Interpreting and predicting with the line

◈ 4 cards

Slope in units per unit, an intercept that usually means nothing, a residual with a sign, and the refusal to extrapolate.

Reading the two numbers

$score = 544.18 + 2.066 \times income ($000)$

The slope is a rate: units of $y$ per unit of $x$ . "For each additional $1,000 of annual income, the fitted credit score is higher by about 2.07 points, on average." Every phrase is doing work — each additional (a change in $x$), \$1,000 (the unit $x$ is measured in), 2.07 points (the unit of $y$ ), on average (a fitted value, not every client), and — because these are observational data — is higher by, not rises by or increases the score by.

The intercept is the fitted value at $x = 0$ : a client with zero income would be fitted a score of 544. No client in the data is anywhere near zero income (the smallest is $35,000), so the number is a mathematical anchor for the line and not a meaningful prediction. Say so. The intercept is interpretable only when $x = 0$ lies inside the range of the data and makes sense.

Predicting

A new client earns $75,000. That is inside the data's income range (35 to 110), so the line may be used:

$\overset{y}{^} (75) = 544.18 + 2.066 \times 75 = 544.18 + 154.94 = 699.1$

A fitted score of about 699. It is a point on the line, not a guarantee — the spread of real scores around the line is the subject of L4.5 and Module 13.

Residuals

A residual is the vertical miss for an observed unit: $e = y - \overset{y}{^}$ . Client 3 earns $52,000 and scores 640. Fitted: $544.18 + 2.066 \times 52 = 651.6 $. R es i d u a l :$ 640 - 651.6 = -11.6$.

The sign is the message. Negative: the line over-predicts — the point sits below the line; this client scores worse than clients of that income typically do. Positive: the line under-predicts; the point is above. Client 5 ($70,000, 700): fitted 688.8, residual +11.2, above the line.

Extrapolation

A prospect earns $200,000. The arithmetic runs — $544.18 + 2.066 \times 200 = 957$ — and produces a credit score that does not exist on any scale in use. The line was fitted to incomes between $35,000 and $110,000; nothing in the data says the pattern is linear beyond that, and the impossible 957 is the symptom. Predict only inside the range of $x$ used to fit. A question that asks for a prediction outside the range expects the refusal, with the reason, more than it expects the number.