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:

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.

Calculate test statistics

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