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Introduction to Analysis of Variance - Coggle Diagram
Introduction to Analysis of Variance
Introduction: An Overview of Analysis of Variance
Statistical Hypotheses for ANOVA
The null hypothesis says all treatment means are equal (no treatment effect).
The alternative hypothesis says at least one mean is different (there is a treatment effect).
The goal of ANOVA is to test whether three or more treatment groups differ in their population means.
Even though many specific differences are possible, ANOVA states the alternative in a general way rather than listing each possible pattern.
Type I Errors and Multiple-Hypothesis Tests
testwise alpha level: the risk of a 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 of the individual tests in the experiment. Typically, the experimentwise alpha level is substantially greater than the value of alpha used for any one of the individual tests.
Terminology in Analysis of Variance
factor: In analysis of variance, the variable (independent or quasi-independent) that designates the groups being compared
levels: The individual conditions or values that make up a factor
two-factor design or a factorial design: a study that combines two factors
single-factor designs: studies that have only one independent variable (or only one quasi-independent variable).
The Test Statistic for ANOVA
t = obtained difference between two sample means/standard error (the difference with no treatment effect)
The Logic of Analysis of Variance
Within-Treatments Variance
Within each treatment group, participants receive the same treatment, yet their scores still differ due to random, unsystematic variation rather than any real treatment effect.
This variability within treatments represents what we expect to see when the null hypothesis is true, and the within-treatments variance provides a measure of how large those random differences typically are.
The F-Ratio: The Test Statistic for ANOVA
F = variance between treatments/variance within treatments
= differences including any treatment effects/differences with no treatment effects
Between-Treatments Variance
There are two possible explanations for between-treatment differences:
The differences between treatments are not caused by any treatment effect but are simply the naturally occurring, random and unsystematic differences that exist between one sample and another. That is, the differences are the result of sampling error.
The differences between treatments have been caused by the treatment effects. For example, if treatments really do affect performance, then scores in one treatment should be systematically different from scores in another condition.
ANOVA Notation and Formulas
Analysis of Sum of Squares (SS)
Within-Treatments Sum of Squares
Between-Treatments Sum of Squares
Total Sum of Squares
The Analysis of Degrees of Freedom (df)
In computing degrees of freedom, there are two important considerations to keep in mind:
Each df value is associated with a specific SS value.
Normally, the value of df is obtained by counting the number of items that were used to calculate SS and then subtracting 1. For example, if you compute SS for a set of n scores, then df = n - 1
ANOVA Formulas
The final calculation for ANOVA is the F-ratio, which is composed of two variances:
F = variance between treatments/variance within treatments
Each of the two variances in the F-ratio is calculated using the basic formula for sample variance:
sample variance = s^2 = SS/df
Calculation of Variances (MS) and the F-Ratio
These two variances form the F-ratio by dividing MS between by MS within.
A large F-ratio means the treatment differences are much bigger than what would be expected by chance, suggesting a real treatment effect (confirmed by comparing to the critical F value).
After computing SS and df, ANOVA divides each SS by its df to get mean squares (MS): MS between and MS within.
Examples of Hypothesis Testing and Effect Size with ANOVA
An Example of Hypothesis Testing and Effect Size with Anova
Step 2: Locate the Critical Region
Step 3: Compute the F-Ratio
Step 1: State the Hypotheses and Select an Alpha Level
Step 4: Compute the F-Ratio
Assumptions for the Independent-Measures ANOVA
The independent-measures ANOVA requires the same three assumptions that were necessary for the independent-measures t hypothesis test:
The populations from which the samples are selected must be normal.
The populations from which the samples are selected must be normal.
The observations within each sample must be independent.
The F Distribution Table
To determine whether an obtained F is large enough to be significant, we compare it to a critical value from the F-distribution table.
To use the table, we need the numerator and denominator degrees of freedom and the selected alpha level.
When the null hypothesis is true, F should be close to 1.00. An F-ratio much larger than 1.00 suggests the null is likely false.
An Example with Unequal Sample Sizes
ANOVA works even when the treatment groups have unequal sample sizes, though it is most accurate with equal n’s.
Researchers typically plan for equal group sizes, but when that isn’t possible, ANOVA can still provide a valid test as long as the sample sizes are not extremely different and are reasonably large.
The Distribution of F-Ratios
Before we examine this distribution in detail, you should note two obvious characteristics:
Because F-ratios are computed from two variances (the numerator and denominator of the ratio), F values always are positive numbers. Remember that variance is always positive.
When H0 is true, the numerator and denominator of the F-ratio are measuring the same variance. In this case, the two sample variances should be about the same size, so the ratio should be near 1. In other words, the distribution of F-ratios should pile up around 1.00.
Measuring Effect Size for ANOVA
To measure effect size, ANOVA uses η² (eta squared), which is the proportion of total variance explained by the treatment (SS_between / SS_total).
It indicates how much of the variability in scores is due to the treatment.
A significant ANOVA result doesn’t show how large the effect is.
Post Hoc Tests
Tukey’s Honestly Significant Difference (HSD) Test
This value, called the honestly significant difference (HSD), is compared to each pair of means: if the difference exceeds HSD, it is significant.
The test uses the q statistic, the within-treatments variance, and the sample size, and requires equal n’s for all groups.
Tukey’s HSD test is a post hoc method that determines the minimum difference between treatment means needed for significance.
The appropriate q value depends on the number of treatments, error degrees of freedom, and the chosen alpha level.
Posttests and Type I Errors
Each comparison carries a risk of a Type I error, and performing multiple comparisons increases the overall, or experimentwise, alpha level.
To control this increased risk, statisticians have developed specific methods for post hoc testing.
A post hoc test allows you to compare treatment groups two at a time (pairwise comparisons) to see which means differ significantly.
The Scheffé Test
The “safety factor” for the Scheffé test comes from the following two considerations:
Although you are comparing only two treatments, the Scheffé test uses the value of k from the original experiment to compute df between treatments. Thus, df for the numerator of the F-ratio is .k-1.
The critical value for the Scheffé F-ratio is the same as was used to evaluate the F-ratio from the overall ANOVA. Thus, Scheffé requires that every posttest satisfy the same criterion that was used for the complete ANOVA.
More about ANOVA
A Conceptual View of ANOVA
The denominator measures variance within treatments; larger variance decreases F.
Larger sample sizes make it easier to detect significant differences but do not affect effect size (η²).
The numerator of F measures differences between treatment means; larger differences increase F.
The Relationship Between
ANOVA and t Tests
For two groups, a t test and a one-way ANOVA give the same result because F = t^2
Both test the same hypotheses, and the t distribution squared produces the F distribution, making ANOVA and t tests equivalent in this case.