Estimacion Error Por Ria

CLIN.CHEM.31/11, 1802-1805 (1985)

Estimation the Response-ErrorRelationship Immunoassay of in
W. A. Sadler1 and M. H. Smith2
Estimation of the response-error relationshipin immunoasestimates of the I3k and she used simulated immunoassay data to demonstrate that a maximum modified likelihood (MML) method produced unbiased estimates for the exponent of an exponential function, with smallersampling variances than a LS estimator. She also showed that the log likelihood ratio (y) for goodness-of-fit is not distributed as x-1 when the r, are small, and derived expressions for the expectation and variance of ‘ in this context. Raab’s paper (6), together with a description of a computer program (7), represents an important theoretical and practical guide to estimating the response-errorrelationship in immunoassay. We have examined MML response-error parameters from many radioimmunoassays and consistently found that differed from ii, by 25) andj = 1,2,..., r1replicates (usually r 3). For the ith sample, the mean and variance of the u are defined respectively by:
Ui
=

[‘

uj

]iri

and s
=

(u

-

fi)2]/(r

-

1)

In general, 52 may vary markedly with U (see,for example, 1), and it is now widely accepted that weighting is required to ensure statistically valid immunoassay data reduction (see, for example, 1,2). The form of the relationship between 2 and U (the response-error relationship) is complex, because many sources of error contribute to 2, but in practice a simple form such as a straight line, parabola, or exponential function is invariablyassumed (1-8), and the inverse of the estimated functional relationship is used as the weighting function for the main analysis. Several unweighted least squares (LS) regression methods have been proposed (1,3,4) for estimating parameters of the assumed response-error function, although the implicit assumption that raw or pooled observations of 2 follow normal distributions with identical variancesis untenable. Finney and Phillips (5) introduced a more realistic model for the data by assuming that the u!j are independently, normally distributed for all i, with expectation and variance of respectively ji and . They used the maximum likelihood (ML) method to estimate all N + p parameters, where the p (i = 1,2,...,N) can be regarded as nuisance parameters and the 13k (k l,2,...,p) are theresponse-error functional parameters of interest. Raab (6) showed that in the presence of many nuisance parameters the ML method yields biased
=

Methods Simulations We drew sets of simulated radioimmunoassay responseerror data from N(; fjz, /3k)) using a uniform pattern for theaccordingthj. 200.0 + 20.0(i -1) i 1,2,...,500. Functional forms and values assigned to the f3k are shown in Table 1, andfor each of these 25 combinations we used four replication patterns: r1 2 throughout, r, 3 throughout, r 4 throughout, and a mixture of r, 2 (i 1,2,...,333) and r1 3 (i 334 335,...,500). In each case 500 sets were drawn and the 13k were estimated by six methods: (a) MML (b) MACL (c) LS regression of s2 upon U (d) Adjacent observations of U1pooled into 10 groups of 50 observations, then LSregression of group-mean values of 2 upon group-mean values of U (e) As for d, but using group median values of 2 (/) MACL applied to the grouped data from d
= = = = = = = = =

‘Department of Nuclear Medicine, Christchurch Hospital, Private Bag, Christchurch, New Zealand. 2Department of Mathematics, University of Canterbury, Christchurch, New Zealand. Received May 17, 1983; accepted July 9, 1985.
1802CLINICAL CHEMISTRY, Vol. 31, No. 11, 1985

Sampling means and variances of the 13k were calculated for each data combination and estimation method. Bias was estimated as the deviation of the observed mean from its expected value, expressed in terms of standard errors.

Table 1. FunctionalForms and ParameterValues
Chosen to Represent Typical Radloimmunoassay Response-Error Data
Paramiter...