Multivariate Assays With Values Below the Lower Limit of Quantitation: Parametric Estimation By Imputation and Maximum Likelihood

Transcription

1 Multivariate Assays With Values Below the Lower Limit of Quantitation: Parametric Estimation By Imputation and Maximum Likelihood Robert E. Johnson and Heather J. Hoffman 2* Department of Biostatistics, Virginia Commonwealth University 2 Department of Epidemiology and Biostatistics, George Washington University * This presentation is based, in part, on Dr. Hoffman s s dissertation directed by Dr. Johnson.

2 Outline Introduction Limit of Quantitation (LOQ) Ignoring values below LOQ problems Solutions Univariate Solutions Multivariate Summary Johnson & Hoffman, BASS XIII, November 7,

3 Purpose To estimate all parameters (means, variances, correlations) associated with a multivariate assay Multiple assays on the same subject Different analytes or repeated measures of one analyte. Do this in the presence of missing data Completely missing due to flawed assay, etc. Some values fall below LOQ Johnson & Hoffman, BASS XIII, November 7,

4 An Example Tobacco Smoke Complex mixture of chemicals released as tobacco products burn Biomarkers include nicotine and its 5 metabolites measured in 24-hour urine samples. Total nicotine sum of Total cotinine sum of Total hydroxycotinine sum of nicotine nicotine-n-glucuronide cotinine cotinine-n-glucuronide trans-3.-hydroxycotinine trans-3.-hydroxycotinine-o-glucuronide Johnson & Hoffman, BASS XIII, November 7,

5 Analysis of Nicotine Nicotine equivalents are calculated according to the following equation: Nicotine Equivalents = 62.23mg mol mg Total Nicotine mg mol mg + Total Cotinine mg mol mg + Total hydroxycotinine mg mol Johnson & Hoffman, BASS XIII, November 7,

6 Analysis of Nicotine Minimize influence of possible incomplete 24-h h urine collections Adjust measured concentration of each subject s s 24-h h urine biomarker by corresponding 24-h h urine creatinine Nicotine Equivalents Creatinine Level Johnson & Hoffman, BASS XIII, November 7,

7 Data Format X X2 X3 X4 X5 X6 X < <0.50 <0.50 < <0.50 < <0.0 <0.50 <0.50 <.20 < <0.0 < <.20. <.00. <0.50 <0.50 <.20 < Total nicotine sum of Total cotinine sum of Total hydroxycotinine sum of Creatinine nicotine nicotine-n-glucuronide cotinine cotinine-nglucuronide trans-3.-hydroxycotinine trans-3.-hydroxycotinine-o-glucuronide Johnson & Hoffman, BASS XIII, November 7, X X2 X3 X4 X5 X6 X7

8 Problems of Interest The general problem Estimate all parameters, mean and standard error of any function of multivariate values A specific problem Estimate mean and standard error of distribution of: T = 6 ai i= bx X 7 i Johnson & Hoffman, BASS XIII, November 7,

9 Primary Questions How does one estimate the parameters of a given set of data containing missing data or values below LOQ? How does one estimate functions of these data? Estimation of means Estimation of standard errors Johnson & Hoffman, BASS XIII, November 7,

10 Methods of Estimation Univariate (single assay) Imputation (Replacement) Cohen s s Method (adjusted mean, std) Maximum Likelihood Estimation (MLE) with left-censored values Extensions of MLE Johnson & Hoffman, BASS XIII, November 7,

11 Methods of Estimation Multivariate (multiple assays) Imputation (Replacement) MLE with left-censoring and correlation structure Pseudo-Likelihood Johnson & Hoffman, BASS XIII, November 7, 2006

12 Parametric Distributions Normal Distribution: Mean= Variance= Coeff Var= σ μ μ σ 2 2 N( μ, σ ) Johnson & Hoffman, BASS XIII, November 7,

13 Parametric Distributions Log-Normal Distribution: 2 exp μ+ σ 2 Mean= Variance= Coeff Var= ( 2 σ ) exp 2 LN( μ, σ ) ( ) exp( 2 2) 2 exp( 2) μ+ σ σ μ = 0 Johnson & Hoffman, BASS XIII, November 7,

14 Parametric Distributions Multivariate Normal Distribution: μ μ ( ) k, μ is the location parameter k of the k th MN ( μ, Σ) variate covariance i Σ ( σ ) ij, σij = variance i = j j Johnson & Hoffman, BASS XIII, November 7,

15 Parametric Distributions Multivariate Log-Normal Distribution: MLN ( μ, Σ) If X MN μσ,, ( ) then Y ( X k ) ( μσ, ) X = e e MLN Johnson & Hoffman, BASS XIII, November 7,

16 Method Validation Validation Criteria Accuracy Precision Specificity Limit of Quantitation/ Limit of Detection Linearity Range Ruggedness/ Repeatability Equivalence USP. General Chapter <225> Validation of Compendial Methods. Pharmacopeial Forum 3(2): , 2005 Johnson & Hoffman, BASS XIII, November 7,

17 Limit of Quantitation (LOQ) The quantitation limit of an individual analytical procedure is the lowest amount of analyte in a sample that can be quantitatively determined with suitable precision and accuracy. Food and Drug Administration, (Draft) Guidance for Industry: Analytical Procedures and Methods Validation: Chemistry, Manufacturing, and Controls Documentation (2000). Accessed online at Johnson & Hoffman, BASS XIII, November 7,

18 Reasons for Values Falling Below LOQ Absence or trace amounts of analyte High degree of noise relative to analyte levels Matrix effects A loss of absolute sensitivity may render some or all of the analytes of interest undetectable Ardrey, R. E., Liquid Chromatography-Mass Spectrometry: An Introduction. John Wiley & Sons, Johnson & Hoffman, BASS XIII, November 7,

19 Determination of LOQ Generally the LOQ is about 0 times the standard deviation of a low standard (blank solutions). The ICH has recognized the ten-to to-one one signal-to to-noise ratio as typical. Johnson & Hoffman, BASS XIII, November 7,

20 Recording Values below LOQ Simple: <LOQ not numeric Numeric values Negative numbers Indicator variable Interval endpoints Johnson & Hoffman, BASS XIII, November 7,

21 Analysis Ignoring values below LOQ Problems The mean will be over-estimated estimated The standard deviation will be under- estimated Type I or Type II error probabilities may be altered in the comparison of means between two groups. Johnson & Hoffman, BASS XIII, November 7,

22 Full curve mean=4 std= Left truncated at LOQ=2 mean=4.055 std=0.945 Full curve mean=4 std= Left truncated at LOQ=3 mean=4.288 std= LOQs Comparison of means: Effect Size Full curve 0 left truncated at LOQs 0.9 Johnson & Hoffman, BASS XIII, November 7,

23 Full curve mean=3 std= left-truncated at LOQ=2 mean=3.288 std= Full curve mean=4 std= left-truncated at LOQ=2 mean=4.055 std=0.945 LOQ Comparison of means: Effect Size Full curve 0.7 left-truncated at LOQ= Johnson & Hoffman, BASS XIII, November 7,

24 Solutions - Univariate Imputation (replacement) Replace values < LOQ with Zero, LOQ/2,, LOQ/ 2, LOQ Treat values < LOQ as left-censored Kaplan-Meier Survival Method Maximum likelihood Cohen s s Method Single LOQ only MLE May have multiple LOQs Robust MLE recommended by Helsel and others Johnson & Hoffman, BASS XIII, November 7,

25 Recommended Book Helsel,, D.R., Nondetects and Data Analysis: Statistics for Censored Environmental Data. John Wiley & Sons, Johnson & Hoffman, BASS XIII, November 7,

26 Replacement: LOQ/2 [Replacement] remains the most commonly used method for computing summary statistics of censored data it is clear that [replacement] must be abandoned it is not an unequivocal method of analysis. Helsel,, 2005 When is replacement good enough? Johnson & Hoffman, BASS XIII, November 7,

27 EPA Guidelines for handling LOD values % values less than LOD < 5% Impute LOD/2 for each nondetect between 5% and 50% Use Cohen's adjustment [ML method] to the sample mean and variance or employ a non-parametric procedure by using the ranks of the observations and by treating all nondetects as tied values greater than 50% use a rank-based procedure. Source: US EPA, Statistical Analysis Of Ground-water Monitoring Data At RCRA Facilities: Addendum To Interim Final Guidance, Office Of Solid Waste, Permits And State Programs Division, U.S. Environmental Protection Agency, July 992. Johnson & Hoffman, BASS XIII, November 7,

28 About the sample mean based on replacement x = ( h) x + hm m [ ] ( ) Ex = Fd ( ) μ + Fdm ( ) m ( Fd ( )) Fd ( ) Var [ x ] m F d n L ( ) 2 ( ) ( ) 2 m = μl + σ L m is the known replacement value h is the observed proportion of values below LOQ d is the LOQ assume a single LOQ was used F ( )) is the underlying CDF L identifies parameters of the left-truncated truncated distribution (truncated at d) Johnson & Hoffman, BASS XIII, November 7,

29 Potential Bias using Imputed Values bias [ ] = E x μ m = Fd ( )( m μ ) R upper bound when bias d F( d) 2 m = d 2 μ is the mean of the underlying distribution R identifies parameters of the right-truncated truncated distribution (truncated at d) Note: μ = Fd ( ) μ + Fd ( ) μ ( ) L R Johnson & Hoffman, BASS XIII, November 7,

30 Potential Bias using Replacement: Log-Normal Distribution bias μ ( 2 zτ τ ) exp 2 = Φ( z) Φ( z τ ) 2 Ratio of bias to mean is a function of Φ(z): Standard normal probability p of values < LOQ τ : Standard deviation of log transformed measure Instead, express τ as a function of the CV : ([ CV ] 2 ) τ = ln + Source: Johnson, R.E., Hoffman, H.J., Estimation in the Presence of Non-Detectables: Imputation and Maximum-Likelihood Approach with Left-Censoring. ASA Proceedings of the Joint Statistical Meetings, , Johnson & Hoffman, BASS XIII, November 7,

31 Bias as Percent of Mean: Log-Normal 5.0% 0.0% % Bias -5.0% -0.0% -5.0% -20.0% % Bias 0.0% -.0% -2.0% -3.0% -4.0% -5.0% -6.0% -7.0% CV -25.0% CV Johnson & Hoffman, BASS XIII, November 7,

32 Full curve mean=4 std= Replace values below LOQ with LOQ/2= Prob LOQ = mean=3.986 std=.0363 Full curve mean=4 std= Replace values below LOQ with LOQ/2=.5 Prob LOQ = 0.59 mean=3.845 std=.2508 LOQs Comparison of means: Effect Size Full curve 0 with replaced values Johnson & Hoffman, BASS XIII, November 7,

33 Full curve mean=3 std= Replace values below LOQ with LOQ/2= Prob LOQ = 0.59 mean=2.925 std=.083 Full curve mean=4 std= Replace values below LOQ with LOQ/2= Prob LOQ = 0.59 mean=3.986 std=.0363 LOQ Comparison of means: Effect Size Full curve 0.7 with replaced values 0.70 Johnson & Hoffman, BASS XIII, November 7,

34 Cohen s s Adjustment Data are assumed to be normally distributed (Transform the data if log-normally distributed) Adjustment of the sample mean and variance ˆλ ˆ ˆ μ = x λ( x d) ˆ ˆ σ = s + λ( x d) is found in Table 2 of Cohen. Can be computed. Cohen, A.C., Jr Simplified estimators for the normal distribution when samples are singly censored or truncated. Technometrics, vol., pp Johnson & Hoffman, BASS XIII, November 7,

35 Parametric estimation in presence of left-censored values: MLE Maximum likelihood estimation (normal distribution) equivalent to Cohen s s adjustment when d i is invariant to i. n δi δi ( θ ) = ( i, θ) ( i, θ) i= L f x F d where δ i = when x i < d i f(x i ) F(d i ) d i Johnson & Hoffman, BASS XIII, November 7,

36 MLE Computations in SAS proc lifereg data=inputdata inputdata; model (Start,End( Start,End)= /dist=lnormal lnormal; run; Regressors may be added to model the mean Johnson & Hoffman, BASS XIII, November 7,

37 Solutions - Multivariate Imputation (replacement) Replace values < LOQ with Zero, LOQ/2,, LOQ/ 2, LOQ Treat values < LOQ as left-censored censored Maximum likelihood EM Algorithm Full Likelihood function Pseudo-likelihood Johnson & Hoffman, BASS XIII, November 7,

38 Extend MLE Method to Multivariate Data Account for correlations among variables Develop MLE tool Construct log-likelihood likelihood Account for left-censored values Maximize using Newton-Raphson Starting value issues Efficiency issues Johnson & Hoffman, BASS XIII, November 7,

39 Likelihood Function Define: Assign: r i s i p = # observed = # censored i X X = i r i + s,..., i,..., X r+ i pi = # non - missing r i X = observed = censored Likelihood for i th subject: L( θ) = f( x,..., x ) F( d,..., d x,..., x ) i r X X r i i r+ i pi i i Johnson & Hoffman, BASS XIII, November 7,

40 Likelihood and Log-Likelihood Likelihood Functions Likelihood n L( θ) = f( x,..., x ) (,...,,..., ) i r F d i X d r i X x x + pi i ri i= Log-Likelihood Likelihood n i= { f x x } r F dx dx x x r ( θ) = ln (,..., ) + ln (,...,,..., ) i i r+ i pi i i Johnson & Hoffman, BASS XIII, November 7,

41 The Crux of the Problem Maximization of likelihood function Must evaluate multivariate normal CDF F( a) Transform the multiple integral Quadrature: : Evaluate the integral = ( 2π ) s Σ a... a s exp 2 ( x μ) Σ ( x μ) dx Genz, A., Numerical Computation of Multivariate Normal Probabilities. Journal of Computational and Graphical Statistics :4-50, 992. Johnson & Hoffman, BASS XIII, November 7,

42 Genz Transformations Compute multivariate normal CDF F( a) = ( 2π ) s Σ a... a s exp 2 ( x μ) Σ ( x μ) dx Transform to integral over unit hyper- cube F( b) = e e2... es dw Johnson & Hoffman, BASS XIII, November 7,

43 Genz Transformations Compute multivariate normal CDF F( a) Standardize data = Define: θ = ( 2π ) s Σ a... ( x μ) Σ ( x μ) Let R be the correlation matrix Standardize limits a,, denoted as b a s exp i i ( θ,..., θ ), where θ =, i s 2 x μ s i () σ = ii dx Johnson & Hoffman, BASS XIII, November 7,

44 Genz Transformations Compute multivariate normal CDF F( b) = ( 2π ) s R b... b s exp θ R 2 θ dθ Johnson & Hoffman, BASS XIII, November 7,

45 Three Transformations Transform to integral over unit hyper- cube () Cholesky decomposition transformation θ = Cy, where CC = Cholesky decomposition of R (2) Transformation of y ( ) y i = Φ z i (3) Transformation converting integral to constant z = ew limit form i i i Johnson & Hoffman, BASS XIII, November 7,

46 () Cholesky Decomposition Transformation Let: Then: θ = Cy, where CC = Cholesky decomposition of R ( 2π ) b 2 b 2 ( y ) 2 b s ( y,..., ys ) 2 y y 2 ys F( b) = exp exp... exp d s 2 2 y 2 where b ( y i,..., yi ) bi = i j= c ii c ij y j Johnson & Hoffman, BASS XIII, November 7,

47 (2) Transformation of y Let: where ( ) y i = Φ z i y Φ y = 2 ( ) exp 2π θ dθ 2 Then: where e e ( z,..., z 2 s s = F( b)... dz e ( z,..., z i 0 i ( z ) 0 e bi ) = Φ 0 i j= c c ij ii Φ ) ( z ) Johnson & Hoffman, BASS XIII, November 7, j

48 (3) Convert Integral to Constant Limit Form Let: Then: z = ew i i i F( b) = e e2... es dw where e i i bi cijφ ( ejwj) j= =Φ cii Johnson & Hoffman, BASS XIII, November 7,

49 Numerically Evaluate Integral We chose to use Legendre-Gauss Quadrature of order 8. Abscissas for quadrature of order n are given by roots of Legendre polynomials Johnson & Hoffman, BASS XIII, November 7,

50 Quadrature Definition: A functional Q n that is defined on C[a,b] ] and has the form Goal: n Qn ( f ) = wv f ( av) for a a... an b, and wv 0 for v = ( )n v= is called a quadrature rule of order n on the fundamental interval [a,b].[ Choose appropriate weights and abscissas Independent of f b For large class of functions f a f ( x) dx Q ( f ) n Johnson & Hoffman, BASS XIII, November 7,

51 Limitations The algorithm works well if only a few observations have 7 or fewer variables with values below their respective LODs. Otherwise the algorithm will consume much time and resources. Johnson & Hoffman, BASS XIII, November 7,

52 Alternate Method: Pseudo-Likelihood Method Instead of applying the ML approach to all variables simultaneously, apply the ML algorithm to pairs of variables. This methodology was recently used by Fieuws to fit mixed models for multivariate longitudinal profiles In the next slides we describe a pseudo- likelihood approach using Fieuws notation. Fieuws, S., Verbeke, G., Pairwise Fitting of Mixed Models for the Joint Modeling of Multivariate Longitudinal Profiles. Biometrics 62: , 2006 Johnson & Hoffman, BASS XIII, November 7,

53 Pseudo-Likelihood Approach Consider the log-likelihood likelihood involving the pair of variables X, X ( ) r n s l = l Θ X, X ( ) rs rsi rs r s i= We have p variables. We may represent rs = k where k =,,P,P and P=p ( p)/2, the number of pairs. Thus l k = l rs Johnson & Hoffman, BASS XIII, November 7,

54 Pseudo-Likelihood Approach We now maximize the pseudo- likelihood function l Θ = P ( ) ( ) k = l p Θ k where Θ= ( Θ ) is the, Θ2,, Θ P stacked parameter vector. Johnson & Hoffman, BASS XIII, November 7,

55 Distribution of Parameter Estimates The estimated stacked parameter vector is asymptotically normal N Θ Θ MN 0, Γ ( ) ( ) Johnson & Hoffman, BASS XIII, November 7,

56 Estimates and Var-Cov of Parameter Estimates The vector of unique parameter estimates is given by and its variance-covariance covariance matrix is given by * Θ = where A is the matrix that calculates the desired averages. ˆ AΘˆ ( Θ) ˆ * ˆ Θ = AΓ A Johnson & Hoffman, BASS XIII, November 7,

57 Back to the Example Estimate mean and standard error of distribution of: T = 6 i= a i bx X 7 i Johnson & Hoffman, BASS XIII, November 7,

58 Back to the Example Replace values below LOQ? Impute missing values? Then compute values of T for each subject and compute the sample mean and standard deviation. Or estimate all parameters with the MLE methods and Johnson & Hoffman, BASS XIII, November 7,

59 Johnson & Hoffman, BASS XIII, November 7, 2006 Johnson & Hoffman, BASS XIII, November 7, Back to the Example Back to the Example Use the delta method Use the delta method Estimate mean Estimate mean Estimate standard error Estimate standard error ( ) + = = = = = ˆ ˆ ˆ ˆ ˆ ˆ 2 ˆ ˆ ) var( μ μ σ σ μ μ σ μ i i i i i i i i i j i ji i j a a a a a b T = ˆ ˆ ˆ ˆ ˆ ˆ ˆ ) ( i i i i a i b T E μ μ σ μ σ μ μ

60 Comparison: based on simulations Multivariate MLE method is generally optimal with respect to bias and MSE assuming either a normal or log-normal distribution. Pseudo-MLE method closely matches the multivariate method with respect to bias and MLE. Pseudo-MLE is much (very much) more efficient with respect to computation resources and time. Johnson & Hoffman, BASS XIII, November 7,

61 Summary Replacement of LOQs is never needed. MLE methods provide for parametric estimation taking censoring and missing values into account. Pseudo-MLE performs better than replacement and essentially as well as the full MLE. When is replacement good enough? Johnson & Hoffman, BASS XIII, November 7,

62 Copy of Presentation Slides An updated and corrected copy of these presentation slides may be found at Johnson & Hoffman, BASS XIII, November 7,