Analiza varijanse i analiza kovarijanse. Jelena Marinković, maj 2012.

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1 Analiza varijanse i analiza kovarijanse Jelena Marinković, maj 2012.

2 A šta ćemo danas i narednih dana učiti? p Koje statističke metode primeniti kada se radi o: n Više od dva uzorka i/ili n Više od dve varijable merene na istim ispitanicima p Zašto u takvim situacijama višestruko ponavljanje statističkih testova za dva uzorka nije ispravno?

3 Kako rešiti ovaj problem? (nastavak) p Potrebne su neke druge, nove, statističke metode. Prva od njih je: p Analiza varijanse (analiza varijansnog količnika, ANOVA - ANalysis Of VAriance).

4 ANOVA Dizajn eksperimenta

5 Opšte okruženje eksperimenta p Istraživač kontroliše jednu ili više nezavisnih varijabli n ime im je faktori ili varijable tretmana n svaki faktor ima bar dva nivoa / gradacije, kategorije, klasifikacije / p Istraživač meri efekte faktora na zavisnoj varijabli p Eksperimentalni dizajn plan za testiranje istraživačke hipoteze

6 Eksperimentalni dizajn / plan, vrste p Potpuno randomizovani faktorijalni plan n eksperimentalne jedinice su slučajno birane i randomizovano se dodeljuju tretmanima p Randomizovani blok dizajn n jedinice se dele u blokove i uparuju se jedinice u različitim uzorcima p Dizajn ponovljenih merenja n jedinice se više puta mere

7 Skala merenja (samo jedne) rezultujuće varijable Eksperimentalni dizajni / Istraživački planovi Randomizovani potpuni blok dizajn Faktorijalni eksperiment 1 faktor 2 i više faktora 1 faktor i 1 kriterijum klasifikacije Dizajn ponovljenih merenja 1 faktor i 1 kriterijum klasifikacije Omerna / Intervalna Jednofaktorska ANOVA Dvofaktorska i višefaktorske ANOVE Dvosmerna ANOVA (ili jednofaktorska ANOVA sa blokovima) Jednofaktorska ANOVA sa ponovljenim merenjima (ili dvosmerna ANOVA) Ordinalna Kraskal-Volisova analiza varijanse sa rangovima Fridmanova dvosmerna analiza varijanse sa rangovima Fridmanova dvosmerna analiza varijanse sa rangovima Nominalna Fišerov varijansni količnik za proporcije χ 2 test za učestalosti / proporcije Loglinearni modeli* Kohrejnov Q test Kohrejnov Q test

8 Opšti model analize varijanse p U opštem modelu analize varijanse odnos variranja uobičajeno se predstavlja sledećim zapisom: Y = X + Z tj. Ukupno variranje (Y) = variranje čiji je izvor u organizovanom delu eksperimenta (X) + variranje čiji je izvor u neorganizovanom delu eksperimenta (Z)

9 Faktorski varijablitet x 1 x 2 x 3 x grupa 1 grupa 2 grupa 3

10 Slučajni varijablitet x 3 x 1 x 2 grupa 1 grupa 2 grupa 3

11 Ukupan varijablitet x grupa 1 grupa 2 grupa 3

12 Šta je ANOVA? p Analiza varijanse (ANOVA) je statistička metoda zaključivanja zasnovana na generalnim linearnim modelima, koja ukupan varijabilitet skupa podataka deli na bar dve komponente (faktorskuorganizovanu i rezidualnu-slučajnu).

13 A koje hipoteze testira? p H 0 : µ 1 = µ 2 =... = µ k p H 1 : sve µ j nisu jednake.

14 Tablica sheme rezultata analize varijanse Izvor variranja Disperzija Broj stepena slobode Varijansa F Između grupa Unutar grupa C x df x = k-1 sd x 2 C z df z = n-k sd z 2 Opšti C y df y = n-1

15 Višestruka poređenja p Uvek kada analiza varijanse dovede do odluke o odbacivanju (neprihvatanju) nulte hipoteze postavlja se pitanje koji je par (ili parovi) prosečnih vrednosti značajno različit, odnosno, koju od (u datom primeru 10 mogućih) pojedinačnih hipoteza treba odbaciti.

16 Metode post hoc testiranja: Tukey - najbolji balans odnosa greške prvog i drugog tipa Scheffe prikladniji nego Tukey ako se veličine grupa razlikuju značajno Dunnett - kada treba porediti nekoliko grupa sa kontrolnom grupom Bonferroni više konzervativan (manja je verovatnoća da će biti nađena statistička značajnost) u odnosu na Tukey i Scheffe Newman-Keuls, Duncan, Fisher s LSD ne kotrolišu sveukupno alfa, više liberalni

17 SAS i DS 2010/2011 Statistika za istraživače Katedra za medicinsku statistiku i informatiku

18 Prirodno proširenje ove statističke metode je MANOVA p Više od jedne zavisne varijable p Zavisne varijable bi trebalo da budu povezane konceptualno p Zavisne varijable bi trebalo da budu slabo do osrednje statistički povezane (korelisane). Ako između njih ne postoji nikakva povezanost, nema razloga da se one analiziraju zajedno u okviru MANOVA

19 SAS i DS 2010/2011 Statistika za istraživače Katedra za medicinsku statistiku i informatiku

20 ANCOVA Analiza kovarijanse

21 ANCOVA analiza kovarijanse p Proverava razlike između aritme0čkih sredina ishodne (rezultujuće, zavisne) varijable dve ili više grupa u situaciji kada znamo da je neka druga pridružena varijabla (kovarijata) ili više njih povezano sa ishodom. p Koris0 se da kontroliše kovarijatu, tj. da otkloni efekat pridružene varijable kao izvora mogućeg objašnjenja razlike tretmana. p Najčešće se koris0 kada grupe nisu formirane randomizacijom, već su formirane same po sebi, npr. pacijen0 koji su primili određeni tretman na osnovu nekog kriterijuma.

22 Varijanse Ukupno variranje podataka Faktorsko variranje Slučajno variranje Kovarijata SV Slide 22

23 Prednosti ANCOVE p Redukuje slučajnu varijansu n Objašnjavanjem dela slučajnog variranja pridruženom varijablom slučajna varijansa može bi0 redukovana. p Veća eksperimentalna kontrola: p Kontrolom poznate pridružene varijable s0čemo bolje objašnjenje o efektu(ima) faktora. p Sta0s0čki metod za kontrolu inicijalnih razlika varijabli u istraživanju.

24 Nulta hipoteza u analizi kovarijanse Jednakost prilagođenih aritmetičkih sredina: H 0 : µ 1 * = µ 2 * =... = µ j * U interpretaciji razmotriti i originalne i prilagođene aritmetičke sredine.

25 Prilagođavanje aritmetičkih sredina p Podrazumeva prilagođavanje aritmetičkih sredina grupa na one vrednosti koje bi inače imale kada bi sve grupe bile jednake na pridruženoj varijabli. Preduslov je jednakost regresionih nagiba za sve grupe

26 Izbor pridružene vrijable p Svaka varijabla koja je povezana sa zavisnom varijablom može biti analizirana kao pridružena varijabla p Ako je u analizu uključeno više od jedne pridružene varijable, poželjno je da povezanost između njih bude slaba, jer tada svaka od njih uklanja relativno različite delove rezidualnog srednjeg kvadrata

27 Primer p Fieldov (2009) Viagra eksperiment: n Ishod (ili ZV) = Učesnikov libido n Faktor / Prediktor (ili NV) = Doza vijagre (Placebo, Niska & Visoka) n Kovarijata = Partnerov libido

28 Odnosi između NV i kovarijate

29 Homogenost regresionih nagiba

30 Dose Par8cipant s Libido Partner s Libido Placebo 3.22 (1.79) 3.44 (2.07) Low Dose 4.88 (1.46) 3.12 (1.73) High Dose 4.85 (2.12) 2.00 (1.63) Y i = + + b0 b1x i b 2Covariate Libido i = b 0 + b1dosei + b2partner's Libido i

31 Placebo Low High Slide 31

32 ANCOVA u SPSS-u

33 Kontrasti

34 Opcije

35 Bez u0caja kovarijate Dependent Variable: Libido Source Corrected Model Intercept DOSE Error Total Corrected Total Tests of Between-Subjects Effects Type III Sum of Squares df Mean Square F Sig a a. R Squared =.152 (Adjusted R Squared =.089) Slide 35

36 Izveštaj Levene's Test of Equality of Error Variances a Dependent Variable: Libido F df1 df2 Sig Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a. Design: Intercept+PARTNER+DOSE Dependent Variable: Libido Source Corrected Model Intercept PARTNER DOSE Error Total Corrected Total Tests of Between-Subjects Effects Type III Sum of Squares df Mean Square F Sig a a. R Squared =.313 (Adjusted R Squared =.234) Slide 36

37 SPSS Output: Contrasts Slide 37

38 Output Slide 38

39 Neprilagođene aritme0čke sredine Placebo Low High Slide 39

40 Glavni efekat Placebo Low High F(2, 26) = 4.14, p <.05

41 SAS i DS 2010/2011 Statistika za istraživače Katedra za medicinsku statistiku i informatiku

42 Prirodno proširenje ove statističke metode je MANCOVA p Dve ili više zavisnih varijabli, jedna ili više pridruženih varijabli p Pretpostavka je da postoji povezanost skupa zavisnih varijabli i skupa pridruženih varijabli

43 SAS i DS 2010/2011 Statistika za istraživače Katedra za medicinsku statistiku i informatiku

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