Independent Variable
- Also called a factor or a treatment variable
- Levels (treatments) = different values or categories of the independent variable/factor
- Example: Instruction method (1. in person, 2. online, 3. hybrid, and 4.
tutoring)
Single-factor (one-way) designs
Involve a single IV with two or more levels:
- One-way independent-groups design
- One-way design with repeated measures
Factorial designs
Involve more than one independent variable with two or more levels. Example: two-way independent-groups designs. When an experimental design has two factors with two levels each, it is called a 2 × 2 factorial design. If two factors, one factor has 2 levels and the other factor has 3 levels, 2 × 3
One-way ANOVA purpose
To test whether the means of k (≥ 2) populations significantly differ.
- H0 : µ1 = µ2 = · · · = µk
- H1 : Not all µ’s are the same. (At least one of the means is different)
One-way ANOVA prior requirements/assumptions
- The population distribution of the DV is normal within each group
- The variance of the population distributions are equal for each group (homogeneity of variance assumption)
- Independence of observations
One-way ANOVA vs. several t-tests
Researchers are often interested in a set of related hypotheses; we call these a family of tests. If we used independent samples t-tests for these research questions, we would have to compare two means at a time:
H01 : µ1 = µ2
H02 : µ1 = µ3
H03 : µ2 = µ3
which would amount to 3 t-tests.
Familywise Type I error rate
The probability of making at least one Type I error in the family of tests if the null hypotheses are true. If we do several t-tests, we would get an inflated familywise Type I error rate.
One-way ANOVA: Basic concepts
ANOVA = ANalysis Of VAriance
1. Divides the observed variance of the dependent variable into parts resulting from different sources
2. Assesses the relative magnitude of the different parts of variance
3. Examines whether a particular part of the variance is greater than expectation under the null hypothesis
Sources of variance
- The variance explained by the model (MSM). MS = mean squares (“mean” of sum of squared deviations). The subscript “M” stands for “model”. This is variance between groups that is due to the IV, or different treatments/levels of a factor
- The variance within groups, or the residual variance (MSR): Within each group, there is some random variation in the scores for the subjects
Properties of MSM and MSR
If group means differ from each other, MSM tends to be large compared to MSR, and in turn F tends to be large. If the observed F statistic is found to be greater than the critical F- value for a given sample size and number of groups, we reject the null hypothesis that group means are equal.
