Week 5 Exercises - One Line, Two Stories

From correlation → line → prediction → residual

NoteOverview

About 55 minutes · Pairs or small groups · 🧰 R (tidyverse, rio, here) or JAMOVI

Every step below is given both ways. Use the tool you want to get better at — the skills being graded (reading a slope, an intercept, a residual, an R²) are identical in both.

• Part A: Follow along (~20 min): We fit our first real regression together — do people watch fewer Transformers movies as they get older?
• Part B: On your own (~25 min): The same routine on a messier question, where the answer depends on one decision you make.
• Part C: Together (~10 min): The punchline from Lab 5 — a t-test is a regression.

By the end you should be able to:

• Fit a simple regression with lm() and find the intercept and slope in the output
• Say what the slope means in the units of the study, out loud, without looking it up
• Decide whether the intercept means anything at all
• Predict a score for a specific person, and compute that person’s residual
• Connect r, R², and the slope’s t-test to each other
• Notice when one strange observation is running the whole model

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Set up (5 min)

1. Download the data and save it in the data folder of your PSYC 640 R Project.

   • Cards Against Humanity: Pulse of the Nation (.csv)

2. Open your R Project (double-click the .Rproj file). The top-right corner of RStudio should show the project name.

3. Create a new Quarto document: File → New File → Quarto Document…

   • Title: Week 5 - One Line, Two Stories
   • Author: you (and your partner)
   • Save it as week5_inclass.qmd in your project folder

4. Delete the example text, then add a setup chunk:

library(tidyverse)
library(rio)
library(here)

cah <- import(here("data", "CAH.csv"))

glimpse(cah)

TipCheck yourself

cah should have 1,000 rows and 16 columns. If here() cannot find the file, run here() in the Console to see where it is looking, and make sure CAH.csv is in that folder’s data subfolder.

Note🖱️ In JAMOVI instead

1. Open JAMOVI → ☰ Menu → Open → Browse → CAH.csv
2. Check the measure-type icon on each column you will use: age, transformers and books should be continuous (ruler); gender should be nominal (three circles). Click a column header to change one.
3. Install the scatr module once: ➕ Modules → jamovi library → scatr (for scatterplots with a regression line).
4. Save your work as a .omv file so you can hand it in or come back to it.

About the data. Cards Against Humanity ran a monthly national poll called Pulse of the Nation. Among the serious demographic items, they asked how many books a person read in the past year and how many Transformers movies they had seen. Today those two questions are our outcomes.

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Part A — Follow along: Transformers and age (20 min)

Research question: Do people see fewer Transformers movies as they get older?

1. Look before you model

cah %>%
  select(age, transformers) %>%
  summary()

age runs from 18 to 99. transformers runs from 0 to 5, with a mean of about 1.3. Note that nobody in this dataset is younger than 18 — we will come back to that.

2. Plot it first

ggplot(cah, aes(x = age, y = transformers)) +
  geom_point(alpha = .3) +
  geom_smooth(method = "lm", se = FALSE) +
  labs(
    title = "Transformers movies seen, by age",
    x = "Age (years)",
    y = "Number of Transformers movies seen"
  ) +
  theme_minimal()

TipTalk about it (2 min)

The line slopes down. Before we fit anything: how strong does that relationship look to you? If you had to guess r, what would you say? Write your guess down — we’ll check it in a minute.

3. Correlation first, then the line

cor.test(cah$age, cah$transformers)

You should get r = −.27, t(936) = −8.46, p < .001.

Now the same relationship as a line:

mod_tf <- lm(transformers ~ age, data = cah)
summary(mod_tf)

Coefficients:
             Estimate Std. Error t value Pr(>|t|)
(Intercept)  2.577000   0.155000  16.626  < 2e-16 ***
age         -0.025398   0.003002  -8.461  < 2e-16 ***

Residual standard error: 1.50 on 936 degrees of freedom
Multiple R-squared:  0.0710,    Adjusted R-squared:  0.0700

Note🖱️ Part A in JAMOVI

• Descriptives: Analyses → Exploration → Descriptives; drag in age and transformers. (Note the two different Ns — 62 people skipped the Transformers question.)
• Scatterplot: Analyses → Exploration → scatr → Scatterplot. X-Axis age, Y-Axis transformers, Regression Line: Linear.
• Correlation: Analyses → Regression → Correlation Matrix; tick Pearson and p-value. You should get r = −0.266, p < .001.
• The line: Analyses → Regression → Linear Regression. Dependent Variable transformers; put age in Covariates (continuous predictors go there). Open Model Fit and tick R²; open Model Coefficients and tick Standardized estimate.

You should see Intercept 2.577 and age −0.0254, R² = 0.071, and a standardized estimate of −0.266 — the same numbers, in a different window.

4. Read the three numbers

The slope: −0.0254. Each additional year of age is associated with 0.025 fewer Transformers movies seen. That sounds like nothing — which is why slopes are almost always more honest when you scale them to a meaningful unit:

Ten years of age is associated with about 0.25 fewer movies; forty years, about one whole movie fewer.

The intercept: 2.577. The model’s predicted number of movies for a person aged 0. Newborns do not watch Transformers, and there is nobody under 18 in the data, so this number is an anchor for the line, not a finding.

R² = .071. Age accounts for about 7% of the variation in movies seen. And notice: (−.27)² = .071. R² is the squared correlation. Also notice the t for the slope (−8.46) is the same t cor.test() gave you. Same test, two outfits.

5. Predict a person, then find the residual

# predicted movies for a 20-year-old
2.577 + (-0.025398 * 20)

The model predicts about 2.07 movies for a 20-year-old.

Now pick a real 20-year-old from the data who has seen 4 movies:

\[\text{residual} = \text{observed} - \text{predicted} = 4 - 2.07 = +1.93\]

Positive residual = above the line = the model under-predicted this person. The residual is the part of this person the model gets wrong — and in a few weeks, residuals become the whole story.

Tip🖱️ Let JAMOVI do it for you

In Linear Regression → Save, tick Predicted values and Residuals. Two new columns appear in the Data tab, one row per person — no arithmetic required. Find a 20-year-old and read their residual straight off the sheet.

❓Questions for Part A

1. Write the fitted equation using the variable names.
2. What does the model predict for a 50-year-old? For a 70-year-old?
3. Would you use this model to predict movies seen for a 10-year-old? Why or why not?
4. Can we say that getting older causes people to see fewer Transformers movies? What else could produce this pattern?

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Part B — On your own: do older people read more books? (25 min)

Research question: Does the number of books a person reads in a year go up with age?

Work through the same five steps in your own document. Write your answers as text between the code chunks, not just as output.

Steps

1. Look first. Run summary() on age and books. Something in books should stop you — what is it?
2. Plot it. Scatterplot with a regression line, labeled properly.
3. Fit it. lm(books ~ age, data = cah) and summary().
4. Read it. Write one sentence interpreting the slope in real units (per year and per decade), one sentence on whether the intercept means anything, and report R².
5. Judge it. Would you report this model in a paper as it stands? Why or why not?

TipCheck yourself

The slope is 0.32 books per year of age, t(975) = 2.21, p = .027, R² = .005.

So it is “statistically significant.” Now look at that R² again, and look back at your scatterplot.

The decision that changes the answer

Someone in this sample reported reading 1,000 books in the past year — about three a day, every day. Five people reported 500 or more.

Re-fit the model without the responses that are not plausible:

cah_trim <- cah %>%
  filter(books <= 365)      # at most one book a day

lm(books ~ age, data = cah_trim) %>% summary()

Note🖱️ The same decision in JAMOVI

Go to the Data tab → Filters (funnel icon) → add a filter and type:

books <= 365

Every analysis updates the moment the filter is active. Toggle the filter’s eye icon off and on to watch the p-value move — which is exactly the point of this part.

Now the slope is 0.10, t(969) = 1.44, p = .150, R² = .002. The “significant” result is gone.

❓Questions for Part B

1. Which of the two models would you report? Defend the choice — there is a defensible answer either way, but you have to make the case.
2. Your cutoff was 365. What would happen with a cutoff of 100? Try it. Does your conclusion depend on a number you picked yourself?
3. What would you have to tell a reader for them to be able to judge your decision? Write the one sentence you would put in a methods section.
4. Does a p-value of .027 mean this is an important finding? Use the R² in your answer.

ImportantThe point of Part B

Both models are “correct.” Neither one is a mistake in R. The difference between them is a judgment call you made about your data, and the p-value flipped because of it. Every analysis you run for the rest of your career will have several of these decisions in it. Making them thoughtfully — and reporting them — is most of what separates trustworthy research from the other kind.

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Part C — Together: the Lab 5 punchline (10 min)

In Lab 5 you ran a t-test and then an lm() and got suspiciously similar numbers. Let’s settle it.

cah_mf <- cah %>%
  filter(gender %in% c("Male", "Female"))

# the t-test you already know
t.test(transformers ~ gender, data = cah_mf, var.equal = TRUE)

# the same question as a line
lm(transformers ~ gender, data = cah_mf) %>% summary()

The t-test gives t(911) = −2.29, p = .022; means of 1.21 (Female) and 1.44 (Male).

The regression gives an intercept of 1.21 and a slope of 0.23, with t = 2.29 and p = .022.

Note🖱️ Part C in JAMOVI

• Filter out “Other”: Data → Filters → gender != "Other"
• The t-test: Analyses → T-Tests → Independent Samples T-Test. Dependent transformers, Grouping gender; tick Student’s and Descriptives.
• The line: Analyses → Regression → Linear Regression. Dependent transformers, and put gender in Factors (categorical predictors go there, not Covariates).
• Bonus: open Reference Levels and switch the reference group from Female to Male. The intercept becomes 1.44, the estimate flips to −0.23, and t and p do not move. Why not?

❓Questions for Part C

1. Where does the intercept 1.21 come from? Which group is it?
2. Where does the slope 0.23 come from?
3. R fit a line through a predictor with only two values. What are the only two predicted values this “line” can produce?
4. In one sentence: why is a t-test a special case of regression?

NoteWhy this matters

This is the reason we are teaching regression first. You do not have to memorize a separate procedure for every design. One model, outcome ~ predictor, handles the continuous case (Parts A and B) and the group case (Part C) — and over the next few weeks we’ll add more predictors to the same machine.

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Exit ticket ✅

Before you leave, write these three things at the bottom of your document (they’ll also help you on Lab 6):

1. A slope, stated in the units of a study, without using the word “slope.”
2. One intercept from today that is meaningless, and why.
3. One thing that is still muddy — put it in your journal entry this week and I will pick it up next class.