Exercise 1 (20 minutes)

You practice: fitting and reading a polynomial — does xenophobia keep rising with age, or level off? When you're done, assist some of the others …

Back to xenophobia and the ESS. Today you ask whether the effect of age on xenophobia is straight-line — or a curve.

1. Prepare the data (Denmark and Bulgaria, as in Lecture 11).

pacman::p_load(tidyverse, haven, estimatr, modelsummary)
ESS <- read_spss("ESS9e03_1.sav") # from your project folder

ESS <- ESS %>%
  filter(cntry == "DK" | cntry == "BG") %>%          # DK = Denmark, BG = Bulgaria
  mutate(
    cntry  = as_factor(cntry) %>% fct_relevel("Denmark"),
    gndr   = as_factor(gndr),
    eduyrs = case_when(eduyrs < 9 ~ 9, eduyrs > 21 ~ 21, TRUE ~ zap_labels(eduyrs)),
    agea    = zap_labels(agea),
    imwbcnt = 10 - zap_labels(imwbcnt)               # reverse: higher = more xenophobic
  ) %>%
  select(idno, cntry, pspwght, gndr, eduyrs, agea, imwbcnt) %>%
  drop_na()

Stuck?

• Did pacman::p_load(...) run without red errors? Fix that first.
• read_spss() can't find the file? ESS9e03_1.sav must sit in your project folder.
• Odd results? Session → Restart R, then rerun from the top.
• Still stuck? Ask in the t-R-ouble forum on Absalon.

2. Predict before you compute. Do you expect the effect of age on xenophobia to be a straight line, or to bend (rise faster at some ages than others)? There is no wrong guess — commit to one, then test it.

3. Fit a quadratic of age. Use poly(I(agea / 10), 2, raw = TRUE) so the coefficients are per decade, and keep country, gender and education as controls, weighted by pspwght.

Hint

lm_robust(imwbcnt ~ cntry + gndr + eduyrs + poly(I(agea / 10), 2, raw = TRUE), data = ESS, weights = pspwght)

R solution — don't peek too early ;) !

ols_poly <- lm_robust(
  imwbcnt ~ cntry + gndr + eduyrs + poly(I(agea / 10), 2, raw = TRUE),
  data = ESS, weights = pspwght
)

modelsummary(list("Xenophobia" = ols_poly),
             coef_rename = c("poly(I(agea/10), 2, raw = TRUE)1" = "Age (per decade)",
                             "poly(I(agea/10), 2, raw = TRUE)2" = "Age² (per decade)"),
             stars = TRUE, gof_map = c("nobs", "r.squared"), output = "kableExtra")

The linear term is and the squared term is . Because the squared term is negative, the age slope gets smaller as people get older — the curve flattensgets steeper as people get olderstays exactly constant.

4. Where would the curve turn around? A quadratic peaks (or bottoms out) at \(-\beta_1 / (2\beta_2)\). With the coefficients per decade, multiply by 10 to get the age in years. The turning point is at about years.

R solution

b1 <- coef(ols_poly)["poly(I(agea/10), 2, raw = TRUE)1"]
b2 <- coef(ols_poly)["poly(I(agea/10), 2, raw = TRUE)2"]

(-b1 / (2 * b2)) * 10   # turning point, in years

5. That turning point is far outside the ages anyone in the data actually has (15–90). So within the real range, xenophobia just keeps rising with age, flattening only slightlyrises and then clearly falls among the elderlyfalls throughout.

6. Write one sentence as a # comment interpreting what the quadratic told you — then compare:

Model answer

Xenophobia rises with age across the whole observed range, but the negative squared term means the rise decelerates — each additional decade adds a little less than the one before. The curve would technically turn around near age 180, which is meaningless for humans, so in practice this is a gently flattening increase, not a hump. A straight line would have been a defensible simplification here; the curvature is real but mild.

7. Bonus, for the fast. Plot the fitted curve to see the mild bend. Overlay a straight-line fit for comparison.

R solution

ggplot(ESS, aes(x = agea, y = imwbcnt)) +
  geom_jitter(alpha = 0.06, height = 0.2) +
  geom_smooth(method = "lm", formula = y ~ x, se = FALSE, aes(colour = "Straight line")) +
  geom_smooth(method = "lm", formula = y ~ poly(x, 2, raw = TRUE), se = FALSE,
              aes(colour = "Quadratic")) +
  scale_colour_manual(values = c("Straight line" = "#425570", "Quadratic" = "#901A1E"),
                      name = NULL) +
  labs(x = "Age", y = "Xenophobia (0–10)") +
  theme_minimal()

The two lines barely diverge — visual confirmation that the curvature is slight.

8. Discuss with your neighbour.

   • Why is a turning point outside the observed data a reason not to make much of the "curve"?
   • A colleague reports "xenophobia peaks at age 180." What has gone wrong in how they read the model?
   • When would you expect a genuine hump in age — a real rise-then-fall — for some social outcome?

Talking points

• A model can only be trusted where it has data. A turning point at 180 is an extrapolation far beyond anyone in the sample; the quadratic is not really claiming a peak, it is just gently flattening within the observed range, and the formula happens to turn around off in fantasy land.
• They have read a mathematical feature of the fitted curve as if it were a finding. Nobody in the data is anywhere near 180, so the "peak" is an artefact of extrapolation, not evidence about old age.
• Genuine humps are common: earnings over the life course (rise, peak in middle age, decline), physical performance, risk-taking, or the crime–age curve. There the turning point sits inside the data and a quadratic earns its second coefficient.