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• PS219: Research Methods in Psychology focuses between-subjects design Part 2, specifically one-way between-subjects ANOVA.
• The lecture took place February 11th, 2025.

Recap of Between-Subjects Design

• In between subjects design, different subjects are used for each experimental condition
• In a non-equivalent group design, participants are not randomly assigned in a controlled manner
• It can be determined if there is a significant difference between groups by checking the p-value (p < .05)
• The larger the t score, the bigger the difference between groups

Topics Covered

• One-factor, multiple group designs
• One-way Analysis of Variance
• Assumptions of ANOVA
• One-way between-subjects ANOVA

Between Groups Designs

• One-factor multiple group designs are research designs where a single measure (the dependent variable) is recorded for more than two levels of one independent variable.
• A typical condition is a control group where the independent variable remains neutral.
• The remaining groups are defined by their respective levels of change introduced for the independent variable

Analysis of Variance Introduction

• Variance in statistics is how spread out the data is.
• When scores are similar, variance is small.
• When scores differ vastly, there is a large variance.

Variance Analysis Rationale

• The variance is analysed to compare means
• For example, If you are comparing three different yoga routines:
  • As a researcher you can ask:
    • Was one routine more successful than another at pain reduction?
    • Were the differences only due to individual difference?
• The first question addresses the variance between groups, the difference in pain between routines.
• The second question addresses overall variance among individuals, how different people are from one another (regardless of group).
• If the difference in pain reduction is due to which group assignment, compared to random individual differences, it suggest there a significant difference between the average of the 3 groups.

Analysis of Variance

• ANOVA compares group and individual variation
• This establishes if groups are actually different on average
• This is done using the F-test
• The F-statistic is the test statistic and a p-value is obtained.
• The F-statistic is constructed by dividing the average amount of group variance by the average amount of individual variance.

One-Way Analysis of Variance

• The null hypothesis in the analysis of variance is that three or more populations being compared all have no differences.
• This is answered by analysing the different types of variances involved:
  • Within group (individual) variance: unaffected by whether the null hypothesis is true.
  • Between group variance: represents differences reported between the three groups.

Advantages of One-Way ANOVA

• ANOVA enables one to compare more than two data sets.
• T-tests used repeatedly have the potential to lose reliability (think 5%).

Assumptions of ANOVA

• Normal distribution of variables.
• Homogeneity of variance
• Equal interval measurement at least
• Independence of observations
• Equal numbers in each cell (or proportionality)

Normal Distribution

• The variable should follow a normal distribution, there should be no skew present in the data

Homogeneity of Variance

• The variances of the groups must be similar
• A Levene's test can be used for homogeneity (or equality) of variances if the group variances are similar enough
• If the p-value yielded from a Levene's test is less than .05, then the assumption of homogeneity of variance has been violated.

Violations of the Assumptions of ANOVA

• The assumption of equal variance is not a problem unless they are very disparate.
• As a rule, the largest variance should not be more than 4 or 5 times the smallest when the sample sizes are equal.
• It assumed scores are at least symmetrical as opposed to normally distributed when using ANOVA.
• ANOVA can still be used if there is a slightly unequal number of scores in each cell/group.

Situations where the ANOVA is Invalid:

• Heterogeneity of variance
• Unequal sample sizes exist

Example ANOVA problem

• Research Question: Is there a difference in frisbee throwing distance between secondary school students, undergraduate students, and postgraduate students?
• In this example the F-statistic is 0.35, p = .709.
• This tells us there was no significant difference in frisbee throwing distance between the groups.

How to writing up a One-Way Between-Subjects ANOVA

• Step 1: State the name and purpose of the test i.e. "A one-way between-subjects ANOVA was conducted to examine if there was a significant difference in frisbee throwing distance between secondary school students, undergraduate students, and postgraduate students."

Writing up results of a One-Way Between-Subjects ANOVA

• Step 2: State the independent (education level with three levels secondary school, undergraduate, and postgraduate) and dependent variables (frisbee throwing distance). Levene's test for homogeneity of variances was not significant (F = 0.19, p = .828), therefore the assumption of homogeneity of variances was met.
• Step 4: Report the results of the ANOVA "Analysis of variance revealed that there was no significant difference in frisbee throwing distance between the three groups (F(2, 29) = 0.35, p = .709)."

Putting It All Together

• "A one-way between-subjects ANOVA was conducted to examine if there was a significant difference in frisbee throwing distance between secondary school students, undergraduate students, and postgraduate students.
• The independent variable was education level (secondary school, undergraduate, and postgraduate).
• The dependent variable was frisbee throwing distance.
• Levene's test for homogeneity of variances was not significant (F = 0.19, p = .828), therefore the assumption of homogeneity of variances was met.
• Analysis of variance revealed that there was no significant difference in frisbee throwing distance between the three groups (F(2, 29) = 0.35, p = .709)."

Post-Hoc Analysis

• A post-hoc analysis is preformed if there had been a significant difference between the three groups to find where exactly those significant differences were.
• Post-hoc analysis includes additional statistical tests conducted after the primary analysis.
• A post-hoc analysis for a one-way between-groups ANOVA is called the Tukey HSD