Two-Way ANOVA

STA4173: Biostatistics
Spring 2025

Introduction: Two-Way ANOVA

  • 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).

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

  • We now partition the SSTrt into the different factors under consideration.

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

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

Introduction: Two-Way ANOVA

  • Let’s discuss some of the language used in two-way ANOVA.

  • Factor A has a levels.

    • Smoking status: non-smoker and smoker

  • Factor B has b levels.

    • Hypertension: normotensive and hypertensive

  • There are ab treatment groups.

    • Non-smoker, normotensive
    • Non-smoker, hypertensive
    • Smoker, normotensive
    • Smoker, hypertensive

Introduction: Two-Way ANOVA

  • Now that we are including two factors, we must consider the interaction between the two.

    • Interaction: The relationship between [outcome] and [factor 1] depends on the level of [factor 2].

  • In our example, suppose we are looking at weight as the outcome. If an interaction is detected,

    • the relationship between weight and hypertension status depends on smoking status, OR
    • the relationship between weight and smoking status depends on hypertension status.

Introduction: Two-Way ANOVA

  • The ANOVA table that we are working to construct:
Source Sum of Squares df Mean Square F
A SSA dfA MSA FA
B SSB dfB MSB FB
AB SSAB dfAB MSAB FAB
Error SSE dfE MSE
Total SSTot dfTot

Computation

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

    • 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)

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*}

Computation

  • To find the MS:

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

  • To find the test statistic:

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

R Syntax

  • We will again use lm() and anova() OR aov() and summary() to analyze the data.

    • For reasons covered in one-way ANOVA, we will focus on using aov().

  • As we are specifying our model, we will:

    • include additional terms with +
    • include interaction terms with :
m <- aov(continuous_variable ~ factor_A + factor_B + factor_A:factor_B,
         data = dataset_name)
summary(m)

Example

  • Suppose we want to examine the effect of diet and age of mothers on the average birth weight (in ounces). Consider the following data,
  • What is the outcome?
  • What are the two factors?
  • What is the interaction term?

Example

library(tidyverse)
example <- tibble(birthweight = c(157.78, 136.79, 138.84, # age 20-29, diet 1
                                  139.72, 125.47, 117.14, # age 20-29, diet 2
                                  129.35, 110.73, 118.38, # age 20-29, diet 3
                                  137.07, 146.28, 130.27, # age 30-39, diet 1
                                  117.46, 128.54, 99.16, # age 30-39, diet 2
                                  97.43, 125.26, 115.42), # age 30-39, diet 3
                  diet = as.factor(c(1, 1, 1, 
                                     2, 2, 2, 
                                     3, 3, 3, 
                                     1, 1, 1, 
                                     2, 2, 2, 
                                     3, 3, 3)),
                  age = as.factor(c(20, 20, 20, 
                                    20, 20, 20, 
                                    20, 20, 20, 
                                    30, 30, 30, 
                                    30, 30, 30, 
                                    30, 30, 30)))

Example

  • Let’s find the ANOVA table,
m <- aov(birthweight ~ diet + age + diet:age, data = example)
summary(m)
            Df Sum Sq Mean Sq F value  Pr(>F)   
diet         2 2104.7  1052.3   7.555 0.00752 **
age          1  332.0   332.0   2.384 0.14855   
diet:age     2   32.5    16.3   0.117 0.89083   
Residuals   12 1671.5   139.3                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Testing Interactions

  • 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}} (pull from ANOVA table)
    • p = P[F_{\text{df}_{\text{AB}}, \text{df}_{\text{E}}} \ge F_{\text{AB}}]

  • Rejection Region

    • Reject H_0 if p<\alpha.

Example

  • Let’s now test the interaction between age and diet. Test at the \alpha=0.05 level.

  • Here’s the information we need:

summary(m)
            Df Sum Sq Mean Sq F value  Pr(>F)   
diet         2 2104.7  1052.3   7.555 0.00752 **
age          1  332.0   332.0   2.384 0.14855   
diet:age     2   32.5    16.3   0.117 0.89083   
Residuals   12 1671.5   139.3                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example

  • Hypotheses

    • H_0: there is not an interaction between age and diet
    • H_1: there is an interaction between age and diet

  • Test Statistic

    • F_{\text{AB}} = 0.117
    • p = 0.891

  • Rejection Region

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

  • Conclusion/Interpretation

    • Fail to reject H_0. There is not sufficient evidence to suggest that the relationship between average birth weight and age depends on the mother’s diet.

Testing Interactions

  • What happens after testing for an interaction?

    • If significant (reject H_0), we can:

      • Construct a profile plot to visualize what’s going on and help explain the effect.
      • Examine posthoc testing to further examine the interaction.

    • If not significant (FTR H_0), we should remove the interaction term so that we can test and interpret the main effects.

  • Remember that we cannot look at main effects (A and B alone) when the interaction is included in the model!

Removing Interactions

  • If the interaction term is not significant, it should be removed from the ANOVA table so that we can test and interpret the main effects.

  • How to remove an interaction:

    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_{\text{AB}} to SS_{\text{E}} and df_{\text{AB}} to df_{\text{E}}.
    3. Recalculate MS_{\text{E}}.
    4. Recalculate F statistics for the main effects and perform appropriate hypothesis tests.

  • In R, we just literally remove the interaction term.

m <- aov(continuous_variable ~ factor_A + factor_B,
         data = dataset_name)
summary(m)

Example

  • Let’s now remove the interaction between diet and maternal age.
#m <- aov(birthweight ~ diet + age + diet:age, data = example)
m <- aov(birthweight ~ diet + age, data = example)
summary(m)
            Df Sum Sq Mean Sq F value  Pr(>F)   
diet         2   2105  1052.3   8.646 0.00359 **
age          1    332   332.0   2.728 0.12085   
Residuals   14   1704   121.7                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Testing Main Effects

  • Now that we have removed the interaction term, we can test for main effects.

  • Hypotheses

    • H_0: there is not a main effect of [factor X]
    • H_1: there not a main effect of [factor X]

  • Test Statistic and p-Value

    • F_{\text{X}} (pull from ANOVA table)
    • p = P[F_{\text{df}_{\text{X}}, \text{df}_{\text{E}}} \ge F_{\text{X}}]

  • Rejection Region

    • Reject H_0 if p<\alpha.

Testing Main Effects

  • A note on hypotheses – we are writing them in sentence form here, however, we can write them mathematically.

  • For Factor A with a levels,

    • H_0: \mu_1 = \mu_2 = ... = \mu_a
    • H_1: at least one is different

  • For Factor B with b levels,

    • H_0: \mu_1 = \mu_2 = ... = \mu_b
    • H_1: at least one is different

Example

  • We now want to determine if there are main effects of age and diet. Test at the \alpha=0.05 level.

  • The ANOVA table without the interaction term:

summary(m)
            Df Sum Sq Mean Sq F value  Pr(>F)   
diet         2   2105  1052.3   8.646 0.00359 **
age          1    332   332.0   2.728 0.12085   
Residuals   14   1704   121.7                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example

  • Hypotheses

    • H_0: there is not a main effect of diet (\mu_1 = \mu_2 = \mu_3)
    • H_1: there is a main effect of diet (at least one \mu_i is different)

  • Test Statistic and p-Value

    • F_{\text{diet}} = 8.646
    • p = 0.004

  • Rejection Region

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

  • Conclusion/Interpretation

    • Reject H_0. There is sufficient evidence to suggest that at least one diet reults in a different birth weight.

Example

  • Hypotheses

    • H_0: there is not a main effect of age (\mu_1 = \mu_2)
    • H_1: there is a main effect of age (\mu_1 \ne \mu_2)

  • Test Statistic and p-Value

    • F_{\text{age}} = 2.728
    • p = 0.121

  • Rejection Region

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

  • Conclusion/Interpretation

    • Fail to reject H_0. There is not sufficient evidence to suggest that there is a difference in birth weights across the age groups.

Introduction: Beyond the F Test

  • In this lecture, we are discussing two-way ANOVA.

  • So far, we have learned how to:

    • Create an ANOVA model with an interaction term
    • Test the interaction term
    • Create an ANOVA model without an interaction term
    • Test the main effects

  • Now we will learn how to best communicate results, whether it’s the resulting interaction term or main effects.

Profile Plots

  • If we detect an interaction term, we must give meaning to it.

  • An easy way to do this is to plot the treatment group means to visualize the interaction. This is called a profile plot.

    • y-axis: always the average outcome

    • x-axis: either factor A or B

      • if only one factor is ordinal, use it for the x-axis

      • if there are two ordinal or two nominal factors, select the factor with the largest number of levels for the x-axis

    • lines on the plot: the factor that was not selected for the x-axis

  • Note that this is just a graph of the means – it’s valid to construct a profile plot even if the interaction is not sigificant.

Example

  • Let’s first find the treatment means for the birth weight data,
means <- example %>%
  group_by(diet, age) %>%
  summarize(mean = mean(birthweight)) %>%
  ungroup()
head(means)

Example

  • The average birth weight will go on our y-axis.

  • Because only maternal age is ordinal, it will go on our x-axis.

  • Thus, the lines will represent the diet

    • This means I want to restructure the dataset with the means to have a column for each of the diets
A <- means %>% filter(diet == 1) %>% rename(d1 = mean) %>% select(-diet)
B <- means %>% filter(diet == 2) %>% rename(d2 = mean) %>% select(-diet)
C <- means %>% filter(diet == 3) %>% rename(d3 = mean) %>% select(-diet)
graph <- full_join(A, B, by = "age")
graph <- full_join(graph, C, by = "age")

Example

graph

Example

  • Building profile plots using ggplot() is a process.

    • First, include a geom_line() for each level of the factor creating the lines.
graph %>% 
  ggplot(aes(x = age, group = 1)) +
  geom_line(aes(y = d1), color = "pink") + # diet 1
  geom_line(aes(y = d2), color = "purple") + # diet 2
  geom_line(aes(y = d3), color = "blue") + # diet 3
  theme_bw()

Example

  • Building profile plots using ggplot() is a process.

    • First, include a geom_line() for each level of the factor creating the lines.

Example

  • Building profile plots using ggplot() is a process.

    • Then, add geom_text() to label each line (use the line colors to make sure everything is labeled properly).
graph %>% 
  ggplot(aes(x = age, group = 1)) +
  geom_line(aes(y = d1), color = "pink") + # diet 1
  geom_line(aes(y = d2), color = "purple") + # diet 2
  geom_line(aes(y = d3), color = "blue") + # diet 3 
  geom_text(aes(x = "30" , y = 137, label = "Diet 1"), color = "pink")  + # diet 1
  geom_text(aes(x = "30" , y = 115, label = "Diet 2"), color = "purple")  + # diet 2
  geom_text(aes(x = "30" , y = 110, label = "Diet 3"), color = "blue")  + # diet 3
  theme_bw()

Example

  • Building profile plots using ggplot() is a process.

    • Then, add geom_text() to label each line (use the line colors to make sure everything is labeled properly).

Example

  • Building profile plots using ggplot() is a process.

    • Then, clean up time: fix axis titles and change line colors to black*
graph %>% 
  ggplot(aes(x = age, group = 1)) +
  geom_line(aes(y = d1), color = "black") + # diet 1
  geom_line(aes(y = d2), color = "black") + # diet 2
  geom_line(aes(y = d3), color = "black") + # diet 3+ 
  geom_text(aes(x = "30" , y = 137, label = "Diet 1"), color = "black")  + # diet 1
  geom_text(aes(x = "30" , y = 115, label = "Diet 2"), color = "black")  + # diet 2
  geom_text(aes(x = "30" , y = 110, label = "Diet 3"), color = "black")  + # diet 3
  labs(x = "Maternal Age",
       y = "Average Birth Weight") +
  theme_bw()

Example

Posthoc Tests

  • We can also apply posthoc testing to two-way ANOVA.

    • If the interaction is significant, we can compare all treatment groups.

    • If the interaction is not significant, we can look at the main effects.

  • For simplicity, we will only look at Tukey’s and Fisher’s.

  • What is the difference between Tukey’s and Fisher’s?

Example

  • Let’s apply Tukey’s to the model with the interaction term.

    • Important!! Remember that we are only doing this for illustration purposes - we should not perform posthoc testing on something non-significant.
m <- aov(birthweight ~ diet + age + diet:age, data = example)
TukeyHSD(m)$"diet:age"
                diff       lwr       upr      p adj
2:20-1:20 -17.026667 -49.39499 15.341659 0.51846118
3:20-1:20 -24.983333 -57.35166  7.384992 0.17257865
1:30-1:20  -6.596667 -38.96499 25.771659 0.98037295
2:30-1:20 -29.416667 -61.78499  2.951659 0.08309462
3:30-1:20 -31.766667 -64.13499  0.601659 0.05550164
3:20-2:20  -7.956667 -40.32499 24.411659 0.95688090
1:30-2:20  10.430000 -21.93833 42.798326 0.87931551
2:30-2:20 -12.390000 -44.75833 19.978326 0.78717713
3:30-2:20 -14.740000 -47.10833 17.628326 0.65386678
1:30-3:20  18.386667 -13.98166 50.754992 0.44210449
2:30-3:20  -4.433333 -36.80166 27.934992 0.99674874
3:30-3:20  -6.783333 -39.15166 25.584992 0.97786517
2:30-1:30 -22.820000 -55.18833  9.548326 0.24087234
3:30-1:30 -25.170000 -57.53833  7.198326 0.16754036
3:30-2:30  -2.350000 -34.71833 30.018326 0.99984711

Example

  • Let’s apply Tukey’s to the model without the interaction term.
m <- aov(birthweight ~ diet + age, data = example)
TukeyHSD(m)$diet
          diff       lwr       upr      p adj
2-1 -19.923333 -36.59450 -3.252169 0.01902981
3-1 -25.076667 -41.74783 -8.405502 0.00397944
3-2  -5.153333 -21.82450 11.517831 0.70372399
  • Which diets are significantly different? Use \alpha=0.05.

Wrap Up

  • We did not explicitly discuss assumptions, but they are the same as in one-way ANOVA.

  • From here, ANOVA can be used to specify any model.

    • We stopped with two-way interactions, but you could have three-way interactions, four-way interactions, etc.

    • As you can see, interactions quickly complicate the interpretation, so I try to stay away from anything higher than a two-way interaction.

  • Our next module is regression analysis, which is the same as ANOVA, but represents the results differently.

    • We can quantify the relationship between two variables using regression.