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1
Department of Clinical Chemistry, Rikshospitalet University Hospital of Oslo, N-0027 Oslo, Norway.
2
Department of Clinical Biochemistry, Odense University Hospital, 5000 Odense C, Denmark.
3
NOKLUS, Norwegian Centre for External Quality Assurance of Primary Care Laboratories, Division for General Practice, University of Bergen, N-5020 Bergen, Norway.
4
Department of Pathology, University of Virginia Health System, Charlottesville, VA 22908.
5
Biochemical Medicine, Tayside University Hospitals NHS Trust, Ninewells Hospital and Medical School, Dundee DD1 9SY, Scotland.
6
Department of Clinical Biochemistry, Sønderborg Hospital, 6400 Sønderborg, Denmark.
aAuthor for correspondence. Fax 47-2307-1080; e-mail ari.lahti{at}rikshospitalet.no.
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Abstract |
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 Top Abstract IntroductionTheoretical BackgroundApplications to Published DataDiscussionconclusionReferences |
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Methods: The proposed criteria relate to percentages of the subgroups outside each of the reference limits of the combined distribution. Critical values suggested as partitioning criteria for these percentages were derived from analytical bias quality specifications for using common reference intervals throughout a geographic area. As alternative partitioning criteria to the actual percentages, these were transformed mathematically to critical distances between the reference limits of the subgroup distributions, to be applied to each pair of reference limits, the upper and the lower, at a time. The new criteria were tested using data on various plasma proteins collected from 
500 reference individuals, and the outcomes were compared with those given by the currently widely applied and recommended partitioning model of Harris and Boyd, the "Harris-Boyd model".
Results: We suggest 4.1% as the critical minimum percentage outside that would justify partitioning into subgroups, and 3.2% as the critical maximum percentage outside that would justify combining them. Percentages between these two values should be classified as marginal, implying that nonstatistical considerations are required to make the final decision on partitioning. The correlation between the critical percentages and the critical distances was mathematically precise in the new model, whereas this correlation is rather approximate in the Harris-Boyd model because focus on the difference between means in this model makes high precision hard to achieve. The application examples suggested that the new model is more radical than the Harris-Boyd model.
Conclusions: New percentage and distance criteria, to be used for partitioning gaussian-distributed data, have been developed. The distance criteria, applied separately to both reference limit pairs of the subgroup distributions, seemed more reliable and correlated more accurately with the critical percentages than the distance criteria of the Harris-Boyd model. As opposed to the Harris-Boyd model, the new model is easily adjustable to new critical values of the percentages, should they need to be changed in the future.
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Introduction |
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TopAbstractIntroductionTheoretical BackgroundApplications to Published DataDiscussionconclusionReferences |
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It might be that the concept of producing reference intervals in every laboratory for every quantity is outdated. The newer idea of establishing common reference intervals for several laboratories that serve a homogeneous population throughout a geographic area is gaining acceptance, both as a theoretical concept and as a practical approach. This idea has been applied to specific plasma proteins in the Nordic countries (8) and to 15 clinical biochemical quantities in Spain (9). Furthermore, a new Nordic project on common reference intervals aims to generate these for 25 clinical biochemical quantities (10). A necessary prerequisite for establishing such common reference intervals is common standardization and common quality control. Every laboratory that takes part in the production of the common reference intervals and wishes to use them must fulfill the required quality specifications (11)(12). Under these conditions, strictly controlled over time and geography to guarantee transferability of the reference intervals, it is possible to produce well-documented common reference intervals based on as many as thousands of reference individuals (8)(9)(10)(11)(12)(13).
Large sample sizes enable partitioning of reference individuals into subgroups according to demographic descriptors, such as gender, age, or ethnic background. Because each subgroup must have a sufficient number of reference individuals to produce high-quality reference intervals alone, small-scale reference interval studies performed by single laboratories seldom give satisfactory scientific bases for proper subgroup reference intervals. In contrast, large-scale studies make partitioning into subgroups feasible and useful, as an additional advantage. The present trend to large-scale studies therefore greatly increases the need for simple and reliable partitioning criteria.
A summary of the current guidelines for partitioning reference values into subgroups is presented in the monograph of Harris and Boyd (14) on the statistical bases of reference values in laboratory medicine. The most often applied approaches at present are those of Sinton et al. (15) and Harris and Boyd (16). According to the partitioning criterion of Sinton et al. (15), the difference between subgroup means should exceed 25% of the reference range calculated for the combined distribution if the subgroups are to be considered separately. The approach of Harris and Boyd (16), called the Harris-Boyd model in this report, is a more sophisticated one, involving the ratio between the subgroup standard deviations, a normal deviate test for comparison of the subgroup means, and calculation of critical decision values dependent on the sample size. Because the approach of Sinton et al. (15) can be described in terms of the Harris-Boyd model (16), we will not consider it separately in this study. An example of the application of the Harris-Boyd model is presented by Harris et al. (17).
The partitioning criteria of the Harris-Boyd model are based on considering the percentages of healthy individuals who would be cut off the subgroup populations by the reference limits of the combined distribution. However, at the time the model was constructed, there was no consensus on what these percentages ought to be to justify partitioning. Attempts to define analytical bias quality specifications for common reference intervals have also focused on percentages of the population outside reference limits (18)(19). A recent approach to the setting of analytical bias quality specifications classifies these into three categories, namely optimum, desirable, and minimum quality, based on different percentages (20). The purpose of the present study is to develop new partitioning criteria using these percentages as a starting point.
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Theoretical Background |
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TopAbstractIntroductionTheoretical BackgroundApplications to Published DataDiscussionconclusionReferences |
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calculation methods
The probability density of a gaussian distribution with mean µ and standard deviation 
for a random variable X (X is the name of the variable, and x values are values adopted by it) is as follows:
 | (1) |
Integrals of f(x), usually denoted as 
(x), are probabilities [P (X 
x)], and the values given by the inverse function -1[P (X x)] are measured as signed distances in standard deviation units, or probits, from the mean. We used the NORMINV function of the Microsoft® Excel 97 program to calculate the probit values.
The limits of the confidence interval (CI) about a percentile of a gaussian distribution are calculated according to IFCC (5) as:
 | (2) |
where m is the sample mean, s is the sample standard deviation, z1 is the probit value corresponding to the percentile, z2 is a covering factor corresponding to the desired confidence level, and n is the sample size. As an example, the limits of the 90% CI around the 97.5 percentile would be m + {1.96 s ± 1.64·[s2/n + (1.962 s2)/(2n)]1/2} 
m + (1.96 s ± 0.25 s), when n = 120 (the accurate value is 0.26 s).
percentages of a gaussian distribution outside the reference limits because of bias (bias case)
By definition, 2.5% of the reference values of a reference distribution lie outside each of its two reference limits because the reference interval is conventionally defined as the central 95% of the population (5). If bias exists, the distribution will move to higher (positive bias) or lower (negative bias) values. Irrespective of the type of bias, >2.5% of the individuals will be outside one limit and <2.5% will be outside the other. However, because of the shape of the distribution, the total percentage outside the limits will always exceed 5.0% in the presence of bias. As an example, the effect of a bias of ± 0.25 s on the percentages outside the reference limits of a gaussian distribution is illustrated in Fig. 1A
. If the distribution is displaced 0.25 s in either direction from its original position, the result will be that 1.4% of the individuals will lie outside one of the original reference limits (smaller areas outside the reference limits in Fig. 1A
) and 4.4% will lie outside the other limit (sums of the areas outside the reference limits in Fig. 1A
), i.e., a total of 5.7% (the sum of 1.4% and 4.4% in fact equals 5.7%, because of rounding errors) will lie outside instead of 5.0%.
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The original IFCC recommendation stated that, ideally, a minimum of 120 reference individuals should be used in reference interval calculations (5). This number is supposed to make even the nonparametric 90% CIs of the reference limits calculable. As was shown above, the 90% confidence limits around the 97.5 percentile of a gaussian distribution lie at 0.25 s on both sides of this percentile when n = 120, and the same is true for the 2.5 percentile. In other words, the choice of n = 120 as the minimum sample size means that the reference limits are allowed to have an uncertainty corresponding to a bias within the interval [-0.25 s, 0.25 s] of the distribution itself.
In the original work on setting analytical quality specifications for bias (18), the desirable analytical performance that would allow the sharing of common reference intervals was based on this IFCC recommendation. Accordingly, the desirable level of analytical bias was defined as corresponding to the 90% CIs of the reference limits for a sample size of 120, or as not exceeding ± 0.25 s. More recent European recommendations support this definition (30)(31). The increase in the percentage of individuals classified as abnormal because of a bias of ± 0.25 s is considered tolerable because 5.7% instead of 5.0% of the population outside the reference limits means 0.7% more, or a proportional increase of 14%. This is analogous to the postulate of Cotlove et al. (32) for monitoring of patients that an increase in the total CV of the order of 10% compared with the within-subject biological variation will not significantly affect clinical decision-making on observed changes within an individual.
In a recent work (20), it was proposed that analytical bias quality could be classified in a hierarchy of three levels, namely, optimum, desirable, and minimum quality. As stated above, to be classified as desirable, the analytical bias should not exceed ± 0.25 s. The optimum quality is considered to be attained when the bias lies within the interval [-0.125 s, 0.125 s]. The percentages outside the reference limits obtained at both end points of this interval are 1.8% and 3.3%, which means that the maximum value for the total percentage outside is 5.1% for this quality specification. Bias of the minimum quality is defined as not exceeding ± 0.375 s. The extreme percentages outside the reference limits would in this case be 0.9% and 5.7%, giving a maximum total percentage of 6.6%. The additional number of individuals classified incorrectly is at most 2% if optimum analytical bias quality is attained, but it may reach 33% if only minimum quality is achieved. The values 0.125 for optimum and 0.375 for minimum are simply 0.5 and 1.5 times the desirable value (0.25), which in turn refers to the IFCC criterion (5). These factors are admittedly somewhat arbitrary, but they seem reasonable and are described by Fraser et al. (20).
The percentages outside the reference limits are plotted as a function of bias in Fig. 2A
. To construct the curves depicted in Fig. 2A
, bias in only one direction needs to be considered because the percentages are equal for both positive and negative bias; only the order of the increasing and the decreasing percentage would be changed at each point of the bias variable. As noted above, the sum of the percentages, shown by the uppermost curve, is always 
5.0%. The increasing and decreasing percentages have 100% and 0% as asymptotes, respectively. These asymptotic percentages will never be reached because the probability density of a gaussian distribution, or f(x) in Eq. 1
, remains above zero at any real value of x. The analytical bias quality specifications, described above, are also indicated in Fig. 2A
.
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percentages of two gaussian distributions outside the reference limits of the combined distribution (partitioning case)
The distributions of two subgroups selected from a reference population sample are often similar but usually not identical. This makes estimation of the percentages outside the reference limits more complicated than it is in the above bias case, where one distribution is simply displaced with respect to itself. However, to make the distinction from the bias case clear, first consider two gaussian subgroup distributions, denoted a and b, that are identical except for different means (Fig. 1B
). Observe that the combined distribution is not a strictly gaussian distribution. We are interested in the percentages (pa and pb) of the subgroup distributions outside the reference limits of the combined distribution as a function of the distance (D) between the subgroup reference limits at each end of the distributions. This distance and these percentages are equal at both ends, or pa at one end equals pb at the other end and vice versa, because the distributions are assumed to be identical. Hence, we can restrict our preliminary analysis to the lower end of the distributions, as illustrated in Fig. 1B
.
Because the lower common reference limit is the 2.5 percentile of the combined distribution, the percentages pa and pb cannot exceed 5.0% of the respective subgroup distributions, and their sum is exactly 5.0%. Fig. 2B
shows the behavior of pa and pb as a function of D, measured in s, the standard deviation of both of the subgroup distributions (in general, we will use the standard deviation of the narrower distribution as a scale unit, but here the distributions have equal standard deviations). As D is increased from 0, the situation where the distributions cover each other, toward infinity, pa, or the increasing percentage, and pb, or the decreasing percentage, start changing from a common value of 2.5% toward their respective asymptotes, 5.0% and 0%.
Panels A and B in Fig. 2
are combined as Fig. 2C
to illustrate the proportions of the percentages outside the reference limits in the bias case (Fig. 1A
) and the partitioning case (Fig. 1B
), as compared with each other. Observe that the curves of Fig. 2A
lie outside of the curves of Fig. 2B
because the effective displacement of the distribution, used to calculate the percentages, is smaller in the partitioning case.
partitioning criteria of the harris-boyd model
The partitioning criteria presented by Harris and Boyd (14)(16) are based both on the ratio between the standard deviations of the subgroups (R) and on a normal deviate testing of the significance of the difference between the means. Because the sample sizes have an effect on the outcome of such tests, Harris and Boyd corrected their decision values for them, as shown below.
Using computer simulations, Harris and Boyd showed that, if R >1.5, the proportion of the wider distribution outside one of the common reference limits exceeds 4.0% regardless of the difference between the subgroup means. They considered this a definite criterion for partitioning. Otherwise, they used the normal deviate test:
 | (3) |
where µi, i, and ni are the mean, the standard deviation, and the size of subgroup i. Initially Harris and Boyd (16) proposed a critical z value of 3, adjusted for the sample sizes, as a decision threshold for partitioning:
 | (4) |
where n is the average of the subgroup sizes (17). If the z value, as calculated from Eq. 3
, exceeded zCrit3, Harris and Boyd recommended partitioning. However, this original criterion turned out to be quite permissive of separate subgroups, and later Harris and Boyd (14) suggested that it could be changed to a value of 5:
 | (5) |
The choice of the critical decision value in the Harris-Boyd model is obviously to some extent arbitrary, a fact that the authors are well aware of (14). We will later discuss the reasons for this latitude of choice.
In applying the Harris-Boyd model in this study, we will use the following three-stage classification:
Partitioning is not recommended (indicated as "nonpartitioning" in Tables 1
and 2
in this report). z < zCrit5 Decision on partitioning must be made using other than simple statistical considerations ("Marginal"). zCrit5 Partitioning is recommended ("Partitioning").
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This classification was not presented by Harris and Boyd themselves, but it seems to be a legitimate derivative of their partitioning criteria, keeping in mind the uncertainty concerning the appropriate critical value of their test parameter.
new model for partitioning criteria
Proposals for partitioning criteria as expressed in percentages of the subgroup distributions outside the common reference limits.
Although the partitioning criteria of the Harris-Boyd model were based on consideration of the percentages outside the common reference limits, Harris and Boyd did not use these percentages explicitly. They have clearly considered 4.0% as being quite high because they recommended R >1.5 as a separate partitioning criterion (see the discussion above), but otherwise they did not give any precise recommendations on these percentages, notwithstanding that their computer simulations aimed specifically at estimation of them. There are two reasons for this. The first reason is that they focused on the distance between the subgroup means, but at a certain distance, the percentages outside the common limits varied considerably at each end of the distributions and between the ends, as R was varied [Figs. 1
and 2
in Ref. (16)]. It was simply not possible to obtain a consistent idea of appropriate percentages as a function of the distance and postulate a certain distance as the critical one, corresponding to certain percentages. The second reason is that at that time (1990), there was no consensus in the clinical chemistry community on what were to be considered as appropriate critical percentages.
We cannot perhaps speak of such consensus today either, but the analytical bias quality specifications described above can give some basis for assessing what percentages might be acceptable as partitioning criteria. As Fig. 2
shows, the bias quality specifications cannot be adopted directly as partitioning criteria because a single percentage may exceed 5.0% in the bias case (Fig. 2A
), whereas 5.0% is the theoretical maximum for a percentage in the partitioning case (Fig. 2B
). However, we might adopt the lower percentage of the minimum bias quality specification, or 0.9% (Fig. 2A
), as the minimum allowed lower percentage for the partitioning case, not implying partitioning. The maximum allowed upper percentage would then be 4.1% because these two percentages sum up to 5.0%. This percentage, 4.1%, derived from the minimum analytical bias quality specification is our suggestion for a critical partitioning percentage, or pCritP. Observe that this percentage is almost equal to 4.0%, which Harris and Boyd regarded as a definite partitioning criterion.
Although the "complementary" percentage of 4.1%, or 0.9%, is also a critical percentage for partitioning, we will not have to consider it separately because, when needed, it is calculable as 5.0% minus 4.1%. Still, it is important to keep in mind that there are actually two percentages to observe. When speaking of 4.1% as a partitioning criterion, we mean indeed that 4.1% of one of the subgroup distributions and 0.9% of the other subgroup distribution then lie outside one of the common reference limits (Fig. 1B
). These percentages at the other end of the distributions will be different from 4.1% and 0.9% if the distributions have different standard deviations.
Similar to the way in which we derived the critical percentage for partitioning from the minimum analytical bias quality specification, we suggest using the optimum bias quality specification, corresponding to 3.3% and 1.8% (Fig. 2A
), as a basis for a nonpartitioning criterion. Of the two potential pairs of percentages, 3.3% and 1.7% on the one hand and 3.2% and 1.8% on the other hand, we choose the latter, because it is slightly more conservative, as our suggestion for a critical nonpartitioning percentage, or pCritNP. Consequently, we can preliminarily summarize our proposals for partitioning criteria as expressed in percentages (p) as follows (p refers to the larger of the percentages outside the common reference limit, i.e., to pa in Fig. 1B
):
Partitioning is not recommended p <4.1% Partitioning is possibly justified 4.1% Partitioning is recommended
If a marginal value, or 3.2% p < 4.1%, for partitioning is obtained, physiologic considerations, clinical judgment, and data from the literature have a central role in making the final decision on partitioning.
Proposals for partitioning criteria as expressed in distances between the reference limits of the subgroup distributions.
The computer programs used to calculate reference intervals do not ordinarily give the percentages of the subgroup distributions outside the common reference limits. To make partitioning easy in practice, it would be desirable to translate these percentages to distances between the distributions. Although we derived our critical percentages from the analytical bias quality specifications, we cannot use these biases as critical distances because they differ from the distances D of the partitioning case at any value of the critical percentage (see Fig. 2C
). Moreover, the percentage curves of Fig. 2B
are not generally valid because they were based on the special case of R = 1.
Harris and Boyd (16) measured the distance between subgroup distributions as a scaled difference between the subgroup means, but this approach gave a rather imprecise correlation between the distances and the percentages. We will show below that if the distance is, in contrast, measured as a difference between the reference limits of the subgroup distributions and that if each pair of reference limits, the lower and the upper one, is considered separately, the distances can be determined more accurately as correlates of the percentages. This precision is in principle perfect for two subgroup distributions with a particular ratio of standard deviations (R), but because we wish to develop easily applicable rule-of-thumb critical distances that cover various values of R simultaneously, there will be some more dispersion in practice.
To account for different standard deviations, we have to calculate percentage-vs-distance curves, similar to the upper curve of Fig. 2B
(we will omit the lower curves because the lower percentages are equal to 5.0% minus the upper ones at each point), for various pairs of subgroup distributions that have different ratios of standard deviation (R). According to Harris and Boyd (14), the ratio between the standard deviations of two subgroup distributions seldom exceeds 1.3 in clinical chemistry. We are also inclined to accept a value exceeding 1.5 of R as a separate partitioning criterion, as suggested by Harris and Boyd (16). Hence, we will consider R values between unity and 1.5. From this interval, Harris and Boyd used R values of 1.0, 1.3, and 1.5 in their computer simulations. We will use the same values to make visual comparison of the two approaches easier. Accordingly, Fig. 3
shows the percentage (pa)-vs-distance curves, illustrating the partitioning case depicted in Fig. 1B
as the ratio of the standard deviation of the displaced distribution (b) to the standard deviation of the fixed distribution (a) adopted values 1.0, 1.3, and 1.5. The fixed distribution is thus a standard gaussian distribution, and the displaced distribution is identical to it or broader than it. The abscissa is given in standard deviations of the narrower distribution, or the fixed one.
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The percentage-vs-distance curves depicted in Fig. 3
are based on calculations that are detailed in the Appendix
.
Hence, they are mathematically precise, as opposed to the curves of Harris and Boyd [Figs. 1
and 2
in Ref. (16)], which are based on computer simulations.
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As we try to translate our proposals for the critical percentages, 3.2% and 4.1%, to distances, compromises are inevitable, because of the dispersion of the percentage-vs-distance curves, if we wish to establish simple rule-of-thumb partitioning distances that cover the relevant range of R in its entirety. However, the task is easier for us than it was for Harris and Boyd because the dispersion of our curves is considerably smaller than the dispersion of the curves that they obtained (16). Moreover, for us it is enough to consider one end of the distributions at a time, whereas Harris and Boyd had to consider both ends simultaneously. It is obvious that we can expect to attain a correlation between critical percentages and critical distances that is more accurate than that given by the Harris-Boyd model. We need, of course, the outcome from both ends before we can make the final decision on partitioning, but the same rule-of-thumb partitioning criteria can in fact be used at both ends, irrespective of the topographic order of two subgroup distributions having different standard deviations. This is shown in the Appendix
.
As confirmed by Fig. 3
, 4.1% translates to a distance D = 0.63 s at R = 1.0 and to D = 0.83 s at R = 1.5 (these distances are actual calculation results and not estimates from Fig. 3
). However, the line D = 0.63 s cuts the curve R = 1.5 at approximately pa = 3.8%. This means that if D = 0.63 s were selected as partitioning criterion, this criterion expressed in percentages would be relaxed from 4.1% to 3.8% if one of the distributions had a 50% larger standard deviation than the other one, as compared with the case of distributions with RIDentical standard deviations. This would probably lead to partitioning that was unnecessary. We feel that the critical distance for partitioning should rather be too stringent than too permissive, and we would therefore prefer looking for a solution at the other end the scale, opened up by the percentage criterion of 4.1%. The line D = 0.83 s cuts the curve R = 1.0 at approximately pa = 4.4%. If 4.4% were accepted, we would have to accept its counterpart, 0.6%, as well, but the counterpart of 0.6% in terms of analytical bias quality specifications would be close to 8.0% outside the limit. Perhaps D = 0.83 s is too stringent, inasmuch as D = 0.63 s is too permissive. As a compromise, the mean of these two distances, or D = 0.73 s, would be a natural choice of course. Because 0.73 s is hard to remember, we suggest D = 0.75 s, which translates to percentages between 4.0% and 4.3%, as the critical distance for partitioning, or DCritP. In other words, using this distance between the reference limits of one end of the subgroup distributions as a criterion, we will have at least 4.0% of one of the distributions outside the common reference limit of that end, irrespective of R 1.5. At R = 1.5, the separate partitioning criterion of Harris and Boyd takes over, guaranteeing that the percentage of the wider distribution beyond one of the common reference limits will exceed 4.0%, irrespective of further increases of R. Observe that we do not have to worry about whether this might happen at lower R values at one end of the distributions while we are considering the other end because a separate analysis will be performed at the other end before final decision on partitioning is made.
The proposed pCritNP, or 3.2%, translates to a distance D = 0.24 s at R = 1.0, and to D = 0.31 s at R = 1.5. The average of these distances would be 0.28 s, but we prefer rounding it slightly downward to obtain a counterpart for DCritP = 0.75 s, which is easy to remember, and suggest D = 0.25 s as a critical distance for nonpartitioning, or DCritNP. This value translates to percentages between 3.1% and 3.2%, as R lies between unity and 1.5.
To make the final decision on partitioning, one has to apply the partitioning criteria, defined above, separately to the lower and the upper reference limit pair of the subgroup distributions. If at least one of the distances between reference limits exceeds DCritP, partitioning is recommended because at least one of the distributions then has a minimum of 4.0% outside one of the common reference limits. On the other hand, a reasonable prerequisite for a recommendation on not to partition seems to be that both distances lie below DCritNP. If, in turn, at least one of the distances is marginal, i.e., the value of the distance lies between DCritNP and DCritP, and neither of them exceeds the value of DCritP, the decision should be made using nonstatistical judgment.
Our recommendations and the application rules for using our distance criteria for partitioning can be summarized as follows:
Partitioning is recommended 1.5 Calculate DL and DU, or the distances between the lower and the upper reference limits, respectively, of the subgroup distributions and use the standard deviation of the narrower subgroup distribution as a scale unit (s) Partitioning is not recommended ("Nonpartitioning") 0.75 s Decision on partitioning must be made using other than simple statistical considerations ("Marginal") 0.75 s Partitioning is recommended ("Partitioning") The nonstatistical considerations needed in the marginal cases might well include considerations such as the following:
It is important to remember that application of these guidelines presupposes that both subgroup distributions are gaussian or have been transformed to gaussian distributions using, e.g., logarithmic transformation.
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Applications to Published Data |
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TopAbstractIntroductionTheoretical BackgroundApplications to Published DataDiscussionconclusionReferences |
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orosomucoid
Three subgroups were previously RIDentified for this quantity (27)(29). The major group consisted of females >50 years of age and men of all ages (n = 269), and the two minor groups were females 50 years of age who either did not use estrogens for contraception (n = 142) or used estrogens (n = 108).
The first section of Table 1
("Data for subgroup distributions") includes two columns that give the reference limits of plasma orosomucoid for the respective two subgroups to be tested and two columns that give the logarithmic values of the data presented in the first two columns, together with means and standard deviations of the logarithmic values, as calculated from the original data. The subgroups are described in footnotes (n = number of individuals in the group). The next section (Partitioning criteria) includes a column, titled "D (s)", that gives the value of the test parameter of the present model or the distance between the respective reference limits measured in the smaller of the two standard deviations. As an example, the first D value, or 0.542, is calculated as |-0.916 - (-0.799)|/0.216. The critical range is 0.25–0.75 s for all subgroups, as shown in parentheses in the title of the column. The column "Conclusion for one end" gives the conclusion on partitioning for the end being examined of the distributions, and the final conclusion taking into consideration the test results from both ends is shown in the column titled "New model" of the section "Conclusions on partitioning". The normal deviate test statistic (z value) and its critical range (zCrit3–zCrit5), as defined in the Harris-Boyd model, are presented in the next two columns of the section "Partitioning criteria". They were calculated using Eqs. 3–5
. As an example, the first z value, or 4.43, is calculated as |-0.484 - (-0.361)|/(0.2192/142+0.2162/108)1/2, and the first critical range zCrit3–zCrit5, or 3.06–5.10, as 3·(125/120)1/2–5·(125/120)1/2, because the mean of 142 and 108 is 125. The conclusion on partitioning based on these values is presented in the column titled "Harris-Boyd model" of the section "Conclusions on partitioning".
Both models classify partitioning between young women who use estrogen and those not using estrogen as a marginal case. Young women who do not use estrogen show a definite difference from the main group for the lower reference limits but a marginal difference for the upper limits. The conclusion suggested by our model is, accordingly, that these two groups should be separated. The Harris-Boyd model, in contrast, considers also this case as marginal. The models agree once again, suggesting partitioning between the main group and the combined group of young women.
To make the final decisions on partitioning, the following considerations should be taken in account:
Thus, we recommend combining the two young female groups without partitioning and keeping this combined group partitioned from the major group.
IgG
Data on immunoglobulins and other plasma proteins are presented in Table 2
. The reference values for IgG consist of two groups, the major group (n = 269) and a group of females 50 years of age (n = 250). Because both our model and the Harris-Boyd model consider IgG a marginal case for partitioning, the decision must be based on nonstatistical considerations:
We do not recommend partitioning, in spite of the lower limit having potential as a decision limit.
IgA
The subgroups of IgA (Table 2
) consist of individuals (both men and women) of either >50 years (n = 132) or 50 years (n = 387) of age, respectively. The test outcomes give no justification for partitioning, except for a marginal value obtained for the upper reference limits by our model. We turn to nonstatistical considerations once again:
Overall, in spite of the supposed age dependence, we do not recommend partitioning.
IgM
The subgroups of IgM (Table 2
) are the same as those of IgG. Our model supports partitioning, whereas the Harris-Boyd model suggests that the case is marginal. We use the following considerations:
In consequence, we recommend separate reference intervals for women under 50 years of age and the rest of the population.
prealbumin
The subgroups are the main group (Table 2
; n = 377) and a group of females 50 years of age who do not use estrogen (n = 142). The differences between both the lower and the upper reference limits as well as the z value of the Harris-Boyd model suggest partitioning. This is in accordance with data in the literature (35) on a difference of 10% between the concentrations of prealbumin of these groups. Hence, we recommend partitioning.
albumin
The subgroups are the main group (Table 2
; n = 240), consisting of individuals >50 years of age and females 50 years of age who use estrogen, and a group of individuals 50 years of age (n = 279). The outcomes of all the tests suggest partitioning, in accordance with the data in the literature (35). However, this case might best be treated using a continuous, age-dependent regression model, which is outside the scope of this study.
1-antitrypsin
1-Antitrypsin (Table 2
) has several genetic subtypes. The most common type is the MM type with a frequency of >90%, whereas the MS type has a frequency of 6.5% in the Nordic countries (28). The MS type does not affect the health of its carriers but leads to slightly lowered plasma concentrations of 1-antitrypsin. As the test results in Table 2
suggest, the MM (n = 356) and the MS (n = 27) type carriers should have separate reference intervals. Because it is usually not possible to perform genetic typing of 1-antitrypsin in clinical routine work, partitioning may have little value in practice. The separate reference intervals should still be available to enable correct interpretation of plasma 1-antitrypsin concentrations measured in patients of known 1-antitrypsin genetic subtypes.
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Discussion |
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4.1% or 0.9% should imply partitioning, whereas all of these percentages should lie within the interval (1.8%, 3.2%) for the subgroups to be considered as one. The basis of the Harris-Boyd model (14)(16) lies on the percentages outside the common reference limits as well. However, Harris and Boyd focused on the distance between the means of the subgroup distributions, whereas we consider the distances between the reference limits of these distributions, one pair of reference limits at a time. The approach of Harris and Boyd was not very precise in terms of the percentages because, by controlling only the distance between the means, they were unable to simultaneously control both ends of the distributions, having different standard deviations. This led to considerable need of heuristics in defining the partitioning criteria, which is reflected by the radically different suggestions made by Harris and Boyd themselves for appropriate critical values of their test parameter.
Another factor that may have contributed to the inaccuracy of the approach of Harris and Boyd is that they were obliged to use computer simulations, rather than mathematic calculations, to construct the percentage-vs-distance curves that were needed for estimation of critical values for the test parameter. The reason that computer simulations were necessary in their approach was specifically their focus on the difference between the means. Simulations were needed to define the reference limits of the nongaussian combined distributions because otherwise the percentages outside these limits could not have been estimated at all in their approach. This is in sharp contrast to the present approach. We do not attempt to establish the common reference limits for the pairs of subgroup distributions being evaluated for partitioning, but we use these limits as variables. Focusing on one pair of reference limits at a time enables us to describe the correlation between distances and percentages in mathematic terms and to perform accurate calculations instead of using computer simulations. In our model, the control over the percentages outside the common reference limits is indeed mathematically precise for each particular value of R (Appendix
Eqs. 3
and 4 ). Moreover, when estimating the critical rule-of-thumb distances, the percentage-vs-distance curves, which describe the other end of the reference intervals, do not interfere with the estimation process at all, either explicitly or implicitly. The dispersion of our percentage-vs-distance curves is therefore considerably smaller than the dispersion of those of Harris and Boyd [compare Figs. 1
and 2
in Ref. (16) and our Fig. 3
], which makes selection of the critical distances that correspond to the desired critical percentages easier and more accurate in our model.
The critical distances suggested by us for general use, 0.25 s for nonpartitioning and 0.75 s for partitioning, are compromises meant to work reasonably well for any R 1.5. If one wishes to use more accurate values corresponding to a particular value of R, these can be calculated from Appendix
Eqs. 3
and 4
, or read approximately from Fig. 3
. As an example, if R = 1, the precise critical distances corresponding to the critical percentages of 3.2% and 4.1% would be 0.24 s and 0.63 s, respectively. Observe, however, that if R >1, the precise values depend on the topographic order of the distributions (see the Appendix
), and Fig. 3
is only approximately valid for the topography depicted in Appendix Fig. 1B
. Appendix Eq. 4 should be used if accurate critical distances for this topography are required.
It seems that the critical distances of our model are more reliable than those of the Harris-Boyd model. However, comparison of the reliabilities of the test parameters between the two models is not that straightforward. The ratio of the standard deviation of the 2.5 and the 97.5 percentiles to that of the mean of a gaussian distribution is 1.7, as can be calculated from Eq. 2
. Hence, the CI for a difference between one pair of reference limits of two gaussian distributions is 1.7 times as large as the CI for the difference between their means. However, because the test parameter of the Harris-Boyd model is not a difference between means but this difference divided by its standard deviation, the standard deviation of the test parameter itself is in fact constant and equal to unity irrespective of the value of z. The 90% CI for this test parameter consequently has a breadth of 3.28 units on the z scale, i.e., the 90% CI is 1.64 times as large as the critical range of z values from 3 to 5 [consider Figs. 1
and 2
in Ref. (16)]. The limits of the 90% CI for our test parameter can be calculated as follows, using Eq. 2
:
 | (6) |
As shown by Eq. 6
, the 90% CI for the test parameter of our model is not constant, but depends on the subgroup standard deviations and on the numbers of reference individuals in these subgroups. However, to get some RIDea of its size, let us calculate the worst-case value for it. In the worst case, we would apparently have R = 1.5 and n1 = n2 = 120 (the recommended minimum sample size). Inserting these values in Eq. 6
, we obtain 0.92 s, a value that is 1.84 times as large as the critical range of the difference D from 0.25 s to 0.75 s in our model (see Fig. 3
) and is in fact proportionally slightly larger than the 90% CI for the test parameter of the Harris-Boyd model compared with its critical range. But this was the worst case. With smaller R values and larger values of ni, the test parameter of our model will have a proportionally smaller CI than that of the Harris-Boyd model. As an example, if we assume equal standard deviations and n1 = n2 = 1000 (as more representative values in real-life large-scale studies), the 90% CI of our test parameter, as calculated from Eq. 6
, would be 0.25 s, i.e., one-half of the critical range. This factor (0.5) is obviously far more reasonable than the constant factor, 1.64, valid for any R and ni, calculated above for the Harris-Boyd model.
We conclude that the imprecision of the test parameter of the Harris-Boyd model is slightly lower than that of our model for small sample sizes, provided that the distributions also have different standard deviations (as an example, R should exceed 1.25 in the case of minimum sample sizes, as can easily be calculated) at the same time, but in all other cases our model performs better in this respect and considerably better in large-scale reference interval studies.
Of the nine examples presented in Applications to Published Data, our model and the Harris-Boyd model agreed in six cases. In each of the three cases with different conclusions, orosomucoid (main group vs females who do not use estrogen), IgA, and IgM, our model was the more radical one. If we are supposed to prefer the results of our model, this might suggest that the new critical value of the Harris-Boyd model, or zCrit5, is too high.
How successful our distance criteria will turn out to be in practice depends to a great extent on how appropriate the critical percentages, derived from the analytical bias quality specifications, are as true partitioning criteria. Our critical percentages are, of course, just proposals that possibly will need to be revised in the future as the clinical chemical community acquires more experience on partitioning. If the critical percentages are in fact changed, our critical distances can easily be changed as well to adjust to the new critical percentages (Appendix Eqs. 3 and 4 ). An important advantage of the present model is indeed that our distance criteria can be made to quite precisely correspond to any desired critical percentages, whereas the effect of changed critical z values in the Harris-Boyd model on these percentages is much harder to control.
We have developed new partitioning criteria to be used for gaussian-distributed data. The focus of the present model lies on estimating the percentages of the subgroup distributions that would fall outside the reference limits of the combined distribution. We first present proposals for critical values of these percentages and develop a method to correlate them to critical distances between the subgroup distributions. These distances are evaluated in terms of differences between reference limits that are a very natural object of interest in reference interval studies. 1 Moreover, we show that correlation between distances measured using reference limits and the percentages is more accurate than the correlation between distances measured using means and these percentages. The new model therefore offers a better guarantee that the percentages chosen as critical really fall outside the combined reference limits when the critical distances are used as partitioning criteria. The model is also easy to adjust to any new critical values of these percentages, should they need to be changed in the future.
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TopAbstractIntroductionTheoretical BackgroundApplications to Published DataDiscussionconclusionReferences |
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Partitioning is not recommended 4.1% Decision on partitioning must be made using other than simple statistical considerations 4.1% Partitioning is recommended Partitioning is recommended 1.5 Calculate DL and DU, or the distances between the lower and the upper reference limits, respectively, of the subgroup distributions and use the standard deviation of the narrower subgroup distribution as a scale unit (s) Partitioning is not recommended 0.75 s Decision on partitioning must be made using other than simple statistical considerations 0.75 s Partitioning is recommended These two criterion categories, percentages or distances, can be applied independently, using the type of data that is more readily available. Application of the distance criteria presupposes that both subgroup distributions are gaussian or have been converted to gaussian distributions by, e.g., logarithmic transformation. If partitioning into more than two subgroups is to be considered simultaneously, these guidelines can be applied by analyzing one pair of the subgroups at a time.
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Appendix |
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were constructed according to the method described in this Appendix
Notation is as in Fig. 1B
. In Appendix Fig. 1A
, the narrower distribution is selected as a reference or the fixed distribution, and the broader distribution is displaced to the right with respect to it. The narrower distribution is the standard normal distribution with a standard deviation (sa) equal to unity, and the broader distribution has a standard deviation (sb) equal to R, or the ratio of the standard deviations. We wish to find a mathematic expression between the percentages pa and pb of the subgroup distributions outside the common reference limit and the distance D between the reference limits of these distributions. Because the sum of pa and pb is 5.0%, we used only the higher percentage, pa, to construct the curves in Fig. 3
. When needed, pb can be calculated as 5.0% minus pa at any point of the distance variable. D and its two components, da and db, can be calculated as follows, using the inverse gaussian function of the Microsoft Excel 97 program or the NORMINV function (the arguments of the NORMINV function are proportion, mean, and standard deviation of the distribution, respectively):
 |
 |
 |
Fig. 3
shows a plot of pa vs D for each of R = 1.0, 1.3, and 1.5.
In addition to the topography of Appendix Fig. 1A
, one has to consider the reversed topography, illustrated in Appendix Fig. 1B
. Here, the broader distribution is chosen as the fixed distribution, and the narrower distribution is displaced to the right of it. D in this case is calculated as follows:
 |
Observe that because sa is now larger than sb, D is measured in sb, i.e., always in the standard deviation of the narrower distribution. In addition, R is equal for both topographies because it is defined as the ratio of the larger standard deviation to the smaller one, i.e., R = sb/sa for the topography of Appendix Fig. 1A
, but R = sa/sb for the topography of Appendix Fig. 1B
.
The percentage-vs-distance curves corresponding to R >1 and illustrating the topography of Appendix Fig. 1B
would lie above the curves of the respective Rs but illustrating the topography of Appendix Fig. 1A
. This is because as the displaced distributions, the broader one in Appendix Fig. 1A
and the narrower one in Appendix Fig. 1B
, move toward the right from the origin, the broader distribution will have to move a larger distance than the narrower one for a similar change of the percentages (the ratio of displacements is not R because the common reference limit is also slightly displaced to the right in each topography). Hence, the percentage-vs-distance curves illustrating the topography of Appendix Fig. 1B
would not add to the dispersion of the curves in Fig. 3
.
If the curves corresponding to the topography of Appendix Fig. 1B
were actually used to determine the critical distances, we would obtain 0.68 s as DCritP and 0.27 s as DCritNP. These values can be rounded to the rule-of-thumb critical distances of 0.75 s and 0.25 s, as was done to the values of 0.73 s and 0.28 s obtained earlier for the topography of Appendix Fig. 1A
, because both calculated pairs of critical distances correspond essentially to the same percentage ranges. Observe that all of these rounding operations are conservative in nature, producing slightly more stringent conditions for the classification of the subgroups as, respectively, separable or nonseparable. Consequently, use of the rule-of-thumb critical distances guarantees a reasonably good correlation between percentages and distances, irrespective of which end of two distributions, having different standard deviations, is being considered.
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Acknowledgments |
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Footnotes |
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References |
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TopAbstractIntroductionTheoretical BackgroundApplications to Published DataDiscussionconclusionReferences |
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rustadp/ref/refprosj.html..
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1-acid glycoprotein (orosomucoid), 1-antitrypsin and haptoglobin: a practical, simple and clinically relevant approach in a large cohort. J Clin Lab Anal 2000;14:284-292.[Web of Science][Medline]
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