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.
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag() masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
# First, download the data to your computer,# save in the same folder as this Rmd file.dat_kang <-read_csv("ADA2_CL_09_kang.csv" , na =c("", ".") ) %>%# subset only our columns of interestselect( sex, species, cw ) %>%# make dose a factor variable and label the levelsmutate(sex =factor(sex , labels =c("M","F")) , species =factor(species, labels =c("Mg", "Mfm", "Mff")) )
Rows: 148 Columns: 11
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (11): sex, species, pow, rw, sopd, cw, ifl, ml, mw, md, arh
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
Removed 148 - 148 = 0 observations with missing values.
# The first few observationshead(dat_kang)
# A tibble: 6 × 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 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 meanskang_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()
`summarise()` has grouped output by 'sex'. You can override using the `.groups`
argument.
kang_mean
# A tibble: 1 × 1
m
<dbl>
1 123.
kang_mean_x
# A tibble: 2 × 2
sex m
<fct> <dbl>
1 M 111.
2 F 136.
kang_mean_s
# A tibble: 3 × 2
species m
<fct> <dbl>
1 Mg 110.
2 Mfm 116.
3 Mff 144.
kang_mean_xs
# A tibble: 6 × 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
Error in e_plot_lm_diagostics(lm_cw_x_s_xs): could not find function "e_plot_lm_diagostics"
Solution
[answer]
(1 p) ANOVA table, test for interaction
Provide your conclusion for the test for interaction.
library(car)
Loading required package: carData
Attaching package: 'car'
The following object is masked from 'package:purrr':
some
The following object is masked from 'package:dplyr':
recode
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
Solution
[answer]
(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.
Solution
[answer]
(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)
