Author: Meike Steinhilber
Date: 2023-07-06

# Sequential One-Way ANOVA

#### Meike Steinhilber

#### 2023-07-06

## Overview

The `sprtt` package is a sequential probability ratio tests toolbox (sprtt). This vignette describes the theoretical background of these tests.

Other recommended vignettes cover:

- a general guide, how to use the package and

- an extended use case.

## What is a sequential test procedure?

With a sequential approach, data is continuously collected and an analysis is performed after each data point, which can lead to three different results (A. Wald, 1945):

- The data collection is terminated because enough evidence has been collected for the null hypothesis (H0).

- The data collection is terminated because enough evidence has been collected for the alternative hypothesis (H1).

- The data collection will continue as there is not yet enough evidence for either of the two hypotheses.

Basically it is not necessary to perform an analysis after each data point — several data points can also be added at once. However, this affects the sample size (N) and the error rates (Schnuerch et al., 2020).

The efficiency of sequential designs has already been examined. Reductions in the sample by 50% and more were found in comparison to analyses with fixed sample sizes (Schnuerch et al., 2020; A. Wald, 1945). Sequential hypothesis testing is therefore particularly suitable when resources are limited because the required sample size is reduced without compromising predefined error probabilities.

## 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 Abraham Wald (1947), 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 (Steinhilber et al., 2023).

## References

Schnuerch, M., Erdfelder, E., & Heck, D. W. (2020). Sequential hypothesis tests for multinomial processing tree models. Journal of Mathematical Psychology, 95, 102326. https://doi.org/10.1016/j.jmp.2020.102326
Steinhilber, M., Schnuerch, M., & Schubert, A.-L. (2023). Sequential one-way ANOVA: Increasing efficiency in psychological hypothesis testing using a variant of sequential probability ratio tests. PsyArXiv. https://doi.org/10.31234/osf.io/m64ne
Wald, A. (1945). Sequential tests of statistical hypotheses. The Annals of Mathematical Statistics, 16(2), 117–186.
Wald, Abraham. (1947). Sequential analysis. Wiley.