Geometric Means and Lognormal Pharmacokinetic Data

Why Summarize AUC on the Original and Log Scales?

AUC is positive, and pharmacokinetic observations often vary proportionally: differences are more naturally described as percentages or fold changes than as fixed additive amounts. A distribution with a long right tail can also make the arithmetic mean sensitive to a few large observations. Geometric means and geometric coefficients of variation (GCVs) provide summaries that reflect this multiplicative structure.

The earlier material on log-scale analysis of pharmacokinetic endpoints motivates taking logarithms before statistical analysis. Here the focus is the meaning and calculation of two descriptive summaries when AUC follows a lognormal model. The summaries do not, by themselves, establish that the model fits the data.

Definition: A positive random variable \(X\) is lognormal if \(Y=\ln(X)\) has a normal distribution, \(Y\sim N(\mu,\sigma^2)\). Its population geometric mean is \(\exp\{E[\ln(X)]\}=\exp(\mu)\). Its geometric coefficient of variation is \(\sqrt{\exp(\sigma^2)-1}\), usually reported as a percentage.

The logarithm here is the natural logarithm. For a sample of positive AUC values \(X_1,\ldots,X_n\), write \(Y_i=\ln(X_i)\). The sample geometric mean is the exponential of the mean of the logged observations. The sample GCV is obtained by substituting the sample standard deviation of those logged observations for \(\sigma\). Both summaries are expressed on the original AUC scale; the GCV is a percentage and therefore has no AUC units.

\bar{Y}=\frac{1}{n}\sum_{i=1}^n \ln(X_i),\qquad s_Y=\sqrt{\frac{\sum_{i=1}^n(\ln(X_i)-\bar{Y})^2}{n-1}}

\widehat{\mathrm{GM}}=\exp(\bar{Y}) =\left(\prod_{i=1}^n X_i\right)^{1/n},\qquad \widehat{\mathrm{GCV}}=100\sqrt{\exp(s_Y^2)-1}\%

The \(n-1\) denominator is used in the usual sample variance \(s_Y^2\). The resulting GCV is a plug-in estimate: it uses the observed log-scale variance to estimate the model’s \(\sigma^2\). It is not generally an unbiased estimate of the population GCV, especially in small samples.

What the Lognormal Model Implies

If \(\ln(X)\sim N(\mu,\sigma^2)\), then \(\exp(\mu)\) is both the population geometric mean and the median of \(X\). This follows because the normal distribution is symmetric about \(\mu\), so half its probability lies below \(\mu\); exponentiation preserves order. The arithmetic mean of \(X\), however, is \(\exp(\mu+\sigma^2/2)\), which exceeds \(\exp(\mu)\) whenever \(\sigma^2>0\). These are different summaries of a right-skewed distribution.

The GCV also has a direct interpretation under the model. It equals the population standard deviation of \(X\) divided by its arithmetic mean, expressed as a proportion. Thus, for an exactly lognormal population, the conventional coefficient of variation and the GCV have the same numerical value. The term GCV emphasizes that the calculation uses the spread of log values to describe multiplicative variability.

E(X)=\exp\left(\mu+\frac{\sigma^2}{2}\right),\qquad \operatorname{Var}(X)=\exp(2\mu+\sigma^2)\left(\exp(\sigma^2)-1\right)

These moment formulas give a short derivation of the GCV. Divide the variance by the squared arithmetic mean, then take the square root:

\frac{\operatorname{Var}(X)}{[E(X)]^2} = \frac{\exp(2\mu+\sigma^2)\{\exp(\sigma^2)-1\}} {\exp(2\mu+\sigma^2)} =\exp(\sigma^2)-1

\frac{\sqrt{\operatorname{Var}(X)}}{E(X)} =\sqrt{\exp(\sigma^2)-1}

This ratio is dimensionless. Multiplying by 100 reports it as a percentage. The derivation also shows why the GCV is not simply \(100\sigma\): \(\sigma\) is the standard deviation of the logarithms, while the GCV is a relative spread on the original scale.

Calculating the Summaries

Worked Example: Geometric Mean and GCV From Log AUC Values

Suppose five hypothetical AUC observations, expressed in mg·h/L, have natural logarithms \(4.20,\ 4.50,\ 4.80,\ 5.10,\ 5.40\). Treat the observations as independent draws from a lognormal population for the model-based interpretation, and calculate the sample geometric mean and GCV.

First calculate the mean log AUC:

\bar{Y}=\frac{4.20+4.50+4.80+5.10+5.40}{5} =\frac{24.00}{5}=4.800

Exponentiating gives the geometric mean:

\widehat{\mathrm{GM}}=\exp(4.800)=121.5\ \text{mg·h/L}

The deviations from 4.800 are \(-0.600,-0.300,0,0.300,0.600\). Their squared sum is \(0.360+0.090+0+0.090+0.360=0.900\). Therefore:

s_Y^2=\frac{0.900}{5-1}=0.2250,\qquad s_Y=\sqrt{0.2250}=0.4743

Substitute the log-scale sample variance into the GCV formula:

\widehat{\mathrm{GCV}} =100\sqrt{\exp(0.2250)-1} =100\sqrt{1.2523-1} =50.23\%

The sample geometric mean is 121.5 mg·h/L, and the plug-in GCV is 50.23%. The GCV is not a statement that each AUC lies within 50.23% of the geometric mean; it is a model-based measure of relative population spread.

Worked Example: A Smaller Log-Scale Standard Deviation

Four hypothetical AUC measurements have natural logarithms \(2.0,\ 2.2,\ 1.8,\ 2.0\). Assume the observations are independent and use the lognormal model to summarize them.

The mean log value is \((2.0+2.2+1.8+2.0)/4=2.000\), so the geometric mean is:

\widehat{\mathrm{GM}}=\exp(2.000)=7.389\ \text{mg·h/L}

The deviations are \(0,0.2,-0.2,0\), with squared sum \(0.0800\). Hence \(s_Y^2=0.0800/3=0.02667\). The estimated GCV is:

\widehat{\mathrm{GCV}} =100\sqrt{\exp(0.02667)-1} =100\sqrt{1.0270-1} =16.44\%

The calculations use the sample variance with \(df=n-1=3\), not the variance divided by \(n\). Using \(n\) would produce a smaller variance and a different plug-in GCV. The chosen estimator here is the conventional sample-variance-based estimate.

Worked Example: Comparing the Geometric and Arithmetic Means

Consider four hypothetical log AUC values \(3.8,\ 4.0,\ 4.2,\ 4.4\), again with AUC in mg·h/L. The mean log is \(4.1\), so the sample geometric mean is \(\exp(4.1)=60.34\) mg·h/L. The corresponding AUC values are approximately \(44.70,\ 54.60,\ 66.69,\ 81.45\) mg·h/L.

Their arithmetic mean is:

\bar{x}=\frac{44.70+54.60+66.69+81.45}{4} =\frac{247.44}{4}=61.86\ \text{mg·h/L}

The arithmetic mean is larger than the geometric mean, as expected for unequal positive observations. For the log-scale sample variance, the deviations from 4.1 are \(-0.3,-0.1,0.1,0.3\), so the squared sum is \(0.200\) and \(s_Y^2=0.200/3=0.06667\). Thus:

\widehat{\mathrm{GCV}} =100\sqrt{\exp(0.06667)-1} =100\sqrt{1.0689-1} =26.26\%

The geometric mean summarizes the center on the multiplicative scale, while the arithmetic mean summarizes the average AUC on the original additive scale. Which is appropriate depends on the scientific question and analysis plan; they are not interchangeable labels for the same quantity.

Assumptions, Diagnostics, and Limitations

For the descriptive sample geometric mean to be calculable, every AUC value must be strictly positive, because \(\ln(X_i)\) is undefined for zero or negative values. The lognormal interpretation adds a model assumption: log AUC values are drawn from a normal distribution with a common mean and variance. For the usual sampling interpretation, observations should also be independent and representative of the population being described. If observations are repeated within subjects or otherwise dependent, the simple sample standard deviation does not separate within- and between-subject variability.

Useful checks include a histogram or normal quantile plot of the logged AUC values, inspection for unusually influential observations, and confirmation that values were recorded in consistent units. The model need not fit perfectly for the geometric mean to be computed, but a strongly skewed or multimodal distribution of log values weakens the lognormal interpretation of the GCV. Small samples make diagnostics inconclusive and make the sample variance, and therefore the GCV estimate, uncertain.

Changing the measurement unit multiplies every AUC by the same positive constant. The geometric mean changes by that constant, as it should, but the GCV does not change: adding a constant to every log value changes \(\bar{Y}\) but leaves \(s_Y\) unchanged. This is one reason the GCV is useful for describing relative variability across measurements expressed in different units.

Summary	Calculation	Interpretation and scale
Geometric mean	\(\exp(\bar{Y})\)	Central summary on the multiplicative scale; has AUC units
Arithmetic mean	\(\bar{x}\)	Average AUC on the original additive scale; has AUC units
Geometric coefficient of variation	\(100\sqrt{\exp(s_Y^2)-1}\%\)	Estimated relative spread under a lognormal model; percentage

A common mistake is to exponentiate the log-scale standard deviation and call the result the GCV, or to report \(100s_Y\) as the GCV. Neither is the defined calculation. The correct expression is \(100\sqrt{\exp(s_Y^2)-1}\%\). Another mistake is to interpret the geometric mean as the arithmetic mean. Under the lognormal model, the arithmetic mean depends on both \(\mu\) and \(\sigma^2\), whereas the geometric mean is \(\exp(\mu)\).

Key takeaway: For positive AUC values, average the natural logarithms and exponentiate to obtain the sample geometric mean. Calculate the sample GCV by transforming the sample variance of those logarithms with \(100\sqrt{\exp(s_Y^2)-1}\%\), and interpret it as a relative-spread summary whose lognormal interpretation depends on model fit.

Check Your Understanding

Answer each question using the definitions and calculations above.

If the mean of four log AUC values is 3.0, what is the sample geometric mean?
Why must AUC observations be positive before taking natural logarithms?
A sample has log-scale sample variance \(0.04\). Write the plug-in GCV formula and approximate its value.
Under a lognormal model with \(\mu=2\) and \(\sigma^2=0.25\), which is larger: the population geometric mean or arithmetic mean? Give both formulas.

Solutions. (1) \(\exp(3.0)=20.09\) in the same units used for AUC. (2) The natural logarithm is defined for positive real values, not zero or negative values. (3) \(100\sqrt{\exp(0.04)-1}\%=100\sqrt{1.0408-1}\%=20.20\%\). (4) The geometric mean is \(\exp(2)=7.389\); the arithmetic mean is \(\exp(2+0.25/2)=\exp(2.125)=8.373\). The arithmetic mean is larger because the log-scale variance is positive.

Geometric Means and Lognormal Pharmacokinetic Data

What You'll Learn