StatTools : Sequential Probability Ratio Test (SPRT) by Wald Explained

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Related link :
Sequential Analysis Introduction and Explained Page
Sequential Probability Ratio Tests for Means Program Page
Sequential Probability Ratio Tests for Proportions Program Page

Introduction SPRT for Means SPRT for Proportions References
General discussions of Sequential Analysis are covered in the Sequential Analysis Introduction and Explained Page and quality statistics in the Quality Statistics Explained Page and will not be repeated here.

This page discusses the Sequential Probability Ratio Test (SPRT). This is the earliest sequential method, developed by Wald, Neyman, and Pearson and their group during the 1930s. These methods were initially developed as a method of quality control, and they form the basis of many subsequent and more sophisticated developments in sequential and quality control methodologies.

The model aims to determine the quality of a batch of products by minimal sampling. The idea is to sample the batch sequentially until a decision can be made whether the batch conforms to specification and can be accepted or that it should be rejected.

These pages offer the 2 major models, for proportion and mean.

Common Terms and Abbreviations

  • α : , also represented as alpha, or p, is the probability of Type I Error. Commonly, p<0.05 or p<0.01 is used as the criteria to reject the null hypothesis
  • β : is the probability of Type II Error. Commonly, β<0.2 is used at the planning stage to determine sample size or calculating borders for sequential analysis.
  • Power : is 1 - β, a concept intuitively easier to understand, and represents the ability to detect a difference, if its really there. A power of 0.8 (80%) is usually used as this is the same as β=0.2
  • Value to accept null hypothesis : is the lower value (proportion or mean) below which the decision to accept the null hypothesis (not significantly different from zero) can be made.
  • Value to reject null hypothesis : is the higher value (proportion or mean) above which the decision to reject the null hypothesis (significantly different from zero) can be made.
  • k : was used by Wald to represent the effect size, and is equivalent to θ which is now more commonly used.
  • Decision borders : are two parallel lines drawn on the sequential chart. Data are plotted as they are sampled. Sampling continues while the plot coordinates are between the two decision lines. Sampling stops when the plot coordinates are outside of the two lines. If it is above the rejection line then the null hypothesis is rejected. If it is below the acceptance line then the null hypothesis is accepted.
  • Truncation : This is the maximum number of samples. Sampling stops at this point even if the coordinate still fall between the two lines. The null hypothesis is usually then accepted. In some algorithms (such as that on these pages, Lines are drawn between the two borders to a midpoint at truncation, and the decision to reject or accept the null hypothesis is made according to which line the last data point crosses.