BUSINFO700 Chap.9 Linear Regression for Business Drivers
Linear Regression for Business Drivers
Linear regression is the course's first supervised learning method and its main tool for estimating business drivers. It is taught in Week 8, after the 4 November test, so it matters for later course work rather than for that test.
This chapter follows the six-step procedure from the lecture: formulate a hypothesis for each independent variable, collect enough data, generate the output in R, interpret the betas, assess model fit and check the assumptions.
It explains how least squares chooses the intercept and slope and what residuals are, how to predict an outcome by substituting values into a fitted equation, and how to read the beta of a 0/1 dummy variable.
It then shows why betas are for prediction while absolute t-values or standardised betas rank drivers, how to use t critical values for the model's degrees of freedom, and how to read R-squared, adjusted R-squared and the F-test. Correlated drivers, the sample-size ratio and the four diagnostic plots complete the picture.
What this chapter covers
- 01
Least squares, intercept and slope
- 02
The six steps of a regression analysis
- 03
Interpreting betas, including a 0/1 dummy
- 04
Ranking drivers by t-statistic
- 05
R-squared, adjusted R-squared and the F-test
- 06
Assumptions, diagnostic plots and multicollinearity
Worked example · free
Fit the least-squares line to flyer spend and sales
- 1Means: x-bar = 21 / 6 = 3.5 and y-bar = 84 / 6 = 14.
- 1Sum of cross-products of deviations Sxy = 7.5 + 6 + 1 − 0.5 + 6 + 15 = 35; sum of squared x deviations Sxx = 17.5.
- 1Slope b1 = 35 / 17.5 = 2 and intercept b0 = 14 − 2 × 3.5 = 7, so y-hat = 7 + 2x.
- 1Prediction at x = 4.5: y-hat = 7 + 2 × 4.5 = 16, about 16,000 dollars of weekly sales.
- 1Week 1 residual: actual 11 minus fitted 9 = +2 thousand dollars, so that week sold more than the line predicts.
Key terms
- Least squares
- The estimation rule that picks the line minimising the sum of squared vertical gaps between points and the line.
- Beta coefficient
- The estimated change in the dependent variable for a one-unit rise in a driver, other drivers held constant.
- Residual
- An observation's actual value minus the value the fitted regression predicts for it.
- Adjusted R-squared
- A fit measure that corrects R-squared for the number of predictors, used to compare competing models.
- Multicollinearity
- Strong correlation among independent variables that destabilises individual betas and inflates their standard errors.
Linear Regression for Business Drivers FAQ
What are the six steps of a regression analysis in this course?
Formulate the null and alternative hypotheses for each driver, collect data, generate the output, interpret it, assess model fit and check that the assumptions hold. Interpretation covers which drivers are significant, their ranking and the meaning of each beta; model fit covers R-squared, adjusted R-squared and the F-test; the last step includes multicollinearity.
How do I decide which driver is most important?
Do not compare raw betas, because they depend on each variable's units. The Week 8 lecture ranks drivers by the absolute value of their t-statistic or by standardised betas. Raw betas are kept for prediction, where you substitute a case's values into the fitted equation.
What is the difference between R-squared and adjusted R-squared?
R-squared is the share of variation in the dependent variable the model explains and never falls when a predictor is added. Adjusted R-squared corrects for the number of predictors, so it is the right figure for comparing models with different numbers of drivers.
How large should my sample be for a multiple regression?
The lecture cites Hair and colleagues: at least five observations per independent variable, and ideally fifteen or twenty. For a model with five drivers that means at least 25 cases, and ideally 75 to 100. Sample size drives the statistical power of each significance test and how well the results generalise to new data.
What does a significant F-test with no significant betas suggest?
The drivers together explain the outcome, but no single one stands out, which is the classic sign of multicollinearity. Correlated predictors share the same information, so their separate effects become unstable and their standard errors grow. Check the variance inflation factors and consider dropping or combining drivers.
Assessment move
Rework the flyer example by hand until you can produce the slope, intercept, a prediction and a residual without notes; it rehearses every piece of regression arithmetic in a few lines. Then practise reading output in the order of the six steps, saying aloud for each driver whether it is significant at 5%, indicative at 10% or not significant, and what its beta means with everything else held constant.
Write one ranking of drivers using absolute t-values and explain why the raw betas could not do that job. For model fit, practise a two-sentence verdict that quotes R-squared and judges it in the context of the application rather than against a fixed cut-off. Finally, sketch the four diagnostic plots from memory and write the healthy pattern under each, so that an unhealthy plot in the lab output stands out immediately.
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