---
title: "Calculated power without simulations"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Calculated power without simulations}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

```{r setup, message=FALSE}
library(anovapowersim)
```

The calculation-only functions provide fast power calculations for balanced
factorial ANOVA designs. They use the noncentral F distribution directly and
do not simulate datasets or fit ANOVA models.

The following calculation-only functions are available in `anovapowersim`:

- `power_n_calc()` finds the sample size required for a target effect size and
  power.
- `power_achieved_calc()` calculates power for a fixed sample size and effect
  size.
- `power_sensitivity_calc()` finds the minimum detectable partial eta squared
  for a fixed sample size and target power.

The result objects use `power_calc` for calculated power. Their `n_sims` and
`power_sim` fields are `NA` because no simulations are run.

## Required sample size

Use `power_n_calc()` when the sample size is unknown. As with `power_n()`, the
reported `n_needed` is the number of subjects per between-subject cell. For a
purely within-subject design, it is the total sample size.

```{r required-sample-size}
power_n_calc(
  between = c(group = 2),
  within = c(time = 2),
  term = "group:time",
  target_pes = 0.08,
  power = 0.90,
  alpha = 0.05
)
```

## Achieved power

Use `power_achieved_calc()` when both the sample size and partial eta squared
are fixed.

```{r achieved-power}
power_achieved_calc(
  between = c(group = 2),
  within = c(time = 2),
  term = "group:time",
  target_pes = 0.08,
  n = 30,
  alpha = 0.05
)
```

## Sensitivity

Use `power_sensitivity_calc()` when the sample size is fixed but the minimum
detectable effect size is unknown. The result's `pes_needed` is the calculated
upper bracket that reaches the requested power.

```{r sensitivity}
power_sensitivity_calc(
  between = c(group = 2),
  within = c(time = 2),
  term = "group:time",
  n = 30,
  power = 0.90,
  pes_tol = 0.001,
  alpha = 0.05
)
```

## Planned nonsphericity

For a term containing a within-subject factor, `epsilon` supplies a planned
population nonsphericity correction. It defaults to `1`, which assumes
sphericity. Values below `1` reduce the numerator degrees of freedom,
denominator degrees of freedom, and noncentrality parameter used in the power
calculation.

The value must be at least the theoretical lower bound for the tested term,
`1 / within_term_df`. It must remain `1` for a purely between-subject term.

```{r epsilon}
power_n_calc(
  between = c(group = 2),
  within = c(time = 3),
  term = "group:time",
  target_pes = 0.08,
  power = 0.90,
  epsilon = 0.70
)
```

Calculation-only functions accept `epsilon`, not a `within_covariance()`
object. Use the simulation-based functions when you want to specify a complete
within-subject covariance structure and generate data from it.

## Comparison with G\*Power

The `gpower` option is shared by the balanced simulation and calculation-only
functions. See the short
[Comparison with G\*Power](comparison-with-gpower.html) guide for matching
G\*Power and the case where the default should be preferred.

## When to use simulations instead

Calculated power is useful for quick planning, sensitivity checks, and
comparison with other software. Prefer the simulation-based functions when
you need:

- a complete within-subject covariance structure;
- simulated ANOVA fitting and Greenhouse--Geisser-adjusted tests;
- direct comparison between calculated and simulated power; or
- exact unequal sample sizes and cell means, with a common standard deviation,
  through `power_unbalanced()`.