Table 1 Model-based simulation scenarios

From: The design and evaluation of hybrid controlled trials that leverage external data and randomization

Scenarios

Distribution of pre-treatment variables in the EC population

Effect of pre-treatment variables on the outcome in the EC (and HT) population

Response rates for the EC, IC, and EXPT

\({p}_{1}\)

\({p}_{2}\)

\({p}_{3}\)

\({\theta }_{S,1}\)

\({\theta }_{S,2}\)

\({\theta }_{S,3}\)

EC

IC and EXPT (TE = 0)

EXPT (TE > 0)

1

0.2

0.8

0.5

0.5

−0.5

0.0

0.43

0.50

0.68

2

0.2

0.8

0.1

0.5

−0.5

1.5

0.46

0.66

0.79

3

0.2

0.8

0.9

0.5

−0.5

1.5

0.73

0.66

0.79

4

0.2

0.8

0.1

0.5

−1.5(1.5)

1.5

0.30

0.66

0.79

5

0.2

0.8

0.9

0.5

−1.5(1.5)

1.5

0.55

0.66

0.79

  1. We consider three binary pre-treatment variables X = (X1, X2, X3). The variable \({X}_{3}\) is not available and is not used in the interim and final analyses. For patients enrolled in the hybrid trial (HT), the three pre-treatment variables are independent, with \({p}(X_{j}=1)=0.5\) for \(j={{{{\mathrm{1}}}}},\,{{{{\mathrm{2}}}}},\,{{{{\mathrm{3}}}}}.\) Columns 2–4 report the distribution \({p}(X_{j}=1)\) of the three independent variables in the external control (EC) population. Patient outcomes Y, given the pre-treatment variables, were randomly generated from a logistic model, \(p\left(\right.Y=1\left|X,\,A,\,S\right)=F\left(\delta A+X^{\prime} {\theta }_{S}\right),\,{A}={{{{\mathrm{0,\,1}}}}}\) and \(S={{{{\mathrm{0,\,1}}}}},\) where \(F\left(t\right)=1/(1+{{\exp }}\{-t\})\). Columns 5–7 show the effects (\({\theta }_{S,j}\), log odds ratio) of the pre-treatment variables \({X}_{j}\) on the expected outcome Y in the EC (S = 1) and HT (S = 0) populations. When \({\theta }_{0,j}={\theta }_{1,j}\) we omit the value in parenthesis (\({\theta }_{0,j}\)). The treatment effect (TE, log odds ratio) for ineffective and effective experimental treatments equals \(\delta=0,\, 0.8\). Columns 8–10 show the average response probability for the EC \((A=0,\, S=1)\), the internal control (IC) \((A=0,\, S=0)\), and the experimental treatment (EXPT, \(A=1,\, S=0\)) populations with and without treatment effects.