Interpreting Comparisons on a Transformed Scale

Why a Difference on the Log Scale Is a Ratio on the Original Scale

When response values are positive and differences in spread tend to grow with the response level, an analysis may use the natural logarithm of each response. Pairwise comparisons then describe differences between group means on the log scale. Those differences can be translated back to the original scale, but not as ordinary additive differences: exponentiating a difference gives a ratio.

This distinction matters for interpretation. A log-scale comparison can support a statement such as “the geometric mean response in group A is 0.67 times that in group B,” or “about 33% lower.” It does not, by itself, say that the arithmetic mean in A is a specified number of response units lower than the arithmetic mean in B.

The direction of a comparison depends on its order. If the contrast is A minus B, then a negative log-scale difference corresponds to a geometric-mean ratio below 1 for A relative to B. A positive difference corresponds to a ratio above 1. Always name the numerator and denominator when reporting a ratio; “the ratio is 0.8” is incomplete unless the order is clear.

Model and Interpretation Conditions

Suppose the analysis uses a one-way ANOVA model for the transformed response \(Z=\ln(Y)\). For group \(i\), the model is \(Z_{ij}=\mu_i+\varepsilon_{ij}\), where \(\mu_i\) is the population mean log response and \(\varepsilon_{ij}\) is the error for observation \(j\) in that group. The usual ANOVA comparisons rely on independent observations, approximately normal errors within groups, and a common error variance across groups. Responses must be positive for the natural logarithm to be defined.

These are conditions for the transformed-scale model, not conditions that must hold for the raw responses to be normally distributed. Diagnostics should therefore be considered on the transformed-model residuals. The analysis also presumes that the comparisons and their multiplicity adjustment have been chosen appropriately for the family of comparisons being reported.

A simultaneous confidence interval for \(\mu_A-\mu_B\) has endpoints on the log scale. Exponentiating each endpoint gives an interval for the ratio \(\exp(\mu_A)/\exp(\mu_B)\). Because the exponential function is strictly increasing, the endpoint order is preserved, and the interval’s coverage carries over to the ratio scale.

For a point estimate \(d=\bar{z}_A-\bar{z}_B\), where \(\bar{z}_A\) and \(\bar{z}_B\) are the sample means of the logged responses, \(\exp(d)\) estimates the ratio of group geometric means. A ratio \(r\) can be expressed as a percentage difference from 1 using \(100(r-1)\%\). For example, \(r=0.80\) means the numerator group’s geometric mean is 20% lower than the denominator group’s; \(r=1.20\) means it is 20% higher.

Worked Examples

Why Exponentiating Works

The translation follows directly from the rules of logarithms. Write the two group geometric means as \(G_A=\exp(\mu_A)\) and \(G_B=\exp(\mu_B)\). Then

This identity establishes that a difference between mean log responses becomes a ratio of geometric means. The same reasoning applies to an interval: if \(L<\mu_A-\mu_B<U\), applying the increasing function \(\exp(\cdot)\) to all three terms gives \(\exp(L)<\exp(\mu_A-\mu_B)<\exp(U)\). The middle term is the geometric-mean ratio, so the transformed endpoints bound that ratio.

The interval is generally not symmetric around the exponentiated point estimate. For instance, equal-sized steps below and above zero on the log scale become reciprocal multiplicative changes when exponentiated. This asymmetry is expected: ratios are multiplicative, not additive.

Geometric Means Are Not Automatically Arithmetic Means

A frequent reporting error is to call \(\exp(\bar{z})\) the ordinary sample arithmetic mean of the original responses. In general it is not. It is the sample geometric mean: if the positive observations are \(y_1,\ldots,y_n\), then

Consequently, the direct interpretation of a log-scale group comparison is a ratio of geometric means. Do not replace that with a difference between arithmetic means in the original units. A separate model-based calculation may target arithmetic means under additional assumptions, but it is not obtained simply by exponentiating a mean log response.

Units also change how results should be stated. A ratio is dimensionless, so a ratio comparison does not have units such as milligrams or seconds. The original response units remain essential context for identifying what was measured, but a statement such as “the geometric mean concentration was 1.28 times as large” is more accurate than “it was 1.28 concentration units larger.”

A Practical Interpretation Sequence

Common Mistakes

• Reversing the comparison. An A-minus-B contrast exponentiates to A divided by B, not B divided by A. If the order is reversed, the ratio becomes its reciprocal.
• Calling a ratio an additive difference. A ratio of 0.80 means 20% lower, not 0.20 response units lower.
• Exponentiating only the point estimate. To report an interval on the original scale, exponentiate both endpoints as well as the estimate.
• Reporting a transformed mean as an arithmetic mean. Exponentiating a mean log response yields a geometric mean, not generally the arithmetic mean.
• Claiming equality or equivalence because an interval includes 1. Such an interval does not establish a difference at the stated confidence level, but it may still include important ratios on either side of 1.
• Ignoring the adjustment. If the interval is simultaneous for a family of comparisons, describe it as such. Its interpretation concerns the familywise procedure, not an unadjusted single comparison.