Curved relationships
Section of Linear regression, part of Statistical inference
Curved relationships
| Type | Section |
| Taught in | Linear regression |
| Part | Statistical inference |
Not every relationship is a straight line. Two remedies keep the least-squares machinery.
The first fits a polynomial, y=b_0+b_1x+\cdots+b_kx^k. The sum of squares is still quadratic in the coefficients, so setting its k+1 partial derivatives to zero gives k+1 linear equations. The model is still linear in its parameters, even though the curve is not a line.
The second transforms the variables. If theory says y=ae^{bx}, then \log y=\log a+bx is linear in x, and a line is fitted to the points (x_i,\log y_i). The same idea linearises y=ax^b, on the log–log scale, and a logistic curve, on the logit scale. The fitted slope is then a slope on the transformed scale, and so are the errors.
Example 1 (Melanoma and latitude) The melanoma rate per 100 000 white males in nine US locations is
| Latitude | 32.8 | 33.9 | 34.1 | 37.9 | 40.0 | 40.8 | 41.7 | 42.2 | 45.0 |
|---|---|---|---|---|---|---|---|---|---|
| Rate | 9.0 | 5.9 | 6.6 | 5.8 | 5.5 | 3.0 | 3.4 | 3.1 | 3.8 |
Question. Compare a straight line, a quadratic and a line for the log rate.
Solution. The straight line is y=20.6-0.399\,x, but the points bend: the southernmost location is far above it. A quadratic fits the bend, y=66.4-2.81\,x+0.0314\,x^2, at the cost of a coefficient with no simple meaning. The log rate is close to linear, \log y=4.51-0.0761\,x, which says the rate falls by a factor e^{-0.0761}=0.93, about 7%, for each degree further north. A constant percentage change per degree is often the more natural reading for a rate.
Used in
See also
Least-squares line · Least-squares estimators are unbiased · Simple linear regression · Sampling distribution of the regression estimators · Slope t pivot · Coefficient of determination · Prediction interval · Regression effect