Two-Way ANOVA

Introduction

• Recall that ANOVA allows us to compare the means of three or more groups.

• In one-way ANOVA, we are only considering one factor (grouping variable).

• We will now discuss two-way ANOVA, which allows us to consider a second factor (grouping variable).

R Setup

library(tidyverse)
library(ssstats)

supplement_heart_rate <- read_csv("https://raw.githubusercontent.com/samanthaseals/STA4173Fa26/refs/heads/main/lectures/data/supplement_heart_rate.csv") 
exercise_recovery <- read_csv("https://raw.githubusercontent.com/samanthaseals/STA4173Fa26/refs/heads/main/lectures/data/exercise_recovery.csv") 

Two-Way ANOVA

• Recall in one-way ANOVA, we partitioned SS_Tot into SS_Trt and SS_E.

• Now that multiple factors are involved, we now partition the SS_Trt into the different factors.

  • Here, that is SS_A, SS_B, and SS_AB.

• Recall that SS_E is the “catch all” for unexplained variance.

  • When we add factors to our model, we are moving part of the SS_E into the SS_Trt.

Two-Way ANOVA

• Terminology in two-way ANOVA.

• Factor A has a levels.

  • Supplement treatment: placebo, supplement

• Factor B has b levels.

  • Pony type: Earth pony, Pegasus, Unicorn

• There are ab “treatment groups.”

  • Every pony belongs to one level of A and one level of B.

  • Here, a=2, b=3, and ab=6 treatment groups.

Interaction Effects

• Fluttershy is testing whether a new dietary supplement affects resting heart rate in adult ponies.

  • The outcome is resting heart rate, measured in beats per minute after a standardized rest period.

• She is considering two factors:

  • Treatment: Placebo or Supplement
  • Pony type: Earth pony, Pegasus, or Unicorn

• Within each pony type, ponies are randomly assigned to a treatment; each pony is measured once.

Interaction Effects

• This design allows us to check for an interaction effect, implying the two factors don’t act independently on the outcome.

  — i.e., the effect of one factor on the outcome depends on the level of the other factor.

• In Fluttershy’s supplement study, an interaction implies both:

  • The relationship between resting heart rate and treatment depends on pony type.

  • The relationship between resting heart rate and pony type depends on treatment.

• Remember, these are two descriptions of the same interaction, not two separate effects.

Two-Way ANOVA: Table

• The table we are working towards:

Two-Way ANOVA: Computation

• Let there be a levels of factor A and b levels of factor B.

• These formulas assume a balanced, fully crossed design with n independent ponies per cell.

  • y_{ijk} is the observation on the k^{th} experimental unit receiving the i^{th} level of factor A and the j^{th} level of factor B
  • y_{i.} is the sum for all observations at the i^{th} level of factor A,
  • y_{.j} is the sum for all observations at the j^{th} level of factor B,
  • y_{ij} is the sum for observations at the i^{th} level of factor A and the j^{th} level of factor B,
  • y_{..} is the sum of all observations,
  • (y^2)_{..} is the sum of the squared observations,
  • n is the number in each group (of a \times b treatments)

Two-Way ANOVA: Computation

• To find the SS and df: \begin{align*} \text{SS}_{\text{A}} &= \frac{\sum_i y_{i.}^2}{bn} - \frac{(y_{..})^2}{abn} & \text{df}_{\text{A}} &= a-1 \\ \text{SS}_{\text{B}} &= \frac{\sum_j y_{.j}^2}{an} - \frac{(y_{..})^2}{abn} & \text{df}_{\text{B}} &= b-1 \\ \text{SS}_{\text{AB}}&= \frac{\sum_{ij} y_{ij}^2}{n} - \frac{(y_{..})^2}{abn} - \text{SS}_{\text{A}} - \text{SS}_{\text{B}} & \text{df}_{\text{AB}} &= (a-1)(b-1) \\ \text{SS}_{\text{E}} &= \text{SS}_{\text{Tot}} - \text{SS}_{\text{A}} - \text{SS}_{\text{B}} - \text{SS}_{\text{AB}} & \text{df}_{\text{E}} &= ab(n-1) \\ \text{SS}_{\text{Tot}} &= (y^2)_{..} - \frac{(y_{..})^2}{abn} & \text{df}_{\text{Tot}} &= abn-1 \end{align*}

Two-Way ANOVA: Computation

• To find the MS:

\text{MS}_{\text{X}} = \frac{\text{SS}_{\text{X}}}{\text{df}_{\text{X}}}

• To find the test statistics:

\text{F}_{\text{X}} = \frac{\text{MS}_{\text{X}}}{\text{MS}_{\text{E}}}

Two-Way ANOVA (R)

• To get the two-way ANOVA table in R,

1dataset_name |> anovas(outcome = response_variable,
2                       A = factor_A,
3                       B = factor_B,
4                       interaction = TRUE)

Hypothesis Test for an Interaction

• Hypotheses

  • H_0: there is not an interaction between [factor A] and [factor B]

  • H_1: there is an interaction between [factor A] and [factor B]

• Test Statistic and p-Value

  • F_{\text{AB}}
  • p = P[F_{\text{df}_{\text{AB}}, \text{df}_{\text{E}}} \ge F_{\text{AB}}]

Hypothesis Test for an Interaction (R)

• We again use the anovas() function,

dataset_name |> anovas(outcome = response_variable,
                       A = factor_A,
                       B = factor_B,
                       interaction = TRUE,
                       alpha = alpha_level)

Example 1

• Fluttershy asks: “Does the supplement affect every pony type in the same way?”

• We will use the supplement_heart_rate dataset to answer:

  • Does the effect of treatment on resting heart rate depend on pony type?

• There are 72 adult ponies: 12 in each of the six treatment-by-type groups.

Example 1

• Exploring the levels in treatment and pony type,

supplement_heart_rate |> n_pct(treatment, pony_type)

Example 1

• Does the effect of treatment on resting heart rate depend on pony type?

supplement_heart_rate |> anovas(outcome = resting_heart_rate_bpm,
                                 A = treatment,
                                 B = pony_type,
                                 interaction = TRUE)

Two-way ANOVA (with interaction)

Source               df         SS        MS       F  p-value 
──────────────────────────────────────────────────────── 
treatment             1   215.2813  215.2813  15.409  < 0.001 
pony_type             2  1383.6803  691.8401  49.519  < 0.001 
treatment:pony_type   2   636.6508  318.3254  22.785  < 0.001 
Residuals            66   922.0942   13.9711                  
──────────────────────────────────────────────────────── 


Hypotheses (treatment:pony_type):
H₀: There is no interaction between treatment and pony_type
H₁: There is an interaction between treatment and pony_type
Test statistic: F(2, 66) = 22.785
p-value: < 0.001
Conclusion: Reject the null hypothesis (p = < 0.001 < α = 0.050)

Example 1

• Hypotheses

  • H_0: there is no interaction between treatment and pony type

  • H_1: there is an interaction between treatment and pony type

• Test Statistic and p-Value

  • F_0 = 22.785
  • p < 0.001

• Rejection Region

  • Reject H_0 if p < \alpha; \alpha = 0.05

• Conclusion and Interpretation

  • Reject H_0. There is sufficient evidence that the supplement’s effect on mean resting heart rate depends on pony type.

Example 2

• Twilight Sparkle asks: “Can the Ponyville clinic help ponies recover faster after exercise? Does the treatment work the same way across pony types?”

• We will use exercise_recovery to test whether the effect of recovery treatment depends on pony type.

• Outcome: minutes until heart rate returns to a prespecified range near the pony’s baseline after a standardized exercise session.

• Within each type, ponies are randomly assigned to Standard care, New treatment, or Enhanced treatment.

• There are 108 different ponies: 12 in each of nine groups; each pony completes one session.

Example 2

• Exploring the levels in treatment and pony type,

exercise_recovery |> n_pct(treatment, pony_type)

Example 2

• Does the effect of treatment on recovery time depend on pony type?

exercise_recovery |> anovas(outcome = recovery_time_min,
                            A = treatment,
                            B = pony_type,
                            interaction = TRUE)

Two-way ANOVA (with interaction)

Source               df         SS         MS       F  p-value 
───────────────────────────────────────────────────────── 
treatment             2  2909.4913  1454.7456  79.154  < 0.001 
pony_type             2    23.1424    11.5712   0.630    0.535 
treatment:pony_type   4    47.5704    11.8926   0.647    0.630 
Residuals            99  1819.4858    18.3786                  
───────────────────────────────────────────────────────── 


Hypotheses (treatment:pony_type):
H₀: There is no interaction between treatment and pony_type
H₁: There is an interaction between treatment and pony_type
Test statistic: F(4, 99) = 0.647
p-value: 0.630
Conclusion: Fail to reject the null hypothesis (p = 0.630 ≥ α = 0.050)

Example 2

• Hypotheses

  • H_0: there is no interaction between treatment and pony type

  • H_1: there is an interaction between treatment and pony type

• Test Statistic and p-Value

  • F_0 = 0.647

  • p = 0.630

• Rejection Region

  • Reject H_0 if p < \alpha; \alpha = 0.05

• Conclusion and Interpretation

  • Fail to reject H_0. There is not sufficient evidence that treatment’s effect on mean recovery time depends on pony type.

Two-Way ANOVA: Testing Main Effects

• What if the interaction is not significant?

  • We remove the nonsignificant interaction and examine the main effects.

  • That is, we want to look at the individual factors.

• We will reconstruct the ANOVA table. If we were doing this by hand:

  1. Rewrite ANOVA table without interaction; do not change SS, df, and MS for the main effects and total.
  2. Update the error term: add the SS_AB to SS_E and df_AB to df_E.
  3. Recalculate MS_E.
  4. Recalculate F_A and F_B, then find the corresponding p-values.

Hypothesis Test for Main Effects

• Hypotheses

  • H_0: \mu_1 = \mu_2 = … = (\mu_a or \mu_b)
  • H_1: At least one \mu_i is different.

• Test Statistic and p-Value

  • F_{\text{X}}
  • p = P[F_{\text{df}_{\text{X}}, \text{df}_{\text{E}}} \ge F_{\text{X}}]

Hypothesis Test for Main Effects (R)

• We again use the anovas() function,

dataset_name |> anovas(outcome = response_variable,
                       A = factor_A,
                       B = factor_B,
                       interaction = FALSE,
                       alpha = alpha_level)

Example 2

• Are either recovery treatment or pony type related to mean recovery time?

exercise_recovery |> anovas(outcome = recovery_time_min,
                            A = treatment,
                            B = pony_type,
                            interaction = FALSE)

Two-way ANOVA (no interaction)

Source      df         SS         MS       F  p-value 
──────────────────────────────────────────────── 
treatment    2  2909.4913  1454.7456  80.254  < 0.001 
pony_type    2    23.1424    11.5712   0.638    0.530 
Residuals  103  1867.0562    18.1268                  
──────────────────────────────────────────────── 


Hypotheses (treatment):
H₀: All means across levels of treatment are equal
H₁: At least one mean across levels of treatment differs
Test statistic: F(2, 103) = 80.254
p-value: < 0.001
Conclusion: Reject the null hypothesis (p = < 0.001 < α = 0.050)

Hypotheses (pony_type):
H₀: All means across levels of pony_type are equal
H₁: At least one mean across levels of pony_type differs
Test statistic: F(2, 103) = 0.638
p-value: 0.530
Conclusion: Fail to reject the null hypothesis (p = 0.530 ≥ α = 0.050)

Example 2: Treatment Main Effect

• Hypotheses

  • H_0: \mu_{\text{Standard}} = \mu_{\text{New}} = \mu_{\text{Enhanced}}
  • H_1: at least one mean recovery time differs across treatments

• Test Statistic and p-Value

  • F_0 = 80.254
  • p < 0.001

• Rejection Region

  • Reject H_0 if p < \alpha; \alpha = 0.05

• Conclusion and Interpretation

  • Reject H_0. There is sufficient evidence that mean recovery time differs among treatments, accounting for pony type in the additive model.

Example 2: Pony-Type Main Effect

• Hypotheses

  • H_0: \mu_{\text{Earth}} = \mu_{\text{Pegasus}} = \mu_{\text{Unicorn}}
  • H_1: at least one mean recovery time differs across pony types

• Test Statistic and p-Value

  • F_0 = 0.638
  • p = 0.530

• Rejection Region

  • Reject H_0 if p < \alpha; \alpha = 0.05

• Conclusion and Interpretation

  • Fail to reject H_0. There is not sufficient evidence of a difference in mean recovery time among pony types, accounting for treatment.

Wrap Up

• In this lecture, we have covered two-way ANOVA.

  • Test for interaction first.

  • If interaction significant \to next lecture.

  • If interaction is not significant \to remove, then reconstruct ANOVA table.

• On Wednesday, we will discuss next steps after initial hypothesis testing in two-way ANOVA.

  • posthoc testing

  • profile plots

Let’s Practice 1: Applejack’s Nutrition Study

• Applejack and the Ponyville clinic are studying whether a high-fiber feed changes postmeal blood glucose. Within each pony type, adult ponies are randomly assigned to Standard feed or High-fiber feed for two weeks. After a standardized meal, glucose is measured once at the same prespecified time for every pony.

• Outcome: Postmeal blood glucose, in mg/dL (postmeal_glucose_mg_dl).

• Factors:

  • treatment
  • pony type

Let’s Practice 2: Fluttershy’s Inflammation Study

• Fluttershy is testing an anti-inflammatory treatment in adult ponies with the same mild inflammatory condition. Within each pony type, ponies are randomly assigned to Placebo, Low dose, or High dose. A blood inflammation marker is measured once after seven days.

• Outcome: Inflammation marker concentration, in mg/L (inflammation_marker_mg_l).

Factors:

• treatment
• pony type

Let’s Practice 3: Nurse Redheart’s Wound-Healing Study

• Nurse Redheart compares dressings in adult ponies receiving care for comparable minor wounds. Within each pony type, ponies are randomly assigned to Standard dressing, Hydrogel, or Enhanced hydrogel. The clinic records the time from first dressing to complete wound closure using a common clinical criterion. Each pony contributes one wound.

• Outcome: Time to wound closure, in days (wound_closure_days).

• Factors:

  • treatment
  • pony type

Let’s Practice 4: Rainbow Dash’s Rehydration Study

• Rainbow Dash helps the Ponyville clinic compare two fluid treatments after a standardized exercise session. Within each pony type, adult ponies are randomly assigned to Standard fluids or Electrolyte fluids. Time to reach a prespecified hydration target is recorded once for each pony.

• Outcome: Time to reach the hydration target, in minutes (rehydration_time_min).

• Factors:

  • treatment
  • pony type