The `sprtt`

package is a **s**equential
**p**robability **r**atio
**t**ests **t**oolbox
(**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.

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 (H_{0}).The data collection is

*terminated*because enough evidence has been collected for the alternative hypothesis (H_{1}).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.

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).

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.