Categories: General

# ANOVA: Types, Assumptions and Hypothesis

## INTRODUCTION

• 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

1. Normality of Sampling Distribution of Mean
2. Independence of Errors
3. Absence of Outliers
4. 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:

## 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

α = 0.05

#### 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.

#### 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

α = 0.05

#### 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.

#### 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

α = 0.05

#### 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.

#### Calculate test statistics

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