--- title: "Logical Vectors, Bayesian power analyses, and ROPEs" author: Phil Chalmers date: "`r format(Sys.time(), '%B %d, %Y')`" output: bookdown::html_document2: base_format: rmarkdown::html_vignette number_sections: false toc: true vignette: > %\VignetteIndexEntry{Logical Vectors, Bayesian power analyses, and ROPEs} %\VignetteEncoding{UTF-8} %\VignetteEngine{knitr::rmarkdown} editor_options: chunk_output_type: console markdown: wrap: 72 --- ```{r include=FALSE} library(Spower) set.seed(42) formals(SpowerCurve)$plotly <- FALSE ``` ```{r include=FALSE} eval <- as.logical(Sys.getenv("SPOWER_EVAL")) if(is.na(eval)) eval <- FALSE # set to FALSE for normal run store <- list() if(!eval) store <- readRDS(system.file("intro2.rds", package = 'Spower')) ``` ```{r include=eval} getwd() ``` # Logical returns In many applications it can be advantageous to directly return `logical` values in the simulation experiment rather than letting `Spower()` perform these threshold transformations internally (e.g., using `sig.level`) as these can include more intricate experimental requirements. The following showcases various ways that returning `logical` values works in the `Spower` package, where the average across the collected `TRUE`/`FALSE` values reflects the target power estimate. Note that returning a `logical` in the simulation experiment necessarily implies that the `sig.level` argument in `Spower()` and friends will not be used, and therefore suitable alternatives must be defined within the context of the simulation experiment code (e.g., including `conf.level` or `sig.level` in the simulation experiment function directly). ## Confidence (and credible) intervals Keeping with the basic $t$-test experiment in the introduction vignette, suppose we're interested in the power to reject the null hypothesis $H_0:\, \mu = \mu_0$ in a one-sample $t$-test, where $P(D|H_0)$ is the probability of the observing 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, ```{r} p_single.t <- function(n, mean, mu=0){ g <- rnorm(n, mean=mean) p <- t.test(g, mu=mu)$p.value p } ``` 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 to define their range, where $CI_\mu=[CI_{\alpha/2},CI_{1-\alpha/2}]$. If one were to take this approach, the defined simulation function should return a `logical` value based on the relation of the parameter estimate to the CI, where the CI is used to evaluate the plausibility of $\mu = \mu_0$. Specifically, in the context of using CI's to reflect $p$-value logic, the CI is used to evaluate whether $\mu_0$ falls *outside* the advertised interval, returning `TRUE` if outside the CI and `FALSE` if within the interval. Alternatively, if one were in a Bayesian analysis context, a *credible interval* could be used 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 the $\mu_0$ parameter falls outside the estimated `CI` returned from `t.test()`. ```{r} 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) ``` Evaluating the power analysis with `Spower()` works out of the box now, noting again that `l_single.t()` will ignore the `Spower(..., sig.level)` information altogether as it is no longer relevant when `logical` information is returned. The following compares both the $p$-value and logical CI approaches, both of which provide identical inferential information in this case (this will not always be true; the $t$-test simply reflects a special case). ```{r eval=eval} p_single.t(n=100, mean=.3) |> Spower() ``` ```{r echo=FALSE} if(eval) store$pPower <- getLastSpower() pPower <- store$pPower print(pPower) ``` ```{r eval=eval} l_single.t(n=100, mean=.3) |> Spower() ``` ```{r echo=FALSE} if(eval) store$lPower <- getLastSpower() lPower <- store$lPower print(lPower) ``` ### Using previouls defined simulation code Note that even in the CI context presented above, writing user-defined functions may not be entirely necessary. This is because the related, internally defined function `p_t.test()` can be used to obtain the same CI information by returning the model itself, and subsequently extracting the `$conf.int` element. The benefit of this, as shown below, is that users do not need to reinvent the data generation and analysis portions of the experiment if this is already available in the package. ```{r} l_single.t <- function(n, mean, mu=0, conf.level=.95){ # return analysis output from t.test() for further extraction out <- p_t.test(n=n, d=mean, mu=mu, type='one.sample', conf.level=conf.level, return_analysis=TRUE) CI <- out$conf.int is.outside_CI(mu, CI) } l_single.t(100, mean=.2) ``` ## Precision criterion Using confidence or credible intervals are also useful in contexts where specific *precision* criteria are important to satisfy. 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 inferences are sufficiently precise, where precision could be based on the *magnitude of the SE*, the *width of the uncertainty interval*, or other relevant precision-based criterion. In this case, one may join the logic of the $p$-value/CI approaches to create a joint evaluation for power, where a result is deemed both "significant and useful" if the null hypothesis is significantly rejected *and* the CI is sufficiently narrow. As a working example, suppose that the above one-sample $t$-test experiment was generalized such that a meaningfully significant result would require a) the rejection of the null, $\mu_0=0$, and b) a CI width less than 1/4 standardized mean units. What value of $N$ would be required to obtain such a significant and sufficiently accurate inference to obtain a power of 80% given, say, the "small" effect size of $d=0.2$? ```{r} l_precision <- function(n, mean, CI.width, mu=0, alpha=.05){ g <- rnorm(n, mean=mean) out <- t.test(g, mu=mu) CI <- out$conf.int width <- CI[2] - CI[1] # return TRUE if significant and CI is sufficiently narrow out$p.value < alpha && width < CI.width } ``` ```{r eval=eval} l_precision(n=interval(10, 500), mean=.2, CI.width=1/4) |> Spower(power=.80) # equivalently: # l_precision(n=NA, mean=.2, CI.width=1/4) |> # Spower(power=.80, interval=c(10, 500)) ``` ```{r echo=FALSE} if(eval) store$lprecision <- getLastSpower() lprecision <- store$lprecision print(lprecision) ``` Compared to the required $N$ from a power analysis that just contains a significant result, this joint practical significance criteria requires a meaningfully higher sample size. Note that in the special case where `CI.width=Inf` then all CI widths will be accepted, which will result in the same power output that would have been obtained using `p_single.t()`. ```{r eval=eval} l_precision(n=interval(10, 500), mean=.2, CI.width=Inf) |> Spower(power=.80) ``` ```{r echo=FALSE} if(eval) store$lprecision2 <- getLastSpower() lprecision2 <- store$lprecision2 print(lprecision2) ``` ## Bayes Factors If one were using a Bayesian analysis criteria rather than the $p$-value approach, 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 would indicate at least "moderate" supporting evidence for the hypothesis of interest compared to some competing hypothesis (often the complementary null, $P(H_0|D)$, though not necessarily), and the average across the independent samples would indicate the degree of power when using this Bayes factor cut-off. 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 strict 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; this is demonstrated in the next section. Nevertheless, such applications are possible with `Spower` if there is sufficient interest in doing so. As a simple example, the following one-sample $t$-test, initially defined above, could be redefined to focus on output from the `BayesFactor` package, which returns the $BF$ criteria in log units (hence, `exp()` is used to return the ratio to its original metric) assuming a non-informative Jeffreys prior for $\mu$. In this case a `TRUE` is returned if the Bayes factor is greater than 3 and `FALSE` if less than or equal to 3. Finally, to ensure that nothing important is lost in the simulation experiment code a `data.frame()` object is returned instead of just the `logical` information, while `Spower()` is informed to only focus on the `logical` information for the purpose of the power computations. ```{r eval=eval} 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 data.frame(largeBF=bf > bf.cut, bf=bf) } ``` Evaluating this simulation with $N=100$, $\mu=.5$, and $\mu_0=.3$ gives the following power estimate. ```{r eval=eval} l_single.Bayes.t_BF(n=100, mean=.5, mu=.3) |> Spower(select='largeBF') -> BFsim BFsim ``` ```{r echo=FALSE} if(eval) store$prospectiveBF3 <- getLastSpower() BFsim <- store$prospectiveBF3 print(BFsim) ``` To view the complete simulation results use `SimResults()` on the resulting output, which if useful could be further plotted. Note that when plotting Bayes factors it is advantageous to present the plot in natural log units. ```{r include=FALSE} library(ggplot2) ``` ```{r} BFresults <- SimResults(BFsim) BFresults # use log-scale for Bayes factors as this is a more useful metric library(ggplot2) ggplot(BFresults, aes(log(bf), fill=largeBF)) + geom_histogram(bins=50) + geom_vline(xintercept=log(3)) + ggtitle('log(BF) distribution') ``` # Bayesian power analysis via posterior probabiltes The canonical way that *Spower* has been designed focuses 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 $\alpha$ (`sig.level`) can be used as the "line-in-the-sand" threshold to flag whether a null hypothesis was rejected in each sample of data as this behaviour is common among popular power analysis software. Bayesian power analysis, on the other hand, are also supported by the package, where instead the posterior probability of the alternative hypothesis, $P(H_1|D)$, is the focus of the simulation experiment. Continuing with the simple one-sample $t$-test example in the introduction vignette and above, 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. For this to work with *Spower* though, the argument `sig.direction = 'above'` should be supplied, where now the `sig.level` indicates that "significance" only occurs when an probability observation is *above* the define `sig.level` cutoff (hence, the default of `.05` is no longer reasonable and should be modified). 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, reflected in the argument `prior_odds`). ```{r eval=eval} # assuming P(H1)/P(H0) are equally likely; hence, prior_odds = 1 pp_single.Bayes.t <- function(n, mean, mu, prior_odds = 1){ g <- rnorm(n, mean=mean) res <- BayesFactor::ttestBF(g, mu=mu) bf <- exp(as.numeric(res@bayesFactor[1])) # Bayes factor posterior_odds <- bf * prior_odds posterior <- posterior_odds / (posterior_odds + 1) posterior # P(H_1|D) } ``` For the Bayesian $t$-test definition 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. ```{r eval=eval} # power cut-off for a significantly supportive posterior is > 0.90 pp_single.Bayes.t(n=100, mean=.5, mu=.3) |> Spower(sig.level = .90, sig.direction = 'above') ``` ```{r echo=FALSE} if(eval) store$prospectiveBayes <- getLastSpower() prospectiveBayes <- store$prospectiveBayes print(prospectiveBayes) ``` With this approach all of the power analysis criteria described in `help(Spower)` are still possible, where for instance solving other experimental components (such as the sample size `n`) are easy to setup by providing suitable `NA` argument flags and search intervals in `Spower()`. # Regions of practical equivalence (ROPEs) This section presents two related concepts for estimating the power where some justifiable equivalence interval is of interest. ## Equivalence testing 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 true differences may exist between groups, the differences are small enough to be considered "practically equivalent" in all subsequent applications. As a running example, suppose that in an independent samples $t$-test the two groups might be considered "equivalent" if the true mean difference in the population is somewhere above $\epsilon_L$ but below $\epsilon_U$, where the $\epsilon$s are used to define the **equivalence interval**. If, for instance, two groups are to be deemed statistically equivalent given these boundary locations then, using a two-one sided hypothesis testing approach (TOST), the two null hypotheses must be evaluated 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 interval is of interest. To put numbers to the above expression, suppose that the true mean difference between the groups was $\mu_d = \mu_2 - \mu_1 = 1$ (labeled `delta`), and each group had an $SD = 2.5$ (labeled `sds`). Furthermore, suppose *any* true difference that fell within the equivalence interval $[-2.5, 2.5]$ (labeled `equiv`) would be deemed practically equivalent a priori. The power to jointly reject the above null hypotheses, and therefore conclude the groups are practically equivalence ($H_1$), is evaluated in the following output for an experiment with $N=100$ observations ($n=50$ for each group). ```{r} 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 } ``` ```{r eval=eval} l_equiv.t(50, delta=1, equiv=c(-2.5, 2.5), sds=c(2.5, 2.5)) |> Spower() ``` ```{r echo=FALSE} if(eval) store$equivT <- getLastSpower() equivT <- store$equivT print(equivT) ``` In this case, the power to conclude that the two groups are equivalent, expressed as a percentage, is `r paste0(round(equivT$power*100), '%')`. You can verify that these computations are correct by comparing to established software for now, such as via the `TOSTER` package. ```{r eval=FALSE} TOSTER::power_t_TOST(n = 50, delta = 1, sd = 2.5, eqb = 2.5, alpha = .025, power = NULL, type = "two.sample") ``` ``` Two-sample TOST power calculation power = 0.8438747 beta = 0.1561253 alpha = 0.025 n = 50 delta = 1 sd = 2.5 bounds = -2.5, 2.5 NOTE: n is number in *each* group ``` Again, the same type of logic can be evaluated using `CI`s alone, and with the built-in `p_t.test()` function, where in this case `TRUE` is returned if the estimated 90% `CI` falls within the defined equivalence interval. ```{r} l_equiv.t_CI <- function(n, delta, equiv, sds = c(1,1), conf.level = .95){ out <- p_t.test(n, delta, sds=sds, conf.level=conf.level, return_analysis=TRUE) is.CI_within(out$conf.int, interval=equiv) # returns TRUE if CI is within equiv interval } ``` ```{r eval=eval} # an equivalent power analysis for "equivalence tests" via CI evaluations l_equiv.t_CI(50, delta=1, equiv=c(-2.5, 2.5), sds=c(2.5, 2.5)) |> Spower() ``` ```{r echo=FALSE} if(eval) store$equivTL <- getLastSpower() equivTL <- store$equivTL print(equivTL) ``` ## Bayesian approach to ROPEs (HDI + ROPE) Finally, though not exhaustively, one could approach the topic of practical equivalence using Bayesian methods using draws from the posterior distribution of interest, such as those available 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()`. ```{r eval=eval} library(bayestestR) library(rstanarm) 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 distribution that falls within the defined ROPE is greater than .95. This can of course be performed manually, returning a `TRUE` when satisfied and `FALSE` otherwise, however in this case it is not necessary. 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. ```{r eval=eval} 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) ``` ```{r echo=FALSE} if(eval) store$ROPE.lm <- getLastSpower() ROPE.lm <- store$ROPE.lm print(ROPE.lm) ``` 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. ```{r eval=eval} rope.lm(n=interval(50, 200), beta0=2, beta1=1, sigma=1/2, range=c(.8, 1.2)) |> Spower(power=.80, sig.level=.95, sig.direction='above', parallel=TRUE) ``` ```{r echo=FALSE} if(eval) store$ROPE.lm_n <- getLastSpower() ROPE.lm_n <- store$ROPE.lm_n print(ROPE.lm_n) ``` ```{r include=FALSE, eval=eval} saveRDS(store, '../inst/intro2.rds') # rebuild package when done ```