One-Way Between-Groups ANOVA: Principles, Calculations, and Interpretation

One-Way Between-Groups ANOVA

Introduction to ANOVA

Analysis of Variance (ANOVA) is a statistical method used to compare means across three or more groups. It is particularly useful when the independent variable is nominal and the dependent variable is measured on a scale. The one-way between-groups ANOVA is designed for situations where different participants are assigned to each group.

• Key Term: F distribution – the probability distribution used in ANOVA to determine statistical significance.

• Application: Used when comparing three or more groups to avoid the increased risk of Type I error associated with multiple t tests.

Why Not Use Multiple t Tests?

Conducting multiple t tests increases the probability of making a Type I error (false positive) and inflates the alpha level, leading to unreliable results.

• Type I Error: The probability of incorrectly rejecting the null hypothesis increases with the number of comparisons.

Number of Means	Number of Comparisons	Probability of Type I Error
2	1	0.050
3	3	0.143
4	6	0.265
5	10	0.401
6	15	0.537
7	21	0.659

z, t, and F Distributions

Statistical distributions such as z, t, and F are increasingly complex variations of the normal curve, each suited for different types of data and sample sizes.

• z distribution: Used when population mean and standard deviation are known.

• t distribution: Used when only the sample mean is known or for small sample sizes.

• F distribution: Used for comparing three or more groups, or the variance between groups.

Distribution	When Used	Links Among Distributions
z	One sample; μ and σ known	Subsumed under t and F distributions
t	One sample (μ only known), two samples	Same as z if sample size is large
F	Three or more samples	Square of z or t if only two samples

The F Distribution

The F statistic is used to analyze variability among group means. It is calculated as the ratio of the variance between groups to the variance within groups.

• Formula: $F = \frac{\text{between-groups variability}}{\text{within-groups variability}}$

• Purpose: To determine if observed differences among group means are statistically significant.

Types of Variance in ANOVA

• Between-groups variance: Estimate of population variance based on differences among group means.

• Within-groups variance: Estimate of population variance based on differences within each group.

Types of ANOVA

• One-way ANOVA: Tests one nominal variable with more than two levels and a scale dependent variable.

• Between-groups ANOVA: Different participants in each sample.

• Within-groups ANOVA: Same participants in each sample (repeated measures).

Assumptions of ANOVA

All ANOVAs share three key assumptions for valid analysis:

• Random selection of samples

• Normally distributed sample

• Homoscedasticity (equal variances across groups)

Steps in One-Way Between-Groups ANOVA

1. Identify the populations, distribution, and assumptions.

2. State the null and research hypotheses.

3. Determine the characteristics of the comparison distribution (degrees of freedom).

4. Determine the critical value or cutoff.

5. Calculate the test statistic.

6. Make a decision (reject or accept the null hypothesis).

Example: ANOVA Steps

• Step 1: Four populations, F distribution, three assumptions.

• Step 2: Null hypothesis: All groups exhibit the same fairness behaviors. Research hypothesis: At least one group differs in fairness behaviors.

• Step 3: Degrees of freedom: $df_{between} = N_{groups} - 1$ $df_{within} = df_1 + df_2 + ... + df_{last}$ $df_i = n_i - 1$

• Step 4: Use F distribution tables to determine critical values (cutoffs).

• Step 5: Calculate the F statistic using sample data.

• Step 6: Compare calculated F to critical F to make a decision.

Logic Behind the F Statistic

• Quantifies overlap between groups.

• Two ways to estimate population variance: between-groups and within-groups.

Source Table in ANOVA

The source table summarizes calculations and results, describing sources of numerical variability.

Source	SS	df	MS	F
Between-groups	SSbetween	dfbetween	MSbetween	F
Within-groups	SSwithin	dfwithin	MSwithin
Total	SStotal	dftotal

Sums of Squared Deviations

• ANOVA analyzes deviations between groups, within groups, and total deviations.

Sum of Squares	Calculation	Formula
Between-groups	Grand mean from sample mean	$SS_{between} = \sum (M - GM)^2$
Within-groups	Sample mean from each score	$SS_{within} = \sum (X - M)^2$
Total	Grand mean from each score	$SS_{total} = \sum (X - GM)^2$

Calculating Effect Size

• R2: Proportion of variance in the dependent variable accounted for by the independent variable.

• Formula: $R^2 = \frac{SS_{between}}{SS_{total}}$

• Conventions: Small = 0.01, Medium = 0.06, Large = 0.14

Effect Size	Convention
Small	0.01
Medium	0.06
Large	0.14

Post Hoc Tests

Post hoc tests are used to determine which groups differ after finding a significant F statistic.

• Tukey HSD: Determines differences between means using standard error. $HSD = \frac{M_1 - M_2}{s_M}$ $s_M = \sqrt{\frac{MS_{within}}{N}}$

• Bonferroni: Adjusts the critical value for multiple comparisons by dividing the p level by the number of comparisons.

• A priori comparisons: Planned comparisons before data collection.

Within-Groups Degrees of Freedom	Alpha Level	k = Number of Treatments (levels): 3	k = Number of Treatments (levels): 4	k = Number of Treatments (levels): 5
8	0.05	4.04	4.53	4.99
9	0.01	5.64	6.20	6.62
10	0.05	3.88	4.33	4.65
10	0.01	5.27	5.77	6.14

Summary and Decision Making

• Compare the calculated F statistic to the critical value from the F distribution table.

• If the calculated F is greater than the critical value, reject the null hypothesis and conclude that there is a significant difference between group means.

Example: If F = 8.27 and the cutoff is 3.86, the null hypothesis is rejected, indicating significant differences among the groups.

Additional info: These notes are based on behavioral science statistics, but the principles and calculations of ANOVA are directly applicable to General Chemistry when comparing means across multiple experimental groups or treatments.