gbayes
Gaussian Bayesian Posterior and Predictive Distributions
gbayes
derives the (Gaussian) posterior and optionally the predictive
distribution when both the prior and the likelihood are Gaussian, and
when the statistic of interest comes from a 2sample problem.
This function is especially useful in obtaining the expected power of
a statistical test, averaging over the distribution of the population
effect parameter (e.g., log hazard ratio) that is obtained using
pilot data. gbayes
is also useful for summarizing studies for
which the statistic of interest is approximately Gaussian with
known variance. An example is given for comparing two proportions
using the angular transformation, for which the variance is
independent of unknown parameters except for very extreme probabilities.
A plot
method is also given. This plots the prior, posterior, and
predictive distributions on a single graph using a nice default for
the xaxis limits and using the labcurve
function for automatic
labeling of the curves.
gbayes2
uses the method of Spiegelhalter and Freedman (1986) to compute the
probability of correctly concluding that a new treatment is superior
to a control. By this we mean that a 1alpha
normal
theorybased confidence interval for the new minus old treatment
effect lies wholly to the right of delta.w
, where delta.w
is the
minimally worthwhile treatment effect (which can be zero to be
consistent with ordinary null hypothesis testing, a method not always
making sense). This kind of power function is averaged over a prior
distribution for the unknown treatment effect. This procedure is
applicable to the situation where a prior distribution is not to be
used in constructing the test statistic or confidence interval, but is
only used for specifying the distribution of delta
, the parameter of
interest.
Even though gbayes2
assumes that the test statistic has a normal distribution with known
variance (which is strongly a function of the sample size in the two
treatment groups), the prior distribution function can be completely
general. Instead of using a stepfunction for the prior distribution
as Spiegelhalter and Freedman used in their appendix, gbayes2
uses
the builtin integrate
function for numerical integration.
gbayes2
also allows the variance of the test statistic to be general
as long as it is evaluated by the user. The conditional power given the
parameter of interest delta
is 1  pnorm((delta.w  delta)/sd + z)
, where z
is the normal critical value corresponding to 1  alpha
/2.
gbayesMixPredNoData
derives the predictive distribution of a
statistic that is Gaussian given delta
when no data have yet been
observed and when the prior is a mixture of two Gaussians.
gbayesMixPost
derives the posterior density, cdf, or posterior
mean of delta
given
the statistic x
, when the prior for delta
is a mixture of two
Gaussians and when x
is Gaussian given delta
.
gbayesMixPowerNP
computes the power for a test for delta
> delta.w
for the case where (1) a Gaussian prior or mixture of two Gaussian priors
is used as the prior distribution, (2) this prior is used in forming
the statistical test or credible interval, (3) no prior is used for
the distribution of delta
for computing power but instead a fixed
single delta
is given (as in traditional frequentist hypothesis
tests), and (4) the test statistic has a Gaussian likelihood with
known variance (and mean equal to the specified delta
).
gbayesMixPowerNP
is handy where you want to use an earlier study in
testing for treatment effects in a new study, but you want to mix with
this prior a noninformative prior. The mixing probability mix
can
be thought of as the "applicability" of the previous study. As with
gbayes2
, power here means the probability that the new study will
yield a left credible interval that is to the right of delta.w
.
gbayes1PowerNP
is a special case of gbayesMixPowerNP
when the
prior is a single Gaussian.
 Keywords
 htest
Usage
gbayes(mean.prior, var.prior, m1, m2, stat, var.stat, n1, n2, cut.prior, cut.prob.prior=0.025)
"plot"(x, xlim, ylim, name.stat='z', ...)
gbayes2(sd, prior, delta.w=0, alpha=0.05, upper=Inf, prior.aux)
gbayesMixPredNoData(mix=NA, d0=NA, v0=NA, d1=NA, v1=NA, what=c('density','cdf'))
gbayesMixPost(x=NA, v=NA, mix=1, d0=NA, v0=NA, d1=NA, v1=NA, what=c('density','cdf','postmean'))
gbayesMixPowerNP(pcdf, delta, v, delta.w=0, mix, interval, nsim=0, alpha=0.05)
gbayes1PowerNP(d0, v0, delta, v, delta.w=0, alpha=0.05)
Arguments
 mean.prior
 mean of the prior distribution
 cut.prior,cut.prob.prior,var.prior

variance of the prior. Use a large number such as 10000 to effectively
use a flat (noninformative) prior. Sometimes it is useful to compute
the variance so that the prior probability that
stat
is greater than some impressive valueu
is onlyalpha
. The correctvar.prior
to use is then((umean.prior)/qnorm(1alpha))^2
. You can specifycut.prior=u
andcut.prob.prior=alpha
(whose default is 0.025) in place ofvar.prior
to havegbayes
compute the prior variance in this manner.  m1
 sample size in group 1
 m2
 sample size in group 2
 stat
 statistic comparing groups 1 and 2, e.g., log hazard ratio, difference in means, difference in angular transformations of proportions
 var.stat

variance of
stat
, assumed to be known.var.stat
should either be a constant (allowed ifn1
is not specified), or a function of two arguments which specify the sample sizes in groups 1 and 2. Calculations will be approximate when the variance is estimated from the data.  x

an object returned by
gbayes
or the value of the statistic which is an estimator of delta, the parameter of interest  sd
 the standard deviation of the treatment effect
 prior
 a function of possibly a vector of unknown treatment effects, returning the prior density at those values
 pcdf

a function computing the posterior CDF of the treatment effect
delta
, such as a function created bygbayesMixPost
withwhat="cdf"
.  delta
 a true unknown single treatment effect to detect
 v

the variance of the statistic
x
, e.g.,s^2 * (1/n1 + 1/n2)
. Neitherx
norv
need to be defined togbayesMixPost
, as they can be defined at run time to the function created bygbayesMixPost
.  n1
 number of future observations in group 1, for obtaining a predictive distribution
 n2
 number of future observations in group 2
 xlim
 vector of 2 xaxis limits. Default is the mean of the posterior plus or minus 6 standard deviations of the posterior.
 ylim
 vector of 2 yaxis limits. Default is the range over combined prior and posterior densities.
 name.stat

label for xaxis. Default is
"z"
.  ...

optional arguments passed to
labcurve
fromplot.gbayes
 delta.w
 the minimum worthwhile treatment difference to detech. The default is zero for a plain uninteristing null hypothesis.
 alpha
 type I error, or more accurately one minus the confidence level for a twosided confidence limit for the treatment effect
 upper
 upper limit of integration over the prior distribution multiplied by the normal likelihood for the treatment effect statistic. Default is infinity.
 prior.aux

argument to pass to
prior
fromintegrate
throughgbayes2
. Inside ofpower
the argument must be namedprior.aux
if it exists. You can pass multiple parameters by passingprior.aux
as a list and pulling off elements of the list insideprior
. This setup was used because of difficulties in passing...
arguments throughintegrate
for some situations.  mix

mixing probability or weight for the Gaussian prior having mean
d0
and variancev0
.mix
must be between 0 and 1, inclusive.  d0

mean of the first Gaussian distribution (only Gaussian for
gbayes1PowerNP
and is a required argument)  v0

variance of the first Gaussian (only Gaussian for
gbayes1PowerNP
and is a required argument)  d1

mean of the second Gaussian (if
mix
< 1)  v1

variance of the second Gaussian (if
mix
< 1). Any of these last 5 arguments can be omitted togbayesMixPredNoData
as they can be provided at run time to the function created bygbayesMixPredNoData
.  what

specifies whether the predictive density or the CDF is to be
computed. Default is
"density"
.  interval

a 2vector containing the lower and upper limit for possible values of
the test statistic
x
that would result in a left credible interval exceedingdelta.w
with probability 1alpha
/2  nsim

defaults to zero, causing
gbayesMixPowerNP
to solve numerically for the critical value ofx
, then to compute the power accordingly. Specify a nonzero number such as 20000 fornsim
to instead have the function estimate power by simulation. In this case 0.95 confidence limits on the estimated power are also computed. This approach is sometimes necessary ifuniroot
can't solve the equation for the critical value.
Value
gbayes
returns a list of class "gbayes"
containing the following
names elements: mean.prior
,var.prior
,mean.post
, var.post
, and
if n1
is specified, mean.pred
and var.pred
. Note that
mean.pred
is identical to mean.post
. gbayes2
returns a single
number which is the probability of correctly rejecting the null
hypothesis in favor of the new treatment. gbayesMixPredNoData
returns a function that can be used to evaluate the predictive density
or cumulative distribution. gbayesMixPost
returns a function that
can be used to evaluate the posterior density or cdf. gbayesMixPowerNP
returns a vector containing two values if nsim
= 0. The first value is the
critical value for the test statistic that will make the left credible
interval > delta.w
, and the second value is the power. If nsim
> 0,
it returns the power estimate and confidence limits for it if nsim
>
0. The examples show how to use these functions.
References
Spiegelhalter DJ, Freedman LS, Parmar MKB (1994): Bayesian approaches to
randomized trials. JRSS A 157:357416. Results for gbayes
are derived from
Equations 1, 2, 3, and 6.
Spiegelhalter DJ, Freedman LS (1986): A predictive approach to selecting the size of a clinical trial, based on subjective clinical opinion. Stat in Med 5:113.
Joseph, Lawrence and Belisle, Patrick (1997): Bayesian sample size determination for normal means and differences between normal means. The Statistician 46:209226.
Grouin, JM, Coste M, Bunouf P, Lecoutre B (2007): Bayesian sample size determination in nonsequential clinical trials: Statistical aspects and some regulatory considerations. Stat in Med 26:49144924.
Examples
# Compare 2 proportions using the var stabilizing transformation
# arcsin(sqrt((x+3/8)/(n+3/4))) (Anscombe), which has variance
# 1/[4(n+.5)]
m1 < 100; m2 < 150
deaths1 < 10; deaths2 < 30
f < function(events,n) asin(sqrt((events+3/8)/(n+3/4)))
stat < f(deaths1,m1)  f(deaths2,m2)
var.stat < function(m1, m2) 1/4/(m1+.5) + 1/4/(m2+.5)
cat("Test statistic:",format(stat)," s.d.:",
format(sqrt(var.stat(m1,m2))), "\n")
#Use unbiased prior with variance 1000 (almost flat)
b < gbayes(0, 1000, m1, m2, stat, var.stat, 2*m1, 2*m2)
print(b)
plot(b)
#To get posterior Prob[parameter > w] use
# 1pnorm(w, b$mean.post, sqrt(b$var.post))
#If g(effect, n1, n2) is the power function to
#detect an effect of 'effect' with samples size for groups 1 and 2
#of n1,n2, estimate the expected power by getting 1000 random
#draws from the posterior distribution, computing power for
#each value of the population effect, and averaging the 1000 powers
#This code assumes that g will accept vectorvalued 'effect'
#For the 2sample proportion problem just addressed, 'effect'
#could be taken approximately as the change in the arcsin of
#the square root of the probability of the event
g < function(effect, n1, n2, alpha=.05) {
sd < sqrt(var.stat(n1,n2))
z < qnorm(1  alpha/2)
effect < abs(effect)
1  pnorm(z  effect/sd) + pnorm(z  effect/sd)
}
effects < rnorm(1000, b$mean.post, sqrt(b$var.post))
powers < g(effects, 500, 500)
hist(powers, nclass=35, xlab='Power')
describe(powers)
# gbayes2 examples
# First consider a study with a binary response where the
# sample size is n1=500 in the new treatment arm and n2=300
# in the control arm. The parameter of interest is the
# treated:control log odds ratio, which has variance
# 1/[n1 p1 (1p1)] + 1/[n2 p2 (1p2)]. This is not
# really constant so we average the variance over plausible
# values of the probabilities of response p1 and p2. We
# think that these are between .4 and .6 and we take a
# further short cut
v < function(n1, n2, p1, p2) 1/(n1*p1*(1p1)) + 1/(n2*p2*(1p2))
n1 < 500; n2 < 300
ps < seq(.4, .6, length=100)
vguess < quantile(v(n1, n2, ps, ps), .75)
vguess
# 75%
# 0.02183459
# The minimally interesting treatment effect is an odds ratio
# of 1.1. The prior distribution on the log odds ratio is
# a 50:50 mixture of a vague Gaussian (mean 0, sd 100) and
# an informative prior from a previous study (mean 1, sd 1)
prior < function(delta)
0.5*dnorm(delta, 0, 100)+0.5*dnorm(delta, 1, 1)
deltas < seq(5, 5, length=150)
plot(deltas, prior(deltas), type='l')
# Now compute the power, averaged over this prior
gbayes2(sqrt(vguess), prior, log(1.1))
# [1] 0.6133338
# See how much power is lost by ignoring the previous
# study completely
gbayes2(sqrt(vguess), function(delta)dnorm(delta, 0, 100), log(1.1))
# [1] 0.4984588
# What happens to the power if we really don't believe the treatment
# is very effective? Let's use a prior distribution for the log
# odds ratio that is uniform between log(1.2) and log(1.3).
# Also check the power against a true null hypothesis
prior2 < function(delta) dunif(delta, log(1.2), log(1.3))
gbayes2(sqrt(vguess), prior2, log(1.1))
# [1] 0.1385113
gbayes2(sqrt(vguess), prior2, 0)
# [1] 0.3264065
# Compare this with the power of a twosample binomial test to
# detect an odds ratio of 1.25
bpower(.5, odds.ratio=1.25, n1=500, n2=300)
# Power
# 0.3307486
# For the original prior, consider a new study with equal
# sample sizes n in the two arms. Solve for n to get a
# power of 0.9. For the variance of the log odds ratio
# assume a common p in the center of a range of suspected
# probabilities of response, 0.3. For this example we
# use a zero null value and the uniform prior above
v < function(n) 2/(n*.3*.7)
pow < function(n) gbayes2(sqrt(v(n)), prior2)
uniroot(function(n) pow(n)0.9, c(50,10000))$root
# [1] 2119.675
# Check this value
pow(2119.675)
# [1] 0.9
# Get the posterior density when there is a mixture of two priors,
# with mixing probability 0.5. The first prior is almost
# noninformative (normal with mean 0 and variance 10000) and the
# second has mean 2 and variance 0.3. The test statistic has a value
# of 3 with variance 0.4.
f < gbayesMixPost(3, 4, mix=0.5, d0=0, v0=10000, d1=2, v1=0.3)
args(f)
# Plot this density
delta < seq(2, 6, length=150)
plot(delta, f(delta), type='l')
# Add to the plot the posterior density that used only
# the almost noninformative prior
lines(delta, f(delta, mix=1), lty=2)
# The same but for an observed statistic of zero
lines(delta, f(delta, mix=1, x=0), lty=3)
# Derive the CDF instead of the density
g < gbayesMixPost(3, 4, mix=0.5, d0=0, v0=10000, d1=2, v1=0.3,
what='cdf')
# Had mix=0 or 1, gbayes1PowerNP could have been used instead
# of gbayesMixPowerNP below
# Compute the power to detect an effect of delta=1 if the variance
# of the test statistic is 0.2
gbayesMixPowerNP(g, 1, 0.2, interval=c(10,12))
# Do the same thing by simulation
gbayesMixPowerNP(g, 1, 0.2, interval=c(10,12), nsim=20000)
# Compute by what factor the sample size needs to be larger
# (the variance needs to be smaller) so that the power is 0.9
ratios < seq(1, 4, length=50)
pow < single(50)
for(i in 1:50)
pow[i] < gbayesMixPowerNP(g, 1, 0.2/ratios[i], interval=c(10,12))[2]
# Solve for ratio using reverse linear interpolation
approx(pow, ratios, xout=0.9)$y
# Check this by computing power
gbayesMixPowerNP(g, 1, 0.2/2.1, interval=c(10,12))
# So the study will have to be 2.1 times as large as earlier thought