Convert Information Target into Sample Size for Fixed N Single-Stage Design

These functions convert an information target into an approximate sample size required for a single-stage fixed sample size design with the same power and type I error requirements under assumed values of nuisance parameters.

Usage

information_to_n_difference_means(
  information,
  sigma_0,
  sigma_1,
  round_up = TRUE
)

information_to_n_risk_difference(
  information,
  pi_0 = NULL,
  pi_1 = NULL,
  delta = NULL,
  round_up = TRUE
)

information_to_n_relative_risk(
  information,
  pi_0 = NULL,
  pi_1 = NULL,
  rr = NULL,
  round_up = TRUE
)

information_to_events_log_hr(
  information,
  allocation_ratio = 1,
  round_up = TRUE
)

Arguments

information: Numeric vector containing the information level
sigma_0: Variance of outcomes in the population of individuals receiving the control intervention
sigma_1: Variance of outcomes in the population of individuals receiving the active intervention
round_up: Logical scalar: should the sample size be rounded up to an integer value?
pi_0: Probability of event in the population of individuals receiving the control intervention
pi_1: Probability of event in the population of individuals receiving the control intervention
delta: The risk difference (i.e. pi_1 - pi_0)
rr: The relative risk of the event in treatment vs. control (i.e. pi_1/pi_0)
allocation_ratio: The allocation ratio of participants receiving treatment to those receiving control

Value

When all parameters are scalars, the result is a scalar, indicating the approximate sample size requirement. When multiple values are specified, a grid of unique parameters are constructed, and the approximate information is computed for each value of the parameters.

Details

The amount of information in a sample of size $N$ depends on nuisance parameters, such as the variance of continuous outcomes, the risk of binary and time-to-event outcomes, rates of misssing data, and the correlation between covariates and the outcomes of interest.

In studies with a fixed sample size, this sample size is chosen based on assumptions about these nuisance parameters, which are incorporated into the effect size. The sample size is chosen to give power $(1 - \beta)$ while maintaining a type I error rate of $(\alpha)$ under some assumed effect size. Inaccurate estimates of nuisance parameters can lead to over-powered or under-powered studies.

In an information-monitored design, investigators choose an estimand of interest, such as the difference in means or proportions, that is free from nuisance parameters. A trial is designed to identify some minimum important difference $\delta_{min}$ in the estimand with power $(1 - \beta)$ while maintaining a type I error rate of $(\alpha)$. Data is collected until the precision of the estimate (i.e. the reciprocal of its variance) reaches a pre-specified threshold $\mathcal{I}$:

$$\mathcal{I} = \left(\frac{Z_{\alpha/s} + Z_{\beta}}{\delta_{min}}\right)^2 \approx \frac{1}{Var(\hat{\delta})} = \frac{1}{\left(SE(\hat{\delta})\right)^2}$$

The sample size required to reach the information target $\mathcal{I}$ depends on nuisance parameters mentioned above.

These functions allow a user to determine an approximate sample size $N$ at which the information target $\mathcal{I}$ would be reached under some assumptions about the nuisance parameters.

References

Mehta, CR, and Tsiatis AA. 2001. "Flexible Sample Size Considerations Using Information-Based Interim Monitoring". Drug Information Journal 35 (4): 1095–1112. https://doi.org/10.1177/009286150103500407

Examples

information_to_n_difference_means(
    information =
      required_information_single_stage(
        delta = 5,
        power = 0.8,
        alpha = 0.05,
        sides = 2
      ),
    sigma_0 = 10,
    sigma_1 = 10,
    round_up = TRUE
  )
#>   n_per_arm n_total
#> 1        63     126

power.t.test(
  delta = 5, sd = 10, sig.level = 0.05, power = 0.8
)
#> 
#>      Two-sample t test power calculation 
#> 
#>               n = 63.76576
#>           delta = 5
#>              sd = 10
#>       sig.level = 0.05
#>           power = 0.8
#>     alternative = two.sided
#> 
#> NOTE: n is number in *each* group
#> 

information_to_n_risk_difference(
  information =
    required_information_single_stage(
      delta = 0.1,
      power = 0.8,
      alpha = 0.05,
      sides = 2
    ),
  pi_0 = 0.25,
  pi_1 = 0.15,
  round_up = TRUE
)
#>   n_per_arm n_total
#> 1       248     496

power.prop.test(p1 = 0.25, p2 = 0.15, sig.level = 0.05, power = 0.8)
#> 
#>      Two-sample comparison of proportions power calculation 
#> 
#>               n = 249.982
#>              p1 = 0.25
#>              p2 = 0.15
#>       sig.level = 0.05
#>           power = 0.8
#>     alternative = two.sided
#> 
#> NOTE: n is number in *each* group
#> 

information_to_n_relative_risk(
  information =
    required_information_single_stage(
      delta = log(0.5),
      power = 0.8,
      alpha = 0.05,
      sides = 2
    ),
  pi_0 = 0.2,
  pi_1 = 0.1,
  round_up = TRUE
)
#>   n_per_arm n_total
#> 1       213     426

rr_design(
  pi_1 = 0.1,
  pi_0 = 0.2,
  power = 0.8,
  alpha = 0.05
)
#>   n_per_arm  n_total
#> 1  212.3734 424.7468

information_to_events_log_hr(
  information =
    required_information_single_stage(
      delta = log(0.5),
      power = 0.8,
      alpha = 0.05,
      sides = 2
    ),
  allocation_ratio = 1,
  round_up = TRUE
)
#> [1] 66

hr_design(
  hazard_ratio = 0.5,
  power = 0.8,
  alpha = 0.05,
  test_sides = 2
)
#> [1] 65.34566