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{{Short description|Measured values that are relatively normal for a particular medical test}}
{{Reference ranges}}
In [[medicine]] and [[health]]-related fields, a '''reference range''' or '''reference interval''' is the [[range (statistics)|range]] or the [[interval (mathematics)|interval]] of values that is deemed normal for a [[physiology|physiologicphysiological]] measurement in healthy persons (for example, the amount of [[creatinine]] in the [[blood]], or the [[blood gas tension|partial pressure of oxygen]]). It is a basis for comparison for a [[physician]] or other [[health professional]] to interpret a set of test results for a particular patient. Some important reference ranges in medicine are [[reference ranges for blood tests]] and [[Urinalysis#Target parameters|reference ranges for urine tests]].
 
The standard definition of a reference range (usually referred to if not otherwise specified) originates in what is most prevalent in a [[reference group]] taken from the general (i.e. total) population. This is the general reference range. However, there are also ''optimal health ranges'' (ranges that appear to have the optimal health impact) and ranges for particular conditions or statuses (such as pregnancy reference ranges for hormone levels).
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Reference ranges that are given by this definition are sometimes referred as ''standard ranges''.
 
Since a range is a defined statistical value ([[Range (statistics)]]) that describes the interval between the smallest and largest values, many, including the International Federation of Clinical Chemistry prefer to use the expression reference interval rather than reference range.<ref>{{cite journal |last1=Dybkaer |first1=R |title=International federation of clinical chemistry (IFCC)1),2) the theory of reference values. Part 6. Presentation of observed values related to reference values. |journal=Journal of Clinical Chemistry and Clinical Biochemistry |date=November 1982 |volume=20 |issue=11 |pages=841–5 |pmid=7153721}}</ref>
 
Regarding the target population, if not otherwise specified, a standard reference range generally denotes the one in healthy individuals, or without any known condition that directly affects the ranges being established. These are likewise established using reference groups from the healthy population, and are sometimes termed ''normal ranges'' or ''normal values'' (and sometimes "usual" ranges/values). However, using the term ''normal'' may not be appropriate as not everyone outside the interval is abnormal, and people who have a particular condition may still fall within this interval.
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===Establishment methods===
Methods for establishing reference ranges can be based on assuming a [[normal distribution]] or a [[log-normal distribution]], or directly from percentages of interest, as detailed respectively in following sections. When establishing reference ranges from bilateral organs (e.g., vision or hearing), both results from the same individual can be used, although intra-subject correlation must be taken into account.<ref>{{cite journal |last1=Davis |first1=C.Q. |last2=Hamilton |first2=R. |title=Reference ranges for clinical electrophysiology of vision |journal=Doc Ophthalmol |date=2021 |volume=143 |issue=2 |pages=155–170 |doi=10.1007/s10633-021-09831-1|pmc=8494724 |pmid=33880667 |doi-access=free }}</ref>
 
====Normal distribution====
{{further|68–95–99.7 rule}}
[[File:Standard deviation diagram.svg|thumb|350px|When assuming a normal distribution, the reference range is obtained by measuring the values in a reference group and taking two standard deviations either side of the mean. This encompasses ~95% of the total population.]]
 
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However, in the real world, neither the population mean nor the population standard deviation are known. They both need to be estimated from a sample, whose size can be designated ''n''. The population standard deviation is estimated by the sample standard deviation and the population mean is estimated by the sample mean (also called mean or [[arithmetic mean]]). To account for these estimations, the 95% [[prediction interval]] (95% PI) is calculated as:
 
: {{math|1= 95% PI = mean ± ''t''{{sub|0.975,''n''&minus;1}}&middot;·{{sqrt|(''n''+1)/''n''}}&middot;·sd}},
 
where <math>t_{0.975,n-1}</math> is the 97.5% quantile of a [[Student's t-distribution]] with ''n''&minus;1 [[Degrees of freedom (statistics)|degrees of freedom]].
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The 90% ''confidence interval of a standard reference range limit'' as estimated assuming a normal distribution can be calculated by:<ref>[https://books.google.com/books?id=p7XwAwAAQBAJ&pg=PA65 Page 65] in: {{cite book|title=Tietz Fundamentals of Clinical Chemistry and Molecular Diagnostics|author=Carl A. Burtis, David E. Bruns|edition=7|publisher=Elsevier Health Sciences|year=2014|isbn=9780323292061}}</ref>
 
: Lower limit of the confidence interval = percentile limit - 2.81 &times;× {{frac|''SD''|{{sqrt|''n''}}}}
 
: Upper limit of the confidence interval = percentile limit + 2.81 &times;× {{frac|''SD''|{{sqrt|''n''}}}},
 
where SD is the standard deviation, and n is the number of samples.
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Taking the example from the previous section, the number of samples is 12 and the standard deviation is 0.42&nbsp;mmol/L, resulting in:
 
:''Lower limit of the confidence interval'' of the ''lower limit of the standard reference range'' = 4.4 - 2.81 &times;× {{frac|0.42|{{sqrt|12}}}} ≈ 4.1
 
:''Upper limit of the confidence interval'' of the ''lower limit of the standard reference range'' = 4.4 + 2.81 &times;× {{frac|0.42|{{sqrt|12}}}} ≈ 4.7
 
Thus, the lower limit of the reference range can be written as 4.4 (90% CI 4.1–4.7) mmol/L.
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====Log-normal distribution====
[[Image:PDF-log normal distributions.svg|thumb|Some functions of [[log-normal distribution]] (here shown with the measurements non-logarithmized), with the same means - ''μ'' (as calculated after logarithmizing) but different standard deviations - ''σ'' (after logarithmizing)]]
In reality, biological parameters tend to have a [[log-normal distribution]],<ref>{{cite book | last = Huxley | first = Julian S. | year = 1932 | title = Problems of relative growth | publisher = London | oclc = 476909537 | isbn = 978-0-486-61114-3 }}</ref> rather than the arithmetical normal distribution (whichor is generally referred to as normalGaussian distribution without any further specification).
 
An explanation for this log-normal distribution for biological parameters is: The event where a sample has half the value of the mean or median tends to have almost equal probability to occur as the event where a sample has twice the value of the mean or median. Also, only a log-normal distribution can compensate for the inability of almost all biological parameters to be of [[negative number]]s (at least when measured on [[absolute scale]]s), with the consequence that there is no definite limit to the size of outliers (extreme values) on the high side, but, on the other hand, they can never be less than zero, resulting in a positive [[skewness]].
 
As shown in diagram at right, this phenomenon has relatively small effect if the standard deviation (as compared to the mean) is relatively small, as it makes the log-normal distribution appear similar to an arithmeticala normal distribution. Thus, the arithmetical normal distribution may be more appropriate to use with small standard deviations for convenience, and the log-normal distribution with large standard deviations.
 
In a log-normal distribution, the [[geometric standard deviation]]s and [[geometric mean]] more accurately estimate the 95% prediction interval than their arithmetic counterparts.
 
=====Necessity=====
Reference ranges for substances that are usually within relatively narrow limits (coefficient of variation less than 0.213, as detailed below) such as [[electrolytes]] can be estimated by assuming normal distribution, whereas reference ranges for those that vary significantly (coefficient of variation generally over 0.213) such as most [[hormones]]<ref name="pmid19758299">{{cite journal| author=Levitt H, Smith KG, Rosner MH| title=Variability in calcium, phosphorus, and parathyroid hormone in patients on hemodialysis. | journal=Hemodial Int | year= 2009 | volume= 13 | issue= 4 | pages= 518–25 | pmid=19758299 | doi=10.1111/j.1542-4758.2009.00393.x | pmc= | s2cid=24963421 | url=https://pubmed.ncbi.nlm.nih.gov/19758299 }}</ref> are more accurately established by log-normal distribution.
The necessity to establish a reference range by log-normal distribution rather than arithmetic normal distribution can be regarded as depending on how much difference it would make to ''not'' do so, which can be described as the ratio:▼
 
▲The necessity to establish a reference range by log-normal distribution rather than arithmetic normal distribution can be regarded as depending on how much difference it would make to ''not'' do so, which can be described as the ratio:
 
:{{math|1=Difference ratio = {{sfrac| {{mabs| Limit{{sub|log-normal}} - Limit{{sub|normal}} }} | Limit{{sub|log-normal}} }} }}
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where:
* ''Limit<sub>log-normal</sub>'' is the (lower or upper) limit as estimated by assuming log-normal distribution
* ''Limit<sub>normal</sub>'' is the (lower or upper) limit as estimated by assuming arithmetically normal distribution.
 
[[File:Diagram of coefficient of variation versus deviation in reference ranges erroneously not established by log-normal distribution.png|thumb|350px|Coefficient of variation versus deviation in reference ranges established by assuming arithmetic normal distribution when there is actually a log-normal distribution.]]
 
This difference can be put solely in relation to the [[coefficient of variation]], as in the diagram at right, where:
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where:
* ''s.d.'' is the arithmetic standard deviation
* ''m'' is the arithmetic mean
 
In practice, it can be regarded as necessary to use the establishment methods of a log-normal distribution if the difference ratio becomes more than 0.1, meaning that a (lower or upper) limit estimated from an assumed arithmetically normal distribution would be more than 10% different from the corresponding limit as estimated from a (more accurate) log-normal distribution. As seen in the diagram, a difference ratio of 0.1 is reached for the lower limit at a coefficient of variation of 0.213 (or 21.3%), and for the upper limit at a coefficient of variation at 0.413 (41.3%). The lower limit is more affected by increasing coefficient of variation, and its "critical" coefficient of variation of 0.213 corresponds to a ratio of (upper limit)/(lower limit) of 2.43, so as a rule of thumb, if the upper limit is more than 2.4 times the lower limit when estimated by assuming arithmetically normal distribution, then it should be considered to do the calculations again by log-normal distribution.
 
Taking the example from previous section, the arithmetic standard deviation (s.d.) is estimated at 0.42 and the arithmetic mean (m) is estimated at 5.33. Thus the coefficient of variation is 0.079. This is less than both 0.213 and 0.413, and thus both the lower and upper limit of fasting blood glucose can most likely be estimated by assuming arithmetically normal distribution. More specifically, the coefficient of variation of 0.079 corresponds to a difference ratio of 0.01 (1%) for the lower limit and 0.007 (0.7%) for the upper limit.
 
=====From logarithmized sample values=====
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=====From arithmetic mean and variance=====
An alternative method of establishing a reference range with the assumption of log-normal distribution is to use the arithmetic mean and arithmetic value of standard deviation. This is somewhat more tedious to perform, but may be useful for example in cases where a study that establishes a reference range presents only the arithmetic mean and standard deviation, while leaving out the source data. If the original assumption of arithmetically normal distribution is shown to be less appropriate than the log-normal one, then, using the arithmetic mean and standard deviation may be the only available parameters to correctdetermine the reference range.
 
By assuming that the [[expected value]] can represent the arithmetic mean in this case, the parameters ''μ<sub>log</sub>'' and ''σ<sub>log</sub>'' can be estimated from the arithmetic mean (''m'') and standard deviation (''s.d.'') as:
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* Instruments and lab techniques used, or how the measurements are interpreted by observers. These may apply both to the instruments etc. used to establish the reference ranges and the instruments, etc. used to acquire the value for the individual to whom these ranges is applied. To compensate, individual laboratories should have their own lab ranges to account for the instruments used in the laboratory.
* [[Risk factor#General determinants(epidemiology)|Determinants]] such as age, diet, etc. that are not compensated for. Optimally, there should be reference ranges from a reference group that is as similar as possible to each individual they are applied to, but it's is practically impossible to compensate for every single determinant, often not even when the reference ranges are established from multiple measurements of the same individual they are applied to, because of [[test-retest reliability|test-retest]] variability.
 
Also, reference ranges tend to give the impression of definite thresholds that clearly separate "good" or "bad" values, while in reality there are generally continuously increasing risks with increased distance from usual or optimal values.
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With this and uncompensated factors in mind, the ideal interpretation method of a test result would rather consist of a comparison of what would be expected or optimal in the individual when taking all factors and conditions of that individual into account, rather than strictly classifying the values as "good" or "bad" by using reference ranges from other people.
 
In a recent paper, Rappoport et al.<ref>{{cite biorxivbioRxiv|last1=Rappoport|first1=Nadav|last2=Paik|first2=Hyojung|last3=Oskotsky|first3=Boris|last4=Tor|first4=Ruth|last5=Ziv|first5=Elad|last6=Zaitlen|first6=Noah|last7=Butte|first7=Atul J.|date=2017-11-04|title=Creating ethnicity-specific reference intervals for lab tests from EHR data|biorxiv=10.1101/213892}}</ref> described a novel way to redefine reference range from an [[electronic health record]] system. In such a system, a higher population resolution can be achieved (e.g., age, sex, race and ethnicity-specific).
 
==Examples==
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==References==
{{Academic peer reviewed|Q44275619|doi-access=free}}
{{Reflist|30em}}
 
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