Sample Size Adjusted for Nonadherence or Variability of Input Parameters

Description

Calculate sample size for two sample difference in means tests. The function, ss.nonadh accounts for known proportions of nonadherence. There are three functions that calculate sample size accounting for the variability that comes from estimating parameters from data.

Details

This package does adjustments for sample size calculations for two-sample difference in means tests. The function ss.nonadh adjusts for nonadherence, where the responses may be continuous, discrete, or binary. There are 3 functions which account for variability of estimating parameters from data: ss.fromdata.nvar– accouts for variability of standard deviation estimate when data are normal and the clincially significant difference is known, ss.fromdata.neff– accounts for variability of estimated standardized difference in means when both groups come from normal distributions with the same variance, ss.fromdata.pois– accounts for variability of means when both means are estimated from data assumed Poisson.

Author(s)

Michael Fay <mfay@niaid.nih.gov>

Maintainer: Michael Fay <mfay@niaid.nih.gov>

References

Fay, M.P., Halloran, M.E., and Follmann, D.A. (2007). 'Accounting for Variability in Sample Size Estimation with Applications to Nonadherence and Estimation of Variance and Effect Size' Biometrics 63: 465-474.

Examples

### Suppose treatment will not work on about 30 percent of subjects 
### then rho0=.3 and clinically significant difference in means is .5 
ss.nonadh(delta=.5,rho0=.3)

### Suppose you estimate standard deviation of .32 from a sample of size 12
### and the clinically significant difference is .2
ss.fromdata.nvar(delta=.2,sdhat=.32,df=11)
### The usual method is to assume that the standard deviation is known 
### this result is the same as df=Inf
ss.fromdata.nvar(delta=.2,sdhat=.32,df=Inf)

Object of class ‘power.htest’

Description

Result from ss.nonadh(mu0=.1,mu1=.4,refinement="Fisher.exact"). It took about 8 seconds to calculate.

Usage

data(example.of.Fisher.exact)

Details

Compared with StatXact, gives same sample sizes.

Find calibrated beta

Description

Called by ss.fromdata.nvar. Finds calibrated beta (1-power) for use when estimating standard deviation from data in sample size calculations for differences in two normal means.

Usage

find.calibrated.beta(beta, df, alpha = 0.05)

Arguments

Details

See ss.fromdata.nvar.

Value

Calibrated beta value = 1 - calibrated power.

Author(s)

Michael P. Fay

References

Fay, M.P., Halloran, M.E., and Follmann, D.A. (2007). 'Accounting for Variability in Sample Size Estimation with Applications to Nonadherence and Estimation of Variance and Effect Size' Biometrics 63: 465-474.

See Also

ss.fromdata.nvar

Examples

## see Table 2 of Fay et al 2006.
1-find.calibrated.beta(.2,12,.025)

Find sample sizes when normal standardized difference is estimated from data

Description

Calculate sample sizes for two-sample differences in normal means when standardized difference is estimated from existing data

Usage

ss.fromdata.neff(thetahat, m0, m1, ss.ratio = 1, thetaB = 0, 
    sig.level = 0.05, real.power = 0.8, nominal.power = NULL, 
    alternative = c("two.sided", "one.sided"), 
    MINN0 = 2, MAXN0 = Inf, subdivisions = 1000)

Arguments

Details

Calculates the sample sizes for a study designed to test the difference between the means of two groups, where it is assumed that the responses from each group are distributed normally with the same variance. The standardized difference in means (thetahat) is estimated from existing data that is assumed to also follow the same normal distribution. The method is inherently conservative, so that with a nominal power of .76 the real power will be about .80, and a nominal power of .88 the real power will be about .90. Other values of nominal power are allowed, but only real powers of .80 or .90 are allowed. The one-sided tests are designed to test either H_0: \theta \leq \theta_B vs. H_1: \theta > \theta_B or to test H_0: \theta \geq \theta_B vs. H_1: \theta < \theta_B. The choice of hypotheses is determined by the value of thetahat; if thetahat > thetaB then the former hypotheses are tested, otherwise the latter are. See Fay, Halloran and Follmann (2007) for details.

Value

Object of class "power.htest", a list of the arguments (including the computed sample sizes) augmented with 'METHOD' and 'NOTE' elements. The values 'n0' and 'n1' are the samples sizes for the two groups, rounded up to the nearest integer.

Note

The function ss.fromdata.neff calls uniroot.integer, a function written for this package that finds the nearest integer to the root.

Author(s)

Michael P. Fay

References

Fay, M.P., Halloran, M.E., and Follmann, D.A. (2007). 'Accounting for Variability in Sample Size Estimation with Applications to Nonadherence and Estimation of Variance and Effect Size' Biometrics 63: 465-474.

See Also

ss.fromdata.nvar, ss.fromdata.pois, ss.nonadh, uniroot.integer

Examples

ss.fromdata.neff(.588,23,25)

Find sample sizes when normal standard deviation is estimated from data

Description

Calculate sample sizes for two-sample differences in normal means when the standard deviation (or variance) is estimated from existing data.

Usage

ss.fromdata.nvar(delta, sdhat = NULL, vhat = NULL, 
    df = Inf, ss.ratio = 1, var.ratio = 1, deltaB = 0, 
    sig.level = 0.05, power = 0.8, 
    alternative = c("two.sided", "one.sided"))

Arguments

Details

Calculates the sample sizes for a study designed to test the difference between the means of two groups, where it is assumed that the responses from each group are distributed normally. The standard deviation (sdhat) or variance (vhat) is estimated from existing data that is assumed to also follow a normal distribution with variance the same as the control group of the proposed study. If sdhat (or vhat) is estimated from one group with a sample size of m, then df=m-1. If sdhat (or vhat) is estimated from two groups with sample sizes of m0 and m1, then df=m0+m1-2.

The one-sided tests are designed to test either H_0: \delta \leq \delta_B vs. H_1: \delta > \delta_B or to test H_0: \delta \geq \delta_B vs. H_1: \delta < \delta_B. The choice of hypotheses is determined by the value of delta; if delta > deltaB then the former hypotheses are tested, otherwise the latter are.

See Fay, Halloran and Follmann (2007) for details.

Value

Object of class "power.htest", a list of the arguments (including the computed sample sizes) augmented with 'METHOD' and 'NOTE' elements. The values 'n0' and 'n1' are the samples sizes for the two groups, rounded up to the nearest integer.

Note

The function ss.fromdata.nvar calls find.calibrated.beta, a function written for this package that finds the calibrated beta value (see Fay, Halloran and Follmann, 2007).

Author(s)

Michael P. Fay

References

Fay, M.P., Halloran, M.E., and Follmann, D.A. (2007). 'Accounting for Variability in Sample Size Estimation with Applications to Nonadherence and Estimation of Variance and Effect Size' Biometrics 63: 465-474.

See Also

ss.fromdata.neff, ss.fromdata.pois, ss.nonadh, find.calibrated.beta

Examples

ss.fromdata.nvar(.4,sdhat=.682,df=46)

Find sample sizes when 2 Poisson means are estimated from data

Description

Calculate sample sizes for two-sample differences in Poisson means when means are estimated from existing data

Usage

ss.fromdata.pois(xbar0, xbar1, m0, m1, ss.ratio = 1, sig.level = 0.05, 
    real.power = 0.8, nominal.power = NULL, 
    alternative = c("two.sided", "one.sided"), MINN0 = 1, MAXN0 = 10^5)

Arguments

Details

Calculates the sample sizes for a study designed to test the difference between the means of two groups, where it is assumed that the responses from both groups are distributed Poisson. The means from each group (xbar0 and xbar1) come from existing data that is assumed to also follow the same Poisson distributions. The method is inherently conservative, so that with a nominal power of .77 the real power will be about .80, and a nominal power of .89 the real power will be about .90. Other values of nominal power are allowed, but only real powers of .80 or .90 are allowed. If mu0 and mu1 are the means from the two groups, the one-sided tests are designed to test either H_0: \mu_0 \leq \mu_1 vs. H_1: \mu_0 > \mu_1 or to test H_0: \mu_0 \geq \mu_1 vs. H_1: \mu_0 < \mu_1. We estimate \mu_0 and \mu_1 with \hat{\mu}_0 = xbar0 + \frac{1}{2m_0} and \hat{\mu}_1 = xbar1 + \frac{1}{2m_1} . The choice of hypotheses is determined by the value of \hat{\mu}_0 and \hat{\mu}_1; if \hat{\mu}_0 > \hat{\mu}_1 then the former hypotheses are tested, otherwise the latter are. See Fay, Halloran and Follmann (2007) for details.

Value

Object of class "power.htest", a list of the arguments (including the computed sample sizes) augmented with 'METHOD' and 'NOTE' elements. The values 'n0' and 'n1' are the samples sizes for the two groups, rounded up to the nearest integer.

Note

The function ss.fromdata.pois calls uniroot.integer, a function written for this package that finds the nearest integer to the root.

Author(s)

Michael P. Fay

References

Fay, M.P., Halloran, M.E., and Follmann, D.A. (2007). 'Accounting for Variability in Sample Size Estimation with Applications to Nonadherence and Estimation of Variance and Effect Size' Biometrics 63: 465-474.

See Also

ss.fromdata.nvar, ss.fromdata.neff, ss.nonadh, uniroot.integer

Examples

ss.fromdata.pois(1.65,.88,23,25)

Find sample sizes accounting for nonadherence

Description

Calculate sample sizes needed for a two-sample difference in means test accounting for nonadherence (i.e., noncompliance) in a proportion of subjects.

Usage

ss.nonadh(mu0 = NULL, mu1 = NULL, delta = NULL, sigma0.sq = 1, 
    rho0 = 0, rho1 = 0, ss.ratio = 1, var.ratio = 1, deltaB = 0, 
    sig.level = 0.05, power = 0.8, alternative = c("two.sided", "one.sided"), 
    refinement = NULL,error.fisher=10^-6)

Arguments

Details

Calculates the sample sizes for a study designed to test the difference between the means of two groups, where a proportion of those randomized to the control regimin switch to the treatment regimin (rho0) and a proportion of those randomized to the treatment regimin switch to the control regimin (rho1).

You must supply either mu0 and mu1 or delta=mu0-mu1. The one-sided tests are designed to test either H_0: \delta \leq \delta_B vs. H_1: \delta > \delta_B or to test H_0: \delta \geq \delta_B vs. H_1: \delta < \delta_B. The choice of hypotheses is determined by the value of delta; if delta > deltaB then the former hypotheses are tested, otherwise the latter are.

The responses may be continuous, discrete, or even binary. The value refinement defines different methods related to the response type. When refinement=NULL, the method is based on asymptotic normal approximation. For refinement='Normal' the method is based on the t-distribution derived when the responses are normal. For refinement='Bernoulli' the sigma0.sq and var.ratio values are defined by mu0 and mu1, and the sample sizes do not use any continuity correction. For refinement='Fisher.exact' only the two-sided alternative is allowed and the sample size related to exact powers for the 2X2 Fisher exact test are calculated.

See Fay, Halloran and Follmann (2007) for details.

Value

Object of class "power.htest", a list of the arguments (including the computed sample sizes) augmented with 'METHOD' and 'NOTE' elements. The values 'n0' and 'n1' are the samples sizes for the two groups, rounded up to the nearest integer.

Note

If you have a two-sided hypotheses with binary data, then refinement='Fisher.exact' is more accurate than refinement='Bernoulli', but it takes considerably longer to run. If 'Fisher.exact' is chosen the method uses the uniroot.integer function, and if you ‘Esc’ during the calculation, intermediate exact powers will print.

Author(s)

Michael P. Fay

References

Fay, M.P., Halloran, M.E., and Follmann, D.A. (2007). 'Accounting for Variability in Sample Size Estimation with Applications to Nonadherence and Estimation of Variance and Effect Size' Biometrics 63: 465-474.

See Also

uniroot.integer, ss.fromdata.neff, ss.fromdata.pois, ss.fromdata.nvar

Examples

### Suppose treatment will not work on about 30 percent of subjects 
### then rho0=.3 and clinically significant difference in means is .5 
ss.nonadh(delta=.5,rho0=.3)

## Binary data, quick calculation
ss.nonadh(mu0=.1,mu1=.4,refinement="B")
# Calculate Fisher.exact before packaging, took about 8 seconds
#example.of.Fisher.exact<-ss.nonadh(mu0=.1,mu1=.4,refinement="Fisher.exact")
data(example.of.Fisher.exact)
example.of.Fisher.exact

Find the root of a function to the nearest integer

Description

Let f be a monotonic function that changes sign within the interval specified. If f(i)=0 for some i within the interval specified (including the ends of the interval), then the root is i. Otherwise if pos.side=TRUE (or FALSE) then uniroot.integer finds the integer i such that f(i) is closest to the sign change and is positive (or negative).

Usage

uniroot.integer(f, interval, lower = min(interval), upper = max(interval), 
    step.power = 6, step.up = TRUE, pos.side = FALSE, print.steps = FALSE, 
    maxiter = 1000, ...)

Arguments

Details

The algorithm evaluates f(i) iteratively, increasing (or decreasing if step.up=FALSE) i by 2^{step.power} until either f(i)=0 or f(i) switches sign. If f(i)=0, then stop. If f(i) switches sign, then the change in 'i' is halved each iteration until convergence.

Value

A list with the following elements:

Note

Unlike uniroot, the function is not automatically evaluated at both extremes. This makes uniroot.integer an efficient method to use when the calculation time of f(i) increases with the value of 'i'. For an example of the importance of this see ss.fromdata.pois.

Author(s)

Michael P. Fay

See Also

uniroot, used by ss.fromdata.neff, ss.fromdata.pois, ss.nonadh

Examples

root.func<-function(i) i - 500.1 
## initial step sizes = 2^2 =4
uniroot.integer(root.func,c(0,Inf),step.power=2)
## more efficient to use bigger initial step sizes = 2^10 =1024
uniroot.integer(root.func,c(0,Inf),step.power=10,print.steps=TRUE)