Predicting, and the limits of the model
Predicting with the model
- To predict a response, substitute the -value into the model and evaluate.
- For , at :
Interpolation vs extrapolation
- Interpolation — predicting for an inside the range of the data. This is reasonable, because the pattern is supported by data there.
- Extrapolation — predicting for an outside the data range. This is unreliable: the relationship may not continue — it could curve or level off.
- Always say whether a prediction is interpolation or extrapolation, and treat extrapolations with caution.
Is the linear model appropriate?
- Check the scatter: if the points follow a curve, a straight line is the wrong model and its predictions will be poor.
- Look for a pattern in the scatter about the line: random scatter suggests the line fits well; a systematic curve means the model is missing something.
- Note limitations: association is not causation, the sample may be small, and a single outlier can pull the line.
The plant model was fitted from data with (fertiliser) between and g. Predict the height at g and at g, and comment on each.
Prediction at
g is within the data range (– g), so this is interpolation — a reasonable, reliable prediction.
Prediction at
g is well beyond the data range, so this is extrapolation. It is unreliable: at very high fertiliser levels the growth may level off (or even reverse), so the linear trend probably does not continue that far.
Test yourself
Practice by grade
One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.
Using , predict when .
A model was fitted from data with between 0 and 50. Predict at and at , stating for each whether it is interpolation or extrapolation.
A linear model predicts a runner's 5 km time keeps falling by a fixed amount for every extra weekly training hour. Explain why using this model to predict the time for someone training 40 hours a week (far beyond the data) is unwise.