that is necessary for significance.
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.
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 (differ
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 betwee
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 treat
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 fa
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
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)
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-measure
Between-subjects variance
measure the size of individual differences
Error variance
The remaining variance in the denominator after the between subjects variance is subtracted out.
Two-factor design
a research design that has two independent variables
Matrix
two factors are used to create a matrix with one factor defining the rows and the other factor defining the columns
Cell
a single box within the matrix of four boxes
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 d
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.