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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.
aAddress correspondence to this author at: Department of Clinical Chemistry, Rikshospitalet University Hospital of Oslo, N-0027 Oslo, Norway. Fax 47-23071080; e-mail ari.lahti{at}rikshospitalet.no.
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Abstract |
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 Top Abstract IntroductionTheoryApplication ExampleDiscussionReferences |
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Methods: We recently proposed a new model for partitioning reference intervals when the underlying data distribution is gaussian. This model is based on controlling the proportions of the subgroup distributions that fall outside each of the common reference limits, using the distances between the reference limits of the subgroup distributions as functions to these proportions. We examine the significance of the unequal prevalence effect for the partitioning problem and quantify it for distance partitioning criteria by deriving analytical expressions to express these criteria as a function of the ratio of prevalences. An application example, illustrating various aspects of the importance of the prevalence effect, is also presented.
Results: Dramatic shrinkage of the critical distances between reference limits of the subgroups needed for partitioning was observed as the ratio of prevalences, the larger one divided by the smaller one, was increased from unity. Because of this shrinkage, the same critical distances are not valid for all ratios of prevalences, but specific critical distances should be used for each particular value of this ratio. Although proportion criteria used in determining the need for reference interval partitioning are not dependent on the prevalence effect, this effect should be accounted for when these criteria are being applied by adjusting the sample sizes of the subgroups to make them correspond to the ratio of prevalences.
Conclusions: The prevalences of subgroups in the reference population should be known and observed in the calculations for every reference interval study, irrespective of whether distance or proportion criteria are being used to determine the need for reference interval partitioning. We present detailed methods to account for the prevalences when applying each of these types of criteria. Analytical expressions for the distance criteria, to be used when high precision is needed, and approximate distances, to be used in practical work, are derived. General guidelines for partitioning gaussian distributed data are presented. Following these guidelines and using the new model, we suggest that partitioning can be performed more reliably than with any of the earlier models because the new model not only offers an improved correspondence between the critical distances and the critical proportions, but also accounts for the prevalence effect.
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Introduction |
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TopAbstractIntroductionTheoryApplication ExampleDiscussionReferences |
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The usefulness of partitioning has recently become more evident with the advent of large-scale reference interval studies involving up to thousands of reference individuals (1)(2)(3), because large reference samples make partitioning into smaller and smaller subgroups feasible. However, existing partitioning models, as reviewed in the monograph of Harris and Boyd on calculation methods of reference values (4), do not seem to give reliable estimates of the proportions of the subgroup distributions falling outside the reference limits of the combined distribution. Because these proportions are considered today as the most relevant criteria for partitioning, we have undertaken to construct a new model that hopefully gives better control over them. Our model, introduced recently (5), is based on correlating these proportions to distances between reference limits of the subgroup distributions, one pair of reference limits being considered at a time.
However, when applying the new model to various partitioning problems, we discovered a complication caused by the subgroup prevalences. Unequal prevalences seem to make the critical distances used as partitioning criteria shorter compared with equal prevalences. Moreover, it appears that subgroup prevalences should also be considered when using proportion criteria or, in practice, whenever reference data are being partitioned. As far as we are aware, the importance of subgroup prevalences on the partitioning problem has not been recognized previously.
Our previous report (5) was intended to be an exhaustive study on partitioning gaussian-distributed data. However, after discovery of the prevalence effect, it appears that the methodology described therein applies only to the development of partitioning criteria for the very special case of equal subgroup prevalences. The present report is meant to be a generalization of our previous work, covering the situation of unequal subgroup prevalences.
Because it is easy to confuse prevalences and sample sizes, we will spend some time at the beginning of this study to make the distinctions between these two concepts clear. Thereafter, we will derive analytical expressions for critical distances as a function of the ratio of prevalences and calculate approximate distances to be used in practical work. An application example, contrived for this particular purpose, is meant to illustrate the consequences on partitioning if the prevalence effect is not accounted for. We will also present general guidelines for partitioning gaussian-distributed data. These guidelines, which are intended to replace those presented in our previous report (5), are an important generalization of the earlier ones.
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Theory |
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TopAbstractIntroductionTheoryApplication ExampleDiscussionReferences |
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background
Proportions of two gaussian distributions, having equal prevalences, outside the reference limits of the combined distribution.
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 (6). Fig. 1A
shows two gaussian distributions, denoted a and b, that are identical except for different means. The lower reference limit, or the 2.5 percentile, is indicated for both distributions, as well as the lower reference limit of the combined distribution (the thick vertical bar).
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The 2.5% of the reference values that lie outside the lower reference limit of the combined distribution in Fig. 1A
are composed of proportions pa and pb of the distributions a and b, respectively, outside this common reference limit:
 | (1) |
. How to interpret the weight factors wa and wb in Eq. 1
is an important point for consideration. One might suggest that these factors be interpreted as the numbers of reference values, or na and nb, in the two subgroups corresponding to the distributions a and b, or subgroups a and b. However, these numbers may be a more or less accidental result of performing the reference interval study, or the designer of the study may have chosen them at will.
It can easily be shown that if the ratio of na/nb changes from one reference interval study to the next, the determined common reference limit will also change. Considering Eq. 1
, if wa and wb were interpreted as na and nb, no fixed values of pa and pb could be valid for both experimental designs simultaneously (except for the trivial case with pa = pb = 2.5%). Because pa and pb define the localization of the lower common reference limit, one would thereby obtain different lower reference limits for different ratios of na and nb. But reference limits intended for general use should obviously not be dependent on such contingencies as the numbers of reference values in a particular reference interval study. Hence, na and nb are not an appropriate interpretation of the weight factors in Eq. 1
. These factors should, in our opinion, be interpreted as prevalences of the subgroup populations, or as fa and fb.
If we specifically assume equal prevalences (fa = fb), Eq. 1
gives the following simple relationship between pa and pb:
 | (2) |
indicates, their sum is always exactly 5.0%. The respective asymptotes of 0.0% and 5.0% cannot be reached because the probability density of a gaussian distribution is positive for any real argument.
The case of equal prevalences was examined in our previous report on partitioning (5). Starting from the currently widely accepted analytical bias quality specifications (7)(8)(9), we derived critical values for the proportions of the subgroup distributions outside the common reference limits as partitioning criteria. Our suggestions were 4.1% as the critical proportion for partitioning, or pCritP, and 3.2% as the critical proportion for nonpartitioning, or pCritNP. Because the sum of pa and pb equals 5.0% (Eq. 2
), these suggestions simultaneously implied that the complementary critical proportions, also abbreviated pCritP and pCritNP, outside the other one of the two subgroup distributions (distribution b in Fig. 1A
) should be 0.9% and 1.8%, respectively. We recommended partitioning if any of the proportions outside the common reference limits was 
4.1% or 
0.9% and nonpartitioning if all of these proportions lay between 1.8% and 3.2%.
We were also able to convert these critical proportions to critical distances between the reference limits of the subgroup distributions. We suggested 0.75 s as the approximate partitioning criterion and 0.25 s as the approximate nonpartitioning criterion, where s is the standard deviation of the narrower one of the two subgroup distributions. These approximate critical distances were considered valid whenever the ratio between the standard deviations of the subgroup distributions (R) is 1.5.5.1
[Note: According to Harris and Boyd (10), larger ratios of R imply partitioning because one of the proportions outside the common reference limits then exceeds 4.0%.]
As stated above, these results were derived under the implicit assumption of equal prevalences, and their applicability is consequently limited to this special case. What follows next is a generalization of our previous study, intended to cover unequal prevalences.
Proportions of two gaussian distributions, having unequal prevalences, outside the reference limits of the combined distribution.
In the section above, we suggested that the weight factors wa and wb of Eq. 1
should be interpreted as prevalences instead of as numbers of reference values. In contrast to these numbers, prevalences cannot be manipulated by the designer of the reference interval study because they are fixed attributes of the subgroup populations.
Consider an example with prevalence fa = 0.33 and prevalence fb = 0.67, i.e., with the ratio of prevalences, or Fr, equal to 2.0. Insertion of these prevalences as wa and wb into Eq. 1
produces the following equation:
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illustrates this example with Fr = 2.0. In Fig. 1B
, distribution a is the same gaussian distribution as in Fig. 1A
, but distribution b has been multiplied by two, which makes the area of distribution b twice as large as that of distribution a. However, the means, the standard deviations, and the lower reference limits of both distributions are as in Fig. 1A
. The enlarged area of distribution b simply indicates the larger prevalence of the subgroup population b, or the weight of distribution b with respect to the weight of distribution a.
Because D (Fig. 1A
) is identical at both values of Fr, Fr = 1.0 and Fr = 2.0, it can be used to determine the required adjustments of pa and pb. First, calculate D for Fr = 1.0 [The quantiles (Q) or the probit values of gaussian distributions were obtained using the NORMINV function of the Microsoft Excel 97 program. Its arguments are proportion, mean, and standard deviation of the distribution, respectively]:
 | (3) |
are made to coincide, or that both pa and pb are set equal to 0.025. As distribution b is moved to the right, i.e., as D is increased from its initial value of zero, there has to be an increase of two units of pa for each decreased unit of pb to keep the proportion of the combined distribution outside the common reference limit equal to 2.5% all the time, because distribution b is twice as "heavy" as distribution a. Hence, as pb is decreased from 0.025 to, e.g., 0.024, pa is increased from 0.025 not to 0.026 but to 0.027. Observing this ratio of changes, the pa and pb for D = 0.40 and Fr = 2.0 can be calculated by solving the following equation:
 | (4) |
0.042, and pb = 0.025 - 0.0084 0.017. Observe that these new proportions fulfill Eq. 1
:
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is not valid for the proportions in the case of unequal prevalences. The second thing that we notice is that, whereas the original pa was below the critical proportion for partitioning, pCritP (4.1% or 0.041), the new pa exceeds this criterion. In other words, if D = 0.40 s, partitioning is recommended at Fr = 2.0 but not at Fr = 1.0. This means, of course, that the critical distance corresponding to the critical partitioning proportion must be shorter for Fr = 2.0 than for Fr = 1.0. This is the case, because if distribution b is "heavier" than distribution a, the lower common reference limit follows closer to distribution b, which makes pa grow faster and reach the critical proportion at a shorter distance. Therefore, if a reference interval study is being carried out without consideration of the prevalences, it will be easy to make false decisions on partitioning. It is obvious that the relationship between the critical distances and the ratio of prevalences needs to be examined systematically.
distance criteria for partitioning with prevalences taken into consideration
To examine the behavior of the critical distances as correlated to the ratio of prevalences, we consider two cases or "topographies of prevalences" separately. In one of these topographies, the distribution with the more extreme value of the reference limit (distribution a in Fig. 1
) has the smaller prevalence (topography of prevalences A, abbreviated TPrevA), and in the other topography this distribution has the larger prevalence (topography of prevalences B, or TPrevB).
However, consideration of these topographies of prevalences is not sufficient because the calculations are complicated by one or more sources of variability. The subgroup distributions may also have different standard deviations, and we need, therefore, to consider both ends of these distributions not only from the point of view of prevalences but also from that of standard deviations. Analogous to the topographies of prevalences, we must distinguish between a case in which the distribution with the more extreme value of the reference limit (distribution a in Fig. 1
) has the smaller standard deviation (topography of standard deviations A, or TStDevA) and the other case, in which this distribution has the larger standard deviation (topography of standard deviations B, or TStDevB).
Taken together, the topographies of prevalences and standard deviations involve four different topographic combinations, indexed AA, AB, BA, BB, where the first letter refers to the topography of prevalences and the second letter to the topography of standard deviations. These four combinations of topographies are depicted schematically in Fig. 2
.
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In Appendix 2, published in a data supplement that accompanies this article at Clinical Chemistry Online (www.clinchem.org/content/vol48/issue11/), we present in detail the rather tedious theoretical work needed to derive analytical expressions relating distances between reference limits and the ratio of prevalences for each of the four topographic combinations depicted in Fig. 2
. In parallel with derivation of these expressions, Appendix 2 describes how the critical distances can be approximated over ratios and topographies of standard deviations as a function of the ratio of prevalences for each of the two topographies of prevalences, TPrevA and TPrevB. The results of this approximation process are summarized graphically in Fig. 3
, covering the range 1.0 Fr 10, and as a list of selected values in Table 1
, covering a larger range of Fr.
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proportion criteria for partitioning with prevalences taken into consideration
The proportion criteria to determine the need for reference interval partitioning are not themselves affected by the prevalences, i.e., the same critical proportions are valid for all Fr. However, the positions of the common reference limits obviously depend on the numbers of reference values included in the calculations from each subgroup. If more reference values are included in either of the subgroups, the common reference limits will tend to be shifted toward the reference limits of the distribution corresponding to this subgroup.
To determine correct positions for the common reference limits, one should therefore not automatically include all of the reference values in the calculations. This would lead to 2.5% of these particular values lying outside the calculated reference limits, whereas we are in fact interested in obtaining such limits that 2.5% of the true source population would fall outside each of them. Hence, the ratio of the numbers of reference values included in the calculations should actually reflect the ratio of prevalences.
To make the ratio of numbers of reference values equal to the ratio of prevalences, one should either exclude an appropriate proportion of reference values from one of the subgroups by use of some randomization procedure or add new reference values to the other subgroup by use of stochastically selected replicates of the original values. It is perhaps better to create new values using replicates than to eliminate a part of the original values, but either approach involves statistical manipulation of the original material. In contrast to the proportion criteria, use of the distance criteria leaves the original material intact and is statistically perhaps more sound. However, if the conclusion is nonpartitioning, the common reference limits will be required in any case, and the problems inherent in determining these limits will be present even when the distance criteria are used.
Another way to determine the common reference limits is to use the numerical calculation method described below in the Application Example section. This method is based on first calculating the proportions outside each common reference limit and then identifying the common reference limits using these proportions.
guidelines for partitioning gaussian-distributed data
These guidelines are a generalization of the guidelines presented earlier (5), intended to cover all ratios of prevalences and standard deviations between the two subgroups, evaluated for partitioning.
Partitioning criteria as expressed in proportions of the subgroup distributions outside the common reference limits.
4.1% or 0.9%, the decision on partitioning should be made based on other than statistical considerations [for some practical examples on such nonstatistical considerations see Ref. (5)]. 4.1% or 0.9%, partitioning is recommended.
Partitioning criteria as expressed in approximate distances between the reference limits of the subgroup distributions (easy-to-use distance criteria read from the nomogram).
1.36, partitioning is recommended.
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or Table 1
, the criteria for TPrevA from Table 1A
, and the criteria for TPrevB from Table 1B
. Interpolate, if necessary.
Partitioning criteria as expressed in accurate distances between the reference limits of the subgroup distributions.
If accurate critical distances are desired, these can be calculated from Eqs. 3, 6, 12, and 13 in Appendix 2 for any particular values of R and Fr. To apply these equations, the topographies of both standard deviations and prevalences of the subgroup distributions need to be considered, as illustrated in Fig. 2
.
If critical proportions other than those recommended by ourselves are to be used, such new critical proportions need to be substituted for pa and pb in Eqs, 3, 6, 12, and 13 of Appendix 2. In addition, the cutoff value for R needs to be recalculated, using the procedure described in Appendix 1.
These two criterion categories, proportions or distances, can be applied independently, using the type of data that is better available. Application of the distance criteria presupposes that both subgroup distributions are gaussian.
As far as partitioning into more than two subgroups is concerned, we wrote earlier (5) that such cases could be solved by applying the guidelines for partitioning to each pair of these subgroups at a time. If the prevalences are equal, this approach is valid in most instances because the approximate DCritP (0.75 s) calculated by us (5) was three times as large as the approximate DCritNP (0.25 s). The case of more than two subgroups having unequal prevalences may be more problematic because inequality of the prevalences may decrease the ratio of DCritP to DCritNP from the value of 3 to <2 at certain values of Fr (Fig. 3
). We will therefore probably return to the case of several subgroups in more detail in a later study.
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Application Example |
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TopAbstractIntroductionTheoryApplication ExampleDiscussionReferences |
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Let the mean and the standard deviation of the subgroup distributions be 100 and 10 for subgroup 1, and 103 and 11 for subgroup 2, and let the prevalences of the subgroup populations in the source population be 0.25 and 0.75, respectively. The distributions are illustrated in Fig. 4
both as probability distributions (Fig. 4A
) and as frequency distributions using the numbers of reference values (Fig. 4B
) and the prevalences (Fig. 4C
) as frequencies. In what follows, these three cases will be called A, B, and C for convenience.
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We will first examine partitioning of subgroups 1 and 2 using the approximate distance criteria as read from Table 1
, and thereafter using the accurate distance criteria, as calculated from the equations of the theory. Finally, we will calculate the proportions of the subgroup distributions outside the common reference limits as well as these limits by solving the pertinent equations numerically.
case a
In Fig. 4A
, the distributions are depicted as probability distributions, determined by their means and standard deviations, i.e., we are ignoring both the numbers of reference values and the prevalences. The reference limits of the subgroup distributions will be the same for the cases A–C because these limits are calculated separately for each subgroup distribution, using the same reference values in all cases. In contrast, the reference limits of the combined distribution depend on the ratio of prevalences. However, in this example we will misuse the ratio of the numbers of reference values in case B and account for the prevalences only in case C. We will therefore obtain different common reference limits for each case. In case A, the common reference limits are primarily of theoretical interest, because in practice the numbers of reference values will most often be considered—either correctly, accounting for the prevalences, or incorrectly, using them as they are—to construct the combined distribution and to calculate the common reference limits. For comparison, we will also identify the common reference limits in case A, using the mathematical properties of gaussian distributions.
Approximate distance criteria.
Following the guidelines for partitioning, presented above, we calculate that R = 11/10 = 1.1 < 1.36 and observe that this ratio is in itself not large enough to support partitioning. Because we are ignoring the prevalences in case A, Fr = 1.0. The reference interval of the subgroup distribution 1 is 100 ± 1.96 x 10 = 80.4–119.6, and that of the subgroup distribution 2 is 103 ± 1.96 x 11 = 81.4–124.6. Hence, we obtain DL = (81.4 - 80.4)/10 s = 0.10 s, and DU = (124.6 - 119.6)/10 s = 0.50 s (observe that the unit is the smaller standard deviation).
Because Fr = 1.0, we do not have to be concerned about the topographies of prevalences in case A, and because we are intending to use the approximate critical distances, we do not have to be concerned about the topographies of standard deviation either. The approximate critical distances can be read from either Table 1A
or Table 1B
because the first rows of these tables, corresponding to Fr = 1.0, are identical. Hence, we obtain the following critical distances for case A: DCritPAave = DCritPBave = 0.68 s and DCritNPAave = DCritNPBave = 0.27 s. Because DL = 0.10 s lies below DCritNPAave (or below DCritNPBave), the conclusion from the lower end is nonpartitioning, and because DU = 0.50 s lies between DCritNPAave and DCritPAave (or between DCritNPBave and DCritPBave), the conclusion from the upper end is equivocal. Following our guidelines, the final conclusion suggested by this approximate method is thereby equivocal, i.e., the decision on partitioning should be made using nonstatistical considerations, such as clinical judgment or literature data.
Accurate distance criteria.
The lower end of the distributions has topography of standard deviations TStDevA, because distribution 1, having the smaller value of the lower reference limit, also has a smaller standard deviation. Hence, the accurate critical distances for this end can be calculated analytically by setting Fr = 1.0 in Eqs. 3 or 12 of Appendix 2 (these are identical for Fr = 1.0) as follows (we will use Eq. 3 of Appendix 2):
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), i.e., they are nearly identical. Observe that all of the accurate critical distances are slightly smaller than the respective approximate ones. This is partly attributable to averaging over R = 1.0 and R = 1.36, because the critical distances tend to increase with increasing R (consider Fig. 2 in Appendix 2), and R = 1.1 lies closer to the lower end of the R scale. Although the conclusions at both ends obtained by applying the approximate critical distances are the same in this case as those obtained for the accurate critical distances, it is possible to obtain different outcomes in borderline cases, as illustrated by case C below.
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Proportion criteria.
We will next apply the proportion criteria to case A. Instead of first localizing the common reference limits and then calculating the proportions of the subgroup distributions outside these limits, as Harris and Boyd (4) did using computer simulations, we will first calculate the proportions and determine the common reference limits thereafter using these proportions. As an example, we will detail the calculations needed to determine the proportions and the common reference limit at the lower end of case A. If we equate the distance between reference limits as expressed in Eq. 3 of Appendix 2 with DL = 0.10 (the unit is s on both sides, so we will omit it), we will obtain:
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, we further calculate that pb = 2.2%. These proportions outside the lower common reference limit are shown in Fig. 4A
. The lower common reference limit can now be calculated as Q(0.028;100;10) = Q(0.022;103;11) 80.9. Using Eq. 13 in Appendix 2 we find in the same way that the upper common reference limit of case A lies at 122.5 and that this limit will cut off 1.2% from distribution 1 and 3.8% from distribution 2. The data for case A are summarized in the first column of Table 3
. We now are ready to apply the proportion criteria, as presented in the guidelines. At the lower end both proportions are smaller than the critical proportion for nonpartitioning (pCritNP = 3.2%), and both of them exceed the complementary pCritNP = 1.8%. Hence, the conclusion from this end is nonpartitioning. At the upper end, one of the proportions lies between pCritNP = 3.2% and pCritP = 4.1%, and the other one lies between the complementary pCritP = 0.9% and the complementary pCritNP = 1.8%. The conclusion from this end is consequently equivocal, and this is also the final conclusion for case A, using the proportion criteria. Hence, the conclusions from both ends, as well as the final conclusion, agree with the conclusions obtained above using the distance criteria for this case.
case b
Case B illustrates the method used most often to solve the partitioning problem today, i.e., by considering the numbers of reference values but ignoring the prevalences. The common reference limits are usually calculated by involving all the reference values that one happens to have in the subgroups. These limits and the proportions outside them thus reflect the ratio of the numbers of reference values instead of that of the prevalences. We will first solve case B by the Harris Boyd model, which is at present the most widely applied partitioning method. It does not account for the prevalences, because originally, when developing their model, Harris and Boyd assumed equal sample sizes and thereby, implicitly, equal sample prevalences (10).
The Harris-Boyd model.
If the Harris-Boyd model is applied to our example by inserting the means, the standard deviations, and the numbers of reference values from Table 2
in the expressions for the test parameter and the critical values of this model (4, 5, 10), we will obtain a z-value of 6.7 and a critical range zCrit3–zCrit5 of 9.3–15.5. Because the z-value lies below the critical range, the conclusion suggested by the Harris-Boyd model is nonpartitioning.
The present model.
In this example, the numbers of reference values in the subgroups are 1300 and 1000, respectively (Table 2
). To account for these numbers in the calculations, we will treat them as weight factors of the distributions, even if this is incorrect, as was discussed in the Background section of this article. Hence, we will have to treat the ratio of these numbers (1300/1000 = 1.3) as if it were a ratio of prevalences and observe the topographies accordingly. Under this assumption, the topographies in case B are obviously TPrevB, TStDevA at the lower end and TPrevA, TStDevB at the upper end. Using Table 1
, B and A, and Eqs. 12 and 6 of Appendix 2, respectively, we can easily obtain the values for the critical distances in case B, reported in Table 3
and, applying the numerical method described above, also determine the proportions and the common reference limits in this case, shown in Table 3
and Fig. 4B
.
Considering the results listed in Table 3
, we notice that the conclusions from both ends are identical to those obtained for case A, whichever criteria we apply. Observe in particular that both the distance criteria, the approximate ones as well as the accurate ones, and the proportion criteria lead to the same conclusion even at the upper end, although this end is clearly a near-to-borderline case with respect to partitioning. We selected the ratio of the numbers of reference values deliberately to make the larger proportion close to the critical value of 4.1% to test the precision of the distance criteria as correlates of the proportion criteria. The accurate critical distance (0.52 s) lies very near to DU = 0.50 s and so does the approximate critical distance (0.54 s). Had DU been only 0.02 s or 0.04 s wider, respectively, or the proportion only 0.1% larger, the conclusion would have been partitioning. Hence, correspondence between proportions and distances seems quite good. Notice also that the approximate and the accurate distances differ from each other by only 0.02 s at most (Table 3
).
Real-life common reference limits as calculated with use of the reference values of the combined distribution would probably differ slightly from those calculated with our numerical method. However, both methods should in principle produce the same common reference limits because both of them weigh distribution 1 in the proportion of 1.3 to 1 with respect to distribution 2. Observe how this weighing has shifted the common reference limits slightly toward the reference limits of distribution 1 from the situation in case A. The changes of the proportions between cases A and B reflect these transitions of the common reference limits in a logical way. All of these tendencies would be more marked if the number of reference values in subgroup 1 were increased further with respect to that of subgroup 2.
In spite of the agreement obtained by use of proportions and distances in both this and the previous case, these results are nonsense; the numbers of reference values should neither be used to define common reference limits nor to make conclusions on partitioning, and the prevalences have not yet been considered.
case c
Case C is intended to be an illustration of how the partitioning problem should, in our opinion, be solved, i.e., by accounting for the prevalences. The ratio of prevalences in this example is equal to 3.0 (Fr = 0.75/0.25 = 3.0), and the topographies in case C are TPrevA, TStDevA at the lower end and TPrevB, TStDevB at the upper end (observing Fig. 4C
, one might suggest that the topography at the lower end should rather be TPrevB, TStDevB, but the distribution having the smaller value of the lower reference limit is distribution 1, as shown in Table 2
). Hence, the results of this case, listed in Table 3
, can be obtained using Table 1
, A and B, and Eqs. 3 and 13 in Appendix 2, respectively.
The decision at the lower end is once again nonpartitioning, as can be concluded by use of either the distance or the proportion criteria. However, the situation at the upper end is precarious because we have intentionally chosen the ratio of prevalences to obtain the critical partitioning value of 0.9% for the smaller proportion. As we see, the accurate DCritP = 0.50 s = DU, as expected, i.e., using the accurate critical distance, we would achieve a perfect correspondence with the proportion criterion. In contrast, the approximate DCritP (0.53 s) exceeds DU slightly, and use of it would accordingly lead to an equivocal conclusion. In spite of this discrepancy in conclusions, the agreement between the approximate and the accurate distances is still good, 0.03 s being the largest difference between these distances (Table 3
).
It is obvious that in borderline cases the approximate distance will most often deviate in either direction from the accurate distance and easily produce a false conclusion. The only way to assure oneself of making a correct decision in such cases is to use the accurate distances.
Observe that in case C the common reference limits have been shifted toward the reference limits of distribution 2, in accordance with this distribution having the larger prevalence, and the changes in proportions are opposite to those seen in case B (Table 3
and Fig. 4C
).
the significance of the numbers of reference values
Apart from case B, which was an example of how the partitioning problem should not be solved, we have not yet had any use of the numbers of reference values in the subgroups. However, the reference limits of each of the subgroup distributions are calculated separately by use of the respective reference values. Moreover, the numbers of reference values determine the statistical quality of reference limits and, thus, the quality of distances between these limits. The 90% confidence intervals (CIs) about the 2.5 and 97.5 percentiles of a gaussian distribution can be calculated as follows (5)(6):
 | (5) |
, we obtain the following 90% CIs for the reference limits of distributions 1 and 2, respectively: ± 0.78 and ± 0.97. The CIs for the distances between reference limit pairs can be calculated as ± (0.782 + 0.972)1/2 = ± 1.24, or ± 0.12 s, where s is the standard deviation of distribution 1.
Conclusion
Although this example was a contrived one and the parameters were manipulated to obtain borderline situations, the characteristics of the distributions (Table 2
) should be rather typical for two subgroups subjected to comparison in a reference interval study. Because the demographic descriptors of these groups are often similar, their means and standard deviations tend to lie close to each other. The need for partitioning in many cases thus is not self-evident but must be examined carefully.
The ratio of prevalences was set to 3.0, by no means an extreme value for this ratio and large enough to make partitioning recommended, in this example, whereas our first guess, when considering Fig. 4A
, may well have been nonpartitioning. Partitioning was indeed the final conclusion based on both the distance and the proportion criteria, when the prevalence effect was accounted for. When this effect was ignored, we obtained other conclusions with both the present model, which suggested an equivocal conclusion, and the Harris-Boyd model, which suggested nonpartitioning. That consideration of the prevalences tends to favor partitioning was an expected result because the prevalence effect makes the critical distances shorter and partitioning as the conclusion more probable.
Proper consideration of the prevalences is obviously of vital importance whenever decisions on partitioning are to be made. Such decisions should, in our opinion, not be made at all if the prevalences are not known. Very clear-cut cases where the distances between the reference limits exceed the critical distances for Fr = 1.0 are an exception from this rule, of course.
As shown by this application example, the accuracy of the approximate distances should be high enough for most practical purposes. The method is relatively easy to use even for hand calculations: one has only to observe the prevalences, consider the topographies of prevalences, calculate the differences between the reference limits, and compare these differences with the critical distances read from Table 1
or Fig. 3
. The analytical calculations are not hard to apply either, and we recommend them when high precision is desired, particularly at borderline situations, as demonstrated in case C. Because imprecision is unavoidable whenever statistically calculated parameters are being used, the accuracy of such analytical calculations is not absolute, however. It is important to keep this fact in mind when performing analysis of reference data in practice.
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Discussion |
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Application of the present model to the partitioning problem is undeniably more laborious than that of any of the earlier models, but as the only one to account for the prevalence effect, the present model hopefully enables more reliable decision-making on partitioning. In contrast to earlier models, it can also guarantee a good correspondence between the proportions of the subgroup distributions outside the common reference limits and the distances between the reference limits of these distributions, as demonstrated in the application example above.
Because the prevalence effect not only concerns the distance criteria but also application of the proportion criteria, the prevalences should be considered whenever the need for partitioning is being evaluated. The easiest way to do so is to use the approximate distance criteria, plotted in Fig. 3
and listed in Table 1
. Both the accurate distance criteria and the proportion criteria require slightly more engagement in the calculations, whereas the accuracy of the approximate distance criteria probably suffices for most practical purposes.
As Fig. 3
illustrates, the critical partitioning distances shrink quite rapidly, particularly at the very beginning of the Fr scale: for TPrevA, DCritPave is 0.49 s at Fr = 1.5, whereas it is 0.68 s at Fr = 1.0. It is likely that most often when a poorly documented assumption or a mere guess on the equality of the prevalences is made, these are in reality not quite equal, but their ratio takes values somewhere near the origin of the Fr scale, i.e., precisely where the critical distances change most unpredictably. Hence, the critical distances may be quite different from what they are supposed to be, and the risk of false decisions on partitioning is considerable even at ratios of prevalences close to unity. How should earlier studies on partitioning, which did not account for the prevalence effect and which may have included an implicit false assumption of equal prevalences, actually be judged? We feel inclined to recommend careful reconsideration of all such studies where the decision on partitioning has not been clear-cut. Obviously, if the distances between the reference limits have been either larger than the largest of our DCritP (0.68 s) or smaller than the smallest of our DCritNP (0.11 s), one can be confident on the adequacy of partitioning and nonpartitioning, respectively, whatever the topographies of the subgroup distributions have been.
As the application example demonstrates, decisions on partitioning may depend on rather small changes in the proportions and the distances in absolute terms. However, these changes are not necessarily small compared with the ranges of the respective parameters. As an example, in the case of equal prevalences, the larger proportion outside the common reference limits has a range that extends from 2.5% to 5.0%. We (5) divided this range of 2.5% into nonpartitioning, equivocal, and partitioning areas, which have widths of 0.7% [2.5–3.2%), 0.9% [3.2–4.1%), and 0.9% [4.1–5.0%), respectively. Divisions other than this one, based on the analytical bias quality specifications (5)(7)(8)(9), may well turn out to be better for partitioning purposes in the future, but whatever these new divisions are, the room for play is in fact not too large. One operates with tenths of a percent at any case. Hence, tenths of a percent may also be significant when decisions on partitioning are being made. How these tenths of a percent convert themselves to changes of distances depends on Fr and on the topographies of the distributions (as an example, in case B of the application example two-hundredths of an s unit corresponded to one-tenth of a percent). Obviously, whichever criteria are being used, even changes that are very small in absolute terms may be significant.
These changes should be compared with the statistical imprecision of the respective test parameters. The CIs of reference limits (Eq. 5
) are probably most often so large that they could change the conclusion on partitioning if the distances between reference limits were extended accordingly. But imprecision of the test parameters is inherent to any statistical model, and as was discussed in our previous report (5), it is in fact usually smaller in the present model than in the Harris-Boyd model. Because of this unavoidable imprecision, common sense should always guide interpretation of the results suggested by these models.
If prevalences have the influence on partitioning that we believe they have, one should carefully observe them in every reference interval study. The prevalences are sometimes evident, for example, the prevalences of both genders are 0.5 in most populations, and sometimes they have been established in epidemiologic studies, but probably most often they are neither evident nor known from the literature. In such cases, they should be determined or estimated by the organizers of the reference interval study. Ideally, one should know even the prevalences of smokers and nonsmokers, birth control pill users and nonusers, or joggers and nonjoggers in the source population when partitioning of these groups is being examined. Because the prevalences of these lifestyle groups vary over time, estimates trying to assess long-term average values for these prevalences might be the only feasible way to go. In contrast, when partitioning of, e.g., defined genetic subgroups is studied, the prevalences should be determined rigorously before partitioning is undertaken.
In conclusion, we have shown that the prevalences of the subgroup populations in the source population of a reference interval study have a significant effect on the partitioning criteria used to determine whether these subgroups need separate reference intervals. The critical distances can be shortened substantially because of this prevalence effect, and negligence of the prevalences may therefore easily lead to false decisions concerning partitioning. The proportion criteria themselves do not depend on the prevalences, but when these criteria are applied, the prevalences should be accounted for by adjusting the sample sizes to make them correspond to the ratio of prevalences. We have developed a nomogram on approximate critical distances to enable an effective and, for most practical purposes, accurate enough way to account for the prevalence effect in the case of gaussian subgroup distributions. We also offer analytical expressions to be used whenever accurate critical distances, corresponding to any desired critical proportions, are needed.
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1.0, which means that partitioning is needed whenever R 1.36. Because a lower value for the upper limit of R leads to better quality of the approximate critical distances, we will use the more precise value of 1.36 for it throughout this report. 
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rustadp/ref/refprosj.html (Accessed May 15, 2002)..
The following articles in journals at HighWire Press have cited this article:
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  A. Lahti, P. Hyltoft Petersen, J. C. Boyd, P. Rustad, P. Laake, and H. E. Solberg Partitioning of Nongaussian-Distributed Biochemical Reference Data into Subgroups Clin. Chem., May 1, 2004; 50(5): 891 - 900. [Abstract] [Full Text] [PDF]
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