Continuous outcomes

1. What does ANOVA compare?

Means of more than two independent groups.

What is the null hypothesis in ANOVA?

All population group means are equal.

What does a significant ANOVA p-value tell us?

At least one group mean differs; it does not identify which group.

What does the F-statistic compare?

Between-group variation relative to within-group variation.

Why do we perform post-hoc tests?

To identify which specific groups differ after ANOVA.

What is the Bonferroni correction?

Adjusted p-value = original p-value × number of comparisons.

What are the three ANOVA assumptions and alternatives for the same

Assumption	What it means	How we check
Normality	Data are approximately bell-shaped within groups	Shapiro–Wilk
Equal variances	Groups have similar spread	Levene's test
Independence	Observations are unrelated	Study design

• Unequal variances → Welch's ANOVA.

• Non-normal outcome → consider Kruskal–Wallis.

Which R function performs standard one-way ANOVA?

aov(outcome ~ group, data = dataset)

Which R function performs Welch's ANOVA?

oneway.test(outcome ~ group, data = dataset, var.equal = FALSE)

Which R function performs the Kruskal–Wallis test?

kruskal.test(outcome ~ group, data = dataset)

What is the basic linear regression equation and what do the variables stand for

Y	Outcome	FEV1
X	Predictor (covariate)	Height
b₀	Intercept	Starting value
b₁	Slope	How much Y changes when X increases by 1

How do you fit/code a linear regression model?

model <- lm(outcome ~ predictor, data = dataset)

How do you display regression results?

summary(model)

How do you obtain 95% confidence intervals for regression coefficients?

confint(model)

How is a residual calculated?

Residual = Observed value − Predicted value.

What are the three linear regression assumptions?

• Linearity

• Homoscedasticity: Residuals have approximately equal variance across predictor values.

• Normally distributed residuals with mean zero

Which plot helps assess residual normality?

A normal Q–Q plot. Points should approximately follow a straight diagonal line.

What does a curved predictor–outcome relationship suggest? How to solve this?

Answer: The linearity assumption may be violated. Put in categories the continuous predictor

What is a reference group in categorical regression? What dummy-variable values represent the reference group?

The baseline category against which other categories are compared. All dummy variables equal zero.

A model is Y = 100 + 12d1 + 20d2. What are the three group means?

• Reference group: 100

• Group 2: 112

• Group 3: 120

How do you compare two non-reference groups using regression coefficients?

Subtract their coefficients; the intercept cancels out.

How do you create four approximately equal-sized predictor groups?

Use quantile() to find quartile cut-offs, then cut() to create categories.

breaks <- quantile(data$x, probs = seq(0, 1, 0.25))</p><p>data$x_group <- cut(data$x, breaks = breaks)

How do you extract residuals from a fitted model?

residuals(model)

Y = 8 + 3X

Answer these three questions:

1. What is the intercept?

2. What is the slope?

3. If X increases by 4 units, how much does Y increase, and what is the final predicted value of Y?

Treatment A is the reference group.

·       Intercept = 3.0

·       Treatment B coefficient = −0.8

·       Treatment C coefficient = −0.3

Calculate:

1.     Predicted mean for A.

2.     Predicted mean for B.

3.     Predicted mean for C.

4.     Difference between B and C.

Question 1. A researcher finds that the relationship between age and cholesterol follows a curved pattern rather than a straight line. Which assumption is violated?

Question 2. The residuals form a bell-shaped distribution centred around zero. Which assumption does this support?

Question 3. Residuals become increasingly spread out as the predictor increases. Which assumption is violated?

A researcher compares tumour volume between patients who received chemotherapy and patients who received a placebo.

• Predictor: Treatment group (binary)

• Outcome: Tumour volume (continuous)

Which statistical test should be used?

A. ANOVA
B. Independent samples t-test
C. Linear regression
D. Cox regression

B. Independent samples t-test

A researcher investigates whether cholesterol differs between patients with low, moderate and high physical activity.

• Predictor: Physical activity (categorical)

• Outcome: Cholesterol (continuous)

Which test is appropriate?

A. t-test
B. ANOVA
C. Linear regression with a continuous predictor
D. Logistic regression

B. ANOVA

A researcher investigates whether age is associated with tumour volume.

• Predictor: Age in years (continuous)

• Outcome: Tumour volume (continuous)

Which method should be used?

A. ANOVA
B. t-test
C. Linear regression
D. Cox regression

C. Linear regression

A researcher wants to investigate whether smoking status (smoker/non-smoker) is associated with the development of cancer (yes/no).

• Predictor: Binary

• Outcome: Binary

Can you identify the appropriate regression approach?

Logistic regression

A researcher investigates whether tumour stage (I, II, III, IV) is associated with patient survival time.

• Predictor: Categorical

• Outcome: Survival

Which type of analysis would be appropriate?

Survival analysis

A researcher investigates whether age (continuous) is associated with whether a patient develops cancer (yes/no).

Which combination is appropriate?

A. Linear regression
B. Logistic regression
C. ANOVA
D. Independent samples t-test

B. Logistic regression

Predictor (X)	Binary outcome	Continuous outcome	Survival outcome
Binary	Logistic regression	t-test	Survival analysis
Categorical	Logistic regression	ANOVA	Survival analysis
Continuous	Logistic regression	Linear regression	Survival analysis