Lecture 1: The General Linear Model: One-way ANOVA

What is the General Linear Model (GLM)?

A flexible statistical approach for testing hypotheses where the outcome (dependent variable) is numeric rather than categorical.

Which statistical tests covered previously are types of GLM?

Regression, correlation, t-tests, and ANOVA.

What do the GLM tests listed in the lecture have in common?

They all have a numeric dependent variable.

How can a GLM be expressed in its simplest form?

Outcomeᵢ = model + errorᵢ.

How do different statistical tests within the GLM differ?

They differ in how the model part is specified.

When is ANOVA useful, according to the lecture?

For factorial designs with categorical predictors, such as placebo versus drug or morning versus evening dose.

What kinds of independent variables can multiple regression include?

Numerical variables, categorical variables, or both.

What kinds of independent variables are used in ANCOVA, according to the lecture?

Categorical and numeric independent variables.

What does ANOVA stand for?

Analysis of Variance.

What two features can ANOVAs differ in?

The number of variables and how subjects are allocated to conditions.

What is a one-way ANOVA?

An ANOVA with a single categorical independent variable (IV).

What is a two-way ANOVA?

An ANOVA with two categorical independent variables (IVs).

What is a factorial ANOVA?

An ANOVA with any arbitrary number of categorical independent variables (IVs).

What is another name for an independent variable in ANOVA?

A factor.

What are the conditions within a factor called?

Levels.

What is a between-subject ANOVA?

Different participants are in each level of the factor(s).

What is a within-subject ANOVA?

The same participants take part in all levels of the factor(s).

What is a mixed ANOVA?

An ANOVA with some between-subject factors and some within-subject factors.

What does a one-way ANOVA test?

Hypotheses about mean group differences when there are two or more groups.

In the lecture example, what is being compared?

Reading speed among children in grades 3, 4, and 5.

What is the null hypothesis for the three-grade reading-speed example?

H₀: μ₁ = μ₂ = μ₃. The population group means are equal.

What does the null model predict about group means?

Each group mean equals the grand mean, μ, which is the overall mean of all data points.

What is the alternative hypothesis in a one-way ANOVA?

There is a difference among the group means; at least one group mean differs from another.

Does the ANOVA alternative hypothesis specify which groups differ?

No. It indicates a group difference but does not identify where the difference is.

What is variance?

Variability in the differences between individual scores and the mean.

What is dispersion?

The extent to which a distribution is stretched or squeezed.

What does ANOVA compare?

Variance between groups (group differences) with variance within groups (error).

What does a large between-group variance relative to within-group variance suggest?

Evidence against the null hypothesis that there are no group differences.

What is the purpose of using the GLM to measure variance?

It provides a method for measuring within- and between-group variance that generalises to designs with more than one factor.

What three components must be estimated to compute the GLM in the lecture's one-way ANOVA?

The grand mean, group effects, and error.

In the lecture's example with 10 children in each of 3 groups, how many GLM equations are needed?

30 equations, one for each child/data point.

In that example, why are there 34 terms to estimate?

There is 1 grand mean, 3 group effects, and 30 error terms (one per participant).

What is a sum of squares (SS), as described in the lecture?

Square each number in a column and add the squared values; it summarises variance associated with a model component.

Why calculate sums of squares in ANOVA?

To summarise the amount of variance associated with each component of the model.

How is mean square (MS) calculated?

Mean square = sum of squares ÷ degrees of freedom.

Why not simply compare sums of squares for between- and within-group variance?

Sums of squares depend on the number of group levels and participants, so they are affected by the design.

What is the formula for dfA in a one-way ANOVA?

dfA = number of groups in the factor − 1.

What is the formula for dfS(A) in a one-way ANOVA?

dfS(A) = number of participants − number of groups.

If a one-way ANOVA has 3 groups, what is dfA?

3 − 1 = 2.

If a one-way ANOVA has 30 participants and 3 groups, what is dfS(A)?

30 − 3 = 27.

What is the F statistic in ANOVA?

The ratio of variance due to differences between groups to variance within groups.

Conceptually, what does a larger F statistic indicate?

Between-group variance is larger relative to within-group variance, providing more evidence against the null hypothesis.

What can be derived from the F statistic?

The p-value.

How are ANOVAs commonly run in practice in the lecture?

Using ready-made functions in R.

What do post-hoc tests help identify?

Where the significant differences between groups are.

What are the three GLM assumptions listed in the lecture?

1. Observations are i.i.d. (independent and identically distributed). 2. Homogeneity of variance. 3. Normality of residuals.

What does i.i.d. stand for?

Independent and identically distributed.

What does independence mean in the i.i.d. assumption?

Knowing one observation does not allow you to predict another observation.

What does identically distributed mean in the i.i.d. assumption?

Observations are generated by the same process.

How is the independence/i.i.d. assumption assessed, according to the lecture?

It is determined by the study design.

What is homogeneity of variance?

Variance is the same across all levels of the factor in a between-subject ANOVA.

How can homogeneity of variance be checked?

Use Levene's test and visual inspection of group violin plots.

How is Levene's test interpreted for homogeneity of variance?

A non-significant result means we can assume homogeneity of variances.

What should you look for in violin plots when checking homogeneity of variance?

The groups should have approximately the same vertical spread.

What does the normality-of-residuals assumption mean?

The residuals (errors) are normally distributed.

How can normality of residuals be checked?

Use a Q–Q plot of the residuals; the points should fall roughly along the line.

What are the options if GLM assumptions are not met?

1. Carry out ANOVA anyway, accepting reduced power and increased Type II error risk. 2. Transform the dependent variable or use a more suitable model. 3. Use a nonparametric test.

What is the consequence of carrying out ANOVA when assumptions are not met, according to the lecture?

It reduces power and increases the risk of a Type II error.

Which nonparametric test does the lecture give as an alternative to one-way ANOVA?

The Kruskal–Wallis test.

Quick recall: factor vs level — what is the difference?

A factor is an independent variable; a level is a condition/category within that factor.

Quick recall: between-subject vs within-subject — what is the difference?

Between-subject designs use different participants in different levels; within-subject designs use the same participants across all levels.

Quick recall: what is the key logic behind the ANOVA F ratio?

Compare variability explained by group differences with variability within groups (error).