gsDesign (version 3.0-1)

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

Description

Support is provided for sample size estimation, power, 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 or power using the method of Farrington and Manning (1990) for 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 routines bsamsize and bpower 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 Miettinen and Nurminen (1985) 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.

varBinomial computes blinded estimates of the variance of the estimate of 1) event rate differences, 2) logarithm of the risk ratio, or 3) logarithm of the odds ratio. This is intended for blinded sample size re-estimation for comparative trials with a binary outcome.

Usage

nBinomial(p1, p2, alpha=.025, beta=0.1, delta0=0, ratio=1,
          sided=1, outtype=1, scale="Difference", n=NULL) 
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")
varBinomial(x,n,delta0=0,ratio=1,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" (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; 1 (default) returns total sample size; 2 returns a data frame with sample size for each group (n1, n2; if n is not input as NULL, power is returned in Power; 3 returns a data frame with total sample size (n), sample size in each group (n1, n2), Type I error (alpha), 1 or 2 (sided, as input), Type II error (beta), power (Power), null and alternate hypothesis standard deviations (sigma0, sigma1), input event rates (p1, p2), null hypothesis difference in treatment group meands (delta0) and null hypothesis event rates (p10, p20).

n

If power is to be computed in nBinomial(), input total trial sample size in n; this may be a vector. This is also the sample size in varBinomial, in which case the argument must be a scalar.

x

Number of “successes” in the combined control and experimental groups.

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

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).

ciBinomial() returns a data frame with 1 row with a confidence interval; variable names are lower and upper.

varBinomial() returns a vector of (blinded) variance estimates of the difference of event rates (scale="Difference"), logarithm of the odds-ratio (scale="OR") or logarithm of the risk-ratio (scale="RR").

With the default outtype=1, nBinomial() returns a vector of total sample sizes is returned. With outtype=2, nBinomial() returns a data frame containing two vectors n1 and n2 containing sample sizes for groups 1 and 2, respectively; if n is input, this option also returns the power in a third vector, Power. With outtype=3, nBinomial() returns a data frame with the following columns:

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

alpha

As input

sided

As input

beta

As input; if n is input, this is computed

Power

If n=NULL on input, this is 1-beta; otherwise, the power is computed for each sample size input

sigma0

A vector containing the standard deviation of the treatment effect difference under the null hypothesis times sqrt(n) when scale="Difference" or scale="OR"; when scale="RR", this is the standard deviation time sqrt(n) for the numerator of the Farrington-Manning test statistic x1-exp(delta0)*x2.

sigma1

A vector containing the values as sigma0, in this case estimated under the alternative hypothesis.

p1

As input

p2

As input

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. This computational method is also used to estimate variances in varBinomial() based on the overall event rate observed and the input treatment difference specified in delta0.

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$<0$.

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.

Miettinen, O and Nurminen, M (1985), Comparative analysis of two rates. Statistics in Medicine; 4 : 213-226.

Examples

# NOT RUN {
# 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.double(x0) <= 
    simBinomial(p1=.078, p2=.078, n1=500, n2=500, nsim=10000)) / 10000
sum(as.double(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)

# 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
n <- nBinomial(p1=.15, p2=.1, beta=.2, alpha=.05)
n
# check if inputing sample size returns the desired power
nBinomial(p1=.15, p2=.1, beta=.2, alpha=.05,n=n)

# re-do with alternate output types
nBinomial(p1=.15, p2=.1, beta=.2, alpha=.05, outtype=2)
nBinomial(p1=.15, p2=.1, beta=.2, alpha=.05, outtype=3)


# 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"))

# compute blinded estimate of treatment effect difference
x1 <- rbinom(n=1,size=100,p=.2)
x2 <- rbinom(n=1,size=200,p=.1)
# blinded estimate of risk difference variance
varBinomial(x=x1+x2,n=300,ratio=2,delta0=0)
# blnded estimate of log-risk-ratio
varBinomial(x=x1+x2,n=300,ratio=2,delta0=0,scale="RR")
# blinded estimate of log-odds-ratio
varBinomial(x=x1+x2,n=300,ratio=2,delta0=0,scale="OR")

# }