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13.11 NONPARAMETRIC REGRESSION ANALYSIS 741
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 = b 0 + b 1 x 1 +P i , i = 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 are mutually independent. The are all distinct
(no ties), and we take x 1 6 x 2 6 . . P . i x i
6 x n .
The data consist of n pairs of sample observations, 1x 1 , y 1 2, 1x 2 , y 2 2, . . . , 1x n , y n 2,
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 =
1y j - y i 2>1x j - x i 2, where i 6 j. There will be N = n C 2 values of S ij . The estimator of
b 1 which we designate by N is the median of values. That is,
b 1
S ij
bN 1 = median5S ij 6
(13.11.1)
The following example illustrates the calculation of
b N 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 = 8 C 2 = 28 ordered values of S ij are shown in Table 13.11.2.
If we let i = 1 and j = 2, the indicators of the first and second values
of Y and X in Table 13.11.1, we may compute as follows:
S 12 = 1175 - 2302>1278 - 4212 = -.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 bN 1 = .4878.
S 12
TABLE 13.11.1
in Adult Males
Plasma Testosterone and Seminal Citric Acid Levels
Testosterone: 230 175 315 290 275 150 360 425
Citric acid: 421 278 618 482 465 105 550 750