---
title: "Conditional Power Analyses via Type M and Type S errors"
author: Phil Chalmers
date: "`r format(Sys.time(), '%B %d, %Y')`"
output: 
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    base_format: rmarkdown::html_vignette
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    toc: true
vignette: >
  %\VignetteIndexEntry{Type M and Type S errors}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
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---

```{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("intro3.rds", package = 'Spower'))
```

```{r include=eval}
getwd()
```


*Gelman, A., & Carlin, J. (2014). Beyond Power
Calculations: Assessing Type S (Sign) and Type M (Magnitude) Errors.
Perspectives on Psychological Science, 9(6), 641-651.
<https://doi.org/10.1177/1745691614551642>*

In the publication above, Gelman and Carlin (2014) present *conditional* power analysis ideas to investigate *Type S* and *Type M* errors, 
which add augmentations to the usual evaluation of power via the average (expected) rejection of $H_0$ or support 
for $H_1$ across independent samples. Specifically:

- **Type S errors** occur when a null hypothesis is confidently
    rejected in light of the alternative being true, however 
    *the rejection is in the wrong direction of the underlying effect*. Hence,
    conditional on a "statistical significance" decision, what is the
    probability that the correct *sign* was inferred?
    -   Ideally Type S errors should be close to 0.
    -   As the magnitude of an effect sizes approaches $H_0$ (i.e., there is no effect) the  
        the probability of a making a Type S error will tend to .50
    -   Type S errors are particularly prevalent in under powered
        research studies, where the sampling variability could result in
        detection of an effect that happens to be *opposite* what should
        have been detected from the population generating structure
- **Type M errors**, on the other hand, occur when the *magnitude* of
    a detected effect is *much larger* than the true/expected effect, and was likely the 
    very reason why "statistical significance" occurred in the first place. These types 
    of errors are effectively errors of exaggeration, occurring due to larger sampling variability, 
    and should be treated as "too large" to be taken seriously (e.g., an observed Cohen's $d$ estimate 
    of 5 is often highly unlikely in practice, such as in psychological research, 
    but surely would be flagged as significant even in the smallest of samples drawn from a population).
    -   Gelman and Carlin (2014) presented this as a ratio idea
        ($M=E(\frac{|estimate|}{parameter})$), where large ratios
        indicate that the estimate had to be on average that many times higher than
        the expected value of the parameter (e.g., a Type M ratio of 8
        indicates that of those effect sizes flagged as significant the
        effect estimate had to be on average 8 times larger than the
        true parameter).

Note that while the ratio for Type M errors gets at the idea nicely as it indicates a type of conditional
"exaggeration" effect, nothing really precludes us from drawing further lines in the sand to better map
this concept into a power framework. For instance, if we would only be
confidence that our sample size is "large enough" if we only treat $M$ ratios 
less than, say, 2 or 3, then `TRUE/FALSE` values
could be used to indicate that the observed sample has passed the desired
$M$ cut-off. 

# Type S errors via simulation

To demonstrate how to estimate Type S errors via simulation, 
the following simulation experiment
performs a two-sample $t$-test with a "small" effect size (Cohen's $d=.2$)
with smaller sample sizes. The first implementation demonstrate the logic with 
a user-defined data generation and analysis, while the second approach demonstrates how
the `p_*` functions defined within `Spower` can be used instead --- provided that they 
are relevant to the power analysis under investigation.


## Manual specification

In the manual implementation, where the user write both the data generation and analysis components themselves into a single function, the use of a `while()` loop is required to generate and analyse the experiment until a significant $p$-value is observed. When observed, the `while()` loop is then terminated as the generated data matches the conditional significance criterion, at which point the subsequent analyses relevant to the compliment of Type S error (the *correct* sign decision) is returned as a `logical` so that conditional power will be reflected in the output.


```{r eval=eval}
l_two.t_correct.sign <- function(n, mean, mu = 0, alpha = .05, ...){
	while(TRUE){
		g1 <- rnorm(n)
		g2 <- rnorm(n, mean=mean)
		out <- t.test(g2, g1, mu=mu, ...)
		if(out$p.value < alpha) break   # if "significant" then break while() loop
	}
	mean_diff <- unname(out$estimate[1] - out$estimate[2])
	mean_diff > mu                      # return TRUE if the correct sign is observed
}
```

```{r eval=eval}
l_two.t_correct.sign(n=15, mean=.2) |> Spower()
```

```{r echo=FALSE, collapse=TRUE}
if(eval) store$typeS <- getLastSpower()
typeS <- store$typeS
print(typeS)
```

From the output from `Spower()`, the `power` information reflects the 
estimated probability that the correct sign
decision was made given that "significance" was observed, while $1-power$
provides the estimate of the associated Type S error (probability of an incorrect sign 
given significance). As can be seen, there is about a `r typeS$power` probability of making the
correct sign decision about the true effect, and a complimentary
probability estimate of `r 1 - typeS$power` of making a Type S error.

With `Spower`, it's of course possible to evaluate other input properties
associated with these power values. For instance, suppose that we wish
to know the requisite sample size such that Type S errors are made with very
little frequency (hence, high conditional sign power). Further suppose we 
only want to make a Type S error, say, 1/100 times a significant effect
is observed. The following code evaluates how to obtain such an $n$ estimate.

```{r eval=eval}
typeS <- .01
l_two.t_correct.sign(n=interval(10, 200), mean=.2) |> Spower(power=1-typeS)
```

```{r echo=FALSE, collapse=TRUE}
if(eval) store$typeS.N <- getLastSpower()
typeS.N <- store$typeS.N
print(typeS.N)
```

Hence, one would need a sample size of approximately
`r ceiling(typeS.N$n)` per group in order to have a Type S error be
approximately 1%.

## Implementation using built-in `p_t.test()` function

Alternatively, and often preferably, if the simulation function and analysis already appear in the context
of the package definition then `Spower`'s internally defined `p_*` functions can be used
in place of the complete manual implementation. This is beneficial as the data generation and analysis components then
do not need to be written by the front-end user, potentially avoiding implementation issues using 
previously defined simulation experiment code.

As a reminder, the default `p_*` functions in the package always return a $p$-value under the 
null hypothesis specified as this is the canonical way in which power analysis via simulation
is explored (cf. posterior probability approaches). However, these simulation experiment functions
also contain a logical argument `return_analysis`, which if set to `TRUE` will return the complete analysis 
object instead of just the extracted $p$-value (most commonly in the element `p.value`, though
please use functions like `str()` to inspect fully). For Type M/S errors multiple components are clearly 
required, and therefore further information should be extracted from the analysis objects directly to accommodate.

To demonstrate, the built-in function `p_t.test()` is used with the `return_analysis = TRUE` argument. 
The simulation experiment still follows the `while()` loop logic to ensure that "significance" is first flagged, 
however the internal function definitions now provide the data generation and analyses so that the front-end 
user does not have to. 

```{r}
l_two.t_correct.sign <- function(n, mean, mu = 0, alpha = .05, ...){
	while(TRUE){
		# return_analysis argument used to return model object
		out <- p_t.test(n=n, d=mean, mu=mu, return_analysis=TRUE, ...)
		if(out$p.value < alpha) break
	}
	mean_diff <- unname(out$estimate[1] - out$estimate[2])
	mean_diff > mu
}
```

```{r}
l_two.t_correct.sign(100, mean=.5)
```

Setting the sample size to $N=30$ (hence, `n=15`) leads to the following power estimates. 

```{r eval=eval}
l_two.t_correct.sign(n=15, mean=.2) |> Spower()
```

```{r echo=FALSE, collapse=TRUE}
if(eval) store$typeS2 <- getLastSpower()
typeS2 <- store$typeS2
print(typeS2)
```

If at all possible it is recommended to use the `return_analysis` approach as the simulation experiments defined within the package have all been well tested. The trade-off, as it is with all higher-level functions in R, is that there will often be slightly more overhead than user-defined functions, though of course the latter approach comes at the cost of safety. 

# Type M errors via simulation

Continuing with the above two-sample $t$-test structure, suppose we're interested in
minimizing Type M errors to ensure that significant results weren't
due to higher sampling variability, ultimately resulting in significance being raised only when 
unreasonably large effect sizes estimates are observed. For the purpose of
this power demonstration, suppose "unreasonably large" is defined such
that the ratio of the absolute (standardized) mean difference in a two-sample $t$-test, 
which was flagged as significant, was larger than three times the value of
the true (standardized) mean difference (hence, `M.ratio = 3`). 

Note that while focusing on an `M.ratio = 3` provides the sufficient means to study 
Type M errors using a cut-off logic to define power, it is also possible to store 
the observed $M$ values for further inspection too, which can be done 
in `Spower` if a `list` or
`data.frame` is returned from the supplied experiment.  For this to behave correctly, 
however, output information relevant to the `power` computations
(probability values/logicals) must be explicitly specified using
`Spower(..., select)` so that other values returned from the simulation are 
stored but not summarized.

As before, the first step is to define the experiment using the conditional
$p$-value logic nested within a `while()` loop, followed by the power (and extra) criteria of interest.


```{r eval=eval}
l_two.t_typeM <- function(n, mean, mu = 0, 
						  alpha = .05, M.ratio = 3, ...){
	while(TRUE){
		# return_analysis argument used to return model object
		out <- p_t.test(n=n, d=mean, mu=mu, return_analysis=TRUE, ...)
		if(out$p.value < alpha) break
	}
	diff <- unname(out$estimate[1] - out$estimate[2])
	M <- abs(diff)/mean
	# return data.frame, where "retain" indicates the (logical) power information
	data.frame(retain=M < M.ratio, M=M)
}
```

With $N=50$ per group and a "small" standardized effect size of .2 gives
the following.

```{r eval=eval}
# only use the "retain" information to compute power, though store the rest
l_two.t_typeM(n=50, mean=.2) |> Spower(select='retain') -> typeM
typeM
```

```{r echo=FALSE, collapse=TRUE}
if(eval) store$typeM <- getLastSpower()
typeM <- store$typeM
print(typeM)
```

In this case, `power` represents the probability that, given a
significant result was observed, the resulting $M$ ratio was less than the
cutoff of 3 (hence, was less than three times the true effect size). The
compliment, which reflects the Type M error, is then `r 1-typeM$power`,
which is indeed quite high as approximately
`r paste0(round(1-typeM$power,2) * 100, '%')` of the samples flagged as
significant would have needed rather large observed effects to indicate that the 
sample was "unusual" in the statistical significance sense.

With respect to the distribution of the observed $M$ values themselves, these can
be further extracted using `SimResults()`. Notice that when plotted, the rejection magnitudes are not normally distributed,
which is to be expected given the nature of the conditional simulation experiment.

```{r}
results <- SimResults(typeM)
results
subset(results, select=M) |> descript()
with(results, hist(M, 30))
```

Finally, increasing the sample size greatly helps with the Type M
issues, as seen below by doubling the sample size below.

```{r eval=eval}
# double the total sample size
l_two.t_typeM(n=100, mean=.2) |> Spower(select='retain') -> typeM2
typeM2
```

```{r echo=FALSE, collapse=TRUE}
if(eval) store$typeM2 <- getLastSpower()
typeM2 <- store$typeM2
print(typeM2)
```

where the Type M error for the $M.ratio = 3$ cutoff is now

```{r eval=FALSE}
last <- getLastSpower()
1 - last$power
```

```{r echo=FALSE}
1 - typeM2$power
```

so approximately `r paste0(round(1-typeM2$power,2) * 100, '%')` of the significant results would have been due to
an overly large effect size estimate. Again, the distributional
properties of the observed $M$ values can be extracted and further
analyzed should the need arise.

```{r}
results <- SimResults(typeM2)
results
subset(results, select=M) |> descript()
with(results, hist(M, 30))
```

Type M errors are of course intimately related to *precision criteria*
in power analyses, and in that sense are an alternative way of looking
at power planning to help from relying on "extreme" observations. 
Personally, the notion of thinking in
terms of "ratios relative to the true population effect size" feels somewhat
unnatural, though the point is obviously important as seeing
significance flagged only when unreasonably large effect sizes occur is
indeed troubling (e.g., replication issues).

In contrast then, I would recommend using precision-based power planning
over focusing explicitly on Type M errors when power planning as this
seems more natural, where obtaining greater precision in the target estimates
will necessarily decrease the number of Type M error in both the "extreme"
and "underwhelming" sense of the effect size estimates (the ladder of
which Type M errors do not address as they do not focus on non-significant results
by definition). More importantly, precis |> ion criteria are framed in the
metric of the parameters relevant to the data analyst rather than on 
a metricless effect size ratio, which in general should be more natural to specify.


```{r include=FALSE, eval=eval}
saveRDS(store, '../inst/intro3.rds') # rebuild package when done
```
