An algorithm to minimize the number of blocks in incomplete block designs

Incomplete block designs are experimental designs in which a subset of treatments are included in each block. The researcher must decide which conditions are assigned to each block. This design concept is quite general. At the level of the experiment, a treatment is a condition in an experiment, blocks are different groups of subjects, and the researcher must decide how to assign a subset of conditions to each block of subjects. At the level of the subject, the treatments correspond to individual stimuli, blocks correspond to experimental trials, and the researcher must decide which subset of stimuli to include in each trial. In this article, we present an efficient algorithm that assigns treatments to blocks in an incomplete block design according to two criteria: Each pair of treatments must appear together in at least one block, and the number of blocks in the experiment is minimized. We discuss details and applications of the algorithm and provide software and a web application to generate designs according to the needs of the researcher.

[1]  L. Tippett Statistical Tables: For Biological, Agricultural and Medical Research , 1954 .

[2]  P. Östergård,et al.  EXISTENCE OF $q$ -ANALOGS OF STEINER SYSTEMS , 2013, Forum of Mathematics, Pi.

[3]  Arryn Robbins,et al.  Simulating the Fidelity of Data for Large Stimulus Set Sizes and Variable Dimension Estimation in Multidimensional Scaling , 2018 .

[4]  Grigori Yourganov,et al.  The Perception of Naturalness Correlates with Low-Level Visual Features of Environmental Scenes , 2014, PloS one.

[5]  Michael C. Hout,et al.  SpAM is convenient but also satisfying: Reply to Verheyen et al. (2016). , 2016, Journal of experimental psychology. General.

[6]  Stephen D. Goldinger,et al.  MM-MDS: A Multidimensional Scaling Database with Similarity Ratings for 240 Object Categories from the Massive Memory Picture Database , 2014, PloS one.

[7]  William L. Hays,et al.  Statistics, 5th ed. , 1994 .

[8]  James R. Clay,et al.  Generating balanced incomplete block designs from planar near rings , 1972 .

[9]  B. N. Mandal,et al.  Efficient Incomplete Block Designs Through Linear Integer Programming , 2014 .

[10]  R. Julian R. Abel Forty-Three Balanced Incomplete Block Designs , 1994, J. Comb. Theory, Ser. A.

[11]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[12]  Marc G. Berman,et al.  Psychological responses to natural patterns in architecture , 2019, Journal of Environmental Psychology.

[13]  R. Goldstone An efficient method for obtaining similarity data , 1994 .

[14]  Michael C. Hout,et al.  The versatility of SpAM: a fast, efficient, spatial method of data collection for multidimensional scaling. , 2013, Journal of experimental psychology. General.

[15]  H. Kimmel,et al.  Balanced incomplete blocks to control individual differences , 1988 .

[16]  Michael C. Hout,et al.  Multidimensional Scaling , 2003, Encyclopedic Dictionary of Archaeology.

[17]  Michael C Hout,et al.  The Novel Object and Unusual Name (NOUN) Database: A collection of novel images for use in experimental research , 2016, Behavior research methods.

[18]  William G. Cochran,et al.  Experimental Designs, 2nd Edition , 1950 .

[19]  Tamaryn Menneer,et al.  Using multidimensional scaling to quantify similarity in visual search and beyond , 2015, Attention, Perception, & Psychophysics.

[20]  R. C. Bose A Note on Fisher's Inequality for Balanced Incomplete Block Designs , 1949 .

[21]  Haim Hanani,et al.  Balanced incomplete block designs and related designs , 1975, Discret. Math..

[22]  M. Colbourn,et al.  On Cyclic Steiner 2-Designs , 1980 .