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Directly testing the linearity assumption for assay validation
Steven Novick / Harry Yang
May, 2013
1
Manuscript accepted March 2013 for
publication in Journal of Chemometrics
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• Steven Novick
– Associate Director @ GlaxoSmithKline
• Harry Yang
– Senior Director of statistics @ MedImmune LLC
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Purpose
• Illustrate a novel method to test for linearity in an analytical assay.
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Analytical assay – standard curve
• We wish to measure the concentration of an analyte (e.g., a protein) in clinical sample.
• Standards = known concentrations of an analyte.
• To estimate the concentration, we create a standard curve.
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Concentration
Assa
y S
ign
al
log10 Concentration
Assay S
ignal
Standards
with a fitted standard curve
5
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Concentration
Assa
y S
ign
al
log10 Concentration
Assay S
ignal
Clinical sample
Estimated concentration
6
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Concentration
Assa
y S
ign
al
log10 Concentration
Assay S
ignal
Clinical sample
Estimated concentration
7
To many, the only
interest lies in the
linear portion of the
curve.
???
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ICH Q2(R1) guideline
• http://www.ich.org/fileadmin/Public_Web_Site/ICH_Products/Guidelines/Quality/Q2_R1/Step4/Q2_R1__Guideline.pdf
• Evaluate linearity by visual inspection
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log10 Concentration
Assay S
ignal
9
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The EP6-A guidelines
• Clinical and Laboratory Standards Institute
– http://www.clsi.org/source/orders/free/ep6-a.pdf
• Compare straight-line to higher-order polynomial curve fits
– Recommendation: Test higher-order coefficients.
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log10 Concentration
Assay S
ignal
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Notation
E[ Yij | Xi ] = a1 + b1 Xi = g1(Xi)
E[ Yij | Xi ] = a2 + b2 Xi + c2 Xi2 = g2(Xi)
E[ Yij | Xi ] = a3 + b3 Xi + c3 Xi2 + d3 Xi
3 = g3(Xi)
• Assume IID normally distributed errors with equal variance.
i = 1, 2, …, L concentrations
j = 1, 2, …, ni replicates
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Orthogonal polynomials • The orthogonal polynomial of degree k is of
the form
r = (k+1) constants (to be estimated)
fr(x) = (k+1) orthogonal polynomials
x = concentration
13
∑r=0
k
θr f r( x ) ,
See: Robson, 1959
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Orthogonal polynomials properties
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∑i=1
L
n i f r (X i )=0
∑i=1
L
ni f r (X i) f s (X i)=0
∑i=1
L
ni f r2(X i)=1 ortho-normal property
See: Robson, 1959
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OLS: orthogonal polynomials
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Y=
(Y 11
Y 12⋮
Y 1n1⋮
⋮
Y L 1Y L 2⋮
Y L nL
),
F=(f 0(X 1)
⋮
f 0(X L nL)
f 1(X 1)⋮
f 1(X L nL)
⋯
f k (X 1)⋮
f k (X Ln L)), θ=(
θ0
θ1
⋮
θk)
E [Y ]=F θ Var [Y ]=σ2 I N , N=total sample size
Var [ θ]=σ2 I k+1 .θ=FTY and
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FT
Y
L
L
L
L
Ln
Ln
Ln
Ln
XfXfXf
XfXfXf
XfXfXf
XfXfXf
32313
22212
12111
02010
cubic][θ
Y linear][θ
L
L
L
L
Ln
Ln
Ln
Ln
XfXfXf
XfXfXf
XfXfXf
XfXfXf
32313
22212
12111
02010
Intercept and slope estimates are same for both!
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Literature • Krouwer and Schlain (1993)
– Assume linearity, except at last concentration
– Ha: max - (a1 + b1 Xmax) 0
• EP6-A
– Ha: One or both of c3 , d3 0
– Ha:
• Kroll et al. (2000)
– Composite statistic, ADL
– Ha: ADL >
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∑i=1
L
{gk (X i)−g1(X i)}2/ X i
Wrong power profile
Assumes linearity?
Wrong power profile
Difficult to choose
∣gk (X i)−g1(X i)∣<δ for all i = 1, 2, …, L Tested without use
of inference
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More literature
• Hsieh and Liu (2008)
– Ha: I-U tests for all i = 1, 2, …, L
• Hsieh, Hsiao, and Liu (2009)
– Composite statistic,
– Ha: SSDL <
– Generalized pivotal quantity (GPQ) method
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SSDL=L−1
∑i=1
L
{gk (X i)−g1(X i)}2.
∣gk (X i)−g1(X i)∣<δ
Choice of concentrations?
Difficult to choose
Generally << 2 !
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Our proposed hypothesis • Ha: for all
• Similar to I-U testing, but instead of individual concentrations, performed across a range of interesting concentrations.
• Bayesian(or GPQ) methods.
– Linear models
– Test is function of linear contrast
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∣g k(x)−g1( x)∣<δ x∈[x L , xU ]
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Our proposed test statistic
• Accept linearity if p > p0 (e.g., p > 0.9).
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p (δ , xL , xU )=Pr{ maxx∈[xL , xU]
∣∑r=2
k
θr f r( x)∣<δ∣data}
∣g k(x)−g1( x)∣<δ
maxx∈[xL , xU ]
∣∑2
k
θr f r(x )∣<δ
x∈[x L , xU ]for all
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log10 Concentration
Assay S
ignal
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How to: with Jeffrey’s prior (or GPQ)
• Given Y (responses) and F (orth poly design matrix). Assume a kth-degree polynomial.
• Let and be OLS estimates
• Error degrees of freedom = N-(k+1)
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• Generate two random variables
Z ~ Nk+1( 0, I )
U ~ 2(N-k-1)
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1
ˆ
kNU
ZBayes
k
rrBayes
xxx
xfUL 2
,
,
)(max
Generate “B” of these
To estimate p(, xL, xU),
count the proportion of times:
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Calcium Assay example • From NCCLS EP6-A,
Appendix C, ex. 2
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Dilution Replicate
1
Replicate
2
1 4.7 4.6
2 7.8 7.6
3 10.4 10.2
4 13 13.1
5 15.5 15.3
6 16.3 16.1
Cubic model = best fit
Compare cubic to linear from Dilution =1 to Dilution = 6: = 0.9 for testing.
NCCLS EP6-A: =0.2 mg/dL
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Dilution
Mean
difference:
Cubic – Linear
1 -0.53
2 -0.13
3 0.42
4 0.74
5 0.42
6 -0.93
+/- =0.9 around cubic fit
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Clear failure at Dilution = 6
• Hsieh and Liu I-U test p-value = 0.61
– for all i = 1, 2, …, 6
– Linear fit not adequate
• p(, xL, xU) = 0.39
– for all 1 x 6
– Linear fit not adequate
• Probability > 0.99
– Linear fit is equivalent to cubic fit 26
26
1
2
136
1
i
ii XgXgSSDL
ii XgXg 13
xgxg 13
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Try again without last dilution • From NCCLS EP6-A,
Appendix C, ex. 2
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Dilution Replicate
1
Replicate
2
1 4.7 4.6
2 7.8 7.6
3 10.4 10.2
4 13 13.1
5 15.5 15.3
6
Quadratic model = best fit
Compare quadratic to linear from Dilution =1 to Dilution = 5 with = 0.9 for testing.
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Dilution
Mean
difference:
Quad – Linear
1 -0.18
2 0.09
3 0.18
4 0.09
5 -0.18
6
+/- =0.9 around quadratic fit
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Linear and quadratic fits equivalent
• Hsieh and Liu I-U test p-value < 0.01
• p(, xL, xU) > 0.99
• SSDL probability > 0.99
• Linear fit is equivalent to quadratic fit for dilutions between 1 and 5.
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Simulation
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Quadratic vs. Linear Simulation
• g2(x) = 10 + (1-40)x + x2, 1 x 40
• = 0 = linear: g2(x) = 10 + x
• = 0.04 = large quadratic component.
• Y ~ N( g2(x), =3 ). 10 g2(x) 50
• 6 concentrations with two replicates each
• Testing limit: = 6.1
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Two sets of concentrations
Set 1: Maximum deviation occurs at x = 1
Set 2: Maximum deviation occurs between points
This case is H0/H1 border
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Test is generally
Too powerful
Maximum difference occurs at x = 1 SSDL can be tuned
with knowledge of
unknown curve.
Because max diff occurs at design point,
these tests are very similar
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Tests are generally
Too powerful
Maximum difference occurs between points SSDL can be tuned
with knowledge of
unknown curve.
Our method retains similar
power profile
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Inverting the test
• Consider the true mean response gk(x) and the reduced model g1(x)=a+bx.
• Let zk(x) = { gk(x) – a }/b
– This is polynomial Y-value back-calculated with best-fitting straight line.
• How close is zk(x) to x ?
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A few hypotheses to consider
• | zk(x) – x | < for all xL x xU
• | 100%{zk(x) – x}/x | <
• | log{zk(x)} – log(x) | <
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Many others!
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• We can compute conditional probability
Pr{| log{zk(x)} – log(x) | < | data }
Or
• Find such that
Pr{| log{zk(x)} – log(x) | < | data } = 0.95
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Pr{| log{z3(x)} – log(x) | < 0.15 | data } = 0.95
Back-calculated values are
within 100%(100.15-1) = 40%
of true value.
Back-calculated values are
within 0.15 log10 units of
true value.
for all 1 x 6
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Pr{| log{z2(x)} – log(x) | < 0.05 | data } = 0.95
Back-calculated values are
within 100%(100.05-1) = 12%
of true value.
Back-calculated values are
within 0.05 log10 units of
true value.
for all 1 x 5
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Extra bits
• When k=2 (quadratic vs. linear), the proposed test statistic is central T distributed.
• When k = 2, by altering the testing limits, the I-U, SSDL, and proposed test methods can be made equal.
• From simulations, test size for the proposed test statistic appears to be , depending on the experimental design.
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Summary
• Test method extends idea of NCCLS EP6-A by computing probability that best-fit curve is equivalent to a linear fit.
• Testing performed across a range of concentrations and not at experimental design points.
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References: Guidelines
• ICH (2005), “Validation of Analytical Procedures: Text and Methodology Q2(R1) – International Conference on Harminisation of Technical Requirements for Registration of Pharmaceuticals for Human Use”, http://www.ich.org/fileadmin/Public_Web_Site/ICH_Products/Guidelines/Quality/Q2_R1/Step4/Q2_R1__Guideline.pdf.
• Clinical Laboratory Standard Institute (2003), “Evaluation of the Linearity of Quantitative Measurement Procedures: A Statistical Approach; Approved Guideline”, http://www.clsi.org/source/orders/free/ep6-a.pdf.
• National Committee for Clinical Laboratory Standards. Evaluation of the linearity of quantitative analytical methods; proposed guideline. NCCLS Publ. EP6-P. Villanova, Pk NCCLS, 1986.
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• Krouwer, J. and Schalin, B. (1993). “A method to quantify deviations from assay linearity”, Clinical Chemistry, 39(8), 1689-1693.
• Kroll MH, Præstgaard J, Michaliszyn E, Styer PE (2000). “Evaluation of the extent of nonlinearity in reportable range studies”, Arch. Pathol. Lab. Med, 124: 1331–1338.
• Hsieh E, Liu JP (2008). “On statistical evaluation of linearity in assay validation”, J. Biopharm. Stat., 18: 677–690.
• Hsieh, Eric, Hsiao, Chin-Fu, and Liu, Jen-pei (2009). “Statistical methods for evaulation the linearity in assay validation”, Journal of chemometrics, 23, 56-63.
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References: Linearity tests
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• Narula, Sabhash (1979). “Orthogonal Polynomial Regression”, International Statistical Review, 47 : 1, 31-36.
• Robson, D.S. (1959). “A simple method for constructing orthogonal polynomials when the independent variable is unequally spaced”, Biometrics, 15 : 2, 187-191.
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References: Orthogonal polynomials
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References: Bayes, GPQ, Fiducial inference
• Gelman, A., Carlin, J., Stern, H., and Rubin, D. (2004). Bayesian Data Analysis Second Edition, New York: Chapman & Hall.
• Hannig, Jan (2009). “On Generalized Fiducial Inference”, Statistica Sinica, 19, 491-544.
• Weerahandi, S. (1993). ―Generalized Confidence Intervals,‖ Journal of the American Statistical Association 88:899-905.
• Weerahandi, S. (1995). Exact Statistical Methods for Data Analysis, New York: Springer-Verlag.
• Weerahandi, S. (2004). Generalized Inference in Repeated Measures: Exact Methods in MANOVA and Mixed Models, New Jersey: John Wiley & Sons.
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Thank you!
Questions?
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