Basic Analysis and Graphing - SAS

124 Performing Bivariate Analysis Chapter 4
Statistical Details for the Bivariate Platform
Statistical Details for Fit Line
The Fit Line command finds the parameters β 0 and β 1 for the straight line that fits the points to minimize
the residual sum of squares. The model for the ith row is written y i
= β 0
+ β 1
x i
+ ε i .
A polynomial of degree 2 is a parabola; a polynomial of degree 3 is a cubic curve. For degree k, the model for
the ith observation is as follows:
y i
=
k
j = 0
β j
x i
j
+ ε i
Statistical Details for Fit Spline
The cubic spline method uses a set of third-degree polynomials spliced together such that the resulting curve
is continuous and smooth at the splices (knot points). The estimation is done by minimizing an objective
function that is a combination of the sum of squares error and a penalty for curvature integrated over the
curve extent. See the paper by Reinsch (1967) or the text by Eubank (1988) for a description of this
method.
Statistical Details for Fit Orthogonal
Standard least square fitting assumes that the X variable is fixed and the Y variable is a function of X plus
error. If there is random variation in the measurement of X, you should fit a line that minimizes the sum of
the squared perpendicular differences. See Figure 4.25. However, the perpendicular distance depends on
how X and Y are scaled, and the scaling for the perpendicular is reserved as a statistical issue, not a graphical
one.
Figure 4.25 Line Perpendicular to the Line of Fit
y distance
orthogonal
distance
x distance
The fit requires that you specify the ratio of the variance of the error in X to the error in Y. This is the
variance of the error, not the variance of the sample points, so you must choose carefully. The ratio
( σ2
y
) ⁄ ( σ2
x
) is infinite in standard least squares because σ2
x is zero. If you do an orthogonal fit with a large
error ratio, the fitted line approaches the standard least squares line of fit. If you specify a ratio of zero, the fit
is equivalent to the regression of X on Y, instead of Y on X.