STATWI0006.01 Cover Page – page 0 2017.03.28

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CVB Quality Manual Chapter 5

USDA Center for Veterinary Biologics

Statistics Section

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Document: STATWI0006.01

Title: Parallelism Analysis in Assay Validation

Author: Statistics Section

Approved by: David Siev on 2017.03.28
Analysis of Parallelism in

RP ELISA Validation

Statistics Section

Center for Veterinary Biologics

Ames, Iowa 50010

March 23, 2017

1 Introduction

Veterinary Services Memorandum 800.112, Appendix III, provides guidance on assay validation

studies for ELISA relative potency assays. (USDA 2011) Section 2.2.5 outlines the design of

the parallelism study conducted during assay validation and gives expectations for satisfactory

parallelism. This note describes a method commonly used for the statistical analysis of data

from such a study. This document is intended for personnel with a substantial background

in mathematical statistics, including familiarity with nonlinear mixed effects models. (e.g.

Pinheiro and Bates 2000, Littell et al. (2006))

2 Preliminaries

2.1 Data Visualization

The first step should always be to visualize the optical density (OD) profiles for suitability of

the data and plausibility of the intended model fit. In particular, verify the following.

• Full curve. The full curves of both preparations from saturation through extinction

must be present.

• Parallelism. Their profiles of the two curves should appear to have the same sigmoid

shape and differ only by a horizontal shift.

• Zero lower asymptote. The ODs are typically corrected by subtracting the mean OD

of a reagent blank or negative control. When this is done, we generally assume that

the lower asymptote is zero. (For some assays, such as some competitive ELISAs, this

assumption may not hold.)

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STATWI0006.01 page 1 of 3
2.2 Data Reduction

Next, for this analysis OD values within each dilution of a preparation on a plate are averaged

prior to analysis. As a result, the data set will now have one mean OD value for each dilution

of a preparation on a plate. This is done for convenience in Section 2.2.5 parallelism studies

only, so that the appropriate random components are used to evaluate the curve fits, without

including the within-plate/prep/dilution residual.

2.3 Model Parameterization

Finally, the analyst must choose a parameterization of the 3-parameter logistic (3PL) curve.

(Ratkowsky and Reedy 1986) Let y denote the (blank corrected) optical density, and x the

dilution. The three parameters of the curve are the upper asymptote, A; a scale factor, B;

and a location parameter, C.

For assay validation studies, use the following parameterization.

y =

A

1 + exp [(C − logd x)/B]

(1)

The covariate is conventionally expressed in base d logarithms, where d is the dilution factor.1

Commercial software packages distributed with plate reading equipment may use another

parameterization, which is acceptable for serial (lot) release testing.

y =

A

1 + ( x

C

)

B

(2)

The A parameter will have the same numerical value in both parameterizations, but B and

C will have different numerical values depending on which parameterization is selected.2

3 Nonlinear mixed effects model

Using the parameterization of Eq.(1), the nonlinear mixed effects model can be formulated

by decomposing each parameter into a fixed effect that depends on preparation (reference or

test serial) and a random effect that depends on plate. (For simplicity we illustrate just a

single random effect. Include others as necessary.) Mathematically, this is written as follows:

y(ij)k =

Ai + aj

1 + exp [(Ci + cj − logd x(ij)k)/(Bi + bj )] + (ij)k (3)

1Any base will work, but the choice will naturally affect the location and scale parameter estimates.

2For additional details about parameterizations see STATWI0005.0#.

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STATWI0006.01 page 2 of 3
where i indexes the preparations (reference or test serial), j indexes the plates, and k indexes

the dilutions. The fixed effects are denoted by capital letters, and the random effects by

lower case letters.

The random effects are a for the upper asymptote, b for the scale factor, and c for the location

parameter. Thus each parameter (A, B, and C) depends on one fixed effect (the preparation)

and one random effect (the plate). A residual error is represented by . The random effects

are typically modeled as follows:

(ij)k ∼ N(0, σ2



) (4)







a

b

c





 ∼ MV N











0

0

0





 ,







σ

2

a

0 0

0 σ

2

b

0

0 0 σ

2

c









 (5)

Here N is the normal distribution and MV N is the multivariate normal distribution. A

diagonal covariance matrix is assumed for the random effects, since it is unlikely there will

be enough data to fit an unstructured one. Sometimes, one or more of the random effect

variance components is estimated to be negligibly small. By convention, when this happens

we have chosen not to remove the random effect from the model. Finally, note that regarding

Eq.(4), it assumes homogeneity and independence of residuals, which may not always be the

case.

Once the model is fitted and parameter estimates are available, parameter ratios and their

confidence intervals may be obtained by the delta method. This requires the estimated

covariance matrix of the parameter estimates from the model fit. Critical values from the

t-distribution should be used; the degrees of freedom will be the number of plates minus

three, which is the number of random effects parameters (a, b, c).

References

Littell, RC, GA Milliken, WW Stroup, RD Wolfinger, and O Schabenberger. 2006. SAS for

Mixed Models. Second. Cary, NC: SAS Institute.

Pinheiro, JC, and DM Bates. 2000. Mixed-Effects Models in S and S-Plus. New York:

Springer.

Ratkowsky, DA, and TJ Reedy. 1986. “Choosing Near-Linear Parameters in the FourParameter Logistic Model for Radioligand and Related Assays.” Biometrics 42: 575–82.

USDA. 2011. Veterinary Services Memorandum No. 800.112: Guidelines for Validation of in

Vitro Potency Assays. Washington DC: U.S. Department of Agriculture.

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