Page 96 - Becoming Metric Wise

86 Becoming Metric-Wise

X (predictor, independent). After a study of possible alternatives,
Dietz (1989) recommends the following nonparametric regression estimators:
For the slope, she recommends the Theil estimator β M , which is the

n
median of the sequence of N 5 sample slopes (assuming all x j , j 5 1,
2
.. ., n are distinct):
y j 2 y i
S ij 5 ; i , j; x i 6¼ x j (4.18)
x j 2 x i
For the intercept, she recommends the median of all y j 2 β M .x j ,
j 5 1, .. ., n.
An example
Consider the points (0,5), (1,1),(2,3), (3,3.5), (4,3.5), (5,5), (6,7).
The parametric linear regression line has equation y 5 0.518 x 1 2.304,
with r 5 0.567. This line is heavily influenced by the first point with coor-
dinates (0,5). The nonparametric regression line (thicker line) has equa-
tion: y 5 0.833 x 1 0.833. Fig. 4.12 clearly shows that this line follows the
general trend much better than the parametric one.

4.9 CONTINGENCY TABLES

The relation between two nominal variables (with a small number of
characteristics) can be represented in a contingency table. A contingency
table is created by counting how often each combination of characteristics

Figure 4.12 Parametric and nonparametric regression lines.