Lecture 12 - Factorial ANOVA of Penguin Mass Balanced

Author

Bill Perry

Lecture 11: Review

ANOVA

Analysis of variance: single and multi-factor designs
Examples: diatoms, circadian rhythms
Predictor variables: fixed vs. random
ANOVA model
Analysis and partitioning of variance
Null hypothesis
Assumptions and diagnostics
Post F Tests - Tukey and others
Reporting the results

Lecture 12: Factorial ANOVA

2-factor designs (2-way ANOVA)

very common in ecology
can have more factors (e.g., 3-way ANOVA) but interpretation gets very challenging…
Most multifactor designs: are factorial or nested
We will cover a 2 Factor Anova - this could be fertilizer and sunlight for plant biomass produced…
- each can have an effect independently
- the effect of one factor may interact with the other factor
- in this example…
  - a low light and high light treatment might affect biomass
  - fertilizer also might affect plant biomass
  - but when you have higher light the high fertilizer treatment may grow better than expected

Factorial ANOVA: Design Structure

Consider two factors:

Factorial/crossed:
- every level of B in every level of A

Factorial ANOVA: Effect Types

In factorial designs

look at two types of factor effects:
- Main effect of each factor (polling across other factor)
- Interaction effects; is there synergistic/ antagonistic effect of factors?

Introduction to Two-Way ANOVA

Factorial Designs

Two-way ANOVA examines:

Main effect of Factor A (Species)
Main effect of Factor B (Sex)
Interaction between A × B
Balanced design: equal sample sizes in all cells
Type III sums of squares for unbiased estimates - if balanced it does not matter

## # A tibble: 6 × 6
##   species   sex        n mean_mass sd_mass se_mass
##   <fct>     <fct>  <int>     <dbl>   <dbl>   <dbl>
## 1 Adelie    female    73     3369.    269.    31.5
## 2 Adelie    male      73     4043.    347.    40.6
## 3 Chinstrap female    34     3527.    285.    48.9
## 4 Chinstrap male      34     3939.    362.    62.1
## 5 Gentoo    female    58     4680.    282.    37.0
## 6 Gentoo    male      61     5485.    313.    40.1

Data Preparation - Creating Balanced Design

Balancing the Penguin Data

The dataframe of penguins is unbalanced - there are unequal samples per cell

## # A tibble: 3 × 3
##   species   female  male
##   <fct>      <int> <int>
## 1 Adelie        73    73
## 2 Chinstrap     34    34
## 3 Gentoo        58    61

## Minimum n = 34
## # A tibble: 3 × 3
##   species   female  male
##   <fct>      <int> <int>
## 1 Adelie        34    34
## 2 Chinstrap     34    34
## 3 Gentoo        34    34

Statistical Mode \(Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \varepsilon_{ijk}\)

Where:

\(\mu\) = grand mean

\(\alpha_i\) = effect of species i

\(\beta_j\) = effect of sex j

\((\alpha\beta)_{ij}\) = interaction effect

\(\varepsilon_{ijk}\) = random error

Descriptive Statistics

Summary Statistics by Groups

## # A tibble: 6 × 6
##   species   sex        n mean_mass sd_mass se_mass
##   <fct>     <fct>  <int>     <dbl>   <dbl>   <dbl>
## 1 Adelie    female    34     3335.    260.    44.5
## 2 Adelie    male      34     4049.    318.    54.5
## 3 Chinstrap female    34     3527.    285.    48.9
## 4 Chinstrap male      34     3939.    362.    62.1
## 5 Gentoo    female    34     4681.    309.    53.0
## 6 Gentoo    male      34     5525     274.    47.0

Factorial ANOVA Models

Factorial designs can be of 3 types:

Model 1 - 2 fixed factors - the focus for today….
Model 2 - 2 random
Model 3 - 1 fixed, 1 random (mixed model) - often nested

Model 1 ANOVA:

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

Model Components Explained

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

\(y_{ijk}\): value of the k_th observation from jth and ith combination of B and A (sex m of species y)
µ: overall mean (overall mass)
αi: effect of the ith level of A, pooling across all levels of B: µi- µ (difference between average mass in all “males” for species x and overall mean)

Interaction Effects

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

Βj: effect of jth level of B, pooling across all levels of A: µj- µ (difference between average mass in all males treatments and overall mean)
(αβ)ij: effect of interaction of ith level of A and jth level of B (µij - µi - µj + µ).
- Does effect of B depend on level of A? (is effect of sex different in the 3 species?)

Model Types and Interpretation

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

Model 2 ANOVA rare in ecology
Model 3 interpretation is different:
- βj: random variable measuring variance in y across all possible levels of B, pooling across all levels of A
- (αβ)ij is random variable measuring variance of interaction between A and B across all possible levels of B (“is effect of A consistent across all possible levels of B that could have been chosen?”)

Estimated Marginal Means

Before we go further we need to define what estimated marginal means are
Balanced data - it is just the means of the groups… easy
Unbalanced data - it is the mean of cells that represent the lowest average of the groups

## Combined table with means and counts:
## # A tibble: 3 × 5
##   species   female_mean male_mean female_n male_n
##   <fct>           <dbl>     <dbl>    <int>  <int>
## 1 Adelie          3369.     4043.       73     73
## 2 Chinstrap       3527.     3939.       34     34
## 3 Gentoo          4680.     5485.       58     61
## 
## 
## Regular means (WRONG for marginal means):
## # A tibble: 3 × 3
##   species   total_n regular_mean
##   <fct>       <int>        <dbl>
## 1 Adelie        146        3706.
## 2 Chinstrap      68        3733.
## 3 Gentoo        119        5092.

## Estimated Marginal Means for Species (CORRECT):
## # A tibble: 3 × 3
##   species   emm_mean calculation                   
##   <fct>        <dbl> <chr>                         
## 1 Adelie       3706. (3368.8 + 4043.5) / 2 = 3706.2
## 2 Chinstrap    3733. (3527.2 + 3939) / 2 = 3733.1  
## 3 Gentoo       5082. (4679.7 + 5484.8) / 2 = 5082.3
## 
## 
## Comparison showing EMM calculation:
## # A tibble: 3 × 6
##   species   cell_mean_female cell_mean_male species_emm regular_mean difference
##   <fct>                <dbl>          <dbl>       <dbl>        <dbl>      <dbl>
## 1 Adelie               3369.          4043.       3706.        3706.   4.55e-13
## 2 Chinstrap            3527.          3939.       3733.        3733.   0       
## 3 Gentoo               4680.          5485.       5082.        5092.  -1.01e+ 1

Estimated Marginal Means

Before we go further we need to define what estimated marginal means are
Balanced data - it is just the means of the groups… easy
Unbalanced data - it is the mean of cells that represent the lowest average of the groups

## Combined table with means and counts:
## # A tibble: 3 × 5
##   species   female_mean male_mean female_n male_n
##   <fct>           <dbl>     <dbl>    <int>  <int>
## 1 Adelie          3369.     4043.       73     73
## 2 Chinstrap       3527.     3939.       34     34
## 3 Gentoo          4680.     5485.       58     61

## 
## Regular means sex (WRONG for marginal means):
## # A tibble: 2 × 3
##   sex    total_n regular_mean
##   <fct>    <int>        <dbl>
## 1 female     165        3862.
## 2 male       168        4546.
## 
## 
## Estimated Marginal Means for Sex (CORRECT):
## # A tibble: 2 × 3
##   sex    emm_mean calculation                            
##   <fct>     <dbl> <chr>                                  
## 1 female    3859. (3368.8 + 3527.2 + 4679.7) / 3 = 3858.6
## 2 male      4489. (4043.5 + 3939 + 5484.8) / 3 = 4489.1
## \n
## Comparison showing difference between regular and marginal means:
## # A tibble: 2 × 5
##   sex    total_n regular_mean emm_mean difference
##   <fct>    <int>        <dbl>    <dbl>      <dbl>
## 1 female     165        3862.    3859.      -3.68
## 2 male       168        4546.    4489.     -56.6

ANOVA Table Structure

SStotal = SSA + SSB + SSAB + SSresidual
SStotal = (Y - Grand Mean Y)^2
SSresidual = (Y -Yhat)^2 or the difference between each observation and the appropriate cell mean, summed over all observations

Source	SS	df	MS
A	\(nq \sum_{i=1 }^{p} (\bar {y}_{i.} - \bar{y})^2\)	\(p-1\)	\(\frac{SS_A} {p-1}\)
B	\(np \sum_{j=1 }^{q} (\bar {y}_{.j} - \bar{y})^2\)	\(q-1\)	\(\frac{SS_B} {q-1}\)
AB	\(n \sum_{i=1 }^{p} \sum_{ j=1}^{q} (\ bar{y}_{ij} - \bar{y}_ {i.} - \bar {y}_{.j} + \bar{y})^2\)	\((p-1)(q-1)\)	\(\frac{SS_{AB }}{(p-1)(q-1) }\)
Residual	\(\sum_{i=1} ^{p} \sum_{ j=1}^{q} \sum_{k=1}^{ n} (y_{ijk} - \bar{y}_{ ij})^2\)	\(pq(n-1)\)	\(\frac{SS_{\ text{Residual }}}{pq(n-1)}\)
Total	\(\sum_{i=1} ^{p} \sum_{ j=1}^{q} \sum_{k=1}^{ n} (y_{ijk} - \bar{y})^2\)	\(pqn-1\)

SSA: Factor A Effects

SSA is SS of differences between each marginal mean of A and overall mean
If A is species then get the emmeans for factor A down and subtract from overall mean

SSB: Factor B Effects

SSB is SS of differences between each marginal mean of B and overall mean
If B is sex then get the emmeans for factor B across and subtract from overall mean

SSAB: Interaction Effects

SSAB is SS of cell means minus marginal means plus overall mean

F-ratio Calculations

SS converted to MS;
F-ratio calculations are different depending on whether factors are fixed, random or mixed

Source	A and B fixed	A and B random	A fixed, B random
A	\(\frac{MS_A}{MS_{ Residual}}\)	\(\frac{MS_A}{MS_{AB}}\)	\(\frac{MS_A}{MS_{AB}}\)
B	\(\frac{MS_B}{MS_{ Residual}}\)	\(\frac{MS_B}{MS_{AB}}\)	\(\frac{MS_B}{MS_{AB}}\)
AB	\(\frac{MS_{AB}}{MS_{ Residual}}\)	\(\frac{MS_{AB}}{MS_{ Residual}}\)	\(\frac{MS_{AB}}{MS_{ Residual}}\)

Hypotheses: Fixed Factors

3 hypotheses are tested in a two-way factorial ANOVA:
A, B, A*B Both factors fixed:
- Ho(A): µ1= µ2= µ3=…. µi= µp (no diff. in marginal means of A, pooling across all levels of B)
- Ho(B): µ1= µ2= µ3=…. µj= µq (no diff. in marginal means of B, pooling across all levels of A)
- Ho(AB): µij- µi - µj + µ = 0 (no effect of interaction)

Hypotheses: Mixed Model

3 hypotheses are tested in a two-way factorial ANOVA: A, B, A*B
One fixed, one random:
- Ho(A): µ1= µ2= µ3=…. µi= µp (no diff. in marginal means of A, pooling across all levels of B)
- Ho(B): σB2= 0 (no added variance due to levels of B that could have been used)
- Ho(AB): σAB2= 0 (no added variance due to interaction between all levels of A and B that could have been used)

Example Study Details

So lets try the example with the penguin data that is in the package penguin

Effect of species and sex on body_mass_g
- 3 species (factor A)
- 2 sexes (factor B)
- 34 replicates in each cell
- This analysis examines the effects of species and sex on the body mass of penguins.

Two-Way ANOVA with Type III Sums of Squares

Fitting the Model

## 
## Call:
## lm(formula = body_mass_g ~ species * sex, data = balanced_df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -827.21 -178.13    6.25  175.00  861.03 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    4175.86      21.23 196.727  < 2e-16 ***
## species1       -484.31      30.02 -16.134  < 2e-16 ***
## species2       -442.77      30.02 -14.750  < 2e-16 ***
## sex1           -328.31      21.23 -15.467  < 2e-16 ***
## species1:sex1   -28.68      30.02  -0.955    0.341    
## species2:sex1   122.43      30.02   4.078 6.57e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 303.2 on 198 degrees of freedom
## Multiple R-squared:  0.8596, Adjusted R-squared:  0.856 
## F-statistic: 242.4 on 5 and 198 DF,  p-value: < 2.2e-16

## Type III Sums of Squares ANOVA:
## Anova Table (Type III tests)
## 
## Response: body_mass_g
##                 Sum Sq  Df    F value    Pr(>F)    
## (Intercept) 3557308900   1 38701.5271 < 2.2e-16 ***
## species       87725999   2   477.2049 < 2.2e-16 ***
## sex           21988483   1   239.2224 < 2.2e-16 ***
## species:sex    1672776   2     9.0994 0.0001657 ***
## Residuals     18199467 198                         
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Understanding Type III SS

Type III sums of squares test each effect after adjusting for all other effects in the model:

Species effect: Tested after adjusting for sex and interaction
Sex effect: Tested after adjusting for species and interaction
Interaction: Tested after adjusting for both main effects

This is especially important for unbalanced designs, but we’re using it here for consistency with the next analysis.

## Effect Sizes (Eta-squared):
##        Effect Eta_Squared
## 1     Species  0.67696748
## 2         Sex  0.16968160
## 3 Interaction  0.01290854

Checking Model Assumptions

Assumption Plots

Formal Tests

These are the formal tests of
- normality of residuals
- homogeneity of variances

## Shapiro-Wilk Normality Test:
## 
##  Shapiro-Wilk normality test
## 
## data:  residuals(anova_model)
## W = 0.99612, p-value = 0.8886
## Levene's Test for Homogeneity:
## Levene's Test for Homogeneity of Variance (center = median)
##        Df F value Pr(>F)
## group   5  0.7841 0.5622
##       198
## Number of outliers (|z| > 3): 0

Estimated Marginal Means (EMMs)

Computing EMMs

## Species EMMs:
##  species   emmean   SE  df lower.CL upper.CL
##  Adelie      3692 36.8 198     3619     3764
##  Chinstrap   3733 36.8 198     3661     3806
##  Gentoo      5103 36.8 198     5030     5175
## 
## Results are averaged over the levels of: sex 
## Confidence level used: 0.95
## Sex EMMs:
##  sex    emmean SE  df lower.CL upper.CL
##  female   3848 30 198     3788     3907
##  male     4504 30 198     4445     4563
## 
## Results are averaged over the levels of: species 
## Confidence level used: 0.95
## Species by Sex EMMs:
##  species   sex    emmean SE  df lower.CL upper.CL
##  Adelie    female   3335 52 198     3232     3437
##  Chinstrap female   3527 52 198     3425     3630
##  Gentoo    female   4681 52 198     4578     4783
##  Adelie    male     4049 52 198     3946     4151
##  Chinstrap male     3939 52 198     3836     4042
##  Gentoo    male     5525 52 198     5422     5628
## 
## Confidence level used: 0.95

Pairwise Comparisons

## [1] "Species pairwise comparisons:\n"
##  contrast           estimate SE  df t.ratio p.value
##  Adelie - Chinstrap    -41.5 52 198  -0.799  0.7041
##  Adelie - Gentoo     -1411.4 52 198 -27.145  <.0001
##  Chinstrap - Gentoo  -1369.9 52 198 -26.346  <.0001
## 
## Results are averaged over the levels of: sex 
## P value adjustment: tukey method for comparing a family of 3 estimates
## [1] "Species comparisons within sex:"
## sex = female:
##  contrast           estimate   SE  df t.ratio p.value
##  Adelie - Chinstrap     -193 73.5 198  -2.620  0.0256
##  Adelie - Gentoo       -1346 73.5 198 -18.310  <.0001
##  Chinstrap - Gentoo    -1154 73.5 198 -15.690  <.0001
## 
## sex = male:
##  contrast           estimate   SE  df t.ratio p.value
##  Adelie - Chinstrap      110 73.5 198   1.490  0.2979
##  Adelie - Gentoo       -1476 73.5 198 -20.079  <.0001
##  Chinstrap - Gentoo    -1586 73.5 198 -21.569  <.0001
## 
## P value adjustment: tukey method for comparing a family of 3 estimates

Post F Test of the interaction

What we need to do if the interaction is significant is to test the overall interaction and ignore the main effects!!!

Interaction tests continued

##  species   sex      emmean       SE  df lower.CL upper.CL .group
##  Adelie    female 3334.559 51.99448 198 3196.378 3472.740  a    
##  Adelie    male   4048.529 51.99448 198 3910.349 4186.710   b   
##  Chinstrap female 3527.206 51.99448 198 3389.025 3665.387  a    
##  Chinstrap male   3938.971 51.99448 198 3800.790 4077.151   b   
##  Gentoo    female 4680.882 51.99448 198 4542.702 4819.063    c  
##  Gentoo    male   5525.000 51.99448 198 5386.819 5663.181     d 
## 
## Confidence level used: 0.95 
## Conf-level adjustment: sidak method for 6 estimates 
## P value adjustment: sidak method for 15 tests 
## significance level used: alpha = 0.05 
## NOTE: If two or more means share the same grouping symbol,
##       then we cannot show them to be different.
##       But we also did not show them to be the same.

interaction plot

Introduction to Unbalanced Designs

The Challenge of Unbalanced Data

Real-world data is often unbalanced:
- Unequal sample sizes across groups
- Missing data patterns
- Natural variation in sampling
Key Issues:
- Type I, II, and III SS give different results
- Order of terms matters for Type I SS
- Marginal means ≠ Simple averages Interpretation becomes complex

Statistical Model (same as balanced): \[Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \varepsilon_{ijk}\]

But parameter estimation differs!

Data Preparation - Using Natural Unbalanced Data

## [1] "Unbalanced sample sizes:"
## # A tibble: 6 × 3
##   species   sex        n
##   <fct>     <fct>  <int>
## 1 Adelie    female    73
## 2 Adelie    male      73
## 3 Chinstrap female    34
## 4 Chinstrap male      34
## 5 Gentoo    female    58
## 6 Gentoo    male      61
## [1] "Total N = 333"
## [1] "Imbalance ratio = 2.15"

Visualizing Imbalance

Understanding Sums of Squares Types

Type I (Sequential) SS

## Type I SS (Species → Sex → Interaction):
## Analysis of Variance Table
## 
## Response: body_mass_g
##              Df    Sum Sq  Mean Sq F value    Pr(>F)    
## species       2 145190219 72595110 758.358 < 2.2e-16 ***
## sex           1  37090262 37090262 387.460 < 2.2e-16 ***
## species:sex   2   1676557   838278   8.757 0.0001973 ***
## Residuals   327  31302628    95727                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Type I SS (Sex → Species → Interaction):
## Analysis of Variance Table
## 
## Response: body_mass_g
##              Df    Sum Sq  Mean Sq F value    Pr(>F)    
## sex           1  38878897 38878897 406.145 < 2.2e-16 ***
## species       2 143401584 71700792 749.016 < 2.2e-16 ***
## sex:species   2   1676557   838278   8.757 0.0001973 ***
## Residuals   327  31302628    95727                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Type I SS depends on order:
##    Effect Order1_SS Order2_SS
## 1 Species 145190219 143401584
## 2     Sex  37090262  38878897

How Type I SS Works

Sequential decomposition:

First term gets all SS it can explain
Second term gets SS after removing first
Third term gets SS after removing first two

Example calculation:
- SS(Species) = reduction in SS from null to species-only model
- SS(Sex|Species) = additional reduction adding sex
- SS(Interaction|Species,Sex) = additional reduction adding interaction
Problems with unbalanced data:
- Order dependency
- Biased if factors are correlated
- Not invariant to coding

Type III Sums of Squares

Type III (Marginal) SS

## [1] "Type III Sums of Squares:"
## Anova Table (Type III tests)
## 
## Response: body_mass_g
##                 Sum Sq  Df   F value    Pr(>F)    
## (Intercept) 5232595969   1 54661.828 < 2.2e-16 ***
## species      143001222   2   746.924 < 2.2e-16 ***
## sex           29851220   1   311.838 < 2.2e-16 ***
## species:sex    1676557   2     8.757 0.0001973 ***
## Residuals     31302628 327                        
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## [1] "Comparison of F-values:"
##        Effect Type_I_F Type_III_F
## 1     Species   758.36     746.92
## 2         Sex   387.46     311.84
## 3 Interaction     8.76       8.76

How Type III SS Works

Marginal decomposition:
- Each effect tested after adjusting for all others:
  - SS(Species|Sex,Interaction)
  - SS(Sex|Species,Interaction)
  - SS(Interaction|Species,Sex)
Advantages:
- Order invariant
- Tests hypotheses about unweighted means
- Standard in most software
Disadvantages:
- Lower power with missing cells
- Tests may not be orthogonal
- Requires careful interpretation

## [1] "Effect Sizes (Type III):"
##        Effect Eta_Squared
## 1     Species      0.6947
## 2         Sex      0.1450
## 3 Interaction      0.0081

Manual SS Calculation Demonstration

Computing SS Step-by-Step

## [1] "Grand mean: 4207.1"
## [1] "Total SS: 215259666"
## [1] "Between-groups SS: 183957038"
## [1] "Within-groups SS: 31302628"
## [1] "Check: Between + Within = 215259666"

Decomposing Between-Group SS

## SS(Species alone): 145190219
## SS(Sex alone): 38878897
## 
## 
## Sum of main effects: 184069116
## Actual between SS: 183957038
## 
## 
## Difference (due to correlation): -112078

Estimated Marginal Means - Unbalanced Data

Computing EMMs

## Species EMMs (model-based):
##  species   emmean   SE  df lower.CL upper.CL
##  Adelie      3706 25.6 327     3656     3757
##  Chinstrap   3733 37.5 327     3659     3807
##  Gentoo      5082 28.4 327     5026     5138
## 
## Results are averaged over the levels of: sex 
## Confidence level used: 0.95
## 
## 
## Sex EMMs (model-based):
##  sex    emmean   SE  df lower.CL upper.CL
##  female   3859 25.3 327     3809     3908
##  male     4489 25.2 327     4440     4539
## 
## Results are averaged over the levels of: species 
## Confidence level used: 0.95
##  contrast                          estimate   SE  df t.ratio p.value
##  Adelie female - Chinstrap female      -193 73.5 198  -2.620  0.0973
##  Adelie female - Gentoo female        -1346 73.5 198 -18.310  <.0001
##  Adelie female - Adelie male           -714 73.5 198  -9.710  <.0001
##  Adelie female - Chinstrap male        -604 73.5 198  -8.220  <.0001
##  Adelie female - Gentoo male          -2190 73.5 198 -29.789  <.0001
##  Chinstrap female - Gentoo female     -1154 73.5 198 -15.690  <.0001
##  Chinstrap female - Adelie male        -521 73.5 198  -7.090  <.0001
##  Chinstrap female - Chinstrap male     -412 73.5 198  -5.600  <.0001
##  Chinstrap female - Gentoo male       -1998 73.5 198 -27.169  <.0001
##  Gentoo female - Adelie male            632 73.5 198   8.600  <.0001
##  Gentoo female - Chinstrap male         742 73.5 198  10.090  <.0001
##  Gentoo female - Gentoo male           -844 73.5 198 -11.480  <.0001
##  Adelie male - Chinstrap male           110 73.5 198   1.490  0.6711
##  Adelie male - Gentoo male            -1476 73.5 198 -20.079  <.0001
##  Chinstrap male - Gentoo male         -1586 73.5 198 -21.569  <.0001
## 
## P value adjustment: tukey method for comparing a family of 6 estimates

Interaction tests continued

##  species   sex      emmean       SE  df lower.CL upper.CL .group
##  Adelie    female 3368.836 36.21222 327 3272.980 3464.692  a    
##  Adelie    male   4043.493 36.21222 327 3947.637 4139.349   b   
##  Chinstrap female 3527.206 53.06120 327 3386.749 3667.662  a    
##  Chinstrap male   3938.971 53.06120 327 3798.514 4079.427   b   
##  Gentoo    female 4679.741 40.62586 327 4572.202 4787.281    c  
##  Gentoo    male   5484.836 39.61427 327 5379.975 5589.698     d 
## 
## Confidence level used: 0.95 
## Conf-level adjustment: sidak method for 6 estimates 
## P value adjustment: sidak method for 15 tests 
## significance level used: alpha = 0.05 
## NOTE: If two or more means share the same grouping symbol,
##       then we cannot show them to be different.
##       But we also did not show them to be the same.

Pairwise Comparisons

Tukey HSD Comparisons

## Species pairwise comparisons (Tukey):
##  contrast           estimate   SE  df t.ratio p.value
##  Adelie - Chinstrap    -26.9 45.4 327  -0.593  0.8241
##  Adelie - Gentoo     -1376.1 38.2 327 -36.007  <.0001
##  Chinstrap - Gentoo  -1349.2 47.0 327 -28.682  <.0001
## 
## Results are averaged over the levels of: sex 
## P value adjustment: tukey method for comparing a family of 3 estimates
## 
## 
## Sex effect within each species:
## species = Adelie:
##  contrast      estimate   SE  df t.ratio p.value
##  female - male     -675 51.2 327 -13.174  <.0001
## 
## species = Chinstrap:
##  contrast      estimate   SE  df t.ratio p.value
##  female - male     -412 75.0 327  -5.487  <.0001
## 
## species = Gentoo:
##  contrast      estimate   SE  df t.ratio p.value
##  female - male     -805 56.7 327 -14.188  <.0001

Interaction Contrasts

## Sex effect (Male - Female) by species:
##     Species Sex_Effect
## 1    Adelie       -675
## 2 Chinstrap       -412
## 3    Gentoo       -805
## 
## 
## Interaction interpretation:
## [1] "Sex effects differ across species"

Diagnostic Plots - Unbalanced Design

Model Diagnostics

Assumption Tests

## [1] "Shapiro-Wilk test:"
## 
##  Shapiro-Wilk normality test
## 
## data:  residuals(model_u)
## W = 0.99776, p-value = 0.9367
## [1] "\nLevene's test:"
## Levene's Test for Homogeneity of Variance (center = median)
##        Df F value Pr(>F)
## group   5  1.3908 0.2272
##       327
## [1] "\nResidual SD by group:"
## # A tibble: 6 × 3
##   species   sex    sd_resid
##   <fct>     <fct>     <dbl>
## 1 Adelie    female     269.
## 2 Adelie    male       347.
## 3 Chinstrap female     285.
## 4 Chinstrap male       362.
## 5 Gentoo    female     282.
## 6 Gentoo    male       313.

Comparing Balanced vs Unbalanced Results

Side-by-Side Comparison

## Balanced vs Unbalanced Results:
##        Effect Balanced_F Balanced_p Unbalanced_F Unbalanced_p
## 1     Species     477.20      0e+00       746.92        0e+00
## 2         Sex     239.22      0e+00       311.84        0e+00
## 3 Interaction       9.10      2e-04         8.76        2e-04
## 
## 
## Sample sizes:
## Balanced total N: 204
## Unbalanced total N: 333
## Data discarded: 129 observations

Visual Comparison

Summary and Best Practices

Key Takeaways

Unbalanced Designs:

Challenges: - Type I SS depends on order - Simple means ≠ EMMs - Reduced power for interactions - Complex interpretation

Solutions: - Use Type III SS for main effects - Report EMMs, not simple means - Check assumptions carefully - Consider impact of imbalance

Recommendations:

Always report:
- Sample sizes per cell
- Type of SS used
- EMMs with CIs
Visualization:
- Show actual data points
- Include error bars
- Note unequal sample sizes
Interpretation:
- Focus on effect sizes
- Consider practical significance
- Acknowledge limitations

Final Recommendations

For Experimental Studies: - Design for balance when possible - Randomize allocation - Plan for attrition

For Observational Studies: - Accept imbalance as reality - Use Type III SS - Report EMMs - Consider covariates (ANCOVA)

Statistical Software Defaults: - R: Type I (sequential) - SAS: Type III (marginal) - SPSS: Type III (marginal)

Always specify which type you’re using!

Other Formats

Lecture 11: Review

ANOVA

Lecture 12: Factorial ANOVA

2-factor designs (2-way ANOVA)

Factorial ANOVA: Design Structure

Consider two factors:

Factorial ANOVA: Effect Types

In factorial designs

Introduction to Two-Way ANOVA

Factorial Designs

Data Preparation - Creating Balanced Design

Balancing the Penguin Data

Descriptive Statistics

Summary Statistics by Groups

Factorial ANOVA Models

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

Model Components Explained

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

Interaction Effects

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

Model Types and Interpretation

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

Estimated Marginal Means

Estimated Marginal Means

ANOVA Table Structure

SSA: Factor A Effects

SSB: Factor B Effects

SSAB: Interaction Effects

F-ratio Calculations

Hypotheses: Fixed Factors

Hypotheses: Mixed Model

Example Study Details

Two-Way ANOVA with Type III Sums of Squares

Fitting the Model

Understanding Type III SS

Checking Model Assumptions

Formal Tests

Estimated Marginal Means (EMMs)

Computing EMMs

Pairwise Comparisons

Post F Test of the interaction

Interaction tests continued

interaction plot

Introduction to Unbalanced Designs

Data Preparation - Using Natural Unbalanced Data

Visualizing Imbalance

Understanding Sums of Squares Types

Type I (Sequential) SS

How Type I SS Works

Type III Sums of Squares

Type III (Marginal) SS

How Type III SS Works

Manual SS Calculation Demonstration

Computing SS Step-by-Step

Decomposing Between-Group SS

Estimated Marginal Means - Unbalanced Data

Computing EMMs

Interaction tests continued

Pairwise Comparisons

Tukey HSD Comparisons

Interaction Contrasts

Diagnostic Plots - Unbalanced Design

Model Diagnostics

Assumption Tests

Comparing Balanced vs Unbalanced Results

Side-by-Side Comparison

Visual Comparison

Summary and Best Practices

Key Takeaways

Unbalanced Designs:

Recommendations:

Final Recommendations