 Open Journal of Statistics, 2012, 2, 198-203 http://dx.doi.org/10.4236/ojs.2012.22023 Published Online April 2012 (http://www.SciRP.org/journal/ojs) Asymmetry Index on Marginal Homogeneity for Square Contingency Tables with Ordered Categories Kouji Tahata, Kanau Kawasaki, Sadao Tomizawa Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Chiba, Japan Email: {kouji_tahata, tomizawa}@is.noda.tus.ac.jp, k.kawasaki123@gmail.com Received December 27, 2011; revised January 25, 2012; accepted February 10, 2012 ABSTRACT For square contingency tables with ordered categories, the present paper considers two kinds of weak marginal homo- geneity and gives measures to represent the degree of departure from weak marginal homogeneity. The proposed meas- ures lie between –1 to 1. When the marginal cumulative logistic model or the extended marginal homogeneity model holds, the proposed measures represent the degree of departure from marginal homogeneity. Using these measures, three kinds of unaided distance vision data are analyzed. Keywords: Marginal Homogeneity; Marginal Cumulative Logistic Model; Measure; Square Contingency Table 1. Introduction Consider an square contingency table with or- dered categories. Let ij denote the probability that an observation will fall in the ith row and jth column of the table (;). Also let RRp1, ,iR1, ,jRX and denote the row and column variables, respectively. The marginal homogeneity (MH) model () is defined by YXYiiFF1, ,1 for iR1ikk where iXFp1iYikk, Fp1kktt1Rkskspp111RXYiii with Rpp, . When the MH model does not hold, we are interested in applying the model that has weaker restriction than the MH model. As such a model, for example, there are the marginal cumulative logistic (ML) model () and the extended marginal homogeneity (EMH) model ([3-5]). We are also interested in considering the other structure of weak MH. The measures to represent the degree of departure from MH are given by, for example, [6,7]. When the structure of weak MH does not hold, we are interested in measuring what degree the departure from weak MH is. The present paper considers two kinds of structures of weak MH and proposes the measures to represent the degree of departure from weak MH. 2. Weak Marginal Homogeneity I and Measure 2.1. Submeasure I Let FF , and *11XiiFF, *21YiiFF, for . 1, ,1iR 1**1211RiiiFFNote that . Assuming that 0XYiiFF , consider the submeasure defined by  11**11214ππ4RiiiiFF , where 1122sinYiiXYiiFFF. Noting that 10π2 1111i, we see that 1) 1 , 2)  0YiF0XF1, ,1R if and only if and i (i11), and 3) 0XF if and only if i and (i0YiF1,,1R1). When the MH model holds,  equals zero. Copyright © 2012 SciRes. OJS K. TAHATA ET AL. 1992.2. Submeasure II Let 1iiSFXX1YYSF 1, ,1XYSS1, , 1, for iR. iiThe MH model may be expressed as ii for iR11RXYiiiSS. Let 21, ,1iR, and for , 21sini22XiXYiiSSS, *12iiSXS, *22YiiSS 1**121RiiSS2. Note that . Assuming that 1i0XYiiSS, we shall define the submeasure  as follows;  124π4Riii**2121πiSS . Noting that 20π2i21 0XS, we see that 1) 2; 2) if and only if i11 and (); and 3) if and only if and (i). When the MH model holds, equals zero. 0YiS1,i0YiSiS, 1R1X1, 0XYFF 0XYiiSS2, 1R022.3. Complete Measure Assume that ii and . Con-sider a measure defined by 12211 1XFXS0YF1YS1. We see that 1) , 2) if and only if i (then ) and (then i10i i) for all , and 3) if and only if 1, ,1iR1 0XiF (then ) and (then ) for all . Thus, indicates that 1R1XiS1, , 1iR1YiF0YiS11p and the other cell probabilities are zero (say, upper-right- marginal inhomogeneity), and indicates that 1R and the other cell probabilities are zero (say, lower-left-marginal inhomogeneity). When , we shall refer to this structure as the weak marginal homo- geneity I (WMH-I). We note that if the MH model holds then the structure of WMH-I holds, but the converse does not hold. 11p0Therefore, using the measure , we can see whether the structure of WMH-I departs toward the upper-right- marginal inhomogeneity or toward the lower-left-margi- nal inhomogeneity. As the measure approaches –1, the departure from WMH-I becomes greater toward the upper-right-marginal inhomogeneity. While as the  approaches 1, it becomes greater toward the lower-left- marginal inhomogeneity. 3. Weak Marginal Homogeneity II and Measure Let 1PriXcikkTXiXYp, 1PriYcikkTYiXYp1, ,1iR, , where for 1ckkkkppp, 1ckkkkppp, ststpXYiiTT1, ,1iR11. The MH model may be expressed by for . We shall consider the submeasure which is de-  replaced XiF and fined by the submeasure YiF by XiT and YiT1XXiiUT 1YYiiUT 1, ,1iRXYiiUU1, ,1iR22, respectively. Let , for . The MH model may be expressed by for . We shall consider the submeasure which is de-  replaced  and XiSfined by the submeasure YiS by XiU and YiU, respectively. 0XYTTii0XYiiUU and Assume that . Consider a measure defined by 121211. We see that . Let cppij ijj (i). In a similar way to , 1 1cp indicates that 1R and the other ij are zero (icpj) (say, conditional upper- right-marginal inhomogeneity), and indicates that 1R11cp and the other ij are zero () (say, condi- tional lower-left-marginal inhomogeneity). When cpij0, we shall refer to this structure as the weak marginal ho- mogeneity II (WMH-II). We note that if the MH model holds then the structure of WMH-II holds, but the con- verse does not hold. Copyright © 2012 SciRes. OJS K. TAHATA ET AL. Copyright © 2012 SciRes. OJS 200  XYLL 1, ,14. Relationships between Measures and Models We shall consider the relationship between the measure (or ) and the ML model. The ML model is given by ii for iR, where log 1XiXiFFXiL, log 1YYiiYiFLF0. A special case of this model obtained by putting is the MH model. The ML model may also be expressed as exp1expiXiFi, exp1expiYiiF1R0XYii, for . Therefore, when the ML model holds, 1, ,i1) if and only if FF0, 2) if and only if XYiiFFYi, and 3) if and only if 0XiFF0 0 00 0 000 0  12iiGG. We obtain the following theorem. Theorem 1. When the ML model holds, 1) if and only if (), 2) if and only if (), 3) (i.e., the MH model holds) if and only if (). Next, we shall consider the relationship between the measure (or ) and the EMH model, defined by 1, ,1iR111iR for , where stistiGp211Ri, stisi tGp 1.  A special case of this model obtained by putting is the MH model. Noting that  12XYiiiiGG FF1, ,1R(i1), we obtain the following theorem. Theorem 2. When the EMH model holds, 0 if and only if 1) 01 (),  if and only if 2) 001 (),  (i.e., the MH model holds) if and only if 3) 00 (). Thus, when the ML (EMH) model holds, the measures  and  are adequate to represent the degree of departure from MH. 5. Approximate Confidence Interval for Measures Let ij denote the observed frequency in the ith row and jth column of the table (;). As- suming that a multinomial distribution applies to the n1, ,iR1, ,jRRR table, we shall consider an approximate standard error and large-sample confidence interval for the meas- ure , using the delta method, as described by . The sample version of , i.e., , is given by ˆ with ˆijp, where ijp replaced by ˆijij nnpnn and ij. Using the delta method, we obtain the fol- lowing theorem. Theorem 3. ˆnn2ˆ has asymptotically (as ) a normal distribution with mean zero and vari-ance , where 22111ˆ4RRij ijijijabp   , with   111221214XYRYXkkijkk kXYkkRijFFaIjkIikF IjkFFF  11πkIik,  122212 2214πXYRYXkkijkk kXYkkkijSSbIik IjkIikSIjkSSS   2 , and I1Iˆp is the indicator function, if true, 0 if not. Also, the sample version of , i.e., , is given by with ij replaced by ˆijp. We obtain the fol-lowing theorem. Theorem 4. ˆnn2ˆ has asymptotically (as ) a normal distribution with mean zero and vari- , where ance 22111ˆ4RRij ijijijjicd p , with 1322131134ππ41 ,XYRXY XYYXkkijkkkkkk kXYkkkRXYkkkTTcIikTIjkT IikTTIjkTTTTIikTI jkT       1 K. TAHATA ET AL. 20114221412144ππ41 ,XYRXY XYkkijkkkk kXYkkkRXYkkkUUdIik UIjk UIik UUIjUUIikUI jkU  YXk kk UUi11XYiiUU,  11RXYiiTT, 4Ri331sinYiiT22XYiiTT, 41sini22XiXYiiUUU. Let 2ˆ denote ˆ 2ˆ with ijp replaced byˆijp. Then ˆn is estimated approximate ˆ, and ˆror for standard er2ˆˆˆpzn  is ap- e 100 1p percent confidence interval for proximatere, wh2pzfromibution coo p. We ales m 7477 women aged 30 to 39 employed in Royal Ordnance factories in Britain from 1943 to 1946. ble 2 that for the data in Table 1(a), the e measure  is –0.0130 and all is the percentage point the stan- dardmal distrrresponding to a two-tail proby equal tlso obtain the similar result for measure . norabilit6. ExampExample 1: Consider the unaided distance vision data in Table 1(a) taken fro . There are data on unaided distance vision ofWe see from Taestimated value of thvalues in confidence interval for  are negative. There- fore, the structure of WMH-I for a woman’s right and left eyes departs toward the upper-right-marginal inhomoge- neity. Also we see from Table 3 that for the data in Ta- ble 1(a), the estimated value of the measure  is –0.0436 and all values in confidene interval for c are negative. Therefore, the structure of WMH-II for a woman’s right and left eyes departs toward the condi- tional upper-right-marginal inhomogeneity. Table 4 gives the values of likelihood ratio chi-squared statistic for testing goodness-of-fit of each of MH,L, and EMH models. We see from Table 4 that each of ML and EMH models fits these data well. Thus the measures  and  would indicate the degree of departure from MH. We can see from these measures that the de Mgree of deinhomogeneity which indicates that the grade of right eye for arbitrary woman is “Best” and the grade of her left eye is “Worst”. Example 2: Consider the unaided vision data in Table 1(b), taken from . We see from Table 2 that for the Br mparture from MH for the vision data in Table 1(a) is estimated to be 1.30 (4.36) percent of the maximum departure toward the (conditional) upper-right-marginal Table 1. Unaided distance vision data of (a) 7477 women in itain fro ; (b) 3242 men in Britain from  and (c) 4746 students in Japan from . (a) Women in Britain Right eye Left eye grade grade (1) (2) (3) (4)Total Best (1) 1520 266 124 66 1976 Second (2) 234 1512 432 78 2256 Third (3) 117 362 1772 2052456 Worst (4) 36 82 179 492789 Total 1907 2222 2507 8417477 (b) Men in Britain Right eye Left eye grade grade (11) ) (4) Total Best (821 35 1053 (2) (3) 112 85 Second (116 494 145 27 T72 151 583 87 W43 106 331 791 919 480 3242 2) 782 hird (3) 893 orst (4) Total 1052 34 514 (c) StudeapaRight eye nts in Jn Left eye grade grade (1) (2(4) Total Be 1291130 4) (3) st (1) 0 22 1483 Second (149 221 114 T64 124 660 W20 25 249 1429 1524500 1063 2)23 507 hird (3) 185 1033 orst (4)1723 Total 16594746Table 2. Estimates of , estimated approximate standard errors for ˆ, and for approximate 95% confidence intervals , applied to Tableˆ 1. Table  S. E. I. C. 1(a) –0.0130 0037 (–0.0 –0.00.203,056) 1(b) 0.0055 0.0064 (–0.0071, 0.010040 (0.00.02081) 1(c) 0.0125 0.048, 3) Copyright © 2012 SciRes. OJS K. TAHATA ET AL. 202 Table 3. Estimpates of timateprox standard erro ˆ for lie, esd apimaters fo, apr , and approximate 95%d to Table confidence intervals 1. Table ˆ S. E. C. I. 1(a) –0.0436 0.0126 (–0.0683, –0.0190) 1(0.0201 (–0.0222, 0.0566) 0.07 3 (0.0198, .0836) b) 0.0172 1(c) 510.0160Table 4. Values of likelihood i-sq the ML, an modlied tratio chels appuared statistic foro Table 1. H, Md EMHApplied Degrees of Table models freedom 1(a) 1(b) 1(c) H 3 M11.99* 3.68 11.18* ML 2 0.39 3.16 .41 E0 21MH 2 .005 .94 0.56 *Meanant at tvel. data in Table 1(b) the estimated value of measure s significhe 0.05 le is H-I in the data in Table 1(b)se Table 3 that for th Table 1(b), estima of measure 72 aor  includes zero. So this mindicatthere structure of WMH-II in the data in Tabl ). Ee 3: C nsider tta inle 1(en om [3,10]. We see from that for the data in value of measure 0.0055 and the confidence interval for  includes zero. So this may indicate that there is a structure of WM. Also we thenfidee fromted valueterval foe data in is 0.01nd the cnce inay e that is ae 1(bxampl ohe daTable 2 Tab c) takfrTable 1(c), the estimated the is 0.0125 and all values in confidence interval for  fare positive. Therefore, the structure of WMH-I for a stu- dent’s right and left eyes departs toward the lower-left- marginal inhomogeneity. Also we see from Table 3 that for the data in Table 1(c), the estimated value of the measure  is 0.0517 and all values in conidence in- terval for  are positive. Therefore, the structure of WMH-II for a student’s right and left eyes departs to- ward the conditional lower-left-marginal inhomogeneity. We see from Table 4 that each of ML and EMH mod-els fits these data well. Thus the measures  and  would indicate the degree of departure from MH. We can see from these measures that the degree of departure from MH for the vision data in Table 1 (c) is estimated to be 1.25 (5.17) percent of the maximum departure toward the (conditional) lower-left-marginal inhomogeneity which indicates tat the grade of right eye for arbitrary student is “Worst” ad the grade of his/her left eye is “Best”. 7. Concluding Remarks hnFot Hr the analysis of square contingency tables with or- dered categories, when the ML model, or he EM model, or other asymmetry models, for example, ’s conditional symmetry model (defined by ijjipp for ij) holds, the proposed measures  and  are adequate to represent the degree of departure from the MH model toward two maximum departures, i.e., toward the (conditional) lower-left-marginal inhomogeneity or toward the (conditional) upper-right-marginal inhomoge- neity. ity (i.e., ther-lefhe-r8. Discussion [6,7] considered the measures to represent the degree of departure from MH. The present paper has considered two types of maximum marginal inhomogene lowet-marginal inhomogeneity and t upperight- marginal inhomogeneity). The measures in [6,7] take the value 1 in two types of maximum marginal inhomoge- neity. The measures  and  in the present paper can distinguish these two kinds of maximum marginal inhomogeneity by the values –1 or 1 although the meas- ures in [6,7] cannot distinguish them. Also the proposed esent the degree of departure from MH tossmeasures can reprwhen the ML or the EMH models, or the other asym- metry models hold. Therefore for the ordinal data, the proposed measures rather than those in [6,7] may be useful to represent the degree of departure from MH. 9. Acknowledgements The authors would like expre their sincere thanks to the editor and a referee for their helpful comments. REFERENCES  A. 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