Biostatistics 103: Qualitative Data Tests of Independence


 Edgar Hancock
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1 Singpore Med J 2003 Vol 44(10) : B s i c S t t i s t i c s F o r D o c t o r s Biosttistics 103: Qulittive Dt Tests of Independence Y H Chn Prmetric & nonprmetric tests (1) re used when the outcome response is quntittive nd our interest is to determine whether there re ny sttisticl differences between/mongst groups (which re ctegoricl). In this rticle, we re going to discuss how to nlyse reltionships between ctegoricl vribles. Tble I shows the first five cses of 200 subjects with their gender nd intensity of snoring (No, At Times, Frequent nd Alwys) nd snoring sttus (Yes or No) recorded. Templte I. Crosstbs. Tble I. Dt structure in SPSS. Subject Gender Snoring Intensity Snoring Sttus 1 Mle No No 2 Mle Alwys Yes 3 Femle Frequent Yes 4 Mle At times Yes It does not mtter whether we put Snoring intensity or Gender into the Row(s) or Columns but for esier interprettion of the results (lter) it is recommended to put the the ctegoricl vrible of outcome interest (in this cse, the Snoring intensity) in the Columns option. Click on the Cells button nd tick the Row Percentges (the Observed Counts is ticked by defult), then Continue. Templte II. Crosstbs: Cell Disply. 5 Femle No No Here, we hve two interests. One is to determine whether there s n ssocition between gender nd snoring intensity nd the other is the ssocition between gender nd snoring sttus. The interprettion of the results for both nlyses is not similr. Let s discuss the 1 st interest. The null hypothesis is: There is No Assocition between gender nd snoring intensity. To test this hypothesis of no ssocition (or independence), the ChiSqure test is performed. With the given dt structure in Tble I, to perform the ChiSqure test in SPSS, use Anlyse, Descriptive Sttistics, Crosstbs nd the following templte is obtined: The crosstbultion tble is shown in Tble II. This tble is 2 X 4 (red s 2 by 4); 2 levels for Gender nd 4 levels for Snoring intensity. Clinicl Trils nd Epidemiology Reserch Unit 226 Outrm Rod Blk A #0202 Singpore Y H Chn, PhD Hed of Biosttistics Correspondence to: Y H Chn Tel: (65) Fx: (65) Emil: cteru.com.sg
2 499 : 2003 Vol 44(10) Singpore Med J Tble II. Crosstbultion tble of Gender nd Snoring intensity. Snoring intensity ALWAYS AT TIMES FREQUENT NO Totl Gender Femle Count % within Gender 8.7% 29.8% 5.8% 55.8% 100.0% Mle Count % within Gender 24.0% 27.1% 6.3% 42.7% 100.0% Totl Count % within Gender 16.0% 28.5% 6.0% 49.5% 100.0% To sk for the ChiSqure test, click on the Sttistics button t the bottom of Templte I nd the Crosstbs:Sttistics templte is shown tick the Chisqure box. Templte III. Crosstbs: Sttistics. lies in the mles being more likely to hve Alwys snoring intensity compred to the femles (24% vs 8.7%). Sometimes it s not so strightforwrd to interpret n ssocition! For the 2 nd interest, the null hypothesis is: There is No Assocition between gender nd snoring sttus. The (2 x 2) crosstbultion tble nd the ChiSqure test results re shown in tbles IV nd V respectively. Tble IV. (2 x 2) crosstbultion tble of Gender nd Snoring sttus. Gender* Snoring sttus Crosstbultion Snoring sttus NO YES Totl Gender Femle Count % within Gender 55.8% 44.2% 100.0% Mle Count % within Gender 42.7% 57.3% 100.0% Tble III gives the result for the ChiSqure test. Tble III. ChiSqure test result for the (2 X 4) Gender nd Snoring intensity. ChiSqure Tests Vlue df Asymp. Sig. (2sided) Person ChiSqure Continuity Correction Likelihood Rtio LinerbyLiner Assocition cells (.0%) hve expected count less thn 5. The minimum expected count is Here the Person ChiSqure vlue is 9.17 with df (degree of freedom) = 3 nd the pvlue is (<0.05) the rest of the sttistics in the tble is of no interest to us! Hence we reject the null hypothesis of no ssocition. The ChiSqure test only tells us whether there is ny ssocition between two ctegoricl vribles but does not indicte wht the ssocition is. From Tble II, by inspection, it is obvious tht the difference Totl Count % within Gender 49.5% 50.5% 100.0% Tble V. Result for ChiSqure test for the (2 X 2) Gender nd Snoring sttus. ChiSqure Tests Vlue df Asymp. Exct Exct Sig. Sig. Sig. (2sided) (2sided) (1sided) Person ChiSqure b Continuity Correction Likelihood Rtio Fisher s Exct Test LinerbyLiner Assocition b Computed only for 2x2 tble. 0 cells (.0%) hve expected count less thn five. The minimum expected count is This hs be 0 for Person s ChiSqure to be vlid
3 Singpore Med J 2003 Vol 44(10) : 500 Here the Person ChiSqure pvlue is (>0.05) which mens tht there ws no ssocition between gender nd snoring sttus. A different conclusion from the bove results on the ssocition between Gender nd Snoring intensity! You my hve observed tht the ChiSqure Tests Tbles of III nd V re different. The reson is tht for (2 x 2) ssocition, SPSS utomticlly gives us the result for the Fisher s Exct Test wheres for non (2 x 2), we hve to sk for it (but we hve to purchse this Exct test module). Why do we need this Fisher s Exct test? The vlidity of the Person s ChiSqure test is violted when there re smll frequencies in the cells. The forml definitions of these ssumptions (not reproduced here) for the vlidity cn be found in ny sttisticl textbook. In SPSS, this vlidity is esily checked by observing the lst line of the ChiSqure Tests Tble (for exmple in Tble V), we wnt 0 cells (.0%) hve expected count less thn five, otherwise we will hve to use the Fisher s Exct test. Tble VI nd VII shows sitution where we should be cutious: Tble VI. 2 x 2 crosstbultion of Gender nd Snoring sttus (n = 56) Gender* Snoring sttus Crosstbultion Snoring sttus NO YES Totl Gender Femle Count % within Gender 95.7% 4.3% 100.0% Mle Count % within Gender 75.8% 24.2% Totl Count % within Gender 83.9% 16.1% 100.0% Tble VII. ChiSqure test for tble VI. ChiSqure Tests Person ChiSqure b Vlue df Asymp. Exct Exct Sig. Sig. Sig. (2sided) (2sided) (1sided) Continuity Correction Likelihood Rtio Fisher s Exct Test LinerbyLiner Assocition N of Vlid Cses 56 b Computed only for 2 x 2 tble. 1 cell (25.0%) hve expected count less thn five. The minimum expected count is From the lst line of tble VII, we observe tht the vlidity of the Person s ChiSqure test is violted (1 cell hs expected count less thn five), thus in this cse the pvlue of for the Fisher s Exct test should be reported (nd not the significnt p = of the Person ChiSqure), signifying no ssocition. For non 2 x 2 tble, we cn sk for Fisher s Exct test by clicking the Exct button (t the left corner of Templte I) nd the following templte is obtined: Templte IV. Exxt Tests. Tick the Exct option. The computtion for this Fisher s Exct test is quite extensive nd sometimes for 4 x 6 tble, sy, most likely the Person s Chi Squre will not be vlid s there s high probbility for some of the cells to hve smll frequencies. After couple of minutes computtion, the only nswer we get from the Fisher s Exct test is Computer memory not enough! Wht should we do? If the pvlue of the violted Person s ChiSqure test is lrge or very smll, we hve no worries s the pvlue of the Fisher s Exct would not be so different. The only time we hve to worry is when this violted Person s pvlue is hovering round 0.04 to 0.06 (nd the Fisher s Exct test did not help), then it is recommended to seek for the help of biosttisticin! There re instnces where we do not hve the rw dt (s given in Tble I) vilble but only the crosstbultion Tble II (perhps ppering in publiction) nd we re interested to perform the ChiSqure test. In this cse, we hve to set up the dtset s shown in Tble VIII (refer to Tble II for the corresponding frequencies). Tble VIII. SPSS dt structure for crosstbultion tble. Gender Snoring Count Mle No 41 Mle At times 26 Mle Frequent 6 Mle Alwys 23 Femle No 58 Femle At times 31 Femle Frequent 6 Femle Alwys 9
4 501 : 2003 Vol 44(10) Singpore Med J Before we crry out the sequence of steps s discussed bove for performing the Chisqure test, we hve to inform SPSS tht this time ech row is not subject but the totl number of cses re being weighted by the Count vrible. In SPSS, go to Dt, Weight Cses nd the following templte ppers: Templte V. Weight Cses. Tble X. Crosstbultion tble for Exposure nd exmple. Exposure* Crosstbultion Yes=1 No=2 Totl Exposure yes=1 Count % within Gender 30.0% 70.0% 100.0% no=2 Count % within Gender 10.0% 90.0% 100.0% Totl Count % within Exposure 20.0% 80.0% 100.0% p<0.001 (Person ChiSqure) Tble XI. Risk estimtes for Exposure nd exmple. Click on the Weight cses by nd bring the Count vrible into the Frequency Vrible box; then perform the sequence of steps for ChiSqure test s described bove. Mesuring the Strength of n Assocition (only for 2 x 2 tbles). The mgnitude of the pvlue does not indicte the strength of ssocition between two ctegoricl vribles s we know tht this vlue is dependent on the smple size. To express the strength of significnt ssocition (only for 2 x 2 tbles), the odds rtio or the reltive risk between the outcomes of the two groups re presented. Tble IX shows the crosstbultion for Exposure nd. Tble IX. 2 x 2 crosstbultion for Exposure nd. YES No Exposed Yes b No c d By definition, the Odds Rtio is given by OR = (d)/(bc): the rtio of the odds hving disese given exposed nd of hving disese given not exposed nd the Reltive Risk (RR) = (c+d)/ c(+b): the rtio of the probbilities of hving disese given exposed nd hving disese given not exposed. How to obtin the odds rtio nd reltive risk from SPSS? From templte III, besides ticking on the Chisqure option, tick the Risk option too. Tbles X XI show the 2 x 2 crosstbultion nd the Risk estimtes for exposure/disese exmple: Odds Rtio for Exposure (yes/no) = yes = no There s significnt ssocition between Exposure nd (p<0.001). Looking t Tble XI, the Odds Rtio for n Yes/No Exposure of hving (the 1 st column of Tble X) is (95% CI to 8.422) which is lso the OR for the No/Yes Exposure for hving No. The Reltive Risk is obtined from the cohort = yes or no. = yes, the Reltive Risk between Exposure nd nonexposure is 3.0 nd is for the cohort = no. This interprettion of the results is rther strightforwrd becuse of the wy we set up the crosstbultion tble. Observe tht the codings for yes = 1 nd no = 2, nd SPSS will disply the yes first nd then the no. Wht if we hve coded yes = 1 nd no = 0 for? Tble XII Exposure* Crosstbultion No=0 Yes=1 Totl Exposure yes=1 Count % within Gender 70.0% 30.0% 100.0% no=2 Count % within Gender 90.0% 10.0% 100.0% Totl Count % within Exposure 80.0% 20.0% 100.0%
5 Singpore Med J 2003 Vol 44(10) : 502 Tble XIII risk estimte tble XVI shows tht the Chinese compred to the nonchinese were less likely to snore (OR = 0.476). Odds Rtio for Exposure (yes/no) = no = = yes = There will be no chnge in the pvlue of the ssocition but from Tble XIII, the OR presented now is for Yes/No Exposure of hving No (the 1 st column of Tble XII) is (which is just the reciprocl of 3.857!). For non 2 x 2 tble, if significnt ssocition exists, we my wnt to find out where the differences re. Let s consider the exmple of Snoring sttus nd Rce. Tble XIV. Crosstbultion of Rce nd Snoring sttus. RACE* Snoring sttus Crosstbultion Snoring sttus Yes No Totl Rce Chinese Count % within RACE 42.3% 57.7% 100.0% Indin Count % within RACE 75.0% 25.0% 100.0% Mly Count % within RACE 61.4% 38.6% 100.0% Others Count % within RACE 28.6% 71.4% 100.0% Totl Count p = (Fisher s Exct test). % within RACE 50.5% 49.5% 100.0% There s n ssocition between Rce nd Snoring sttus (p=0.013) nd from Tble XIV, it s not obvious where this ssocition is. Since Rce is nominl ctegoricl vrible, we cn crete four dummy vribles: Chinese vs nonchinese, Mly vs non Mlys, etc. Tht is the new vrible Chinese hs only two levels: Chinese or nonchinese nd then we perform the ChiSqure test using these four dummy vribles with Snoring sttus. Tble XV shows the crosstbultion for the Chinese nd Snoring Sttus nd the pvlue for this ssocition is which is sttisticlly significnt even fter we djusted for the type 1 error for multiple comprison (1) (p<0.05/4 = 0.125). The Tble XV. Crosstbultion of Chinese vs NonChinese with Snoring sttus. Crosstb Snoring Sttus Yes No Totl Chinese Chinese Count % within Chinese 42.3% 57.7% 100.0% Other Count % within Chinese 60.7% 39.3% 100.0% Totl Count % within Chinese 50.5% 49.5% 100.0% p = (ChiSqure test) Tble XVI. Risk estimte for Chinese vs nonchinese nd Snoring sttus. Odds Rtio for Exposure (Chinese/Other) Snoring sttus = yes Snoring sttus = no Tbles XVII nd XVIII indicte tht the Mlys compred to the nonmlys hd higher likelihood to snore but we hve to be cutious bout this conclusion fter we hve tken into considertion the djustment of the type 1error for multiple comprison! Tble XVII. Crosstbultion of Mly vs nonmly nd Snoring sttus. Crosstb Snoring Sttus Yes No Totl Mly Mly Count % within Mly 61.4% 38.6% 100.0% Other Count % within Mly 44.6% 55.4% 100.0% Totl Count % within Mly 50.5% 49.5% 100.0% p = (ChiSqure test)
6 503 : 2003 Vol 44(10) Singpore Med J Tble XVIII. Risk estimte tble for Mly vs nonmly nd Snoring sttus. Tble XXI. McNemr test. Vlue df Asymp. Sig. Exct Sig. (2sided) (2sided) McNemr Test N of Vlid Cses Odds Rtio for Exposure (Mly/Other) Snoring sttus = yes Snoring sttus = no There were no significnt ssocition for the Indins (p = 0.080) nd the Other rce (p = 0.277) with Snoring sttus. MCNEMAR TEST The McNemr test is used when we hve mtched csecontrol study. For exmple, we re interested to determine whether there s ny ssocition between dibetes nd AMI. One study design is to mtchbyge, sy, 50yerold dibetic with nother 50yerold non dibetic nd follow them up for length of time. Four possible outcomes could be obtined. See Tble XIX (which is lso the SPSS dt structure for McNemr test) Tble XIX. Possible outcomes of the mtched csecontrol study. Dibetic Person Non Dibetic Person Count Hd AMI Hd AMI 9 Hd AMI No AMI 37 No AMI Hd AMI 16 No AMI No AMI 82 Binomil distribution used. In totl, we hve 144 pirs of prticipnts. There is significnt ssocition between dibetes nd AMI (p=0.005). The McNemr test compres the observtions of the discordnt pirs (Dibetic hving AMI nd NonDibetic not hving AMI) vs (Dibetic not hving AMI nd NonDibetic hving AMI) which is 37/144 (25.7%) vs 16/144 (11.1%). CONCLUSIONS We hve covered the nlysis of both quntittive (1) nd qulittive type of dt (in this rticle) nd tble XXII summrises the vrious techniques vilble. Tble XXII. Summry of Univrite Sttisticl techniques. Quntittive dt Qulittive dt Prmetric test NonPrmetric Independent Mtched test smples smples 1 Smple Ttest Wilcoxon Signed ChiSqure McNemr Pired Ttest Rnk test test/fisher s test 2 Smple Ttest Mnn Whitney Exct test U test / Wilcoxon Rnk Sum test ANOVA Kruskl Wllis test The next rticle will be Biosttistics 104: Correltionl nlysis. REFERENCES 1. Chn YH. Biosttistics 102: Quntittive Dt: Prmetric & Nonprmetric tests. Singpore Medicl Journl 2003; Vol 44(8): To crry out McNemr test in SPSS is exctly the sme s performing ChiSqure test, except tht t Templte III, we tick the McNemr option. Tbles XX nd XXI show the crosstbultion nd McNemr test respectively. Tble XX. Dibetic* NonDibetic Crosstbultion AMI = No AMI = Yes Totl Dibetic AMI = No AMI = Yes Totl
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