Sociology 106: Quantitative Sociological Methods
March 31, 2026
Before running any regression, be explicit about your variables and expectations:
Your research question should name both roles:
“How does occupational prestige impact household income?”
income91 — household income, in dollars — what we’re explainingprestg80 — occupational prestige score — our main theoretical variableWrite out your hypothesis before looking at the data:
\[H_0: \beta_{\text{prestige}} = 0 \quad \text{(prestige has no effect on income)}\]
\[H_1: \beta_{\text{prestige}} > 0 \quad \text{(higher prestige → higher income)}\]
This is a one-tailed test — theory strongly predicts the direction (positive).
In most cases, we’ll default to two-tailed unless we have strong prior theoretical reason for directionality.
Always plot your variables before running a regression:
ggplot(attain_reg, aes(x = prestg80, y = income91)) +
geom_point(alpha = 0.2, color = "gray50", size = 1.2) +
geom_smooth(method = "lm", se = TRUE, color = "#4E79A7",
fill = "#4E79A7", alpha = 0.15) +
scale_y_continuous(labels = scales::dollar_format()) +
labs(title = "Occupational Prestige and Household Income (GSS, 1991)",
x = "Occupational Prestige Score (Key Predictor)",
y = "Household Income — $ (Outcome)") +
theme_minimal(base_size = 11)
Two visualizations worth including in hw9:
Use geom_smooth(method = "lm") to overlay the regression line on a scatterplot:
ggplot(attain_reg, aes(x = prestg80, y = income91)) +
geom_point(alpha = 0.2, color = "gray50", size = 1) +
geom_smooth(method = "lm", se = TRUE,
color = "#4E79A7", fill = "#4E79A7", alpha = 0.2) +
scale_y_continuous(labels = scales::dollar_format()) +
labs(title = "Effect of Occupational Prestige on Household Income",
x = "Occupational Prestige Score", y = "Household Income ($)") +
theme_minimal(base_size = 11)
modelplot() from modelsummary shows coefficients with 95% confidence intervals. If the CI does not cross 0, the coefficient is significant at \(\alpha = 0.05\):
So far our key predictor has been continuous (occupational prestige). What if the key predictor is categorical — like educational degree, sex, or race?
The challenge: We can’t multiply a category label by a number. Regression needs numbers.
The solution: Replace a categorical variable with a set of dummy (indicator) variables — one for each category except the baseline.
We can see groups differ — but how much? And is the difference significant? Regression tells us.
With degree as the key predictor (baseline = lt high [less than HS]):
\[\widehat{\text{income}} = \alpha + b_{\text{hs}} X_{\text{hs}} + b_{\text{jc}} X_{\text{jc}} + b_{\text{ba}} X_{\text{ba}} + b_{\text{grad}} X_{\text{grad}}\]
Each \(X\) is a dummy variable (0/1) for that degree level; each coefficient = the income difference relative to the baseline group.
# Predicted income for each degree level — ordered lowest to highest
tibble(
Degree = c("Less than HS (baseline)", "High School", "Junior College",
"Bachelor's Degree", "Graduate Degree"),
degree_f = factor(
c("lt high", "high sch", "junior c", "bachelor", "graduate"),
levels = levels(attain_cat$degree_f)
)
) |>
mutate(Predicted_Income = scales::dollar(
round(predict(model_cat, newdata = pick(degree_f)), 0)
)) |>
select(Degree, Predicted_Income)# A tibble: 5 × 2
Degree Predicted_Income
<chr> <chr>
1 Less than HS (baseline) $23,181
2 High School $37,541
3 Junior College $47,288
4 Bachelor's Degree $57,315
5 Graduate Degree $64,003 # Rename and reverse order so lowest degree appears at top (matching table above)
modelplot(
model_cat,
coef_omit = "Intercept",
coef_rename = c(
"degree_fhigh sch" = "High School",
"degree_fjunior c" = "Junior College",
"degree_fbachelor" = "Bachelor's Degree",
"degree_fgraduate" = "Graduate Degree"
)
) +
scale_y_discrete(limits = rev) + # lowest degree (HS) at top
geom_vline(xintercept = 0, linetype = "dashed", color = "#E15759") +
labs(
title = "Income Gaps by Degree (vs. Less Than HS baseline)",
x = "Estimated Income Difference ($)"
) +
theme_minimal(base_size = 10)
Observations are independent — typically satisfied by random sampling.
There is a linear relationship between the continuous IV and the outcome.
Always check this with a scatter plot of Y vs X before running the regression.
The errors are approximately normally distributed — most important in small samples. In large samples, violations are usually not a big problem (Central Limit Theorem applies).
The spread of residuals is constant across all values of X (equal variance).
Violations bias standard errors — important for t-tests. Fix: robust standard errors.
A residuals vs. fitted plot diagnoses both linearity and homoskedasticity at once:
