Pa/7/77, A General Microcomputer Program for
Parallel-Line Analysis of Bioassays
JOLYON JESTY, PH.D. AND HENRY P. GODFREY, M.D., PH.D.
Commonly used manual and calculator methods for analysis of
clinically important parallel-line bioassays are subject to operator
bias and provide neither confidence limits for the results nor any
indication of their validity. To remedy this, the authors have
written a general program for statistical analysis of these bioassays for the IBM 9 Personal Computer and its compatibles. The
program has been used for analysis of bioassays for specific
coagulation factors and inflammatory lymphokines and for radioimmunoassays for prostaglandins. The program offers a choice
of no transform, logarithmic, or logit transformation of data,
which are fitted to parallel lines for standard and unknown. It
analyzes the fit for parallelism and linearity with an F test, and
calculates the best estimate of the result and its 95% confidence
limits. Comparison of results calculated by PARLIN with those
previously obtained manually shows excellent correlation (r
> 0.99). Results obtained using PARLIN are quickly available
with current assay technics and provide a complete evaluation
of the bioassay at no increase in cost. (Key words: Microcomputer; Parallel-line assay; Clotting factor assay; Radioimmunoassay; Lymphokine assay; BASIC) Am J Clin Pathol 1986;
85: 485-489
PARALLEL-LINE BIOASSAYS occupy an increasingly
important place in clinical and research laboratories in a
wide range of disciplines. Major examples of this type of
bioassay include radioimmunoassays (RIA), enzymelinked immunosorbent assays, specific coagulation factor
assays, and lymphokine assays. In these assays, the doseresponse curve is characteristically hyperbolic and can be
transformed into acceptably linear form by plotting log
(response) or logit (response) against log (dose). If this is
done for both standard and unknown samples, two parallel lines are obtained, and their separation along the
x-axis can be used to calculate the concentration of active
material in the unknown sample.2,7
In practice, parallel-line assays usually are calculated
by fitting the transformed data obtained from dilutions
of the standard material to a straight line and then reading
off the doses of unknown from this line. If multiple dilutions of the unknown sample are assayed, a final result
may be obtained by averaging the data. This method of
Received May 10, 1985; received revised manuscript and accepted for
publication September 9, 1985.
Supported in part by NIH research grants HL 21113 and CA 34141,
and by Research Career Development Award K04-HL 00477.
Address reprint requests to Dr. Jesty: Division of Hematology, Department of Medicine, Health Sciences Center, State University of New
York, Stony Brook, New York 11794.
Division of Hematology, Department of Medicine, State
University of New York, Stony Brook; and Department of
Pathology, New York Medical College, Valhalla, New York
calculation can be done manually with graph paper and
a calculator or with a calculator alone. It can also be done,
and in programmed laboratory instruments frequently is,
by a microcomputer or microprocessor. There are a number of inherent difficulties in this particular way of calculating parallel-line assays, whether it is done by hand
or with the aid of a computer: (1) it assumes that the
standard line is error-free or less subject to error than the
line for the unknown; (2) it lacks any objective means of
testing whether the assay result is valid; and (3) it provides
no sound method for calculating confidence limits of the
result.
Data from an average parallel-line assay are usually
quite sufficient for a thorough statistical analysis. Methods
exist for simultaneous fitting of data for standard and unknown samples to two parallel lines, performance of an
analysis of variance to determine whether the lines are
acceptably linear and parallel, and calculation of the result
and 95% confidence limits.2,4"6,9 Although the data manipulation required to calculate assay results and 95%
confidence limits can be done on a programmable, handheld calculator, the full analysis of variance complete with
F test probably requires more memory. Previously published programs for analyzing parallel-lines assays have
been directed at specific assays such as clotting assays 310
or interleukin-2 assays,5 appear to be machine-dependent
in some cases,10 are designed for a particular formats of
data entry,3,5 and, in general, do not appear to be set up
for easy data entry and editing. Furthermore, only some
of these programs are readily available. We therefore have
written a general program for parallel-line assay analysis
by microcomputer that can handle the wide range of these
assays encountered in clinical laboratories. This program
does not require any particular data structure or order of
data entry, supports full editing, looks after most problems
caused by complications of unitage, and is easy to use.
We describe the use of the program in three different assays—a specific clotting factor assay, a lymphokine assay,
and an RIA—and compare results calculated with the
program to those previously calculated manually.
485
486
JESTY AND GODFREY
FACTOR IX ASSAY.
LOG(RESPONSE) TRANSFORM.
STANDARD: PNP LOT 326
CONCENTRATION FACTOR IX STANDARD = 1 UNIT/ML
DILUTION
5
10
20
40
80
RAW DATA FILED
RESPONSE (SEC)
44.4
45.1
50.8
51.7
58.6
57.8
68.7
65
73.7
75.1
IN C:FIXST02.DAT
mmmmmttm
UNKNOWN: UK23 vs. PNP LOT 326
ASSAY ENTERED 09-04-1985
DILUTION
2
5
10
20
RAW DATA FILED
RESPONSE (SEC)
50.7
51
58.9
60.1
66.2
68
75.6
78.8
IN C:FIXUK23.DAT
UNKNOWN = .2170347 UNIT/ML
957.-C0NFIDENCE LIMITS ARE .1972313 AND .2383214 UN IT/ML
THE ASSAY IS VALID:
P(PARALLEL) = 0.6
P(LINEAR) = 0.1
mmtmmttmt*
FlG. 1. Printed output of clotting assay analysis depicted in Figure 2.
Information is entered in response to menus and prompts; the entire
output is under program control.
Materials and Methods
Assay of Coagulation Factor IX
Assay of clotting Factor IX was done using appropriate
factor-deficient plasma (George King Biomedical, Overland, KS) in a modified partial thromboplastin time1 with
Platelin-Plus® (General Diagnostics, Morris Plains, NJ)
as activator and lipid source. Pooled normal plasma
(George King Biomedical), used as the standard, was arbitrarily assigned an activity of 1 unit/mL. Most assays
were done on a Coag-A-Mate® instrument (General Diagnostics), but some were done with Fibrometer® timers
(Becton-Dickinson, Cockeysville, MD). Duplicate assays
were done on each dilution of standard and unknown.
Assay of Macrophage Agglutination Factor
Macrophage-agglutination factor (MAggF) was assayed
on serially diluted aliquots of coded samples of antigenstimulated, T-lymphocyte culture supernatants, using
Marcol® 52-elicited guinea pig peritoneal exudate cells as
indicator cells.8 After 4 to 6 hours at 37 °C, the tubes
were gently swirled to dislodge loosely adherent cells and
scored as — to ++, depending on the number of macroscopically visible clumps of cells on the culture tube walls.
Titer was defined as the reciprocal of the highest dilution
A.J.C.P.- April 1986
to show a + response. The MAggF standard was arbitrarily
assigned an activity of 100 units/mL. Randomized quadruplicate measurements were done on each dilution of
standard and unknown.
RIA of Prostaglandin E2
Prostaglandin E2 (PGE2) was assayed by the method of
Dray and Seis6 using anti-PGE2 (Sigma Chemical Co., St.
Louis, MO), PGE2 (Upjohn Diagnostics, Kalamazoo, MI),
and 3H-PGE2 (Amersham, Needham Heights, IL). Duplicate measurements were done on each dilution of standard and unknown.
Description of Program (PARLIN) and Parallel-Line
Analysis
PARLIN was written in IBM PC BASIC and is available
in compiled form. This occupies 45K bytes of memory,
and, depending on the operating system, will usually require more than 64 Kbytes to run (users restricted to 64
Kb may request the BASIC source version, which runs a
little more slowly, but occupies about 20 Kb). The program also needs a printer. PARLIN is compatible with
all versions of IBM DOS and should run on most IBMcompatible microcomputers (only one compatible has
been expressly tested—the Compac®). Copies of PARLIN,
program notes, and documentation, and related materials
are available from the authors on 5.25-inch diskettes (DOS
1.1-format, double-density, single-sided).
After starting the program, the user is asked to choose
whether doses are to be entered as dilutions or as amounts
added to the assay. The program automatically formats
the printed output in the two cases. The user then chooses
whether responses are to be untransformed, logarithmically transformed, or logit transformed. (To use logit
transformation, the user must enter a maximum response.) In the present studies, log transforms were chosen
for Factor IX and MAggF assay data, and logit transforms
for RIA data. The program then requests description of
the assay, identification of sample, and quantitation of
the standard units (mg, ng, moles, etc.). Raw (untransformed) data can be either entered from the keyboard or
recalled from previously stored data files. PARLIN has
capacity for 12 replicate responses at each of 15 doses,
permits extensive editing and also supports the storage
and subsequent recall of raw data to and from standard
sequential ASCIIfiles.All printed output is under program
control (Fig. 1). Because of the complexities of programming graphics and the lack of standardization among
computers, PARLIN does not plot the data. This should
be done by the user to confirm that they fall on a straight
line and not in the sigmoid region of the curve. The simple
structure of the data files should permit easy rearrangement into a format that can be read by commercially
PROGRAM FOR PARALLEL-LINE ANALYSIS
Vol. 85 • No. 4
available graphics programs {e.g., VisiPlot®, Lotus 1-23®). However, in general, such software is not capable of
handling the double-log axes or the plotting of fitted lines
that users of PARLIN probably would like.
Our general approach in developing the program has
been to restrict the printed statistical output as far as possible and print only those results from the analysis of variance that we believe are of immediate interest to the average clinical user: (1) the best estimate of the result; (2)
its 95% confidence jimits; (3) /"(linear); and (4) P (parallel).
Thus, the program prints neither the full analysis of variance nor a table of residuals, even though both are readily
available within the program. Users who wish to add this
function can obtain a copy of the BASIC source code
from us for modification.
Doses are log transformed and responses are appropriately transformed by the program before fitting. The
method used'in PARLIN is essentially that described by
Colquhoun for a general asymmetrical paral}el-line assay.2
The total sum of squared deviations (SSD) from the fitted
lines is divided to obtain the SSD for deviations from
linearity and deviations from parallelism. These each are
compared with the error variance calculated from the SSD
within groups to obtain a value of F, and the probability
is then determined for the likelihood that observed deviations from linearity and parallelism are accounted for
by the variance of the data within groups. Because of the
prohibitive size of the F table that would be needed for
all cases, PARLIN uses an approximation to the F distribution to determine P. The errof variance obtained in
the analysis is used to calculate 95% confidence limits
using Student's Mest. On completion of an analysis, the
user can enter another unknown for assay against the
standard already in memory, enter a new standard, or
finish.
Results
Use of PARLIN for Parallel-Line Analysis
A typical printed output from a Factor IX clotting assay
is shown in Figure 1. Data entry for this assay took less
than two minutes, the time needed for calculation was
insignificant, and printing took just a few seconds. PARLIN was evaluated using three different types of parallelline assay: (1) a specific coagulation factor assay for Factor
IX, using a log (response) transform; (2) a lymphokine
assay for MAggF with either log or logit transform; and
(3) an RIA of PGE 2 , using a logit transform. Because
PARLIN uses an unweighted fit to the data, it was first
necessary to determine whether the standard deviations
of the transformed responses were reasonably constant
across the response range examined, i.e., whether the
transformed data were homoscedastic. Figures 2 to 4 show
487
5
10
FACTOR IX DILUTION (-fold)
100
FIG. 2. Factor IX assay. Pooled normal plasma was defined as containing 1 unit/mL. Doses were entered as dilutions of standard of unknown. In the assay shown, one dilution of each sample was used for
the measurement of the duplicate responses, both of which are shown.
A log (response) transform was used to fit the data. For figure clarity,
untransformed response is shown on a log axis. The results of this assay
are as follows: [IX] = 01217 unit/mL, with 95% confidence limits of
0.197 and 0.238 unit/mL. /'(parallel) = 0.6; P(linear) = 0.1. Note that
PARLIN does not plot assay data.
typical plots of each type of assay. In each case, the standard deviations of the transformed data are reasonably
constant, even in the case of the transformed MAggF titers
where the measured response varied over several orders
of magnitude.
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MAggF DILUTION (-fold)
FIG. 3. MAggF assay. MAggF was assayed using guinea pig peritoneal
macrophages as indicator cells. A standard MAggF preparation was defined as containing 100 units/^L. Doses were entered as dilutions of
standard and unknown. Separate dilutions were made in randomized
fashion to obtain quadruplicate responses. A log(titer) transform was
used to fit the data. The results of this assay were as follows: [MAggF]
= 15.0 units/mL, with 95% confidence limits of 6.5 and 29.0 unit/mL.
/'(parallel) = 0.5; P(linear) = 0.7.
JESTY AND GODFREY
488
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A.J.C.P. • April 1986
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FIG. 4. PGE2 assay. PGE2 was assayed by RIA using anti-PGE2 and
labeled and unlabeled PGE2. Doses of standard PGE2 were entered as
pg, doses of unknown as ^L. In this assay one dilution of sample was
used for each pair of duplicate responses shown. A logit (response) transform was used to fit the data. The results of this assay were as follows:
[PGE2] = 6.14 pg/nL, with 95% confidence limits of 5.30 and 7.08 pg/
iiL. P (parallel) = 0.08; P (linear) = 0.005.
5 4 T3
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Comparison of Program and Manual Results
<v\
Assay results werefirstcalculated by plotting the transformed data, drawing a straight line by eye, and reading
the transformed unknown responses off the standard line
to obtain the unknown dose. They then were calculated
using PARLIN. Correlation for Factor IX and MAggF
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Discussion
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assays are shown in Figures 5 and 6, which also show 95%
confidence limits given by the program. The correlation
coefficients for the two methods of calculation were 0.996
for Factor IX assays and 0.990 for MAggF assays.
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FIG. 6. MAggF—comparison of manual and program results. Results
derived by PARLIN show 95% confidence limits. The slope of the regression line is 0.991, the intercept 0.164.
I
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2 _
150
MANUAL RESULT (%)
FIG. 5. Factor IX—comparison of manual and program results. Results
derived by PARLIN show 95% confidence limits. The slope of the regression line is 1.057,' the intercept 0.91%.
Any evaluation of a program such as PARLIN for
analysis of parallel-line assays has to examine the assumptions used in constructing the program as well as
the utility of the results obtained using it. PARLIN fits
straight lines to unweighted transformed data. This is a
reasonable approach only if the transformed data are
homoscedastic (Figs. 2-4). We must emphasize that users
of PARLIN must confirm that their transformed data are
in fact homoscedastic.
The usefulness of the information provided by PARLIN
on assay validity, especially as regards the test of parallelism, is obvious and agrees well with subjective estimates
from assay plots. For example, in the PGE2 assay (Fig. 4),
Vol. 85 • No. 4
PROGRAM FOR PARALLEL-LINE ANALYSIS
489
there is by eye a small but fairly clear tendency towards
convergence of the data at high PGE 2 doses, and P (parallel) = 0.08, i.e., on the borderline of what usually is
taken as defining validity. However, in contrast to the
clear utility of the analysis of parallelism, users find that
the program often reports nonlinearity in cases where,
seeing an excellent pair of parallel lines with good agreement within replicates, they would expect a report of excellent validity. In Figure 2, for example, /"(parallel) = 0.6,
as one would expect, but /"(linear) = 0.1 {i.e., barely valid).
A worse situation is seen in Figure 4, where P (linear)
< 0.01. In both of these cases, however, assays were done
without randomizing samples over the course of the assay,
and duplicate measurements were generally made on single sample dilutions. The program calculates P (linear)
by comparing the variance within all replicate groups with
the SSD of the data points from the fitted lines. In simple
terms, therefore, if an assay gives good agreement within
replicates but shows greater deviation of the points from
the fitted lines, analysis will show significant nonlinearity.
This is often not so much caused by the magnitude of the
deviations of the points from the fitted lines as by the
close agreement between replicates.
to within a factor of 10. We were particularly interested
in whether PARLIN could be.Used to analyze these assays.
As can be seen from the results of a typical assay (Fig. 3),
the responses at each dose varied over a wide range (up
to 1000-fold differences in titer), but with randomized
quadruplicate measurements, they gave a quite acceptable
continuous response after transformation. The transformed data were homoscedastic, not only in the particular set shown, but throughout our experience with these
assays.
Analysis of bioassays using PARLIN can be seen to
have major advantages over manual calculation. The
method is relatively free from operator bias, while its statistical analyses permit objective measurement of both
the validity and accuracy of the assay. The design of this
program for general application should enable it to provide
rapid, accurate, and convenient analysis of a large class
of bioassays in current clinical use with little change in
assay procedures and cost.
Unexpectedly good agreement within replicates usually
occurs when they are done together without sample randomization. In the worst case, replicates are done oil the
same sample dilution, which even eliminates variation
caused by dilutional errors. In the absence of randomization or, at the very least, testing of standard and unknown
samples in some balanced (palindromic) order,4 the estimate of assay error obtained from variation within replicate groups will clearly be urirealistically small because
it will not include errors arising from dilution, instrument
drift, reagent drift, and operator bias. When assays are
performed using intermixed or randomized samples, the
frequency of nonlinearity drops to an acceptable level and
PARLIN's analysis of variance becomes a reliable part of
its function. True randomization of assay samples probably is unjustified in general use. However, as a.minimal
precaution, replicates should be dispersed over the course
of an assay and replicate measurements on the same sample dilution should be avoided.
The use of PARLIN to calculate MAggF potencies deserves some comment. MAggF assays use titers as the
measured response, which in this case were determined
References
Acknowledgments. The authors thank Ms. Dorene Turi for access to
Factor IX assay data, and Dr. Kathleen Madden for access to RIA data.
They also thank Ms. Lorraine Ostrubak and Shirley Murray for manuscript preparation.
1. BrinkHous K.M, Dpmbrose FA: Partial thromboplastin time, CRC
Handbook of Hematology, vol III. Edited by RM Schmidt. Boca
Raton, CRC Press, 1980, pp 221-246
2. Colqhoun D: Lectures in Biostatistics. Oxford, Clarendon Press, 1971,
pp 279-332
3. Counts RB, Hays JE: A computer program for analysis of clotting
factor assays and other parallel-line bioassays. Am J Clin Pathol
1979;71:167-171
4. Curtis AD: The statistical evaluation of factor VIII clotting assays.
Scand J Haematol (Suppl) 1984; 33:55-68
5. Davis B, Huffmann M, Krioblock K, Baker P: Logit analysis of
interleukin-2 activity using a BASIC program. Comput Appl Lab
1984;2:269-275
6. Dray F, Seiss W: Radioimmunoassay of prostanoids, Principles of
competitive protein binding assays. Edited by WD Odell, P Franchimon. New York, John Wiley, 1983, pp 225-242
7. Finney DJ: Statistical Method in Biological Assay, 3rd ed. London,
Charles Griffin, 1978, pp 148-156
8. Godfrey HP, Purohit A: Reversible binding of a guinea pig lymphokine to gelatin andfibrinogen:Possible relationship of macrophage agglutination factor andfibronectin.Immunology 1982;
46:515-526
9. Kirkwbod TBL, Snape TJ: Biometric principles in clotting and clot
lysis assays. Clin Lab Haematol 1980; 2:155-167
10. Williams KN, Davidson JMF, Ingram GIC: A computer program
for the analysis of parallel-line assays of clotting factors. Br J
Haematol 1975;31:13-23