1-WAY ANALYSIS OF VARIANCE (ANOVA)

Purpose

Compare three or more population means, m, using sample means, , when data is collected in an independent groups/between participants type of design.

BACKGROUND

1. Participants can be randomly chosen from the population, or obtained by some other, more convenient means (haphazard/convenient sampling).

2. Participants are either randomly assigned to 1 level of the independent variable, such as in an experiment/manipulated design, or can be categorized based on membership to some type of subject variable (academic major, for example), such as in a non-manipulated/correlational design).

THE NULL HYPOTHESIS

The null hypothesis for ANOVA states that the population means from which the samples are taken are equal. In other words:

m₁₌ m_{2 =} m_{3 =…=} m_k

The mean of the population, m, refers to the hypothetical value that would result if all the individuals from the population had been observed.

LOGIC OF THE TEST

An ANOVA tests mean differences among 3 or more groups by analyzing the variance in the data collected. The estimate for the total amount of variance in the data can be divided into estimates for between and within groups variance.

Estimate of the total variance in the data

=

Estimate of the between subjects variance

+

Estimate of the within subjects variance

The F-value is the ratio of between groups variance to within groups variance. Conceptually, the F ratio can be thought of using the following formula:

Variance is a measure of how disperse scores are about some central value. Recall, to estimate the population variance based on scores from a sample, we use the formula:

DEGREES OF FREEDOM

ANOVA estimates two population parameters (between and within groups variance) therefore two types the degrees of freedom are used.

The degrees of freedom calculated for the between subjects variance estimate is:

k - 1

Where:

k = the number of conditions

The degrees of freedom calculated for the within subjects variance estimate is:

N - k

Where:

N = the total number of participants in the study

k = the number of conditions

The between subjects degrees of freedom is often referred to as the degrees of freedom in the numerator, while the within subjects degrees of freedom is often referred to as the degrees of freedom in the denominator because the estimate for the between subjects variance is in the numerator of the F ratio and the estimate for the within subjects variance is in the denominator of the F ratio.

1-WAY ANOVA FORMULA

Where:

ms_between = the estimated between subjects variance

ms_within = the estimated within subjects variance

THE BETWEEN SUBJECTS VARIANCE ESTIMATE:

THE WITHIN SUBJECTS VARIANCE ESTIMATE:

SAMPLING DISTRIBUTION OF F

The probability distribution of F-values that would occur if all possible samples of fixed sizes for N and k were drawn from the null population. Gives two pieces of information:

1. All possible values for F of sample size N with groups k

2. Probability of getting each values of F if sampling was random from the null hypothesis population.

If the null hypothesis was true (i.e., if m₁₌ m₂₌ m_3=…= m_k) then the F-value should equal 1. However, due to sampling fluctuation, occasionally one mean will be slightly lower (or higher) than the others. In rare cases (i.e., on the tail of the sampling distribution) the means are very different, merely due to chance fluctuation.

The means may be very different even if the null is true. It is not likely, but it can occur.

STATING A CONCLUSION

Your conclusion should contain several pieces of information. First, state if the alternative hypothesis was supported (if null was rejected) or refuted (if null was retained). Next state the statistics, both descriptive (i.e., means) and inferential (i.e., F-ratio, df, alpha level, critical value) using proper APA format. Third, tell if any causal relationship exists (only when null is rejected and experimental design used). Lastly, calculate effect sizes and discuss the strength of the relationship between the variables. Example:

The hypothesis that type of beverage consumed was related to perceived attractiveness was supported. The omnibus F test of perceived attractiveness ratings for participants who consumed non-alcoholic beer (M = 7), regular beer (M = 8) and water (M = 4) was significant, F (2, 9) = 6.49, p < .05, given F_crit = 4.26. The type of beverage caused differences in perceived attractiveness. The strength of the relationship between type of beverage and perceived attractiveness was medium (h² = .59, w² = .48).

CALCULATING AN ANOVA FOR INDEPENDENT SAMPLES FROM RAW DATA

Step 1: Calculate the ms_between

Step 1a: Calculate SS_between

Step 1b: Determine df_between

Step 1c: Divide SS_between by df_between

Step 2: Calculate the ms_within

Step 1a: Calculate SS_within

Step 1b: Determine df_within

Step 1c: Divide SS_within by df_within

Step 3: Calculate the F-ratio

Step 4: Create an F-table

Source SS df ms F

Between

Within

Total

Step 5: Evaluate the F-value

Step 6: Interpret results using group means