By the end of today you can

• explain how ANOVA splits variation, and what F compares;
• compare three or more groups, and find which differ;
• analyse two factors and their interaction;
• fit and interpret simple and multiple linear regression;
• separate a real effect from a confounder;
• model curves and interactions;
• read diagnostic plots;
• interpret logistic regression as odds ratios and probabilities;
• report ANOVA and regression in a thesis.

One idea behind both methods

Students differ in stress, grades, and wellbeing.

Which part of those differences is connected with faculty, sleep, or support, and which part remains unexplained?

The data for this chapter

scores <- questionnaire |>
  mutate(stress_4 = 6 - stress_4,
         stress   = rowMeans(pick(stress_1:stress_6), na.rm = TRUE),
         support  = rowMeans(pick(support_1:support_6), na.rm = TRUE)) |>
  select(student_id, stress, support)

study <- students |>
  left_join(scores, join_by(student_id)) |>
  left_join(semesters |> filter(semester == 1), join_by(student_id))

One row per student: background, first semester, and questionnaire scores.

Why not ten t-tests?

Five faculties give 10 pairs.

Each test has a 5% false alarm risk, and across 10 tests the chance of at least one is about 40%.

ANOVA asks one question first: do the faculty averages differ at all?

Variation between and within groups

Group averages (copper) vs the overall average: between. Points vs their own average: within.

The F statistic

\[ F = \frac{\text{variation between groups}}{\text{variation within groups}} \]

summary(aov(stress ~ group, data = group_scores))

F near 1: averages differ no more than chance produces. Large F: they differ more.

The ratio matters, not only the gaps

tight <- group_scores |> mutate(stress = group_avg + 0.5 * offsets)
wide  <- group_scores |> mutate(stress = group_avg + 4 * offsets)

Half a point is convincing when students within a group hardly differ, and unremarkable when they differ by two.

Stress by faculty

Much overlap. Health Sciences sits a little higher, Humanities a little lower.

One-way ANOVA

stress_anova <- aov(stress ~ faculty, data = study)
summary(stress_anova)

p = 0.003: the faculties differ in stress.

The next question is how much.

Eta squared: the size of the effect

ss <- summary(stress_anova)[[1]][["Sum Sq"]]
ss[1] / sum(ss)

\(\eta^2\) ≈ 0.03: only about 3% of the differences in stress lie between faculties.

Faculty matters, but says very little about one student’s stress.

Post-hoc tests: which groups differ?

TukeyHSD(stress_anova)

Tukey’s test compares every pair while keeping the overall false alarm rate at 5%.

Only one pair differs: Health Sciences is 0.34 points more stressed than Humanities.

Assumptions of ANOVA

kruskal.test(stress ~ faculty, data = study)

Kruskal-Wallis compares ranks when assumptions fail. Here it agrees.

Two-way ANOVA

An interaction plot

Parallel lines: no interaction. Part-time is lower in both programmes.

Testing main effects and interaction

summary(aov(wellbeing ~ programme * study_mode, data = study))

programme * study_mode asks for both main effects and their interaction.

Study mode matters; programme and the interaction do not.

A significant interaction is interpreted before the main effects.

Regression models the average outcome

ANOVA compares groups. Regression models how an outcome changes with predictors, numeric or categorical.

ANOVA is a special case of regression.

Average GPA at each level of sleep

sleep_bands <- study |>
  filter(!is.na(sleep_hours)) |>
  mutate(sleep_band = round(sleep_hours * 2) / 2) |>
  summarise(mean_gpa = mean(gpa), students = n(), .by = sleep_band)

The average GPA of students who sleep 5 hours, 5.5 hours, 6 hours, and so on.

The line summarises conditional averages

The spread of grey points around the line is what the model does not explain.

Simple linear regression

\[ \text{GPA} = b_0 + b_1 \times \text{sleep} + \text{error} \]

Least squares makes the squared residuals as small as possible.

Fitting a line to five students

five <- tibble(sleep = c(5, 6, 6.5, 7, 8),
               gpa   = c(2.8, 3.0, 3.2, 3.1, 3.4))
lm(gpa ~ sleep, data = five)

Each extra hour goes with a GPA about 0.19 higher.

Nobody sleeps zero hours, so the intercept rarely means anything on its own.

Three results in summary()

gpa_simple <- lm(gpa ~ sleep_hours, data = study)
summary(gpa_simple)

Sleep matters, but it is far from the whole story.

Multiple regression

gpa_model <- lm(gpa ~ sleep_hours + study_hours + stress + support,
                data = study)

Each coefficient holds the other predictors constant.

How much the model explains

R-squared ≈ 25%: modest, and normal for grades, which depend on much no survey measures.

• adjusted R-squared corrects for the number of predictors;
• lm() drops students with any missing value: 13 here.

Always report the number of cases each model is based on.

Categorical predictors

lm(gpa ~ sleep_hours + study_hours + stress + support + gender,
   data = study)

R compares each category with a reference category, alphabetically first: Female.

genderMale is tiny and far from significant: no evidence that gender affects GPA. That is a finding too.

relevel(factor(faculty), ref = "Humanities") chooses another reference.

Two explanations for caffeine and GPA

flowchart LR
  subgraph A["Caffeine as a cause"]
    C1[Caffeine] --> G1[GPA]
  end
  subgraph B["Sleep as a confounder"]
    S2[Sleep] --> C2[Caffeine]
    S2 --> G2[GPA]
    C2 -. "apparent link" .- G2
  end

Regression can tell them apart by comparing students who sleep the same amount.

Adjusting for the confounder

lm(gpa ~ caffeine_mg, data = study)
lm(gpa ~ caffeine_mg + sleep_hours, data = study)

Among students who sleep the same, caffeine makes no difference to grades.

Regression adjusts only for what you include

A confounder is removed only if it was measured and put in the model.

Outside a randomised experiment, regression shows associations adjusted for the variables in the model, not proof of cause.

Curved relationships

gpa_curve <- lm(gpa ~ study_hours + I(study_hours^2), data = study)

I() calculates study_hours^2 before fitting.

The squared term is negative and significant: the line bends downwards.

Diminishing returns

The first hours make a real difference. The curve peaks at about 32 hours a week.

Interactions in regression

study <- study |> mutate(support_c = support - mean(support))
lm(gpa ~ support_c * programme + sleep_hours + study_hours, data = study)

Centring: zero now means average support.

Support matters about 2.3 times as much for PhD students.

What violated assumptions look like

A shapeless band is good. A U shape means a missed curve. A funnel makes p-values unreliable.

Checking the GPA model

par(mfrow = c(1, 2))
plot(gpa_model, which = 1:2)

Residuals against fitted: a shapeless band, no curve, no funnel.

Q-Q plot of residuals: close to the line, so roughly normal. The model looks good.

Tidy results with broom

library(broom)
tidy(gpa_model, conf.int = TRUE)   # coefficients with intervals
glance(gpa_model)                  # overall fit

Model results as data frames, ready for a thesis table.

Writing it up

A multiple regression of first-semester GPA on sleep, study hours, stress, and supervisor support explained 25% of the variance, F(4, 582) = 49.14, p < .001, N = 587. Holding the other predictors constant, each additional hour of sleep was associated with a GPA 0.11 points higher (95% CI [0.08, 0.13]).

The null hypothesis of a zero sleep coefficient was rejected.

Yes-or-no outcomes need logistic regression

Linear regression could predict probabilities below 0 or above 1.

Logistic regression models the probability of “yes” in a way that stays between 0 and 1.

Odds and odds ratios

jobs <- table(students$employment, students$considering_dropout)
odds_full <- jobs["Full-time job", "Yes"] / jobs["Full-time job", "No"]
odds_none <- jobs["None", "Yes"] / jobs["None", "No"]

1 means no difference; above 1 higher odds; below 1 lower.

A model for dropout

study <- study |> mutate(dropout = as.integer(considering_dropout == "Yes"))

dropout_model <- glm(
  dropout ~ stress + support + financial_worry + employment + study_mode,
  data = study, family = binomial)

exp(cbind(odds_ratio = coef(dropout_model),
          confint.default(dropout_model)))

The outcome is 0 or 1. exp() turns log-odds into odds ratios.

Reading the odds ratios

Stress raises the odds; support cuts them by about 67%. The stress hypothesis is supported.

What happened to employment?

Alone, a full-time job about doubled the odds of considering dropout.

In the model, both employment intervals include 1.

A job likely matters because of the stress and money pressure it brings. Multiple regression makes that visible.

Predicted probabilities are easier to read

examples <- tibble(stress = c(2.5, 4.0), support = c(4.0, 2.0),
                   financial_worry = 3, employment = "None",
                   study_mode = "Full-time")
predict(dropout_model, newdata = examples, type = "response")

Explaining is not predicting

This model explains who considers dropping out.

Whether it can predict that for new students, and how to test that fairly, is a different question.

It is the subject of Chapters 11 and 12.

In your field: forestry and agriculture

tree_model <- lm(Volume ~ Girth + Height, data = trees)
summary(tree_model)$r.squared

Timber volume of 31 black cherry trees from measurements taken without felling them.

Girth and height explain 95% of volume, against 25% for GPA: physics is more predictable than people.

Practical lab: the Chapter 8 playground

Work through the playground exercises in your browser, with hints and solutions.

Every exercise also runs in RStudio, from the downloadable chapter project.

Practical exercises 1–3: ANOVA and regression

1. Wellbeing by faculty: ANOVA, \(\eta^2\), and Tukey if significant.
2. Wellbeing on sleep: interpret the slope and R-squared.
3. Add stress and support: what happens to the sleep slope and R-squared?

Practical exercises 4–6: categories, odds, confounders

4. Add study mode: name the reference and interpret its coefficient.
5. Dropout on burnout alone: report and explain the odds ratio.
6. Draw a confounder from your own field, and say how regression would separate it.

Try this yourself

Choose one outcome in the student wellbeing data.

• plot it against one predictor;
• fit a simple regression, then add two plausible confounders;
• watch how the first coefficient changes;
• check the residual plot;
• write one sentence reporting the adjusted effect with its interval.

Troubleshooting guide (Part 1)

Troubleshooting guide (Part 2)

Completion checklist

• groups are compared with one ANOVA, then post-hoc tests;
• every ANOVA reports \(\eta^2\);
• interactions are checked and interpreted first;
• each coefficient is read as “holding the others constant”;
• plausible confounders are in the model;
• residual plots are checked;
• logistic results are reported as odds ratios or probabilities;
• the number of cases is reported for each model.

Misconceptions to leave behind (Part 1)

Misconceptions to leave behind (Part 2)

The chapter in one sentence

ANOVA and regression split variation into what the predictors explain and what remains, and each coefficient holds the others constant, but only the ones you measured.

Next: Chapter 9

The next chapter looks at many variables at once:

• constructs and latent variables;
• principal component analysis;
• factor analysis of the questionnaire;
• reporting a questionnaire’s structure;
• cluster analysis for student profiles.

Questions

Which confounder could explain the main association in your own research?

Did you measure it?