![]() ![]() Birnbaum, “An inequality for Mill’s ratio”, Annals of Math. 12 (1941).Ībraham Wald, “On cumulative sums of random variables”, Annals of Math. Harold Hotelling, “Experimental determination of the maximum of a function”, Annals of Math. Walter Bartky, “Multiple sampling with constant probability”, Annals of Math. Romig, “A method of sampling inspection,” The Bell System Tech. This process is experimental and the keywords may be updated as the learning algorithm improves. These keywords were added by machine and not by the authors. This process is continued until either the first or the second decision is made. Again on the basis of the first two trials one of the three decisions is made and if the third decision is reached a third trial is performed, etc. If the third decision is made, a second trial is performed. If the first or the second decision is made, the process is terminated. On the basis of the first trial, one of the three decisions mentioned above is made. Thus, such a test procedure is carried out sequentially. 20 or less), andĬohen´s d (either as the expected effect size or as the lower limit for a substantial effect).By a sequential test of a statistical hypothesis is meant any statistical test procedure which gives a specific rule, at any stage of the experiment (at the n-th trial for each integral value of n), for making one of the following three decisions: (1) to accept the hypothesis being tested (null hypothesis), (2) to reject the null hypothesis, (3) to continue the experiment by making an additional observation. The □ error probability (usually 0.05 or less), In summary, three specifications are required to calculate a sequential t-test: The A and B boundaries are calculated with the previously defined error rates □ (Type I error) and □ (Type II error) as follows: Wald (1945) defined the following rules for the SPRT: Condition To account for the fact that the algebraic sign is unknown in a two-sided test, the t-value is squared (Rushton, 1952).Īfter the calculation of the test statistic, the decision will be either to continue sampling or to terminate the sampling and accept one of the hypotheses. More specifically, it is the ratio of the likelihood of the alternative hypothesis to the likelihood of the null hypothesis at the m-th step of the sampling process (LR m). The test statistic of the SPRT is based on a likelihood ratio, which is a measure of the relative evidence in the data for the given hypotheses. In the SPRT the null and alternative hypotheses are defined as follows, with □ representing the model parameter : The basic idea is to transform the sequence of observations (which is dependent on the variance) into a sequence of the associated t-statistic (which is independent of the variance). Rushton (1950, 1952) and Hajnal (1961) have further developed the SPRT using the t-statistic. However, the usage of Wald´s SPRT is limited in the case of normally distributed data, because the variance has to be known or specified in the hypothesis. The sequential t-test is based on the Sequential Probability Ratio Test (SPRT) by Abraham Wald (1947), which is a highly efficient sequential hypothesis test. Sequential hypothesis testing is therefore particularly suitable when resources are limited because the required sample size is reduced without compromising predefined error probabilities. Reductions in the sample by 50% and more were found in comparison to analyses with fixed sample sizes (Schnuerch & Erdfelder, 2020 Wald, 1945). The efficiency of sequential designs has already been examined. However, this affects the sample size (N) and the error rates (Schnuerch & Erdfelder, 2020). The data collection will continue as there is not yet enough evidence for either of the two hypotheses.īasically it is not necessary to perform an analysis after each data point - several data points can also be added at once. The data collection is terminated because enough evidence has been collected for the alternative hypothesis (H 1). The data collection is terminated because enough evidence has been collected for the null hypothesis (H 0). With a sequential approach, data is continuously collected and an analysis is performed after each data point, which can lead to three different results (Wald, 1945): ![]()
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