Introduction

Effect modification is specified inside dose() with the modifier argument. Formula modifiers can be binary numeric/logical variables or factors. Character variables are not accepted as modifiers; convert them to factors first so the level order is explicit.

There are two useful ways to code effect modification:

modifier = ~ M uses reference-plus-contrast coding.
modifier = ~ 0 + M or modifier = ~ M - 1 uses subgroup-specific coding.

These are reparameterizations of the same model. They give the same fitted likelihood when the same model and data are used, but the coefficients answer different questions directly.

Reference-plus-contrast coding

Reference-plus-contrast coding is the default formula-modifier coding. For a binary modifier M1, the coefficient named dose is the dose effect in the reference group, and dose:M1 is the difference between the non-reference group and the reference group.

fit_contrast <- ameras(
  Y.gaussian ~ dose(V1:V10, modifier = ~ M1) + X1 + X2,
  data = data,
  family = "gaussian",
  methods = "RC"
)

coef(fit_contrast)
#>                     RC
#> (Intercept) -1.3602199
#> X1           0.4788215
#> X2          -0.5192733
#> dose         1.0836911
#> dose:M1      0.1566173
#> sigma        1.1021197

This coding is convenient when the effect-modification contrast is the main quantity of interest.

Subgroup-specific coding

Subgroup-specific coding reports one dose-effect parameter per subgroup. For the same binary modifier M1, the coefficient names are dose[M1=0] and dose[M1=1].

fit_subgroup <- ameras(
  Y.gaussian ~ dose(V1:V10, modifier = ~ 0 + M1) + X1 + X2,
  data = data,
  family = "gaussian",
  methods = "RC"
)

coef(fit_subgroup)
#>                     RC
#> (Intercept) -1.3602200
#> X1           0.4788212
#> X2          -0.5192734
#> dose[M1=0]   1.0836912
#> dose[M1=1]   1.2403085
#> sigma        1.1021195

The two codings are equivalent. The non-reference subgroup effect is the reference effect plus the contrast:

contrast_coef <- fit_contrast$RC$coefficients
subgroup_coef <- fit_subgroup$RC$coefficients

data.frame(
  term = c("reference group", "non-reference group"),
  from_contrast = c(
    contrast_coef["dose"],
    contrast_coef["dose"] + contrast_coef["dose:M1"]
  ),
  from_subgroup = c(
    subgroup_coef["dose[M1=0]"],
    subgroup_coef["dose[M1=1]"]
  ),
  row.names = NULL
)
#>                  term from_contrast from_subgroup
#> 1     reference group      1.083691      1.083691
#> 2 non-reference group      1.240308      1.240309

Subgroup-specific coding is usually the cleaner choice when subgroup-specific effect estimates and confidence intervals are the main goal. For linear ERR and linear-exponential models, it can also be more numerically robust because default bounds are applied directly to each subgroup dose-effect parameter. With reference-plus-contrast coding, contrast parameters are not bounded by the default transformation.

Factor modifiers

Factor modifiers use the existing level order of the factor. The first level is the reference level.

data$M_factor <- factor(
  ifelse(data$M1 == 0, "unexposed modifier group", "modifier group"),
  levels = c("unexposed modifier group", "modifier group")
)

With modifier = ~ M_factor, ameras reports contrast terms for the non-reference levels.

fit_factor_contrast <- ameras(
  Y.gaussian ~ dose(V1:V10, modifier = ~ M_factor) + X1 + X2,
  data = data,
  family = "gaussian",
  methods = "RC"
)

coef(fit_factor_contrast)
#>                                      RC
#> (Intercept)                  -1.3602199
#> X1                            0.4788215
#> X2                           -0.5192733
#> dose                          1.0836911
#> dose:M_factor=modifier group  0.1566173
#> sigma                         1.1021197

With modifier = ~ 0 + M_factor, ameras reports subgroup-specific effects for all levels.

fit_factor_subgroup <- ameras(
  Y.gaussian ~ dose(V1:V10, modifier = ~ 0 + M_factor) + X1 + X2,
  data = data,
  family = "gaussian",
  methods = "RC"
)

coef(fit_factor_subgroup)
#>                                                 RC
#> (Intercept)                             -1.3602200
#> X1                                       0.4788212
#> X2                                      -0.5192734
#> dose[M_factor=unexposed modifier group]  1.0836912
#> dose[M_factor=modifier group]            1.2403085
#> sigma                                    1.1021195

Multi-level factors are supported. With reference-plus-contrast coding, ameras reports one contrast for each non-reference level. With subgroup-specific coding, ameras reports one dose-effect parameter for each factor level.

data$M3 <- factor(
  rep(c("low", "middle", "high"), length.out = nrow(data)),
  levels = c("low", "middle", "high")
)

fit_three_level <- ameras(
  Y.gaussian ~ dose(V1:V10, modifier = ~ 0 + M3) + X1 + X2,
  data = data,
  family = "gaussian",
  methods = "RC"
)

coef(fit_three_level)
#>                         RC
#> (Intercept)     -1.3619937
#> X1               0.4813357
#> X2              -0.5196477
#> dose[M3=low]     1.1433969
#> dose[M3=middle]  1.1715892
#> dose[M3=high]    1.1807593
#> sigma            1.1073548

Confidence intervals

For subgroup-specific confidence intervals, fit the model using subgroup-specific coding and then call confint() as usual. For RC, ERC, and MCML, profile likelihood intervals treat subgroup effects as ordinary coefficient rows.

fit_subgroup <- confint(
  fit_subgroup,
  type = "proflik",
  parm = "dose"
)

summary(fit_subgroup)

For FMA and BMA, sample-based confidence or credible intervals are computed from the sampled coefficients. When subgroup-specific coding is used, BMA priors and MCMC sampling remain on the internal reference-plus-contrast scale, but stored samples, summaries, diagnostics, trace plots, and sample-based intervals are reported on the subgroup-specific scale.

Choosing a coding

Use reference-plus-contrast coding when the contrast itself is the main estimand. Use subgroup-specific coding when subgroup effects are the main estimands, when subgroup-specific profile likelihood intervals are desired, or when ERR/LINEXP optimization is sensitive to the unbounded contrast parameterization.

Testing whether effect modification improves the model is a model-comparison question. Fit a model without the modifier and a model with the modifier, then compare the fitted likelihoods using the appropriate degrees of freedom. dose_lrt() tests dose-related parameters, but its global dose test is not the same as a test for effect modification.