Editing Reference range (section)

====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 normal distribution or Gaussian distribution.

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 a normal distribution. Thus, the 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 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}} }} }}

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 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 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:

:{{math|1=Coefficient of variation = {{sfrac|s.d.|m}}}}

where:
* ''s.d.'' is the 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 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 normal distribution, then it should be considered to do the calculations again by log-normal distribution.

Taking the example from previous section, the 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 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=====
A method to estimate the reference range for a parameter with log-normal distribution is to logarithmize all the measurements with an arbitrary [[base of a logarithm|base]] (for example [[e (mathematical constant)|''e'']]), derive the mean and standard deviation of these logarithms, determine the logarithms located (for a 95% prediction interval) 1.96 standard deviations below and above that mean, and subsequently [[exponentiation|exponentiate]] using those two logarithms as exponents and using the same base as was used in logarithmizing, with the two resultant values being the lower and upper limit of the 95% prediction interval.

The following example of this method is based on the same values of [[fasting plasma glucose]] as used in the previous section, using [[e (mathematical constant)|''e'']] as a [[base of a logarithm|base]]:<ref name=Keevil1998/>

{|class="wikitable"
|-
! !! [[Fasting plasma glucose]]<br> (FPG) <br>in mmol/L !! log<sub>[[e (mathematical constant)|''e'']]</sub>(FPG) !! log<sub>e</sub>(FPG) deviation from<br> mean ''μ''<sub>log</sub> !! Squared deviation<br>from mean
|-
| Subject 1 || 5.5 || 1.70 || 0.029 || 0.000841
|-
| Subject 2 || 5.2 || 1.65 || 0.021 || 0.000441
|-
| Subject 3 || 5.2 || 1.65 || 0.021 || 0.000441
|-
| Subject 4 || 5.8 || 1.76 || 0.089 || 0.007921
|-
| Subject 5 || 5.6 || 1.72 || 0.049 || 0.002401
|-
| Subject 6 || 4.6 || 1.53 || 0.141 || 0.019881
|-
| Subject 7 || 5.6 || 1.72 || 0.049 || 0.002401
|-
| Subject 8 || 5.9 || 1.77 || 0.099 || 0.009801
|-
| Subject 9 || 4.7 || 1.55 || 0.121 || 0.014641
|-
| Subject 10 || 5.0 || 1.61 || 0.061 || 0.003721
|-
| Subject 11 || 5.7 || 1.74 || 0.069 || 0.004761
|-
| Subject 12 || 5.2 || 1.65 || 0.021 || 0.000441
|-
| || '''Mean: 5.33''' <br> (''m'') || '''Mean: 1.67'''<br> (''μ''<sub>log</sub>) ||  || Sum/(n-1) : 0.068/11 = 0.0062 <br> <math>
    \sqrt{0.0062} = 0.079</math><br>= '''standard deviation of log<sub>e</sub>(FPG)'''<br> (''σ''<sub>log</sub>)
|}

Subsequently, the still logarithmized lower limit of the reference range is calculated as:

: <math>\begin{align} \ln (\text{lower limit}) &= \mu_{\log} - t_{0.975,n-1} \times\sqrt{\frac{n+1}{n}} \times \sigma_{\log}\\ 
&= 1.67 - 2.20\times\sqrt{\frac{13}{12}} \times 0.079 = 1.49, \end{align}</math>

and the upper limit of the reference range as:

: <math>\begin{align} \ln (\text{upper limit}) &= \mu_{\log} + t_{0.975,n-1} \times\sqrt{\frac{n+1}{n}} \times \sigma_{\log}\\ 
&= 1.67 + 2.20\times\sqrt{\frac{13}{12}} \times 0.079 = 1.85 \end{align}</math>

Conversion back to non-logarithmized values are subsequently performed as:

: <math> \text{Lower limit} = e^{\ln (\text{lower limit})} = e^{1.49} = 4.4</math>

: <math> \text{Upper limit} = e^{\ln (\text{upper limit})} = e^{1.85} = 6.4</math>

Thus, the standard reference range for this example is estimated to be 4.4 to 6.4.

=====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 standard deviation. This is somewhat more tedious to perform, but may be useful in cases where a study presents only the arithmetic mean and standard deviation, while leaving out the source data. If the original assumption of normal distribution is less appropriate than the log-normal one, then, using the arithmetic mean and standard deviation may be the only available parameters to determine 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:
: <math> \mu_{\log} = \ln(m) - \frac12 \ln\!\left(1 + \!\left(\frac\text{s.d.}{m}\right)^2 \right) </math>

: <math> \sigma_{\log} = \sqrt{\ln\!\left(1 + \!\left(\frac\text{s.d.}{m}\right)^2 \right)} </math>

Following the exampled reference group from the previous section:

: <math> \mu_{\log} = \ln(5.33) - \frac12 \ln\!\left(1 + \!\left(\frac{0.42}{5.33}\right)^2 \right) = 1.67</math>

: <math> \sigma_{\log} = \sqrt{\ln\!\left(1 + \!\left(\frac{0.42}{5.33}\right)^2 \right)} = 0.079 </math>

Subsequently, the logarithmized, and later non-logarithmized, lower and upper limit are calculated just as by logarithmized sample values.