CoRpower’s Algorithms for Simulating Placebo Group and Baseline Immunogenicity Predictor Data

Trichotomous  X and  S(1) Using Approach 1

1. Specify P₀^lat, P₂^lat, P₀, P₂, risk₀, n_cases, 0, n_{controls, 0}, K
   • N_{complete, 0} = n_cases, 0 + n_{controls, 0}
2. Specify Sens, Spec, FP⁰, and FN²
3. Number of observations in each latent subgroup: N_x = N_{complete, 0}P_x^lat
4. Simulate X under the assumption of homogeneous risk in the placebo group:
   • Cases: (n_cases, 0(0), n_cases, 0(1), n_cases, 0(2)) ∼ Mult(n_cases, 0, (p₀, p₁, p₂)), where
   • Controls: (n_{controls, 0}(0), n_{controls, 0}(1), n_{controls, 0}(2)) ∼ Mult(n_{controls, 0}, (p₀, p₁, p₂)), where
   • n_{controls, 0}(x) = N_x − n_cases, 0(x)
5. Simulate Y: Vector with n_cases, 0(0) 1’s, followed by n_{controls, 0}(0) 0’s, followed by n_cases, 0(1) 1’s, etc.
6. Simulate S(1): For each of the N_x subjects, generate S(1) by a draw from Mult(1, (p₀, p₁, p₂)), where p_k = P(S(1) = k|X = x) is given by Sens, Spec, etc.
   

Trichotomous  X and  S(1) Using Approach 2

1. Specify P₀^lat, P₂^lat, P₀, P₂, risk₀, N_{complete, 0}, n_cases, 0, n_cases^S, K
2. Specify ρ and σ_obs²
3. Calculation of (Sens, Spec, FP⁰, FP¹, FN¹, FN²):
   1. Assuming the classical measurement error model, where X^* ∼ N(0, σ_tr²), solve P₀^lat = P(X^* ≤ θ₀)  and  P₂^lat = P(X^* > θ₂) for θ₀ and θ₂
   2. Generate B realizations of X^* and S^* = X^* + e, where e ∼ N(0, σ_e²), and X^* independent of e + B = 20, 000 by default

   3. Using θ₀ and θ₂ from Step i., define

      Estimate Spec(ϕ₀) by $$\widehat{Spec}(\phi_0) = \frac{\#\{S^{\ast}_b \leq \phi_0, X^{\ast}_b \leq \theta_0\}}{\#\{X^{\ast}_b \leq \theta_0\}}\,$$ etc.
   4. Find ϕ₀ = ϕ₀^* and ϕ₂ = ϕ₂^* that numerically solve and compute $$ Spec = \widehat{Spec}(\phi^{\ast}_0),\; Sens = \widehat{Sens}(\phi^{\ast}_2),\; \textrm{etc.} $$
4. Follow Steps 3–6 under Approach 1

Continuous  X^* and  S^*(1)

1. Specify P_lowestVE^lat, ρ, σ_obs², VE_lowest, risk₀, n_cases, 0, n_{controls, 0}, n_cases^S, K
   • N_{complete, 0} = n_cases, 0 + n_{controls, 0}
2. Simulate Y by creating a vector with n_cases, 0 1’s followed by n_{controls, 0} 0’s.
3. Simulate X^* under the assumption of homogeneous risk in the placebo group:
   • Cases: from a grid of values ranging from -3 to 3, sample n_cases, 0 with replacement from:
   • Controls: from a grid of values ranging from -3 to 3, sample n_{controls, 0} with replacement from:
   • f_X^*(x^*) is fully specified because X^* ∼ N(0, σ_tr²)
4. Simulate S^*(1): S^*(1) = X^* + ϵ, where ϵ ∼ N(0, σ_e²) and σ_e² = (1 − ρ)σ_obs². ϵ is independent of X^* and is simulated by rnorm(Ncomplete, mean=0, sd=sqrt(sigma2e))

Trichotomous  X, S(1), and  BIP Using Approach 1

1. The user specifies a classification rule defined by P(BIP = i ∣ S(1) = j), i, j = 0, 1, 2.
2. For a subject with biomarker measurement S_k(1), generate BIP_k by a draw from Mult(1, (q₀, q₁, q₂)), where q_i = P(BIP_k = i ∣ S(1) = S_k(1)), i = 0, 1, 2.

Trichotomous  X, S(1), and  BIP Using Approach 2

Note: All variables with * are continuous.

1. The user specifies corr (BIP^*, S^*(1)).
2. Assuming that BIP^* follows an additive measurement error model, i.e., BIP^* := S^*(1) + δ, where δ ∼ N(0, σ_δ²) with an unknown σ_δ², and δ, ϵ, and X^* are independent, solve the following equation for var δ = σ_δ²: $$ \mathop{\mathrm{corr}}(BIP^*, S^*(1)) = \sqrt\frac{\mathop{\mathrm{var}}X^* + \mathop{\mathrm{var}}\epsilon}{\mathop{\mathrm{var}}X^* + \mathop{\mathrm{var}}\epsilon + \mathop{\mathrm{var}}\delta} $$
3. For the fixed ϕ₀^* and ϕ₂^* derived above, define
4. Using the same technique as in the derivation of ϕ₀^* and ϕ₂^* above, find ξ₀ = ξ₀^* and ξ₂ = ξ₂^* that numerically solve and compute $$ Spec_{BIP} = \widehat{Spec}_{BIP}(\xi^{\ast}_0),\; Sens_{BIP} = \widehat{Sens}_{BIP}(\xi^{\ast}_2),\; \textrm{etc.} $$
5. For a subject with biomarker measurement S_k(1), generate BIP_k by a draw from Mult(1, (q₀, q₁, q₂)), where q_i, i = 0, 1, 2, are determined by Sens_BIP, Spec_BIP, etc. obtained in Step 4.

Continuous  X^*, S^*(1), and  BIP^*

1. The user specifies corr (BIP^*, S^*(1)).
2. Assuming that BIP^* follows an additive measurement error model, i.e., BIP^* := S^*(1) + δ, where δ ∼ N(0, σ_δ²) with an unknown σ_δ², and δ, ϵ, and X^* are independent, solve the following equation for var δ = σ_δ²: $$ \mathop{\mathrm{corr}}(BIP^*, S^*(1)) = \sqrt\frac{\mathop{\mathrm{var}}X^* + \mathop{\mathrm{var}}\epsilon}{\mathop{\mathrm{var}}X^* + \mathop{\mathrm{var}}\epsilon + \mathop{\mathrm{var}}\delta} $$
3. For a subject with biomarker measurement S_k^*(1), generate BIP_k^* as BIP_k^* = S_k^*(1) + δ using σ_δ² = var δ obtained in Step 2.