ads2-correlation-regression

name

ads2-correlation-regression

description

Use this skill for ADS/ADS2 correlation, simple linear regression, multiple regression, regression assumptions, residual diagnostics, slope interpretation, R-squared, nested model comparison, interaction terms, VIF/multicollinearity, time-window/local regressions, and ICA trend modeling. Trigger on correlation, cor.test, regression, lm, slope, residuals, R-squared, model comparison, VIF, trend, predictor, or association.

ADS2 Correlation And Regression

Correlation and regression describe association. Do not claim causation unless the study design supports it.

Correlation

Use when both variables are numeric and the question asks about association.

plot(dat$x, dat$y, xlab = "x", ylab = "y")
cor(dat$x, dat$y, use = "complete.obs")
cor.test(dat$x, dat$y)

Interpret:

The correlation was r = [r], indicating a [weak/moderate/strong] [positive/negative] association between [x] and [y]. This does not show causation.

Simple Linear Regression

fit <- lm(y ~ x, data = dat)
summary(fit)
plot(dat$x, dat$y)
abline(fit, col = "red")
plot(fit)
confint(fit)

Slope:

For each one-unit increase in [x], the model predicts [y] changes by [slope] units on average.

Multiple Regression

fit <- lm(y ~ x1 + x2 + group, data = dat)
summary(fit)
plot(fit)

Interaction:

fit_int <- lm(y ~ x1 * group + x2, data = dat)
summary(fit_int)

If interaction is significant, the slope/effect of one predictor depends on the other.

Assumptions

Check:

• independence of observations/errors from design;
• linear relationship;
• homoscedastic residuals;
• approximately normal residuals;
• no severe multicollinearity in multiple regression.

R checks:

plot(fit, which = 1)  # residuals vs fitted
plot(fit, which = 2)  # QQ plot
hist(resid(fit))
shapiro.test(resid(fit))

Optional VIF if car is installed:

library(car)
vif(fit)

Rule of thumb from course notes: VIF > 5 suggests potentially severe multicollinearity.

Nested Model Comparison

fit_simple <- lm(y ~ x1 + x2, data = dat)
fit_complex <- lm(y ~ x1 * x2 + x3, data = dat)
anova(fit_simple, fit_complex)

Interpret:

The nested model comparison tests whether the additional terms improve fit enough to justify the more complex model. If p < 0.05, the complex model fits significantly better; otherwise prefer the simpler model.

Local/Time-Window Regression

Useful for COVID/ICA trend tasks:

library(dplyr)

models <- dat |>
  group_by(age) |>
  do(model = lm(value ~ year, data = .))

slopes <- sapply(models$model, function(m) coef(summary(m))[2, 1])

Compare slopes between periods:

wilcox.test(slopes_late, slopes_early, paired = TRUE, alternative = "greater")

State that this is a trend analysis in observational data.