Biostatistics 102: Quantitative Data Parametric & Nonparametric Tests


 Everett Sutton
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1 Singpore Med J 2003 Vol 44(8) : B s i c S t t i s t i c s F o r D o c t o r s Biosttistics 102: Quntittive Dt Prmetric & Nonprmetric Tests Y H Chn In this rticle, we re going to discuss on the sttisticl tests vilble to nlyse continuous outcome vribles. The prmetric tests will be pplied when normlity (nd homogeneity of vrince) ssumptions re stisfied otherwise the equivlent nonprmetric test will be used (see tble I). TbleI. Prmetric vs NonPrmetric tests. Prmetric NonPrmetric Figure 1. Histogrms of Systolic & Distolic blood pressures. Histogrm Frequency Smple Ttest Sign Test/Wilcoxon Signed Rnk test Pired Ttest Sign Test/Wilcoxon Signed Rnk test 2 Smple Ttest Mnn Whitney U test/wilcoxon Sum Rnk test ANOVA Kruskl Wllis test We shll look t vrious exmples to understnd when ech test is being used Systolic blood pressure SAMPLE TTEST The 1Smple T test procedure determines whether the men of single vrible differs from specified constnt. For exmple, we re interested to find out whether subjects with cute chest pin hve bnorml (norml = 120 mmhg) nd/or distolic (norml = 80 mmhg) blood pressures. 500 subjects presenting themselves to n emergency physicin were enrolled. Assumption for 1 smple T test: Dt re normlly distributed. We hve discussed in the lst rticle (1) on how to check the normlity ssumption of quntittive dt. One issue being highlighted ws tht these forml normlity tests re very sensitive to the smple size of the vrible concerned. As seen here, tble II shows tht the normlity ssumptions for both the nd distolic blood pressures re violted but bsing on their histogrms (see figure 1), normlity ssumptions re fesible. Tble II. Forml normlity tests. Tests of Normlity KolmogorovSmirnov ShpiroWilk Sttistic df Sig. Sttistic df Sig. blood pressure distolic blood pressure Lilliefors Significnce Correction. Frequency Distolic blood pressure So with the normlity ssumptions stisfied, we could use the 1 Smple Ttest to check whether the nd distolic blood pressures for these subjects re sttisticlly different from the norms of 120 mmhg nd 80 mmhg respectively. Firstly, simple descriptive would give us some ide, see tble III. Tble III. Descriptive sttistics for the & distolic BP. Men Std Minimum Mximum Medin Devition blood pressure distolic blood pressure 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
2 392 : 2003 Vol 44(8) Singpore Med J To perform 1 Smple Ttest, in SPSS, use Anlyze, Compre Mens, OneSmple T test. For, put test vlue = 120 nd for distolic put test vlue = 80 (we hve to do ech test seprtely). Tbles IV & V shows the SPSS output. Tble IV. 1 Smple Ttest for testing t 120 mmhg. OneSmple Test Test Vlue = 120 Intervl of the t df Sig. Men Lower Upper (2tiled) blood pressure Tble V. 1 Smple Ttest for distolic BP testing t 80 mmhg. OneSmple Test Test Vlue = 80 Intervl of the t df Sig. Men Lower Upper (2tiled) distolic blood pressure These subjects hd much higher (p<0.001, difference = 20.5, 95% CI 18.4 to 22.6) compred to the norm of 120 mmhg. This difference is cliniclly relevnt too. For the distolic BP, though there ws sttisticl significnce of 1.35 (95% CI 0.22 to 2.5, p = 0.019) lower thn the norm of 80 mmhg, this difference my not be of clinicl significnce. By now, we should relize tht the pvlue is significntly ffected by smple size (2), thus we should be looking t the clinicl significnce first then the sttisticl significnce. If the normlity ssumptions were not stisfied, then the equivlent nonprmetric Sign test or Wilcoxon Signed Rnk test would be used. In SPSS, before we could perform the nonprmetric nlysis, we will hve to crete new vrible in the dtset, sy, sysnorm (which is just column of 120). Use the Trnsform, Compute commnd to do this (likewise, we hve to crete new vrible, sy, dinorm which is just column of 80). Then go to Anlyze, Non Prmetric tests, 2 relted smples to do the tests (we cn do both tests for nd distolic simultneously, Tbles VI & VII show the SPSS outputs). In this cse, we re nlyzing the medins of the vribles rther thn their mens. Tble VI. Wilcoxon Signed Rnk tests. Test Sttistics c SYSTOLIC  DIASTOLI  blood distolic blood pressure pressure z Asymp. Sig. (2tiled) c Wilcoxon Signed Rnks test. Tble VII. Sign test. Test Sttistics SYSTOLIC  DIASTOLI  blood distolic blood pressure pressure z Asymp. Sig. (2tiled) Sign Test. In the Sign test, the mgnitude of the differences between the vrible nd the norm is not tken into considertion when deriving the significnce. It uses the number of positives nd negtives of the differences. Thus if there were nerly equl numbers of positives nd negtives, then no sttisticl significnce will be found regrdless of the mgnitude of the positives/negtives. The Wilcoxon Signed Rnk test, on the other hnd, uses the mgnitude of the positives/negtives s rnks in the clcultion of the significnce, thus more sensitive test. PAIRED TTEST When the interest is in the before nd fter responses of n outcome (within group comprison), sy, the before nd fter n intervention, the pired Ttest would be pplied. Tble VIII shows the descriptive sttistics for the before nd fter intervention s of 167 subjects. Tble VIII. Descriptive sttistics for the before & fter intervention. Std Men Devition Minimum Mximum Medin BP before BP fter Assumption for the Pired T test: The difference between the before & fter is normlly distributed We will hve to compute new vrible for the difference between the before & fter nd then check it s normlity ssumption. Tble IX shows the forml tests for the checking of the normlity
3 Singpore Med J 2003 Vol 44(8) : 393 ssumption nd figure 2 shows the corresponding histogrm. Tble IX. Normlity ssumption checks. KolmogorovSmirnov ShpiroWilk Sttistic df Sig. Sttistic df Sig. Systolic BP before Figure 2. Histogrm of the difference between the Before & After intervention. 30 Tble X. Pired Ttest for the Before & After intervention. Pir 1 before fter Pired s Intervl of the Std. Std. Error Sig. (2 Men Devition Men Lower Upper t df tiled) Tble XI. 1 Smple T test for the difference between the Before & After intervention. Frequency before  fter OneSmple Test Test Vlue = 0 Intervl of the t df Sig. Men Lower Upper (2tiled) before  fter Since the normlity ssumption is stisfied, we cn use the pired Ttest to perform the nlysis: In SPSS, use Anlyze, Compre Mens, Pired Smples T test. Tble X shows the SPSS output for the pired Ttest. Cliniclly there ws men reduction of 5.17 mmhg but this ws not sttisticlly significnt (p = 0.052). Should we then increse the smple size to chse fter the pvlue? We shll discuss this issue t the end of this rticle. Alterntively, we cn use the 1Smpe T test (with test vlue = 0) on the difference between the Before & After to check whether there ws sttisticl significnce; see Tble XI. In the event tht the normlity ssumption ws not stisfied, we will use the Wicoxon Signed Rnk test to perform the comprison on the medins. In SPSS, use Anlyze, Non Prmetric tests, 2 relted smples: tble XII shows the SPSS output. 2 SAMPLE TTEST When our interest is the BetweenGroup comprison, the 2 Smple T test would be pplied. For exmple, we wnt to compre the between the norml weight nd the overweight ( proper power nlysis should be done before embrking on the study (2) ). 250 subjects for ech group were recruited. Tble XIII gives the descriptive sttistics. Assumptions of the 2 Smple T test: 1. Observtions re normlly distributed in ech popultion. Tble XII. Wilcoxon Signed Rnk test on the difference on the Before & After intervention. Test Sttistics b fter  before z Asymp. Sig. (2tiled).071 Bsed on positive rnks b Wilcoxon Signed Rnks Test Tble XIII. Descriptive sttistics of Systolic BP by group. Men Std Minimum Mximum Medin Devition overweight normlweight Homogeneity of vrince (The popultion vrinces re equl). 3. The 2 groups re independent rndom smples. The 3 rd ssumption is esily checked from the design of the experiment ech subject cn only be in one of the groups or intervention. The 1 st ssumption of normlity is lso esily checked by using the Explore option in SPSS (with group declred in the Fctor list this will produce normlity checks for ech group seprtely). Normlity ssumptions must be stisfied for both groups for the 2 Smple T test to be pplied. Lstly, the 2 nd ssumption of homogeneity of vrince will be given in the 2 Smple T test nlysis.
4 394 : 2003 Vol 44(8) Singpore Med J Tble XIV. 2 Smple T test. Independent Smples Test Levene s Test for Equlity of Vrinces ttest for Equlity of Mens F Sig. t df Sig. Men Std. Error Intervl of the (2tiled) Lower Upper Equl vrinces ssumed Equl vrinces not ssumed To perform 2 Smple T test, in SPSS, use Anlyze, Compre Mens, Independent Smples Ttest. Tble XIV shows the SPSS output. The Levene s Test for equlity of vrinces checks the 2 nd ssumption. The Null hypothesis is: Equl Vrinces ssumed. The Sig vlue (given in the 3 rd column) shows tht the Null hypothesis of equl vrinces ws rejected nd SPSS djusts the results for us. In this cse we hve to red off the pvlue (Sig 2tiled) from the 2 nd line (equl vrinces not ssumed) rther thn from the 1 st line (equl vrinces ssumed). As expected, there ws significnt difference in the between the overweight nd norml (p<0.001, difference = 44.53, 95% CI to mmhg) When normlity ssumptions re not stisfied for ny one or both of the groups, the equivlent nonprmetric Mnn Whitney U/Wilcoxon Rnked Sum tests should be pplied. In SPSS, use Anlyze, Non Prmetric tests, 2 Independent Smples. Tble XV shows the results for the nonprmetric test. Tble XV. Mnn Whitney U & Wilcoxon Rnk Sum tests. Test Sttistics BPSYS MnnWhitney U Wilcoxon W z Asymp. Sig. (2tiled).000 Grouping Vrible: TRT. Observe tht only 1 pvlue will be given for both Mnn Whitney U nd Wilcoxon Rnk Sum tests. Tble XVI. Descriptive sttistics of Systolic BP by weight groups. Men Std Minimum Mximum Medin Devition overweight underweight normlweight After checking for the normlity ssumptions, to perform n ANOVA, in SPSS, use Anlyze, Compre Mens, OneWy ANOVA. Click on Options nd tick the Homogeneity of Vrince test. Tbles XVII & XVIII shows the results for the homogeneity of vrince nd ANOVA tests respectively. Tble XVII. Homogeneity of Vrince test. Test of Homogeneity of vrinces Levene sttistic df1 df2 Sig The Null hypothesis is: Equl Vrinces ssumed. Since p = 0.09>0.05, we cnnot reject the null hypothesis of equl vrince. Tble XVIII. ANOVA results. ANOVA Sum of df Men F Sig. Squres Squre Between Groups Within Groups Totl ANOVA (ANALYSIS OF ONE WAY VARIANCE) The ANOVA is just n extension of the 2Smple T test when there re more thn 2 groups to be compred. The 3 ssumptions for the 2Smple T test lso pply for the ANOVA. Let s sy, this time we hve 3 weight groups (norml, under nd over weight), the descriptive sttistics is given in Tble XVI. The Null Hypothesis: All the groups mens re equl. Since p<0.001, not ll the groups mens re equl. We would wnt to crry out posthoc test to determine where the differences were. In SPSS, under the ANOVA, click on the Post Hoc button nd tick Bonferroni (3) (this method is most commonly used nd rther conservtive in testing for multiple
5 Singpore Med J 2003 Vol 44(8) : 395 Tble XIX. ANOVA Bonferroni djustment for multiple comprisons. Dependent vrible: Bonferroni Multiple comprisons Intervl (1) GROUP (J) GROUP Men (IJ) Std. Error Sig. Lower Bound Upper Bound overweight overweight underweight * normlweight * underweight overweight * underweight normlweight normlweight overweight * underweight normlweight * The men difference is significnt t the.05 level. comprisons). Tble XIX shows the posthoc multiple comprisons using Bonferroni djustments. The of the overweights were sttisticlly (nd cliniclly) higher thn the other 2 weight groups but there ws no sttisticl difference between the norml nd under weights (p = 0.099). If we hve crried out multiple 2 Smple T tests on our own, we hve to djust the type 1 error mnully. By Bonferroni, we hve to multiply the pvlue obtined by the number of comprisons performed. For 3 groups, there will be 3 comprisons (ie. A vs B, B vs C & A vs C). Tble XX shows the 2 Smple Ttest between the norml nd under weights. It seems tht there s lso sttisticl difference between the 2 groups in but tking into ccount multiple comprison nd djusting for type 1 error, we will hve to multiply the pvlue (= 0.033) by 3 which gives the sme result s in ANOVA posthoc. Tble XX. 2 Smple T test for Norml vs Under weight. Independent Smples Test Levene s Test for Equlity of Vrinces ttest for Equlity of Mens Sig. (2 F Sig. t df tiled) Equl vrinces ssumed Equl vrinces not ssumed When normlity nd homogeneity of vrince ssumptions re not stisfied, the equivlent nonprmetric Kruskl Wllis test will be pplied. In SPSS, use Anlyze, Non Prmetric tests, k Independent Smples. Tble XXI shows the SPSS results. Tble XXI. Kruskl Wllis test on for the 3 groups. Test Sttistics,b ChiSqure df 2 Asymp. Sig..000 Kruskl Wllis Test b Grouping Vrible: GROUP There ws sttisticl significnt difference mongst the groups. In Kruskl Wllis, there s no posthoc option vilble, we will hve to do djust for the type 1 error mnully for multiple comprisons. TYPE 1 ERROR ADJUSTMENTS A type 1 error is committed when we reject the Null Hypothesis of no difference is true. If we tke the conventionl level of sttisticl significnce t 5%, it mens tht there is 0.05 (5%) probbility tht result s extreme s the criticl vlue could occur just by chnce, i.e. the probbility of flse positive is There re few scenrios when djustments for type 1 error is required: Multiple comprisons When we re compring between 2 tretments A & B with 5% significnce level, the chnce of true negtive in this test is But when we perform A vs B nd A vs C (in three tretment study), then the probbility tht neither test will give significnt result when there is no rel difference is 0.95 x 0.95 = 0.90; which mens the type 1 error hs incresed to 10%.
6 396 : 2003 Vol 44(8) Singpore Med J Tble XXII. Tble XXII shows the probbility of getting flse positive when repeted comprisons t 5% level of significnce re performed. Thus for 3 pirwise comprisons for 3tretment groups study (generlly, number of pirwise comprisons for ngroup study is given by n(n1)/2), without performing type 1 error djustment, the probbility of flse positive is 14%. Number of comprisons Probbility of flse positive 5% 10% 14% 19% 23% 27% 30% 34% 37% 40% As mentioned in ANOVA, Bonferroni djustments (multiplying the pvlue obtined in ech multiple testing by the number of comprisons) would be the most convenient nd conservtive test. But this test hs low power (the bility to detect n existing significnt difference) when the number of comprisons is lrge. For exmple with 4 tretment groups, we will hve 6 comprisons which mens tht for every pirwise pvlue obtined, we hve to multiply by 6. In such sitution, other multiple comprison techniques like Tukey or Scheffe would be pproprite. Miller (1981) (4) gve comprehensive review of the pros nd cons of the vrious methods vilble for multiple comprisons. In ANOVA, this multiple comprison is utomticlly hndled by the posthoc option but for Kruskl Wllis test, mnul djustments needed to be crried out by the user which mens tht the Bonferroni method would normlly be used becuse of it s simplicity. Chsing fter the pvlue In the exmple of the Pired Ttest, the before & fter tretment nlysis gve pvlue of with n=167. Perhps this pvlue will be significnt if we increse the smple size. If the smple size ws indeed incresed, then the obtined pvlue will hve to be multiplied by 2! The reson being tht we re lredy bised by the positive trend of the findings nd the type 1 error needed to be controlled. Interim nlysis Normlly in lrge smple size clinicl trils, interim nlyses re crried out t certin time points to ssess the efficcy of the ctive tretment over the control. This is crried out usully on the ethicl bsis tht perhps the ctive tretment is relly superior by lrger effect difference thn expected (thus smller smple size would be sufficient to detect sttisticl significnce) nd we do not wnt to put further subjects on the control rm. These plnned interim nlyses with documenttions of how the type 1 error djustments for multiple comprisons must be specificlly writeup in the protocol. CONCLUSIONS The concentrtion of the bove discussions hve been on the ppliction of the relevnt tests for different types of designs. The theoreticl spects of the vrious sttisticl techniques could be esily referenced from ny sttisticl book. The next rticle (Biosttistics 103: Qulittive Dt Test of Independence), we will discuss on the techniques vilble to nlyse ctegoricl vribles. REFERENCES 1. Chn YH. Biosttistics 101: Dt presenttion, Singpore Medicl Journl 2003; Vol 44(6): Chn YH. Rndomised Controlled Trils (RCTs) smple size: the mgic number? Singpore Medicl Journl 2003; Vol 44(4): Blnd, JM & DG. Altmn. Multiple significnce tests: the Bonferroni method, 1995 British Medicl Journl 310: Miller RG Jr. Simultneous Sttisticl Inference, 2nd ed, 1981, New York, SpringerVerlg.
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