Impact of Censoring Data Below an Arbitrary Quantification

Impact of Censoring Data Below an Arbitrary Quantification

Impact of Censoring Data Below an Arbitrary Quantification Limit on Structural Model Misspecification
W. Byon 1, C. V. Fletcher 2, R. C. Brundage 3
Pfizer Global Research and Development, New London Connecticut, USA 1, University of Colorado Denver, Colorado, USA 2. University of Minnesota, Minneapolis, Minnesota, USA 3
ABSTRACT

5.000

Full data
BQL censoring
BQL data

NONMEM VI 5FOCEI / YLO with LAPLACIAN )
$EST Advan1 Trans2 for 1 Compartment model

1.50

2.00

2.50

3.00

3.50

4.00

Time After Dose

Figure 1. Simulation flow chart using a typical simulated concentrationtime profile from scenario 2

10.2

0.0

2

5

0.0132

0.0884

< 20 17.3 0.2 3 5 0.0187 0.1250 < 20 26.7 0.4 4 5 0.0265 0.1768 < 20 37.6 0.9 5 5 0.0374 0.2500 < 20 49.1 1.8 Simulation plan 2 10 6 5 0.0132 0.2640 < 10 51.3 0.2 7 5 0.0132 0.0294 < 50 3.1 0.2 8 5 0.0132 0.0139 < 100 1.1 0.2 SIMULATION & ESTIMATION Each simulated data set was designated as Full data 5no BQL censoring). This data set was then used to generate a second data set that excluded data below the relevant LLOQ to the scenario, which was designated BQL data. The simulations and population analyses were performed using a nonlinear mixed effects model implemented in NONMEM VI [7] using Compaq Visual Fortran version 6.5. The preparation of BQL censored datasets was performed using SPLUS 7.0 5Insightful Corporation). BQL data and Full data were analyzed with a one-compartment model using ADVAN1 and TRANS2 and a two-compartment model using ADVAN3 and TRANS4. When the twocompartment model was tested, the peripheral volume of distribution and the intercompartmental clearance were added into the model without between subject variability on them. The FOCEI was used for this estimation. Additionally, a new conditional likelihood estimation feature in NONMEM VI 5YLO) was evaluated.. 20 30 40 50 1.0 BQL data data from from all all 500 runs runs Full data data from all 500 runs runs 0.8 0.6 < 20 CONCLUSION 10 20 Percent of median censored data 30 40 50 Percent of median censored data Figure 2. Type I error rates at the 5 % 5left) and 1 % 5right) alpha levels in simulation plan 1 for BQL data 5solid lines) and Full data 5dashed lines). BQL data from all 500 runs Table 2. Type I error rates when testing YLO at the 5 % alpha level BQL data from all 500 runs 10 20 50 CV at LLOQ 5%) 100 Scenario No. 10 20 50 CV at LLOQ 5%) 100 Figure 3. Type I error rates at the 5 % 5left) and 1 % 5right) alpha levels in simulation plan 2 The censoring of concentrations as BQL can lead to structural model misspecification in population PK analyses. Furthermore, relaxing the current practice of censoring data with less than 20% precision can help prevent this misspecification. With the nave cut-off values in the 2-distribution at two degrees of freedom, the type I error rates from Full data 5without any BQL censoring) in simulation plan 1 were lower than the nominal value at both the 5% and 1% alpha levels in scenario 1 3. This is a known result under the constrained one-sided test using log likelihood ratio test under a boundary condition [8]. A trend in type I error rate in Full data was that it increased across scenarios 1 through 5. This is suspected to result from the simulated non-positive data which were removed from the parent datasets. The maximum conditional likelihood estimation minimized the elevation of type I error across all scenarios. Therefore, in a PK analysis that includes a substantial fraction of data being censored, the use of YLO options should be strongly considered to avoid any model misspecification 0.4 0.0625 Type I error rate at the alpha level of 0.01 0.0093 BQL data from from all 500 500 runs Full data data from from all 500 500 runs Type I error rate at the alpha level of 0.01 0.2 0.4 0.6 0.8 1.0 5 1.0 Median of percent data Median of percent data set censored as set censored as BQL and negative negative Concentrations concentrations 1 RESULTS Simulation plan 1 1.00 0.8 error CV at LLOQ 5%) 0.25 0.5 $EST Advan3 Trans4 for 2 Compartment model 0.6 LLOQ LLOQ 0.4 Scenario No. Additive Type I error rates were elevated when datasets included BQL censoring compared to when all the data were available across all the scenarios. The increasing trend in type I error rate was observed as the median of percent censored data increased when BQL data were estimated at both alpha levels. When the rules of successful minimization, successful covariance step, and reasonable results were applied, the type I error rates were nearly identical to the results from all 500 runs. Type I error rates in Full data without BQL censoring generally stayed close or lower than the expected 5 or 1%. However, the trend was observed that the error rate slightly increased as the median of percent censored data increased. Restricting the CV to 10% caused a higher type I error rate compared to the 20% CV, while the error rate was reduced to the nominal value as the CV increased to 100% When the YLO option was implemented with both one-compartment and twocompartment models for BQL data, the type I error rate for structural model misspecification was close to nominal values. DISCUSSIONS 0.500 Simulated Concentrations $SIM SUBPROBLEMS=500 Table 1. Summary of simulation plans for eight scenarios Proportional Error 5%) 0.2 0.0 Scenario 2 5LLOQ=0.0884 AE=0.0132) 0.050 NONMEM VI In simulation plan 1, scenarios 1 to 5 examined the influence of the percentage of data censored on the structural model decision when the LLOQ had no greater than a 20% CV. Five different LLOQ values were defined as the concentration at 2, 2.5, 3, 3.5, and 4 half-lives using typical parameter values. Once the LLOQ was decided for each scenario, an additive error was chosen so that the CV at the LLOQ was no more than 20%. In simulation plan 2, scenarios 6 to 8 evaluated the impact of allowing more and less precise CVs at the LLOQ than the current practice of 20%. This was conducted as variations of scenario 2. Three different CV values were chosen as 10, 50, and 100%, and these were analyzed in addition to the 20% CV which was tested as scenario 2. For each scenario, 500 simulations were conducted. Each simulation consisted of 50 subjects with 9 PK observations at 0.25, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, and 4 units of time. 0.005 SCENARIOS 0.001 OBJECTIVE Assess the impact of the percentage of data censored as BQL on the PK structural model decision Evaluate the effect of different CV values to define the LLOQ on the structural model decision Evaluate the use of a maximum conditional likelihood estimation method available in NONMEM VI 5YLO/LAPLACIAN). exp 0.693t The between-subject variability on CL and V were assumed to follow a log-normal distribution with an exponential error model, and both were set to a 20% CV. The residual unexplained variability was chosen as a combined proportional/additive error model to represent an analytical proportional component 5constant CV), and an absolute additive component 5constant standard deviation) of measurement noise. The proportional error component was set to a 5% CV. A different additive error was chosen for each scenario to control the CV at the LLOQ according to the following plans and scenarios. 0.0 CL t V 0.2 Dose C(t) exp V A type I error was declared when the OFV was significantly lower for the two-compartment model compared to the one-compartment model using the standard likelihood ratio test. The error rate was determined at a level of significance of 5% and 1%; with two degrees of freedom, the associated drops in OFV from a 2 table were 5.99 and 9.21, respectively. The type I error rate was determined from 500 simulations per each scenario with the following rules. 1. All runs 2. Runs with a successful minimization 3. Runs with a successful minimization and a successful covariance step 4. Runs with reasonable results for the two-compartment model in addition to a successful minimization and covariance step where a reasonable result was defined as the alphaphase half-life 56t1/2) had to be greater than 0.25 5the first sampling time), and the betaphase half-life 59t1/2) had to be less than 10 units 5considering concentrations were sampled over 4 units of time). An intravenous one-compartment pharmacokinetic model was chosen for the simulation. The clearance 5CL) and volume of distribution 5V) were 0.693 and 1, respectively. A single unitvalued dose was administered at time zero. The PK model becomes and the units of time can be regarded as half-lives 54). 0.0 It is not uncommon that some concentrations are censored by the bioanalytical laboratory since those concentrations are below the lower limit of quantification 5LLOQ). The acceptance of a LLOQ in analytical methods development is nearly universal. In an effort to report only those concentrations that are considered to have acceptable precision, the laboratory determines a LLOQ and truncates a standard curve so that no concentrations are reported below that limit. The LLOQ is often defined in practice as the lowest concentration on the standard curve that is associated with a CV 5coefficient of variation) of no more than 20%. The 20% CV is suggested by the FDA Guidance for Industry Bioanalytical Method Validation and other reports [1, 2]. Typically, any sample associated with a signal less than LLOQ is not reported quantitatively, but as below the quantification limit 5BQL). Although these standard operating procedures are well intentioned, the policy of censoring observations below the LLOQ violates one of the assumptions PK/PD modelers often make. When using the maximum likelihood estimation method in fitting models to data, it is assumed that residual errors are independent and normally distributed with zero mean and a variance. However, censoring data below the LLOQ truncates the tail of this normal distribution and violates the assumption of residual errors. The impact of censoring has been examined in population PK settings and procedures for handling BQL information have been suggested [3-6]. However, these references have focused on bias and precision of parameter estimates when some data were censored as BQL. Since the BQL censoring occurs more frequently at later time points, a visual examination of the cloud of concentration-time data can appear to be associated with a multiple-compartment drug. To our knowledge, structural model misspecification related to BQL censoring has not been examined. TYPE I ERROR RATE Type I error rate at the alpha level of 0.05 PK MODEL Type I error rate at the alpha level of 0.05 0.2 0.4 0.6 0.8 1.0 INTRODUCTION RESULTS 0.0 Objectives: The current simulation study investigated the impact of the percentage of data censored as BQL on the PK structural model decision; evaluated the effect of different coefficient of variation 5CV) values to define the LLOQ; and tested the maximum conditional likelihood estimation method in NONMEM VI 5YLO). Methods: Using a one-compartment intravenous model, data were simulated with 10 to 50% BQL censoring, while maintaining a 20% CV at LLOQ. In another set of experiments, the LLOQ was chosen to attain CVs of 10, 20, 50 and 100%. Parameters were estimated with both one- and twocompartment models using NONMEM VI 5GloboMax LLC, Hanover, MD). A type I error was defined as a significantly lower objective function value for the two-compartment model compared to the one-compartment model using the standard likelihood ratio test at alpha=0.05 and alpha=0.01. Results: The type I error rate substantially increased to as high as 96% as the median of percent censored data increased at both the 5% and 1% alpha levels. Restricting the CV to 10% caused a higher type I error rate compared to the 20% CV, while the error rate was reduced to the nominal value as the CV increased to 100%. The YLO option prevented the type I error rate from being elevated. Conclusions: This simulation study has shown that the practice of assigning a LLOQ during analytical methods development, although well intentioned, can lead to incorrect decisions regarding the structure of the pharmacokinetic model. The standard operating procedures in analytical laboratories should be adjusted to provide a quantitative value for all samples assayed in the drug development setting where sophisticated modeling may occur. However, the current level of precision may need to be maintained when laboratory results are to be used for direct patient care in a clinical setting. Finally, the YLO option should be considered when more than 10% of data are censored as BQL. METHODOLOGY Median of percent data set Type I censored as error rate BQL and negative concentrations 1 10.2 0.00 2 17.3 0.02 3 26.7 0.02 4 37.6 0.05 5 49.1 0.06 This simulation study has shown that the practice of assigning a LLOQ during analytical methods development, although well intentioned, can lead to incorrect decisions regarding the structure of the pharmacokinetic model. The standard operating procedures in analytical laboratories should be adjusted to provide a quantitative value for all samples assayed in the drug development setting where sophisticated modeling may occur. However, the current level of precision may need to be maintained when laboratory results are to be used for direct patient care in a clinical setting. Finally, the YLO option should be considered when more than 10% of data are censored as BQL. REFERENCES 1. FDA guidance for industry bioanalytical method validation. Available from http://www.fda.gov/cder/guidance/4252fnl.pdf Accessed on may 2001 2. Shah VP, Midha KK, Findlay JW, Hill HM, Hulse JD, McGilveray IJ, McKay G, Miller KJ, PatnaikRN, Powell ML, Tonelli A, Viswanathan CT, Yacobi A 52000) Bioanalytical method validationa revisit with a decade of progress. Pharm Res 17512):15511557 3. Hing JP, Woolfrey SG, Greenslade D, Wright PM 52001) Analysis of toxicokinetic data using NONMEM:impact of quantification limit and replacement strategies for censored data. J PharmacokinetPharmacodyn 2855):465479 4. Beal SL 52001) Ways to fit a PK model with some data below the quantification limit. J Pharmacokinet Pharmacodyn 2855):481504 5. Duval V, Karlsson MO 52002) Impact of omission or replacement of data below the limit of quantification on parameter estimates in a two-compartment model. Pharm Res 19512):18351840 6. Beal SL 52005) Conditioning on certain random events associated with statistical variability in PK/PD. J Pharmacokinet Pharmacodyn 3252):213243 7. Beal SL, Sheiner LB, Boeckmann AJ 5eds) 519892006) NONMEM users guides. Icon development solutions. Ellicott City 8. Stram DO, Lee JW 51994) Variance components testing in the longitudinal mixed effects model. Biometrics5054):11711177

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