Part 6 · Module 8

Two-Way and Factorial Analysis of Variance

Learn how factorial ANOVA estimates main effects and interactions between two categorical factors, with cell means and reproducible R output.

Level: Advanced
Audience: Medical personnel, Master of Medicine students
Reading time: 50 minutes
Updated: 3 August 2026
Last reviewed: 3 August 2026

Learning outcomes

After completing this lecture, you should be able to:

Describe a two-factor factorial design using cells, main effects and interaction.
Interpret a treatment-by-factor interaction before interpreting marginal main effects.
Fit a balanced two-way ANOVA using linear regression in R.
Calculate and interpret simple effects and interaction contrasts.
Explain the importance of factor coding, balance and sums-of-squares conventions.
Report cell means, interaction estimates and confidence intervals.

Overview

A factorial design studies two or more factors simultaneously and has a long history in efficient experimental design [1].

In a two-by-two design, two factors each have two levels, creating four cells.

Factorial ANOVA estimates:

the main effect of factor A;
the main effect of factor B;
the interaction between factors A and B.

An interaction means that the effect of one factor differs according to the level of the other factor [2].

When interaction is present, marginal main effects can conceal clinically important differences. Cell means and simple effects should be prioritised.

Why this matters

Suppose telemonitoring lowers blood pressure substantially without coaching but has little additional effect when intensive coaching is already present.

The average telemonitoring effect across coaching levels may appear modest. The interaction reveals that the effect is not constant.

Testing one subgroup and not another separately is not an interaction test. The correct question is whether the subgroup effects differ from each other.

When to use this method

Use factorial ANOVA when:

the outcome is numerical;
two or more categorical factors define independent groups;
cell-level means and interactions are scientifically meaningful;
observations are independent;
and the linear-model assumptions are sufficiently reasonable.

If one factor is repeated within participants, use a mixed or repeated-measures factorial model rather than ordinary between-participant ANOVA.

Research question and variables

The outcome is 12-week systolic blood pressure.

Factors are:

treatment;
coaching.

The four population cell means are the basic quantities.

The interaction null states that the telemonitoring effect is the same under coaching and no coaching.

Data structure

Two-by-two factorial structure

simulated data

treatment	coaching	participants	Mean systolic blood pressure
Usual care	No coaching	15	144.87 mmHg
Usual care	Coaching	15	136.40 mmHg
Telemonitoring	No coaching	15	135.27 mmHg
Telemonitoring	Coaching	15	138.13 mmHg

Note: Means are simulated.

A plot of the four means is essential. Non-parallel lines suggest interaction; crossing lines indicate qualitative interaction but are not required for statistical interaction.

Statistical assumptions

Balanced designs simplify interpretation because factor contrasts are orthogonal. In unbalanced observational data, sequential, type II and type III sums of squares can test different hypotheses. The analysis plan must specify the estimand and coding.

How the method works

A two-factor model is:

If interaction is clinically important, report:

cell means;
simple effect of treatment within each coaching level;
interaction contrast and CI.

Worked example

Interpreting a treatment-by-coaching interaction

Observed cell means are:

usual care, no coaching: 144.87 mmHg;
telemonitoring, no coaching: 135.27 mmHg;
usual care, coaching: 136.40 mmHg;
telemonitoring, coaching: 138.13 mmHg.

Simple telemonitoring effects are:

without coaching: $135.27-144.87=-9.60$ mmHg;
with coaching: $138.13-136.40=+1.73$ mmHg.

The difference in treatment effects is:

1.73-(-9.60)=11.33\text{ mmHg}

The interaction coefficient is 11.33 mmHg, 95% CI 4.30 to 18.36, $p=0.002$ .

The ordinary ANOVA table gives:

treatment main effect: $F=5.03$ , $p=0.029$ ;
coaching main effect: $F=2.55$ , $p=0.116$ ;
interaction: $F=10.43$ , $p=0.002$ .

Because the interaction is present, the marginal treatment and coaching main effects should not be interpreted as constant effects.

(Calculations for the factorial ANOVA tests, interaction p-value and 95% CI are not shown step by step here.)

Reproducible R analysis

Base R fits the factorial model with lm() and produces an ANOVA table with anova() [3], [4].

factorial <- data.frame(
  treatment = factor(
    rep(c("Usual care", "Telemonitoring"), each = 30),
    levels = c("Usual care", "Telemonitoring")
  ),
  coaching = factor(
    rep(rep(c("No coaching", "Coaching"), each = 15), 2),
    levels = c("No coaching", "Coaching")
  ),
  sbp_week12 = c(
    141,148,133,139,146,149,137,147,153,160,144,142,144,141,149,
    123,144,133,137,137,141,143,124,133,142,130,130,147,140,142,
    141,131,135,126,133,135,148,126,141,132,131,129,148,132,141,
    135,145,141,144,138,146,140,136,134,133,146,125,130,140,139
  )
)

# `lm()` fits the factorial linear model [3]
fit <- lm(
  sbp_week12 ~ treatment * coaching,
  data = factorial
)

# `anova()` provides the sequential ANOVA table for this balanced design [4]
anova(fit)
summary(fit)
confint(fit)

R ANOVA output

                   Df Sum Sq Mean Sq F value Pr(>F)
treatment           1  232.1   232.1   5.025 0.0290
coaching            1  117.6   117.6   2.547 0.1162
treatment:coaching  1  481.7   481.7  10.431 0.0021
Residuals          56 2586.0    46.2

R coefficient output

                                              Estimate Std. Error t value Pr(>|t|)
(Intercept)                                   144.8667     1.754    82.6  <2e-16
treatmentTelemonitoring                        -9.6000     2.481    -3.87  0.00029
coachingCoaching                               -8.4667     2.481    -3.41  0.00120
treatmentTelemonitoring:coachingCoaching       11.3333     3.510     3.23  0.00208

Cell means and simple effects

cell_means <- with(
  factorial,
  tapply(sbp_week12, list(treatment, coaching), mean)
)

tele_effect_no_coaching <-
  cell_means["Telemonitoring", "No coaching"] -
  cell_means["Usual care", "No coaching"]

tele_effect_coaching <-
  cell_means["Telemonitoring", "Coaching"] -
  cell_means["Usual care", "Coaching"]

interaction <-
  tele_effect_coaching - tele_effect_no_coaching

cell_means
interaction

Interpreting the output

Presenting the results

Treatment	No coaching mean (SD)	Coaching mean (SD)
Usual care	144.9 (6.7)	136.4 (7.3)
Telemonitoring	135.3 (7.1)	138.1 (6.1)

Treatment-by-coaching interaction 11.3 mmHg (95% CI 4.3 to 18.4), $F(1,56)=10.43$ , $p=0.002$ .

Suggested reporting

Factorial efficiency and interaction power MMed extension

Factorial designs can estimate two intervention main effects efficiently when interaction is negligible and treatments can be combined safely.

However, the study may have much less power for interaction than for each main effect. A non-significant interaction does not prove absence of effect modification.

The primary sample-size calculation should reflect whether interaction is a confirmatory objective; general design principles are developed in standard experimental-design texts [5].

Common mistakes

Knowledge check

What does interaction mean in this study?

Answer and reasoning

It means the telemonitoring effect on mean blood pressure differs according to whether coaching is present.

Why are separate subgroup p-values not an interaction test?

Answer and reasoning

The question is whether subgroup effects differ. That requires a direct difference-in-differences estimate and its standard error.

What is the simple telemonitoring effect without coaching?

Answer and reasoning

It is 135.27 minus 144.87, or −9.60 mmHg.

Why should marginal main effects be interpreted cautiously when interaction is present?

Answer and reasoning

They average over levels where the effect differs and may not represent any clinically relevant subgroup.

What changes if coaching is repeated within the same participants?

Answer and reasoning

The observations become dependent, requiring a mixed or repeated-measures factorial model rather than ordinary between-participant ANOVA.

Key take-home messages

References

Fisher RA. The design of experiments. Edinburgh: Oliver and Boyd; 1935.
VanderWeele TJ, Knol MJ. A tutorial on interaction. Epidemiol Methods. 2014;3(1):33–72. https://doi.org/10.1515/em-2013-0005
R Core Team. Fitting linear models. R documentation, package stats version 4.6.1. Vienna: R Foundation for Statistical Computing; 2026. https://search.r-project.org/R/refmans/stats/html/lm.html
R Core Team. ANOVA for linear model fits. R documentation, package stats version 4.6.1. Vienna: R Foundation for Statistical Computing; 2026. https://search.r-project.org/R/refmans/stats/html/anova.lm.html
Montgomery DC. Design and analysis of experiments. 10th ed. Hoboken (NJ): Wiley; 2019.

These notes support learning and research planning and do not replace statistical consultation for a specific study.

Report an error in this lecture

Learning outcomes

Overview

Why this matters

When to use this method

Research question and variables

Data structure

Statistical assumptions

How the method works

Worked example

Reproducible R analysis

R ANOVA output

R coefficient output

Cell means and simple effects

Interpreting the output

Presenting the results

Suggested reporting

Common mistakes

Knowledge check

Key take-home messages

References

Related resources

Lectures