---
title: "Sequential One-Way ANOVA"
author: "Meike Steinhilber"
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    toc: true
    toc_depth: 4
description: >
  This vignette describes the sequential one-way ANOVA.
vignette: >
  %\VignetteIndexEntry{Sequential One-Way ANOVA}
  %\VignetteEncoding{UTF-8}{inputenc}
  %\VignetteEngine{knitr::rmarkdown}
bibliography: references.bib
csl: "apa.csl"
---

```{r, echo = FALSE, message = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE
)
```

```{r load-intro, child="sprts.Rmd"}

```

## What is the sequential one-way ANOVA?

The sequential one-way fixed effects ANOVA is based on the Sequential Probability Ratio Test (SPRT) by Abraham @wald1947, which is a highly efficient sequential hypothesis test.
It can be used instead of *t*-tests if the means of two or more groups are compared.
For detailed information see the public preprint [@steinhilber2023a].
Note: keep in mind that the repeated measurement ANOVA is not implemented yet in the sprtt package.

## How to plot the ANOVA results

### Scenario 1: Perfect data
In this case, we have data that are perfectly balanced and in a perfect sampling order.
Here, we can use the default value of the 'plot' argument 'single' or we can choose 'balanced'.

```{r example-1}
set.seed(333)
data <- sprtt::draw_sample_normal(3, f = 0.25, max_n = 30)

# calculate the SPRT -----------------------------------------------------------
anova_results <- sprtt::seq_anova(y~x, f = 0.25, data = data, plot = TRUE)
anova_results <- sprtt::seq_anova(y~x, f = 0.25, data = data, plot = TRUE, seq_steps = "single")
anova_results <- sprtt::seq_anova(y~x, f = 0.25, data = data, plot = TRUE, seq_steps = "balanced")

# plot the results -------------------------------------------------------------
sprtt::plot_anova(anova_results)
```


### Scenario 2: Unbalanced data in an imperfect order
In this case, we have a data set with unbalanced sample sizes between the groups and the data points are not in a perfect order.
Because the order is not perfect, it does not make sense to use the 'balanced' option.
Because the first data points (2*k_groups) are not equally distributed between the groups (some groups have less than 2 data points), the option 'single' would output an error.

**Thus, we need to define the sequential steps ourselves.**

Here, we start later (after 12 data points) but then calculate the LR after every single data point.
```{r example-2}
set.seed(333)
data <- sprtt::draw_sample_normal(3, f = 0.25, max_n = 50, sample_ratio = c(1,2,2))
data <- data[sample(nrow(data)),] # destroy the perfect order of the data

# calculate the SPRT -----------------------------------------------------------
anova_results <- sprtt::seq_anova(y~x, f = 0.25, data = data,
                                  plot = TRUE,
                                  seq_steps = 12:nrow(data) # we start with the first 12 data points instead of the first 6
                                  )

# plot the results -------------------------------------------------------------
sprtt::plot_anova(anova_results,
                 labels = TRUE,
                 position_labels_x = 0.2,
                 position_labels_y = 0.2,
                 position_lr_x = 0.1,
                 position_lr_y = -1.8,
                 font_size = 20,
                 line_size = 1,
                 highlight_color = "darkblue"
                 )
```


## References