gsDesign (version 2.4-01)

Binomial: 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

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"

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 returns additional interim calculations); 3 and delta0=0 returns a list with total sample size (n), s

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 Nur

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 no

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

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
• When delta0=0, instead of pbar, the following 2 vectors are returned (see details):
• p10
  group 1 event rate used for null hypothesis
• p20
  group 2 event rate used for null hypothesis

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$

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% of 
# 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)

# provide 95% CI (Miettinen and Nurminen method)
ciBinomial(x1=410, x2=400, n1=500, n2=500)

# 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"))