html5-img
1 / 72

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

sitara
Télécharger la présentation

Contingency Table Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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…

More Related