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Using the Gamma Distribution to Determine Anti-drug antibody Screening Assay Cut Points with Non-normal Data Brian Schlain Nonclinical Statistics for Midwest Biopharmaceutical Statistics Workshop May 24-26, 2010

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high volume? ANTI-DRUG ANTIBODY (ADA) TESTING yes screening assay confirmation assay no positive? yes positive? titration assay output titer yes

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NEUTRALIZING ANTIBODY ASSAY (NAB) TESTING high volume? screening assay yes titration assay/ output titer positive? yes no Anti-drug Anti- bodies (ADA) detected? yes

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Types of Immunogenicity Assays Radioimmunoprecipitation Assay (RIPA) Enzyme-Linked Immunosorbent Assay (ELISA) *Standard sandwich ELISA *Bridging ELISA Electrochemiluminescence *Electrochemiluminescence (ECL; IGEN) Optical Sensor-based * Surface Plasmon Resonance (SPR; Biacore) *Guided Mode Resonance Filter (BIND; BD Biosci) Anti-Drug Antibody (ADA) Testing with Binding Assays Neutralizing Antibody (NAB) Testing with Blocking Assays Related to drug mechanism of action Converted PK assays Cell based: Some of the reagents genetically engineered and stored as cell lines

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Example ADA Assay Positive Control Response Curve

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Example NAB Assay Positive Control Response Curve

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8×12 Example Screening/Confirmation Assay Plate Layout 123456789101112 A1S1C5S5CNCLPC9S9C13S13C17S17C B1S1C5S5CNCLPC9S9C13S13C17S17C C2S2C6S6CNCLPC10S10C14S14C18S18C D2S2C6S6CNCHPC10S10C14S14C18S18C E3S3C7S7CNCHPC11S11C15S15C19S19C F3S3C7C NCHPC11S11C15S15C19S19C G4S4C8S8CNCHPC -C 12S12C16S16C20S20C H4S4C8S8CNCHPC -C 12S12C16S16C20S20C

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Screening Assay Cut Points Fixed: mean and sd do not change across plates. – CP=mean X ± a×sd X –X=Sample assay signal response Floating: mean changes, but sd remains constant across plates. –Multiplicative: CF=mean Y ± a×sd Y Y=X/NC CP=CF×NC –Additive: CF=mean Z ± a×sd Z Z=X – NC CP=CF + NC Dynamic: mean and sd both change with every plate.

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Screening Multiplicative Cut Point CF Determination Screen data for outliers Normalize:Y = Sample/NC Model of normalized values Model distribution of normalized values non-normal normal or log- normal Gamma Weibull Nonparametric percentile Validation Phase CP=CF×NC Estimate CF

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CUT POINT RULE sample>cut point, judge screening positive. cut point=CF avg(NC). NC: pooled negative sera control NORMAL THEORY CORRECTION FACTOR (CF) ESTIMATION (Guttman I, Statistical Tolerance Regions: Classical and Bayesian, 1970) n negative sera samples. R (or normalized value)=Sample/Avg(NC). mean (M R ) and sd (S R ) of R. CF=M R + t df,α S R ×(1 + 1/n) 0.5 no transformation. CF=exp{M R + t df, α S R ×(1 + 1/n) 0.5 } log transformation. df=n – 1. α=targeted false positive rate=0.05. ADA Screening Assay Cut Point CF Estimation under Normal Theory

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CUT POINT RULE sample < cut point, judge screening positive. cut point=CF avg(NC). NC: pooled negative sera control NORMAL THEORY CORRECTION FACTOR (CF) ESTIMATION (Guttman I, Statistical Tolerance Regions: Classical and Bayesian, 1970) n negative sera samples. R (or normalized value)=Sample/Avg(NC). mean (M R ) and sd (S R ) of R. CF=M R - t df,α S R ×(1 + 1/n) 0.5 no transformation. CF=exp{M R - t df, α S R ×(1 + 1/n) 0.5 } log transformation. df=n – 1. α=targeted false positive rate=0.05. NAB Screening Assay Cut Point CF Estimation under Normal Theory

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2-Parameter Gamma Density Functions Gamma approaches normal as k increases.

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2-parameter Gamma

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3-parameter Gamma (threshold parameter) Some common gamma fitting methods: maximum likelihood estimation (MLE) -SAS UNIVARIATE, R OPTIM. -becomes unstable when k approaches 1 method of moments (ME) modified moment estimation (MME) (Cohen and Whitten)

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Maximum Likelihood Estimation (MLE) of 2- parameter Gamma Unstable when estimate of k close to 1. ADA distributions tend to be unimodal with k>1. NAB distributions tend to be unimodal with k<1. -Fit gamma to 1/R=Avg(NC)/Sample.

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Estimation of Gamma Parameters by Method of Moments (ME) Gamma central moment Empirical central moment Mean S + θk =m 1 (mean) Variance θ2kθ2kθ2kθ2k =m 2 (variance) Third standard moment =m 3 /(m 2 ) 3/2 (std. skewness)

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ME for 3-parameter Gamma Gamma parameter Moment estimator k (shape) =4m 2 3 /m 3 2 θ (scale) =m 3 / (2m 2 ) s (threshold) =m 1 - (2m 2 2 )/m 3

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Modified Moment Estimators (MME) for 3-parameter Gamma (using SAS PROBGAM function) Cohen AC, Whitten BJ, Modified moment estimation for the 3-parameter Gamma distribution, J. of Qual. Tech., 18, 1, 1986, 53-62

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Calculating the Cut Point Correction Factor as a Gamma Percentile Cut point correction factor (CF) calculated using SAS GAMINV function: Cut point correction factor (CF) calculated using SAS GAMINV function: P=.95 if false positive rate targeted at 5% (ADA assay) -NAB assay: Model reciprocal of normalized values so that skewness > 0. θ=scale parameter s=threshold parameter k=shape parameter CP=CF×NC

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3-parameter Gamma Estimation MLE preferred to MME or ME. –SAS UNIVARIATE –R OPTIM MME generally better than ME. MLE unstable for k near 1. (Johnson and Kotz recommend k>2.5). MME can be calculated for any k. MME comparable to MLE with increasing n or α 3 (Cohen and Whitten). For < 0.10, consider normal distribution (Cohen and Whitten).

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Standard Gamma Simulations Comparing MLE and Nonparametric Percentile Cut Point Estimators (targeted false positive rate=5%) Abbreviations: n=sample size; η=gamma shape parameter; MLE=maximum likelihood estimator; se=standard error; FP rate=false positive rate; Nonpar. Est.=nonparametric estimator.

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Standard Gamma Simulations Comparing MLE with Normal Based Cut Point Estimators (targeted false positive rate=5%) Abbreviations: n=sample size; η=gamma shape parameter; MLE=maximum likelihood estimator; se=standard error; FP rate=false positive rate; normal z-method= mean +1.645×sd; Normal Predict. Interval=upper limit of a 1-sided prediction interval based on normal theory.

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Standard Gamma Distribution Simulations Comparing MLE with Log-normal Cut Point Estimators (targeted false positive rate=5%)

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Histogram of ADA ELISA Trial Pre-dose Screening Assay Normalized Values (n=175) Sample #29 Sample #34 outlying values

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MME Gamma Q-Q Plot (with outlying sample 34 excluded) Sample # 29 outlying value

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MME Gamma Q-Q Plot (with outlying samples 29 and 34 excluded)

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Histogram of ADA Elisa Trial Pre-dose Normalized Values (with outlying samples 29 and 34 excluded) skewed to the right

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SAS UNIVARIATE EDF Goodness of Fit Test p-values (outlying samples 29 and 34 excluded) testgammanormallog-normal Shapiro-Wilk-<.0001.031 Kolmogorov- Smirnov.146.010.025 Cramer-von Mises.089.0050.073 Anderson- Darling.151.0050.060

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Gamma Parameter and CF Estimates (outlying samples 29 and 34 excluded) MMEMLE mean (of norm. values) 1.270 sd (of norm. values) 0.170 skewness (of norm. values) 0.902 W 1 (smallest stand. value) -2.102 n173 α3α3α3α3.474.727 θ (scale).040.061 s (threshold).553.809 k (shape) 17.7967.578 CF (upper 3.9% percentile of gamma) 1.601.60 CP=CF×NC target upper gamma percentile= 5% - 100%×(2/175)=3.9%

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Gamma Distribution CF Determination Target false positive (fp) rate=5% Estimated percentage of outlying samples is 1.1% (=100%×2/175) Target fp rate – percentage of outlying samples =5% - 1.1%=3.9% CF=upper 3.9% percentile of fitted gamma=1.60 –CF=θ×GAMINV(p,k) + s=1.60. p=1- 0.039=.961; k= 17.796; θ=.040; s=.553. CP=CF×NC Empirical fp rate=5.7% (=100%×10/175)

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References Schlain B et al., A novel gamma-fitting statistical method for anti-drug antibody assays to establish assay cut points for data with non-normal distribution, JIM, V. 352, Issues 1-2, 31Jan. 2010, pp. 161-168. Guttman I, Statistical Tolerance Regions: Classical and Bayesian, 1970). Cohen AC, Whitten BJ, Modified moment estimation for the 3- parameter Gamma distribution, J. of Qual. Tech., 18, 1, 1986, 53-62. Cohen AC, Whitten BJ, Modified moment and maximum likelihood estimators for parameters of the three-parameter gamma distribution, commun. Statist.-Simula. Computa., 11(2), 197-216 (1982). Bowman KO and Shenton LR, Properties of Estimators for the Gamma Distribution, Marcel Dekker, 1988. Johnson NL and Kotz S, continuous Univariate Distributions, Vol. 1, Houghton Mifflin company, 1970. Krishnamoorthy K, Mathew T, and Mukherjee S, Normal-based methods for a gamma distribution: prediction and tolerance intervals and stress-strength reliability, Technometrics, Vol. 50, No. 1, pp. 69- 78.

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BACKUP SLIDES

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Standard Gamma Simulations Comparing MLE, Nonparametric, and WH Cut Point Estimators (targeted false positive rate=5%) Abbreviations: n=sample size; η=gamma shape parameter; MLE=maximum likelihood estimator; se=standard error; FP rate=false positive rate; Nonpar. Est.=nonparametric estimator; WH=Wilson-Hilferty estimator.

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Need for further research on WH How well does it perform when the real distribution is not quite a gamma, but the gamma is the best approximation that can be found?

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