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Chapter 12: Analysis of Variance - Coggle Diagram
Chapter 12: Analysis of Variance
Analysis of Variance (ANOVA): A hypothesis-testing procedure that is used to evaluate mean differences between two or more treatments (or populations)
Unlike t testing, it can compare more than two treatments
Because many tests are occurring, error piles up from each of them
Experimentwise Alpha Level: Total probability of a Type I error that is accumulated from all of the individual tests in the experiment
Testwise Alpha Level: The risk of a Type I error for an individual hypothesis test
Terminology
Factor: An independent or quasi-indepent variable
Two-Factor Design: A study which combines two factors, also called factorial design
Single-Factor Design: A study with one factor
Levels: The individual conditions or values that make up a factor (different doses, test types, etc)
Hypotheses
Null Hypothesis: u1=u2=u3 (and so forth), similar to previous null hypotheses
Alternative Hypothesis:
There is at least one mean difference among the populations
, unlike before the alternative hypothesis is represented by a sentence because we are not looking for change in a specific location
Can have many forms, like u1!=u2!=u3 or u1=u2!=u3. Usually a researcher does choose a few hypotheses like these to consider
Test Statistic
Based on variance instead of sample mean difference
Between-Treatments Variance: Variability that is occurring because of differences between sample means, or differences in scores
between treatments
Within-Treatments Variance: Variability that is occurring because of differences between scores
within treaments
Within-Treatment variance is caused by randomness, so we want to remove it from the total variance to see if the treatment effect is significant
Also could be described as variance between treatments divided by variance within treatments
You can find the sum of squares for these two values using the computational formula (Chapter 4)
Degrees of freedom
Total df is N-1
df within-treatments is the sum of the degrees of freedom of each treatment
df between-treatments is k-1
Formula: F=variance between sample means divided by variance expected with no treatment effect (in other words, if the null hypothesis is true)
Called the F-Ratio
Essentially is the intended treatment effect plus the random differences divided by the random differences. If there is no treatment effect, then F=1
Error Term: The denominator of the F-Ratio which measures random and unsystematic variability
Assumptions
Observations must be independent
Populations sampled from must be normal
Populations must have equal variances
The usual suspects again...
Recall Hartley's F-Max from Chapter 10 to test for homogeneity of variance
Notation
k=Number of treatment conditions, or levels of factor
n=Scores in each treatment, but if treatment sizes vary then denote which sample is being discussed using subscripts (as mentioned in Chapter 10)
N=Total scores in the entire study, so either k*n for a study with uniform sizes or the sum of n across all samples for one without
T=Sum of the scores for each treatment condition (level)
G=The sum of all the scores (the sum of all "T")
Notation is not uniform, good luck out there kid
eta (which looks like an n with a long right arm) squared=percentage of variance accounted for by treatment effect
This one is more uniform
Use instead of r^2