Log-Scale Interaction for Quantitative Outcomes

Why the Scale Matters

For a positive quantitative outcome, a regression model can describe how the outcome changes on its original scale or how its logarithm changes. These choices answer different questions. In a model for the logarithm of the outcome, a genotype-by-exposure product term describes how the exposure-associated geometric-mean ratio differs across genotype values. Exponentiating that product-term coefficient gives a ratio of ratios.

This interpretation is useful when changes are naturally expressed proportionally, such as a percentage increase in a biomarker. It is important to state the scale explicitly: the interaction described here is on the log-outcome scale, not a difference between exposure-associated changes in the outcome’s arithmetic mean.

Model and Definitions

Suppose \(Y\) is a strictly positive quantitative outcome, \(G\) is a genotype code, and \(E\) is an exposure. For clarity, begin with binary coding: \(G=0\) or \(1\), and \(E=0\) or \(1\). A log-linear interaction model with covariates \(C\) is

\log(Y)=\beta_0+\beta_G G+\beta_E E+\beta_{GE}(G E)+\gamma^{T}C+\varepsilon .

The model assumes that the conditional mean of \(\log(Y)\), given \(G\), \(E\), and \(C\), follows the displayed linear expression, so that \(E(\varepsilon\mid G,E,C)=0\). It also assumes the outcome is positive, the chosen coding represents the contrasts of interest, and the model’s functional form is adequate for the data. To estimate a genotype-by-exposure contrast from observed data, there must be enough support for the relevant genotype and exposure combinations. Valid uncertainty estimates additionally require an appropriate standard-error method; they need not rely on normally distributed errors if a suitable robust method is used.

Definition: At fixed covariate values \(C\), the conditional geometric mean of \(Y\) is \(\exp\{E[\log(Y)\mid G,E,C]\}\). For binary \(E\), the exposure-associated geometric-mean ratio at genotype \(g\) is the conditional geometric mean at \(E=1\) divided by that at \(E=0\), holding \(C\) fixed.

The logarithm can formally be taken of a dimensionless outcome such as \(Y/y_0\), where \(y_0\) is a fixed reference value in the outcome’s units. Changing that reference changes the intercept, not the genotype, exposure, or interaction coefficients. In reporting results, name the original outcome and its units, and make clear when a result is a ratio rather than an absolute change.

Deriving the Ratio-of-Ratios Interpretation

Write the conditional mean of the log outcome at genotype \(g\), exposure \(e\), and fixed covariates \(C\) as \(\eta(g,e,C)\). The model gives

\eta(g,e,C)=\beta_0+\beta_Gg+\beta_Ee+\beta_{GE}ge+\gamma^TC .

For a binary exposure, subtract the log-scale prediction at \(E=0\) from the prediction at \(E=1\). Exponentiating this difference converts it into the exposure-associated geometric-mean ratio:

\begin{aligned} \eta(g,1,C)-\eta(g,0,C)&=\beta_E+\beta_{GE}g,\\ \mathrm{GMR}_{E\mid G=g,C}&=\exp(\beta_E+\beta_{GE}g). \end{aligned}

Thus the exposure-associated geometric-mean ratios for \(G=0\) and \(G=1\) are \(\exp(\beta_E)\) and \(\exp(\beta_E+\beta_{GE})\), respectively. Their ratio is

\frac{\mathrm{GMR}_{E\mid G=1,C}}{\mathrm{GMR}_{E\mid G=0,C}} =\frac{\exp(\beta_E+\beta_{GE})}{\exp(\beta_E)} =\exp(\beta_{GE}).

This derivation shows precisely what the product term means: \(\beta_{GE}\) is the difference between the two log exposure-associated geometric-mean ratios, while \(\exp(\beta_{GE})\) is the ratio of those geometric-mean ratios. If \(\beta_{GE}=0\), the model gives the same exposure-associated geometric-mean ratio at both genotype values. If \(\exp(\beta_{GE})>1\), the exposure-associated ratio is larger at \(G=1\) than at \(G=0\); if it is below 1, it is smaller.

Worked Example: Binary Genotype and Binary Exposure

Hypothetical model. Suppose \(Y\) is a positive biomarker measured in mg/L, and a fitted model with binary \(G\) and \(E\) has \(\beta_0=2.3026\), \(\beta_G=0.10\), \(\beta_E=0.20\), and \(\beta_{GE}=0.30\). For this calculation, take \(C\) to be fixed at its reference value, so its contribution is zero.

At \(G=0,E=0\), the fitted log value is \(2.3026\), and the conditional geometric mean is \(\exp(2.3026)\approx 10.00\) mg/L. At \(G=0,E=1\), the fitted log value is \(2.3026+0.20=2.5026\); the corresponding geometric mean is approximately \(12.21\) mg/L. Therefore, the exposure-associated geometric-mean ratio at \(G=0\) is

\exp(\beta_E)=\exp(0.20)\approx 1.221.

This is a 1.221-fold geometric-mean ratio, or about a 22.1% higher conditional geometric mean at \(E=1\) than at \(E=0\), at \(G=0\). At \(G=1\), the exposure-associated log contrast is \(0.20+0.30=0.50\), so

\mathrm{GMR}_{E\mid G=1}=\exp(0.50)\approx 1.649.

The ratio of the exposure-associated geometric-mean ratios is \(\exp(0.30)\approx1.350\). Equivalently, \(1.649/1.221\approx1.350\). In this hypothetical fitted model, the exposure-associated geometric-mean ratio at \(G=1\) is about 1.35 times the ratio at \(G=0\). This is an association described by the model; the product term by itself does not establish a causal effect of exposure.

Continuous Exposures and Genotype Dosage

The same algebra applies when \(E\) is continuous. In that case, \(\beta_E+\beta_{GE}g\) is the change in the conditional mean log outcome per one-unit increase in exposure at genotype value \(g\). Exponentiating it gives the geometric-mean ratio for a one-unit exposure increase. The units of that increase must be stated. If genotype is coded as allele dosage \(G=0,1,2\), the interaction coefficient describes the change in the log exposure contrast per one-unit increase in dosage.

Worked Example: Dosage and a Continuous Exposure

Hypothetical model. Let \(E\) be a daily exposure measured in hours and \(G\) be allele dosage coded \(0,1,2\). Suppose the fitted log-outcome coefficients are \(\beta_E=0.040\) per hour and \(\beta_{GE}=-0.015\) per dosage unit per hour. The geometric-mean ratio for a one-hour increase at dosage \(g\) is \(\exp(0.040-0.015g)\).

At \(G=0\), this ratio is \(\exp(0.040)\approx1.041\), about a 4.08% higher geometric mean per additional hour. At \(G=1\), it is \(\exp(0.025)\approx1.025\), about 2.53% higher per hour. At \(G=2\), it is \(\exp(0.010)\approx1.010\), about 1.01% higher per hour.

For each one-unit increase in genotype dosage, the exposure-associated geometric-mean ratio is multiplied by \(\exp(-0.015)\approx0.9851\). Thus, in this model, the ratio for a one-hour exposure increase is about 1.49% smaller for each additional dosage unit. That statement concerns the specified one-hour contrast; a \(k\)-hour increase instead has ratio \(\exp\{k(0.040-0.015g)\}\). A linear exposure term assumes the log-outcome contrast changes linearly with exposure over the range being interpreted.

Uncertainty on the Ratio Scale

A confidence interval for the product-term coefficient can be transformed to the ratio-of-ratios scale by exponentiating both endpoints. The transformation is increasing, so the endpoint order is preserved. For an approximate 95% Wald interval, the coefficient estimate is combined with its standard error using the normal critical value 1.96. The interval’s interpretation is about repeated-sampling coverage under the model and inference procedure, not a probability that a fixed parameter lies in this particular interval.

Worked Example: Confidence Interval for the Ratio of Ratios

Hypothetical estimate. In a binary-genotype, binary-exposure model, suppose \(\hat{\beta}_{GE}=0.30\) with \(SE(\hat{\beta}_{GE})=0.12\). The estimated ratio of exposure-associated geometric-mean ratios is \(\exp(0.30)\approx1.350\). An approximate 95% Wald interval for the log interaction is

0.30\ \pm\ 1.96(0.12) =0.30\ \pm\ 0.2352 =(0.0648,\ 0.5352).

Exponentiating the endpoints gives

\bigl(\exp(0.0648),\exp(0.5352)\bigr)\approx(1.067,\ 1.708).

The estimated exposure-associated geometric-mean ratio at \(G=1\), relative to the corresponding ratio at \(G=0\), is 1.350; the approximate 95% confidence interval for this ratio-of-ratios is 1.067 to 1.708. Since the interval excludes 1, it is inconsistent with a zero product-term coefficient at the corresponding two-sided 5% level under the Wald approximation. The test statistic is \(0.30/0.12=2.50\), giving a two-sided normal-approximation \(p\)-value of about 0.0124. This \(p\)-value is not the probability that the null hypothesis is true.

Retransformation: Geometric Means Are Not Automatically Arithmetic Means

The model is fitted to \(\log(Y)\). Exponentiating its fitted conditional log mean gives \(\exp\{E[\log(Y)\mid G,E,C]\}\), the conditional geometric mean as defined above. It does not in general give \(E[Y\mid G,E,C]\), the conditional arithmetic mean. In fact, writing the model as \(\log(Y)=\eta+\varepsilon\), with conditional mean-zero error, gives

E(Y\mid G,E,C)=\exp(\eta)\,E\{\exp(\varepsilon)\mid G,E,C\}.

The second factor depends on the residual distribution and can depend on genotype and exposure. If \(\varepsilon\) is conditionally normal with mean zero and constant variance \(\sigma^2\), then

E(Y\mid G,E,C)=\exp(\eta+\sigma^2/2).

Under that particular assumption, \(\exp(\eta)\) is the conditional geometric mean and median, while the arithmetic mean includes the variance correction \(\exp(\sigma^2/2)\). If the conditional residual variance changes across genotype or exposure groups, that correction also changes; simply exponentiating the fitted log mean will not recover the arithmetic mean. Without a suitable residual-distribution assumption or an appropriate retransformation method, an arithmetic-mean contrast cannot be read directly from the log-scale coefficients.

The product-term interpretation derived above is specifically about geometric-mean ratios. Under the normal, constant-variance assumption, the same variance correction cancels when comparing arithmetic means across groups, so the model’s ratios also describe arithmetic-mean ratios. That conclusion requires the stated assumption; it is not a general property of log-outcome regression.

Common Pitfalls and Interpretation

Calling \(\beta_{GE}\) itself the ratio of ratios. It is the difference between log exposure contrasts. The ratio of exposure-associated geometric-mean ratios is \(\exp(\beta_{GE})\).
Describing a ratio as an absolute change. A geometric-mean ratio of 1.22 means a 22% relative increase, not an increase of 1.22 outcome units. Absolute differences require predictions on the original scale and a clearly specified target quantity.
Calling an exponentiated prediction the arithmetic mean without justification. Exponentiating a fitted log mean gives a geometric mean. Recovering an arithmetic mean requires accounting for the residual distribution.
Assuming the interaction is causal. A product term estimates a model-based association. Causal interpretation requires additional design and identification assumptions; it does not follow from the coefficient or its \(p\)-value.
Ignoring coding and units. The exposure contrast depends on whether exposure is binary or continuous and, if continuous, on its unit. Genotype coding determines the genotype contrast represented by the interaction.

The log scale is valuable when proportional changes are scientifically meaningful, but an interaction is always tied to a scale and a contrast. Report the outcome, genotype and exposure coding, the exposure increment, the ratio-of-ratios, and the assumptions behind any arithmetic-mean interpretation.

Check Your Understanding

Answer these questions using the model and definitions in this tutorial.

For binary genotype and exposure, what does \(\exp(\beta_{GE})\) compare?
If \(\beta_{GE}=-0.20\), is the exposure-associated geometric-mean ratio larger or smaller at \(G=1\) than at \(G=0\), and by what multiplicative factor?
For a continuous exposure, what additional information is needed to interpret its geometric-mean ratio?
What quantity does exponentiating the fitted conditional mean of \(\log(Y)\) produce?
Under what residual assumption does a lognormal variance correction recover the conditional arithmetic mean?