Sample size / power for a randomized trial with an ordered categorical outcome, analysed by the proportional-odds model

Source: R/artcatr.R

Implements the methods of White et al (2023), including their new ologit method (which allows a non-proportional-odds anticipated effect via pe, is more accurate for large effects, and supports non-inferiority / substantial-superiority trials via margin), and Whitehead's (1993) original method for comparison.

artcatr(
  pc,
  cumulative = FALSE,
  pe = NULL,
  or = NULL,
  rr = NULL,
  margin = 1,
  favourable = NULL,
  power = NULL,
  n = NULL,
  aratio = c(1, 1),
  alpha = 0.05,
  onesided = FALSE,
  method = c("ologit", "whitehead"),
  variance = c("NA", "NN", "AA"),
  quiet = FALSE
)

Arguments

pc

Numeric vector. Anticipated probabilities (or cumulative probabilities, if cumulative = TRUE) in the control arm, one per outcome level, including the right-most level (probabilities must sum to 1, or cumulative probabilities must reach 1).

cumulative

Logical. Are pc/pe cumulative probabilities?

pe

Numeric vector. Anticipated (cumulative) probabilities in the experimental arm, specified like pc. Exactly one of pe, or, rr must be given (or none, if margin != 1, in which case or = 1 is assumed).

or

Numeric scalar. Anticipated common odds ratio applied to the control-arm cumulative probabilities (assumes proportional odds). An odds ratio below 1 shifts probability towards the right-most level.

rr

Numeric scalar. Anticipated common risk ratio (applied to every level except the right-most).

margin

Numeric scalar odds ratio defining the null hypothesis. margin = 1 (default) gives a superiority trial; otherwise a non-inferiority or substantial-superiority trial (see favourable).

favourable

Logical or NULL. Is the left-most outcome level the most favourable outcome? If NULL (default), inferred from the anticipated average odds ratio vs margin.

power

Numeric scalar in (0, 1). Power to achieve; sample size is calculated. Default 0.8 if neither power nor n given.

n

Numeric scalar. Total sample size; power is calculated.

aratio

Length-2 numeric vector, allocation ratio control:experimental (e.g. c(1, 2) means 2 experimental participants per 1 control).

alpha

Numeric scalar significance level. Default 0.05.

onesided

Logical. Is alpha one-sided? Default FALSE (two-sided).

method

One of "ologit" (default, the new method) or "whitehead".

variance

For method = "ologit", one of "NA" (default, recommended by White et al 2023), "NN" (equivalent to Whitehead's method), or "AA". "NA" uses the null-hypothesis SE for the alpha term and the alternative-hypothesis SE for the beta (power) term; "NN"/"AA" use one SE for both.

quiet

Logical. Suppress the printed summary?

Value

A list with the computed sample size or power, per-group sample sizes, the anticipated probability table, and other design parameters (mirroring the Stata command's stored results).

References

White IR, Marley-Zagar E, Morris TP, Parmar MKB, Royston P, Babiker AG (2023). artcat: Sample-size calculation for an ordered categorical outcome. Stata Journal 23(1):3-23. https://doi.org/10.1177/1536867X231161934

Whitehead J (1993). Sample size calculations for ordered categorical data. Statistics in Medicine 12:2257-2271.

Examples

# FLU-IVIG trial (Davey et al 2019), reproducing White et al (2023) sec 4:
# default (new) ologit method
artcatr(pc = c(.018, .036, .156, .141, .39, .259), or = 1 / 1.77, power = .8,
       favourable = FALSE)
#> artcatr: sample size for an ordered categorical outcome (proportional-odds model)
#> 
#>       based on the Stata implementation from White et al. (2023)----------------------------------------------------------------------
#> Type of trial                         superiority
#> Favourable/unfavourable outcome       unfavourable
#> Null hypothesis                       odds ratio = 1
#> Superiority region                    odds ratio < 1
#> Allocation ratio C:E                  1:1
#> Anticipated probabilities, control    0.018 0.036 0.156 0.141 0.390 0.259
#>                       experimental    given by odds ratio = 0.565
#> 
#> Table of anticipated probabilities
#>                                C       E
#> 1   least favourable       0.018   0.010
#> 2                          0.036   0.021
#> 3                          0.156   0.099
#> 4                          0.141   0.103
#> 5                          0.390   0.384
#> 6   most favourable        0.259   0.382
#> 
#> Alpha                                 0.05 (two-sided)
#> Power (designed)                      0.8
#> Method                                ologit (variance NA)
#> 
#> Total sample size (calculated)        322
#> Sample size per group (calculated)    161 161
#> ----------------------------------------------------------------------

# Whitehead method, for comparison
artcatr(pc = c(.018, .036, .156, .141, .39, .259), or = 1 / 1.77, power = .8,
       favourable = FALSE, method = "whitehead")
#> artcatr: sample size for an ordered categorical outcome (proportional-odds model)
#> 
#>       based on the Stata implementation from White et al. (2023)----------------------------------------------------------------------
#> Type of trial                         superiority
#> Favourable/unfavourable outcome       unfavourable
#> Null hypothesis                       odds ratio = 1
#> Superiority region                    odds ratio < 1
#> Allocation ratio C:E                  1:1
#> Anticipated probabilities, control    0.018 0.036 0.156 0.141 0.390 0.259
#>                       experimental    given by odds ratio = 0.565
#> 
#> Table of anticipated probabilities
#>                                C       E
#> 1   least favourable       0.018   0.010
#> 2                          0.036   0.021
#> 3                          0.156   0.099
#> 4                          0.141   0.103
#> 5                          0.390   0.384
#> 6   most favourable        0.259   0.382
#> 
#> Alpha                                 0.05 (two-sided)
#> Power (designed)                      0.8
#> Method                                Whitehead
#> 
#> Total sample size (calculated)        320
#> Sample size per group (calculated)    160 160
#> ----------------------------------------------------------------------

# Non-inferiority trial (margin expressed as an odds ratio)
artcatr(pc = c(.010, .021, .099, .103, .384, .383), or = 1, margin = 1.33,
       power = .8, favourable = FALSE)
#> artcatr: sample size for an ordered categorical outcome (proportional-odds model)
#> 
#>       based on the Stata implementation from White et al. (2023)----------------------------------------------------------------------
#> Type of trial                         non-inferiority
#> Favourable/unfavourable outcome       unfavourable
#> Null hypothesis                       odds ratio = 1.33
#> Non-inferiority region                odds ratio < 1.33
#> Allocation ratio C:E                  1:1
#> Anticipated probabilities, control    0.010 0.021 0.099 0.103 0.384 0.383
#>                       experimental    given by odds ratio = 1
#> 
#> Table of anticipated probabilities
#>                                C       E    E null
#> 1   least favourable       0.010   0.010     0.013
#> 2                          0.021   0.021     0.028
#> 3                          0.099   0.099     0.125
#> 4                          0.103   0.103     0.122
#> 5                          0.384   0.384     0.394
#> 6   most favourable        0.383   0.383     0.318
#> 
#> Alpha                                 0.05 (two-sided)
#> Power (designed)                      0.8
#> Method                                ologit (variance NA)
#> 
#> Total sample size (calculated)        1314
#> Sample size per group (calculated)    657 657
#> ----------------------------------------------------------------------