*! version 0.9 31 Jan 2019

Title

des_ma -- Single-stage multi-arm trial simulation for normally distributed outcomes

Syntax

des_ma, [options]

Optional
n(#) Stage-wise group size in the control arm. Default value is 34.

tau(numlist) Treatment effects. Default value is (0, 0, 0).

sd(#) Standard deviation of the responses. Default value is 1.

ratio(#) Allocation ratio between the experimental arms and the shared control arm. Default value is 1.

alpha(#) Chosen significance level. Default value is 0.05.

correction(string) Multiple comparison correction. Default value is dunnett.

replicates(#) Number of replicate simulations. Default value is 10000.

Description

sim_ma simulated multi-arm trials with K experimental (see option k) and a single control treatment.
The outcome variable of interest is assumed to be normally distributed.

It is assumed that K null hypotheses are to be tested, given by H_k: τ_k = μ_k - μ_0 ≤ 0, k = 1,..., K, where μ_k is the mean response on arm k.
Thus, arm 0 is assumed to be the control arm.
A set of (computed) p-values are used in combination with a chosen multiple comparison procedure in order to determine which of the null hypotheses to reject.

Note that it is assumed n patients will be recruited to the control arm, and rn patients to each experimental arm (see option ratio).
It is also assumed that the standard deviation of the accrued responses is known (see option sd).

The operating characteristics are evaluated at a specified vector of treatment effects (τ_1, ..., τ_K) (see option tau), using a particular number of replicate simulations (see option replicates).

Several multiple comparison procedures are supported (see correction), which in general work to control a particular error-rate to a specified significance level (see option alpha):

• benjamini: Control the false-discovery rate using the Benjamini-Hochberg procedure (Benjamini and Hochberg, 1995).

• bonferroni: Control the familywise error-rate using Bonferroni's correction (Bonferroni, 1936; Dunn, 1958; Dunn, 1961).

• dunnett: Control the familywise error-rate using Dunnett's correction (Dunnett, 1955; Dunnett, 1964).

• holm: Control the familywise error-rate using the Holm-Bonferroni method (Holm, 1979).

• none: Do not apply a multiple comparison procedure (i.e., test each hypothesis at level alpha).

• sidak: Control the familywise error-rate using Šidák's correction (Šidák, 1967).

• step_down_dunnett: Control the familywise error-rate using the step-down Dunnett correction (Naik, 1975; Marcus et al, 1976).

Options

n(#) An integer giving the group size in the control arm.
Should be greater than or equal to 1.
The default value is 68.

tau(#) A numlist (vector) giving the treatment effects.
It can be of length 1.
Thus, sim_ma will simulate single-stage two-arm trials.
However, if such simulations are required, the user is recommended to use sim_fixed (available from the author) instead, which is optimised for such scenarios.
The (internally specified) default value is (0, 0, 0).

sd(#) A real giving the assumed value of the standard deviation of the responses in the control and experimental arms.
It should be strictly greater than 0.
The default value is 1.

ratio(#) A real giving the allocation ratio between the experimental arms and the shared control arm.
That is, ratio patients are allocated to each experimental arm for every one patient allocated to the control arm.
It should be strictly positive.
The default value is 1.

alpha(#) A real giving the chosen significance level to use in the chosen multiple comparison procedure (see option correction).
It should be strictly between 0 and 1.
The default value is 0.05.

correction(string) A string giving the multiple comparison procedure to use.
It should be one of benjamini, bonferroni, dunnett, holm, none, sidak, or step_down_dunnett.
The default value is dunnett.
Note used if option k is equal to 1.

replicates(#) An integer giving the number of replicate simulations to use to evaluate required error-rates and sample sizes.
It should be an integer greater than or equal to 1.
The default value is 10000.

Examples

Note: Depending on the available processing power, each of the following examples could take several minutes to run.

/// Example 1: Find a 3-experimental treatment multi-arm design, and simulate
/// its operating characteristics under the global null hypothesis.

des_ma

sim_ma

/// Example 2: Find a design with modified multiplicity correction, and
/// simulate its operating characteristics under the least
/// favourable configuration.

des_ma, correction(holm)

sim_ma, n(71) tau(0.5, 0, 0) correction(holm)

/// Example 3: Find a 2-experimental treatment multi-arm design using
/// Bonferroni's correction, with unequal allocation to the
/// experimental and control arms, and simulate its operating
/// characteristics under the global null hypothesis.

des_ma, k(2) ratio(0.5) correction(bonferroni)

sim_ma, n(96) tau(0, 0) ratio(0.5) correction(bonferroni)

Stored results

des_ma stores the following in r():

scalars:

r(P_HA) Estimated probability that at least one null hypothesis is rejected.

r(P_H1) Estimated probability that H_1 is rejected.

r(FDR_HG) Estimated false discovery rate.

Author

Dr Michael J Grayling
Institute of Health & Society, Newcastle University, UK
Email: michael.grayling@newcastle.ac.uk

See also

References:

Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J Roy Stat Soc B 57(1):289–300.

Bonferroni CE (1936) Teoria statistica delle classi e calcolo delle probabilità. Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze 8:3-62.

Dunn OJ (1958) Estimation of the means for dependent variables. Ann Math Stat 29(4):1095–111.

Dunn OJ (1961) Multiple comparisons among means. J Am Stat Assoc 56(293):52–64.

Dunnett CW (1955) A multiple comparison procedure for comparing several treatments with a control. J Am Stat Assoc 50(272):1096–121.

Dunnett CW (1964) New tables for multiple comparisons with a control. Biometrics 20(3):482–91.

Holm S (1979) A simple sequentially rejective multiple test procedure. Scand J Stat 6(2):65–70.

Marcus R, Peritz E, Gabriel KR (1976) On closed testing procedures with special reference to ordered analysis of variance. Biometrika 63(3):655–60.

Naik UD (1975) Some selection rules for comparing p processes with a standard. Commun Stat A 4(6):519–35.

Šidák ZK (1967) Rectangular confidence regions for the means of multivariate normal distributions. J Am Stat Assoc 62(318):626–33.

Related commands:

des_dtl (a command for designing multi-stage drop-the-losers trials)
des_ma (a command for designing single-stage multi-arm trials)
des_mams (a command for designing multi-arm multi-stage trials)
sim_dtl (a command for simulating multi-stage drop-the-losers trials)
sim_fixed (a command for simulating single-stage two-arm trials)
sim_mams (a command for simulating multi-arm multi-stage trials)