Measuring Association Among Ordinal Categorical Variables

Measuring association among ordinal categorical variables-Goodman and Kruskal’s Gamma1 An exampleEffect of smoking at 45 years of age on self reported health five years laterVariable Categories Smoking 1 Never smoked 2 Stopped smoking 3 1-14 cigarettes/day 4 15-24 cigarettes/day 5 25+ cigarettes/daySRH 1 Very good 2 Good 3 Fair 4 Bad Ordinal categoriesExpected monotonous association: Increasing codes on Smoking  Increasing code on SRH2 Data on males from the Glostrup surveys+ SMOKE45 | | B:HEALTH51 | J | Vgood Good Fair Bad | TOTAL | ------+------+------+ Never | 16 73 6 1 | 96 | row%| 16.7 76.0 6.3 1.0 | 100.0 | No mo | 15 75 6 0 | 96 | row%| 15.6 78.1 6.3 0.0 | 100.0 | 1-14 | 13 59 7 1 | 80 | row%| 16.3 73.8 8.8 1.3 | 100.0 | 15-24 | 10 81 17 3 | 111 | row%| 9.0 73.0 15.3 2.7 | 100.0 | 25+ | 1 29 3 1 | 34 | row%| 2.9 85.3 8.8 2.9 | 100.0 | ------+ TOTAL | 55 317 39 6 | 417 | row%| 13.2 76.0 9.4 1.4 | 100.0 | ------+ 2 = 16.2 df = 12 p = 0.182No evidence of association even though the expected monotonous relationship is plain as the nose on your face3 Correlation coefficients for ordinal categorical variablesPearson’s correlation coefficient 1) Measures linear association which is not meaningful for ordinal variables2) Evaluation of significance requires normal distributionsRank correlations (Kendall’s  and Spearman’s ) are more appropriate but require continuous data with very little risk of ties.4 Goodman and Kruskal’s  for ordinal categorical data1) Similar to Kendall’s .2) Related to the odds ratio for 22 tables.3) Well-known asymptotic properties.4) A partial  coefficient measuring conditional monotonous relationship among ordinal variables is available.5 Monotonous relationshipsY increases when X increases/decreasesTwo variables: X,YProbabilities: pxy = Prob(X=x,Y=y)X and Y are independent  pxy = Prob(X=x)P(Y=y)What exactly do we mean when we say that there is a monotonous relationship between X and Y?6 Concordance and discordanceCompare outcomes on (X,Y) for two stochastically independent cases(X1,Y1) and (X2,Y2)Concordance (C) if X1<X2 and Y1<Y2 or X1>X2 and Y1>Y2Discordance (D) if X1<X2 and Y1>Y2 or X1>X2 and Y1<Y2Tie (T) if X1=X2 or Y1=Y27 Concordance = same trend in X and Y Discordance = different trends in X and YProbability of concordance p= P P C x1 y 1 x 2 y 2 (x1 ,y 1 ,x 2 ,y 2 ) C p= P P D x1 y 1 x 2 y 2 (x1 ,y 1 ,x 2 ,y 2 ) DPositive relationship: PC > PDNegative relationship: PC < PD8 The gamma coefficientA measure of the strength of the monotonous relationshipP- P g = D C PD+ P CSatisfies all conventional requirements of correlationcoefficients:-1    +1  = 0 if X and Y are independent Positive association if  > 0 Negative association if  < 0 Change the order of Y categories:  after recoding = - before recoding9 Interpretation of P(C) P(C | C� D) P(C)+ P(D)P(D) P(D | C� D) P(C)+ P(D) such that = P(C|CD) – P(D|CD) is the difference between two conditional probabilities10 Estimation of Pairwise comparison of all persons in the data set nC = number of concordances nD = number of discordances nT = number of tiesRelative frequences nC nD hC = hD = nC+ n D + n T nC+ n D + n T nS hT = nC+ n D + n TThe estimate of  h- h n - n G =C D = C D hC+ h D n C + n D11 A little bit of notation nxy = number of persons with X=x and Y=yAij=邋 n xy + n xy x> i,y > j x < i,y < jDij=邋 n xy + n xy x> i,y < j x < i,y > jX 1 . x . c1 Axy Dxy .Y y nxy . . Dxy Axy RNumber of concordances and discordances n= n A n= n D C xy xy D xy xy x,y x,y12 The  coefficient for 22 tables a b c d nC = ad nD = bc g = nC- n D nC+ n D = ad 1 ad- bc- OR - 1 =bc = ad bcad OR 1 ++ 1 + bc13 OR- 1 1 + g g = OR = OR+ 1 1 - gGamma, odds-ratio og logit values gamma oddsratio logit LN(odds-ratio -1,00 ,00 - -,90 ,05 -2,94 -,80 ,11 -2,20 -,70 ,18 -1,73 -,60 ,25 -1,39 -,50 ,33 -1,10 -,40 ,43 -,85 -,30 ,54 -,62 -,20 ,67 -,41 -,10 ,82 -,20 ,00 1,00 ,00 ,10 1,22 ,20 ,20 1,50 ,41 ,30 1,86 ,62 ,40 2,33 ,85 ,50 3,00 1,10 ,60 4,00 1,39 ,70 5,67 1,73 ,80 9,00 2,20 ,90 19,00 2,94 1,00 + +Note: logit  2gamma in the interval [-0,30 til 0,30] 14 Properties of the estimate of The estimate is unbiased E(G) =  and asymptotically normally distributed with standard error, s1, given by2 2 16 轾 s1=4 犏 n xy( n D A xy - n C D xy ) (nC+ n D ) 臌x,yIf X and Y are independent, then the standard error, s0, of G is given by2 4 轾 2 (n- n ) s2 =犏 n A - D - C D 02 xy( xy xy ) n (nC+ n D ) 臌犏x,y15 Statistical inference95 % confidence intervalsG  1.96se1Test of significanceIf X and Y are independent then G z =  Norm(0,1) se0Notice that confidence intervals and assessment of significance uses different estimates of the standard errors16 The example+ SMOKE45 | | B:HEALTH51 | J | Vgood Good Fair Bad | TOTAL | ------+------+------+ Never | 16 73 6 1 | 96 | row%| 16.7 76.0 6.3 1.0 | 100.0 | No mo | 15 75 6 0 | 96 | row%| 15.6 78.1 6.3 0.0 | 100.0 | 1-14 | 13 59 7 1 | 80 | row%| 16.3 73.8 8.8 1.3 | 100.0 | 15-24 | 10 81 17 3 | 111 | row%| 9.0 73.0 15.3 2.7 | 100.0 | 25+ | 1 29 3 1 | 34 | row%| 2.9 85.3 8.8 2.9 | 100.0 | ------+ TOTAL | 55 317 39 6 | 417 | row%| 13.2 76.0 9.4 1.4 | 100.0 | ------+ 2 = 16.2 df = 12 p = 0.182  = 0.24 p < 0.0005 Very strong evidence of an effect of smoking on healthFor ordinal variables,  is much more powerful than 2 distributed test statistics17 Exact conditional inferenceThe problem:Can asymptotic distributions of estimates and test statistics be approximated by asymptotic distributions in small and moderate samples.The small number of persons with bad health would result in warnings from most statistical programs that asymptotics probably do not work.If in doubt use exact conditional tests instead of asymptotic tests.18 The hypergeometric distributionThe contingency table: nxy, x = 1,..c y =1,,.rThe margins of the table: n= n n= n x+ xy n+ y= n xy xy y x x,yThe probability of the table n 骣 nxy P(n11,…,ncr) = 琪 pxy 桫n11 ...n rc x,y19 H0: pxy = px+p+y骣 n n nx+ + y P(n11,…,ncr) = 琪照px+ p + y 桫n11 ...n rc x yThe marginal tables, nx+ and n+y, are sufficient under H0P(n11,…,ncr | n1+,..,nc+, n+1,..,n+r) = 骣骣琪照nx+ !琪 n + y ! 桫x桫 y n! nxy ! x,y does not depend on unknown parameters20 The exact conditional test procedure 1Find all tables with the same marginal tables as the observed table.For each of these tables calculate:The conditional probability of the table The tests statistics of interestThe exact p-value = the sum of probabilities for tables with test statistics that are more extreme than the test statistic of the observed table21 The exact conditional test procedureTest statistic T(M) Where M is a rc tableObserved teststatistic = tobsThe exact p-value pexact= P(M | n 1+ ,..,n c + ,n + 1 ,..,n + r ) M: mx+= n x + , m+y= n + y , T(M) tobsFisher’s exact test for 22 tables Also appropriate for rc tables, but may be very time consuming due to the number of tables fitting the margins The Monte Carlo test22 Since the conditional probabilities are known exactly we may ask the computer to generate a random sample consisting of a large number of independent tables from this distribution:The MC test procedure:Generate tables: M1,…,MNsim Calculate test statistics for each table: Ti = T(Mi), i = 1,..NsimCount the number of random test statistics which are as extreme as the observed statisticNsim S= 1 {Ti t obs} i= 1 S p = is an unbiased estimate of pexact MC NsimThe standard error of pMC depends on Nsim Sequential Monte Carlo tests23 Interrupts the Monte Carlo procedure when it becomes obvious that the test statistic will not be significantNsim = 10,000 Critical level of the test = 5 %, The sequential Monte Carlo test interrupts the Monte Carlo procedure when the number of tables with T(Mi) is equal to 501S  501 pMC  501/10000 > 0.0524 Repeated Monte Carlo testsThe repeated Monte Carlo test interrupts the Monte Carlo procedure when the “risk” of a significant pMC-value has become very smallParameters of the Repeated Monte Carlo test:Nsim = the total number of tables to be generatedNstart = the minimum number of tables to be Generated Critical value Max risk of stopping too soon (default = 0.1 %)25 Smoking and Self reported health+ SMOKE45 | | B:HEALTH51 | J | V.goo Good Fair Bad | TOTAL | ------+------+------+ Never | 16 73 6 1 | 96 | row%| 16.7 76.0 6.3 1.0 | 100.0 | No mo | 15 75 6 0 | 96 | row%| 15.6 78.1 6.3 0.0 | 100.0 | 1-14 | 13 59 7 1 | 80 | row%| 16.3 73.8 8.8 1.3 | 100.0 | 15-24 | 10 81 17 3 | 111 | row%| 9.0 73.0 15.3 2.7 | 100.0 | 25+ | 1 29 3 1 | 34 | row%| 2.9 85.3 8.8 2.9 | 100.0 | X² = 16.2 ------+ df = 12 TOTAL | 55 317 39 6 | 417 | p = 0.182 row%| 13.2 76.0 9.4 1.4 | 100.0 | Gam = 0.24 ------+ p = 0.000 Confounding?26 Analysis of the conditional association given self reported health at 45 years HEALTH45 = Very good+HEALTH45 | + SMOKE45 | | | B:HEALTH51 | G J | V.goo Good Fair Bad | TOTAL | ------+------+------+ 1 Never | 4 12 0 0 | 16 | row%| 25.0 75.0 0.0 0.0 | 100.0 | No mo | 5 7 0 0 | 12 | row%| 41.7 58.3 0.0 0.0 | 100.0 | 1-14 | 9 6 0 0 | 15 | row%| 60.0 40.0 0.0 0.0 | 100.0 | 15-24 | 2 3 0 0 | 5 | row%| 40.0 60.0 0.0 0.0 | 100.0 | 25+ | 0 3 0 0 | 3 | row%| 0.0 100.0 0.0 0.0 | 100.0 | X² = 6.0 ------+ df = 4 TOTAL | 20 31 0 0 | 51 | p = 0.196 row%| 39.2 60.8 0.0 0.0 | 100.0 | Gam = -0.18 ------+ p = 0.18827 HEALTH45 = Good +HEALTH45 | + SMOKE45 | | | B:HEALTH51 | G J | V.goo Good Fair Bad | TOTAL | ------+------+------+ 2 Never | 11 55 5 1 | 72 | row%| 15.3 76.4 6.9 1.4 | 100.0 | No mo | 10 59 5 0 | 74 | row%| 13.5 79.7 6.8 0.0 | 100.0 | 1-14 | 3 50 4 1 | 58 | row%| 5.2 86.2 6.9 1.7 | 100.0 | 15-24 | 6 76 8 1 | 91 | row%| 6.6 83.5 8.8 1.1 | 100.0 | 25+ | 1 25 1 0 | 27 | row%| 3.7 92.6 3.7 0.0 | 100.0 | X² = 9.8 ------+ df = 12 TOTAL | 31 265 23 3 | 322 | p = 0.636 row%| 9.6 82.3 7.1 0.9 | 100.0 | Gam = 0.17 ------+ p = 0.041HEALTH45 = Fair +HEALTH45 | + SMOKE45 | | | B:HEALTH51 | G J | V.goo Good Fair Bad | TOTAL | ------+------+------+3 Never | 1 6 1 0 | 8 | row%| 12.5 75.0 12.5 0.0 | 100.0 | No mo | 0 6 1 0 | 7 | row%| 0.0 85.7 14.3 0.0 | 100.0 | 1-14 | 1 3 3 0 | 7 | row%| 14.3 42.9 42.9 0.0 | 100.0 | 15-24 | 2 1 6 2 | 11 | row%| 18.2 9.1 54.5 18.2 | 100.0 | 25+ | 0 1 2 1 | 4 | row%| 0.0 25.0 50.0 25.0 | 100.0 | X² = 17.5 ------+ df = 12 TOTAL | 4 17 13 3 | 37 | p = 0.131 row%| 10.8 45.9 35.1 8.1 | 100.0 | Gam = 0.52 ------+ p = 0.00128 HEALTH45 = Bad+HEALTH45 | + SMOKE45 | | | B:HEALTH51 | G J | V.goo Good Fair Bad | TOTAL | ------+------+------+4 Never | 0 0 0 0 | 0 | row%| 0.0 0.0 0.0 0.0 | 0.0 | No mo | 0 3 0 0 | 3 | row%| 0.0 100.0 0.0 0.0 | 100.0 | 1-14 | 0 0 0 0 | 0 | row%| 0.0 0.0 0.0 0.0 | 0.0 | 15-24 | 0 1 3 0 | 4 | row%| 0.0 25.0 75.0 0.0 | 100.0 | 25+ | 0 0 0 0 | 0 | row%| 0.0 0.0 0.0 0.0 | 0.0 | X² = 3.9 ------+ df = 1 TOTAL | 0 4 3 0 | 7 | p = 0.047 row%| 0.0 57.1 42.9 0.0 | 100.0 | Gam = 1.00 ------+ p = 0.001------** Local testresults for strata defined by HEALTH45 (G) ** p-values p-values (1-sided) G: HEALTH45 X² df asympt exact Gamma asympt exact ------1: V.good 6.04 4 0.1960 0.1880 -0.18 0.1884 0.2050 2: Good 9.77 12 0.6358 0.6310 0.17 0.0410 0.0290 3: Fair 17.52 12 0.1311 0.1430 0.52 0.0010 0.0030 4: Bad 3.94 1 0.0472 0.1540 1.00 0.0006 0.1220 ------29 Tests of conditional independence H0: P(X,Y|Z=z) = P(X|Z=z)P(Y|Z=z) for all zZ X Y Concordance Test statistics and discordance 1 2 3 1 1 2 12 N1C and N1D N1C  N1D N  N 1= 1C 1D2 1 2 22 N2C and N2D N2C  N2D N  N 2= 2C 2D ...... 2 kZ 1 k2 N2C and N2D NkC  NkD N  N k= kC kD All test statistics must be insignificant Global tests of conditional independenceThe global  22 2     z z30 df  dfz zThe partial  coefficient N  N N  N C  zC D  zD z zNzC  NzD  NC  ND z  partial   NC  ND NzC  NzD  z = NzC  NzD   z z   w z  z  NzC  NzD  z zNzC  NzD where wz = N  N i  iC iD  partial   w z  z z Weighted mean  Asymptotic normal distribution31 Monte Carlo approximation of exact conditional p-values as for 2-way tables------** Local testresults for strata defined by HEALTH45 (G) ** p-values p-values (1-sided) G: HEALTH45 X² df asympt exact Gamma asympt exact ------1: V.good 6.04 4 0.1960 0.1880 -0.18 0.1884 0.2050 2: Good 9.77 12 0.6358 0.6310 0.17 0.0410 0.0290 3: Fair 17.52 12 0.1311 0.1430 0.52 0.0010 0.0030 4: Bad 3.94 1 0.0472 0.1540 1.00 0.0006 0.1220 ------2 Global  = 37.3 df = 29 p = 0.139 pexact = 0.148partial = 0.17 p = 0.034 pexact = 0.027 32 Are the local  coefficients homogenous?Least square estimate: Gamma = 0.1998 s.e. = 0.0777G: HEALTH45 Gamma variance s.e. weight residual ------1: V.good -0.18 0.0397 0.1993 0.152 -2.060 2: Good 0.17 0.0097 0.0987 0.620 -0.411 3: Fair 0.52 0.0264 0.1625 0.228 2.237 4: Bad 1.00 standard error is not available ------Incomplete set of GammasTest for partial association: X² = 7.5 df = 2 p = 0.023 Pairwise comparisons of strata:Comparison of strata 1+2 - p = 0.11Significant difference between 1+2 and 3 P = 0.025Notice the similarity of the analysis of  coefficients and Mantel-Haenszel analysis of odds-ratios33

Measuring Association Among Ordinal Categorical Variables

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