The canonical way that Spower has been written by focusing primarily on \(p\)-values involving the null hypothesis to be tested (\(P(D|H_0)\)). The reason for setting the package up this way is so that the parameter \(\alpha\) parameter can be used as the “line-in-the-sand” threshold to flag whether a hypothesis was rejected (under the null hypothesis) as this behaviour is common among popular power analysis software.
However, Bayesian power analysis are also supported by the package, where for instance the posterior probability of the alternative hypothesis, \(P(H_1|D)\), is the outcome of the simulation experiment. For instance, and continuing with the simple one-sample \(t\)-test example in the introduction vignette, were the power analysis context be that of a Bayesian analysis the conditional probability of the alternative, \(P(H_1|D)\), may be used instead. However, for this to work with Spower the argument sig.direction = 'above' should be supplied, where sig.level now indicts that “significance” only occurs when an observation is above this cutoff.
Below is one such Bayesian approach using posterior probabilities using the BayesFactor package, which is obtained by translating the Bayes factor output into a suitable posterior probability and focusing on the alternative hypothesis (hence, the posterior probability returned corresponds to \(P(\mu \ne \mu_0|D)\)). The following also assumes that the competing hypotheses are equally likely when obtaining the posterior probability (hence, prior odds are 1:1).
p_single.Bayes.t <- function(n, mean, mu){
g <- rnorm(n, mean=mean)
res <- BayesFactor::ttestBF(g, mu=mu)
bf <- exp(as.numeric(res@bayesFactor[1])) # Bayes factor
posterior <- bf / (bf + 1) # assuming hypotheses are equally likely
posterior # P(H_1|D)
}For the Bayesian \(t\)-test definition defined in the next code chunk evaluation, “significance” is obtained whenever the sample posterior is greater than sig.level = .90, demonstrating strong support of \(H_1\). Note that this is a more strict criteria than the null hypothesis criteria presented in the introduction vignette, and therefore has notably lower power.
# power cut-off for a significantly supportive posterior is > 0.90
p_single.Bayes.t(n=100, mean=.5, mu=.3) |>
Spower(sig.level = .90, sig.direction = 'above')##
## Design conditions:
##
## # A tibble: 1 × 5
## n mean mu sig.level power
## <dbl> <dbl> <dbl> <dbl> <lgl>
## 1 100 0.5 0.3 0.9 NA
##
## Estimate of power: 0.143
## 95% Confidence Interval: [0.136, 0.150]With this approach, the power analysis criteria described in help(Spower) are now possible, where for instance solving other experimental components (such as the sample size n) are easy to setup by providing suitable NA argument flags.
In many applications, it can be advantageous to directly return logical values direction in the simulation experiment rather than letting Spower() perform these threshold transformations as these can include more intricate experimental result requirements. The following showcases various ways that returning logical values works in the Spower package. Note that returning a logical in the simulation experiment necessarily means that the sig.level argument in Spower() and friends will not be used, and therefore suitable alternatives will have to be defined within the context of the simulation experiement code (e.g., including conf.level).
Keeping with the basic t-test experiment in the introduction vignetted, suppose we’re interested in the power to reject the null hypothesis \(H_0:\, \mu = \mu_0\) in a one-sample \(t\)-test scenario, where \(P(X|H_0\)) is the probability of the data given the null hypothesis. Normally, one could simply write an experiment that returns a \(p\)-value in this context, such as the following,
However, an equivalent way to explore power in this context would be to investigate the same null hypothesis via confidence intervals given a specific \(\alpha\) level, where \(CI_\mu=[CI_{\alpha/2},CI_{1-\alpha/2}]\). If one were to take this approach, the define function must return a logical value instead, where the CI is utilize to evaluate the plausibility of \(\mu = \mu_0\) by testing whether \(\mu_0\) falls outside the advertised interval. Alternatively, if one were in a Bayesian context,
one could use the credible interval instead of the confidence interval to construct the same logical output.
The following code demonstrates this logic, assuming that \(\alpha = .05\) (and therefore a two-tailed, 95% CI is used), and uses the is.outside_CI() function to evaluate whether mu falls within the estimated CI returned from t.test().
l_single.t <- function(n, mean, mu=0, conf.level=.95){
g <- rnorm(n, mean=mean)
out <- t.test(g, mu=mu, conf.level=conf.level)
CI <- out$conf.int
is.outside_CI(mu, CI) # equivalent to: !(CI[1] < mu && mu < CI[2])
}
l_single.t(100, mean=.2)## [1] TRUERunning a power analysis with Spower() works out of the box now, where p_single.t() utilizes the sig.level argument in Spower() to construct an equivalent TRUE/FALSE expression internally, while l_single.t() ignores sig.level altogether.
##
## Design conditions:
##
## # A tibble: 1 × 4
## n mean sig.level power
## <dbl> <dbl> <dbl> <lgl>
## 1 100 0.3 0.05 NA
##
## Estimate of power: 0.851
## 95% Confidence Interval: [0.844, 0.858]##
## Design conditions:
##
## # A tibble: 1 × 4
## n mean sig.level power
## <dbl> <dbl> <dbl> <lgl>
## 1 100 0.3 0.05 NA
##
## Estimate of power: 0.845
## 95% Confidence Interval: [0.838, 0.852]Using confidence or credible intervals are also useful in contexts where specific precision criteria are importance in power analysis contexts. Suppose that, in addition to detecting a particular effect of interest in a given sample, the results are only deemed “practically useful” if the resulting effect size inference are sufficiently precise, where precision is based on the width of the uncertainty interval. In this case, one may join the logic of the \(p\)-value and CI approaches presented thus far to create a joint evaluation for power, where a result is deemed “significant” if the tests are null hypothesis test is “surprisingly” rejected (in the \(p\)-value context) and the CI is sufficiently narrow.
As a working example, suppose that the above experiment was generalized such that a meaningfully significant result would require the rejection of the null, \(\mu_0=0\), and that the width of the CI must be less than 1/4 units. What value of \(N\) would be required to obtain such a significant and accurate inference to obtain a power of 80%?
l_precision <- function(n, mean, CI.width, mu=0, alpha=.05){
g <- rnorm(n, mean=mean)
out <- t.test(g, mu=mu)
CI <- out$conf.int
width <- CI[2] - CI[1]
out$p.value < alpha && width < CI.width
}##
## Design conditions:
##
## # A tibble: 1 × 4
## n mean sig.level power
## <dbl> <dbl> <dbl> <dbl>
## 1 NA 0.2 0.05 0.8
##
## Estimate of n: 273.3
## 95% Prediction Interval: [272.7, 273.9]Compared the required \(N\) to an power analysis that just contains a significant result below this estimate is notably higher. Note that this can be evaluated in the above simulation experiment by setting CI.width=Inf as all widths will be accepted, though this could have equivalently been evaluated using p_single.t() defined earlier.
##
## Design conditions:
##
## # A tibble: 1 × 5
## n mean CI.width sig.level power
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 NA 0.2 Inf 0.05 0.8
##
## Estimate of n: 198.0
## 95% Prediction Interval: [195.6, 200.5]If one were using a Bayesian analysis criteria, the Bayes factor (BF) ratio could be used in the logical return context too. For example, returning whether the observed \(BF>3\) in a given random sample indicates at least “moderate” supporting evidence for the hypothesis of interest compared to some competing hypothesis (often the complementary null, though not necessarily).
The downside of focusing on BFs is that they require the computation of the marginal likelihoods, typically via bridge sampling (e.g., via the bridgesampling package), in addition to fitting the model using Markov chain Monte Carlo (MCMC) methods (e.g., brms, rstan, rstanarm). Though not a limitation per se, it is often more natural to focus directly on the sample from posterior distribution for power analysis applications rather than on the marginal Bayes factors. Nevertheless, such applications are possible with Spower if there is interest in doing so.
As a simple example, the following one-sample \(t\)-test initially defined above could be redefined as
l_single.Bayes.t_BF <- function(n, mean, mu=0, bf.cut=3){
g <- rnorm(n, mean=mean)
res <- BayesFactor::ttestBF(g, mu=mu)
bf <- exp(as.numeric(res@bayesFactor[1])) # Bayes factor
bf > bf.cut
}where by default a TRUE is returned if the Bayes factor is greater than 3 and FALSE otherwise. Evaluating this simulation with \(N=100\), \(\mu=.5\), and \(\mu_0=.3\) gives the following power estimate when evaluating each sample against a Bayes factor cut-off of 3.
##
## Design conditions:
##
## # A tibble: 1 × 5
## n mean mu sig.level power
## <dbl> <dbl> <dbl> <dbl> <lgl>
## 1 100 0.5 0.3 0.05 NA
##
## Estimate of power: 0.258
## 95% Confidence Interval: [0.249, 0.266]This section presents two related concepts for estimating the power where some justifiable equivalence interval is of interest.
As an alternative approach to the rejection of the null hypothesis via the \(p\)-value or CI approaches, there may be interest in evaluating power in the context of establishing equivalence, or in directional cases superiority or non-inferiority. The purpose of an equivalence tests is to establish that, although exact effects may exist, they are small enough to be considered practically equal in applications. For example, in an independent samples \(t\)-test, two groups might be considered “equivalent” if the true mean difference in the population is somewhere above \(\epsilon_L\) but below \(\epsilon_U\), where \(\epsilon\)s are used to define the equivalence interval. If, for instance, two groups are equivalent then the null hypotheses to test using the two one-sided test (TOST) logic are \[H_{0a}:\, (\mu_1 - \mu_2) \le -\epsilon_L\] and \[H_{0b}:\,(\mu_1 - \mu_2) \ge \epsilon_U\] Rejecting both of these null hypotheses leads to the induced complementary hypothesis of interest \[H_1:\, \epsilon_L < (\mu_1 - \mu_2) < \epsilon_U\] or in words, the population mean difference falls within the defined region of equivalence. Superiority testing and non-inferiority testing follow the same type of logic, however rather than defining a region of equivalence only one tail of the equivalence is of interest.
To put numbers to the above expression, suppose that the true mean difference between the groups was \(\mu_d = \mu_1 - \mu_2 = 0.05\), however this effect is quite small and effectively equivalent to 0. Moreover, suppose any true difference that fell between \([-0.2, 0.2]\) were determined to be practically equivalent a priori.
l_equiv.t <- function(n, delta, equiv, sds = c(1,1),
sig.level = .025){
g1 <- rnorm(n, mean=0, sd=sds[1])
g2 <- rnorm(n, mean=delta, sd=sds[2])
outL <- t.test(g2, g1, mu=-equiv[1], alternative = "less")$p.value
outU <- t.test(g2, g1, mu=equiv[2], alternative = "less")$p.value
outL < sig.level && outU < sig.level
}##
## Design conditions:
##
## # A tibble: 1 × 4
## n delta sig.level power
## <dbl> <dbl> <dbl> <lgl>
## 1 100 1 0.05 NA
##
## Estimate of power: 0.988
## 95% Confidence Interval: [0.986, 0.990]You can verify that these computations are correct by comparing to established software for now, such as via the TOSTER package.
TOSTER::power_t_TOST(n = 100,
delta = 1,
sd = 2.5,
eqb = 2.5,
alpha = .025,
power = NULL,
type = "two.sample") Two-sample TOST power calculation
power = 0.9881517
beta = 0.01184831
alpha = 0.025
n = 100
delta = 1
sd = 2.5
bounds = -2.5, 2.5
NOTE: n is number in *each* groupAgain, the same type of logic can be evaluated using CIs alone.
l_equiv.t_CI <- function(n, delta, equiv,
sds=c(1,1), conf.level = .95){
g1 <- rnorm(n, mean=0, sd=sds[1])
g2 <- rnorm(n, mean=delta, sd=sds[2])
CI <- t.test(g2, g1, conf.level=conf.level)$conf.int
is.CI_within(CI, interval=equiv) # returns TRUE if CI is within equiv interval
}# an equivalent power analysis for "equivalence tests" via CI evaluations
l_equiv.t_CI(100, delta=1, equiv=c(-2.5, 2.5),
sds=c(2.5, 2.5)) |> Spower()##
## Design conditions:
##
## # A tibble: 1 × 4
## n delta sig.level power
## <dbl> <dbl> <dbl> <lgl>
## 1 100 1 0.05 NA
##
## Estimate of power: 0.987
## 95% Confidence Interval: [0.984, 0.989]Finally, though not exhaustively, one could approach the topic of equivalence using Bayesian methods using draws from the posterior distribution of interest, such as from BUGS or HMC samplers (e.g., stan). This approach is highly similar to the equivalence testing approach described above, but uses highest density interval + ROPE in Bayesian modeling instead. Below is one such example that constructs a simple linear regression model with a binary \(X\) term that is analysed with rstanarm::stan_glm().
library(bayestestR)
library(rstanarm)
p_rope.lm <- function(n, beta0, beta1, range, sigma=1, ...){
# generate data
x <- matrix(rep(0:1, each=n))
y <- beta0 + beta1 * x + rnorm(nrow(x), sd=sigma)
dat <- data.frame(y, x)
# run model, but tell stan_glm() to use its indoor voice
model <- quiet(rstanarm::stan_glm(y ~ x, data = dat))
rope <- bayestestR::rope(model, ci=1, range=range, parameters="x")
as.numeric(rope)
}In the above example, the proportion of the sampled posterior distribution that falls within the ROPE is returned, which works well with the sig.level argument coupled with sig.direction = 'above') in Spower() to define a suitable accept/reject cut-off. Specifically, if sig.level = .95 and sig.direction = 'above') then the ROPE will only be accepted when the percentage of the posterior that falls within the defined ROPE is greater than 95%. This can of course be performed manually instead, returning a TRUE when satisfied and FALSE otherwise.
Below reports a power estimate given \(N=50\times 2=100\), where the ROPE criteria is deemed satisfied/significant if 95% of the posterior distribution for the \(\beta_1=1\) falls within the defined range of \(1 \pm .2\rightarrow [.8,1.2]\). Due to the slower execution speeds of the simulations the power evaluations are computed using parallel=TRUE to utilize all available cores.
p_rope.lm(n=50, beta0=2, beta1=1, sigma=1/2, range=c(.8, 1.2)) |>
Spower(sig.level=.95, sig.direction='above', parallel=TRUE)##
## Design conditions:
##
## # A tibble: 1 × 5
## n beta0 beta1 sig.level power
## <dbl> <dbl> <dbl> <dbl> <lgl>
## 1 50 2 1 0.95 NA
##
## Estimate of power: 0.142
## 95% Confidence Interval: [0.135, 0.149]Finally, to demonstrate why this might be useful, the following estimates the required sample size to achieve 80% power when using a 95% HDI-ROPE criteria.
p_rope.lm(n=NA, beta0=2, beta1=1, sigma=1/2, range=c(.8, 1.2)) |>
Spower(power=.80, sig.level=.95, sig.direction='above',
interval=c(50, 200), parallel=TRUE)##
## Design conditions:
##
## # A tibble: 1 × 5
## n beta0 beta1 sig.level power
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 NA 2 1 0.95 0.8
##
## Estimate of n: 107.7
## 95% Prediction Interval: [107.0, 108.4]