Ch. 12: Introduction to Analysis of Variance

Analysis of variance (ANOVA)

A statistical test used to compare three or more group means to see if at least one group is significantly different. It looks at how much the total variability in scores can be explained by group differences versus random error

Factor

In analysis of variance, the variable (independent or quasi-independent) that designates the groups being compared is called a factor

Levels

The individual conditions or values that make up a factor are called the levels of the factor

Two-factor design (factorial design)

A design with two independent variables (factors) studied at the same time. It allows researchers to see main effects for each factor and whether they interact (combine in unique ways)

Single-factor design

A study that examines only one independent variable (one factor) with two or more levels

Single-factor independent-measures design

A type of single-factor design where different participants are in each group (between-subjects). Example: one group studies with music, another without

Testwise alpha level

The testwise alpha level is the risk of a Type I error, or alpha level, for an individual hypothesis test

Experimentwise alpha level

When an experiment involves several different hypothesis tests, the experimentwise alpha level is the total probability of a Type I error that is accumulated from all of the individual tests in the experiment. Typically, the experimentwise alpha level is substantially greater than the value of alpha used for any one of the individual tests

Between-treatments variance

How much the group means differ from each other. This captures both the treatment effect (real differences caused by the experiment) and random error

Treatment effect

The actual change or difference in scores caused by the experimental manipulation (not by chance)

Within-treatments variance

The variabiity of scores inside each group, caused by individual differences, measurement error, or chance—not by treatment itself

F-ratio

The statistic calculated in ANOVA: F=between treatments variance/within-treatments variance. If F is large, it suggests the group means are more different than chance alone can explain

Error term

Represents the amount of variability due to random, unexplained factors (individual differences, chance, etc.) It’s used in the denominator of the F-ratio

Mean square (MS)

An average of squared deviations (variances). In ANOVA, MS is calculated for both between-treatments and within-treatments: MS = SS/df (SS= sum of squares, df= degrees of freedom)

Distribution of F-ratios

A theoretical distribution showing the range of possible F-values you could get if the null hypothesis were true. Most F-values are near 1.0 (no difference), but larger values become less likely and may indicate a real effect

ANOVA summary table

A table that organizes all the key calculations in an ANOVA—showing sources of variance (between, within, total), their SS, df, MS, F, and significance

Eta squared (n²)

A measure of effect size for ANOVA. It shows the proportion of total variability in the data that’s explained by the treatment. Ranges from 0 to 1 — largers values mean a stronger effect

Post hoc tests (or Prottests)

Additional tests done after finding a significant ANOVA result to figure out which specific groups differ from each other

Pariwise comparisons

Comparisons made between every possible pair of group means to see which ones are significantly different

Tukey’s HSD test (Honest significant difference)

A post hoc test that compares all possible pairs of means while controlling for overall error rate. It’s used when all groups have equal sample sizes and variances

Scheffé test

A very cautious post hoc test that controls for Type I error very strictly. It’s flexible (works with unequal group sizes or complex comparisons) but less powerful — meaning it’s harder to find a significant difference unless it’s strong