1 Kangaroos skull measurements: crest width

What effect does sex and species have on the crest width of a kangaroo skull?

The data to be analyzed here are selected skull measurements on 148 kangaroos of known sex and species. There are 11 columns of data, corresponding to the following features. The measurements are in meters/10000 (mm/10).

column Variable name Description
1 * sex sex (1=M, 2=F)
2 * species species (0=M. giganteus, 1=M.f. melanops, 2=M.f. fuliginosus)
3 pow post orbit width
4 rw rostal width
5 sopd supra-occipital - paroccipital depth
6 * cw crest width
7 ifl incisive foramina length
8 ml mandible length
9 mw mandible width
10 md mandible depth
11 arh ascending ramus height

Some of the observations in the data set are missing (not available). These are represented by a period ., which in the read_csv() function is specified by the na = "." option.

library(tidyverse)

# load ada functions
source("ada_functions.R")

# First, download the data to your computer,
#   save in the same folder as this Rmd file.

dat_kang <-
  read_csv(
    "ADA2_WS_09_kang.csv"
  , na = c("", ".")
  ) %>%
  # subset only our columns of interest
  select(
    sex, species, cw
  ) %>%
  # make dose a factor variable and label the levels
  mutate(
    sex     = factor(sex    , labels = c("M","F"))
  , species = factor(species, labels = c("Mg", "Mfm", "Mff"))
  )

# remove observations with missing values
n_start  <- nrow(dat_kang)
dat_kang <- na.omit(dat_kang)
n_keep   <- nrow(dat_kang)
n_drop   <- n_start - n_keep
cat("Removed", n_start, "-", n_keep, "=", n_drop, "observations with missing values.")
Removed 148 - 148 = 0 observations with missing values.
# The first few observations
head(dat_kang)
# A tibble: 6 x 3
  sex   species    cw
  <fct> <fct>   <dbl>
1 M     Mg        153
2 M     Mg        141
3 M     Mg        144
4 M     Mg        116
5 M     Mg        120
6 M     Mg        188

1.1 (1 p) Interpret plots of the data, distributional centers and shapes

The side-by-side boxplots of the data compare the crest widths across the 6 combinations of sex and species. Comment on the distributional shapes and compare the typical crest widths across groups.

# Calculate the cell means for each (sex, species) combination
# Group means
kang_mean     <- dat_kang %>%                            summarise(m = mean(cw))
kang_mean_x   <- dat_kang %>% group_by(sex)          %>% summarise(m = mean(cw)) %>% ungroup()
kang_mean_s   <- dat_kang %>% group_by(species)      %>% summarise(m = mean(cw)) %>% ungroup()
kang_mean_xs  <- dat_kang %>% group_by(sex, species) %>% summarise(m = mean(cw)) %>% ungroup()

kang_mean
# A tibble: 1 x 1
      m
  <dbl>
1  123.
kang_mean_x
# A tibble: 2 x 2
  sex       m
  <fct> <dbl>
1 M      111.
2 F      136.
kang_mean_s
# A tibble: 3 x 2
  species     m
  <fct>   <dbl>
1 Mg       110.
2 Mfm      116.
3 Mff      144.
kang_mean_xs
# A tibble: 6 x 3
  sex   species     m
  <fct> <fct>   <dbl>
1 M     Mg       103.
2 M     Mfm      102.
3 M     Mff      128.
4 F     Mg       117.
5 F     Mfm      128.
6 F     Mff      161 
# Interaction plots, ggplot
library(ggplot2)
p1 <- ggplot(dat_kang, aes(x = sex, y = cw, colour = species))
p1 <- p1 + geom_hline(aes(yintercept = 0), colour = "black"
                  , linetype = "solid", size = 0.2, alpha = 0.3)
p1 <- p1 + geom_boxplot(alpha = 0.5, outlier.size=0.1)
p1 <- p1 + geom_point(data = kang_mean_xs, aes(y = m), size = 4)
p1 <- p1 + geom_line(data = kang_mean_xs, aes(y = m, group = species), size = 1.5)
p1 <- p1 + labs(title = "Kangaroo interaction plot, species by sex")
#print(p1)

p2 <- ggplot(dat_kang, aes(x = species, y = cw, colour = sex))
p2 <- p2 + geom_hline(aes(yintercept = 0), colour = "black"
                  , linetype = "solid", size = 0.2, alpha = 0.3)
p2 <- p2 + geom_boxplot(alpha = 0.5, outlier.size=0.1)
p2 <- p2 + geom_point(data = kang_mean_xs, aes(y = m), size = 4)
p2 <- p2 + geom_line(data = kang_mean_xs, aes(y = m, group = sex), size = 1.5)
p2 <- p2 + labs(title = "Kangaroo interaction plot, sex by species")
#print(p2)

library(gridExtra)
grid.arrange(grobs = list(p1, p2), nrow=1, top="Kangaroo crestwidth plots")

1.1.1 Solution

[answer]

1.2 (1 p) Do the plots above suggest there is an interaction?

Do the lines for each group seem to be very different from parallel?

1.2.1 Solution

[answer]

1.3 Fit the two-way interaction model

Here it is.

lm_cw_x_s_xs <-
  lm(
    cw ~ sex * species
  , data = dat_kang
  , contrasts = list(sex = contr.sum, species = contr.sum)
  )

1.4 (1 p) Check model assumptions for full model

Recall that we assume that the full model is correct before we perform model reduction by backward selection.

Non-constant Variance Score Test 
Variance formula: ~ fitted.values 
Chisquare = 3.094663, Df = 1, p = 0.078549


    Anderson-Darling normality test

data:  lm_cw_x_s_xs$residuals
A = 0.38405, p-value = 0.3907

1.4.1 Solution

[answer]

1.5 (1 p) ANOVA table, test for interaction

Provide your conclusion for the test for interaction.

library(car)
Anova(lm_cw_x_s_xs, type=3)
Anova Table (Type III tests)

Response: cw
             Sum Sq  Df  F value    Pr(>F)    
(Intercept) 2244042   1 1643.795 < 2.2e-16 ***
sex           22556   1   16.523 7.927e-05 ***
species       34195   2   12.524 9.788e-06 ***
sex:species    2367   2    0.867    0.4224    
Residuals    193853 142                       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

1.5.1 Solution

[answer]

1.6 (4 p) Reduce to final model, test assumptions

If the model can be simplified (because interaction is not significant), then refit the model with only the main effects. Test whether the main effects are significant, reduce further if sensible. Test model assumptions of your final model.

1.6.1 Solution

[answer]

1.7 (2 p) Summarize the differences

Summarize differences, if any, in sexes and species using relevant multiple comparisons. Give clear interpretations of any significant effects.

This code is here to get you started. Determine which comparisons you plan to make and modify the appropriate code. Make the code chunk active by moving the {R} to the end of the initial code chunk line.

{R}
library(emmeans)
# Contrasts to perform pairwise comparisons
cont_kang <- emmeans(lm_object, specs = "sex")
cont_kang <- emmeans(lm_object, specs = "species")
cont_kang <- emmeans(lm_object, specs = "sex", by = c("species"))
cont_kang <- emmeans(lm_object, specs = "species", by = c("sex"))
# Means and CIs
cont_kang
# Pairwise comparisons
cont_kang %>% pairs()

EMM plot interpretation

This EMM plot (Estimated Marginal Means, aka Least-Squares Means) is only available when conditioning on one variable. The blue bars are confidence intervals for the EMMs; don’t ever use confidence intervals for EMMs to perform comparisons – they can be very misleading. The red arrows are for the comparisons among means; the degree to which the “comparison arrows” overlap reflects as much as possible the significance of the comparison of the two estimates. If an arrow from one mean overlaps an arrow from another group, the difference is not significant, based on the adjust setting (which defaults to “tukey”).

{R, fig.height = 5, fig.width = 6}
# Plot means and contrasts
p <- plot(cont_kang, comparisons = TRUE)
p <- p + labs(title = "Tukey-adjusted contrasts")
p <- p + theme_bw()
print(p)

1.7.1 Solution

[answer]