Biostatistics

728 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS
Theil’s Slope Estimator Theil (12) proposes a method for obtaining a point
estimate of the slope coefficient b. We assume that the data conform to the classic
regression model
y i ¼ b 0 þ b 1 x 1 þ e i ;
i ¼ 1; ...; n
where the x i are known constants, b 0 and b 1 are unknown parameters, and Y i is an observed
value of the continuous random variable Y at x i . For each value of x i , we assume a
subpopulation of Y values, and the e i are mutually independent. The x i are all distinct (no
ties), and we take x 1 < x 2 < < x n .
The data consist of n pairs of sample observations, ðx 1 ; y 1 Þ; ðx 2 ; y 2 Þ; ...; ðx n ; y n Þ,
where the ith pair represents measurements taken on the ith unit of association.
To obtain Theil’s estimator of b 1 we first form all possible sample slopes
S ij ¼ y j y i = xj x i , where i < j. There will be N ¼ n C 2 values of S ij . The estimator
of b 1 which we designate by ^b 1 is the median of S ij values. That is,
^b 1 ¼ median S ij
The following example illustrates the calculation of ^b 1 .
(13.11.1)
EXAMPLE 13.11.1
In Table 13.11.1 are the plasma testosterone (ng/ml) levels (Y) and seminal citric acid
(mg/ml) levels in a sample of eight adult males. We wish to compute the estimate of the
population regression slope coefficient by Theil’s method.
Solution: The N ¼ 8 C 2 ¼ 28 ordered values of S ij are shown in Table 13.11.2.
If we let i ¼ 1 and j ¼ 2, the indicators of the first and second values of
Y and X in Table 13.11.1, we may compute S 12 as follows:
S 12 ¼ ð175 230Þ= ð278 421Þ ¼ :3846
When all the slopes are computed in a similar manner and ordered as
in Table 13.11.2, :3846 winds up as the tenth value in the ordered
array.
The median of the S ij values is .4878. Consequently, our estimate of the
population slope coefficient ^b 1 ¼ :4878.
TABLE 13.11.1 Plasma Testosterone and Seminal Citric Acid
Levels in Adult Males
Testosterone: 230 175 315 290 275 150 360 425
Citric acid: 421 278 618 482 465 105 550 750