#### Degree Name

EdD (Doctor of Education)

#### Program

Educational Leadership

#### Date of Award

8-2007

#### Committee Chair or Co-Chairs

W. Hal Knight

#### Committee Members

Edith Seier, Richard E. Osborn, Robert M. Price Jr., Terrence A. Tollefson

#### Abstract

To be effective, a linear congruential random number generator (LCG) should produce values that are (a) uniformly distributed on the unit interval (0,1) excluding endpoints and (b) substantially free of serial correlation. It has been found that many statistical methods produce inflated Type I error rates for correlated observations. Theoretically, independently seeding an LCG under the following conditions attenuates serial correlation: (a) simple random sampling of seeds, (b) non-replicate streams, (c) non-overlapping streams, and (d) non-adjoining streams. Accordingly, 4 algorithms (each satisfying at least 1 condition) were developed: (a) zero-leap, (b) fixed-leap, (c) scaled random-leap, and (d) unscaled random-leap. Note that the latter satisfied all 4 independent seeding conditions.

To assess serial correlation, univariate and multivariate simulations were conducted at 3 equally spaced intervals for each algorithm (N=24) and measured using 3 randomness tests: (a) the serial correlation test, (b) the runs up test, and (c) the white noise test. A one-way balanced multivariate analysis of variance (MANOVA) was used to test 4 hypotheses: (a) omnibus, (b) contrast of unscaled vs. others, (c) contrast of scaled vs. others, and (d) contrast of fixed vs. others. The MANOVA assumptions of independence, normality, and homogeneity were satisfied.

In sum, the seeding algorithms did not differ significantly from each other (omnibus hypothesis). For the contrast hypotheses, only the fixed-leap algorithm differed significantly from all other algorithms. Surprisingly, the scaled random-leap offered the least difference among the algorithms (theoretically this algorithm should have produced the second largest difference). Although not fully supported by the research design used in this study, it is thought that the unscaled random-leap algorithm is the best choice for independently seeding the multiplicative congruential random number generator. Accordingly, suggestions for further research are proposed.

#### Document Type

Dissertation - Open Access

#### Recommended Citation

Stewart, Robert Grisham, "A Statistical Evaluation of Algorithms for Independently Seeding Pseudo-Random Number Generators of Type Multiplicative Congruential (Lehmer-Class)." (2007). *Electronic Theses and Dissertations.* Paper 2049. http://dc.etsu.edu/etd/2049

#### Copyright

Copyright by the authors.