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

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**ANTI-DRUG ANTIBODY (ADA) TESTING**

high volume? yes screening assay no confirmation assay yes positive? -ADA -Does Ab attach to drug? This may or may not result in neutralizing the drug. -Titer used as a quantitative measure for patient monitoring. yes positive? titration assay output titer

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**NEUTRALIZING ANTIBODY ASSAY (NAB) TESTING**

Anti-drug Anti- bodies (ADA) detected? yes yes screening assay high volume? no yes -NAB -Does the Ab neutralize the drug? titration assay/ output titer positive?

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**Types of Immunogenicity Assays**

Anti-Drug Antibody (ADA) Testing with Binding Assays Radioimmunoprecipitation Assay (RIPA) Enzyme-Linked Immunosorbent Assay (ELISA) *Standard sandwich ELISA *Bridging ELISA *Electrochemiluminescence (ECL; IGEN) Optical Sensor-based * Surface Plasmon Resonance (SPR; Biacore) *Guided Mode Resonance Filter (BIND; BD Biosci) -Cell based: reagents are genetically engineered and then stored as cell lines. -Have to be concerned about consistency across cell passages. 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**

1 2 3 4 5 6 7 8 9 10 11 12 A 1S 1C 5S 5C NC LPC 9S 9C 13S 13C 17S 17C B C 2S 2C 6S 6C 10S 10C 14S 14C 18S 18C D HPC E 3S 3C 7S 7C 11S 11C 15S 15C 19S 19C F G 4S 4C 8S 8C HPC-C 12S 12C 16S 16C 20S 20C H -Drug and no-drug sample within same row, because row effects dominate. This reduces plate effects. -Get each ratio rep. And then average ratios. Removes row effects -For screening assay, no-drug sample replicates straddle rows to capture more variability. -NC placed down a middle column to represent all of the rows, which are the major source of variability. -Lately, we have been placing the negative control to minimize false negatives, which might mean putting in the most depressed row.

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**Screening Assay Cut Points**

Fixed: mean and sd do not change across plates. CP=meanX ± a×sdX X=Sample assay signal response Floating: mean changes, but sd remains constant across plates. Multiplicative: CF=meanY ± a×sdY Y=X/NC CP=CF×NC Additive: CF=meanZ ± a×sdZ 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 distribution of normalized values non-normal normal or log-normal Gamma Weibull Nonparametric percentile Estimate CF Validation Phase CP=CF×NC

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**ADA Screening Assay Cut Point CF Estimation under Normal Theory**

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 (MR) and sd (SR) of R. CF=MR + tdf,α SR×(1 + 1/n)0.5 no transformation . CF=exp{MR + tdf,α SR×(1 + 1/n) 0.5 } log transformation. df=n – 1. α=targeted false positive rate=0.05. -For inhibition assays, cut point rule is sample>cut point -Assumption that 30 neg. sera are representative of the target population. Don’t have resources to run more than 30 sera. -An alternative would be to not normalize and to use neg. control as proxy as mean, but have found that the neg. control is sometimes higher than the mean of the sera composing it. -Among-sample matrix variability is often a sizeable component of the SD -Don’t have luxury of having many plates. -Test is conditional on NC reps. Uses within-plate sd. False positive rate is a random variable which should be centered around some value near to the target. Never have enough plates to bring in among-plate variability into sd, and that would complicate the problem with intra-plate correlation. The plate is flooded with NC reps. To obtain a good estimate of the mean of the normalized values for those 30 neg. sera.

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**NAB Screening Assay Cut Point CF Estimation under Normal Theory**

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 (MR) and sd (SR) of R. CF=MR - tdf,α SR×(1 + 1/n)0.5 no transformation . CF=exp{MR - tdf,α SR×(1 + 1/n) 0.5 } log transformation. df=n – 1. α=targeted false positive rate=0.05. -For inhibition assays, cut point rule is sample>cut point -Assumption that 30 neg. sera are representative of the target population. Don’t have resources to run more than 30 sera. -An alternative would be to not normalize and to use neg. control as proxy as mean, but have found that the neg. control is sometimes higher than the mean of the sera composing it. -Among-sample matrix variability is often a sizeable component of the SD -Don’t have luxury of having many plates. -Test is conditional on NC reps. Uses within-plate sd. False positive rate is a random variable which should be centered around some value near to the target. Never have enough plates to bring in among-plate variability into sd, and that would complicate the problem with intra-plate correlation. The plate is flooded with NC reps. To obtain a good estimate of the mean of the normalized values for those 30 neg. sera.

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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 =m1 (mean) Variance θ2k =m2 (variance) Third standard moment =m3 /(m2 )3/2 (std. skewness)

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**ME for 3-parameter Gamma**

Gamma parameter Moment estimator k (shape) =4m23 /m3 2 θ (scale) =m3 / (2m2 ) s (threshold) =m1 - (2m22 )/m3

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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: 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 α3 (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 ×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)**

outlying values Sample#34 Sample#29

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**MME Gamma Q-Q Plot (with outlying sample 34 excluded)**

outlying value Sample # 29

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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)**

gamma normal log-normal Shapiro-Wilk - <.0001 .031 Kolmogorov-Smirnov .146 .010 .025 Cramer-von Mises .089 .0050 .073 Anderson-Darling .151 .060

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**Gamma Parameter and CF Estimates (outlying samples 29 and 34 excluded)**

MME MLE mean (of norm. values) 1.270 sd (of norm. values) 0.170 skewness (of norm. values) 0.902 W1 (smallest stand. value) -2.102 n 173 α3 .474 .727 θ (scale) .040 .061 s (threshold) .553 .809 k (shape) 17.796 7.578 CF (upper 3.9% percentile of gamma) 1.60 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= =.961; k= ; 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 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, Cohen AC, Whitten BJ, Modified moment and maximum likelihood estimators for parameters of the three-parameter gamma distribution, commun. Statist.-Simula. Computa., 11(2), (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

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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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