Optimal Statistical Design of Radioimmunoassays and Competitive Protein-Binding Assays POTTER C. CHANG,* ROBERT T. RUBIN,! AND MIMI YU* * Division of Bio statistics, U.C.L.A. School of Public Health, and f Department of Psychiatry, U.C.LA. School of Medicine, Los Angeles, California 90024 sion by either simple linear regression analysis or by the more complex weighted regression technique. Unknown estimates of highest precision are obtained when 1) the percent counts of the standard doses covers a range of approximately 80% to 20%, 2) the number of standard dose levels is eight or more, 3) the number of replicates at each dose level is two or more, and 4) the percent counts of the unknowns also are within the range 80% to 20%. Under these conditions, also, simple linear regression yields unknown estimates of comparable precision to weighted regression and therefore may be safely used. (Endocrinology 96: 973, 1975)
ABSTRACT. The statistical analysis of radioimmunoassay and competitive protein binding assay data is complex. Because the response variable (percent counts) is not linearly related to log dose, a logit transformation of the response variable usually is performed to permit linear regression analysis. This transformation induces marked heterogeneity of variance, so that iterative weighted regression programs have been used to achieve the best standard curve and the most precise dose estimates of unknowns. In this study several parameters of assay design are investigated in order to establish those designs yielding antigen concentration estimates of highest precision as well as estimates of comparable preci-
T
HE high precision and sensitivity of radioimmunoassays (RIA) and competitive protein-binding assays (CPBA) have resulted in their widespread use. The mathematical theory and statistical methods for these techniques have been developed by Berson and Yalow (1,2), Meinert and McHugh (3), Ekins (4), and particularly by Rodbard and his coworkers (5-8). Ekins and Newman (9) and Feldman and Rodbard (10) have considered the chemical kinetic aspects of optimal assay systems such as the concentrations of tracer and antibody, the duration of incubation, etc. However, the statistical aspects of assay design, such as the range and number of standard doses and the number of repliReceived June 17, 1974. Supported by ONR Contract N00014-73-C-0127 and NIMH Research Scientist Development Award KlMH47363 (to R.T.R.). Computing assistance was obtained from the U.C.L.A. Health Sciences Computing Facility under NIH Special Research Resources Grant RR-3. Address reprint requests to Dr. Potter C. Chang, Division of Biostatistics, U.C.L.A. School of Public Health, Los Angeles, California 90024.
cates at each dose level, have not received sufficient attention. Some investigators establish the dose-response curve by using only a few levels of standard doses in the range where the curve is approximately linear, whereas others use a greater number of standard doses which cover a much wider portion of the dose-response curve. Although sophisticated iterative regression procedures for fitting the standard dose-response curve have been suggested by Meinert and McHugh (3) and Rodbard (5-8), some investigators still fit the doseresponse curve graphically by hand or by simple linear regression. Because of these major differences, it becomes important to use the well developed theoretical statistical models (1-8) to establish statistical assay designs that will yield antigen concentration estimates of highest precision. From a practical standpoint it is also important to identify those assay designs that yield estimates of comparable precision by simple linear regression or the more complicated iterative procedures. Based on the data obtained from the RIA's of insulin, angiotensin I, and growth 973
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974
Endo • 1975 Vol 96 • No 4
CHANG, RUBIN AND YU
hormone, Rolleri et al. (11) suggested that if the number of standard dose levels and the number of replicates are properly chosen, then the simple least squares procedure, which may be done on a desk-top calculator, is sufficient. However, their study only demonstrated that the slope and intercept of the regression lines obtained by the simple and weighted least squares procedures were equivalent for the assays they studied. They did not consider the precision (variance) of the antigen concentration estimates, the improvement of which is a major reason for recommending the weighted regression procedure. The precision of the antigen concentration estimates was the main focus of our study and was examined by comparing the variances of the antigen concentration estimates obtained from the two regression procedures when different statistical designs were used. In order to be independent from specific assay data, our theoretical variances were compared under the assumption that the model suggested by Rodbard and Cooper (7) is accurate. The heteroscedasticity postulated by this model is much greater than- that shown in the assay data of Rolleri et al. (11). The statistical aspects of assay design investigated in our study include the following: 1) the range of the levels of standard doses 2) the number of standard dose levels, given a fixed total number of standard tubes in the assay 3) the number of standard tubes (replicates) at each dose level 4) the effects of the response level of the unknown sample The Model
proposed for the dose-response curve and the variance of the response variables. Among these, the log-logit model suggested by Rodbard and Cooper (7) seems to be well received by many practicing assayists. The estimation method based on this model and particularly the computer program using this method have been widely used. Therefore, the log-logit transformation is used here as the basic model to investigate the effects of the four statistical aspects of assay design on the precision of the antigen concentration estimates. For the model proposed by Rodbard and Cooper (7) the response variable is defined on a scale from zero to one as: Y=
B-N Bo-N
(1)
where B is the number of bound counts per unit time in the presence of standard or unknown unlabeled antigen; Bo is the mean number of bound counts in the absence of unlabeled antigen; and N is the mean number of nonspecific counts in the absence of antibody. Bo and N also are referred to as the 100% and the background (0%) counts, respectively. Based on both empirical and theoretical considerations, Rodbard and Cooper (7) established that E{Y|x}, the mean of the response variable Y at a given standard dose x, can be related to z = In x, the natural logarithm of the standard dose, by the following doseresponse regression curve:
E{Y|z} = 1 -
1 exp{-a + /3z)}
(2)
where a and /3 are unknown parameters. This dose-response regression curve implies that the asymptotic mean of the logit of Y at a log standard dose z, E{logit(Y|z)}, can be related to log dose z by the following dose-response regression line: E{logit(Y|z)}=a + /3z. (3) / Y \ Here logit (Y) = In . The parameters a and /3 are the intercept and slope of this dose response regression line. The asymptotic variance of logit (Y) at a given dose can be approximated by:
The statistical problems of estimating the unknown antigen concentrations by RIA and CPBA are primarily that the standard doseVAR{Y|z} VAR{logit(Y|z)} = (4) response curve is not linear and that the var(E{Y|z})2(l - E{Y|z})2 iances of the response variable at different dose levels are not equal. Several models have been where E{Y|z} and VAR{Y|z} are respectively
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975
STATISTICAL DESIGN OF RADIOLIGAND ASSAYS the mean and variance of Y at a given log standard dose z. Taking into account the variations in pipetting, separating bound from free, and the Poisson variation in radioactive counts, Rodbard and Cooper observed that VAR{Y|z} can be approximated by: VAR{Y|z} = a0 + a,E{Y|z}
(5)
where (6) and SBo
a, =
(Bo - N) 2
-a0
(7)
Here sBo2 and sN2 are the observed variances of the 100% and 0% counts, respectively. This model specifies both the mean and variance of logit (Y) as functions of z, the logarithm of the dose level. While the asymptotic dose response regression curve is reduced to a linear function by the logit transformation, the problem of unequal variances at different dose levels is exacerbated. It is toward this problem of heteroscedasticity that the iterative weighted least squares procedure has been applied (12).
The Precision of the Antigen Concentration Estimates In an assay the estimates of a and /3, the intercept and slope of the dose-response regression line (Eq. 3), are used to estimate the antigen concentrations of the samples. Either the simple or the weighted least squares procedure may be used to determine the estimates of a and j8. For convenience, we use a and b to denote the estimates of a and /3 obtained by either procedure and write the dose-response regression line obtained from the data as: 1 = a + bz
(8)
where 1 is the estimate of E{ logit Y|z}. If l0 is the logit of the response of an unknown sample, then z0, the estimate of the natural logarithm of the antigen concentration of the sample, may be obtained as: b
(9)
The estimate of the antigen concentration of this sample is then: x0 = exp(z0)
(10)
The variance of z0 may be approximated by: VAR{z0} =_(VAR{1O} +VAR{I} + (z o -z) 2 VAR{b}). (11) Here VAR{1O} is determined by substituting x0 in Equations 4, 5, 6 and 7. The expressions for I, z, VAR{I} and VAR{b}, when the simple and the weighted least squares procedures are used, are given in the Appendix. In this study, VAR{zo}, the variance of the estimate of the logarithm of an antigen concentration, was used as the measure of precision. From the expressions given in the Appendix it is evident that the values of VAR{I} and VAR[b} depend both on the statistical design (the range and number of standard dose levels; the number of replicates at each level) and on the assay system (the characteristics of the antibody; the radioactivity of the antigen; other laboratory factors that determine the values of a, /3, ao, and a j . The values of VARflo} and ( v z ) , on the other hand, depend on the antigen concentration of the unknown sample.
Method For the purpose of elucidating those statistical designs that yield estimates of highest precision and/or estimates of comparable precision for either of the two least-squares procedures, the values of VAR{zo} were compared when £,, was determined by both least squares procedures, using different standard dose ranges, numbers of standard dose levels, and number of replicates at each dose. This comparison of VARfio} was carried out for various values of ao, a^ and a. The values of ao, a! and a were determined by using as reference the average values obtained from a number of assay systems used in our laboratory (Table 1). These assay systems include human GH, LH, FSH, and TESTO (13, 14). Since the theoretical values of /3 when all TABLE 1. Average values of ao, B.U a and /3 calculated from four different radioimmunoassay systems Type of assay HGH HLH HFSH TESTO
2.2 1.9 3.2 1.6
X
10"
4
X 10" 5 X
X
a
a,1
aci
10" 4
io- 5
5.8 6.5 1.2 5.9
3
X 10" X 10" 3 X io- 3 4 X 10"
0.95 1.4 0.67 1.8
-1.10 -1.09 -1.16 -1.01
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22
6.5
6.1
66
290
150
78
8
1
2
4
22
22
23
22
22
.54
.48 .46
.47
.43 .41
.74
.59 .51
.79
.59
.49
140
69
35 18
97
49
24
1
2
4 8
1
2
4
2.1
2.1
2.0
2.1
2.2
6.3
8
2.3 2.1
2.5 2.2
2.2
2.3
2.7
2.2 2.1
2.3 2.2
2.6 2.3
19 11
2 4
.90
2.5
.94
3.0
.97
3.3
2.3
2.7
.89
.91
.96
3.3
7.1
36
1
.94
1.0
1.1
8
.40
1.2
.41
1.4
.91
.96
1.1
13
4
.41
1.0
.43
.48
.48
.44
.58
.48
.46 .45
.49 .46
.54
.55
.67
1.2
.96
1.1
1.2
26
2
1.7
.63 .51
.87
1.3
.85 .64
1.3
2.1
.90
1.2
.87
.95 .90
1.1 .96
1.3
1.0
1.3
.44
1.6
12
51
1
.40 .40
.42
.46
.46 .44
.48
.53
8
.55
.64
.86
.89
.98
24
8
1.1
.93 .89
1.0 .94
1.4 1.1
95 48
2 4
2.3 1.6
6.2
5.8
1.1
5.9
1.0
1.2
1.3
6.0
6.5
5.9
5.9
2.0
42
190
1
3.7
6.5
5.9
6.0
6.1
7.3
5.9
6.1
2.2
2.8 2.4
2.1 2.1 2.0
3.7
.98
1.1
1.3
1.8
.43
.48
.57
.75
.56 .50
.70
.96
.97
7.7 6.8
6.0
6.1
6.3 6.1
6.5
.43
.39
.46
.49
.90
1.0
1.1 .99
1.3
5.2 3.6 2.8 2.4
2.5
1.1
2.2 2.1 2.1
2.4 2.2 2.1
2.7 2.3 2.2
2.1 2.2 2.3
8.0 5.0 3.5
2.5 2.2
2.9 2.4 3.4 2.7
1.7 14
3.0 3.8 4.8
.93
4.2
1.1
2.5
7.5
.40 1.4
2.4
1.8 1.4
.42
4.7 3.3
7.3
13
1.6
2.4
3.8
6.8
.92
.65 . .62 .52 .50
.43 .41
.47 .43 .41
.53 .46
.85
.45 1.3
.46 1.5
.48
.56
.67
.46
.49
.54
.55 .49
.45 .44
.46 .45
.67
.98
1.4 1.1
1.9
6.6
7.4
9.0
12
23
8
.44 .44
.45
.46
.50
1.1 .90
.92 2.8
3.4
.96
.93
.98
1.1
.86
.87
.96
1.3
1.8
.99
1.1
1.3
.91 .87
.97 .91
1.1
.56
1.6 1.2
.98
1.1
1.0
2.3
5.9
6.0
1.3
1.6
2.6
1.1
.40
.58 .48
.41 .40
1.3
1.6
.43 .41
.44
.77
.44
.49
.57 .41
.51 .47
.44 .44
.48
.59
.45
.47 .45 .44
.74
.47 .45
.51
.60
.47
.94
.85
.86
.89 .51
1.3 1.0
.89 .87
.94 .89
6.2
9.6
6.1
6.4
6.8
1.0 .94
1.4 1.1
13
6.5
7.0
7.9
1.7
1.3
24
25
22
22 22
23 25
33 28
29
35
23
22
24
4
2
3.7 2.9
5.4
8.9
1.3
1.8
2.8
4.7
.46
.53
.68
.98
.46 .45 .44
.49
.92
1.2 1.0
1.5
6.3
6.8
7.8
9.9
23
24
26
29
(0.60,0.40)
16
23 22
8
23
25 24
4
(0.70,0.30)
7.3
8.8
.94
5.9
6.0
5.9 5.9
6.0
6.2
1.0
6.0
6.2
6.1
6.5
7.2
6.5
23
22
22
22
22
23
27 24
23 22 22
23
2
16
23 22
8
23
24
4
(0.80,0.20)
1.9
2.3
.88
•90
.94
1.0
.39
.40
.41
.43
.44 .44
.46
.48
.85
.89 .87
.95
7.2
6.1
6.5
8.7 6.0
6.5
7.1
12
6.3
6.0
8.6
22
22 6.1
23
26 24
2
22
22 22
16
6.3
6.8
22 '
22
23 22
22
8
23 22
24
4
30 26 24
2
(0.90,0.10)
23 22
23 23
16
8
7.2
8.5
23
110
4
26 24
380 200
1
2
8
See Table 2 for explanation.
0.05
0.10
0.30
0.50
0.70
0.90
0.95
4
(0.99,0.01)
3.0 2.5
3.9
5.9
1.1
1.4
1.9
3.0
.43
.47
.55
.72
.44
.45 .44
.46
.89
1.0 .93
1.2
6.1
6.4
6.9
8.1
22
23
26 24
16
TABLE 3. VAR {ZQ} X 100 calculated by the weighted least-squares method
4.6 3.3
7.3
13
1.6
2.4
3.8
6.8
.53
.67
.96
1.5
.49
.65 .54
.86
1.0
1.6 1.2
2.3
6.6
7.5
9.2
13
23
25
28
34
2
8
2.2 2.1
2.5
3.0
.93
.99
1.1
1.4
.41
.44
.49
.60
.48
.53
.62
.82
.94
1.3 1.0
1.7
6.1
6.3
6.9
7.9
22
23
24
25
4
2.1 2.1
2.2
2.5
.89
.92
.98
1.1
.40
.41
.44
.50
.46
.55 .49
.67
.91
1.1 .97
1.4
6.0
6.1
6.5
7.1
22
22
23
24
(0.50,0.01)
2.0
2.1
2.1
2.2
.88
.89
.92
.97
.39
.40
.41
.44
.45
.47
.50
.57
.88
.99 .91
1.1
5.9
6.0
6.2
6.5
22
22
22
23
16
56 29
110
220
16
31
61
120
5.0 2.7
9.6
19
.49
.65 .54
.86
3.1
9.9 5.4
19
21
36
66
130
49
76
130
240
2
4.2 3.1
6.3
11
1.5
2.2
3.5
6.1
.56
.74
1.1
1.8
.49
.54
.64
.86
.91
.98
1.1
1.4
6.2
6.6
7.4
9.0
23
23
28 25
48
3.4 2.7
4.8
7.6
1.3
1.8
2.6
4.4
.52
.65
.92
1.5
.48
.60 .52
.78
.87
.96 .90
1.1
6.0
6.2
6.5
7.2
22
23
23
25
(0.99,0.50)
2.8 2.4
3.7
5.3
1.1
1.4
1.9
3.0
.56 .47
.73
1.1
.46
.49
.54
.65
.85
.90 .87
.96
5.9
6.0
6.1
6.5
22
22
23
23
16
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140 72 36 19
140 69 35 18
140 72 36 18
170 85 43 22
200 99 51 26
1
1 2 4 8
1 2 4 8
1 2 4 8
1 2 4 8
0.70
0.50
0.30
0.10
0.05
82 42 22 12
8.4 4.7 2.8 1.8
4.2 2.3 1.3 .86
3.6 2.0 1.2 .82
4.6 2.7 1.8 1.3
31 13 17 7.6 94 4.8 5.7 3.4
21 61 11 31 16 6.0 8.4 3.4
11 39 5.9 20 10 3.1 5.2 1.8
9.8 5.1 2.8 1.6
12 39 20 6.4 10 3.6 5.6 2.2
35 18 9.1 4.8
33 28 25 23
26 13 16 9.6 11 7.7 8.4 6.7
51 37 29 26
16
6.7 4.3 3.2 2.6
4.2 2.5 1.7 1.3
2.3 1.3 .86 .62
2.1 1.3 .85 .64
2.8 1.8 1.3 1.1
9.1 7.5 6.6 6.2
27 24 23 22
5.0 3.5 2.7 2.4
2.7 1.8 1.3 1.1
1.1 .73 .56 .47
.92 .68 .55 .49
1.5 1.2 1.0 .93
7.7 6.7 6.3 6.0
25 23 23 22
3.4 2.7 2.4 2.2
1.7 1.3 1.1 .97
.67 .53 .46 .42
.61 .52 .48 .45
1.1 .98 .91 .87
6.7 6.2 6.0 5.9
23 23 22 22
(0.90, 0.10)
2.7 2.3 2.2 2.1
1.3 1.1 .97 .92
.51 .45 .42 .40
.51 .47 .45 .44
.97 .90 .87 .86
6.2 6.0 5.9 5.9
23 22 22 22
16
4.9 3.4 2.7 2.4
2.7 1.8 1.3 1.1
1.1 .75 .57 .48
.96 .70 .56 .50
1.6 1.2 1.0 .93
7.7 6.7 6.3 6.0
25 23 23 22
4.2 3.1 2.5 2.3
2.2 1.5 1.2 1.0
.74 .57 .48 .43
.62 .53 .48 .46
1.2 1.0 .93 .88
7.1 6.5 6.1 6.0
24 23 22 22
3.2 2.6 2.3 2.2
1.6 1.2 1.0 .96
.57 .48 .43 .41
.52 .47 .45 .44
1.0 .93 .88 .86
6.5 6.2 6.0 5.9
23 23 22 22
(0.80, 0.20)
2.6 2.3 2.2 2.1
1.2 1.1 .96 .91
.48 .43 .41 .40
.47 .45 .44 .44
.93 .88 .86 .85
6.2 6.0 5.9 5.9
23 22 22 22
16
6.0 4.0 3.0 2.5
3.2 2.0 1.5 1.2
1.0 .69 .54 .46
.74 .59 .51 .47
1.5 1.1 .99 .92
8.2 7.0 6.4 6.1
26 24 23 22
5.4 3.7 2.8 2.4
2.8 1.8 1.3 1.1
.79 .59 .49 .44
.56 .50 .47 .45
1.2 1.0 .94 .89
7.7 6.8 6.3 6.0
25 24 23 22
4.1 3.0 2.5 2.3
2.0 1.5 1.2 1.0
.61 .50 .44 .41
.50 .46 .45 .44
1.1 .95 .90 .87
7.0 6.4 6.1 5.9
24 23 22 22
(0.70, 0.30)
3.1 2.6 2.3 2.1
1.5 1.2 1.0 .94
.51 .45 .42 .40
.46 .45 .44 .44
.96 .90 .87 .85
6.4 6.1 6.0 5.9
23 23 22 22
16
15 8.3 5.1 3.6
8.0 4.4 2.6 1.8
1.6 1.0 .70 .54
.67 .55 .49 .46
2.1 1.5 1.2 1.0
13 9.4 7.6 6.7
35 28 25 24 9.8 7.8 6.8 6.3
29 26 24 23
.49 .46 .45 .44
13 7.6 4.8 3.4
7.1 4.0 2.4 1.6
9.1 5.6 3.8 2.9
4.9 2.9 1.9 1.4
1.4 1.0 .90 .71 .64 .55 .51 .47
.55 .49 .46 .45
1.9 1.6 1.4 1.2 1.1 1.0 .97 .92
12 8.9 7.4 6.6
33 28 25 23
(0.60,0.40)
6.0 4.0 3.0 2.5
3.1 2.0 1.4 1.1
.74 .56 .47 .43
.46 .45 .44 .44
1.2 1.0 .93 .88
8.0 6.9 6.4 6.1
26 24 23 22
51 37 29 26
9.0 5.5 3.7 2.9
7.4 4.1 2.5 1.7
9.5 4.9 2.7 1.5
13 6.9 3.7 2.1
4.2 3.1 2.5 2.3
2.7 1.8 1.3 1.1
4.3 2.3 1.4 .88
7.5 4.0 2.2 1.3
19 13 10 6.7 5.5 3.8 3.2 2.3
37 28 21 17 14 11 9.7 8.6
62 42 32 27
2.8 2.4 2.2 2.1
1.5 1.2 1.0 .94
2.1 1.2 .81 .60
3.7 2.1 1.3 .84
6.4 3.6 2.2 1.5
17 11 8.5 7.1
36 29 26 24
(0.50, 0.01)
T
Ranges of standard dose (as percent of total) are listed across the top of the table, below which are the numbers of standard dose levels (k). Along the left margin are listed the values of the response variable (unknown) (Yo), next to which are the numbers of replicates (n) at each standard dose level. Within each rectangular grouping (specific dose range and Y9), equal total numbers of standard tubes (kn) lie on the diagonals from lower left to upper right.
4 8
2
180 90 48 27
1 2 4 8
0.90
66 36 21 13
220 100 120 62 71 42 46 32
1 2 4 8
0.95
8
(0.99, 0.01)
y
TABLE 2. VAR {ZQ} X 100 calculated by the simple least-squares method
V
2.4 2.2 2.1 2.0
1.1 1.0 .93 .90
1.2 .77 .58 .48
2.0 1.2 .81 .62
3.5 2.2 1.5 1.2
11 8.3 7.1 6.4
29 25 24 23
16
^
78 40 20 11
41 23 15 10
65 44 33 27
64 32 16 8.3
33 17 8.9 4.8
17 12 8.7 7.2
37 30 26 24
820 590 280 410 300 140 210 150 72 100 76 37
640 450 210 320 220 110 160 110 53 80 56 27
380 240 110 190 120 55 95 59 27 48 30 14
260 140 130 71 66 36 33 18
180 93 47 24
140 72 39 22
160 93 57 40
(0.99, 0.50)
130 67 34 18
97 49 25 13
50 25 13 6.6
29 15 7.6 4.0
15 7.9 4.4 2.6
10 8.1 7.0 6.4
28 25 24 23
16
978
CHANG, RUBIN AND YU
antibodies (or ligand-binding sites) are saturated is —1.0, and the estimates of/3 obtained from all the assay systems were close to -1.0 (Table 1), /? was set to be —1.0 for all computations. The specific statistical designs (range and number of standard dose levels; number of replicates at each standard dose level) are detailed in Tables 2 and 3. In these computations, the standard dose levels were set for each given dose range so that the ratio of adjacent standard dose levels was constant. It also was assumed that equal numbers of replicates were counted at each standard dose. For each assay system and each statistical design, the values of VAR{zo) for antigen concentrations whose responses in terms of Y equaled 0.95, 0.90, . . . , 0.10, 0.05 were computed. First, Equation 3 was used to determine ZQ, the corresponding log antigen concentration of the logit of a given response Y. Then the aforementioned values of ao, aly a and )3, and the value of ZQ were used to determine the magnitudes of (T[2, Wi, and VARfz,,} for both the simple and the weighted least squares procedure.
Results Initially, the values of VAR{z0} were computed for a wide range of values for a©, ax, and a. Specifically, these values were ao: 2 x 10~7, 2 x 10"6, . . . , 2 x 10' 3 ; ax: 5 x 1CT6, 5 x 10~5, . . . , 5 x 10~2; and a:
0.5, 1.0, . . . , 5.0. The ranges for ao, a1} and a were chosen to bracket widely the values of these parameters found with actual assays used in our laboratory (13,14) as indicated in Table 1. Using combinations of these values of ao, a1} and a, VAR{z0} was computed for all unknowns whose response levels in terms of Y ranged from 0.95 to 0.05 and for all statistical designs (ranges of standard curve). With these various combinations, the relative relationships of VAR{z0} between the simple and the weighted least squares procedures were identical. Therefore, in this paper only the values of VAR{z0} obtained by using the average values of a0, ax and a from the eight TESTO assays (Table 1) are presented in Tables 2 and 3. Table 2 gives the results for the simple least squares
Endo • 1975 Vol 96 • No 4
procedure, and Table 3 gives the results for the weighted least squares procedure. In these tables the dose ranges are expressed in terms of the expected value of the response variable Y for the smallest and the largest standard doses; i.e., for the dose range (0.80, 0.20) the expected values of the response variable Y for the smallest and largest standard dose levels are 0.80 and 0.20, respectively. The response levels of the unknown samples also are expressed in terms of the response variable Y. In order to keep the sizes of these tables reasonable, only the values of VAR{z0} obtained for the unknown samples whose response levels are 0.95, 0.90, 0.70, 0.50, 0.30, 0.10 and 0.05 using the more important dose ranges are included. The dose ranges investigated but not included in the tables were (0.25, 0.05), (0.95, 0.50) and (0.50, 0.05). The values of VAR{z0} for a given dose range and a given unknown response level using various numbers of standard doses (k = 2,4,8,16) and various numbers of replicates at each dose (n = 1,2,4,8) are grouped in the tables. Dose range and the two least-squares procedures The effects of the dose range can be examined by comparing the values of VAR{z0} for the different dose ranges, with number of dose levels and number of replicates held constant. For the simple least squares procedure (Table 2) the values of VAR{z0} are much greater if the standard dose levels cover a very wide range (expected response levels varying from 0.99 to 0.01) or are shifted to either side of the standard curve (expected responses ranging from 0.99 to 0.50 or from 0.50 to 0.01). Also, when the standard dose levels are all small (0.99 to 0.50), the values of VAR{z0} are larger than when the standard dose levels are all large (0.50 to 0.01). By comparing the corresponding values of VAR{z0} in Tables 2 and 3 it can be observed that when the expected response
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STATISTICAL DESIGN OF RADIOLIGAND ASSAYS levels for the standard doses are in the range of (0.90, 0.10), (0.80, 0.20), (0.70, 0.30) or (0.60, 0.40) the values of VAR{z0} when the estimates of the intercept and slope are obtained by the simple least squares procedure are very close to the values of VAR{z0} when the estimates of the intercept and slope are obtained by the weighted least squares procedure. On the other hand, when the expected response levels of the standard doses range from 0.99 to 0.01, or are shifted to (0.99, 0.50) or (0.50, 0.01), the values of VAR{z0} are much greater for the simple least-squares procedure. Number of standard dose levels (k) and number of replicates (n) at each dose
979
creases considerably as n increases for the same number of dose levels (k). With a fixed total number of standard tubes (kn), however, the number of dose levels (k) becomes the major influence on VAR{z0}. Response level of the unknown sample The effects of the response level of the unknown sample can be seen by comparing the values of VAR{z0} in each single column of the table. The precision of the antigen concentration estimate is greatly affected by the response level of the unknown sample. Given the same design, VAR{z0} is the smallest if the response level of the unknown sample falls in the middle of the expected response levels of the smallest and largest standard dose levels, and conversely VAR{z0} is much increased when the response level of the unknown sample is outside of the range of the expected response level of the smallest and the largest standard dose levels.
The value of VAR{z0} is decreased when the total number of replicates (kn) is increased. This can be observed by moving from top to bottom or from left to right within each grouping of the table. The effects of the number of standard dose levels and number of replicates at Discussion each dose may be examined by comparing From the results of this study, it is clear the values of VAR{z0} for a given dose that more precise antigen concentration range and a given unknown sample when the total number of replicates (kn) is fixed. estimates are obtained if the response These can be found by moving diagonally levels of the standard doses vary from 0.90 from the lower left corner to the upper to 0.10, 0.80 to 0.20, or 0.70 to 0.30. When right corner of each grouping. For instance, the standard dose levels used in an assay when the expected response levels of the are within these ranges, the complicated standard dose vary from 0.80 to 0.20, when weighted least-squares procedure, which the response level of the unknown sample requires a sophisticated computing facility, is 0.50, and when the total number of is no longer necessary. The precision of the replicates (kn) is 16, the values of VAR{z0} antigen concentration estimates obtained for k = 2, 4, 8, 16 are 0.50, 0.48, 0.47 and by using the simple least-squares proce0.47 using the simple least squares proce- dure to fit the linear dose-response regresdure (Table 2). This is a typfcal pattern, in sion line is as good as that of the weighted that the values of VAR{z0} are quite close procedure. This is consistent with the for most dose ranges when the number of results of the theoretical investigation by standard doses (k) is eight or more. The Jacquez and Norusis (15), in which the exceptions are when the dose range is simple and weighted regression proce(0.99, 0.01) or (0.99, 0.50). When the dures were compared for several patterns number of standard dose levels is small, of heteroscedasticity. However, the pattern larger values of VAR{z0} occur frequently. of heteroscedasticity stipulated by the With reference to number of replicates (n) model of Rodbard and Cooper (7) was not at each dose level, VAR{z0} always de- examined by Jacquez and Norusis. Our
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980
Endo • 1975 Vol 96 • No 4
CHANG, RUBIN AND YU
results also are consistent with those of Rolleri et al. (11), which were derived from empirical assay data containing a lesser degree of heteroscedasticity. If the standard doses are not properly chosen, so that the response levels for the standard doses cover the wide range from 0.99 to 0.01 or are shifted to one side of the curve (0.99 to 0.50 or 0.50 to 0.01), the weighted least-squares procedure such as the one suggested by Rodbard and Lewald (8) considerably improves the precision of the antigen concentration estimates. However, the weighted least-squares procedure will not improve the precision of a badly designed assay compared with the precision obtained by using the simple leastsquares procedure with a well designed assay (compare extreme groupings in Table 3 with "good" groupings in the middle of Table 2). Therefore, the selection of dose range clearly remains the major consideration. The use of dose levels so that the response levels of the standard doses range from 0.90 to 0.10 or 0.80 to 0.20 is a specific a priori method of truncation. Using these dose ranges, it becomes reasonable to make the requisite assumption of homoscedasticity for simple linear regression. The possibility of the use of truncation also has been discussed by Rodbard (12), and Vivian and LaBella (16) have used empirical sequential truncation to achieve acceptable homogeneity of variance. The next most important factor affecting the precision of the antigen concentration estimates appears to be the response level of the unknown sample. If the amounts of standard antigen and antibody used in the assay system are such that the response levels of the unknown samples are in the middle of the response levels of the smallest and the largest standard dose levels, then the precision of the unknown estimates is at its greatest. As expected, the precision of the antigen concentration estimates is improved when more replicates are used to determine the standard regression line. If the total
number of standard tubes that can be used in an assay is fixed, the precision of the antigen concentration estimates is not much affected by changing the number of standard dose levels, as long as it is not too small. On the other hand, too many levels of the standard dose may very well increase the cumulative errors of diluting and pipetting. It appears to be generally advisable to use eight or more standard dose levels, with at least two replicates at each dose. Based on the above considerations, a good statistical design for an assay is one in which (1) the average responses (in terms of Y) of the standard dose levels vary approximately from 0.80 to 0.20, (2) the response levels (in terms of Y) of all unknown samples also fall within 0.80 to 0.20, and (3) at least eight standard dose levels with at least two replicates at each level are used. With this design, a simple least-squares method for fitting the standard dose-response regression line may be safely used. Prior information is required in designing experiments. RIA and CPBA have the advantage that they usually are used repeatedly. Thus, past experience can be used in selecting the optimal standard dose levels. When a new assay system is used for the first time in a laboratory, experience with the assay in other laboratories may guide the initial design. Since the standard dose-response regression line usually is used to estimate the antigen concentration of many unknown samples, it would appear worthwhile to conduct a small assay of a few unknowns to obtain the information necessary for selecting the desired standard dose levels for subsequent assays. References 1. Berson, S. A., and R. S. Yalow, Adv Biol Med Phys 6: 349, 1958. 2. , and ,J Clin Invest 38: 1996, 1959. 3. Meinert, C. L., and R. B. McHugh, Math Biosci 2: 319, 1968. 4. Ekins, R. P., In In Vitro Procedures with Radioisotopes in Medicine, Int. Atomic Energy Agency, Vienna, 1970, p. 325.
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981
STATISTICAL DESIGN OF RADIOLIGAND ASSAYS 5. Rodbard, D., P. L. Rayford, J. A. Cooper, and G. T. RossJ Clin Endocrinol Metab 28: 1412, 1968. 6. , W. Bridson, and P. L. Rayford J Lab Clin Med 74: 770, 1969. 7. , and J. A. Cooper, In In Vitro Procedures with Radioisotopes in Medicine, Int. Atomic Energy Agency, Vienna, 1970, p. 6§9. 8. , and J. E. Lewald, Ada Endocrinol (Kbh) (Suppl. 147) 64: 79, 1970. 9. Ekins, R. P., and B. Newman, Ada Endocrinol (Kbh) (Suppl. 147) 64: 11, 1970. 10. Feldman, H., and D. Rodbard, In Odell, W. D., and W. H. Daughaday (eds.), Principles of Competitive Protein-Binding Assays, Lippincott, Philadelphia, 1971, p. 158. 11. Rolleri, E., P. G. Novario, and B. PaglianoJ Nucl Biol Med 17: 128, 1973. 12. Rodbard, D., In Odell, W. D., and W. H. Daughaday (eds.), Principles of Competitive Protein-Binding Assays, Lippincott, Philadelphia, 1971, p. 204. 13. Odell, W. D., P. L. Rayford, and G. T. RossJ Lab Clin Med 70: 973, 1967. 14. , R. S. Swerdloff, J. Bain, F. Wollesen, and P. K. Grover, Endocrinology 95: 1380, 1974. 15. Jacquez, J. A., and M. Norusis, Biometrics 29: 771, 1973. 16. Vivian, S. R., and F. S. LaBellaJ Clin Endocrinol Metab 33: 225, 1971.
Appendix In this appendix, the expressions for z, I, VAR{1} and VAR{b} are derived for the simple and the weighted least squares procedures. We first define the notations as follows: k = the number of standard dose levels x, = the i-th standard dose level, i = 1, ••-, k Zj = In Xi = the natural logarithm of the i-th standard dose level n = the number of replicates at each standard dose level y u = the response of the j-th replicate at i-th standard dose level, j = 1, •••, n. y^ is expressed in terms of the response variable Y defined by Eq. 1. l u = logit{y,j} = In
1J
cri2 = VAR{logit(Y|zi)} = the variance of l u , j = 1, •••, n at the i-th dose level. For the model used in this study,
ABSTRACT. The statistical analysis of radioimmunoassay and competitive protein binding assay data is complex. Because the response variable (percent counts) is not linearly related to log dose, a logit transformation of the response variable usually is performed to permit linear regression analysis. This transformation induces marked heterogeneity of variance, so that iterative weighted regression programs have been used to achieve the best standard curve and the most precise dose estimates of unknowns. In this study several parameters of assay design are investigated in order to establish those designs yielding antigen concentration estimates of highest precision as well as estimates of comparable preci-
T
HE high precision and sensitivity of radioimmunoassays (RIA) and competitive protein-binding assays (CPBA) have resulted in their widespread use. The mathematical theory and statistical methods for these techniques have been developed by Berson and Yalow (1,2), Meinert and McHugh (3), Ekins (4), and particularly by Rodbard and his coworkers (5-8). Ekins and Newman (9) and Feldman and Rodbard (10) have considered the chemical kinetic aspects of optimal assay systems such as the concentrations of tracer and antibody, the duration of incubation, etc. However, the statistical aspects of assay design, such as the range and number of standard doses and the number of repliReceived June 17, 1974. Supported by ONR Contract N00014-73-C-0127 and NIMH Research Scientist Development Award KlMH47363 (to R.T.R.). Computing assistance was obtained from the U.C.L.A. Health Sciences Computing Facility under NIH Special Research Resources Grant RR-3. Address reprint requests to Dr. Potter C. Chang, Division of Biostatistics, U.C.L.A. School of Public Health, Los Angeles, California 90024.
cates at each dose level, have not received sufficient attention. Some investigators establish the dose-response curve by using only a few levels of standard doses in the range where the curve is approximately linear, whereas others use a greater number of standard doses which cover a much wider portion of the dose-response curve. Although sophisticated iterative regression procedures for fitting the standard dose-response curve have been suggested by Meinert and McHugh (3) and Rodbard (5-8), some investigators still fit the doseresponse curve graphically by hand or by simple linear regression. Because of these major differences, it becomes important to use the well developed theoretical statistical models (1-8) to establish statistical assay designs that will yield antigen concentration estimates of highest precision. From a practical standpoint it is also important to identify those assay designs that yield estimates of comparable precision by simple linear regression or the more complicated iterative procedures. Based on the data obtained from the RIA's of insulin, angiotensin I, and growth 973
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974
Endo • 1975 Vol 96 • No 4
CHANG, RUBIN AND YU
hormone, Rolleri et al. (11) suggested that if the number of standard dose levels and the number of replicates are properly chosen, then the simple least squares procedure, which may be done on a desk-top calculator, is sufficient. However, their study only demonstrated that the slope and intercept of the regression lines obtained by the simple and weighted least squares procedures were equivalent for the assays they studied. They did not consider the precision (variance) of the antigen concentration estimates, the improvement of which is a major reason for recommending the weighted regression procedure. The precision of the antigen concentration estimates was the main focus of our study and was examined by comparing the variances of the antigen concentration estimates obtained from the two regression procedures when different statistical designs were used. In order to be independent from specific assay data, our theoretical variances were compared under the assumption that the model suggested by Rodbard and Cooper (7) is accurate. The heteroscedasticity postulated by this model is much greater than- that shown in the assay data of Rolleri et al. (11). The statistical aspects of assay design investigated in our study include the following: 1) the range of the levels of standard doses 2) the number of standard dose levels, given a fixed total number of standard tubes in the assay 3) the number of standard tubes (replicates) at each dose level 4) the effects of the response level of the unknown sample The Model
proposed for the dose-response curve and the variance of the response variables. Among these, the log-logit model suggested by Rodbard and Cooper (7) seems to be well received by many practicing assayists. The estimation method based on this model and particularly the computer program using this method have been widely used. Therefore, the log-logit transformation is used here as the basic model to investigate the effects of the four statistical aspects of assay design on the precision of the antigen concentration estimates. For the model proposed by Rodbard and Cooper (7) the response variable is defined on a scale from zero to one as: Y=
B-N Bo-N
(1)
where B is the number of bound counts per unit time in the presence of standard or unknown unlabeled antigen; Bo is the mean number of bound counts in the absence of unlabeled antigen; and N is the mean number of nonspecific counts in the absence of antibody. Bo and N also are referred to as the 100% and the background (0%) counts, respectively. Based on both empirical and theoretical considerations, Rodbard and Cooper (7) established that E{Y|x}, the mean of the response variable Y at a given standard dose x, can be related to z = In x, the natural logarithm of the standard dose, by the following doseresponse regression curve:
E{Y|z} = 1 -
1 exp{-a + /3z)}
(2)
where a and /3 are unknown parameters. This dose-response regression curve implies that the asymptotic mean of the logit of Y at a log standard dose z, E{logit(Y|z)}, can be related to log dose z by the following dose-response regression line: E{logit(Y|z)}=a + /3z. (3) / Y \ Here logit (Y) = In . The parameters a and /3 are the intercept and slope of this dose response regression line. The asymptotic variance of logit (Y) at a given dose can be approximated by:
The statistical problems of estimating the unknown antigen concentrations by RIA and CPBA are primarily that the standard doseVAR{Y|z} VAR{logit(Y|z)} = (4) response curve is not linear and that the var(E{Y|z})2(l - E{Y|z})2 iances of the response variable at different dose levels are not equal. Several models have been where E{Y|z} and VAR{Y|z} are respectively
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975
STATISTICAL DESIGN OF RADIOLIGAND ASSAYS the mean and variance of Y at a given log standard dose z. Taking into account the variations in pipetting, separating bound from free, and the Poisson variation in radioactive counts, Rodbard and Cooper observed that VAR{Y|z} can be approximated by: VAR{Y|z} = a0 + a,E{Y|z}
(5)
where (6) and SBo
a, =
(Bo - N) 2
-a0
(7)
Here sBo2 and sN2 are the observed variances of the 100% and 0% counts, respectively. This model specifies both the mean and variance of logit (Y) as functions of z, the logarithm of the dose level. While the asymptotic dose response regression curve is reduced to a linear function by the logit transformation, the problem of unequal variances at different dose levels is exacerbated. It is toward this problem of heteroscedasticity that the iterative weighted least squares procedure has been applied (12).
The Precision of the Antigen Concentration Estimates In an assay the estimates of a and /3, the intercept and slope of the dose-response regression line (Eq. 3), are used to estimate the antigen concentrations of the samples. Either the simple or the weighted least squares procedure may be used to determine the estimates of a and j8. For convenience, we use a and b to denote the estimates of a and /3 obtained by either procedure and write the dose-response regression line obtained from the data as: 1 = a + bz
(8)
where 1 is the estimate of E{ logit Y|z}. If l0 is the logit of the response of an unknown sample, then z0, the estimate of the natural logarithm of the antigen concentration of the sample, may be obtained as: b
(9)
The estimate of the antigen concentration of this sample is then: x0 = exp(z0)
(10)
The variance of z0 may be approximated by: VAR{z0} =_(VAR{1O} +VAR{I} + (z o -z) 2 VAR{b}). (11) Here VAR{1O} is determined by substituting x0 in Equations 4, 5, 6 and 7. The expressions for I, z, VAR{I} and VAR{b}, when the simple and the weighted least squares procedures are used, are given in the Appendix. In this study, VAR{zo}, the variance of the estimate of the logarithm of an antigen concentration, was used as the measure of precision. From the expressions given in the Appendix it is evident that the values of VAR{I} and VAR[b} depend both on the statistical design (the range and number of standard dose levels; the number of replicates at each level) and on the assay system (the characteristics of the antibody; the radioactivity of the antigen; other laboratory factors that determine the values of a, /3, ao, and a j . The values of VARflo} and ( v z ) , on the other hand, depend on the antigen concentration of the unknown sample.
Method For the purpose of elucidating those statistical designs that yield estimates of highest precision and/or estimates of comparable precision for either of the two least-squares procedures, the values of VAR{zo} were compared when £,, was determined by both least squares procedures, using different standard dose ranges, numbers of standard dose levels, and number of replicates at each dose. This comparison of VARfio} was carried out for various values of ao, a^ and a. The values of ao, a! and a were determined by using as reference the average values obtained from a number of assay systems used in our laboratory (Table 1). These assay systems include human GH, LH, FSH, and TESTO (13, 14). Since the theoretical values of /3 when all TABLE 1. Average values of ao, B.U a and /3 calculated from four different radioimmunoassay systems Type of assay HGH HLH HFSH TESTO
2.2 1.9 3.2 1.6
X
10"
4
X 10" 5 X
X
a
a,1
aci
10" 4
io- 5
5.8 6.5 1.2 5.9
3
X 10" X 10" 3 X io- 3 4 X 10"
0.95 1.4 0.67 1.8
-1.10 -1.09 -1.16 -1.01
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22
6.5
6.1
66
290
150
78
8
1
2
4
22
22
23
22
22
.54
.48 .46
.47
.43 .41
.74
.59 .51
.79
.59
.49
140
69
35 18
97
49
24
1
2
4 8
1
2
4
2.1
2.1
2.0
2.1
2.2
6.3
8
2.3 2.1
2.5 2.2
2.2
2.3
2.7
2.2 2.1
2.3 2.2
2.6 2.3
19 11
2 4
.90
2.5
.94
3.0
.97
3.3
2.3
2.7
.89
.91
.96
3.3
7.1
36
1
.94
1.0
1.1
8
.40
1.2
.41
1.4
.91
.96
1.1
13
4
.41
1.0
.43
.48
.48
.44
.58
.48
.46 .45
.49 .46
.54
.55
.67
1.2
.96
1.1
1.2
26
2
1.7
.63 .51
.87
1.3
.85 .64
1.3
2.1
.90
1.2
.87
.95 .90
1.1 .96
1.3
1.0
1.3
.44
1.6
12
51
1
.40 .40
.42
.46
.46 .44
.48
.53
8
.55
.64
.86
.89
.98
24
8
1.1
.93 .89
1.0 .94
1.4 1.1
95 48
2 4
2.3 1.6
6.2
5.8
1.1
5.9
1.0
1.2
1.3
6.0
6.5
5.9
5.9
2.0
42
190
1
3.7
6.5
5.9
6.0
6.1
7.3
5.9
6.1
2.2
2.8 2.4
2.1 2.1 2.0
3.7
.98
1.1
1.3
1.8
.43
.48
.57
.75
.56 .50
.70
.96
.97
7.7 6.8
6.0
6.1
6.3 6.1
6.5
.43
.39
.46
.49
.90
1.0
1.1 .99
1.3
5.2 3.6 2.8 2.4
2.5
1.1
2.2 2.1 2.1
2.4 2.2 2.1
2.7 2.3 2.2
2.1 2.2 2.3
8.0 5.0 3.5
2.5 2.2
2.9 2.4 3.4 2.7
1.7 14
3.0 3.8 4.8
.93
4.2
1.1
2.5
7.5
.40 1.4
2.4
1.8 1.4
.42
4.7 3.3
7.3
13
1.6
2.4
3.8
6.8
.92
.65 . .62 .52 .50
.43 .41
.47 .43 .41
.53 .46
.85
.45 1.3
.46 1.5
.48
.56
.67
.46
.49
.54
.55 .49
.45 .44
.46 .45
.67
.98
1.4 1.1
1.9
6.6
7.4
9.0
12
23
8
.44 .44
.45
.46
.50
1.1 .90
.92 2.8
3.4
.96
.93
.98
1.1
.86
.87
.96
1.3
1.8
.99
1.1
1.3
.91 .87
.97 .91
1.1
.56
1.6 1.2
.98
1.1
1.0
2.3
5.9
6.0
1.3
1.6
2.6
1.1
.40
.58 .48
.41 .40
1.3
1.6
.43 .41
.44
.77
.44
.49
.57 .41
.51 .47
.44 .44
.48
.59
.45
.47 .45 .44
.74
.47 .45
.51
.60
.47
.94
.85
.86
.89 .51
1.3 1.0
.89 .87
.94 .89
6.2
9.6
6.1
6.4
6.8
1.0 .94
1.4 1.1
13
6.5
7.0
7.9
1.7
1.3
24
25
22
22 22
23 25
33 28
29
35
23
22
24
4
2
3.7 2.9
5.4
8.9
1.3
1.8
2.8
4.7
.46
.53
.68
.98
.46 .45 .44
.49
.92
1.2 1.0
1.5
6.3
6.8
7.8
9.9
23
24
26
29
(0.60,0.40)
16
23 22
8
23
25 24
4
(0.70,0.30)
7.3
8.8
.94
5.9
6.0
5.9 5.9
6.0
6.2
1.0
6.0
6.2
6.1
6.5
7.2
6.5
23
22
22
22
22
23
27 24
23 22 22
23
2
16
23 22
8
23
24
4
(0.80,0.20)
1.9
2.3
.88
•90
.94
1.0
.39
.40
.41
.43
.44 .44
.46
.48
.85
.89 .87
.95
7.2
6.1
6.5
8.7 6.0
6.5
7.1
12
6.3
6.0
8.6
22
22 6.1
23
26 24
2
22
22 22
16
6.3
6.8
22 '
22
23 22
22
8
23 22
24
4
30 26 24
2
(0.90,0.10)
23 22
23 23
16
8
7.2
8.5
23
110
4
26 24
380 200
1
2
8
See Table 2 for explanation.
0.05
0.10
0.30
0.50
0.70
0.90
0.95
4
(0.99,0.01)
3.0 2.5
3.9
5.9
1.1
1.4
1.9
3.0
.43
.47
.55
.72
.44
.45 .44
.46
.89
1.0 .93
1.2
6.1
6.4
6.9
8.1
22
23
26 24
16
TABLE 3. VAR {ZQ} X 100 calculated by the weighted least-squares method
4.6 3.3
7.3
13
1.6
2.4
3.8
6.8
.53
.67
.96
1.5
.49
.65 .54
.86
1.0
1.6 1.2
2.3
6.6
7.5
9.2
13
23
25
28
34
2
8
2.2 2.1
2.5
3.0
.93
.99
1.1
1.4
.41
.44
.49
.60
.48
.53
.62
.82
.94
1.3 1.0
1.7
6.1
6.3
6.9
7.9
22
23
24
25
4
2.1 2.1
2.2
2.5
.89
.92
.98
1.1
.40
.41
.44
.50
.46
.55 .49
.67
.91
1.1 .97
1.4
6.0
6.1
6.5
7.1
22
22
23
24
(0.50,0.01)
2.0
2.1
2.1
2.2
.88
.89
.92
.97
.39
.40
.41
.44
.45
.47
.50
.57
.88
.99 .91
1.1
5.9
6.0
6.2
6.5
22
22
22
23
16
56 29
110
220
16
31
61
120
5.0 2.7
9.6
19
.49
.65 .54
.86
3.1
9.9 5.4
19
21
36
66
130
49
76
130
240
2
4.2 3.1
6.3
11
1.5
2.2
3.5
6.1
.56
.74
1.1
1.8
.49
.54
.64
.86
.91
.98
1.1
1.4
6.2
6.6
7.4
9.0
23
23
28 25
48
3.4 2.7
4.8
7.6
1.3
1.8
2.6
4.4
.52
.65
.92
1.5
.48
.60 .52
.78
.87
.96 .90
1.1
6.0
6.2
6.5
7.2
22
23
23
25
(0.99,0.50)
2.8 2.4
3.7
5.3
1.1
1.4
1.9
3.0
.56 .47
.73
1.1
.46
.49
.54
.65
.85
.90 .87
.96
5.9
6.0
6.1
6.5
22
22
23
23
16
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140 72 36 19
140 69 35 18
140 72 36 18
170 85 43 22
200 99 51 26
1
1 2 4 8
1 2 4 8
1 2 4 8
1 2 4 8
0.70
0.50
0.30
0.10
0.05
82 42 22 12
8.4 4.7 2.8 1.8
4.2 2.3 1.3 .86
3.6 2.0 1.2 .82
4.6 2.7 1.8 1.3
31 13 17 7.6 94 4.8 5.7 3.4
21 61 11 31 16 6.0 8.4 3.4
11 39 5.9 20 10 3.1 5.2 1.8
9.8 5.1 2.8 1.6
12 39 20 6.4 10 3.6 5.6 2.2
35 18 9.1 4.8
33 28 25 23
26 13 16 9.6 11 7.7 8.4 6.7
51 37 29 26
16
6.7 4.3 3.2 2.6
4.2 2.5 1.7 1.3
2.3 1.3 .86 .62
2.1 1.3 .85 .64
2.8 1.8 1.3 1.1
9.1 7.5 6.6 6.2
27 24 23 22
5.0 3.5 2.7 2.4
2.7 1.8 1.3 1.1
1.1 .73 .56 .47
.92 .68 .55 .49
1.5 1.2 1.0 .93
7.7 6.7 6.3 6.0
25 23 23 22
3.4 2.7 2.4 2.2
1.7 1.3 1.1 .97
.67 .53 .46 .42
.61 .52 .48 .45
1.1 .98 .91 .87
6.7 6.2 6.0 5.9
23 23 22 22
(0.90, 0.10)
2.7 2.3 2.2 2.1
1.3 1.1 .97 .92
.51 .45 .42 .40
.51 .47 .45 .44
.97 .90 .87 .86
6.2 6.0 5.9 5.9
23 22 22 22
16
4.9 3.4 2.7 2.4
2.7 1.8 1.3 1.1
1.1 .75 .57 .48
.96 .70 .56 .50
1.6 1.2 1.0 .93
7.7 6.7 6.3 6.0
25 23 23 22
4.2 3.1 2.5 2.3
2.2 1.5 1.2 1.0
.74 .57 .48 .43
.62 .53 .48 .46
1.2 1.0 .93 .88
7.1 6.5 6.1 6.0
24 23 22 22
3.2 2.6 2.3 2.2
1.6 1.2 1.0 .96
.57 .48 .43 .41
.52 .47 .45 .44
1.0 .93 .88 .86
6.5 6.2 6.0 5.9
23 23 22 22
(0.80, 0.20)
2.6 2.3 2.2 2.1
1.2 1.1 .96 .91
.48 .43 .41 .40
.47 .45 .44 .44
.93 .88 .86 .85
6.2 6.0 5.9 5.9
23 22 22 22
16
6.0 4.0 3.0 2.5
3.2 2.0 1.5 1.2
1.0 .69 .54 .46
.74 .59 .51 .47
1.5 1.1 .99 .92
8.2 7.0 6.4 6.1
26 24 23 22
5.4 3.7 2.8 2.4
2.8 1.8 1.3 1.1
.79 .59 .49 .44
.56 .50 .47 .45
1.2 1.0 .94 .89
7.7 6.8 6.3 6.0
25 24 23 22
4.1 3.0 2.5 2.3
2.0 1.5 1.2 1.0
.61 .50 .44 .41
.50 .46 .45 .44
1.1 .95 .90 .87
7.0 6.4 6.1 5.9
24 23 22 22
(0.70, 0.30)
3.1 2.6 2.3 2.1
1.5 1.2 1.0 .94
.51 .45 .42 .40
.46 .45 .44 .44
.96 .90 .87 .85
6.4 6.1 6.0 5.9
23 23 22 22
16
15 8.3 5.1 3.6
8.0 4.4 2.6 1.8
1.6 1.0 .70 .54
.67 .55 .49 .46
2.1 1.5 1.2 1.0
13 9.4 7.6 6.7
35 28 25 24 9.8 7.8 6.8 6.3
29 26 24 23
.49 .46 .45 .44
13 7.6 4.8 3.4
7.1 4.0 2.4 1.6
9.1 5.6 3.8 2.9
4.9 2.9 1.9 1.4
1.4 1.0 .90 .71 .64 .55 .51 .47
.55 .49 .46 .45
1.9 1.6 1.4 1.2 1.1 1.0 .97 .92
12 8.9 7.4 6.6
33 28 25 23
(0.60,0.40)
6.0 4.0 3.0 2.5
3.1 2.0 1.4 1.1
.74 .56 .47 .43
.46 .45 .44 .44
1.2 1.0 .93 .88
8.0 6.9 6.4 6.1
26 24 23 22
51 37 29 26
9.0 5.5 3.7 2.9
7.4 4.1 2.5 1.7
9.5 4.9 2.7 1.5
13 6.9 3.7 2.1
4.2 3.1 2.5 2.3
2.7 1.8 1.3 1.1
4.3 2.3 1.4 .88
7.5 4.0 2.2 1.3
19 13 10 6.7 5.5 3.8 3.2 2.3
37 28 21 17 14 11 9.7 8.6
62 42 32 27
2.8 2.4 2.2 2.1
1.5 1.2 1.0 .94
2.1 1.2 .81 .60
3.7 2.1 1.3 .84
6.4 3.6 2.2 1.5
17 11 8.5 7.1
36 29 26 24
(0.50, 0.01)
T
Ranges of standard dose (as percent of total) are listed across the top of the table, below which are the numbers of standard dose levels (k). Along the left margin are listed the values of the response variable (unknown) (Yo), next to which are the numbers of replicates (n) at each standard dose level. Within each rectangular grouping (specific dose range and Y9), equal total numbers of standard tubes (kn) lie on the diagonals from lower left to upper right.
4 8
2
180 90 48 27
1 2 4 8
0.90
66 36 21 13
220 100 120 62 71 42 46 32
1 2 4 8
0.95
8
(0.99, 0.01)
y
TABLE 2. VAR {ZQ} X 100 calculated by the simple least-squares method
V
2.4 2.2 2.1 2.0
1.1 1.0 .93 .90
1.2 .77 .58 .48
2.0 1.2 .81 .62
3.5 2.2 1.5 1.2
11 8.3 7.1 6.4
29 25 24 23
16
^
78 40 20 11
41 23 15 10
65 44 33 27
64 32 16 8.3
33 17 8.9 4.8
17 12 8.7 7.2
37 30 26 24
820 590 280 410 300 140 210 150 72 100 76 37
640 450 210 320 220 110 160 110 53 80 56 27
380 240 110 190 120 55 95 59 27 48 30 14
260 140 130 71 66 36 33 18
180 93 47 24
140 72 39 22
160 93 57 40
(0.99, 0.50)
130 67 34 18
97 49 25 13
50 25 13 6.6
29 15 7.6 4.0
15 7.9 4.4 2.6
10 8.1 7.0 6.4
28 25 24 23
16
978
CHANG, RUBIN AND YU
antibodies (or ligand-binding sites) are saturated is —1.0, and the estimates of/3 obtained from all the assay systems were close to -1.0 (Table 1), /? was set to be —1.0 for all computations. The specific statistical designs (range and number of standard dose levels; number of replicates at each standard dose level) are detailed in Tables 2 and 3. In these computations, the standard dose levels were set for each given dose range so that the ratio of adjacent standard dose levels was constant. It also was assumed that equal numbers of replicates were counted at each standard dose. For each assay system and each statistical design, the values of VAR{zo) for antigen concentrations whose responses in terms of Y equaled 0.95, 0.90, . . . , 0.10, 0.05 were computed. First, Equation 3 was used to determine ZQ, the corresponding log antigen concentration of the logit of a given response Y. Then the aforementioned values of ao, aly a and )3, and the value of ZQ were used to determine the magnitudes of (T[2, Wi, and VARfz,,} for both the simple and the weighted least squares procedure.
Results Initially, the values of VAR{z0} were computed for a wide range of values for a©, ax, and a. Specifically, these values were ao: 2 x 10~7, 2 x 10"6, . . . , 2 x 10' 3 ; ax: 5 x 1CT6, 5 x 10~5, . . . , 5 x 10~2; and a:
0.5, 1.0, . . . , 5.0. The ranges for ao, a1} and a were chosen to bracket widely the values of these parameters found with actual assays used in our laboratory (13,14) as indicated in Table 1. Using combinations of these values of ao, a1} and a, VAR{z0} was computed for all unknowns whose response levels in terms of Y ranged from 0.95 to 0.05 and for all statistical designs (ranges of standard curve). With these various combinations, the relative relationships of VAR{z0} between the simple and the weighted least squares procedures were identical. Therefore, in this paper only the values of VAR{z0} obtained by using the average values of a0, ax and a from the eight TESTO assays (Table 1) are presented in Tables 2 and 3. Table 2 gives the results for the simple least squares
Endo • 1975 Vol 96 • No 4
procedure, and Table 3 gives the results for the weighted least squares procedure. In these tables the dose ranges are expressed in terms of the expected value of the response variable Y for the smallest and the largest standard doses; i.e., for the dose range (0.80, 0.20) the expected values of the response variable Y for the smallest and largest standard dose levels are 0.80 and 0.20, respectively. The response levels of the unknown samples also are expressed in terms of the response variable Y. In order to keep the sizes of these tables reasonable, only the values of VAR{z0} obtained for the unknown samples whose response levels are 0.95, 0.90, 0.70, 0.50, 0.30, 0.10 and 0.05 using the more important dose ranges are included. The dose ranges investigated but not included in the tables were (0.25, 0.05), (0.95, 0.50) and (0.50, 0.05). The values of VAR{z0} for a given dose range and a given unknown response level using various numbers of standard doses (k = 2,4,8,16) and various numbers of replicates at each dose (n = 1,2,4,8) are grouped in the tables. Dose range and the two least-squares procedures The effects of the dose range can be examined by comparing the values of VAR{z0} for the different dose ranges, with number of dose levels and number of replicates held constant. For the simple least squares procedure (Table 2) the values of VAR{z0} are much greater if the standard dose levels cover a very wide range (expected response levels varying from 0.99 to 0.01) or are shifted to either side of the standard curve (expected responses ranging from 0.99 to 0.50 or from 0.50 to 0.01). Also, when the standard dose levels are all small (0.99 to 0.50), the values of VAR{z0} are larger than when the standard dose levels are all large (0.50 to 0.01). By comparing the corresponding values of VAR{z0} in Tables 2 and 3 it can be observed that when the expected response
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STATISTICAL DESIGN OF RADIOLIGAND ASSAYS levels for the standard doses are in the range of (0.90, 0.10), (0.80, 0.20), (0.70, 0.30) or (0.60, 0.40) the values of VAR{z0} when the estimates of the intercept and slope are obtained by the simple least squares procedure are very close to the values of VAR{z0} when the estimates of the intercept and slope are obtained by the weighted least squares procedure. On the other hand, when the expected response levels of the standard doses range from 0.99 to 0.01, or are shifted to (0.99, 0.50) or (0.50, 0.01), the values of VAR{z0} are much greater for the simple least-squares procedure. Number of standard dose levels (k) and number of replicates (n) at each dose
979
creases considerably as n increases for the same number of dose levels (k). With a fixed total number of standard tubes (kn), however, the number of dose levels (k) becomes the major influence on VAR{z0}. Response level of the unknown sample The effects of the response level of the unknown sample can be seen by comparing the values of VAR{z0} in each single column of the table. The precision of the antigen concentration estimate is greatly affected by the response level of the unknown sample. Given the same design, VAR{z0} is the smallest if the response level of the unknown sample falls in the middle of the expected response levels of the smallest and largest standard dose levels, and conversely VAR{z0} is much increased when the response level of the unknown sample is outside of the range of the expected response level of the smallest and the largest standard dose levels.
The value of VAR{z0} is decreased when the total number of replicates (kn) is increased. This can be observed by moving from top to bottom or from left to right within each grouping of the table. The effects of the number of standard dose levels and number of replicates at Discussion each dose may be examined by comparing From the results of this study, it is clear the values of VAR{z0} for a given dose that more precise antigen concentration range and a given unknown sample when the total number of replicates (kn) is fixed. estimates are obtained if the response These can be found by moving diagonally levels of the standard doses vary from 0.90 from the lower left corner to the upper to 0.10, 0.80 to 0.20, or 0.70 to 0.30. When right corner of each grouping. For instance, the standard dose levels used in an assay when the expected response levels of the are within these ranges, the complicated standard dose vary from 0.80 to 0.20, when weighted least-squares procedure, which the response level of the unknown sample requires a sophisticated computing facility, is 0.50, and when the total number of is no longer necessary. The precision of the replicates (kn) is 16, the values of VAR{z0} antigen concentration estimates obtained for k = 2, 4, 8, 16 are 0.50, 0.48, 0.47 and by using the simple least-squares proce0.47 using the simple least squares proce- dure to fit the linear dose-response regresdure (Table 2). This is a typfcal pattern, in sion line is as good as that of the weighted that the values of VAR{z0} are quite close procedure. This is consistent with the for most dose ranges when the number of results of the theoretical investigation by standard doses (k) is eight or more. The Jacquez and Norusis (15), in which the exceptions are when the dose range is simple and weighted regression proce(0.99, 0.01) or (0.99, 0.50). When the dures were compared for several patterns number of standard dose levels is small, of heteroscedasticity. However, the pattern larger values of VAR{z0} occur frequently. of heteroscedasticity stipulated by the With reference to number of replicates (n) model of Rodbard and Cooper (7) was not at each dose level, VAR{z0} always de- examined by Jacquez and Norusis. Our
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980
Endo • 1975 Vol 96 • No 4
CHANG, RUBIN AND YU
results also are consistent with those of Rolleri et al. (11), which were derived from empirical assay data containing a lesser degree of heteroscedasticity. If the standard doses are not properly chosen, so that the response levels for the standard doses cover the wide range from 0.99 to 0.01 or are shifted to one side of the curve (0.99 to 0.50 or 0.50 to 0.01), the weighted least-squares procedure such as the one suggested by Rodbard and Lewald (8) considerably improves the precision of the antigen concentration estimates. However, the weighted least-squares procedure will not improve the precision of a badly designed assay compared with the precision obtained by using the simple leastsquares procedure with a well designed assay (compare extreme groupings in Table 3 with "good" groupings in the middle of Table 2). Therefore, the selection of dose range clearly remains the major consideration. The use of dose levels so that the response levels of the standard doses range from 0.90 to 0.10 or 0.80 to 0.20 is a specific a priori method of truncation. Using these dose ranges, it becomes reasonable to make the requisite assumption of homoscedasticity for simple linear regression. The possibility of the use of truncation also has been discussed by Rodbard (12), and Vivian and LaBella (16) have used empirical sequential truncation to achieve acceptable homogeneity of variance. The next most important factor affecting the precision of the antigen concentration estimates appears to be the response level of the unknown sample. If the amounts of standard antigen and antibody used in the assay system are such that the response levels of the unknown samples are in the middle of the response levels of the smallest and the largest standard dose levels, then the precision of the unknown estimates is at its greatest. As expected, the precision of the antigen concentration estimates is improved when more replicates are used to determine the standard regression line. If the total
number of standard tubes that can be used in an assay is fixed, the precision of the antigen concentration estimates is not much affected by changing the number of standard dose levels, as long as it is not too small. On the other hand, too many levels of the standard dose may very well increase the cumulative errors of diluting and pipetting. It appears to be generally advisable to use eight or more standard dose levels, with at least two replicates at each dose. Based on the above considerations, a good statistical design for an assay is one in which (1) the average responses (in terms of Y) of the standard dose levels vary approximately from 0.80 to 0.20, (2) the response levels (in terms of Y) of all unknown samples also fall within 0.80 to 0.20, and (3) at least eight standard dose levels with at least two replicates at each level are used. With this design, a simple least-squares method for fitting the standard dose-response regression line may be safely used. Prior information is required in designing experiments. RIA and CPBA have the advantage that they usually are used repeatedly. Thus, past experience can be used in selecting the optimal standard dose levels. When a new assay system is used for the first time in a laboratory, experience with the assay in other laboratories may guide the initial design. Since the standard dose-response regression line usually is used to estimate the antigen concentration of many unknown samples, it would appear worthwhile to conduct a small assay of a few unknowns to obtain the information necessary for selecting the desired standard dose levels for subsequent assays. References 1. Berson, S. A., and R. S. Yalow, Adv Biol Med Phys 6: 349, 1958. 2. , and ,J Clin Invest 38: 1996, 1959. 3. Meinert, C. L., and R. B. McHugh, Math Biosci 2: 319, 1968. 4. Ekins, R. P., In In Vitro Procedures with Radioisotopes in Medicine, Int. Atomic Energy Agency, Vienna, 1970, p. 325.
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981
STATISTICAL DESIGN OF RADIOLIGAND ASSAYS 5. Rodbard, D., P. L. Rayford, J. A. Cooper, and G. T. RossJ Clin Endocrinol Metab 28: 1412, 1968. 6. , W. Bridson, and P. L. Rayford J Lab Clin Med 74: 770, 1969. 7. , and J. A. Cooper, In In Vitro Procedures with Radioisotopes in Medicine, Int. Atomic Energy Agency, Vienna, 1970, p. 6§9. 8. , and J. E. Lewald, Ada Endocrinol (Kbh) (Suppl. 147) 64: 79, 1970. 9. Ekins, R. P., and B. Newman, Ada Endocrinol (Kbh) (Suppl. 147) 64: 11, 1970. 10. Feldman, H., and D. Rodbard, In Odell, W. D., and W. H. Daughaday (eds.), Principles of Competitive Protein-Binding Assays, Lippincott, Philadelphia, 1971, p. 158. 11. Rolleri, E., P. G. Novario, and B. PaglianoJ Nucl Biol Med 17: 128, 1973. 12. Rodbard, D., In Odell, W. D., and W. H. Daughaday (eds.), Principles of Competitive Protein-Binding Assays, Lippincott, Philadelphia, 1971, p. 204. 13. Odell, W. D., P. L. Rayford, and G. T. RossJ Lab Clin Med 70: 973, 1967. 14. , R. S. Swerdloff, J. Bain, F. Wollesen, and P. K. Grover, Endocrinology 95: 1380, 1974. 15. Jacquez, J. A., and M. Norusis, Biometrics 29: 771, 1973. 16. Vivian, S. R., and F. S. LaBellaJ Clin Endocrinol Metab 33: 225, 1971.
Appendix In this appendix, the expressions for z, I, VAR{1} and VAR{b} are derived for the simple and the weighted least squares procedures. We first define the notations as follows: k = the number of standard dose levels x, = the i-th standard dose level, i = 1, ••-, k Zj = In Xi = the natural logarithm of the i-th standard dose level n = the number of replicates at each standard dose level y u = the response of the j-th replicate at i-th standard dose level, j = 1, •••, n. y^ is expressed in terms of the response variable Y defined by Eq. 1. l u = logit{y,j} = In
1J
cri2 = VAR{logit(Y|zi)} = the variance of l u , j = 1, •••, n at the i-th dose level. For the model used in this study,
