COMM5000 Chap.6 Simple Regression and Prediction
Simple Regression and Prediction
Interpret simple-regression coefficients with units, examine residuals, separate coefficient inference from prediction, and avoid extrapolation and unsupported causal language. A fitted line describes the estimated conditional mean of an outcome at each regressor value. The intercept is the fitted mean at zero and deserves a substantive interpretation only when zero is meaningful and represented by the data.
The slope is the estimated change in mean outcome associated with a one-unit increase in the regressor. Both units must appear in the sentence. Residuals are observed outcomes minus fitted values and should be inspected for curvature, changing spread and influential observations. A confidence interval for the slope addresses uncertainty about the population association.
A confidence interval for a mean response differs from a prediction interval for one future outcome; the prediction interval is wider because individual outcomes vary around the mean. Predictions outside the observed regressor range are extrapolations and rely on an untested continuation of the fitted form. A nonzero slope does not establish causality.
Omitted variables, reverse direction, selection and measurement error can explain the association. Use regression to describe or predict according to the study purpose, then state which design evidence would be needed before recommending an intervention based on the slope.
What this chapter covers
- 01
Fitted line
- 02
Slope and intercept
- 03
Residual diagnostics
- 04
Coefficient inference
- 05
Mean and individual prediction
Worked example · free
Handling-time forecast
- +1The four marks used here are not an official university marking scheme. The intercept is an 18-minute fitted mean at zero special items.
- +1Each additional item is associated with 2.4 more minutes in fitted mean time.
- +1At x = 4, prediction = 18 + 2.4 x 4 = 27.6 minutes.
- +1Treat values outside the observed zero-to-six range as extrapolations.
Key terms
- Regression slope
- The fitted change in mean outcome associated with a one-unit change in the regressor.
- Regression intercept
- The fitted mean outcome when the regressor equals zero.
- Prediction interval
- A range for one future outcome that includes residual and fitted-mean uncertainty.
Simple Regression and Prediction FAQ
When is the intercept meaningful?
The intercept is practically meaningful when zero is sensible and represented by the data. Otherwise it remains part of the fitted equation but should not receive an unsupported substantive story.
Why is a prediction interval wider?
One future outcome varies around the fitted mean, so its interval includes individual residual variation as well as uncertainty in the estimated conditional mean overall.
What should I check before using a regression forecast?
Confirm the outcome and predictor units, the observed predictor range and the population and time period used to fit the model. Decide whether the target is a mean response or one future outcome, because the uncertainty interval differs. Inspect residuals for curvature, changing spread and influential observations. Avoid extrapolation beyond the represented range unless a defensible mechanism supports it.
Compare the forecast with a simple baseline and verify that all predictors are available at the intended prediction time. Describe the result as a prediction or association unless the study design justifies an intervention effect.
Assessment move
For every fitted equation, write the outcome unit above y and the regressor unit above x. Interpret the intercept and slope in complete sentences before making any prediction. Substitute a value inside the observed range and show each arithmetic step, then explain that the result is a fitted conditional mean. Label the target as a coefficient, mean response or individual future response before choosing an uncertainty interval.
Draw a residual plot and look for curves, funnels and influential points; describe what each pattern suggests about the simple linear model. Mark the smallest and largest observed x values and flag any requested prediction outside them. Practise replacing causal verbs with associated language, then list one plausible omitted variable and one stronger design that could investigate an intervention effect.
In assessment writing, place coefficient units, interval and substantive magnitude together. For a decision, say whether the predicted difference is operationally important and which process conditions must remain comparable for the forecast to travel to new cases. Reconstruct a line from two fitted values and check that the change agrees with the reported slope.
Compare a confidence interval for the mean response with a prediction interval at the same x and explain the additional individual variability. Use a residual plot checklist: centre around zero, curvature, changing spread, isolated points and clusters. When a pattern appears, name the model feature it challenges rather than merely saying the residuals look bad.
For forecasting, record when the predictor becomes available and ensure no future information leaks into the model. Keep a baseline forecast for comparison. A more elaborate regression should demonstrate better relevant performance or a clearer explanatory purpose, not just a longer equation.
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