Lecture notes. Written from the material of GTPB Psls20, licensed CC BY 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.

31. Hypothesis Testing on the Cuckoo Data¶

What this covers¶

This chapter works through a single tutorial exercise: a hypothesis test comparing the length of cuckoo eggs raised by two different host ("foster") species. It assumes the reader already has the two-sample \(t\)-test — its null and alternative hypotheses, its assumptions, and how to read a diagnostic plot for them — since that machinery is what the exercise asks you to apply, not what it teaches. What this tutorial adds is the practical step in between: deciding which test question a multi-group dataset actually poses, and preparing the data so that the test is testing what you think it is.

The cuckoo dataset¶

The common cuckoo does not build its own nest. It lays its eggs in another bird's nest and lets that species raise the chick. This has been known since 1892, and a 1940 study showed something more specific: a cuckoo returns to the same nesting area every year and always parasitises the same foster species. Over generations this produced geographically distinct cuckoo subspecies, each with eggs that mimic the eggs of their particular foster species as closely as possible.

The dataset records 120 cuckoo eggs, sampled from randomly selected foster nests. For each egg it gives:

length — the egg length, in mm
type — the foster species, coded 1–6: meadow pipit, tree pipit, dunnock, European robin, white wagtail, Eurasian wren

The research question is whether the foster species has an effect on egg length — consistent with the mimicry story above, different foster species should be associated with different mean egg lengths.

One test, or many?¶

A \(t\)-test compares the means of exactly two groups. Here there are six. Two ways to get from "six groups" down to something a \(t\)-test can answer:

Run every pairwise comparison. Six groups give \(\binom{6}{2} = 15\) pairs, so 15 separate \(t\)-tests.
Run a single ANOVA across all six groups at once.

The tutorial notes that the second approach is more efficient and has higher statistical power, and defers it to the chapter on ANOVA. This exercise instead does the simplest possible version of option 1: a single pairwise comparison, between the European robin (type=4) and the Eurasian wren (type=6), leaving the full six-group analysis to a later tutorial on the same dataset.

Preparing the comparison¶

Two things have to happen to the raw data before a two-sample test makes sense on it.

Subsetting. Since only the robin/wren comparison is being made, every other foster type is filtered out:

Cuckoo <- Cuckoo %>%
  filter(type %in% c("4","6")) %>%
  mutate(type = as.factor(type))

Recoding type as a factor. The type column is stored as the numeric codes 1–6, but it does not represent a quantity — it names a species. Left as a number, software would treat type=6 as three times type=2, which is meaningless here; a two-sample test needs it as a categorical grouping variable with exactly two levels, 4 and 6. Hence the explicit as.factor(type) above.

With the data subset and type correctly typed, the tutorial then asks you to check the group sizes (how many robin eggs, how many wren eggs) and to look at the distribution of length in each group before running any test — the usual order of operations: explore first, then test.

Exercises¶

Using the two-group cuckoo subset above (European robin, type=4, vs Eurasian wren, type=6):

What do you observe, having tabulated the group sizes and visualised the two distributions of egg length?
How will you model the data?
Translate the research question into a null and an alternative hypothesis.
Which test will you use to assess the research hypothesis?
Formulate the assumptions of the test, and assess them using diagnostic plots.
If the assumptions for the test hold, complete the analysis and formulate a proper conclusion.

Sources¶

Tutorial exercise sheet: Tutorial 2.4: Hypothesis testing on the cuckoo dataset, GTPB PSLS20, tutorialScripts/excercises/05_statisticalInference/Hypothesis_testing_cuckoo_half.Rmd (converted copy at docs/omics-statistics/gtpb/psls20/tutorialScripts/excercises/05_statisticalInference/Hypothesis_testing_cuckoo_half.md in the knowledge-base-library).
No slide deck or lecture transcript was supplied for this tutorial; the code chunks for loading libraries, importing the data, the first data-exploration step, and the visualisation step are blank in the source (left for the student to fill in), so they are not reproduced here as worked code. The ANOVA treatment of all six foster species that this tutorial defers to is likewise not contained in this source and is not covered in this chapter.

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