MGMT90280 · Managerial Decision Analytics
Regression Modelling
This chapter of University of Melbourne MGMT90280 Managerial Decision Analytics fits regression models by least squares and tests what the data support: estimating the slope b₁ and intercept b₀, reporting R² and adjusted R², and using the overall F-test and coefficient t-tests (reject H₀ when p < α) to judge significance. It also covers dummy and interaction variables, confidence intervals for coefficients, and the multicollinearity warning. This toolkit anchors the exam's time-series regression question and the group Assignment 2 analytics report.
What this chapter covers
- 01Simple and multiple regression models: ŷ = b₀ + b₁x₁ + … + b_qx_q
- 02Least-squares estimates: b₁ = Σ(xᵢ−x̄)(yᵢ−ȳ)/Σ(xᵢ−x̄)² and b₀ = ȳ − b₁x̄, and interpreting slope/intercept
- 03Decomposing variation: SST = SSR + SSE and the coefficient of determination R² = SSR/SST
- 04Adjusted R² = 1 − (1−R²)(n−1)/(n−p−1) as the fair way to compare models of different sizes
- 05Overall F-test (upper-tailed): H₀ all slopes zero, F = MSR/MSE, df (p, n−p−1), reject when p < α
- 06Coefficient t-test (two-tailed): t = bₖ/SE(bₖ), df = n−p−1, reject when p < α; use t not z
- 07Confidence interval for a coefficient bₖ ± t_(α/2, n−p−1)·SE(bₖ); a CI that excludes 0 ⇔ reject H₀
- 08Dummy variables (k−1 for k levels, baseline) and interaction terms (one effect depends on another)
- 09Multicollinearity (|r| > 0.7 among predictors) inflates standard errors and distorts individual t-tests
Simple regression by hand: fit, R²/adjusted R², F-test, t-test and 95% CI
- +1Means: x̄ = (2+4+6+8+10)/5 = 6; ȳ = (14+18+23+25+30)/5 = 110/5 = 22.
- +2Deviations (xᵢ−x̄) = −4,−2,0,2,4 and (yᵢ−ȳ) = −8,−4,1,3,8. Σ(xᵢ−x̄)(yᵢ−ȳ) = 32+8+0+6+32 = 78; Σ(xᵢ−x̄)² = 16+4+0+4+16 = 40. So b₁ = 78/40 = 1.95 and b₀ = 22 − 1.95·6 = 10.3, giving ŷ = 10.3 + 1.95x.
- +2Sums of squares: SST = Σ(yᵢ−ȳ)² = 64+16+1+9+64 = 154; SSR = b₁·Σ(xᵢ−x̄)(yᵢ−ȳ) = 1.95·78 = 152.1; SSE = 154 − 152.1 = 1.9.
- +1R² = SSR/SST = 152.1/154 = 0.988 (the model explains ≈ 98.8% of the variation in sales). With n = 5, p = 1: adjusted R² = 1 − (1−0.988)(5−1)/(5−1−1) = 1 − 0.01234·(4/3) = 0.984.
- +2Overall F-test. H₀: β₁ = 0 vs H_A: β₁ ≠ 0. MSR = SSR/p = 152.1; MSE = SSE/(n−p−1) = 1.9/3 = 0.6333; F = MSR/MSE = 240.2, df = (1, 3). Critical F₀.₀₅,₁,₃ = 10.13; since 240.2 > 10.13 (p ≈ 0.0006 < 0.05) → reject H₀: the regression is significant overall.
- +2Slope t-test and CI. SE(b₁) = √(MSE/Σ(xᵢ−x̄)²) = √(0.6333/40) = 0.1258; t = b₁/SE(b₁) = 1.95/0.1258 = 15.50, df = 3, t₀.₀₂₅,₃ = 3.182; since |15.50| > 3.182 → reject H₀, the slope is significant (check: t² ≈ 240 = F). 95% CI for β₁ = 1.95 ± 3.182·0.1258 = [1.55, 2.35], which excludes 0 — consistent with rejecting H₀.
Key terms
- Least-squares estimates (b₀, b₁)
- The intercept and slope that minimise the sum of squared residuals Σ(yᵢ − ŷᵢ)². For simple regression b₁ = Σ(xᵢ−x̄)(yᵢ−ȳ)/Σ(xᵢ−x̄)² and b₀ = ȳ − b₁x̄; the fitted line always passes through the centroid (x̄, ȳ).
- R² (coefficient of determination)
- The fraction of variation in y explained by the model, R² = SSR/SST = 1 − SSE/SST, between 0 and 1. It can only rise as predictors are added, so it must not be used alone to compare models.
- Adjusted R²
- R² penalised for the number of predictors, 1 − (1−R²)(n−1)/(n−p−1) where n = observations and p = predictors. It can fall when a weak variable is added, making it the fair yardstick for comparing models of different sizes.
- Overall F-test
- Tests H₀: β₁ = … = β_q = 0 (no predictor explains y) against at least one βⱼ ≠ 0, using F = MSR/MSE with df (p, n−p−1). It is upper-tailed: reject H₀ when Significance-F (the p-value) < α, meaning the model is significant overall.
- Coefficient t-test
- A two-tailed test of H₀: βₖ = 0 against βₖ ≠ 0, using t = bₖ/SE(bₖ) with df = n−p−1. Reject H₀ when the p-value < α (equivalently |t| > t_(α/2)); t (not z) is used because the error variance is estimated by MSE.
- Confidence interval for a coefficient
- bₖ ± t_(α/2, n−p−1)·SE(bₖ); a 95% interval means we are 95% confident it contains the true βₖ. A CI that excludes 0 is equivalent to rejecting H₀ at the 5% level; one that contains 0 is equivalent to failing to reject.
- Dummy and interaction variables
- A dummy codes a category as 0/1; a k-level category needs k−1 dummies with one omitted baseline, and each coefficient is the mean-y difference versus that baseline. An interaction term x₁·x₂ lets the effect of one predictor depend on the level of the other.
- Multicollinearity
- Strong correlation among the predictors themselves (a warning when a pairwise |r| > 0.7). It inflates standard errors and distorts the individual t-tests — a classic symptom is a significant overall F-test while no single coefficient is significant.
Regression Modelling FAQ
When do I reject the null hypothesis in a regression test?
Use one consistent rule: reject H₀ when the p-value is less than α (typically 0.05), and fail to reject when it is not. The overall F-test is upper-tailed (a large F, small p, means the model explains real variation). Each coefficient t-test is two-tailed with df = n−p−1, testing βₖ = 0 against βₖ ≠ 0. 'Fail to reject' means there is not enough evidence that the coefficient differs from zero — it is not proof the effect is exactly zero. Equivalently, a 95% confidence interval for a coefficient that excludes 0 corresponds to rejecting H₀ at the 5% level.
What is the difference between R² and adjusted R², and which should I use to compare models?
R² is the share of variation in y the model explains (SSR/SST). Its weakness is that it can only rise when you add a predictor, even a useless one, so a bigger model always looks better on R². Adjusted R² divides each sum of squares by its degrees of freedom, so it penalises extra predictors and can fall when a weak variable is added. Compare competing models on adjusted R² (or a lower MSE / standard error of the estimate), not on plain R² — if adding a term raises adjusted R² it earned its place, and if it lowers it the term is dead weight.
Can AI help me with regression modelling in MGMT90280?
Yes — Sia can explain the method step by step: how to compute the least-squares slope and intercept, interpret R² and adjusted R², set up and read the ANOVA F-test and each coefficient's t-test and p-value, build a confidence interval, and reason about dummy variables, interactions and multicollinearity. It works through practice problems with you and checks your reasoning, but it does not sit assessments for you, hand over exam or assignment answers, or guarantee any grade — and Assignment 1 is a Respondus quiz that prohibits generative AI, so use Sia only for study and revision.
Exam move
Drill one regression workflow until it is automatic: compute b₁ and b₀, write the fitted equation, and interpret the slope as the change in mean y per unit of x with the other predictors held constant. Then split the variation (SST = SSR + SSE), report R² and — for model comparison — adjusted R². For inference, run every test as hypotheses → statistic → critical value/p-value → explicit decision: the overall F-test (F = MSR/MSE, df p and n−p−1, upper-tailed) asks whether the model explains anything, and each two-tailed coefficient t-test (t = bₖ/SE(bₖ), df n−p−1) asks whether one predictor matters; reject H₀ when p < α, and remember to use t, not z, because σ is estimated. Practise reading Excel Regression output, handling dummy (k−1) and interaction terms, and spotting multicollinearity when F is significant but no t is. Budget time by marks — with 100 marks over 120 writing minutes, about 1.2 minutes per mark (a 20-mark question ≈ 24 minutes) — and confirm the exact date and permitted materials on the LMS for the open-book (Casio FX-82) end-of-semester (~November 2026) sitting.
Working through Regression Modelling in MGMT90280? Sia is AskSia’s AI Statistics tutor — ask any MGMT90280 Regression Modelling question and get a clear, step-by-step explanation grounded in how MGMT90280 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
