Relative Linkage Disequilibrium An intersection between evolution algebraic

Relative Linkage Disequilibrium: An intersection between evolution, algebraic statistics, text mining and contingency tables Ron S. Kenett KPA Ltd. , Raanana, Israel and Department of Statistics and Applied Mathematics "Diego de Castro", University of Torino, Italy in collaboration with Silvia Salini Department of Economics, Business and Statistics, University of Milan, Italy

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Evolution Background 1930 - 1975 Linkage Disequilibrium R. A. Fisher Sam Karlin The Fundamental Theorem of Natural Selection: “the rate of increase of fitness of any organism at any time is equal to its genetic variance at that time” FISHER, R. A. 1930, The Genetical Theory of Natural Selection. Clarendon, Oxford, U. K. . LEWONTIN, R. , and KOJIMA, K. , 1960, The evolutionary dynamics of complex polymorphisms. Evolution 14: 458 -472. KARLIN, S. and FELDMAN, M. , 1970, Linkage and selection: Two locus symmetric viability model. Theoretical Population Biology 1 : 39 -71. KARLIN, S. , 1975, General two-locus models: some objectives, results and interpretations. Theoretical Population Biology 7 : 364 -398. KARLIN, S. and KENETT, R. 1977, Variable Spatial Selection with Two Stages of Migration and Comparisons Between Different Timings, Theoretical Population Biology, 11, pp. 386 -409.

Contingency Tables Background - 1975 Discrete multivariate analysis: theory and practice “The first comprehensive treatise on the analysis of categorical data using loglinear and related statistical models…”

Graphical Display Background - 1983

Algebraic Statistics Background - 2008

Contingency Tables The Data LHS RHS ^LHS RHS ^RHS e RHS ^RHS d RHS ^RHS e RHS ^RHS LHS 57 40 ^LHS 109 48 h RHS ^RHS LHS x 1=. 224 x 2=. 157 LHS x 1=. 389 x 2=. 056 LHS x 1=. 057 x 2=. 321 ^LHS x 3=. 429 x 4=. 189 ^LHS x 3=. 50 ^LHS x 3=. 189 x 4=. 434 x 4=. 056

Contingency Tables “…a man who has carefully investigated a printed table, finds, when done, that he has only a very faint and partial idea of what he has read; and that like a figure imprinted on sand, is soon totally erased and defaced. ” William Playfair (1786), The Commercial and Political Atlas, from Edward R. Tufte (1983), The Visual Display of Quantitative Information.

Graphical Display The Simplex x 1 x 2 x 3 x 4

Graphical Display The Simplex

“Algebraic Statistics” Two loci, two alleles each, four genotypes: AB, Ab, a. B, ab Linkage Disequilibrium RHS ^RHS x 1 x 2 ^LHS x 3 x 4 LHS D can be extended to more dimensions…

“Algebraic Statistics” Linkage Disequilibrium An algebraic observation…

“Algebraic Statistics” Relative Linkage Disequilibrium D is the distance from the point corresponding to the contingency table in the simplex, to the surface D=0 in the e e direction. DM is the distance from the point corresponding to the contingency table on the surface D=0 in the e e direction, to the surface of the simplex, in that direction.

Graphical Display

Contingency Tables RLD Example RLD e RHS ^RHS d RHS ^RHS h RHS ^RHS LHS x 1=. 224 x 2=. 157 LHS x 1=. 389 x 2=. 056 LHS x 1=. 057 x 2=. 321 ^LHS x 3=. 429 x 4=. 189 ^LHS x 3=. 50 ^LHS x 3=. 189 x 4=. 434 x 4=. 056

Text Mining Association Rules • Association rules are one of the most popular unsupervised data mining methods used in applications such as Market Basket Analysis, to measure the associations between products purchased by each consumer, or in web clickstream analysis, to measure the association between the pages seen (sequentially) by a visitor of a site. • Mining frequent itemsets and association rules is a popular and well researched method for discovering interesting relations between variables in large databases. The structure of the data to be analyzed is typically referred to as transactional. • Once obtained, the list of association rules extractable from a given dataset is compared in order to evaluate their importance level. The measures commonly used to assess the strength of an association rule are the indexes of support, confidence, and lift.

Text Mining Support • The support for a rule A => B is obtained dividing the number of transactions which satisfy the rule, NA=>B, by the total number of transactions, N. support {A=>B} = NA=>B / N

Text Mining Support The higher the support the stronger the information that both type of events occur together. support {A=>B} = {NA=>B / N} = x 1 RHS ^RHS LHS x 1 x 2 g ^LHS x 3 x 4 1 -g f 1 -f 1

Text Mining Confidence • The confidence of the rule A => B is obtained by dividing the number of transactions which satisfy the rule, NA=>B , by the number of transactions which contain the body of the rule, A. confidence {A=>B} = NA=>B / NA

Text Mining Confidence A high confidence that the LHS event leads to the RHS event implies causation or statistical dependence. confidence {A=>B} = {NA=>B / NA}= x 1/g RHS ^RHS LHS x 1 x 2 g ^LHS x 3 x 4 1 -g f 1 -f 1

Text Mining Lift • The lift of the rule A => B is the deviation of the support of the whole rule from the support expected under independence given the supports of the LHS (A) and the RHS (B). lift {A=>B} = confidence{A=>B} / support{B} = support{A=>B}/support{A}support{B}

Contingency Tables Relative Linkage Disequilibrium and other measures RLD e RHS LHS 57 ^LHS 109 ^RHS 40 48 Sup = 57/254 =. 224 Conf = 57/97 =. 588 Sup (RHS) = 166/254 =. 654 lift =. 588/. 654 =. 90

Association Rules The groceries example First 20 rules for groceries data, sorted by Lift

Association Rules The groceries example For “Lift” > 2. 5, RLD varies between 1%-40% RLD Lift Plot of Relative Disequilibrium versus Lift for the 430 rules of groceries data set

Association Rules The groceries example

Association Rules The groceries example RLD shows more variability than “support” RLD Support Plot of Relative Disequilibrium versus Support for the 430 rules of groceries data set

Association Rules The groceries example For RLD of 20%, “Confidence” varies between 1%-40% RLD Confidence Plot of Relative Disequilibrium versus Confidence for the 430 rules of groceries data set

Association Rules The groceries example First 20 rules for groceries data, sorted by Lift

Association Rules The groceries example For “Lift” > 2. 5, RLD varies between 1%-40% RLD Lift Plot of Relative Disequilibrium versus Lift for the 430 rules of groceries data set

Association Rules RLD The groceries example Chi. Squared Plot of RLD versus Chisquare for the top 20 rules of groceries data set, sorted by RLD

Association Rules RLD The groceries example Odds Ratio Plot of RLD versus Odds Ratio for the top 20 rules of groceries data set, sorted by RLD

Summary • RLD is intuitive • RLD yields different answers from “usual” measures • RLD can be extended to higher dimensions • There are opportunities in considering the relationship between Association Rules and Contingency Tables

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References • • • Bishop, Y. M. M. , Fienberg, S. E. and Holland, P. W. (1975). Discrete Multivariate Analysis: Theory and Practice. M. I. T. Press, Cambridge, MA. Paperback edition (1977). Reprinted, by Springer. Verlag, New York (2007). Fisher, R. A. (1930). The Genetical Theory of Natural Selection. Clarendon, Oxford, U. K. . Hahsler, M. , Grun, B. , and Hornik, K. (2005). arules – A computational environment for mining association rules and frequent item sets. Journal of Statistical Software, 14(15): 1– 25. ISSN 1548 -7660. URL http: //www. jstatsoft. org/v 14/i 15/. Karlin, S. and Feldman, M. (1970). Linkage and selection: Two locus symmetric viability model. Theoretical Population Biology 1 : 39 -71. Karlin, S. and Kenett, R. S. (1977). Variable Spatial Selection with Two Stages of Migration and Comparisons Between Different Timings, Theoretical Population Biology, 11, pp. 386 -409. Kenett, R. (1983). On an Exploratory Analysis of Contingency Tables. The Statistician, 32, pp. 395 -403. Kenett, R. and Salini, S. (2008). Relative Linkage Disequilibrium: A New measure for association rules UNIMI - Research Papers in Economics, Business, and Statistics http: //services. bepress. com/unimi/statistics/art 32/ Lewontin, R, . C. , and Kojima, K. (1960). The evolutionary dynamics of complex polymorphisms. Evolution 14: 458 -472. Omiecinski, E. (2003). Alternative interest measures for mining associations in databases. IEEE Transactions on Knowledge and Data Engineering, 15(1): 57– 69. Piatetsky-Shapiro, G. (1991). Discovery, analysis, and presentation of strong rules. In: Knowledge Discovery in Databases, pages 229– 248. Shimada, K. , Hirasawa K, and Hu J. (2006) Association Rule Mining with Chi-Squared Test Using Alternate Genetic Network Programming, ICDM 2006. Tan, P-N, Kumar, V. , and Srivastava, J. (2004). Selecting the right objective measure for association analysis. Information Systems, 29(4): 293– 313.