Description Usage Arguments Details Value Author(s) References Examples
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.
1 2 3 4 5 6 7 8 | 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")
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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 varBinomial()
, x
may be a vector.
For nBinomial()
, testBinomial()
and varBinomial()
, when one or more arguments is a vector, the routines return a vector of sample sizes or powers (nBinomial
), test statistics (testBinomial
), or variance estimates (varBinomial
).
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 |
beta |
type II error |
delta0 |
A value of 0 (the default) always represents no difference between treatment groups under the null hypothesis.
|
ratio |
sample size ratio for group 2 divided by group 1 |
sided |
2 for 2-sided test, 1 for 1-sided test |
outtype |
|
n |
If power is to be computed in |
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 |
adj |
With |
scale |
“Difference”, “RR”, “OR”; see the |
nsim |
The number of simulations to be performed in |
tol |
Default should probably be used; this is used to deal with a rounding issue in interim calculations |
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
$<0$.
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 |
Power |
If |
sigma0 |
A vector containing the standard deviation of the treatment effect difference under the null hypothesis times |
sigma1 |
A vector containing the values as |
p1 |
As input |
p2 |
As input |
p10 |
group 1 event rate used for null hypothesis |
p20 |
group 2 event rate used for null hypothesis |
Keaven Anderson keaven\_anderson@merck.com
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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | # 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")
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