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Chapter 12: Introduction to Analysis of Variance - Coggle Diagram
Chapter 12:
Introduction to Analysis of Variance
ANOVA
ANOVA is a hypothesis-testing procedure that is used to evaluate the mean difference between two or more treatments.
Factor: an independent or quasi-independent variable
Level: the individual groups or treatment conditions that are used to make up a factor
Factorial design: a study that combines two factors
Null hypothesis: the treatment conditions have no effect on the participant's scores
A specific alternative hypothesis is not stated because there are so many possibilities.
Experimentwise alpha level: the total probability of a Type I error accumulated from all of the separate tests in an experiment.
F-Ratio: The Test Statistic for ANOVA
To demonstrate that there really is a treatment effect, we must establish that the differences between treatments are bigger than would be expected by sampling error alone.
F = variance between sample means /
variance expected with no treatment effect
An F-ratio near 1.00 indicates that the differences between treatments are random and unsystematic.
The denominator of the F-ratio is also known as the error term because it measures only random and unsystematic variability.
F values are always positive the the distribution of F ratios should pile up around 1.00
The exact shape of the F distribution depends on the degrees of freedom for the two variances in the F-ratio.
Smaller df values ---> the F distribution is more spread out
Assumptions for the independent-measures ANOVA: 1) the observations in each sample must be independent
2) the populations from which the samples are selected must be normal
3) the populations must have equal variance (homogeneity of variance)
Notation & Formulas
T = the sum of the scores for each treatment condition
k = the number of treatment conditions or number of levels of the factor
n = number of scores in each treatment
(if unequal sample sizes, use a subscript for distinction)
N = total number of scores in the entire study
G = the sum of all the scores in the research study or the grand total
F = variance between treatments / variance within treatments
Each variance is computed as SS/df
Post Hoc Tests
These tests are done after ANOVA when you reject the null hypothesis and there are three or more treatments.
Tukey's HSD Test
allows you to compute a single value that determines the minimum difference between treatment means that is necessary for significance
If the mean difference exceeds Tukey's HSD, you conclude that there is a significant difference between the treatments.
The Scheffe' Test
uses and F-ratio and is a safe test due to limited risk of a Type I error
the numerator is an MS between treatments that is calculated using only the two treatments you want to compare
the denominator is the same MS within that was used for the overall ANOVA
Pairwise comparisons involve comparing the individual treatments two at a time looking for a significant mean difference.
