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The Statistical Basis of Laboratory Data Normalization Juha Karvanen, DSc (Tech) Signal Processing Laboratory, Helsinki University of Technology, Helsinki, Finland
Key Words Normalization; Reference range; Reference limits; Location-scale family; Scale family Reprint Address Juha Karvanen, DSc (Tech), Signal Processing Laboratory, Helsinki University of Technology, P.O. Box 300, FIN–02015 HUT, Finland (e-mail:
[email protected]).
INTRODUCTION In clinical trials laboratory data are collected as a part of the assessment of the safety of the treatments. Abnormal assay values are indicators of possible toxicities. International Conference on Harmonisation (ICH) topic E9 (1) recommends two ways to analyze safety laboratory data from clinical trials. In the qualitative analysis individual measurements are compared with the reference limits and subjects who have abnormal values are listed. The reference limits are provided by the laboratory and they typically depend on age and gender. The identification of patients with abnormal laboratory results is easy to carry out and usually does not cause any statistical problems. ICH topic E9 also recommends comparison of means between treatments. This approach is usually much more problematic. The calculation of means demands that the measurements are commensurable. Problems arise at least in the following situations: 1. Several laboratories are used in the study, for example, every center has its own laboratory or the method of measurement is changed during the study, or 2. Several central laboratories are used in different
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The problem of laboratory data normalization in clinical trials is considered in the statistical framework. Leading principles for laboratory data normalization are proposed and practical normalization formulas are derived using as general assumptions as possible. The locationscale family and the scale family are used to model the assay value distributions. These assumptions lead to straightforward normalization formulas that have been proposed as ad hoc solutions. The preferred normalization is found to depend on the distribution of an assay value. We also consider the case where some normalized values become negative and propose that the negative values indicate that an unsuitable normalization is applied.
studies and we want to summarize safety over these studies.
The use of central laboratories has become a common practice and thus case 1 is not so usual nowadays. Conversely, case 2 occurs in almost every clinical program. If the clinical program is global, it is unrealistic to assume that the same central laboratory could be employed in all studies. Laboratories use different methods and thus the results are not directly comparable. Overall results are, however, needed for the safety summaries. If the results from different laboratories are summarized directly calculating the mean, it is obvious that variance is artificially increased and small differences between treatments are more likely to be interpreted as random noise. Normalization means that the assay values from different laboratories are transformed in such a way that they are directly comparable. For normalization we must choose a standard laboratory that can be either a real laboratory or a theoretical convention. After normalization, the assay values may be treated as if they were obtained from the standard laboratory. In practice, we have limited information for the normalization: in addition to the data, usually only reference ranges are available. It might be appealing to utilize the data in normalization, but
Drug Information Journal, Vol. 37, pp. 101–107, 2003 • 0092-8615/2003 Printed in the USA. All rights reserved. Copyright © 2003 Drug Information Association, Inc.
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in clinical trials this is not acceptable because we have no guarantee that the populations in the different studies are homogeneous. Despite the practical importance of laboratory data normalization, the topic has not been popular in literature. Two papers by ChuangStein (2,3) introduce the problem and propose practical formulas for normalization. The notation in this paper is similar to the notation used in by Chuang-Stein (3). After submitting the first version of this paper, Huang and Brunelle proposed a nonparametric method for laboratory data normalization (4). Their starting point is quite similar to this paper but the conclusions drawn are different. They propose that laboratory data from different studies should be normalized by matching the empirical cumulative distribution functions of the samples. The key assumption is that two samples are random samples from the same population. This assumption is unrealistic in many clinical programs where clinical trials are conducted in different countries. The studies may have different inclusion/exclusion criteria and the ratio of active and placebo patients may differ. Thus, normalization based on empirical distributions may mask significant differences between the studies. The reference ranges have their drawbacks but their undisputable advantage is that they come from the central laboratory, not from the statistician. In this paper, the normalization problem is considered in the statistical framework. We start from general principles and proceed to practical normalization formulas. We propose two normalization formulas and suggest guidelines for their use. The cases where some normalized values are negative can be avoided using an appropriate normalization formula. This is illustrated in the fourth section.
S TAT I S T I C A L M O D E L S F O R L A B O R AT O RY M E A S U R E M E N T S Let X represent an assay value from a certain laboratory. In the population, random variable X has a distribution characterized by cumulative distribution function (cdf) FX. Further, let S be an assay value from the standard laboratory with
the population cdf FS. The normalization problem can now be described as follows: Assume that the same sample can be analyzed in both laboratories. What is the expected value of S on the condition X = x? In other words, we want to estimate the assay value of the standard laboratory from the observed assay value. As a solution, we propose such s that FX(x) = FS(s).
(1)
In practice, this condition means that if a collection of blood samples is analyzed in two laboratories the assay values may differ but their order is the same in both laboratories. Even though this condition may be false in some cases, the condition itself is very desirable and related to the quality of the laboratories. To proceed, some assumptions on FX and FS are needed. On the other hand, we try to keep the presentation as general as possible. Two models are proposed: 1. FX and FS belong to the same location-scale family, and 2. FX and FS belong to the same scale family (but they do not belong to the same location family).
Location-scale family is a family of distributions characterized by a location parameter µ and a scale parameter σ (not necessarily mean and standard deviation). For instance, the following distributions are location-scale families: Normal distribution with mean µ and standard deviation σ f ( x, µ , σ ) =
1
σ 2π
e
−
( x − µ )2 2σ 2
,
(2)
logistic distribution with mean µ and scaling parameter σ f ( x; µ , σ ) =
1 σ
e
−
(1 + e
x−µ σ
−
x−µ σ
)
2
,
(3)
and Laplace distribution with mean µ and scaling parameter σ |x − µ|
f ( x; µ , σ ) =
1 − e 2σ
σ
.
(4)
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The Statistical Basis of Laboratory Data Normalization
If FX and FS belong to the same location-scale family, they may be expressed as follows v − µS FS ( v ) = F( v; µ S , σ S ) = F ; 0, 1 , σ S v − µX FX ( v ) = F( v; µ X , σ X ) = F ; 0, 1 . σX
A scale family is a family of distributions characterized by a scale parameter λ. For instance, the following distributions are scale families: exponential distribution f ( x; λ ) = λ −1e
−x λ
, x ≥ 0,
(6)
gamma distribution with scaling parameter λ and fixed shape parameter α f ( x; α , λ ) =
1 α −1 −α − λx x λ e , x ≥ 0, Γ (α )
where Γ() denotes gamma function and Weibull distribution with scaling parameter λ and fixed shape parameter α f ( x; α , λ ) = xα −1λ −α e
(5)
(7)
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(− λ ) x
α
, x ≥ 0.
(8)
Note that these are asymmetric distributions that can have only positive values. If FX and FS belong to the same scale family they may be expressed as follows v FS ( v ) = F( v; λ S ) = F ; 1 , λS v FX ( v ) = F( v; λ X ) = F ; 1 . λX
(9)
An illustration of the concepts of location-scale family and scale family is presented in Figure 1. An examination of typical laboratory data reveals that the population distributions of the as-
FIGURE 1 An illustration of the concepts of location-scale family and scale family.
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say values are different. Most variables can be modeled using a symmetric unimodal distribution. For example, hemoglobin measurements seem to be approximately normally distributed when males and females are considered separately. The assumption of the location-scale family is suitable for these distributions. In clinical chemistry, however, the distributions are clearly skewed. Typically, most of the values are relatively close to zero while there are a few very large values. The assumption of the scale family is suitable for these distributions.
N O R M A L I Z AT I O N B A S E D O N REFERENCE RANGES Now we establish a connection between the reference ranges and the statistical models from the previous section. Let assay values X and S have reference ranges (LX, UX) and (LS, US), respectively. This is usually the only information available from distributions FX and FS. If (LX, UX) and (LS, Us) are determined using the same rule, assumption (1) leads to the following equations FX (LX) = FS (LS),
(10)
FX (UX) = FS (US).
(11)
THE LOCATION-SCALE MODEL If the location-scale model (5) is assumed for FX and FS, their relationship may be expressed as follows x − µX F ( x; µ X , σ X ) = F µ S + σ S ; µ S , σ S . (12) σX
F ( x; µ X , σ X ) U − LS = F LS + ( x − LX ) S ; µS , σ S , U X − LX
leading to the location-scale normalization formula s = LS + ( x − LX )
(16)
SCALE MODEL If the scale model (9) is assumed for FX and FS their relationship may be expressed as follows λ F ( x; λ X ) = F x S ; λ S . λX
(17)
From assumptions (10) and (11) it follows F( LX ; λ X ) = F( LS ; λ S ) F(U X ; λ X ) = F(U S ; λ S )
(18)
λS LX λ = LS X λS U X = US . λX
(19)
(13)
and further
and further LX − µX µ S + σ S = LS σX µ + U X − µX σ = U . S S σ X S
U S − LS . U X − LX
This is exactly the normalization formula proposed by Chuang-Stein (2). We have now shown that under the general assumption of a location-scale family this normalization is statistically valid. As discussed in the end of the second section, the location-scale assumption is suitable for the majority of typical laboratory variables. However, problems may occur if normalization (16) is applied for variables that do not follow the location-scale model. Such a problem is the case where some normalized values become negative. This problem is also discussed by Chuang-Stein (3) and an ad hoc solution is proposed. We propose that negative values indicate that an unsuitable normalization is applied. The problem of negative values can be avoided using the normalization based on the scale model.
From assumptions (10) and (11) it follows F( LX ; µ X , σ X ) = F( LS ; µ S , σ S ) F(U X ; µ X , σ X ) = F(U S ; µ S , σ S ),
(15)
(14)
Substituting (14) into equation (12) we obtain
The normalization can be based on either the upper reference limits or the lower reference limits. We choose the upper reference limits be-
The Statistical Basis of Laboratory Data Normalization
cause they have more clinical relevance in the cases where the scale model is appropriate, for example, in clinical chemistry. We obtain U F ( x; λ X ) = F x S ; λ S , UX
(20)
leading to the scale normalization formula s= x
US . UX
(21)
This normalization is also used by SoglieroGilbert, Mosher, and Zubkoff (5) in constructing Genie scores.
EXAMPLES The importance of the correct normalization (location-scale normalization versus scale normalization) is demonstrated in examples. In the
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first example, the examined laboratory variable is bilirubin, which typically has an asymmetric distribution with a heavy tail. Central laboratory A has a reference range of 2 to 20 µmol/l and central laboratory B has a reference range of 4 to 22 µmol/l. In the data from laboratory A, the sample minimum is 1 µmol/l and the sample maximum is 35 µmol/l. In the data from laboratory B, these values are 1 µmol/l and 38 µmol/l, respectively. Both location-scale normalization and scale normalization are illustrated in Figure 2. It can be seen that in the location-scale normalization some normalized values are negative. This problem is avoided in the scale normalization, which is the preferred type of normalization for bilirubin. In the second example, hemoglobin is considered. Since the reference ranges for hemoglobin are different for males and females, the normal-
FIGURE 2 Normalization of bilirubin values. The values from laboratory B are normalized to correspond with the values from laboratory A. The original data ranges as well as the data range for the normalization of B → A are presented. The lines present the range of data. The upper and lower reference limits are marked by ‘x.’ Location-scale normalization is not suitable for bilirubin because it makes some normalized values negative. This problem is avoided in scale normalization.
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B
FIGURE 3 Normalization of hemoglobin values. The values from laboratory B are normalized to correspond with the values from laboratory A. The original data ranges as well as the data range for the normalization of B → A are presented. The lines present the range of data. The upper and lower reference limits are marked by ‘x.’ Location-scale normalization is preferable for hemoglobin because the distribution in a population (separately for males and females) is approximately normal.
A
B
A
110
120
130
140
150 160 Hemoglobin g/l
170
180
190
180
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(a) Location-scale normalization
B
A
B
A
110
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140 150 160 Hemoglobin g/l
170
(b) Scale normalization
ization must be carried out separately for each gender. Central laboratory A has a reference range of 135 to 180 g/l for males. For central laboratory B, the reference range is 130 to 170 g/l. In the data (males only) from laboratory A, the sample minimum is 121 g/l and the sample maximum is 184 g/l. In the data from laboratory B, these values are 117 g/l and 176 g/l, respectively. Again, both location-scale normalization and scale normalization are illustrated in Figure 3. The difference between the normalizations can be seen comparing the normalization of the sample minimum. In scale normalization the distance between the lower reference limit and the sample minimum is shrunken. In locationscale normalization the effect of normalization is similar for both low and high values. Locationscale normalization is preferable because the
assay values of hemoglobin are approximately normally distributed.
CONCLUSION Practical normalization formulas were derived starting from general statistical assumptions. The assumption on the locationscale family leads to normalization where both the upper and the lower reference limit are utilized. The assumption on the scale family leads to normalization where only the upper reference limit is utilized. From the theoretical point of view the results justify the use of the normalization formulas that have been proposed earlier. From the practical point of view the main result is that the preferred normalization formula depends on the laboratory variable. Location-scale normalization
The Statistical Basis of Laboratory Data Normalization
is suitable for most variables but scale normalization is preferable especially in clinical chemistry. The results suggest that normalization should not be a blind procedure. A qualified statistician working with laboratory data knows his or her variables. The laboratory variables are “individual” with different statistical properties and clinical interpretations. Consequently, they also require different normalization formulas.
REFERENCES 1.
European Agency for the Evaluation of Medical Products. ICH Topic E9, Statistical Principles for Clinical Trials. London, United Kingdom: Euro-
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3.
4.
5.
pean Agency for the Evaluation of Medical Products; 1998. Chuang-Stein C. Summarizing laboratory data with different reference ranges in multi-center clinical trials. Drug Inf J. 1992;26(1):77–84. Chuang-Stein C. Some issues concerning the normalization of laboratory data based on reference ranges. Drug Inf J. 2001;35(1):153–156. Huang J, Brunelle R. A nonparametric method for combining multilaboratory data. Drug Inf J. 2002;36(2):395–406. Sogliero-Gilbert G, Mosher K, Zubkoff L. A procedure for the simplification and assesment of lab parameters in clinical trials. Drug Inf J. 1986; 20:279–296.
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