Analisis varian mutlivariat (MANOVA) satu arah dengan uji rasio likelohood wilks

Abstract

Analisis varian multivariat (MANOVA) satu arah digunakan untuk menguji ada atau tidaknya perbedaan yang nyata tentang pengaruh perlakuan terhadap p variabel respon (p>1), atau menguji kesamaan vektor rata-rata dari beberapa (k) populasi. Pada penulisan ini ditentukan mengenai analisis varian multivariat (MANOVA) satu arah untuk model tetap. Hipotesis tidal( ada perbedaan vektor rata-rata perlakuan adalah Ho : = = likatau ekuivalen dengan Ho : ti = 0, i = 1,2,...,k. Hipotesis dapat diuji dengan Wilks' A A = IGI + HI di mana, G adalah matrik error berukuran pxp berdistribusi Wp(kn-k,E), dan H adalah matrik hipotesis berukuran pxp berdistribusi Wp(k-1,E). Ho ditolak jika A nilai kritis dari Wilks' A. Matrik G dan G H definit positif jika (kn-k) p, oleh karena itu A terdefinisi hanya dalarn kondisi (kn-k) > p. Jika Ho ditolak berarti ada perbedaan antar vektor rata-rata perlakuan. Dalam hal ini perbandingan antar kelompok vektor rata-rata diperlukan. Kontras multivariat dapat digunakan untuk perbandingan antar kelompok vektor rata-rata. Hipotesis untuk kontras multivariat adalah Ho : ciµi + c2g2 + + clam( = 0, hipotesis dapat diuji dengan statistik Hotelling's T2, di mana -I k T2 - kn k EciYi. Ec Y. i=i berdistribusi Tpkn-k Ho ditolak jika T2 > T2 a;; kritiskn-k nilakritisBari distribusi 2 T2. Oneway multivariate analysis of variance (MANOVA) is used for testing there are no significantly differences about treatment effect with p respon variates, or testing the equality of mean vector of several populations. In this paper, is developed the oneway multivariate analysis of variance (MANOVA) for the fixed effect model. Hypothesis of no defferences in treatment means vector is Ho : = = µk or equivalently Ho : Tl = 0, i = 1,2,...,k. The hypothesis can be tested with Wilks' A 1G1 1G + HI Where, G is the pxp error matrix distributed Wp(lui-k,E), and II the pxp hypothesis matrix distributed Wp(k-1,E). Ho is rejected if /15. AcE;p;k_1;kn_k, exact critical value for Wilks' A. The matrix G and G + II is positive definit if (kn-k) p, hence A is defined only under condition (kn-k) p. If Ho is rejected, thus there are differences between the treatment means vector. In this situation, further comparisons between goups of treatment means vector may be useful. Contrast multivariate can be used to compare between groups of treatment means vector. Hypothesis of contrasts is Ho + c2p.2 +...+ colt = 0, hypothesis can be tested with Hotelling's T2, Where n r k G -1( k T2 — E • ct yi k) Eciyi 2 \j=i n— i=i z, c; 1=1' Which is distributed as T2p;kn-k' Ho is rejected if T2 > Ta2;pjai_k , exact critical value of T2 distribution.