Interpreting the interaction output in Multivariable Survival Analysis

This guide is a practical companion to Interaction terms in Multivariable Survival Analysis. That vignette shows how to build an interaction; this one shows how to read the results — what the two interaction tables mean, how the within-subgroup hazard ratios are computed, and why they differ slightly from running separate Cox models in each subgroup.

Throughout we use a concrete breast-cancer example: a set of candidate markers (AR and others) tested for whether their effect on overall survival depends on ER status.

Prognostic vs. predictive

Two different clinical claims hide behind the word “biomarker”:

A prognostic marker predicts outcome overall — higher AR is associated with worse (or better) survival, regardless of anything else.
A predictive marker’s effect depends on another factor — AR matters in ER-positive tumours but not in ER-negative ones (or vice versa).

A single Cox model fitted per marker cannot separate these. The interaction test can: it asks whether the marker’s hazard ratio is the same in both ER groups, or different. A different effect across groups is effect modification, and it is the statistical signature of a predictive biomarker.

The model

When you cross ER with a marker in the Interaction Terms builder, the module fits a single Cox model with the crossed term:

survival::coxph(survival::Surv(OS_days, OS_event) ~ ER * Marker, data = df)
# ER * Marker expands to:  ER + Marker + ER:Marker

The ER:Marker coefficient is the interaction. Exponentiated, it is a ratio of hazard ratios: how much the marker’s HR in ER-positive patients differs from its HR in ER-negative patients. If it equals 1, the marker behaves identically in both groups (not predictive). If it departs from 1, the marker’s effect is modified by ER.

Table 1 — Interaction (Effect-Modification) Test

This table has one row per crossed term and reports the formal test of whether the effect differs.

Interaction	HR	95% CI	p
`AR : ER`	0.42	0.22 – 0.80	0.008

HR is the ratio of AR’s hazard ratio between the ER groups. 0.42 means AR is far more protective in one ER stratum than the other.
HR = 1 would mean AR does the same thing regardless of ER — no modification, not predictive.
p is the p-for-interaction. Here p = 0.008, so AR’s effect on survival genuinely differs by ER status.

Read this table as the answer to the predictive-biomarker question: does the marker’s effect depend on ER? It tells you that the effect differs; the next table tells you how.

Table 2 — Within-Subgroup Hazard Ratios

This table gives the marker’s actual effect size inside each ER subgroup — the clinically usable numbers.

Interaction	Subgroup	Focal	HR	95% CI	p
`AR * ER`	ER0 (ER-negative)	AR	1.02	0.71 – 1.47	0.90
`AR * ER`	ER1 (ER-positive)	AR	0.43	0.26 – 0.72	<0.001

AR is inert in ER-negative tumours (HR 1.02, p = 0.90) but strongly protective in ER-positive tumours (HR 0.43, p < 0.001). AR is therefore a predictive marker within ER-positive disease — exactly what the significant interaction in Table 1 promised. Reporting only a pooled AR HR would have averaged these two very different realities into a misleading middle.

How the within-subgroup HRs are computed

The subgroup HRs are read straight out of the one interaction model — not from separate Cox models fitted in each ER group. The module uses relevel-and-refit:

Fit the full model Surv ~ AR * ER (plus any covariates you added) on all patients.
Relevel ER so a chosen level (ER0, then ER1) is the reference. This changes only the parameterisation, not the fit.
In the releveled model, the AR main-effect coefficient now is AR’s effect within that ER level. Its HR = exp(coef), with the CI and p from the model’s own variance.

Because releveling does not alter the fit, this is mathematically identical to reading β(AR) (within ER0) and β(AR) + β(AR:ER) (within ER1) off the single model — releveling is just the clean way to obtain each one together with its correct standard error. These are the model’s implied conditional effects.

Why they differ slightly from separate subgroup Cox models

If you fit AR in each ER subgroup independently, the point estimates (HRs) will be almost, but not exactly, the same, and the confidence intervals and p-values will differ a little. This is expected, and it follows from what each method assumes:

	Interaction model (what the module reports)	Separate Cox per ER group
Baseline hazard	common across ER (ER is a covariate)	each subgroup its own
Other covariates	effects assumed shared, estimated once	re-estimated within each group
Standard errors	borrow strength across the whole sample	use only that subgroup’s data
Consequence	narrower CIs, more power	wider CIs, less power

HRs nearly identical: with only ER * Marker, they differ only because the interaction model uses a common baseline hazard across ER while a subgroup-only fit uses an ER-specific baseline. Add other covariates and the small gap also reflects the interaction model assuming common covariate effects.
CIs and p differ more: the interaction model pools variance across both ER groups, so its intervals are tighter — the small difference you may notice against a manual subgroup check.

Both are valid, but they answer subtly different questions. The interaction-model (conditional) subgroup effect is the preferred, more efficient estimand for predictive-biomarker subgroup analysis, and it is what the module reports (noted in the table caption). Separate subgroup fits are a fine sanity check: expect matching point estimates and slightly wider intervals.

Getting the subgroup table to fill — focal and moderator

The builder reads a crossed term as focal : moderator. The moderator is the variable that defines the subgroups, and it must be categorical. The focal is the effect measured within those subgroups.

The module auto-detects the moderator: for a term A:B it uses whichever of A / B is categorical as the moderator (preferring the second variable when both are categorical). So:

AR : ER (ER categorical) → subgroups by ER, AR effect shown within each — the marker-by-ER question. Fills the table.
Marker : ER with a continuous marker → still fills: ER is the moderator, and the marker’s per-unit HR is shown within each ER subgroup.
If both variables in the term are continuous, there are no discrete subgroups to form, so the Within-Subgroup table is intentionally left empty (an on-screen note explains this). The interaction-test table (Table 1) fills either way.

To be sure a categorical variable is used as the moderator, simply include it in the crossed term; the module will assign the roles so the subgroups are formed by the categorical variable.

A reproducible check in R

To confirm the interaction HR for one marker against the module (matching the module’s complete-case handling for that marker):

library(survival); library(broom)

fit <- coxph(Surv(OS_days, OS_event) ~ ER * AR, data = na.omit(df[, c("OS_days","OS_event","ER","AR")]))

# interaction HR / CI / p (matches Table 1)
broom::tidy(fit, exponentiate = TRUE, conf.int = TRUE)[grepl(":", broom::tidy(fit)$term), ]

# within-subgroup AR HR, ER-positive as reference (matches Table 2, ER1 row)
df$ER <- relevel(factor(df$ER), ref = "ER1")
coxph(Surv(OS_days, OS_event) ~ ER * AR, data = df)   # read the AR coefficient

Cautions before concluding “predictive”

Power. Interaction tests need substantially more events than main-effect tests. A non-significant interaction in a modest sample is weak evidence of no modification, not proof of its absence.
Multiplicity. Scanning many markers inflates false positives. Adjust the p-for-interaction (for example, with an FDR correction) before calling any single marker predictive.
Complete cases. If you compare the module to a per-marker R script, note that the module uses complete cases across all selected variables, whereas a per-marker script drops rows missing that marker only. Match the row set when comparing a single marker, or N — and therefore the CI and p — can differ for that reason alone.

Summary

Table 1 tells you whether a marker’s effect on survival depends on ER (the predictive-biomarker test). Table 2 tells you how big the effect is inside each ER group, estimated from the same interaction model as its implied conditional effects. Include a categorical variable in the crossed term and the module forms the subgroups by it automatically.

Prognostic vs. predictive, the two interaction tables, and how the within-subgroup hazard ratios are estimated

ClinicoPath

2026-08-14