UCLA STAT 110B Applied Statistics for Engineering and the Sciences

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1 UCLA SA 0B Applied Statistics for Egieerig ad the Scieces Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology eachig Assistats: Bria Ng, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig Why Use Noparametric Statistics? Parametric tests are based upo assumptios that may iclude the followig: he data have the same variace, regardless of the treatmets or coditios i the experimet. he data are ormally distributed for each of the treatmets or coditios i the experimet. What happes whe we are ot sure that these assumptios have bee satisfied? Slide Slide How Do Noparametric ests Compare with the Usual z, t, ad F ests? Studies have show that whe the usual assumptios are satisfied, oparametric tests are about 95% efficiet whe compared to their parametric equivalets. Whe ormality ad commo variace are ot satisfied, the oparametric procedures ca be much more efficiet tha their parametric equivalets. he Wilcoxo Rak Sum est Suppose we wish to test the hypothesis that two distributios have the same ceter. We select two idepedet radom samples from each populatio. Desigate each of the observatios from populatio as a A ad each of the observatios from populatio as a B. If H 0 is true, ad the two samples have bee draw from the same populatio, whe we rak the values i both samples from small to large, the A s ad B s should be radomly mixed i the rakigs. Slide 3 Slide What happes whe H 0 is true? Suppose we had 5 measuremets from populatio ad 6 measuremets from populatio. If they were draw from the same populatio, the rakigs might be like this. ABABBABABBA I this case if we summed the raks of the A measuremets ad the raks of the B measuremets, the sums would be similar. What happes if H 0 is ot true? If the observatios come from two differet populatios, perhaps with populatio lyig to the left of populatio, the rakig of the observatios might take the followig orderig. AAABABABBB I this case the sum of the raks of the B observatios would be larger tha that for the A observatios. Slide 5 Slide 6
2 How to Implemet Wilcoxo s Rak est Rak the combied sample from smallest to largest. Let represet the sum of the raks of the first sample (A s). he, defied below, is the sum of the raks that the A s would have had if the observatios were raked from large to small. ( + + ) he Wilcoxo Rak Sum est H 0 :: the two populatio distributios are the same H a a :: the two populatios are i i some way differet he test statistic is the smaller of ad. Reject H 0 if the test statistic is less tha the critical value foud i able (a). able (a) is idexed by lettig populatio be the oe associated with the smaller sample size, ad populatio as the oe associated with, the larger sample size. Slide Slide 8 he wig stroke frequecies of two species If If several of measuremets bees were are are recorded tied, for a sample of each gets from the species the average of of the ad the raks 6 from species. they would have gotte, if if they Ca were you coclude that the distributios of wig ot tied! (See x 80) strokes differ for these two species? Use α.05. Species 35 (0) 5 (9) 90 (8) 88 () H Species 0 : the are the H 80 a : a : the the two species are are i i some way differet (3.5) 69 (). he sample with the smaller sample 80 size is called sample. (3.5) 85 (6). We rak the 0 observatios from 8 () smallest to largest, show i 8 (5) paretheses i the table. Slide 9 he Bee Problem Ca you coclude that the distributios of wig strokes differ for these two species? α.05. Species Reject Species H 0.. Data Calculate provides sufficiet evidece (0) 80 (3.5) idicatig a differece i i 5 the (9) 69 () ( + + ) distributios of of wig stroke 90 frequecies. (8) 80 (3.5) ( ) () (6) () (5). he test statistic is 0.. he critical value of from able (b) for a twotailed test with α/.05 is ; H 0 is rejected if. Slide 0 Miitab Output Recall 3; 0. MaWhitey est ad CI: Species, Species Species N Media 0.50 Species N 6 Media Poit estimate for EAEA is Percet CI for EAEA is (5.99,56.0) W 3.0 est of EA EA vs EA ot EA is sigificat at 0.0 he test is sigificat at (adjusted for ties) Miitab calls the the procedure the the MaWhitey U est, equivalet to to the the Wilcoxo Rak Sum est. he test statistic is is W 3 3 ad has pvalue.0. Do Do ot reject H 0 for 0 for α.05. Slide Large Sample Approximatio: Wilcoxo Rak Sum est Whe ad are large (greater tha 0 is large eough), a ormal approximatio ca be used to approximate the critical values i able.. Calculate ad. Let mi,. he statistic z σ z distributio with µ ( + + ) ad σ has a approximate sum of the raks of sample (A s). ( + + ) Slide ). ( + + ) µ
3 Some Notes Whe should you use the Wilcoxo Rak Sum test istead of the twosample t test for idepedet samples? whe the resposes ca oly be raked ad ot quatified (e.g., ordial qualitative data) whe the F test or the Rule of humb shows a problem with equality of variaces whe a ormality plot shows a violatio of the ormality assumptio he Sig est he sig test is a fairly simple procedure that ca be used to compare two populatios whe the samples cosist of paired observatios. It ca be used whe the assumptios required for the paireddifferece test are ot valid or whe the resposes ca oly be raked as oe better tha the other, but caot be quatified. Slide 3 Slide he Sig est he Sig est For each pair, measure whether the first respose say, A exceeds the secod respose say, B. he test statistic is is x, the umber of times that A exceeds B i the pairs of observatios. Oly pairs without ties are icluded i the test. Critical values for the rejectio regio or exact pvalues ca be foud usig the cumulative biomial distributio (SOCR resource olie). Slide 5 H 0 : the two populatios are idetical versus H a : oe or twotailed alterative is is equivalet to H 0 : p P(A exceeds B).5 versus H a : p (, <, or >).5 est statistic: x umber of plus sigs Rejectio regio, pvalues from Bisize, p). Slide 6 he Gourmet Chefs wo gourmet chefs each tasted ad rated eight differet meals from to 0. Does it appear that oe of the chefs teds to give higher ratigs tha the other? Use α.0. Meal Chef A Chef B Sig H 0 : 0 : the the ratig distributios are are the the same (p (p.5).5) H a : a : the the ratigs are are differet (p (p.5).5) Slide Meal Chef A Chef B Sig + + pvalue.5 0 is too large to pvalue.5 is too large to reject H 0. H 0 : 0 : p.5 0. here is is isufficiet.5 evidece to to idicate that oe H a : a : p.5.5 with (omit chef the the tied pair) teds to to rate oe meal est Statistic: x umber of of higher plus sigs tha the the other. Use able with ad p pvalue P(observe x or or somethig equally as as ulikely) P(x ) ) + P(x 5) 5) (.).5 k P(x k) Slide 8
4 Large Sample Approximatio: he Sig est Y~Bi, p) E(Y) p Var(Y) p(p) Whe 5, a ormal approximatio ca be used to approximate the critical values of Biomial distributio.. Calculate x. he statistic z distributio. umber of plus sigs. x.5 z.5 has a approximate You record the umber of accidets per day at a large maufacturig plat for both the day ad eveig shifts for 00 days. You fid that the umber of accidets per day for the eveig shift x E exceeded the For correspodig a two tailed umber test, we we of reject accidets H 00 i the day shift x D o 63 of the if if 00 z z > days..96 (5% Do these level). results provide sufficiet evidece to idicate H that 0 is more accidets ted to occur 0 is rejected. here is is evidece o oe shift tha o the other? of of a differece betwee the the day ad H 0 : 0 : the the distributios (# (# of of accidets) ight shifts. are are the the same (p (p.5).5) H a : a : the the distributios are are differet (p (p.5).5) est statistic: z x (00) Slide 9 Slide 0 Which test should you use? Which test should you use? We compare statistical tests usig Defiitio: Power β P(reject H 0 whe H a is true) he power of the test is the probability of rejectig the ull hypothesis whe it is false ad some specified alterative is true. he power is the probability that the test will do what it was desiged to do that is, detect a departure from the ull hypothesis whe a departure exists. If all parametric assumptios have bee met, the parametric test will be the most powerful. If ot, a oparametric test may be more powerful. If you ca reject H 0 with a less powerful oparametric test, you will ot have to worry about parametric assumptios. If ot, you might try more powerful oparametric test or icreasig the sample size to gai more power Slide Slide he Wilcoxo SigedRak est differet form Wilcoxo Rak Sum est he Wilcoxo SigedRak est is a more powerful oparametric procedure that ca be used to compare two populatios whe the samples cosist of paired observatios. It uses the raks of the differeces, d x x that we used i the paireddifferece test. Slide 3 he Wilcoxo SigedRak est differet form Wilcoxo Rak Sum est For each pair, calculate the differece d x x x.. Elimiate zero differeces. Rak the absolute values of of the differeces from to to.. ied observatios are assiged average of of the raks they would have gotte if if ot tied. + rak sum for positive differeces rak sum for egative differeces If the two populatios are the same, + ad should be early equal. If If either + or or is is uusually large, this provides evidece agaist the ull hypothesis. Slide
5 he Wilcoxo SigedRak est H 0 : the two populatios are idetical versus H a : oe or twotailed alterative est statistic: mi ( + ad ) Critical values for a oe or twotailed rejectio regio ca be foud usig Wilcoxo SigedRak est able. o compare the desities of cakes usig mixes A ad B, six pairs of pas (A ad B) were baked sidebyside i six differet ove locatios. Is there evidece of a differece i desity for the two cake mixes? Locatio Cake Mix A Cake Mix B d x A x B H : 0 : the the desity distributios are are the the same 0 H a : a : the the desity distributios are are differet Slide 5 Slide 6 Locatio Cake Mix A Cake Mix B d x A x B Rak Cake Desities Do ot reject H 0.here is Do ot reject H 0.here is isufficiet..3 evidece. to to idicate.9.0. that.5there.35is is a differece.63 i i desities.0.00 for for the the.09 two cake mixes Rak Calculate: the the 6 + ad differeces, he test statistic is is mi ( ( +,, )).. without Rejectio regard regio: Use able For a twotailed test with to α to sig..05, reject H 0 if 0 if.. Large Sample Approximatio: he SigedRak est Whe 5, a ormal approximatio ca be used to approximate the critical values i able 8.. Calculate + ad. Let mi, µ. he statistic z hasa approximat e σ z distributio with + ) + )( + ) µ ad σ + ). Slide Slide 8 he KruskalWallis H est he KruskalWallis H est he KruskalWallis H est is a oparametric procedure that ca be used to compare more tha two populatios i a completely radomized desig. Noparametric equivalet to ANOVA Ftest! All k measuremets are joitly raked. We use the sums of the raks of the k samples to compare the distributios. Slide 9 Rak the total measuremets i i all k samples from to to.. ied observatios are assiged average of of the raks they would have gotte if if ot tied. Calculate i i rak sum for the ii th th sample i i,,,,k k Ad the test statistic H is is (aalog to: F MSS/MSSE) k H i 3( + ) ( + ) i i Slide 30
6 he he KruskalWallis H est H est H 0 : the k distributios are idetical versus H a : at least oe distributio is differet est statistic: KruskalWallis H Whe H 0 is true, the test statistic H has a approximate χ distributio with df k. Use a righttailed rejectio regio or pvalue based o the Chisquare distributio. Four groups of studets were radomly assiged to be taught with four differet techiques, ad their achievemet test scores were recorded. Are the distributios of test scores the same, or do they differ i locatio? Slide 3 Slide 3 i eachig Methods 3 65 (3) 5 () 59 () 9 (6) 8 (3) 69 (5) 8 (8) 89 (5) 3 (6) 83 () 6 () 80 (0) 9 (9) 8 () 6 () 88 () 3 Rak the the 6 6 H : 0 : the the distributios of of scores are are the the same 0 measuremets H a : a : the the distributios differ i i locatio from to to 6, 6, ad calculate i eststatistic: H 3( + ) the the four rak + ) i sums () () Slide eachig Methods H : 0 : the the distributios of of scores are are the the same 0 H a : a : the the distributios differ i i locatio i eststatistic: H 3( + ) + ) i () () Rejectio regio: Use able For a righttailed chisquare test with α ad df df 3, reject H 0 if 0 if H.8. Slide 3 Reject H here is is sufficiet evidece to to idicate that there is is a differece i i test scores for for the the four teachig techiques. I. Noparametric Methods. hese methods ca be used whe the data caot be measured o a quatitative scale, or whe. he umerical scale of measuremet is arbitrarily set by the researcher, or whe 3. he parametric assumptios such as ormality or costat variace are seriously violated. II. Wilcoxo Rak Sum est: Idepedet Radom Samples. Joitly rak the two samples: Desigate the smaller sample as sample. he Rak of sample ( + + ) Slide. Use to test for populatio to the left of populatio Use to test for populatio to the right of populatio. Use the smaller of ad to test for a differece i the locatios of the two populatios. 3. able of Appedix I has critical values for the rejectio of H 0.. Whe the sample sizes are large, use the ormal approximatio: µ z σ ( + + ) ( + + ) µ ad σ Slide 8
7 III. Sig est for a Paired Experimet. Fid x, the umber of times that observatio A exceeds observatio B for a give pair.. o test for a differece i two populatios, test H 0 : p 0.5 versus a oe or twotailed alterative. 3. Use able of Appedix I to calculate the pvalue for the test.. Whe the sample sizes are large, use the ormal approximatio: x.5 z.5 IV. Wilcoxo SigedRak est: Paired Experimet. Calculate the differeces i the paired observatios. Rak the absolute values of the differeces. Calculate the rak sums ad + for the positive ad egative differeces, respectively. he test statistic is the smaller of the two rak sums.. able 8 of Appedix I has critical values for the rejectio of for both oe ad twotailed tests. 3. Whe the samplig sizes are large, use the ormal approximatio: [ + ) ] z [ + )( + ) ] Slide 9 Slide 50 V. KruskalWallis H est: Completely Radomized Desig. Joitly rak the observatios i the k samples. Calculate the rak sums, i rak sum of sample i, ad the test statistic i H 3( + ) + ) i. If the ull hypothesis of equality of distributios is false, H will be uusually large, resultig i a oetailed test. 3. For sample sizes of five or greater, the rejectio regio for H is based o the chisquare distributio with (k ) degrees of freedom. VI. he Friedma F r est: Radomized Block Desig. Rak the resposes withi each block from to k. Calculate the rak sums,,, k, ad the test statistic Fr i 3b( k + ) bk( k + ). If the ull hypothesis of equality of treatmet distributios is false, F r will be uusually large, resultig i a oetailed test. 3. For block sizes of five or greater, the rejectio regio for F r is based o the chisquare distributio with (k ) degrees of freedom. Slide 5 Slide 5 VII. Spearma's Rak Correlatio Coefficiet. Rak the resposes for the two variables from smallest to largest.. Calculate the correlatio coefficiet for the raked observatios: r s S S S xx xy yy or r s 6 di ) 3. able 9 i Appedix I gives critical values for rak correlatios sigificatly differet from 0.. he rak correlatio coefficiet detects ot oly sigificat liear correlatio but also ay other mootoic relatioship betwee the two variables. Slide 53 if thereareo ties Sesitivity vs. Specificity Sesitivity is a measure of the fractio of gold stadard kow examples that are correctly classified/idetified. Sesitivity P/(P+FN) Specificity is a measure of the fractio of egative examples that are correctly classified: Specificity N/(N+FP) P rue Positives H o : o effects (µ0) rue Reality FN False Negatives H o true H o false N rue Negatives Ca t reject N FN FP False Positives Reject H o FP P est Results Slide 5
8 he ROC Curve Receiver Operatig Characteristic (ROC) curve.0 Fractio rue Positives Sesitivity Better est Worse est Bigger area higher test accuracy ad better discrimiatio i.e. the ability of the test to correctly classify those with ad without the disease. Fractio False Positives Specificity Slide 55.0 ReceiverOperatig Characteristic curve ROC curve demostrates several thigs: It shows the tradeoff betwee sesitivity ad specificity (ay icrease i sesitivity will be accompaied by a decrease i specificity). he closer the curve follows the left border ad the the top border of the ROC space, the more accurate the test. he closer the curve comes to the 5degree diagoal of the ROC space, the less accurate the test. he slope of the taget lie at a cutpoit gives the likelihood ratio (LR) for that value of the test. You ca check this out o the graph above. he area uder the curve is a measure of test accuracy. Slide 56
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