Use of predictive statistics using the linear model
Use of predictive statistics using the linear model
This domain tests correlation and least-squares regression: describing a linear relationship, predicting a response, and judging whether the model is appropriate. Items ask for the slope or intercept, the meaning of r and r², residual interpretation, and the assumptions behind inference on the slope.
Core rules
- Least-squares line: ŷ = b0 + b1x, with b1 = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)² and b0 = ȳ − b1x̄.
- b1 is the predicted change in y per one-unit increase in x, measured in y-units per x-unit. b0 is the predicted y when x = 0, often meaningless outside the data range.
- Pearson r measures the strength and direction of linear association, −1 ≤ r ≤ 1; r² is the proportion of variation in y explained by the linear model.
- Residual = observed − predicted. Residuals should show no pattern, roughly constant spread, and approximate normality for inference.
- Assumptions: linearity, independence, normal residuals, equal variance (LINE), plus no influential outliers.
- Correlation does not establish causation; a lurking variable or reverse causation can produce r ≠ 0.
- Multiple regression: each coefficient is the effect of that predictor holding the others constant; collinear predictors inflate standard errors.
Worked example
Data: (1,2), (2,4), (3,5), (4,8). x̄ = 2.5, ȳ = 4.75. Σ(x−x̄)(y−ȳ) = 4.125 + 0.375 + 0.125 + 4.875 = 9.5. Σ(x−x̄)² = 2.25 + 0.25 + 0.25 + 2.25 = 5. So b1 = 9.5/5 = 1.9 and b0 = 4.75 − 1.9(2.5) = 0. The fitted line is ŷ = 1.9x. At x = 5 the prediction is 9.5, but x = 5 lies outside the observed range, so this is extrapolation and should be flagged.
Traps
- Saying r² = 0.81 means '81% chance the model is correct'; it is explained variation.
- Regressing y on x and then inverting the line to predict x - the inverse regression is not the same least-squares fit.
- Reporting a slope without its units or ignoring that b0 may be outside the data range.
- Trusting a strong r when the scatterplot is curved; r captures linear association only.
- Confusing a residual with the model error term.