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Table 2 Average absolute bias \(\left|{\widehat{\beta }}_{TM}-{\beta }_{TM}\right|\) for the different confounding adjustment methods for confounding adjustment models (b)-(d)

From: The impact of moderator by confounder interactions in the assessment of treatment effect modification: a simulation study

   Adjustment model
Adjustment method (a) (b) (c) (d)
\({{\varvec{\beta}}}_{{\varvec{T}}{\varvec{M}}}=0.3\)
Prev.\({\varvec{M}}=0.5\)
(1) 0.066664 0.000773 0.000791 0.000791
(2) 0.082535 0.059600 0.058446 0.000765
(3) 0.063750 0.005581 0.001400 0.001400
(4) 0.055594 0.008806 0.004108 0.004226
\({{\varvec{\beta}}}_{{\varvec{T}}{\varvec{M}}}=0.3\)
Prev.\({\varvec{M}}=0.1\)
(1) 0.075085 0.001873 0.001715 0.001715
(2) 0.093259 0.062743 0.322948 0.001586
(3) 0.083261 0.017981 0.011239 0.011239
(4) 0.096792 0.038555 0.043617 0.038939
\({{\varvec{\beta}}}_{{\varvec{T}}{\varvec{M}}}=0.6\)
Prev.\({\varvec{M}}=0.5\)
(1) 0.066454 0.000816 0.000749 0.000749
(2) 0.081596 0.059213 0.056933 0.000941
(3) 0.060059 0.003119 0.002269 0.002269
(4) 0.052857 0.006547 0.007896 0.005279
\({{\varvec{\beta}}}_{{\varvec{T}}{\varvec{M}}}=0.6\)
Prev.\({\varvec{M}}=0.1\)
(1) 0.073871 0.000848 0.000925 0.000925
(2) 0.092722 0.056906 0.319295 0.00106
(3) 0.074819 0.013062 0.012955 0.012955
(4) 0.079721 0.036695 0.046072 0.041207
  1. Confounding models: (a) adjusting for no moderator-confounder interactions, (b) adjusting for the one moderator-confounder interaction, (c) adjusting for all possible moderator-confounder interactions, (d) subgroup-specific confounding adjustment
  2. Confounding methods: (1) Regression adjustment, (2) Propensity score covariate adjustment, (3) IPTW, (4) Propensity score matching