When testing the difference between two groups, if previous data indicate non-normality, then either transform the data if they comprise percentages, integers or scores or use a non-parametric test. If there is uncertainty whether the data are normally distributed, then deviations from normality are likely to be small if the data are measurements to three significant figures. Unless there is clear evidence that the distribution is non-normal, it is more efficient to use the conventional t-tests. It is poor statistical practice to carry out both the parametric and non-parametric tests on a set of data and then choose the result that is most convenient to the investigator

Armstrong, Richard A.

Hilton, Anthony

English

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4Stat NoteIn the fourth of a series of articles about statistics for biologists, Anthony Hilton andRichard Armstrong analyse non-parametric data involving two groupsWhat if the data are not normal?www.sfam.org.uk34 March 2006www.sfam.org.uk 35March 2006Featuresmode but there are tests, to bedescribed later, that essentiallytest the differences betweenthe medians of two groups. An important property ofnon-normal distributions isthat the standard deviation(SD) is no longer an accuratedescriptor of the spread of adistribution with a givenmean. Hence, ‘z’ and ‘t’ tablescannot be used to predict theproportion of observationsthat fall a given distance fromthe mean. On reportingfrequency distributions fromlarge samples that are notnormally distributed,investigators often quote thepercentiles of thedistribution, e.g., the 90%percentile of a distribution isthe score such that 90% of theobservations fall short of and10% exceed the score(Snedecor and Cochran,1980).What is datatransformation?One method of analysingnon-normal data is to convertor transform the originalmeasurements so that they areexpressed on a new scale thatis more likely to be normallydistributed than the original.The usual parametric ‘t’ testscan then be carried out on thetransformed values. There arethree common circumstancesin which such atransformation should beconsidered. First, if the dataare percentages and especiallyif the majority of theobservations are close to zeroor 100%. Percentage data canbe transformed to an angularor arcsin scale defined asfollows:Angular measurement= sin-1 √ %/100 Statistical software willoften provide thistransformation or see Table Xin Fisher and Yates (1963).Percentage data can besignificantly skewed when themean is small or large andconsequently, the effect of thetransformation is thatFig. 1. Frequency distribution of the bacterial counts on clothsand sponges. Fitted curves are those of the normal distribution.asymmetrical and the mean isdisplaced either to the left(positive skew) or to the right(negative skew). By contrast,distributions that exhibitkurtosis are either more ‘flat-topped’ (negative kurtosis) orhave longer tails than normal(positive kurtosis). Fig. 1shows the frequencydistribution of the bacterialcounts on 54 sponges and 46cloths introduced in Statnote 1(Microbiologist, June 2005).In both cases the distributionsare clearly asymmetrical withthe means located to the leftof the histogram and thereforeexhibit a degree of positiveskew. As a result, thearithmetic mean is no longer agood description of thecentral tendency of such adistribution. There are twoadditional statistics that canbe used to describe the centraltendency of a skeweddistribution. First, is themode, the value of thevariable ‘x’ with the highestfrequency, i.e., the maximumpoint of the curve. Second, isthe median, the middle valueof ‘x’, i.e., if all the values of‘x’ were listed in ascending ordescending order, the medianwould be the middle value ofthe array. Little progress hasbeen made in devisingstatistical tests based on theinferences about means areless susceptible to thisproblem. Third, moderatedepartures from normality donot significantly affect thevalidity of parametric tests.Consideration of these pointsmay lead to the conclusionthat despite somereservations, the data may notdepart radically enough fromnormality to question thevalidity of a parametricanalysis. In othercircumstances, however, itmay be clear that the datadepart significantly fromnormality and a differentapproach required.Deviations from anormal distributionThe two most commonways in which a distributionmay deviate from normalityare called skew and kurtosis.Most statistical software willprovide tests of theseproperties and tables ofsignificance (Table A20) of therelevant statistics are given bySnedecor and Cochran (1980).It is important to note thatsome distributions maydeviate from normal in morecomplex ways and therefore,absence of skew and kurtosisdoes not guarantee that adistribution is normal. Askewed distribution isHE STATISTICALtests described inprevious Statnotes(see MicrobiologistSeptember and December2005) make a number ofassumptions about theexperimental data.The most important ofthese assumptions is that thequantity analysed, whether anindividual measurement,treatment mean, or differencebetween two means, must be aparametric variable, i.e., amember of a normallydistributed population. Whenthis assumption is met, the ‘z’and ‘t’ distributions can beused to make statisticalinferences from the data. Insome circumstances, however,a variable may not benormally distributed and thisStatnote is concerned with theanalysis of non-parametricdata involving two groups.How do we know if thedata are not normallydistributed? An investigator may knowin advance from previousstudies whether or not aquantity comes from a normaldistribution. In othercircumstances, data may havebeen collected to specificallytest whether the data comefrom a normal distribution, aprocedure that was describedin Statnote 1 (Microbiologist,June 2005). In manyexperimental situations,however, there may beinsufficient data available tocarry out a test of normalityand to obtain such data maybe either too expensive ortime-consuming. In situationssuch as these, the followingpoints should be considered.First, many measurements inthe biosciences made to atleast three significant figureshave a reasonable chance ofbeing normally distributed.Second, the distribution ofsample means taken from apopulation is more likely to benormal than the individualmeasurements and therefore,Twww.sfam.org.ukMarch 200636percentages near 0% or 100%are spread out to a greaterdegree than those near themean so as to increase theirvariance. A paired or unpaired‘t’ test can then be carried outusing the transformed valuesas described previously(Microbiologist, December2005). Second, data thatcomprise small wholenumbers or quantitiesassessed using a score thathas a limited scale, e.g., ifbacterial abundance wasscored from 0 to 5, areunlikely to be normallydistributed. In this case, atransformation to √ x (or √ x+ 1 if many zeroes arepresent) may make the scoresmore normally distributed.Third, the ‘t’ test described inStatnote 3 also assumeshomogeneity of variance,i.e., that the degree ofvariability is similar for bothgroups of observations. It isnot unusual, however, forresults from a ‘control’ groupto be more consistent thanvalues from an experimentallytreated group. In this case, atransformation of the originalmeasurements to a logarithmicscale may equalise thevariance and in addition, mayalso improve the degree ofnormality of the data. How are non-parametrictests done?An alternative approach totransformation in the analysisof non-normal data is to use anon-parametric test. As anillustration, we return to thescenario described in Statnote1 (Microbiologist, June2005). To recapitulate, giventhe intrinsic structural andcompositional differencesbetween cloths and sponges, astudy was envisaged toinvestigate if one materialprovided a more favourableenvironment for bacterialsurvival than the other. A totalof 54 ‘in-use’ dishcloths and46 sponges were collectedfrom domestic kitchens andthe aerobic colony count ofeach determined in thelaboratory. The frequencydistributions of the countsfrom both materials are shownin Fig. 1. In Statnote 1, thesedistributions were tested fornormality and it wasconcluded that the cloth dataexhibited a marked deviationfrom normal whereas thesponge data were closer to anormal distribution. However,it may be prudent to concludethat the data as a whole donot conform closely enough toa normal distribution to usethe parametric ‘t’ testsdescribed in Statnote 3. Analternative approach is to usea distribution-free or non-parametric test. These testscan be used regardless of theshape of the underlyingdistribution as long as thesamples being compared canbe assumed to come fromdistributions of the samegeneral shape.The Mann-Whitney U-Test ( for unpaired data)To illustrate this test and tosimplify the calculations wewill use data from a sample of10 cloths and 10 sponges only.The Mann-Whitney U-test canbe carried out on twoindependent groups of data(A,B) and is the non-parametric equivalent of theunpaired ‘t’ test (Statnote 3).Although most statisticalsoftware will carry out thistest, it is still useful tounderstand its ‘mechanics’(Table 1). First, ranks 1, 2, 3,... are assigned to the wholeset of observations, regardlessof group. A rank of 1 is givento the lowest count, 2 to thenext lowest etc. with repeatedvalues, called ‘ties’, given themean of the ranks within thatrun. The ranks of each groupare then added togetherseparately (RA, RB). Thequantities UA and UB are thencalculated as shown in Table1. Whichever is the smaller ofUA and UB, is taken to the tableof Wilcoxon’s U to judge thesignificance of the differencebetween cloths and sponges(Snedecor and Cochran, 1980;Table A10). The lesser U hasto be equal to or less than thetabulated value forsignificance, i.e., low values ofU indicate a significantdifference between thegroups. In the presentexample, a value of U = 16.5was obtained which is lessthan the value tabulated at P= 0.05. Hence, there isevidence that the spongesharbour considerably morebacteria than the cloths. Forlarger samples, outside therange of the statistical table,the data may approach anormal distribution moreclosely and a value of Z can becalculated (Table 1), thestatistic being referred totables of the normaldistribution.The Wilcoxon signedrank test (for paireddata)If the data in the twogroups are paired(Microbiologist, December2005), then the appropriatenon-parametric test is theWilcoxon signed rank test. Toillustrate this test (Table 2),we collected data on thenumber of bacteria on a singlepair of cloths and sponges on10 separate occasions. Hence,we do not have twoindependent samples as in theprevious example. In this case,there is a link between aparticular cloth and sponge inthat the data for each pairwere collected on a specificoccasion. Essentially, the dataare subtracted for each pair ofobservations (A - B). Omittingzero differences, ranks (r) areapplied to all of the remainingvalues of A - B regardless ofwhether the difference ispositive or negative. If tiesoccur between positive andnegative columns, the ranksare amended in any such runof ties to the mean rank withinthe run. The positive andnegative signs are restored tothe ranks and the positive andnegative ranks added up. R isthe smaller of the two sums ofranks and is taken to the tableof the Wilcoxon signed rankstatistic T to obtain a P-value(Snedecor and Cochran 1980;Count1.1 x 1082.2 x 1084.6 x 1069.8 x 1071.3 x 1081.3 x 1081.5 x 1084.7 x 1071.4 x 1081.2 x 108Rank1320511.515.515.518917141. Add up the ranks for each group: RA = 71.5, RB = 138.52. UA = {nA(nA + 1)/2 + (nAnB)} – RA = 83.5 where nA and nB are the numberof observations in each group3. UB = {nB(nB + 1)/2 + (nAnB)} – RB = 16.54. The smaller U (in this case 16.5) is the test statistic5. Lesser U must be ≤ Wilcoxon’s tabulated U for significant difference6. For larger samples: Z = (φµ – Tφ –1/2)σ where σ = √nB µ/6 and µ = nA(nA + nB + 1)/2Count1.8 x 1061.8 x 1072.0 x 1075.9 x 1071.6 x 1082.0 x 1059.8 x 1071.1 x 1066.9 x 1043.0 x 107Rank4671019211.5318Table 1. Comparison of the number of bacteria on 10 cloths and sponges (two independent groups, Mann-Whitney test)Clothes (A) Sponges (B)www.sfam.org.uk March 2006 37FeaturesReferences■ Fisher RA and Yates F (1963)Statistical tables. Longman,London.■ Hilton A & Armstrong RA(2005a) Are the data normal?Microbiologist June 2005. 6: 2,pp34-35■ Hilton A & Armstrong RA(2005b) Describing the normal.Microbiologist September 2005.6: 3, pp30-33.■ Hilton A & Armstrong RA(2005c) Testing the differencebetween two groups.Microbiologist December 2005.6: 4, pp30-33.■ Snedecor GW and CochranWG (1980) Statistical methods,7th Ed. Iowa State UniversityPress, Ames Iowa. Chapters 5,6.Rank-2-6-7-9+1-5-3-8-10-41. Subtract each pair of counts A - B2. Assign ranks (r) to differences ignoring the sign of the difference3. Restore the signs and add up the positive and negative ranks 4. Compare the lesser R ( in this case +R = 1) with the tabulatedWilcoxon’s signed rank statistic T, R ≤ T for significance5. For larger samples Z = (µ – T –1/2)σ where T is the smaller rank sumand σ = √(2n + 1)µ/6 where n = number of pairs and µ = n(n + 1)/4Occasion12345678910Cloth (A)1 x 1043.3 x 1075.7 x 1071.9 x 1071.2 x 1048.8 x 1022.6 x 1063.3 x 1078.7 x 1067.6 x 107Sponge (B)4.6 x 1069.8 x 1071.3 x 1081.3 x 1086.0 x 1024.7 x 1071.4 x 1081.2 x 1082.1 x 1081.1 x 108A - B-4.5 x 106-6.5 x 107-7.3 x 1071.11 x 108+1.1 x 104-4.7 x 107-1.14 x 107-8.7 x 107-2.0 x 108-3.4 x 107Table 2. Comparison of bacteria on pairs of cloths and spongessampled on 10 occasions (two dependent groups, Wilcoxonsigned rank test)Table A9). The value of R hasto be equal to or LESS thanthe value of T in the P = 0.05column to demonstrate asignificant difference betweenthe two groups. In this case,our value of R = 1 was lessthan the tabulated valueindicating that spongesharbour more bacteria thanthe cloths. With largernumbers of observations, avalue of Z can be calculatedand referred to tables of thenormal distribution.Comparison of theparametric and non-parametric testsIt is reasonable to ask whatis the relative sensitivity ofparametric and non-parametric tests and whathappens if they are usedincorrectly? If a t-test is usedon non-normal data, thesignificance probabilities arechanged and the sensitivity orpower of the test is alteredand this can result inerroneous conclusionsespecially if treatment effectsare of borderline significance.With non-parametric tests, thesignificance levels remain thesame for any continuousdistribution with the exceptionthat they are affected by thenumber of zeros and tiedvalues in the Wilcoxon signedrank test (Snedecor andCochran 1980). With largenormal samples, the efficiencyof the non-parametric tests isabout 95% compared with thet-test. With non-normal datafrom a continuousdistribution, however, theefficiency of the non-parametric tests relative to ‘t’never falls below 86% in largesamples and may be greaterthan 100% for distributionsthat are highly skewed.ConclusionsWhen testing the differencebetween two groups, ifprevious data indicate non-normality, then eithertransform the data if theycomprise percentages,integers or scores or use anon-parametric test. If there isuncertainty whether the dataare normally distributed, thendeviations from normality arelikely to be small if the dataare measurements to threesignificant figures. Unlessthere is clear evidence that thedistribution is non-normal, itis more efficient to use theconventional t-tests. It is poorstatistical practice to carry outboth the parametric and non-parametric tests on a set ofdata and then choose theresult that is most convenientto the investigator!Did you know thatprevious Stat Notes areavailable for downloadfrom the website inAdobe Actobat PDFformat?Simply click the articlesyou wish to view and/orright click a link to savea copy of the PDF toyour hard disk. Simply visit: http://www.sfam.org.uk/features.phpInstantaccess!Dr Anthony* Hilton and Dr Richard Armstrong***Pharmaceutical Sciences and**Vision Sciences, AstonUniversity, Birmingham, UK

Statnote 4: what if the data are not normal?

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Statnote 4: what if the data are not normal?

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