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Operational characteristics

Several measures of imbalance and randomness have been implemented in the package:

Measures of imbalance

Let us consider the following quantity as a measure of imbalance:

\[D(j) = N_E(j)-N_C(j), \: \text{for two-arm trial and 1:1 randomization}, \: j = 1, 2, \ldots, n;\]

and

\[D(j) = \sqrt{\sum\limits_{k=1}^K \left(N_k(j)-j\rho_k\right)^2}, \: \text{otherwise}, \: j = 1, 2, \ldots, n,\]

where the latter is the Euclidean distance between the observed and the targeted allocation.

The following measures of imbalance are available in the package.

Final imbalance

$D(n)$ – final imbalance/distance (between the observed and targeted sample sizes) after all treatment assignments made.

Incertus.calc_final_imb — Method

Function calculates final imbalance

Call

calc_final_imb(sr)

Arguments

  • sr::SimulatedRandomization: an instance of SimulatedRandomization, an object, representing simulation output.

Result

  • A Vector of final imbalance values obtained via simulations.
source
Incertus.calc_final_imb — Method

Function calculates final imbalance

Call

calc_final_imb(sr)

Arguments

  • sr::Vector{SimulatedRandomization}: a vector of instances of SimulatedRandomization, representing simulation output.

Result

  • A DataFrame of simulated final imbalances' values obtained via simulations.
source

Expected absolute imbalance

$\mathbf{E}\left[|D(j)|\right]$ – expected absolute imbalance/distance (between the observed and targeted sample sizes) at the $j^\text{th}$ allocation step.

Incertus.calc_expected_abs_imb — Method

Function calculates expected absolute imbalance vs. allocation step.

Call

calc_expected_abs_imb(sr)

Arguments

  • sr::SimulatedRandomization: an instance of SimulatedRandomization, an object, representing simulation output.

Result

  • A Vector of expected absolute imbalance values summarized via simulations.
source
Incertus.calc_expected_abs_imb — Method

Function calculates expected absolute imbalance vs. allocation step.

Call

calc_expected_abs_imb(sr)

Arguments

  • sr::Vector{SimulatedRandomization}: a vector of instances of SimulatedRandomization, representing simulation output.

Result

  • A DataFrame of expected absolute imbalances' values summarized via simulations.
source

Variance of imbalance

$\mathbf{E}\left[|D(j)|^2\right]=\mathbf{var}\left[N_E(j)-N_C(j)\right]$ or $=\sum_{k=1}^K\mathbf{var}\left[N_k(j)\right]$ – variance of imbalance/total variance of the treatment sample sizes at the $j^\text{th}$ allocation step.

Incertus.calc_variance_of_imb — Method

Function calculates variance of imbalance vs. allocation step.

Call

calc_variance_of_imb(sr)

Arguments

  • sr::SimulatedRandomization: an instance of SimulatedRandomization, an object, representing simulation output.

Result

  • A Vector of variance of imbalance values summarized via simulations.
source
Incertus.calc_variance_of_imb — Method

Function calculates variance of imbalance vs. allocation step.

Call

calc_variance_of_imb(sr)

Arguments

  • sr::Vector{SimulatedRandomization}: a vector of instances of SimulatedRandomization, representing simulation output.

Result

  • A DataFrame of variances' of imbalance values summarized via simulations.
source

Expected maximum imbalance over first allocation steps

$\mathbf{E}\left[\max\limits_{1\leq m \leq j}|D(m)|\right]$ – expected maximum imbalance/distance (between the observed and the targeted treatment sample sizes) over the first $j$ allocation steps.

Incertus.calc_expected_max_abs_imb — Method

Function calculates expected maximum absolute imbalance over first allocations vs. allocation step.

Call

calc_expected_max_abs_imb(sr)

Arguments

  • sr::SimulatedRandomization: an instance of SimulatedRandomization, an object, representing simulation output.

Result

  • A Vector of expected maximum absolute imbalance over firat allocations values summarized via simulations.
source
Incertus.calc_expected_max_abs_imb — Method

Function calculates expected maximum absolute imbalance over first allocations vs. allocation step.

Call

calc_expected_max_abs_imb(sr)

Arguments

  • sr::Vector{SimulatedRandomization}: a vector of instances of SimulatedRandomization, representing simulation output.

Result

  • A DataFrame of expected maximum absolute imbalances' values (over first allocations) summarized via simulations.
source

A cumulative average loss over the first allocation steps

$Imb(j) = \frac{1}{j}\sum\limits_{m=1}^j\frac{\mathbf{E}\left[|D(m)|^2\right]}{m}$ – a cumulative average loss at the $j^\text{th}$ allocation step.

Incertus.calc_cummean_loss — Method

Function calculates cumulative average loss vs. allocation step.

Call

calc_cummean_loss(sr)

Arguments

  • sr::SimulatedRandomization: an instance of SimulatedRandomization, an object, representing simulation output.

Result

  • A Vector of cumulative average loss' values summarized via simulations.
source
Incertus.calc_cummean_loss — Method

Function calculates cumulative average loss vs. allocation step.

Call

calc_cummean_loss(sr)

Arguments

  • sr::Vector{SimulatedRandomization}: a vector of instances of SimulatedRandomization, representing simulation output.

Result

  • A DataFrame of cumulative average losses' values summarized via simulations.
source

Measures of randomness

Cumulative average of expected proportions of correct guesses over first allocation steps under two different guessing strategies

$EPCG_{conv}(j)$ – cumulative average of expected proportions of correct guesses over first ($j$) allocation steps under the convergence guessing strategy.

  • For a two-arm trial with $1:1$ target allocation, it is defined as

\[\begin{aligned} EPCG_{conv}(j) &= \frac{1}{j}\sum\limits_{m=1}^j\mathbf{E}\left[G_m = \delta_m\right], \: j = 1, 2, \ldots, n, \\ \text{where }G_m &= \left\{ \begin{array}{rl} 1, & D(m-1) < 0; \\ \sim Bernoulli(0.5), & D(m-1) = 0; \\ 0, & D(m-1) > 0. \end{array} \right. \end{aligned} \]

  • For a multu-arm trial, it is defined as

\[\begin{aligned} EPCG_{conv}(j) &= \frac{1}{j}\sum\limits_{m=1}^j\mathbf{E}\left[\mathbf{G}_m = \boldsymbol{\delta}_m\right], \: j = 1, 2, \ldots, n, \\ \text{where }G_m &\sim Multinomial\left(1, \mathbf{P}\right), \\ \text{where }\mathbf{P} &= \left(\frac{\left\{\Delta_1 = \min\limits_{1\leq i \leq K}\Delta_i\right\}}{\sum_{k=1}^K\left\{\Delta_k = \min\limits_{1\leq i \leq K}\Delta_i\right\}}, \ldots, \frac{\left\{\Delta_K = \min\limits_{1\leq i \leq K}\Delta_i\right\}}{\sum_{k=1}^K\left\{\Delta_k = \min\limits_{1\leq i \leq K}\Delta_i\right\}}\right), \\ \text{where }\Delta_i &= \frac{N_i(m)}{m}-\rho_i, \: i = 1, 2, \ldots, K. \end{aligned} \]

$G_m$ (or $\mathbf{G}_m$) is a random variable taking values based on the investigator's guess; if $G_m = \delta_m$ ($\mathbf{G}_m = \boldsymbol{\delta_m}$), the guess is correct at the $m^\text{th}$ allocation step.

$EPCG_{max}(j)$ – cumulative average of expected proportions of correct guesses over first ($j$) allocation steps under the maximum probability guessing strategy.

  • For a two-arm trial with $1:1$ target allocation, it is defined as

\[\begin{aligned} EPCG_{max}(j) &= \frac{1}{j}\sum\limits_{m=1}^j\mathbf{E}\left[\widetilde{G}_m = \delta_m\right], \\ \text{where }\widetilde{G}_m &= \left\{ \begin{array}{rl} 1, & \phi_m > 0.5; \\ \sim Bernoulli(0.5), & \phi_m = 0.5; \\ 0, & \phi_m < 0.5. \end{array} \right. \end{aligned}\]

  • For a multu-arm trial, it is defined as

\[\begin{aligned} EPCG_{max}(j) &= \frac{1}{j}\sum\limits_{m=1}^j\mathbf{E}\left[\widetilde{\mathbf{G}}_m = \boldsymbol{\delta}_m\right], \: j = 1, 2, \ldots, n, \\ \text{where }G_m &\sim Multinomial\left(1, \mathbf{P}\right), \\ \text{where }\mathbf{P} &= \left(\frac{\left\{P_1(m) = \max\limits_{1\leq i \leq K}P_i(m)\right\}}{\sum_{k=1}^K\left\{P_k(m) = \max\limits_{1\leq i \leq K}P_k(m)\right\}}, \ldots, \frac{\left\{P_K(m) = \max\limits_{1\leq i \leq K}P_i(m)\right\}}{\sum_{k=1}^K\left\{P_k(m) = \max\limits_{1\leq i \leq K}P_i(m)\right\}}\right). \end{aligned} \]

$\widetilde{G}_m$ (or $\widetilde{\mathbf{G}}_m$) is a random variable taking values based on the investigator's guess; if $\widetilde{G}_m = \delta_m$ ($\widetilde{\mathbf{G}}_m = \boldsymbol{\delta_m}$), the guess is correct at the $m^\text{th}$ allocation step.

Incertus.calc_cummean_epcg — Method

Function calculates cumulative averages of expected proportions of correct guesses vs. allocation step.

Call

calc_cummean_epcg(sr, gs)

Arguments

  • sr::SimulatedRandomization: an instance of SimulatedRandomization, an object, representing simulation output.
  • gs::String: guessing strategy; accepts two values: "C" (corresponds to the convergence guessing strategy) of "MP" (corresponds to the maximum probability guessing strategy).

Result

  • A Vector of cumulative averages of the expected proportions of correct guesses summarized via simulations.
source
Incertus.calc_cummean_epcg — Method

Function calculates cumulative averages of expected proportions of correct guesses vs. allocation step.

Call

calc_cummean_epcg(sr, gs)

Arguments

  • sr::Vector{SimulatedRandomization}: a vector of instances of SimulatedRandomization, representing simulation output.
  • gs::String: guessing strategy; accepts two values: "C" (corresponds to the convergence guessing strategy) of "MP" (corresponds to the maximum probability guessing strategy).

Result

  • A vector of cumulative averages of the expected proportions of correct guesses summarized via simulations.
source

Cumulative average of expected proportions of deterministic assignments over first allocation steps

$PD(j)$ – cumulative average of expected proportions of deterministic assignments over first ($j$) allocation steps.

  • For a two-arm trial with $1:1$ target allocation, it is defined as

\[PD(j) = \frac{1}{j}\sum\limits_{m=1}^j\Pr(\phi_m\in\{0, 1\})\]

  • For a multu-arm trial, it is defined as

\[\begin{aligned} PD(j) &= \frac{1}{j}\sum\limits_{m=1}^j\Pr(\mathbf{P}(m)\in\{\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_K\}), \\ \text{where }\mathbf{e}_k &= (0, \ldots, \underbrace{1}_{k^\text{th}\:\text{element}}, \ldots, 0), \: k = 1, 2, \ldots, K. \end{aligned}\]

Incertus.calc_cummean_pda — Method

Function calculates cumulative averages of the proportions of deterministic assignments vs. allocation step.

Call

calc_cummean_pda(sr)

Arguments

  • sr::SimulatedRandomization: an instance of SimulatedRandomization, an object, representing simulation output.

Result

  • A vector of the cumulative averages of the proportions of deterministic assignmnets values summarized via simulations.
source
Incertus.calc_cummean_pda — Method

Function calculates cumulative averages of the proportions of deterministic assignments vs. allocation step.

Call

calc_cummean_pda(sr)

Arguments

  • sr::Vector{SimulatedRandomization}: a vector of instances of SimulatedRandomization, representing simulation output.

Result

  • A DataFrame of the cumulative averages of the proportions of deterministic assignmnets summarized via simulations.
source

Forcing index

$FI(j)$ – forcing index, which takes values on a scale 0–1.

  • For a two-arm trial with $1:1$ target allocation, it is defined as

\[FI(j) = \frac{4}{j}\sum\limits_{m=1}^j\mathbf{E}\left[|\phi_m-0.5|\right].\]

Note that $FI(j) = 0, \forall j$ for CRD and $FI(j) = 1$ for PBD with a block size $bs=2$, assuming $j$ is even (most balanced design).

  • For a multu-arm trial, it is defined as

\[FI(j) = \frac{1}{j}\sum\limits_{m=1}^j\mathbf{E}\left[\sqrt{\sum\limits_{k=1}^K\left(P_k(m)-\rho_k\right)^2}\:\right].\]

Note that $FI(j) = 0, \forall j$ for CRD.

Incertus.calc_fi — Method

Function calculates forcing index vs. allocation step.

Call

calc_fi(sr)

Arguments

  • sr::SimulatedRandomization: an instance of SimulatedRandomization, an object, representing simulation output.

Result

  • A Vector of the forcing index values summarized via simulations.
source
Incertus.calc_fi — Method

Function calculates forcing index vs. allocation step.

Call

calc_fi(sr)

Arguments

  • sr::Vector{SimulatedRandomization}: a vector of instances of SimulatedRandomization, representing simulation output.

Result

  • A DataFrame of the forcing index values summarized via simulations.
source

Balance-randomness trade-off

$G(j) = \sqrt{\left\{Imb(j)\right\}^2 + \left\{FI(j)\right\}^2}$ represents a balance-randomness trade-off at the $j^\text{th}$ allocation step. Lower values of $G(j)$ indicate better balance–randomness trade-off.

The following functions are available to deal with the characteristic.

Incertus.calc_brt — Method

Function calculates balance-randomness trade-off vs. allocation step.

Call

calc_brt(sr)

Arguments

  • sr::SimulatedRandomization: an instance of SimulatedRandomization, an object,

representing simulation output.

Result

  • A Vector of balance-randomness trade-off measurement values summarized via simulations.
source
Incertus.calc_brt — Method

Function calculates balance-randomness trade-off vs. allocation step.

Call

calc_brt(sr)

Arguments

  • sr::Vector{SimulatedRandomization}: a vector of instances of SimulatedRandomization, representing simulation output.

Result

  • A DataFrame of balance-randomness trade-off measurements' values summarized via simulations.
source

Allocation ratio preserving property

A procedure has an ARP property if

\[\mathbf{E}\left[P_k(j)\right] = \rho_k, \: k = 1, 2, \ldots, K; \: j = 1, 2, \ldots, n,\]

where $P_k(j)$ is the conditional randomization probability for the $k^\text{th}$ treatment group at the $j^\text{th}$ allocation step, and $\rho_k$ is a target allocation proportion for the $k^\text{th}$ tretament.

The following functionality is avalable to evaluate ARP property:

Incertus.ARP — Type

A data structure to deal with Allocation Ratio Preserving (ARP) property.

It has the following fields:

  • label::String: a label for a randomization procedure that has been simulated.
  • ρ::Vector{Float64}: a vector of target allocation proportions.
  • expected_prb::Matrix{Float64}: expected values of $P_k(j)$ evaluated via simulations.
source
Incertus.eval_arp — Method

Function evaluates allocation ratio preserving (ARP) property, i.e., calculates unconditional allocation probabilities. A procedure has an ARP property if

Call

eval_arp(sr)

Arguments

  • sr::SimulatedRandomization: an instance of SimulatedRandomization, an object, representing simulation output.

Result

  • An instance of ARP type, summarizing the expected values of $P_k(j)$ via simulations.
source
Incertus.eval_arp — Method

Function evaluates allocation ratio preserving (ARP) property, i.e., calculates unconditional allocation probabilities.

Call

eval_arp(sr)

Arguments

  • sr::Vector{SimulatedRandomization}: a vector of instances of SimulatedRandomization, representing simulation output.

Result

  • A Vector{ARP} object – a vector of instances of ARP, each summarizing expected values of $P_k(j)$ via simulations.
source