Binomial {gsDesign}R Documentation

3.2: Testing, Confidence Intervals and Sample Size for Comparing Two Binomial Rates

Description

Support is provided for sample size estimation, testing confidence intervals and simulation for fixed sample size trials (that is, not group sequential or adaptive) with two arms and binary outcomes. Both superiority and non-inferiority trials are considered. While all routines default to comparisons of risk-difference, options to base computations on risk-ratio and odds-ratio are also included.

nBinomial() computes sample size using the method of Farrington and Manning (1990) to derive sample size required to power a trial to test the difference between two binomial event rates. The routine can be used for a test of superiority or non-inferiority. For a design that tests for superiority nBinomial() is consistent with the method of Fleiss, Tytun, and Ury (but without the continuity correction) to test for differences between event rates. This routine is consistent with the Hmisc package routine bsamsize for superiority designs. Vector arguments allow computing sample sizes for multiple scenarios for comparative purposes.

testBinomial() computes a Z- or Chi-square-statistic that compares two binomial event rates using the method of Miettinen and Nurminen (1980). This can be used for superiority or non-inferiority testing. Vector arguments allow easy incorporation into simulation routines for fixed, group sequential and adaptive designs.

ciBinomial() computes confidence intervals for 1) the difference between two rates, 2) the risk-ratio for two rates or 3) the odds-ratio for two rates. This procedure provides inference that is consistent with testBinomial() in that the confidence intervals are produced by inverting the testing procedures in testBinomial(). The Type I error alpha input to ciBinomial is always interpreted as 2-sided.

simBinomial() performs simulations to estimate the power for a Miettinin and Nurminen (1980) test comparing two binomial rates for superiority or non-inferiority. As noted in documentation for bpower.sim() in the HMisc package, by using testBinomial() you can see that the formulas without any continuity correction are quite accurate. In fact, Type I error for a continuity-corrected test is significantly lower (Gordon and Watson, 1996) than the nominal rate. Thus, as a default no continuity corrections are performed.

Usage

nBinomial(p1, p2, alpha=.025, beta=0.1, delta0=0, ratio=1,
          sided=1, outtype=1, scale="Difference") 
testBinomial(x1, x2, n1, n2, delta0=0, chisq=0, adj=0,
             scale="Difference", tol=.1e-10)
ciBinomial(x1, x2, n1, n2, alpha=.05, adj=0, scale="Difference")
simBinomial(p1, p2, n1, n2, delta0=0, nsim=10000, chisq=0, adj=0,
            scale="Difference")

Arguments

For simBinomial() and ciBinomial() all arguments must have length 1.
For testBinomial(), x2, x2, n1, n2, delta0, chisq, and adj may be vectors.
For nBinomial(), p1, p2, beta, delta0 and ratio may be vectors.
For nBinomial() or testBinomial(), when one or more arguments is a vector, the routines return a vector of sample sizes and powers, respectively. Where vector arguments are allowed, there may be a mix of scalar and vector arguments. All arguments specified using vectors must have the same length.

p1 event rate in group 1 under the alternative hypothesis
p2 event rate in group 2 under the alternative hypothesis
alpha type I error; see sided below to distinguish between 1- and 2-sided tests
beta type II error
delta0 A value of 0 (the default) always represents no difference between treatment groups under the null hypothesis. delta0 is interpreted differently depending on the value of the parameter scale. If scale="Difference" (the default), delta0 is the difference in event rates under the null hypothesis (p10 - p20). If scale="RR", delta0 is the logarithm of the relative risk of event rates (p10 / p20) under the null hypothesis. If scale="LNOR", delta0 is the difference in natural logarithm of the odds-ratio under the null hypothesis log(p10 / (1 - p10)) - log(p20 / (1 - p20)).
ratio sample size ratio for group 2 divided by group 1
sided 2 for 2-sided test, 1 for 1-sided test
outtype nBinomial only; (default) returns total sample size; 2 returns sample size for each group (n1, n2); 3 and delta0=0 returns a list with total sample size (n), sample size in each group (n1, n2), null and alternate hypothesis variance (sigma0, sigma1), input event rates (p1, p2) and null hypothesis event rates (p10, p20).
x1 Number of “successes” in the control group
x2 Number of “successes” in the experimental group
n1 Number of observations in the control group
n2 Number of observations in the experimental group
chisq An indicator of whether or not a chi-square (as opposed to Z) statistic is to be computed. If delta0=0 (default), the difference in event rates divided by its standard error under the null hypothesis is used. Otherwise, a Miettinen and Nurminen chi-square statistic for a 2 x 2 table is used.
adj With adj=1, the standard variance with a continuity correction is used for a Miettinen and Nurminen test statistic This includes a factor of n / (n - 1) where n is the total sample size. If adj is not 1, this factor is not applied. The default is adj=0 since nominal Type I error is generally conservative with adj=1 (Gordon and Watson, 1996).
scale “Difference”, “RR”, “OR”; see the scale parameter documentation above and Details. This is a scalar argument.
nsim The number of simulations to be performed in simBinomial()
tol Default should probably be used; this is used to deal with a rounding issue in interim calculations

Details

Testing is 2-sided when a Chi-square statistic is used and 1-sided when a Z-statistic is used. Thus, these 2 options will produce substantially different results, in general. For non-inferiority, 1-sided testing is appropriate.

You may wish to round sample sizes up using ceiling().

Farrington and Manning (1990) begin with event rates p1 and p2 under the alternative hypothesis and a difference between these rates under the null hypothesis, delta0. From these values, actual rates under the null hypothesis are computed, which are labeled p10 and p20 when outtype=3. The rates p1 and p2 are used to compute a variance for a Z-test comparing rates under the alternative hypothesis, while p10 and p20 are used under the null hypothesis.

Sample size with scale="Difference" produces an error if p1-p2=delta0. Normally, the alternative hypothesis under consideration would be p1-p2-delta0$>0$. However, the alternative can have p1-p2-delta0$<0$.

Value

testBinomial() and simBinomial() each return a vector of either Chi-square or Z test statistics. These may be compared to an appropriate cutoff point (e.g., qnorm(.975) for normal or qchisq(.95,1) for chi-square).
With the default outtype=2, nBinomial() returns a list containing two vectors n1 and n2 containing sample sizes for groups 1 and 2, respectively. With outtype=1, a vector of total sample sizes is returned. With outtype=3, nBinomial() returns a list as follows:

n A vector with total samples size required for each event rate comparison specified
n1 A vector of sample sizes for group 1 for each event rate comparison specified
n2 A vector of sample sizes for group 2 for each event rate comparison specified
sigma0 A vector containing the variance of the treatment effect difference under the null hypothesis
sigma1 A vector containing the variance of the treatment effect difference under the alternative hypothesis
p1 As input
p2 As input
pbar Returned only for superiority testing (delta0=0), the weighted average of p1 and p2 using weights n1 and n2
p10 group 1 event rate used for null hypothesis
p20 group 2 event rate used for null hypothesis

Author(s)

Keaven Anderson keaven_anderson@merck.com

References

Farrington, CP and Manning, G (1990), Test statistics and sample size formulae for comparative binomial trials with null hypothesis of non-zero risk difference or non-unity relative risk. Statistics in Medicine; 9: 1447-1454.

Fleiss, JL, Tytun, A and Ury (1980), A simple approximation for calculating sample sizes for comparing independent proportions. Biometrics;36:343-346.

Gordon, I and Watson R (1985), The myth of continuity-corrected sample size formulae. Biometrics; 52: 71-76.

Miettinin, O and Nurminen, M (1980), Comparative analysis of two rates. Statistics in Medicine; 4 : 213-226.

Examples

# Compute z-test test statistic comparing 39/500 to 13/500
# use continuity correction in variance
x <- testBinomial(x1=39, x2=13, n1=500, n2=500, adj=1)
x
pnorm(x, lower.tail=FALSE)

# Compute with unadjusted variance
x0 <- testBinomial(x1=39, x2=23, n1=500, n2=500)
x0
pnorm(x0, lower.tail=FALSE)

# Perform 50k simulations to test validity of the above
# asymptotic p-values 
# (you may want to perform more to reduce standard error of estimate)
sum(as.real(x0) <= 
    simBinomial(p1=.078, p2=.078, n1=500, n2=500, nsim=10000)) / 10000
sum(as.real(x0) <= 
    simBinomial(p1=.052, p2=.052, n1=500, n2=500, nsim=10000)) / 10000

# Perform a non-inferiority test to see if p2=400 / 500 is within 5
# p1=410 / 500 use a z-statistic with unadjusted variance
x <- testBinomial(x1=410, x2=400, n1=500, n2=500, delta0= -.05)
x
pnorm(x, lower.tail=FALSE)

# since chi-square tests equivalence (a 2-sided test) rather than
# non-inferiority (a 1-sided test), 
# the result is quite different
pchisq(testBinomial(x1=410, x2=400, n1=500, n2=500, delta0= -.05, 
                    chisq=1, adj=1), 1, lower.tail=FALSE)

# now simulate the z-statistic witthout continuity corrected variance
sum(qnorm(.975) <= 
    simBinomial(p1=.8, p2=.8, n1=500, n2=500, nsim=100000)) / 100000

# compute a sample size to show non-inferiority
# with 5% margin, 90% power
nBinomial(p1=.2, p2=.2, delta0=.05, alpha=.025, sided=1, beta=.1)

# assuming a slight advantage in the experimental group lowers
# sample size requirement
nBinomial(p1=.2, p2=.19, delta0=.05, alpha=.025, sided=1, beta=.1)

# compute a sample size for comparing 15% vs 10% event rates
# with 1 to 2 randomization
nBinomial(p1=.15, p2=.1, beta=.2, ratio=2, alpha=.05)

# now look at total sample size using 1-1 randomization
nBinomial(p1=.15, p2=.1, beta=.2, alpha=.05)

# look at power plot under different control event rate and
# relative risk reductions
p1 <- seq(.075, .2, .000625)
p2 <- p1 * 2 / 3
y1 <- nBinomial(p1, p2, beta=.2, outtype=1, alpha=.025, sided=1)
p2 <- p1 * .75
y2 <- nBinomial(p1, p2, beta=.2, outtype=1, alpha=.025, sided=1)
p2 <- p1 * .6
y3 <- nBinomial(p1, p2, beta=.2, outtype=1, alpha=.025, sided=1)
p2 <- p1 * .5
y4 <- nBinomial(p1, p2, beta=.2, outtype=1, alpha=.025, sided=1)
plot(p1, y1, type="l", ylab="Sample size",
     xlab="Control group event rate", ylim=c(0, 6000), lwd=2)
title(main="Binomial sample size computation for 80 pct power")
lines(p1, y2, lty=2, lwd=2)
lines(p1, y3, lty=3, lwd=2)
lines(p1, y4, lty=4, lwd=2)
legend(x=c(.15, .2),y=c(4500, 6000),lty=c(2, 1, 3, 4), lwd=2,
       legend=c("25 pct reduction", "33 pct reduction",
                "40 pct reduction", "50 pct reduction"))

[Package gsDesign version 2.0-5 Index]