---
title: "powergrid"
output:
 rmarkdown::html_vignette:
  toc: true
  toc_depth: 3
vignette: >
  %\VignetteIndexEntry{powergrid}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  dpi=100
)
```

```{r setup}
library(powergrid)
```

# Introduction
The powergrid package is made to facilitate the exploration of statistical power
of a study. In a typical use case (also shown in this vignette) you want to
explore how sample size, the effect size of interest and the assumed variability
of the data influence the power of your planned analysis.

This package does not dictate any of the above ingredients for you. It just
allows you to evaluate a function at a grid of parameters and visualize the
interrelation in plots that are fine-tuned for analyses of power and sample
size. 

You may, however use powergrid for very different purposes. An example close to
a power analysis would be an analysis of the precision you may expect of a study
design. As a more remote application, you can think of a coverage study for
different methods to calculate confidence intervals. Or an exploration for
planning recruitment of study participants under various scenarios.

## Functions covered
This vignette covers the following functions of `powergrid`

- `PowerGrid` for evaluating the power across a grid, with and without
  iterations
- `Example` for inspecting the relation between power and sample size in a
  specific scenario
- `Powerplot` Graphical exploration of power under different scenarios
- `GridPlot` Graphical exploration of power under even more scenarios
- `AddExample` Add an example to an existing PowerPlot or GridPlot
- `FindTarget` For a range of parameters, searching along one other parameter
  for a setting yielding a target power (or other value)

## Outline
In the next section, I start by showing a technical example that should
illustrate the essence of PowerGrid and FindTarget. Then, in the third section,
I show 4 actual use cases exploiting the flexibility of powergrid.

# Technical example
Before turning to a real-world example, I show an example illustrating general
working of the core functions of powergrid: PowerGrid and FindTarget. Note that
you will hardly ever explicitly call FindTarget.

PowerGrid is in essence similar to `tapply` (or `dplyr::group_by1` +
`dplyr::summarise`). FindTarget is in essence a combination of `apply` +
`match` + `min` + `which.min`.

```{R, bare functionality}
## I aim to create an array with the product of three numbers

## A function taking the product of three numbers:
Prod = function(x, y, z){
  x * y * z
}
## A grid of numbers I want to calculate the products of. Note that the names of
## the list equal the argument names in Prod()
par = list(x = 1:10,
	       y = 1:3,
           z = c(1, 10))
products = PowerGrid(par, Prod)
print(products)

## Now, I can ask: for each combination of y and z, what is the lowest x where
## the product is at least 20?
FindTarget(products,
           par_to_search = 'x',
           find_lowest = TRUE,
           target_at_least = TRUE,
           target_value = 20)
## Note: when both y and z are 1, there is no x so that the product becomes 20.

```

If you understand the R chunk above, you understand the essence of powergrid.

### Now, forget about FindTarget!

In typical use cases, you will use Example(), PowerPlot(), and GridPlot() and
AddExample(), which rely on FindTarget internally.

# Four use cases
I describe two basic use cases here, and two slightly more fancy ones. The first
two concern the relation between power, sample size, and two further
parameters. In the [first situation](#ex1), a function for calculating the power
is available. In the [second situation](#ex2), the power needs to be found
through (re-)sampling. The [third example](#ex3) concerns a situation where the
objective is not a minimal power, but a CI-width that must be at most a certain
value. In the [fourth example](#ex4), we do not search the *lowest required*
sample size, but the *highest permissible* standard deviation to achieve a
target power.

## Power and sample size for a t-test {#ex1}
Assume we aim to collect data in a two-armed RCT and plan to perform a simple
t-test. In this situation, the situation concerning power can be summarized in
the following ingredients:

1. total sample size
2. effect size of interest
3. expected standard deviation in the study arms
4. the objective: to achieve a significant t-test


### Calculate and inspect

I use the function `PowerGrid` to evaluate the situation sketched
above. This is done as follows

```{r, basic PowerGrid, fig.width=5.5, fig.height=5.5, fig.align='center'}
## A function that returns the power as a function of three input parameters
PowFun <- function(n, delta, sd){
  ptt = power.t.test(n = n/2,
                     delta = delta,
                     sd = sd,
                     sig.level = 0.03) # the typical 3% alpha threshold
  return(ptt$power)
}
## A list of values of input parameters to study
pars = list( # a normal list
  n = seq(from = 10, to = 60, by = 5), # sample size
  delta = seq(from = 0.5, to = 1.7, by = 0.1), # effect size
  sd = seq(.5, 1, .1) # variability
)
## Apply PowFun to all crossings of the parameters in pars
power = PowerGrid(pars = pars, fun = PowFun)
summary(power)
PowerPlot(power,
          slicer = list(sd = .7))
```

In the code above, note that the names of the elements in the list `pars` match
the names of the function arguments in PowFun. This is a requirement for
PowerGrid to work.

### Focus on an example situation
Now, say, you want to be pretty sure (say, power = 90%) to detect an effect size
as small as 1.1, and the best guess for SD = .7. We can calculate this example:
```{R, basic Example}
Example(power,
        example = list(delta = 1.1, sd = .7),
        target_value = .9) # power = 90%
```

#### Draw a figure with example

```{R, basic PowerPlot, fig.width=5.5, fig.height=5.5, fig.align='center'}
PowerPlot(power,
          slicer = list(sd = .7),
          example = list(delta = 1.1),
          target_value = .9
          )
```

Some things to note in the figure code: 

* You need to "slice out" one plain from your power_array. In this case, this
  the slice where `sd = .9`. The slice has the form of the figure: delta
  by n.
* Note that the example is a bit above the power = 90% line. This is because of
  the resolution of parameter n: at the example value of delta, the arrow points
  at the lowest n in pars, for which the power is at least 90%.
* You could also slice out a plain where `delta = .8` and show how the relation
  between power and n depends on the standard deviation. Or slice out a plain
  where `n = 50`, and see how power behaves as a function of delta and sd.
* You can add additional examples, either by increasing the the length of the
  vector in the argument to example, (e.g., `example = list(delta = c(1, 1.2)))`
  or by using the higher level plotting function `AddExample`. The latter allows
  you more flexibility, like setting different colors or line types.
* There are many options in PowerPlot and AddExample that you may want to learn
  about in the help files.

#### Why target, not power?
The wording "target_value" in both the function argument and the printed result may be
a bit confusing at first. 'Why not "power"?', you may ask. The reason is, that
there is nothing that keeps you from optimizing other things with the
functionality of `powergrid`. Indeed, instead of finding a target power, you may
be looking for a target precision.

### More elaborate exploration of power
The PowerPlot already gives some insight into how the relation between sample
size and power depends on a third parameter, effect size. Now, GridPlot offers a
figure to explore one extra parameter.

#### GridPlot
The figure created by `PowerPlot` can only show the interplay of two variables
and power. GridPlot often offers a more insightful picture, in particular when,
as in this example, we have more than 2 dimensions in our `pars` argument.

The code below shows how to plot the interplay between n, delta and sd when the
goals is the achieve 90% power.

```{r, GridPlot, fig.width=5.5, fig.height=5.5, fig.align='center'}
GridPlot(power,
         target_value = .9, # you need to choose one target level of power
         example = list(delta = 1, sd = .7)) # defined by two parameters now.
```

Note that there are many options in this plot. See the help file of `GridPlot`
for more info. Importantly, any dimension of the the `power_array` in argument x
may be mapped to the x- or y-axis, or to the different lines.

## Power evaluation using simulation and resampling {#ex2}
Assume we have about the same situation as above, but we do not have a
simple solution to calculate the power: we only have a limited pilot data set
that looks as follows:
```{r, pilot data}
pilot_scores = c(2.1, 4.3, 2.3, 5.2, 1.9, 8.3, 7, 2.6, 2.4, 3.2, 2.1, 2.8, 3.4)
```

Since we do not really understand the distribution (it looks pretty
right-skewed), we plan to perform a Mann-Whitney U-test. We do not want to
simply simulate, but draw from our pilot sample to mimic the variability and
distributional form. We do have an idea about effect size (somewhere in the
range of .5 and 2). The following code my be our approach to the exploration of
power:

```{r, resample MannU, fig.width=5.5, fig.height=5.5, fig.align='center', warning = FALSE}
sse_pars = list(
  n = seq(10, 100, 20),
  delta = seq(.5, 2, .2)) # only effect size
PowFun = function(n, delta, pilot_data){
  arm_1 = sample(pilot_data, n, replace = TRUE)
  arm_2 = sample(pilot_data, n, replace = TRUE) + delta
  significant = wilcox.test(arm_1, arm_2)$p.value < .03 # the typical 3% alpha threshold
  return(significant) # each call of this function gives significant either TRUE
                      # or FALSE
  }
power = PowerGrid(pars = sse_pars,
                  fun = PowFun,
                  more_args = list(pilot_data = pilot_scores), # pass the pilot
                                                               # data on to the
                                                               # fun argument
                  n_iter = 99) # we need to iterate over simulated experiemtns
                               # to get a power. I would take a higher value
                               # than 99; this is to keep the example quick.
summary(power)
PowerPlot(power)
```

A couple of notes:

* The power in the example above is calculated by simulating TRUE's and FALSE's
  for significance. These were then automatically summarized by `mean` to yield
  the power. 
* You may want to keep the outcomes of individual iterations of your
  function. To do so, set `summarize = FALSE`. In this situation, the resulting
  array will have one additional dimension, "iter".
* If you choose to keep the individual iterations, be aware that plotting
  functions and `Example` automatically summarize these taking the mean. You
  can, however, choose a different `summary_function`.
* As in PowerPlot, any dimension of the `power_array` may be represented by the
  x-axis, y-axis.
  
## Target a maximum CI95-width {#ex3}
The powergrid package allows to do more than finding a power of at least some
value. Instead, we can target, for example, a 95% confidence interval that
should have at most a certain width. Say, we have the following situation, where
we plan to compare two groups with a t-test and we want the CI not to be wider
than .7 points on our outcome scale. Maybe, 7 is considered a clinically
important difference according to the literature and when our estimates are not
further than .7 points apart, we can't really conclude anything. We can use the
powergrid functionality to find the sample size that gives us an expected CI of
at most that value:

```{r, CI example, fig.width=5.5, fig.height=5.5, fig.align='center'}
CIFun = function(n, delta, sd){
  x1 = rnorm(n, mean = 0, sd = sd)
  x2 = rnorm(n, mean = delta, sd = sd) 
  abs(diff(t.test(x1, x2)$conf.int)) # return the CI-width
}
pars = list( # a normal list
  n = seq(from = 10, to = 60, by = 5), # sample size
  delta = seq(from = 0.5, to = 1.7, by = 0.1), # effect size
  sd = seq(.5, 1, .1) # variability
)
set.seed(1)
CI_array = PowerGrid(pars, CIFun, n_iter = 20) 
summary(CI_array)
## This object now contains, for each parameter combination, the CI-width
## averaged over 20 iterations.

Example(CI_array,
        example = list(delta = .7, sd = .8),
        target_value = .7,
        target_at_least = FALSE,
        find_lowest = TRUE)
## Show results
PowerPlot(CI_array, slicer = list(delta = .7),
          target_levels = c(.6, .7, .8), # this defines the lines
          title = "CI-width as a funtion of sd and n,\nassuming delta = .7",
          shades_of_grey = FALSE) # Grey scale is optimized for situation where
                                  # the array contains power.
AddExample(CI_array,
           slicer = list(delta = .7),
           example = list(sd = .8),
           target_value = .7,
           target_at_least = FALSE)
```

A couple of things to observe: 
* We set "target_at_least" to FALSE, since we are looking for a CI of *at most*
  .7
* The lines in the figure now connect situations with the same CI-width (not
  power, as in the earlier examples)
* We changed the title accordingly.

## Find the upper bound {#ex4}
We have the same situation as in the first use case, where we plan a t-test and
aim for a power of 90%. In the current situation, however, we have a very
limited flexibility in the number of participants that we can recruit. We do
have some way of decreasing the variability by improving our measurements,
however. We want to study how small our SD must be for our study to have
desirable power.

<summary in numbers here>
* we look for the largest acceptable SD (not the smallest n) where we can
  achieve the target value of .7. Therefore, we set `find_lowest = FALSE`.

```{R, maximum SD, fig.width=5.5, fig.height=5.5, fig.align='center'}
sse_pars = list(
  n = c(30, 40),
  delta = seq(from = 0.4, to = 1.2, by = 0.01), ## effect size
  sd = seq(.3, .9, .01)) ## Standard deviation
PowFun <- function(n, delta, sd){
  ptt = power.t.test(n = n/2, delta = delta, sd = sd,
                     sig.level = 0.05)
  return(ptt$power)
}
power_array = PowerGrid(pars = sse_pars, fun = PowFun, n_iter = NA)
Example(power_array,
        example = list(n = 30, delta = .8),
        find_lowest = FALSE,
        target_value = .9)
```

We see that, by setting <code>find_lowest = FALSE</code> Example() shows us the
"maximal permissible" value that sd may take in order to achieve the set target.

Below, inspect the situation in graphic form. Note that the argument
`par_to_search` is set to "sd", putting that parameter on the y-axis.
```{R, maximum SD PowerPlot, fig.width=5.5, fig.height=5.5, fig.align='center'}
PowerPlot(power_array,
          slicer = list(n = 30),
          par_to_search = 'sd',
          example = list(delta = .8),
          find_lowest = FALSE,
          target_value = .9)
```


