BASIC NON-PARAMETRIC STATISTICAL TOOLS*

Transcription

1 BAIC NON-PARAMRIC AIICAL OOL* Pepaed fo GCMA 00 Pete M. Quesada Gegoy. Rash * xamples peseted these otes wee obtaed fom Pme of Bostatstcs by tato. Glatz (McGaw Hll ext; IBN: )

2 Odal Data valuatg wo Itevetos o wo Dffeet Goups Ma-Whtey Rak-um est Based o akg of all obsevatos wthout egad to goup assocated wth each obsevato Ca also be used wth teval o ato data that ae ot omally dstbuted est statstc,, s sum of all aks fo the smalle goup R whee R s the ak of the th obsevato of the smalle goup ad s the umbe of obsevatos the smalle goup o deteme must fst ak all obsevatos fom both goups togethe ed aks eceve aveage of aks that would have bee spaed (e.g. f 3 obsevatos ae ted followg ak 4, the each of the ted obsevatos would eceve the aveage of aks 5, 6 ad 7, o (567)/ 6; the ext obsevato would eceve ak 8) Ctcal values of ae based o the tals of the dstbuto of all possble values (assumg o tes) xample: goups wth 3 obsevatos oe ad 4 obsevatos the othe Goup Goup Obseved Value Oveall Rak Obseved Value Oveall Rak (based o Goup ) o deteme pobablty of obtag a patcula value cosde all possble akgs fo goup obsevatos. st Obsevato Raks d Obsevato Raks d Obsevato Raks um of Raks st Obsevato Raks d Obsevato Raks d Obsevato Raks um of Raks hty fve possble combatos of aks If all obsevatos ae actually daw fom the same populato, the each combato s equally possble, ad the dstbuto s as show below

3 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X If all obsevatos wee tuly fom a sgle populato the thee would be a / (5.7%) pobablty of obtag oe of the two exteme values (6 o 8). mlaly, thee would be a 4/ (.4%) pobablty fo a value 7 o 7. Note that these pobabltes ae dscete atue. I peset example 9 s assocated wth pobablty of 4/ (40%), whch would ot be exteme eough to eject ull hypothess that all obsevatos wee daw fom the same populato. Whe the lage sample cotas eght o moe obsevatos, dstbuto of appoxmates a omal dstbuto wth mea µ ( ) B whee B s the umbe of samples the bgge goup, ad stadad devato θ B ( ) B Ca the costuct test statstc, z z µ θ whch ca be compaed wth t-dstbuto wth fte degees of feedom (d.o.f.) hs compaso s moe accuate wth a cotuty coecto whee z µ θ

4 Odal Data valuatg hee o moe Itevetos o Dffeet Goups of Idvduals Kuskal-Walls tatstc Based o akg of all obsevatos wthout egad to goup assocated wth each obsevato est statstc, H, s a omalzed, weghted sum of squaed dffeeces betwee each goup s mea ak ad the oveall mea ak o deteme H must fst ak all obsevatos wthout egad fo goups ed aks eceve aveage of aks that would have bee spaed Mea ak s detemed fo each goup, j, as R j j R j j whee R j s the ak of the th obsevato of the j th goup ad j s the umbe of goup j obsevatos Oveall mea ak s R N N N whee N s the total umbe of obsevatos ( N m j j whee m s the umbe of goups) he weghted sum of squaed dffeeces s D m j j ( R R ) j H s computed by dvdg D by ( N ) deped o sample sze N whch esults a test statstc value that does ot H N D ( N ) N( N ) m j j ( R R ) j If o eal dffeece exsts betwee tevetos the mea goup aks should be close to oveall mea ak; D ad, subsequetly, H should be smalle values that would peclude ejecto of the ull hypothess Ctcal values of H ae based o the tals of the dstbuto of all possble H values (assumg o tes) If sample szes ae suffcetly lage ( j 5 fo m 3; N > 0 whe m 4) the the dstbuto of H appoxmates the dstbuto wth d.o.f., ν m-. 3

5 xample: 3 goups wth dffeet umbe obsevatos Goup Goup Goup 3 Obseved Value Oveall Rak Obseved Value Oveall Rak Obseved Value Oveall Rak Rak 46 Rak 8 Rak um Mea Rak um.3 Mea Rak um 4. Mea Rak Oveall Mea Rak (3 ) / H N ( R R ) j j ( N ) j [ 3(.3 6) 9( 4. 6) 9( ) ] ( ) 3 3 m.07 >.0, ν 9.0 Reject ull hypothess that all obsevatos fom a sgle populato o deteme whee dffeeces exst pefom pa-wse Ma-Whtey tests wth Bofeo adjustmets ad cotuty coectos. 4

6 Odal Data valuatg wo Itevetos o the ame Goup of Idvduals Wlcoxo ged-rak est Based o akg of absolute dffeeces betwee two obsevatos fo each dvdual est statstc, W, s sum of all aks of dffeeces W R whee s the umbe of dvduals, s the dffeece betwee obsevatos fo the th dvdual, ad R s the ak of the absolute dffeece fo the th dvdual (ote: the facto fot of the aks wll always have magtude,, ad wll have the sg of the dffeece) If o eal dffeece exsts betwee dvduals obsevatos, the the sgs of the obseved dffeeces should occu by adom chace; W would the compute to a umbe close to zeo. xteme values of W ethe postve o egatve sese, thus, lead to ejecto of the ull hypothess that o dffeece exsts betwee obsevatos. xample: goup wth obsevatos fo each dvdual Idvdual Obsevato Oe Obsevato wo Dffeece Rak of Dffeece ged Rak of Dffeece W -3 Fo sx dvduals thee ae 65 possble combatos of sged aks (assumg o tes). If o eal dffeece exsts betwee obsevatos, the each combato s equally possble, ad the dstbuto s as show below X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Fo ths dstbuto thee s a 4/ (6.5%) chace of obtag a value of W at o beyod 9 (o 9) f o eal dffeece exsts. Fo peset example W -3 s ot exteme eough to eject ull hypothess. As wth othe paametc methods, p-values fo the Wlcoxo ged-rak est ae dscete atue. Fo lage umbe of dvduals, howeve, dstbuto of W values appoxmate a omal dstbuto wth mea µ W 0 5

7 ad stadad devato θ W ( )( ) 6 Fom whch test statstc, z W ca be computed as z W W µ θ W W W ( )( ) 6 whch ca be compaed wth t-dstbuto wth fte degees of feedom (d.o.f.) whch wth a cotuty coecto becomes z W W ( )( ) 6 o hadle ted aks, must fst detfy type of te. es whch dffeece s zeo esult dvdual beg dopped fom sample etely. es whch dffeece s o-zeo ae hadled as befoe. 6

8 Odal Data valuatg hee o Moe Itevetos o the ame Goup of Idvduals Fedma tatstc Based o akgs of each dvdual s obsevatos assocated wth each teveto Itally, s detemed as the sum of squaed dffeeces betwee each teveto s obseved ak sum, R j R j ad k s the umbe of tevetos., ad the expected ak sums, ( k ) whee s the umbe of dvduals k ( Rj ( k ) ) j Fedma statstc s the fomed by dvdg by ( k ) dstbuto appoxmates a dstbuto wth ν k-. k to obta a statstc whose k j ( R ( k ) ) j k ( k ) If o eal dffeece exsts betwee dvduals obsevatos, the obseved ak sums should be close to expected ak sums; thus squaed dffeeces should be small, ad & should be close to zeo. If < 9 fo k 3 o < 4 fo k 4, dstbuto of Fedma statstc does ot appoxmate dstbuto; must use actual dstbuto of Fedma statstc to deteme dscete ctcal values (see able below) xample: lage goup wth 6 obsevatos fo each dvdual st Obsevato d Obsevato 3 d Obsevato 4 th Obsevato 5 th Obsevato 6 th Obsevato Idvdual Value Rak Value Rak Value Rak Value Rak Value Rak Value Rak R ( k ) 0( 6 ) k ( R ( k ) ) j j k > ( k ).00, ν Reject ull hypothess that o dffeece exsts betwee tevetos [( 44 35) ( 53 35) ( 39 35) ( 0 35) ( 49 35) ( 0 35) ] ( 0)( 6)( 6 ) 7

9 xample: small goup wth 3 obsevatos fo each dvdual st Obsevato d Obsevato 3 d Obsevato Idvdual Value Rak Value Rak Value Rak R 5 7 ( k ) 4( 3 ) 8 k ( R ( k ) ) j j k Fom table below ( k ) [( 8) ( 5 8) ( 7 8) ] ( 4)( 3)( 3 ) 6.5 matches value wth p.04 fo 4 ad k 3 Reject ull hypothess k 3 tevetos k 4 tevetos p p

10 Odal Data valuatg Assocato Betwee wo Vaables peama Rak Coelato Coeffcet Based o assocato betwee akgs of each vaable Itally, must ak each vaable ethe ascedg o descedg ode peama Rak Coelato Coeffcet, s the essetally detemed as the Peaso poductmomet coelato betwee the aks, athe tha the actual values of the vaables. Alteatvely, ca be computed usg the equato 6 3 d whee d s the dffeece betwee vaable aks fo the th dvdual ad s the umbe of dvduals. If o eal assocato exsts betwee vaables, the the sum of squaed dffeeces wll ted towad lage values, ad wll ted towad zeo. As appoaches, t becomes less lkely that value was obtaed by adom chace fo two vaables wth o assocato betwee them. Ctcal values fo ae detfed fom peama Rak Coelato Coeffcet table depedg o acceptable p-value (.e. chace of falsely cocludg that a assocato exsts) ad umbe of dvduals (samples). If > 50, howeve, ca compute a t-value as t ( ) ( ) whch ca be evaluated fo sgfcace based o v. xample: Vaable Vaable Idvdual Value Rak Value Rak Rak Dff d > 6 p.00, 0 [( ) ( ) ( 0.5) ] Reject ull hypothess that o assocato exsts betwee vaable ad vaable 9

11 Pobablty of Geate Value P

12 Nomal Data valuatg wo o Moe Itevetos o Dffeet Goups Ch-squae Aalyss of Cotgecy Based o cotgecy tables cotag cells wth umbes of dvduals matchg ow ad colum specfcatos wo types of cotgecy tables Obseved (actual) xpected Ch-quae test statstc,, s a sum of omalzed squaed dffeeces betwee coespodg cells of obseved ad expected tables m ( O ) j whee s the ow dex, j s the colum dex, O s the umbe of obsevatos cell, s the expected umbe of obsevatos cell, s the umbe of ows, ad m s the umbe of colums. he expected umbe of obsevatos fo a gve cell s detemed fom the ow, colum ad oveall obsevato totals fom the obseved table as R C j whee R s the total umbe of obsevatos ow, C j s the total umbe obsevatos colum j, ad s the total umbe of obsevatos the ete table. gets lage as obseved table devates moe fom expected table If o eal dffeece exsts betwee cell o ow codtos, the lage values ae less lkely to occu due to adom chace. values assocated wth adom chace pobabltes less tha a ctcal value (p ct ) cause ejecto of the ull hypothess. pobabltes obtaed fom table based o d.o.f. (ν) ( )( ) ν m Whe ν (.e. fo X cotgecy table), should apply Yates coecto such that j O m xample: outcomes, 3 classfcatos (goups, tevetos) Obseved able Outcome Outcome Row otals Classfcato Classfcato Classfcato Colum otals

13 xpected able Outcome Outcome Row otals Classfcato (54)(69)/65.58 (54)(96)/ Classfcato (3)(69)/ (3)(96)/ Classfcato 3 (88)(69)/ (88)(96)/ Colum otals m ( O ) j ( 4.58) ( ) ( 9 9.6) ( ) ( ) ( 4 5.) > ( ν ) Reject ull hypothess ad coclude that thee s a dffeece outcomes betwee the classfcatos Note that esults do ot yet dcate whee the dffeeces ae; oly that they exst Ca subdvde cotgecy table to pefom pa-wse compasos xample cotued: Obseved able Outcome Outcome Row otals Classfcato Classfcato Colum otals xpected able Outcome Outcome Row otals Classfcato (3)(55)/.40 (3)(56)/.60 3 Classfcato 3 (88)(55)/ (88)(56)/ Colum otals m j O < ( ν ) Caot eject ull hypothess, so classfcatos ad 3 ae deemed to be a sgle classfcato Classfcatos ad 3 ae combed to fom a ew classfcato (4) whch ca the be compaed wth classfcato Obseved able Outcome Outcome Row otals Classfcato Classfcato Colum otals

14 xpected able Outcome Outcome Row otals Classfcato (54)(69)/65.58 (54)(96)/ Classfcato 4 ()(55)/ (88)(56)/ Colum otals m j O > ( ν ) Reject ull hypothess, classfcato dffes sgfcatly fom the combato of classfcatos & 3 3

15 Nomal Data valuatg wo Itevetos o the ame Goup of Idvduals McNema s est fo Chages Based o cells X cotgecy table that epeset dvduals wth dffeet outcomes fo each teveto (cell that epeset smla outcomes fo each teveto ae goed) Ch-quae test statstc,, s sum of omalzed squaed dffeeces betwee coespodg obseved ad expected table cells that ae ot goed (wth ν ) O xpected value fo the emag cells s computed as the aveage of the emag cells O If o eal dffeece exsts betwee tevetos, the lage values ae less lkely to occu due to adom chace. values assocated wth adom chace pobabltes less tha a ctcal value (p ct ) cause ejecto of the ull hypothess. xample: outcomes, 3 classfcatos (goups, tevetos) Obseved able Outcome Outcome Outcome 8 48 Outcome 3 xpected able Outcome Outcome Outcome (483)/ 35.5 Outcome (483)/ 35.5 Couts of dvduals wth outcome fo both tevetos o outcome fo both tevetos ae goed; calculato based o emag cells. O > ( ν ) Reject ull hypothess ad coclude that thee s a dffeece outcomes betwee the classfcatos 4