Two-Way ANOVA

Several Treatment Factors
Math 219

Experiments with two factors

We have considered experiments with one treatment factor and one blocking factor
Now we will consider experiments with two treatment factors
The approach generalizes to more than two factors as well

Experimental Designs

Randomized complete block design (RCBD): 1 treatment factor with \(t\) levels, 1 blocking factor with \(b\) levels, exactly \(t\) experimental units in each block (one for each treatment level)
Factorial design: 2 or more factors, 1 or more observations per cell (replications)
Replication is necessary if you need to evaluate interaction between factors
A factorial design is called balanced if there is the same number of replications in each cell
Our focus will be on balanced factorial designs

Iron Content and Pot Type

Anemia, caused by iron deficiency, is a common form of malnutrition in developing countries
Does iron content in cooked food depend on the type of pot?
Do the results depend on the type of food?
Data from Adish et al. (1999) ¹
ironpot dataset from HH package

Response: Iron (mg per 100 g of food)
Two Treatment factors:
- Pot with 3 levels (\(a=3\)): “Aluminum”, “Clay”, “Iron”
- Food with 3 levels (\(b=3\)): “meat”, “legumes”, “vegetables”
Meat, legume, and vegetable dishes cooked according to recipes from Ethiopian Nutritional Institute
Each dish was cooked 4 times in each type of pot (4 experimental units (replicants) per cell, 9 cells)

Additive Statistical Model

Additive model for the \(k\)th observation at the \(i\)th level of factor \(A\) and the \(j\)the level of factor \(B\)

\[y_{ijk}=\mu+\alpha_i+\beta_j+\varepsilon_{ijk}\]

\(\mu\) is the overall/grand mean
\(\alpha_i\) is the differential effect of the \(i\)th level of treatment \(A\)
\(\beta_j\) is the differential effect of the \(j\)th level of treatment \(B\)
\(\varepsilon_{ijk}\sim N(0,\sigma^2)\) represents the error

Hypothesis Tests

There are two null-hypotheses of interest
For type of pot:
- \(H_0: \alpha_1 = \alpha_2 = \alpha_3 = 0\)
- \(H_A:\) At least one \(\alpha_i\) is different
For type of food:
- \(H_0: \beta_1 = \beta_2 = \beta_3 = 0\)
- \(H_A:\) At least one \(\beta_i\) is different

Iron content in food cooked in different types of pots.

Iron content of different types of cooked food.

Iron content of different types of food cooked in different types of pots.

ANOVA Table (Additive Model)

lm(Iron ~ Pot + Food, data = ironpot) |>
  anova() |>
  tidy()

# A tibble: 3 × 6
  term         df sumsq meansq statistic   p.value
  <chr>     <int> <dbl>  <dbl>     <dbl>     <dbl>
1 Pot           2 24.9  12.4        61.4  1.65e-11
2 Food          2  9.30  4.65       22.9  7.71e- 7
3 Residuals    31  6.28  0.203      NA   NA

Does order matter?

lm(Iron ~ Pot + Food, data = ironpot) |>
  anova() |>
  tidy()

# A tibble: 3 × 6
  term         df sumsq meansq statistic   p.value
  <chr>     <int> <dbl>  <dbl>     <dbl>     <dbl>
1 Pot           2 24.9  12.4        61.4  1.65e-11
2 Food          2  9.30  4.65       22.9  7.71e- 7
3 Residuals    31  6.28  0.203      NA   NA

lm(Iron ~ Food + Pot, data = ironpot) |>
  anova() |>
  tidy()

# A tibble: 3 × 6
  term         df sumsq meansq statistic   p.value
  <chr>     <int> <dbl>  <dbl>     <dbl>     <dbl>
1 Food          2  9.30  4.65       22.9  7.71e- 7
2 Pot           2 24.9  12.4        61.4  1.65e-11
3 Residuals    31  6.28  0.203      NA   NA

Due to the experimental design, there is no association between them
There are the same number of replicates of each dish cooked with each type of pot
As a result, the order that the treatment variables enter the model does not matter

Conclusions based on additive model

Using the additive model, we would conclude that there is convincing evidence of an effect for both food type and pot type
However, the additive model assumes that the effect of pot type is the same for each food type
This does not appear to be the case
We need to consider interaction between two treatment factors (which means the effect of one of those variables on a third variable is not constant - the effect differs at different values of the other)

Iron content of different types of food cooked in different types of pots.

Statistical Model with Interaction

Statistical model with interaction

\[y_{ijk}=\mu+\alpha_i+\beta_j+(\alpha\beta)_{ij}+\varepsilon_{ijk}\]

\(\mu\) is the overall/grand mean
\(\alpha_i\) is the differential effect of the \(i\)th level of treatment \(A\)
\(\beta_j\) is the differential effect of the \(j\)th level of treatment \(B\)
\((\alpha\beta)_{ij}\) models the possible interaction between treatment \(A\) and \(B\)
\(\varepsilon_{ijk}\sim N(0,\sigma^2)\) represents the error

For example, if cooking a dish in an iron pot (\(i=3\)) raises the iron level more for meat (\(j=2\)) than it does for legumes or vegetables, then we would expect \((\alpha\beta)_{3,2}>0\)
Treatments \(A\) and \(B\) interact if the difference in response between two levels of \(A\) depends on the level of \(B\)
If an interaction is present, it is inappropriate to use an additive model

Properties of Interaction Model

The interaction coefficients satisfy \[\sum_{i=1}^a(\alpha\beta)_{ij}=\sum_{j=1}^b(\alpha\beta)_{ij}=0\]
The model can be written more simply as \[y_{ijk}=\mu_{ij}+\varepsilon_{ijk}\] where \(\mu_{ij}\) is the mean for level \(i\) of treatment \(A\) and level \(j\) of treatment \(B\):\[\mu_{ij}=\mu + \alpha_i+\beta_j+(\alpha \beta)_{ij}\]

Testing for an Interaction

Test for an interaction using the hypotheses
- \(H_0: (\alpha\beta)_{ij} = 0,\) for all \(i=1\ldots a\), \(j=1\ldots b\)
- \(H_A:\) At least one \((\alpha\beta)_{ij}\) is different
If interaction is present, we do not test for main effects
If there is not a significant interaction, we can drop the interaction and use the additive model instead
We can only test for an interaction if there is replication, otherwise there are not enough degrees of freedom

ANOVA table with interaction

lm(Iron ~ Pot * Food, data = ironpot) |>
  anova() |>
  tidy()

# A tibble: 4 × 6
  term         df sumsq meansq statistic   p.value
  <chr>     <int> <dbl>  <dbl>     <dbl>     <dbl>
1 Pot           2 24.9  12.4       92.3   8.53e-13
2 Food          2  9.30  4.65      34.5   3.70e- 8
3 Pot:Food      4  2.64  0.660      4.89  4.25e- 3
4 Residuals    27  3.64  0.135     NA    NA

ANOVA table key (\(n\) = number of observations in each cell)

term	df	sumsq	meansq	statistic
Treatment A	\(df_A=a-1\)	\(SSA\)	\(MSA=SSA/df_A\)	\(F=MSA/MSE\)
Treatment B	\(df_B=b-1\)	\(SSB\)	\(MSB=SSB/df_B\)	\(F=MSB/MSE\)
Interaction AB	\(df_{AB}=(a-1)(b-1)\)	\(SSAB\)	\(MSAB=SSAB/df_{AB}\)	\(F=MSAB/MSE\)
Residuals (error)	\(df_E=ab(n-1)\)	\(SSE\)	\(MSE=SSE/df_E\)

There is a significant interaction between food type and plot type
There is convincing evidence that the response of iron content to pot type depends on the type of food

Interaction sum of squares

The model predicts the response (iron content) for an observation with level \(i\) for treatment \(A\) and level \(j\) for treatment \(B\) using the corresponding cell sample mean \[\widehat{y_{ijk}}=\bar{y}_{ij}\]
The sum of squares for the interaction term is \[SSAB = n\sum_{i=1}^{a}\sum_{j=1}^{b}\left(\bar{y}_{ij}-\bar{\bar{y}}\right)^2-SSA - SSB\]

Cell Means

When an interaction is present we report the cell means
We can make a table of the cell means in R

ironpot |>
  group_by(Pot, Food) |>
  summarize(mean = mean(Iron)) |>
  pivot_wider(names_from = Pot, names_prefix = "Pot_",
              values_from = mean)

# A tibble: 3 × 4
  Food       Pot_Aluminum Pot_Clay Pot_Iron
  <fct>             <dbl>    <dbl>    <dbl>
1 legumes            2.33     2.47     3.67
2 meat               2.06     2.18     4.68
3 vegetables         1.23     1.46     2.79

Interaction plot

An interaction plot shows how the cell means vary with respect to the levels of one of the treatments
Separate line plots for each level of the other treatment
Note that if there was no interaction then lines would be parallel to each other

Interaction plot for food type and pot type.

# A tibble: 3 × 4
  Food       Pot_Aluminum Pot_Clay Pot_Iron
  <fct>             <dbl>    <dbl>    <dbl>
1 legumes            2.33     2.47     3.67
2 meat               2.06     2.18     4.68
3 vegetables         1.23     1.46     2.79

Interaction effects compare cell means across levels of both factors
For example, \[(\bar{y}_{12}-\bar{y}_{32})-(\bar{y}_{13}-\bar{y}_{33})=(2.06-4.68)-(1.23-2.79) = -1.06\] is the interaction effect that compares the difference in cell means for aluminum (\(i=1\)) and iron (\(i=3\)) pots across the meat (\(j=2\)) and vegetable (\(j=3\)) food types

Toothbrushes

Is there a difference in the effectiveness of different types of toothbrushes in removing plaque?
Do the results depend on sex of the toothbrusher?
tootbrush dataset from GLMsData package

Response: plaque_rem (difference in a plaque index after brushing - before brushing)
Independent variables:
- Sex with 2 levels (\(a=2\)): “F”, “M”
- Toothbrush with 2 levels (\(b=2\)): “Conventional”, “Hugger”

There is not a significant interaction
We should consider an additive model instead

lm(plaque_rem ~ Sex * Toothbrush, data = toothbrush) |>
  anova() |>
  tidy()

# A tibble: 4 × 6
  term              df  sumsq meansq statistic   p.value
  <chr>          <int>  <dbl>  <dbl>     <dbl>     <dbl>
1 Sex                1  6.45   6.45      15.5   0.000268
2 Toothbrush         1  0.751  0.751      1.80  0.186   
3 Sex:Toothbrush     1  1.13   1.13       2.70  0.107   
4 Residuals         48 20.0    0.416     NA    NA

# A tibble: 2 × 3
  Toothbrush       F     M
  <fct>        <dbl> <dbl>
1 Conventional 0.664  1.66
2 Hugger       1.18   1.59

There is convincing evidence of an association between sex and the amount of plaque removed
However, we failed to reject the null hypothesis for the effect of toothbrush type

lm(plaque_rem ~ Sex + Toothbrush, data = toothbrush) |>
  anova() |>
  tidy()

# A tibble: 3 × 6
  term          df  sumsq meansq statistic   p.value
  <chr>      <int>  <dbl>  <dbl>     <dbl>     <dbl>
1 Sex            1  6.45   6.45      15.0   0.000325
2 Toothbrush     1  0.751  0.751      1.74  0.193   
3 Residuals     49 21.1    0.431     NA    NA

Main effects

The main effects compare the marginal means for one factor

# A tibble: 1 × 2
      F     M
  <dbl> <dbl>
1  0.92 1.626

For example, \(\bar{y}_{1\cdot}-\bar{y}_{2\cdot}=0.92-1.626=-0.706\) compares means for females (\(i=1\)) and males (\(i=2\))

# A tibble: 1 × 2
  Conventional Hugger
         <dbl>  <dbl>
1        1.126  1.366

And \(\bar{y}_{\cdot 1}-\bar{y}_{\cdot 2}=1.126-1.366=-0.24\) compares means for conventional toothbrushes (\(j=1\)) and hugger toothbrushes (\(j=2\))
Note that it is inappropriate to consider main effects when there is an interaction

We can use the TukeyHSD function calculate the main effects, including confidence intervals
This is also appropriate for follow-up tests comparing pairs of marginal means (when there are more than 2 levels)

aov(plaque_rem ~ Sex + Toothbrush, data = toothbrush) |>
  TukeyHSD()

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = plaque_rem ~ Sex + Toothbrush, data = toothbrush)

$Sex
       diff       lwr      upr     p adj
M-F 0.70625 0.3392677 1.073232 0.0003245

$Toothbrush
                         diff        lwr       upr     p adj
Hugger-Conventional 0.2403846 -0.1255103 0.6062796 0.1928874

There is significant difference in the removed plaque index between males and females
There is no significant difference in the removed plaque index between different types of toothbrushes