library(tidyverse)
library(easystats)
penguins <- read_csv("penguins.csv")
fit1 <- lm(body_mass ~ flipper_len, data = penguins)
fit2 <- lm(body_mass ~ flipper_len + I(flipper_len^2), data = penguins)
model_parameters(fit2, digits = 4)
r2(fit1)
r2(fit2)
plot(estimate_relation(fit2, by = "flipper_len"), show_data = TRUE)Lecture 09a Activity
Unit C · Chapter 09
An in-class activity. Nothing to turn in and no answer key – this one needs other people, which is why it happens in class rather than at home.
The idea. Whether a curve is real is a judgment call, and the fastest way to find out how strong your case is, is to be assigned the other side of it.
Format. About 12 minutes · pairs · one laptop per pair.
1. Get the evidence (4 minutes)
Then two things on paper, from the output. A parabola \(b_0 + b_1 x + b_2 x^2\) turns around at \(x = -b_1 / (2 b_2)\): take the two flipper_len coefficients from the table and compute where this one turns. And from the plot, read off the smallest and largest flipper lengths that actually occur. Is the turning point inside the data?
2. Take a side (5 minutes)
One of you is the author: the curvature is real and belongs in the paper. The other is the reviewer: the quadratic term should come out. You do not get to pick, and you both have exactly this output.
Build your case in three minutes, then argue it for two. Each side must use the vertex, the range, and the \(R^2\) at least once.
3. Whole class (3 minutes)
- Reviewers: what was your strongest point? Authors: what was your best answer?
- The room votes. Then: what additional evidence would have settled it, and could it have come from this dataset at all?
If you have more time: which of these theories predicts a turning point and which only predicts a bend? (a) anxiety and performance follow an inverted U; (b) study hours help with diminishing returns; (c) income rises with age until retirement. A quadratic gives you a turn whether you asked for one or not.
If you were not in class
The vertex falls near 158 mm; the observed flipper lengths run from 172 to 231. So the fitted parabola turns around outside the data, and within the data the curve is monotonic – the quadratic term is bending the line, not producing a peak. The reviewer’s strongest point is that no penguin in this dataset is anywhere near the vertex, so “body mass peaks at 158 mm” is a claim about the algebra of a parabola rather than about penguins. The author’s best answer is that they never claimed a peak: the term is buying a small, real improvement in fit (\(R^2\) goes from .759 to .775) in the region where data exist.
Both are right, which is why this is a judgment call and not a test. The useful habit is the one the argument forces: compute the extremum, then check it against the observed range before writing a sentence about it.
On the extension: (a) and (c) genuinely predict turning points, so a quadratic is a reasonable shape. (b) predicts diminishing returns – a bend with no peak – and a quadratic will impose a peak anyway, then predict that studying more eventually makes you worse. A log transformation of the predictor gives the bend without the turn.