Interaction Scales For Drug Response

Why the Scale Matters

A treatment can have different effects across genotype groups, but the answer to “different” depends on how the effect is measured. For a binary outcome, the treatment effect might be a change in probability, a ratio of probabilities, or a ratio of odds. For a quantitative outcome, it might be a difference in mean response or a ratio of mean responses. These are distinct summaries, not interchangeable descriptions of one universal interaction.

This matters in pharmacogenomics because a conclusion about effect modification can change with the scale. For example, treatment may produce the same absolute increase in response probability across genotypes while producing different relative increases. Scale choice should therefore reflect the scientific question and the outcome’s meaningful measurement scale, and it should be stated alongside any interaction result.

Definition: Let $T=1$ denote treatment and $T=0$ denote a comparator, and let $G=1$ and $G=0$ denote two genotype groups. For a binary outcome, write $p_{tg}=P(Y=1\mid T=t,G=g)$. The additive interaction is the difference between the two genotype-specific treatment risk differences: $$ \delta_{\mathrm{add}}=(p_{11}-p_{01})-(p_{10}-p_{00}). $$ When all risks in the ratios are positive, the multiplicative interaction is the ratio of the two genotype-specific treatment risk ratios: $$ \delta_{\mathrm{mult}}=\frac{p_{11}/p_{01}}{p_{10}/p_{00}}. $$ No additive interaction means $\delta_{\mathrm{add}}=0$; no multiplicative interaction means $\delta_{\mathrm{mult}}=1$.

The first subscript on $p_{tg}$ is treatment status and the second is genotype group. Thus, $p_{11}$ is the outcome risk among treated people in genotype group 1. These contrasts concern treatment effects within genotype groups; they do not imply that genotype itself is a treatment that can be assigned. If the outcome is an adverse event rather than a beneficial response, the signs and direction of the interpretations should be described accordingly.

Binary Outcomes: Risk Differences and Ratios

The additive interaction asks whether the treatment-associated change in outcome probability is the same in both genotype groups. Its units are probability points. For example, a risk difference of $0.10$ is an increase of 10 percentage points. The multiplicative interaction asks whether the treatment risk ratio is the same in both groups. Its scale is a ratio of ratios and has no units.

A ratio-based analysis needs positive denominator risks for the risk ratios to be defined. When risks are zero, the multiplicative contrast may be undefined; this is a limitation of that estimand, not a reason to silently substitute a different scale. The risk difference remains defined for risks between zero and one.

Worked Example: One Binary Outcome, Three Scales

In a hypothetical randomized trial, response is recorded for 100 people in each treatment-by-genotype group. Suppose the counts are: genotype 0, 10 responses among 100 comparator recipients and 20 among 100 treated recipients; genotype 1, 40 among 100 comparator recipients and 60 among 100 treated recipients. These are illustrative data, not results from a real study.

The four risks are $p_{00}=10/100=0.10$, $p_{10}=20/100=0.20$, $p_{01}=40/100=0.40$, and $p_{11}=60/100=0.60$. For genotype 0, the treatment risk difference is $0.20-0.10=0.10$; for genotype 1, it is $0.60-0.40=0.20$. Therefore:

\delta_{\mathrm{add}}=(0.60-0.40)-(0.20-0.10)=0.20-0.10=0.10.

On the additive scale, the treatment-associated increase in response risk is 10 percentage points larger in genotype 1 than in genotype 0. The treatment risk ratios are $0.20/0.10=2.00$ and $0.60/0.40=1.50$, so:

\delta_{\mathrm{mult}}=\frac{1.50}{2.00}=0.75.

On the multiplicative risk scale, the treatment risk ratio in genotype 1 is 0.75 times the ratio in genotype 0. This does not mean treatment reduces response in genotype 1: both observed risk ratios exceed 1. It means the relative increases differ.

A third scale gives another answer. The odds are $p/(1-p)$. The treatment odds ratio in genotype 0 is $(0.20/0.80)/(0.10/0.90)=2.25$. In genotype 1 it is $(0.60/0.40)/(0.40/0.60)=2.25$. Thus the ratio of odds ratios is $2.25/2.25=1$: there is no interaction on the odds-ratio scale in these data. The additive, risk-ratio, and odds-ratio conclusions are all arithmetically compatible because they measure different contrasts.

For binary outcomes, logistic regression uses the log odds as its modeled scale. With a binary treatment and genotype, a model of the form

\log\left(\frac{P(Y=1\mid T,G)}{1-P(Y=1\mid T,G)}\right) =\beta_0+\beta_TT+\beta_GG+\beta_{TG}TG

has $\exp(\beta_{TG})$ equal to the ratio of the treatment odds ratio in genotype 1 to the treatment odds ratio in genotype 0. It is not, in general, the ratio of risk ratios or the additive interaction. In the worked example, that odds-ratio interaction is 1 even though the risk-difference interaction is $0.10$. When outcomes are common, odds ratios can also differ substantially in magnitude from risk ratios, so the modeled scale should be named explicitly.

Quantitative Outcomes: Differences and Ratios

For a quantitative outcome, let $\mu_{tg}=E(Y\mid T=t,G=g)$ be the mean outcome in treatment-by-genotype group $t,g$. The additive interaction is

\delta_{\mathrm{add}}=(\mu_{11}-\mu_{01})-(\mu_{10}-\mu_{00}).

It compares treatment-associated mean differences and is expressed in the outcome’s units. If $Y$ is a score measured in points, the interaction is measured in points. For strictly positive means, a multiplicative interaction can instead compare treatment mean ratios:

\delta_{\mathrm{mult}}=\frac{\mu_{11}/\mu_{01}}{\mu_{10}/\mu_{00}}.

This ratio of ratios is dimensionless. It requires positive comparator means in both genotype groups. It is meaningful only when a ratio scale is scientifically interpretable; a change in the outcome’s units by a positive constant leaves ratios unchanged, but adding a constant to every outcome generally changes them.

Worked Example: Equal Mean Differences, Unequal Mean Ratios

Suppose a hypothetical continuous drug-response measure is recorded in units, with group means of 10 and 12 in comparator and treatment groups for genotype 0, and 20 and 22 for genotype 1.

For genotype 0, the treatment mean difference is $12-10=2$ units. For genotype 1, it is $22-20=2$ units. Thus the additive interaction is $(22-20)-(12-10)=2-2=0$ units: there is no interaction on the mean-difference scale.

The treatment mean ratios are $12/10=1.20$ for genotype 0 and $22/20=1.10$ for genotype 1. The multiplicative interaction is $1.10/1.20=0.9167$. On the ratio scale, the treatment-associated mean ratio in genotype 1 is about 8.33% lower than in genotype 0. This is not a contradiction with equal two-unit mean increases; the comparator means differ.

Worked Example: Equal Mean Ratios, Unequal Mean Differences

Now suppose the hypothetical group means are 10 and 12 for genotype 0, and 20 and 24 for genotype 1, again in the same positive response units.

The treatment mean differences are $12-10=2$ units in genotype 0 and $24-20=4$ units in genotype 1. The additive interaction is $4-2=2$ units, so treatment-associated mean differences are not equal across genotype groups.

The treatment mean ratios are $12/10=1.20$ and $24/20=1.20$. Their ratio is $1.20/1.20=1$, indicating no multiplicative interaction. Thus a 20% increase in each group corresponds to different absolute increases because the comparator means differ.

Models and a Scale-Specific Justification

For a quantitative outcome, a linear mean model with an interaction term is

E(Y\mid T,G)=\beta_0+\beta_TT+\beta_GG+\beta_{TG}TG.

The mean in group $T=0,G=0$ is $\beta_0$; the treatment mean difference for genotype 0 is $\beta_T$. For genotype 1, the treatment mean difference is $\beta_T+\beta_{TG}$. Subtracting these differences gives $\beta_{TG}$, exactly the additive interaction. Therefore, the interaction coefficient in this identity-link model is measured in outcome units and represents a difference of mean differences.

For positive quantitative means, a log-mean model, $\log E(Y\mid T,G)=\beta_0+\beta_TT+\beta_GG+\beta_{TG}TG$, makes $\exp(\beta_{TG})$ the ratio of treatment mean ratios. The interpretation differs because the model is additive on the log-mean scale, which corresponds to a multiplicative scale on the original mean scale.

The same logic applies to binary risks if a log link is used to model the log risk: the interaction coefficient exponentiates to a ratio of risk ratios, provided the model specifies valid risks. A logistic model instead models log odds and yields a ratio of odds ratios. The link function determines the scale of the interaction coefficient; including a product term alone does not define a scale-independent interaction.

Justification: Suppose the treatment mean ratio is the same in both genotype groups, equal to $r$. Then $\mu_{10}=r\mu_{00}$ and $\mu_{11}=r\mu_{01}$. The additive interaction is $$ (\mu_{11}-\mu_{01})-(\mu_{10}-\mu_{00}) =(r-1)\mu_{01}-(r-1)\mu_{00} =(r-1)(\mu_{01}-\mu_{00}). $$ This is generally nonzero when $r\ne1$ and the comparator means differ. Hence no multiplicative interaction need not imply no additive interaction. Conversely, equal additive treatment differences can yield different mean ratios when the comparator means differ.

Interpretation, Assumptions, and Common Pitfalls

Interaction estimates describe the population and outcome definition to which they refer. Interpretation as causal treatment-effect modification requires a well-defined treatment and outcome, consistency between assigned and received treatment as specified by the study, and no interference between participants. In a randomized trial, randomization supports unbiased comparisons of treatment within genotype groups, assuming treatment assignment and outcome measurement are properly implemented. In nonrandomized data, treatment comparisons within genotype groups additionally require exchangeability after conditioning on appropriate covariates and positivity: each relevant covariate-genotype group must have a positive probability of receiving either treatment. These conditions do not make genotype itself an intervention.

The interaction scale should be selected before interpreting the result, based on the target question. Absolute differences can be useful for understanding changes in event frequency or response units; ratios can express proportional changes. Report genotype-specific treatment effects as well as the interaction contrast, with uncertainty measures when performing inference. A small interaction estimate is not automatically evidence of no meaningful heterogeneity, and a numerically large contrast should be judged in context and with its uncertainty.

Common mistake: “There is an interaction, so treatment works differently.” This statement is incomplete unless it names the scale. State, for example, that treatment risk differences differ, or that treatment odds ratios differ.
Common mistake: treating odds ratios as risk ratios. An odds ratio is a ratio of odds, not a ratio of probabilities. For common outcomes, the numerical difference can be substantial.
Common mistake: assuming a product term has one universal interpretation. Its meaning depends on the outcome model and link function. Identify the model scale and translate the coefficient accordingly.
Common mistake: comparing interaction estimates across studies without checking scales. A difference of risk differences and a ratio of risk ratios are not directly comparable. Align the outcome definition, genotype contrast, treatment contrast, and interaction scale first.

Key takeaway: Interaction is scale-dependent. For binary outcomes, distinguish differences in risks, ratios of risks, and ratios of odds. For quantitative outcomes, distinguish differences in means from ratios of means. Report the scale, the group-specific treatment effects, and the units or ratio interpretation.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

In the binary worked example, what does an additive interaction of $0.10$ mean in percentage-point terms?
Why can a logistic interaction coefficient not generally be interpreted as a ratio of risk ratios?
For a positive quantitative outcome, what does a mean ratio of 1.25 represent, and what are its units?
Can there be no interaction on the mean-difference scale but interaction on the mean-ratio scale? Explain using the relevant contrasts.
Name two assumptions needed to interpret treatment comparisons within genotype groups causally.