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Contingency Table Analysis

Contingency Table Analysis. contingency tables show frequencies produced by cross-classifying observations e.g., pottery described simultaneously according to vessel form & surface decoration. most statistical tests for tables are designed for analyzing 2-dimensions

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Contingency Table Analysis

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  1. Contingency Table Analysis

  2. contingency tables show frequencies produced by cross-classifying observations • e.g., pottery described simultaneously according to vessel form & surface decoration

  3. most statistical tests for tables are designed for analyzing 2-dimensions • only examine the interaction of two variables at one time… • most efficient when used with nominal data • using ratio data means recoding data to a lower scale of measurement (ordinal) • means ignoring some of the information originally available…

  4. still, you might do this, particularly if you are interested in association between metric and non-metric variables • e.g.: variation in pot size vs. surface decoration… • may decide to divide pot size into ordinal classes…

  5. small medium large small large

  6. rim diameter: slip: • other options may let you retain more of the original information content

  7. could use a “t-test” to test the equality of the means • makes full use of ratio data… non-specularslip specularslip

  8. why do we work with contingency tables??

  9. because we think there may be some kind of interaction between the variables… • basic question: can the state of one variable be predicted from the state of another variable? • if not, they are independent

  10. expected counts • a baseline to which observed counts can be compared • counts that would occur by random chance ifthe variables are independent, over the long run • for any cell E = (col total * row total)/table total

  11. significance • = probability of getting, by chance, a table as or more deviant than the observed table, if the variables are independent • ‘deviant’ defined in terms of expected table • no causality is necessarily implied by the outcome • but, causality may well be the reason for observed association… • e.g.: grave goods and sex

  12. Fisher’s Exact Test • just for 2 x 2 tables • useful where chi-square test is inappropriate • gives the exact probability of all tables with • the same marginal totals • as or more deviant than the observed table…

  13. P = (a+b)!(a+c)!(b+d)!(c+d)! / (N!a!b!c!d!) P = 5!5!6!6! / 11!4!1!1!5! = 5*6!6! / 11! P = 5*6!6! / 11! = 5*6! / 11*10*9*8*7 P = 5*6! / 11*10*9*8*7 = 3600 / 55440 P = .065

  14. P = .065 use R (or Excel) if the counts aren’t too large… > fisher.test(x)

  15. (observed) (expected) 0.013 0.325 • P = 0.065+0.002 = 0.067 or • P = 0.067+0.013 = 0.080 0.162 0.065 0.433 0.002

  16. in R: > fisher.test(x, alt = "two.sided") > fisher.test(x, alt = “greater”) [i.e.: H1: odds ratio > 1] • 2-tailed test = 0.067+0.013 = 0.080 • 1-tailed test = 0.065+0.002 = 0.067

  17. Chi-square Statistic • an aggregate measure (i.e., based on the entire table) • the greater the deviation from expected values, the larger (exponentially!) the chi-square statistic… • one could devise others that would place less emphasis on large deviations  |o-e|/e

  18. X2 is distributed approximately in accord with the X2 probability distribution • X2 probabilities are traditionally found in a table showing threshold values from a CPD • need degrees of freedom • df = (r-1)*(c-1) • just use R…

  19. (43*24)91 =2 (7-11.8)211.8   .025

  20. 5 6 5 6 X2 assumptions & problems • must be based on counts: • not percentages, ratios or weighted data • fails to give reliable results if expected counts are too low: obs. exp. X2=0.74 P(Fishers)=1.0

  21. rules of thumb • no expected counts less than 5 • almost certainly too stringent • no exp. counts less than 2, and 80% of counts > 5 • more relaxed (but more realistic)

  22. collapsing tables • can often combine columns/rows to increase expected counts that are too low • may increase or reduce interpretability • may create or destroy structure in the table • no clear guidelines • avoid simply trying to identify the combination of cells that produces a “significant” result

  23. obs. counts exp. counts obs. counts exp. counts

  24. chi-square is basically a measure of significance • it is not a good measure of strength of association • can help you decide if a relationship exists, but not how strong it is

  25. X2=1.07alpha=.30 X2=2.13alpha=.14

  26. also, chi-square is a ‘global statistic’ • says nothing (directly) about which parts of a table may be ‘driving’ a large chi-square statistic • ‘chi-square contributions’ from individual cells can help:

  27. Monte Carlo test of X2 significance • based on simulated generation of cell-counts under imposed conditions of independence • randomly assign counts to cells:

  28. significance is simply the proportion of outcomes that produced a X2 statistic >= observed • not based on any assumptions about the distribution of the X2 statistic • overcomes the problems associated with small expected frequencies

  29. G Test • a measure of significance for any r x c table • look up critical values of G2 in an ordinary chi-square table; figure out degrees of freedom the same way • conforms to chi-square distribution better than the chi-square statistic

  30. an R function for G2 gsq.test  function(obs) { df  (nrow(obs)-1) * (ncol(obs)-1) exp  chisq.test(obs)$expected G  2*sum(obs*log(obs/exp)) 2*dchisq(G, df) }

  31. Measures of Association

  32. Phi-Square (2) • an attempt to remove the effects of sample size that makes chi-square inappropriate for measuring association • divide chi-square by n • 2=X2/n • limits: 0: variables are independent 1: perfect association in a 2x2 table; no upper limit in larger tables

  33. 2=0.18 2=0.18

  34. Cramer’s V • also a measure of strength of association • an attempt to standardize phi-square (i.e., control the lack of an upper boundary in tables larger than 2x2 cells) • V=2/m where m=min(r-1,c-1) ; i.e., the smaller of rows-1 or columns-1) • limits: 0-1 for any size table; 1=highest possible association

  35. Yule’s Q • for 2x2 tables only • Q = (ad-bc)/(ad+bc)

  36. Yule’s Q • often used to assess the strength of presence / absence association • range is –1 (perfect negative association) to 1 (perfect positive association); values near 0 indicate a lack of association Q = -.72

  37. Yule’s Q • not sensitive to marginal changes (unlike Phi2) • multiply a row or column by a constant; cancels out… (Q=.65 for both tables)

  38. Yule’s Q • can’t distinguish between different degrees of ‘complete’ association • can’t distinguish between ‘complete’ and ‘absolute’ association

  39. “odds” ratio • easiest with 2 x 2 tables • what are the ‘odds’ of a man being buried on his right side, compared to those of a woman?? • if there is a strong level of association between sex and burial position, the odds should be quite different…

  40. c odds ratio = b a d

  41. 29/11=2.64 14/33=0.42 2.64/0.42=6.21 if there is no association, the odds ratio=1departures from 1 range between 0 and infinity >1 =‘positive association’ <1 =‘negative association’

  42. Goodman and Kruskal’s Tau () • “proportional reduction of error” • how are the probabilities of correctly assigning cases to one set of categories improved by the knowledge of another set of categories??

  43. Goodman and Kruskal’s Tau () • limits are 0-1; 1=perfect association • same results as Phi2 w/ 2x2 table • sensitive to margin differences • asymmetric • get different results predicting row assignments based on columns than from column assignments based on rows

  44. =[P(error|rule 1)-P(error|rule 2)] / P(error|rule 1) • rule 1: random assignments to one variable are made with no knowledge of 2nd variable • rule 2: random assignments to one variable are made with knowledge of 2nd variable

  45. Table Standardization • even very large and highly significant X2 (or G2) statistics don’t necessarily mean that all parts of the table are equally “deviant” (and therefore interesting) • usually need to do other things to highlight loci of association or ‘interaction’ • which cells diverge the most from expected values? • very difficult to decide when both row and column totals are unequal…

  46. Percent standardization • highly intuitive approach, easy to interpret • often used to control the effects of sample-size variation • have to decide if it makes better sense to standardize based on rows, or on columns

  47. usually, you want to standardize whatever it is you want to compare • i.e., if you want to compare columns, base percents on column totals • you may decide to make two tables, one standardized on rows, the other on columns…

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