Prediction and intervals

In depth

Why the intervals widen away from x̄

The term $(x_0 - \bar{x})^2 / S_{xx}$ is zero at the mean of x and grows with the square of the distance from it. The least-squares line always passes through $(\bar{x}, \bar{y})$; an error in its slope swings it round that point like a seesaw, so the line is pinned most firmly in the middle of the data and least at the ends. At $x_0 = \bar{x}$ the standard error of the mean response is simply $s/\sqrt{n} = 15.24/7 = 2.18$, the standard error of a mean; at 80°F it is 2.99 and at the warmest day in the data, 94°F, 4.45.

The interval for the mean more than doubles in width from x̄ to 94°F, while the prediction interval hardly moves: the day’s own error, s, dwarfs the uncertainty in the line. That is typical of business data with a modest R² (here 0.56). Better estimates of the line help the mean much more than the single case; only a better model, one that explains more of each day’s scatter, narrows the prediction interval substantially.

The seesaw also says where data are most useful. A prediction at the warm end is helped by more warm days, which pull $\bar{x}$ towards 80°F, and by a wide spread of highs, which raises $S_{xx}$. A sample of days all near 64°F would predict 64°F well and 90°F badly, even with the same n and the same s. If a model will be used mostly at one end of its range (Ana rosters with care on hot weekends, rarely on cold ones), that end is where the data should be densest.

The prediction interval’s floor

As n grows, $1/n$ and $(x_0 - \bar{x})^2/S_{xx}$ both shrink towards 0, so $SE_{\text{mean}} \to 0$ but $SE_{\text{pred}} \to s$. With ten years of weekends Ana would know the average at 80°F almost exactly, and a single Saturday would still range about ±1.96 × 15 transactions. Penn State’s STAT 462 puts it this way: the prediction interval is always wider than the confidence interval at the same x, and the extra term never goes away. The only way to narrow it is to reduce s itself. Rain is the obvious candidate: with the day’s rain in the model as well (multiple regression, module 3), s falls from 15.24 to 12.91.

Extrapolation

The intervals assume the straight line holds at $x_0$. Inside the data’s range, 33°F to 94°F, the data can check that; outside it, nothing can. A forecast of 100°F gives ŷ = 126.6 with a prediction interval of 94.2 to 159.0, but the interval only measures uncertainty about a line that is assumed to continue. Real trade may level off as the waterfront becomes too hot to sit in, or the café may simply run out of seats. That is extrapolation, and NIST’s e-Handbook lists poor extrapolation among the main weaknesses of linear least squares. OpenStax’s statistics texts draw the same line between interpolation (predicting inside the observed x values) and extrapolation (outside them).

Extrapolation is not only about x. Ana’s 49 days are Harbourside weekends from July, after the café had built up its regulars. Applied to March, inside the range of temperatures but outside the period, the line overpredicts every one of the eight March weekend days, by 16 transactions on average: the café was still in its first months. The same model is equally unfit for a weekday, for Millbrook, or for a festival day. A prediction is safe only for cases like the ones the model was fitted to, in x and in everything the model leaves out.

What the prediction interval assumes

The interval for the mean leans on the same conditions as the slope’s inference; the prediction interval leans on them harder, because it describes where one error will fall, not where an average will.

• Equal spread. The interval has the same half-width (apart from the small seesaw term) at every x. If the spread grows with the mean, it is too wide for cool days and too narrow for warm ones. The case’s generator is multiplicative, so this is the direction to expect; in these 49 days the residuals’ standard deviation is 15.8 on days of 70°F or more against 13.2 below 55°F, a modest difference the Breusch–Pagan test can’t distinguish from chance.
• Roughly normal errors. The ±t × SE recipe sets the 95% limits where a normal error would put them. For the slope, the central limit theorem rescues the interval as n grows; for one new case it doesn’t, because a single error is not an average. Skewed errors make one tail of the prediction interval too short at any n. Module 4’s normal Q–Q plot checks this.
• Independence and the right model. The day must be like those the model was fitted to, and the model must not have left out a trend that has moved since.

A rough check is how many of the fitted days fall outside their own 95% prediction intervals: here 2 of 49 (4%), about the 5% expected. That is reassuring, though not a test: the intervals were computed from the same days.

Reporting a prediction

A prediction is only useful with four things beside it: the point value, the interval and which kind it is, the confidence level, and the scope (the range of x and the kind of case the model was fitted to). Software names the two intervals differently: OpenStax’s business statistics text calls them the confidence interval and the prediction interval, Penn State’s course the confidence interval for the mean response and the prediction interval for a new response, and some statistical packages label them “mean” and “obs” (observation). Whatever the label, ask whether it contains the 1 under the square root. If a report gives a single interval for a forecast without saying which, ask; one much narrower than the residual standard error suggests is probably the interval for the mean.

In Excel

The ToolPak’s Regression tool prints no interval for a new x, so build them from functions. With the highs in B2:B50, transactions in C2:C50 and the new high in F1:

• Point prediction: =FORECAST.LINEAR(F1,C2:C50,B2:B50) (new x first, then known_y’s, then known_x’s; Microsoft’s page notes it replaced FORECAST in Excel 2016). =TREND(C2:C50,B2:B50,F1) gives the same value.
• The pieces: s is =STEYX(C2:C50,B2:B50), Sxx is =DEVSQ(B2:B50), n is =COUNT(B2:B50) and the multiplier is =T.INV.2T(0.05,COUNT(B2:B50)-2).
• Half-width of the prediction interval: =T.INV.2T(0.05,COUNT(B2:B50)-2)*STEYX(C2:C50,B2:B50)*SQRT(1+1/COUNT(B2:B50)+(F1-AVERAGE(B2:B50))^2/DEVSQ(B2:B50)). Drop the 1+ for the interval for the mean.

One-sided bounds for planning

A 95% prediction interval leaves 2.5% in each tail. A roster only worries about the upper tail: too few baristas is the costly mistake. A one-sided upper bound puts all the risk on that side: $\hat{y}_0 + t_{0.90,\,n-2} \times SE_{\text{pred}}$ is exceeded on about 10% of such days, if the model and its conditions hold. At 80°F that is $106.5 + 1.300 \times 15.53 = 126.7$ transactions, with 1.300 from =T.INV(0.9,47). Which tail probability to accept is a business choice, the cost of an idle barista against the cost of long waits; the statistics only price each choice.