ANOVA (analysis of variance) is a statistical method used to compare the means of two or more groups.
FACTORS
LEVELS
The main idea of analysis of variance
How much of the total variance comes from
The variance between the groups
The variance within the groups
ASSUMPTIONS IN ANOVA
Normality of Sampling Distribution of Mean
Independence of Errors
Absence of Outliers
Homogeneity of variance
HYPOTHESIS IN ANOVA
Analysis of variance with 1 factor ( A with three levels)
Analysis of variance with 2 factors( A & B with 3 levels)
MAIN EFFECT IN ANALYSIS OF VARIANCE
Pretend we are comparing the test scores of people who have received a medication ( 100 mg dosage group) and people who have not received a medication ( 0mg dosage group). The 0 mg condition has a mean of 60 while 100 mg condition has a mean of 80. this could be represented in a graph like this:
INTERACTION EFFECTS IN ANALYSIS OF VARIANCE
POST HOC ANALYSIS IN ANOVA
If we reject the null hypothesis, all we know is that there is a difference somewhere among the groups.
Additional tests called POST HOC TESTS can be done to determine where differences lie.
THE F DISTRIBUTION IN ANALYSIS OF VARIANCE
If there are no treatment differences ( no actual effect), we expect F to be 1.
If there are treatment differences, we expect F to be greater than 1.
TYPES OF ANOVA
One-way ANOVA
Repeated-measure ANOVA
Factorial ANOVA
ONE-WAY ANOVA
One factor with at least TWO levels.
Levels are INDEPENDENT.
REPEATED-MEASURES ANOVA
One factor with at least TWO levels.
Levels are DEPENDENT.
FACTORIAL ANOVA
Two or more factors
Each factor with at least TWO levels.
Levels can be either INDEPENDENT, DEPENDENT or both (mixed).
ONE WAY ANOVA
STEPS
Define Null and alternate hypothesis
State Alpha
Calculate the degree of freedom
State decision rule
Calculate test statistics
State results
State conclusion
Define null and alternate hypothesis
State alpha
α = 0.05
Degree of Freedom
State decision rule
To look up the critical value, we need to use two different degrees of freedom.
If F is greater than 3.55 then reject the null hypothesis.
Calculate test statistics
State results
If F is greater than 3.55 then reject the null hypothesis.
F= 86.56
So, reject the null hypothesis.
State conclusion
The three conditions differed significantly on anxiety levels.
REPEATED-MEASURES ANOVA
One factor with at least TWO levels.
Levels are DEPENDENT ( they share variability)
Identical to one way ANOVA, except for one additional calculation for this shared variability.
Steps
Define Null and alternate hypothesis
State Alpha
Calculate the degree of freedom
State decision rule
Calculate test statistics
State results
State conclusion
Define null & alternate hypothesis
State alpha
α = 0.05
Calculate degrees of freedom
State decision rule
To look up the critical value, we use two different degrees of freedom.
If F is greater than 3.89, reject the null hypothesis.
Calculate test statistics
State results
If F is greater than 3.89, reject the null hypothesis.
F = 224.27
So, reject the null hypothesis.
FACTORIAL ANOVA
Two or more factors
Each factor with at least TWO levels.
Levels can be either INDEPENDENT, DEPENDENT or both (mixed).
FACTORIAL ANOVA ( two independent factors)
Two factors with at least two levels each, levels are independent.
Factorial ANOVA with independent factor is like one way ANOVA except you are dealing with more than one independent variable.
Steps
Define Null and alternate hypothesis
State Alpha
Calculate the degree of freedom
State decision rule
Calculate test statistics
State results
State conclusion
Define null & alternate hypothesis
State alpha
α = 0.05
Degrees of freedom
State decision rule
[ school ] if F is greater than 4.17, then reject the null hypothesis.
[ Dosage ] if F is greater than 3.32, then reject the null hypothesis.
[interaction] if F is greater than 3.32, then reject the null hypothesis.