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Proteomics & Bioinformatics

Short Communication Volume 6 Issue 4

Correct use of percent coefficient of variation (%CV) formula for log-transformed data

Jesse A Canchola, Shaowu Tang, Pari Hemyari, Ellen Paxinos, Ed Marins

Roche Molecular Systems, Inc., USA

Correspondence: Jesse A. Canchola, Roche Molecular Systems, Inc., 4300 Hacienda Drive, Pleasanton, CA 94588, USA

Received: October 30, 2017 | Published: November 16, 2017

Citation: Canchola JA, Tang S, Hemyari P, et al. Correct use of percent coefficient of variation (%CV) formula for log-transformed data. MOJ Proteomics Bioinform. 2017;6(4):316-317. DOI: 10.15406/mojpb.2017.06.00200

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Abstract

The coefficient of variation (CV) is a unit less measure typically used to evaluate the variability of a population relative to its standard deviation and is normally presented as a percentage.1 When considering the percent coefficient of variation (%CV) for log-transformed data, we have discovered the incorrect application of the standard %CV form in obtaining the %CV for log-transformed data. Upon review of various journals, we have noted the formula for the %CV for log-transformed data was not being applied correctly. This communication provides a framework from which the correct mathematical formula for the %CV can be applied to log-transformed data.

Keywords: coefficient of variation, log-transformation, variances, statistical technique

Abbreviations

CV, coefficient of variation; %CV, CV x 100%

Introduction

  1. The percent coefficient of variation, %CV, is a unitless measure of variation and can be considered as a “relative standard deviation” since it is defined as the standard deviation divided by the mean multiplied by 100 percent:

%CV=100% σ μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca GGLaGaam4qaiaadAfacqGH9aqpcaaIXaGaaGimaiaaicdacaGGLaGa eyyXIC9aaSaaaeaacqaHdpWCaeaacqaH8oqBaaaaaa@4436@ (1)

This formula (1) holds true for non-transformed data. The %CV calculation will be different mathematically depending on the mean and variance of the transformation.

If the untransformed %CV is used on log-normal data, the resulting %CV will be too small and give an overly optimistic, but incorrect, view of the performance of the measured device.

For example, Hatzakis et al.,1 Table 1, showed an assessment of inter-instrument, inter-operator, inter-day, inter-run, intra-run and total variability of the Aptima HIV-1 Quant Dx in various HIV-1 RNA concentrations. In Table 1, below, we recreate their total SD and %CV columns (the latter for which they use Formula (1), and calculate the correct log-normal %CV from Formula (7) below. From the Table 1, it can be seen that using the incorrect %CV formula for lognormally distributed data will give abnormally smaller %CVs.

Formula (1)

Formula (7)

Log Normal

Log Normal

Published
Incorrect

Correct

Level

N

Mean

Total SD

%CV

%CV

5.00E+01

41

1.66

0.144

8.67

34.1

1.00E+02

74

1.82

0.180

9.91

43.3

1.00E+03

81

2.75

0.112

4.08

26.2

1.00E+04

81

3.81

0.067

1.77

15.5

1.00E+05

81

4.96

0.067

1.35

15.5

1.00E+06

78

6.00

0.055

0.92

12.7

1.00E+07

81

6.89

0.062

0.90

14.3

Table 1 Recreation of portions of Table 5 from Hatzakis et al. (2016) and the correct calculation of lognormal %CV

To estimate variances of transformations of raw values, we use a statistical technique called the method of moments. Table 2 shows the variances standard deviations and %CVs for the untransformed and log-transformation one may consider.

The formula has been published previously in Nelson.2 The next section derives the correct percent coefficient of variation formula for the log-transformation in Table 2.

Transformation

Var[ f( x ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiaahAfacaWHHbGaaCOCa8aadaWadaqaa8qacaWHMbWd amaabmaabaWdbiaahIhaa8aacaGLOaGaayzkaaaacaGLBbGaayzxaa aaaa@40D5@

SD[ f( x ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiaahofacaWHebWdamaadmaabaWdbiaahAgapaWaaeWa aeaapeGaaCiEaaWdaiaawIcacaGLPaaaaiaawUfacaGLDbaaaaa@3FBA@

%CV %CV

None: X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiaadIfaaaa@3936@

Var( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiaadAfacaWGHbGaamOCamaabmaapaqaa8qacaWG4baa caGLOaGaayzkaaaaaa@3DB7@

Var( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbmaakaaapaqaa8qacaWGwbGaamyyaiaadkhadaqadaWd aeaapeGaamiEaaGaayjkaiaawMcaaaqabaaaaa@3DE6@

%CV=100% σ μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca GGLaGaam4qaiaadAfacqGH9aqpcaaIXaGaaGimaiaaicdacaGGLaGa eyyXIC9aaSaaaeaacqaHdpWCaeaacqaH8oqBaaaaaa@4436@

Log: lo g 10 X or ln( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiaadYeacaWGVbGaam4zaiaacQdacaqGGaGaamiBaiaa d+gacaWGNbWdamaaBaaabaWdbiaaigdacaaIWaaapaqabaWdbiaadI facaqGGaGaam4BaiaadkhacaqGGaGaamiBaiaad6gapaWaaeWaaeaa peGaamiwaaWdaiaawIcacaGLPaaaaaa@4996@

%CV(Y)=100% 10 ln(10) σ log 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiyjai aadoeacaWGwbGaaiikaiaadMfacaGGPaGaeyypa0JaaGymaiaaicda caaIWaGaaiyjaiabgwSixpaakaaabaGaaGymaiaaicdadaahaaqcfa sabeaaciGGSbGaaiOBaiaacIcacaaIXaGaaGimaiaacMcacqaHdpWC juaGdaWgaaqcfasaaiGacYgacaGGVbGaai4zaaqabaqcfa4aaWbaaK qbGeqabaGaaGOmaaaaaaqcfaOaeyOeI0IaaGymaaqabaaaaa@50F6@

Var( x ) ( ln( 10 ).E( x ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbmaalaaapaqaa8qadaGcaaWdaeaapeGaamOvaiaadgga caWGYbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaeqaaaWdae aapeWaaeWaa8aabaWdbiGacYgacaGGUbWaaeWaa8aabaWdbiaaigda caaIWaaacaGLOaGaayzkaaGaaiOlaiaadweadaqadaWdaeaapeGaam iEaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaaaaa@48FE@

%CV(Y)=100% 10 ln(10) σ log 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca GGLaGaam4qaiaadAfacaGGOaGaamywaiaacMcacqGH9aqpcaaIXaGa aGimaiaaicdacaGGLaGaeyyXIC9aaOaaaeaacaaIXaGaaGimamaaCa aabeqaaKqbGiGacYgacaGGUbGaaiikaiaaigdacaaIWaGaaiykaiab eo8aZLqbaoaaBaaabaGaciiBaiaac+gacaGGNbaabeaadaahaaqcfa sabeaacaaIYaaaaaaajuaGcqGHsislcaaIXaaabeaaaaa@51EF@

Table 2 Variances, SDS and %CV of log-transformation

σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiabeo8aZbaa@3A1C@ = standard deviation; μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiabeY7aTbaa@3A0F@ =mean; ln(•)=natural logarithm; σ log MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiabeo8aZ9aadaWgaaqcfasaa8qacaWGSbGaam4Baiaa dEgaaKqba+aabeaaaaa@3DF8@ = standard deviation of the log-transformed data; E( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiaadweapaWaaeWaaeaapeGaamiEaaWdaiaawIcacaGL Paaaaaa@3BD7@ is the expected value of x.

%CV for the log-normally distributed random variable (RV)

We show the derivation of the percent coefficient of variation (%CV) for a log-normally distributed random variable. The coefficient of variation for log-normally distributed random variable Y=ln(X) is estimated using the following formula:

% C V ( Y ) = 100 % e [ ln ( 10 ) ] 2 σ 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca GGLaGaam4qaiaadAfacaGGOaGaamywaiaacMcacqGH9aqpcaaIXaGa aGimaiaaicdacaGGLaGaeyyXIC9aaOaaaeaacaWGLbWaaWbaaeqaba WaamWaaeaaciGGSbGaaiOBaiaacIcacaaIXaGaaGimaiaacMcaaiaa wUfacaGLDbaadaahaaqabKqbGeaacaaIYaaaaKqbakabeo8aZnaaCa aabeqcfasaaiaaikdaaaaaaKqbakabgkHiTiaaigdaaeqaaaaa@5143@ Or its equivalent l o g b ( X ) = l o g c ( X ) l o g c ( b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiaadYgacaWGVbGaam4za8aadaWgaaqcfasaa8qacaWG IbaajuaGpaqabaWdbmaabmaapaqaa8qacaWGybaacaGLOaGaayzkaa Gaeyypa0ZaaSaaa8aabaWdbiaadYgacaWGVbGaam4za8aadaWgaaqc fasaa8qacaWGJbaajuaGpaqabaWdbmaabmaapaqaa8qacaWGybaaca GLOaGaayzkaaaapaqaa8qacaWGSbGaam4BaiaadEgapaWaaSbaaKqb GeaapeGaam4yaaqcfa4daeqaa8qadaqadaWdaeaapeGaamOyaaGaay jkaiaawMcaaaaaaaa@4FC2@

Where ln is the natural log and σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiabeo8aZ9aadaahaaqcfasabeaapeGaaGOmaaaaaaa@3B47@ is the variance. The derivation of the formulae follows.

Since the random variable X is log‑normally distributed, then Y = ln ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGzbGaeyypa0JaciiBaiaac6gadaqadaqaaiaadIfaaiaawIcacaGL Paaaaaa@3E67@ is distributed as a Normal probability distribution with mean µ and variance λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcq aH7oaBdaahaaqcfasabeaacaaIYaaaaaaa@3AF9@ , that is, Y ~ N ( μ , λ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGzbGaaiOFaiaad6eadaqadaqaaiabeY7aTjaacYcacqaH7oaBdaah aaqabKqbGeaacaaIYaaaaaqcfaOaayjkaiaawMcaaaaa@4229@ .

Now, the moment generating function for a Normal probability distribution is:3

M ( t ) = E ( e t Y ) = e μ t + λ 2 t 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGnbGaaiikaiaadshacaGGPaGaeyypa0JaamyraiaacIcacaWGLbWa aWbaaeqajuaibaGaamiDaiaadMfaaaqcfaOaaiykaiabg2da9iaadw gadaahaaqabeaacqaH8oqBcaWG0bGaey4kaSYaaSaaaeaacqaH7oaB daahaaqcfasabeaacaaIYaaaaKqbakaadshadaahaaqabKqbGeaaca aIYaaaaaqcfayaaiaaikdaaaaaaaaa@4E75@ (2)

Therefore, it follows by substitution:

C V ( Y ) = S D ( Y ) E ( Y ) = E ( e 2 Y ) [ E ( e Y ) ] 2 E ( e Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGdbGaamOvaiaacIcacaWGzbGaaiykaiabg2da9maalaaabaGaam4u aiaadseacaGGOaGaamywaiaacMcaaeaacaWGfbGaaiikaiaadMfaca GGPaaaaiabg2da9maalaaabaWaaOaaaeaacaWGfbGaaiikaiaadwga daahaaqabKqbGeaacaaIYaGaamywaaaajuaGcaGGPaGaeyOeI0Yaam WaaeaacaWGfbGaaiikaiaadwgadaahaaqabKqbGeaacaWGzbaaaKqb akaacMcaaiaawUfacaGLDbaadaahaaqabKqbGeaacaaIYaaaaaqcfa yabaaabaGaamyraiaacIcacaWGLbWaaWbaaeqajuaibaGaamywaaaa juaGcaGGPaaaaaaa@58B8@ = M ( 2 ) [ M ( 1 ) ] 2 M ( 1 ) = e 2 μ + 2 λ 2 e 2 μ + λ 2 e μ + λ 2 2 = e λ 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcq GH9aqpdaWcaaqaamaakaaabaGaamytaiaacIcacaaIYaGaaiykaiab gkHiTmaadmaabaGaamytaiaacIcacaaIXaGaaiykaaGaay5waiaaw2 faamaaCaaabeqcfasaaiaaikdaaaaajuaGbeaaaeaacaWGnbGaaiik aiaaigdacaGGPaaaaiabg2da9maalaaabaWaaOaaaeaacaWGLbWaaW baaeqabaqcfaIaaGOmaiabeY7aTjabgUcaRiaaikdacqaH7oaBjuaG daahaaqcfasabeaacaaIYaaaaaaajuaGcqGHsislcaWGLbWaaWbaae qabaqcfaIaaGOmaiabeY7aTjabgUcaRiabeU7aSLqbaoaaCaaajuai beqaaiaaikdaaaaaaaqcfayabaaabaGaamyzamaaCaaabeqaaKqbGi abeY7aTLqbakabgUcaRmaalaaabaGaeq4UdW2aaWbaaeqajuaibaGa aGOmaaaaaKqbagaacaaIYaaaaaaaaaGaeyypa0ZaaOaaaeaacaWGLb WaaWbaaeqajuaibaGaeq4UdWwcfa4aaWbaaKqbGeqabaGaaGOmaaaa aaqcfaOaeyOeI0IaaGymaaqabaaaaa@6A7C@ (3)

using the general statistical property that defines the variance as

V a r ( Y ) = E [ ( Y E [ Y ] ) 2 ] = E ( Y 2 ) [ E ( Y ) ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiaadAfacaWGHbGaamOCamaabmaapaqaa8qacaWGzbaa caGLOaGaayzkaaGaeyypa0Jaamyramaadmaapaqaa8qadaqadaWdae aapeGaamywaiabgkHiTiaadweadaWadaWdaeaapeGaamywaaGaay5w aiaaw2faaaGaayjkaiaawMcaa8aadaahaaqabKqbGeaapeGaaGOmaa aaaKqbakaawUfacaGLDbaacqGH9aqpcaWGfbWaaeWaa8aabaWdbiaa dMfapaWaaWbaaKqbGeqabaWdbiaaikdaaaaajuaGcaGLOaGaayzkaa GaeyOeI0YaamWaa8aabaWdbiaadweadaqadaWdaeaapeGaamywaaGa ayjkaiaawMcaaaGaay5waiaaw2faa8aadaahaaqcfasabeaapeGaaG Omaaaaaaa@57E6@ (4)

such that the standard deviation becomes

S D ( Y ) = E ( Y 2 ) [ E ( Y ) ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiaadofacaWGebWaaeWaa8aabaWdbiaadMfaaiaawIca caGLPaaacqGH9aqpdaGcaaWdaeaapeGaamyramaabmaapaqaa8qaca WGzbWdamaaCaaabeqcfasaa8qacaaIYaaaaaqcfaOaayjkaiaawMca aiabgkHiTmaadmaapaqaa8qacaWGfbWaaeWaa8aabaWdbiaadMfaai aawIcacaGLPaaaaiaawUfacaGLDbaapaWaaWbaaeqajuaibaWdbiaa ikdaaaaajuaGbeaaaaa@4AC6@ (5)

To simplify expression (5), above, we use the logarithm base change rule result4 that shows

l o g b ( X ) = l o g c ( X ) l o g c ( b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiaadYgacaWGVbGaam4za8aadaWgaaqcfasaa8qacaWG IbaajuaGpaqabaWdbmaabmaapaqaa8qacaWGybaacaGLOaGaayzkaa Gaeyypa0ZaaSaaa8aabaWdbiaadYgacaWGVbGaam4za8aadaWgaaqc fasaa8qacaWGJbaajuaGpaqabaWdbmaabmaapaqaa8qacaWGybaaca GLOaGaayzkaaaapaqaa8qacaWGSbGaam4BaiaadEgapaWaaSbaaKqb GeaapeGaam4yaaqcfa4daeqaa8qadaqadaWdaeaapeGaamOyaaGaay jkaiaawMcaaaaaaaa@4FC2@ for any logarithm base b and c. If b=10 and c = the “natural log base e” = e,then

log 10 ( X ) = log e ( X ) log e ( 10 ) = ln ( X ) ln ( 10 ) = Y ln ( 10 ) ~ N ( μ , σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGci GGSbGaai4BaiaacEgadaWgaaqcfasaaiaaigdacaaIWaaajuaGbeaa caGGOaGaamiwaiaacMcacqGH9aqpdaWcaaqaaiGacYgacaGGVbGaai 4zamaaBaaajuaibaGaamyzaaqcfayabaGaaiikaiaadIfacaGGPaaa baGaciiBaiaac+gacaGGNbWaaSbaaKqbGeaacaWGLbaajuaGbeaaca GGOaGaaGymaiaaicdacaGGPaaaaiabg2da9maalaaabaGaciiBaiaa c6gacaGGOaGaamiwaiaacMcaaeaaciGGSbGaaiOBaiaacIcacaaIXa GaaGimaiaacMcaaaGaeyypa0ZaaSaaaeaacaWGzbaabaGaciiBaiaa c6gacaGGOaGaaGymaiaaicdacaGGPaaaaiaac6hacaWGobGaaiikai abeY7aTjaacYcacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaKqbakaa cMcaaaa@6852@ ,     (6)

since Y = ln ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGzbGaeyypa0JaciiBaiaac6gadaqadaqaaiaadIfaaiaawIcacaGL Paaaaaa@3E67@ and, given that Y is distributed as a Normal probability distribution with mean µ and variance λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcq aH7oaBdaahaaqcfasabeaacaaIYaaaaaaa@3AF9@ , that is, Y ~ N ( μ , λ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGzbGaaiOFaiaad6eadaqadaqaaiabeY7aTjaacYcacqaH7oaBdaah aaqabKqbGeaacaaIYaaaaaqcfaOaayjkaiaawMcaaaaa@4229@ , this implies that λ 2 = [ ln ( 10 ) ] 2 σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcq aH7oaBdaahaaqcfasabeaacaaIYaaaaKqbakabg2da9maadmaabaGa ciiBaiaac6gacaGGOaGaaGymaiaaicdacaGGPaaacaGLBbGaayzxaa WaaWbaaKqbGeqabaGaaGOmaaaajuaGcqaHdpWCdaahaaqabKqbGeaa caaIYaaaaaaa@479A@ [using the statistical property that V A R ( a X )   =   a 2 V a r X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqa aaaaaaaaWdbiaadAfacaWGbbGaamOua8aadaqadaqaa8qacaWGHbGa amiwaaWdaiaawIcacaGLPaaapeGaaeiiaiabg2da9iaabccacaWGHb WdamaaCaaajuaibeqaa8qacaaIYaaaaKqbakaackcicaWGwbGaamyy aiaadkhacaWGybaaaa@47A1@ where a is a constant and X is a random variable].

Next, substituting this result into the formula for the %CV involving λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcq aH7oaBaaa@39ED@ and multiplying by 100% we obtain the final %CV expression:

% C V ( Y ) = 100 % e [ ln ( 10 ) ] 2 σ 2 1 = 100 % 10 ln ( 10 ) σ 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca GGLaGaam4qaiaadAfacaGGOaGaamywaiaacMcacqGH9aqpcaaIXaGa aGimaiaaicdacaGGLaGaeyyXIC9aaOaaaeaacaWGLbWaaWbaaeqaju aibaqcfa4aamWaaKqbGeaaciGGSbGaaiOBaiaacIcacaaIXaGaaGim aiaacMcaaiaawUfacaGLDbaajuaGdaahaaqcfasabeaacaaIYaaaai abeo8aZLqbaoaaCaaajuaibeqaaiaaikdaaaaaaKqbakabgkHiTiaa igdaaeqaaiabg2da9iaaigdacaaIWaGaaGimaiaacwcacqGHflY1da GcaaqaaiaaigdacaaIWaWaaWbaaKqbGeqabaGaciiBaiaac6gacaGG OaGaaGymaiaaicdacaGGPaGaeq4Wdmxcfa4aaWbaaKqbGeqabaGaaG OmaaaaaaqcfaOaeyOeI0IaaGymaaqabaaaaa@64FD@ (7)

Conclusion

The authors have shown that it is easy for the researcher to be confused with respect to which is the correct formula to use for log-transformed data when calculating the percent coefficient of variation (%CV). When using the incorrect formula, the researcher may be faced with abnormally low %CV values. With that in mind, the authors have shown the correct formula to use for calculating %CV for log-transformed data.

Acknowledgements

The authors thank Enrique Marino, Merlin Njoya and Jeff Vaks for reviewing the earlier work and providing useful comments. This work is supported by Roche Molecular Systems, Inc.

Conflict of interest

The author declares no conflict of interest.

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