PSYU2248 Lecture 2: Simple Linear Regression

What is the main purpose of regression?

To explain variability in the dependent variable (Y) using independent variable(s) (X), and to predict values of Y based on X.

What is the key difference between correlation and regression?

Correlation describes the strength of a relationship, whereas regression predicts one variable based on another and identifies direction (IV to DV).

Can regression imply causation?

No, regression suggests prediction, not causation. Prediction does not guarantee a causal relationship.

What is the general form of the simple linear regression equation?

Y = a + bX, where:

• a = intercept (Y when X = 0)

• b = slope (rate of change in Y for each unit change in X)

What do the intercept and slope represent in regression?

Intercept (a) is the predicted Y when X = 0; slope (b) indicates how much Y changes per unit increase in X.

What is the 'best-fitting line' in regression?

The regression line that minimizes the sum of squared errors (residuals) between actual and predicted Y values.

How is total variability in Y decomposed in regression?

Total SS = Model SS + Residual SS

What is a residual in regression?

The difference between the observed value and the predicted value of Y; residual = Y - Ŷ.

What does R² represent?

The proportion of variance in Y explained by the regression model. R² = SS_model / SS_total or (correlation)².

What does it mean when a predictor is significant in a regression model?

There is a statistically significant relationship between the IV and DV.

What are the thresholds for interpreting R² effect sizes?

• 2–12% = small

• 13–25% = medium

• 26%+ = large

What is the hypothesis tested in simple regression?

H₀: The predictor (b) = 0; H₁: The predictor (b) ≠ 0

What are the 4 assumptions of simple linear regression?

• Independence of observations

• Normal distribution of residuals

• Homoscedasticity (equal variance)

• Linearity

How is normality of residuals tested in Stata?

• Use: swilk residuals (Shapiro-Wilk test)

• histogram residuals

• pnorm residuals

How is homoscedasticity tested in Stata?

• Visual check: RVF plot (rvfplot)

• Look for no patterns or fanning in residuals

What is linearity and how is it checked?

Linearity means a straight-line relationship between X and Y. Check with scatterplots or RVF plots.

What is the Stata command to run a simple linear regression?

regress DV IV

How do you save residuals in Stata?

predict newvar, resid

What does adding , beta to a regression command in Stata do?

Produces standardised beta coefficients.

What is an unstandardised coefficient?

Indicates the change in Y per one unit of X, using the original measurement units.

What is a standardised beta coefficient?

The effect size expressed in standard deviation units; allows comparisons across variables with different scales.

How is the regression equation used for prediction?

Substitute X values into the regression equation to calculate predicted Y values (e.g., Ŷ = 30 + 5X).

Why is prediction useful in regression?

Allows estimation of Y for values of X not in the sample, aiding generalisation to the population.

What types of research designs use regression?

Non-experimental designs like cross-sectional and longitudinal studies.

What makes a research question suitable for simple regression?

It involves prediction, with a continuous IV and DV, and assumes a directional (predictive) relationship.