Menna Arafat

19 Log-scale ANCOVA

CRP is positive and strongly right-skewed, so a log-scale model is more appropriate than modeling raw CRP directly.

\[ \log(CRP_{12}) = \beta_0+\beta_1\log(CRP_0)+\beta_2I(A)+\beta_3I(B)+\varepsilon. \]

crp_anc <- dat |>
  filter(!is.na(crp_baseline_mg_L),!is.na(crp_week12_mg_L),
         crp_baseline_mg_L>0,crp_week12_mg_L>0)

crp_common <- lm(log(crp_week12_mg_L)~ log(crp_baseline_mg_L)+treatment_arm,data=crp_anc)

crp_int <- lm(log(crp_week12_mg_L)~ log(crp_baseline_mg_L)*treatment_arm,data=crp_anc)

hc3_interaction_test(crp_int)
#> 
#> Linear hypothesis test:
#> log(crp_baseline_mg_L):treatment_armDrug_A = 0
#> log(crp_baseline_mg_L):treatment_armDrug_B = 0
#> 
#> Model 1: restricted model
#> Model 2: log(crp_week12_mg_L) ~ log(crp_baseline_mg_L) * treatment_arm
#> 
#> Note: Coefficient covariance matrix supplied.
#> 
#>   Res.Df Df    F Pr(>F)
#> 1    332               
#> 2    330  2 1.05   0.35

The interaction is not needed, so the common-slope log model is retained.

19.1 Treatment effects as geometric-mean ratios

Because CRP is modeled on the log scale, emmeans() estimates baseline-adjusted treatment means on that scale. contrast() then compares each active treatment with Control.

crp_emm <- emmeans::emmeans(
  crp_common,~treatment_arm,
  vcov.=sandwich::vcovHC(crp_common,type="HC3")
)

crp_ctrl <- summary(
  contrast(crp_emm,method="trt.vs.ctrl",ref=1,adjust="holm"),
  infer=c(TRUE,TRUE)
) |>
  mutate(ratio=exp(estimate),
         ratio_low=exp(lower.CL),
         ratio_high=exp(upper.CL)) |>
  dplyr::select(contrast,ratio,ratio_low,ratio_high,p.value)

kbl(crp_ctrl,digits=3,
    caption="Adjusted CRP geometric-mean ratios")
Table 19.1: Adjusted CRP geometric-mean ratios
contrast ratio ratio_low ratio_high p.value
Drug_A - Control 0.904 0.853 0.958 0
Drug_B - Control 0.804 0.758 0.854 0

The treatment contrasts are first estimated as differences on the log scale. Exponentiating these differences converts them to ratios of adjusted geometric means:

\[ \text{Ratio} = \frac{\text{adjusted geometric mean in treatment}} {\text{adjusted geometric mean in Control}}. \]

A ratio of 1 indicates no difference between groups. A ratio of 0.80 means the active-treatment group has about 20% lower CRP, while a ratio of 1.25 means about 25% higher CRP, after adjustment for baseline CRP.

19.2 Diagnostics

par(mfrow=c(1,2)); plot(crp_common,which=1); plot(crp_common,which=2); par(mfrow=c(1,1))

19.3 Key interpretation

The meaning of a treatment effect depends on the model:

  • Standard ANCOVA: adjusted mean difference.
  • Interaction model: treatment effect depends on baseline value.
  • Log-scale ANCOVA: exponentiated treatment effect is a ratio of adjusted geometric means.

Always interpret the treatment effect according to the model that was fitted.

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