Procedures implemented in the package

Completely Randomized Design (CRD)

For a two-arm trial and 1:1 target allocation, every subject is allocated to treatments with a fixed probability

\[\phi_j = 0.5, \: j = 1, \ldots, n.\]

In case of two-arm trial with unequal allocation or multi-arm trial with equal/unequal allocation, every subject is allocated to treatments with fixed probabilities that are equal to the target allocation proportions:

\[P_k(j) = \rho_k, \: k = 1, \ldots, K; \: j = 1, \ldots, n.\]

julia> using Incertus
julia> crd =  CRD()     # complete randomization, targeting 1:1 allocationCRD: complete randomization procedure, targeting 1:1 allocation in 2-arm trial.
julia> w = [1, 2, 3, 4]4-element Vector{Int64}:
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julia> crd =  CRD(w)    # complete randomization, targeting allocation specified by wCRD: complete randomization procedure, targeting 1:2:3:4 allocation in 4-arm trial.

Permuted Block Design (PBD)

Treatment assignments are made in blocks of size $bs$ (for a two-arm trial and 1:1 target allocation, $bs= 2\lambda$; otherwise, $bs=\lambda W$, where $W=w_1 + \ldots + w_K$ is a sum of elements of vector $\mathbf{w}$, target allocation vector. Here, $\lambda$ is a parameter of the PBD, representing the number of minimal balanced sets in the block of size $bs$).

At the $j^\text{th}$ allocation step, let $k^{(j-1)} = \left \lfloor \frac{j-1}{bs}\right\rfloor$ ($\lfloor x \rfloor$ is a floor function that returns the greatest integer less than or equal to $x$). In essence, $k^{(j-1)}$ is the number of complete blocks among the first $j-1$ assignments.

The probabilities of treatment assignments within each block are changed according the current imbalance in a block:

• for a two-arm trial and 1:1 target allocation,

\[\phi_j = \frac{0.5bs(1+k^{(j-1)})-N_1(j-1)}{bs(1+k^{(j-1)})-(j-1)}, \: j = 1, \ldots, n;\]

• for a two-arm trial with unequal allocation or multi-arm trial with equal/unequal allocation,

\[P_k(j) = \frac{w_k\lambda(1+k^{(j-1)})-N_k(j-1)}{bs(1+k^{(j-1)})-(j-1)}, k = 1, \ldots, K; \: j = 1, \ldots, n.\]

See [Zhao and Weng (2011), page 955, equation (5)]

julia> using Incertus
julia> pbd = PBD(1)      # PBD, targeting 1:1 allocation, with a block size equal to 2*1 = 2PBD(1): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.
julia> pbd = PBD(3)      # PBD, targeting 1:1 allocation, with a block size equal to 2*3 = 6PBD(3): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.
julia> w = [1, 2, 3, 4]4-element Vector{Int64}:
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julia> pbd =  PBD(w, 1)  # PBD, targeting allocation specified by w, with a block size sum(w)PBD(1): restricted randomization procedure, targeting 1:2:3:4 allocation in 4-arm trial.
julia> pbd =  PBD(w, 3)  # PBD, targeting allocation specified by w, with a block size 3*sum(w)PBD(3): restricted randomization procedure, targeting 1:2:3:4 allocation in 4-arm trial.

Random Allocation Rule (Rand)

A version of PBD, when the block size $bs$ equals to the total sample size $n$. At the $j^\text{th}$ allocation step, probabilities of treatment assignments are calculated as:

• for a two-arm trial and 1:1 target allocation,

\[\phi_j = \frac{0.5n-N_1(j-1)}{n-(j-1)}, \: j = 1, \ldots, n;\]

• for a two-arm trial with unequal allocation or multi-arm trial with equal/unequal allocation,

\[P_k(j) = \frac{nw_k/W-N_k(j-1)}{n-(j-1)}, k = 1, \ldots, K; \: j = 1, \ldots, n,\]

where $W=w_1 + \ldots + w_K$ is a sum of elements of vector $\mathbf{w}$, target allocation vector.

julia> using Incertus
julia> rnd = RAND(50)     # RAND, targeting 1:1 allocation, in a trial with 50 subjectsRAND: restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.
julia> w = [1, 2, 3, 4]4-element Vector{Int64}:
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julia> rnd =  RAND(w, 50) # RAND, targeting allocation specified by w, in a trial with 50 subjectsRAND: restricted randomization procedure, targeting 1:2:3:4 allocation in 4-arm trial.

Truncated Binomial Design (TBD)

Treatment assignments are made with probability 0.5 until one of the treatments receives its quota of $\frac{n}{2}$ subjects; thereafter all remaining assignments are made deterministically to the opposite treatment.

At the $j^\text{th}$allocation step, let $N_1$ and $N_2$ be the numbers of subjects allocated to treatments s.t. $N_1+N_2 = j-1.$ Then,

\[\phi_j = \left\{\begin{array}{rl} 0.5, & \max(N_1, N_2) < \frac{n}{2} \\ 1, & N_1 < N2 \\ 0, & N_1 > N2 \end{array}\right. ,\: j = 1, \ldots, n.\]

julia> using Incertus
julia> tbd = TBD(50)  # TBD, targeting 1:1 allocation, in a trial with 50 subjectsTBD: restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.

Truncated Multinomial Design (TMD)

Treatment assignments are made according to CRD with probabilities $P_k(j) = \rho_k$, $j=1, 2, \ldots$; $k = 1, 2, \ldots, K$, until one of the treatments ($k_1$) receives its quota of $nw_{k_1}$ subjects. Next, the randomization process switches to the remaining $K-1$ treatments with adjusted multinomial probabilities until some other treatment ($k_2$) receives its target quota of $nw_{k_2}$ subjects, and so on, until only one incomplete treatment is left. The remaining subjects in the sequence are deterministically assigned to this treatment.

For $1:1$ target allocation, TMD is simular to TBD.

julia> using Incertus
julia> tmd = TMD(50)     # TMD, targeting 1:1 allocation, in a trial with 50 subjectsTMD: restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.
julia> w = [1, 2, 3, 4]4-element Vector{Int64}:
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julia> rnd =  TMD(w, 50) # TMD, targeting allocation specified by w, in a trial with 50 subjectsTMD: restricted randomization procedure, targeting 1:2:3:4 allocation in 4-arm trial.

Efron's Biased Coin Design (EBCD)

At any allocation step, if treatment numbers $N_1$ and $N_2$ are balanced, the next assignment is made with probability 0.5; otherwise, the underrepresented treatment is assigned with probability $p$, where $0.5 \leq p \leq 1$ is a fixed and pre-specified parameter that determines the trade-off between balance and randomness.

At the $j^\text{th}$ allocation step, given treatment numbers $N_1$ and $N_2$, s.t. $N_1+N_2 = j-1$, and imbalance $d = N_1-N_2$,

\[\phi_j = \left\{\begin{array}{rl} 0.5, & N_1 = N_2\\ p, & N_1 < N2 \\ 1-p, & N_1 > N2 \end{array}\right. ,\: j = 1, \ldots, n.\]

Note that $p=1$ corresponds to PBD with block size $b=2$.

julia> using Incertus
julia> ebcd = EBCD(2//3) # EBCD, targeting 1:1 allocation, with parameter p=2/3EBCD(2/3): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.

Adjustable Biased Coin Design (ABCD)

An extension of Efron’s BCD. At the $j^\text{th}$ allocation step, given treatment numbers $N_1$ and $N_2$, s.t. $N_1+N_2 = j-1$, and imbalance $d = N_1-N_2$,

\[\phi_j = \left\{\begin{array}{rl} 0.5, & |d| <= 1 \\ \frac{|d|^a}{1+|d|^a}, & d < -1 \\ \frac{1}{1+|d|^a}, & d > 1 \end{array}\right. ,\: j = 1, \ldots, n.\]

julia> using Incertus
julia> abcd = ABCD(2) # ABCD, targeting 1:1 allocation, with parameter a=2ABCD(2): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.

Generalized Biased Coin Design (GBCD)

A generalization of Efron’s BCD. At the $j^\text{th}$ allocation step, given treatment numbers $N_1$ and $N_2$, s.t. $N_1+N_2 = j-1$, and imbalance $d = N_1-N_2$,

\[\phi_j = \left\{\begin{array}{rl} 0.5, & j = 1 \\ \frac{N_2^\gamma}{N_1^\gamma+N_2^\gamma}, & j = 1, \ldots, n. \end{array}\right. \]

julia> using Incertus
julia> gbcd = GBCD(2) # GBCD, targeting 1:1 allocation, with parameter γ=2GBCD(2): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.

Big Stick Design (BSD)

An example of maximum tolerated imbalance (MTI) procedures. It makes prediction of the future treatment allocations more difficult (even knowing the current sizes of the treatment groups) and controls treatment imbalance at a predefined threshold throughout the experiment. A general MTI procedure specifies a certain boundary for treatment imbalance, say $mti$, that cannot be exceeded.

At the $j^\text{th}$ allocation step, given treatment numbers $N_1$ and $N_2$, s.t. $N_1+N_2 = j-1$, and imbalance $d = N_1-N_2$,

\[\phi_j = \left\{\begin{array}{rl} 0.5, & |d| < mti \\ 0, & d = mti \\ 1, & d = -mti \end{array}\right. ,\: j = 1, \ldots, n.\]

julia> using Incertus
julia> bsd = BSD(3) # BSD, targeting 1:1 allocation, with parameter mti=3BSD(3): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.

Biased Coin Design With Imbalance Tolerance (BCDWIT)

A combination of Efron’s BCD and BSD. At the $j^\text{th}$ allocation step, given treatment numbers $N_1$ and $N_2$, s.t. $N_1+N_2 = j-1$, and imbalance $d = N_1-N_2$,

\[\phi_j = \left\{\begin{array}{rl} 0.5, & |d| < mti\: \& \: d = 0 \\ p, & |d| < mti \: \& \: d < 0 \\ 1-p, & |d| < mti \: \& \: d > 0 \\ 0, & d = mti \\ 1, & d = -mti \end{array}\right. ,\: j = 1, \ldots, n.\]

julia> using Incertus
julia> bcdwit = BCDWIT(2//3, 3) # BCDWIT with p = 2/3 and mti=3, targeting 1:1 allocationBCDWIT(2/3, 3): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.

Block Urn Design (BUD)

This design was proposed by Zhao and Weng (2011), to provide a more random design than the PBD. Let $N_k(j-1)$ denote the number of treatment $k$ assignments among first $j-1$ subjects, and $k^{(j-1)}=\min\limits_{1 \leq k \leq K} \left \lfloor \frac{N_k(j-1)}{w_k}\right \rfloor$ denote the number of minimal balanced sets among the first $j-1$ assignments. Then, at the $j^\text{th}$ allocation step, probabilities of treatment assignments are calculated as:

• for a two-arm trial and 1:1 target allocation,

\[\phi_j = \frac{\lambda+\min(N_1(j-1), N_2(j-1))-N_1(j-1)}{2(\lambda+\min(N_1(j-1), N_2(j-1)))-(j-1)}, \: j = 1, \ldots, n;\]

• for a two-arm trial with unequal allocation or multi-arm trial with equal/unequal allocation,

\[P_k(j) = \frac{w_k(\lambda+k^{(j-1)})-N_k(j-1)}{W(\lambda+k^{(j-1)})-(j-1)}, k = 1, \ldots, K; \: j = 1, \ldots, n,\]

where $W=w_1 + \ldots + w_K$ is a sum of elements of vector $\mathbf{w}$, target allocation vector.

See [Zhao and Weng (2011), page 955, equations (2) and (3)].

julia> using Incertus
julia> bud = BUD(2)      # BUD, targeting 1:1 allocation (λ=2)BUD(2): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.
julia> w = [1, 2, 3, 4]4-element Vector{Int64}:
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julia> bud =  BUD(w, 2)  # BUD, targeting allocation specified by w (λ=2)BUD(2): restricted randomization procedure, targeting 1:2:3:4 allocation in 4-arm trial.

Ehrenfest Urn Design (EUD)

Another example of the maximum tolerated imbalance (MTI) procedure. At the $j^\text{th}$ allocation step, given treatment numbers $N_1$ and $N_2$, s.t. $N_1+N_2 = j-1$, and imbalance $d = N_1-N_2$,

\[\phi_j = \frac{1}{2}\left(1-\frac{d}{mti}\right) ,\: j = 1, \ldots, n,\]

where $mti$ ($>0$) is a parameter of the procedure.

julia> using Incertus
julia> eud = EUD(2) # EUD, targeting 1:1 allocation (mti=2)EUD(2): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.

Bayesian Biased Coin Design (BBCD)

A special class of biased coind designs (BCDs). At the $j^\text{th}$ allocation step, given treatment numbers $N_1$ and $N_2$, s.t. $N_1+N_2 = j-1$,

\[\phi_j = \left\{\begin{array}{rl} 0.5, & j = 1 \\ 1, & j = 2 \: \& \: N_1 = 0 \\ 0, & j = 2 \: \& \: N_1 = 1 \\ \frac{\left(1 + \frac{N_2}{(j-1)N_1}\right)^\frac{1}{\gamma}}{\left(1 + \frac{N_2}{(j-1)N_1}\right)^\frac{1}{\gamma} + \left(1 + \frac{N_1}{(j-1)N_2}\right)^\frac{1}{\gamma}}, & j \geq 3 \end{array}\right. ,\: j = 1, \ldots, n.\]

where $\gamma$ ($>0$) is a parameter of the procedure.

julia> using Incertus
julia> bbcd = BBCD(0.1) # BBCD, targeting 1:1 allocation (γ=0.1)BBCD(0.1): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.

Mass Weighted Urn Design (MWUD)

A randomization procedure that can be used fo for two- and multi-arm trials, targeting both equal and unequal allocation. At the $j^\text{th}$ allocation step, given $K$ treatments and corresponding treatment numbers $N_1(j-1) \ldots N_K(j-1)$, s.t. $\sum_{k = 1}^K{N_k(j-1)} = j-1$,

\[P_k(j) = \frac{\max\left\{\alpha\rho_k-N_k(j-1) +(j-1)\rho_k , 0\right\}}{\sum_{k=1}^K\max\left\{\alpha\rho_k-N_k(j-1) +(j-1)\rho_k , 0\right\}} ,\: j = 1, \ldots, n.\]

where $\alpha$ ($>0$) is a parameter of the procedure that controls maximum tolerated imbalance.

julia> using Incertus
julia> mwud = MWUD(2)     # MWUD, targeting 1:1 allocation (α=2)MWUD(2): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.
julia> w = [1, 2, 3, 4]   # target allocation ratio4-element Vector{Int64}:
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julia> mwud =  MWUD(w, 2) # MWUD, targeting 1:2:3:4 allocation (α=2)MWUD(2): restricted randomization procedure, targeting 1:2:3:4 allocation in 4-arm trial.

Doubly-Adaptive Biased Coin Design (DBCD)

A randomization procedure that can be used fo for two- and multi-arm trials with $K$ treatments, targeting both equal and unequal allocation. Initial treatment assignments ($j=1, 2, \ldots, m_0$) are made completely at random ($P_k(j)=\rho_k$, $k=1, 2, \ldots, K$) until each group has at least one subject (i.e., $N_k(m_0)>0$). Subsequent treatment assignments are made as follows:

\[P_k(j) = \frac{\rho_k\left(\rho_k/\frac{N_k(j-1)}{j-1}\right)^\gamma}{\sum_{k=1}^K\rho_k\left(\rho_k/\frac{N_k(j-1)}{j-1}\right)^\gamma} ,\: j = m_0+1, \ldots, n.\]

where $\gamma$ ($>0$) is a parameter of the procedure.

julia> using Incertus
julia> dbcd = DBCD(2)     # DBCD, targeting 1:1 allocation (γ=2)DBCD(2): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.
julia> w = [1, 2, 3, 4]   # target allocation ratio4-element Vector{Int64}:
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julia> dbcd =  DBCD(w, 2) # DBCD, targeting 1:2:3:4 allocation (γ=2)DBCD(2): restricted randomization procedure, targeting 1:2:3:4 allocation in 4-arm trial.

Maximum Entropy Constraint Balance Randomization (MaxEnt)

A randomization procedure that can be used fo for two- and multi-arm trials with $K$ treatments, targeting both equal and unequal allocation.

Consider a point in the trial when $j-1$ subjects have been randomized among the $K$treatments, and let denote the corresponding treatment numbers as $N_k(j-1)$ ($k=1, \ldots, K$ and $\sum_{k_1}^KN_k(k-1) = j-1$). At the $j^\text{th}$ allocation step, the randomization rule is defined as follows:

a) For $k=1, 2, \ldots, K$, compute $B_k$, the hypothetical "lack of balnce", which results from assigning the $j^\text{th}$ subject to treatment $k$:

\[B_k = \max\limits_{1\leq i \leq K}\left|\frac{N^{(k)}_i(j)}{k}-\rho_k\right|, \text{ where }N^{(k)}_i(j) = \left\{ \begin{array}{rl} N_i(j-1) + 1, & i = k \\ N_i(j-1), & i \ne k \end{array} \right. .\]

b) The treatment randomization probabilities for the $j^\text{th}$ subject $\left(P_1(j), P_2(j), \ldots, P_K(j)\right)$ are determined as a solution to the constrained optimization problem:

\[\begin{aligned} &\text{minimize}\sum_{k=1}^KP_k(j)\log\left(\frac{P_k(j)}{\rho_k}\right) \\ &\text{subject to}\sum_{k=1}^KB_kP_k(j) \leq \eta B_{(1)} + (1-\eta)\sum_{k=1}^KB_k\rho_k \\ &\text{and}\sum_{k=1}^KP_k(j) = 1; \: 0\leq P_k(j) \leq 1, \: k = 1, 2, \ldots, K, \end{aligned}\]

where $B_{(1)}=\min\limits_{1\leq k \leq K}B_k$, and $\eta$ $\left(\in [0; 1]\right)$ is a parameter of the procedure that controls degree of randomness.

julia> using Incertus
julia> maxent = MaxEnt(0.5)     # MaxEnt, targeting 1:1 allocation (η=2)MaxEnt(0.5): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.
julia> w = [1, 2, 3, 4]   # target allocation ratio4-element Vector{Int64}:
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julia> maxent = MaxEnt(w, 0.5)  # MaxEnt, targeting 1:2:3:4 allocation (η=2)MaxEnt(0.5): restricted randomization procedure, targeting 1:2:3:4 allocation in 4-arm trial.

Drop-the-Loser Urn Design (DLUD)

julia> using Incertus
julia> dlud = DLUD(2)     # DLUD, targeting 1:1 allocation (a=2)DLUD(2): restricted randomization procedure, targeting 1:1 allocation in 2-arm trial.
julia> w = [1, 2, 3, 4]   # target allocation ratio4-element Vector{Int64}:
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julia> dlud = DLUD(w, 2)  # DLUD, targeting 1:2:3:4 allocation (a=2)DLUD(2): restricted randomization procedure, targeting 1:2:3:4 allocation in 4-arm trial.

Funcions implemented to calculate allocation probabilities