Determine the information level required for a one-stage, fixed sample size design

In information monitored designs, data are accrued until the information (i.e. precision) reaches the level required to achieve the desired power when testing the null hypothesis against a specific alternative while maintaining a specified Type I Error Rate. required_information_uninflated_mann_whitney is an alias to required_information_single_stage for ordinal outcomes, where the null value of the Mann-Whitney estimand is 0.5.

Usage

required_information_single_stage(
  delta,
  delta_0 = 0,
  alpha = 0.05,
  sides = 2,
  power = 0.8
)

required_information_mw_single_stage(mw, alpha = 0.05, sides = 2, power = 0.8)

Arguments

delta: Numeric vector containing the estimand under the alternative hypothesis
delta_0: Numeric vector containing the estimand under the null hypothesis
alpha: Desired Type I Error Rate of the test
sides: (Scalar - 1 or 2): Type of Test, either 1-sided or 2-sided.
power: Desired power of the test (1 - Type II Error Rate)
mw: Numeric vector containing the estimand under the alternative hypothesis: the value of the Mann-Whitney under the null is always 0.5.

Value

When all parameters are scalars, the result is a scalar, indicating the required information. When multiple values are specified, a grid of unique parameters are constructed, and the required 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.

This function allows a user to determine the target level of information $\mathcal{I}$ needed to detect a difference of $\delta_{min}$ in the estimand with power $(1 - \beta)$ while maintaining a type I error rate of $(\alpha)$.

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

# When a single value is supplied for each parameter, a scalar is returned:
required_information_single_stage(
  delta = 5,
  delta_0 = 0,
  alpha = 0.05,
  sides = 2,
  power = 0.8
)
#> [1] 0.3139552

# When multiple values are supplied for one or more parameters, the grid of
# parameters are created, and a data.frame is returned.
required_information_single_stage(
  delta = c(5, 7.5),
  delta_0 = 0,
  alpha = 0.05,
  sides = 2,
  power = c(0.8, 0.9)
)
#>   delta delta_0 alpha sides power information
#> 1   5.0       0  0.05     2   0.8   0.3139552
#> 2   7.5       0  0.05     2   0.8   0.1395356
#> 3   5.0       0  0.05     2   0.9   0.4202969
#> 4   7.5       0  0.05     2   0.9   0.1867986