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Pertemuan 15

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Pertemuan 15. Analisis Ragam Peubah Ganda (MANOVA III). Matakuliah: I0214 / Statistika Multivariat Tahun: 2005 Versi: V1 / R1. Learning Outcomes. Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : - PowerPoint PPT Presentation
  • Pertemuan 15Matakuliah: I0214 / Statistika MultivariatTahun: 2005Versi: V1 / R1

    Analisis Ragam Peubah Ganda(MANOVA III)

  • Learning OutcomesPada akhir pertemuan ini, diharapkan mahasiswa akan mampu :

    Mahasiswa dapat menerangkan konsep dasar analisis ragam peubah ganda (manova) C2

    Mahasiswa dapat menghitung manova satu klasifikasi C3

    Mahasiswa dapat melakukan uji Fisher dan uji Bartlette C3

  • Outline Materi

    Konsep dasar analisis ragam peubah ganda (manova)

    Analisis ragam peubah ganda satu klasifikasi

    Uji Fisher

    Uji Bartlette

  • Null Hypothesis

    Univariate t-test:

    H0 : (1 = (2 (population means are equal)

    Multivariate case (2-group MANOVA):

    H0 :

    (population mean vectors are equal)

    Main assumptions: normally distributed DVs, equal covariance matrices across groups


  • Test Statistic for 2-group MANOVA

    Hotellings T2 : T2 =

    n1 : sample size in first group

    n2 : sample size in second group

    : vector of means of DVs in first group

    : vector of means of DVs in second group

    S : pooled within-group covariance matrix




  • Hotellings T2 measures the between-group difference

    , which is weighted by the within-group covariance matrix S-1. The test works as follows: From Hotellings T2, form

    F =

    F is the test statistic for testing whether there is a significant group difference with respect to the whole vector y of dependent variables. F-distributed with p and (n1+n2p1) degress of freedom



  • Rao's F Approximation

    degrees of Freedom

    Special Note Concerning s

    If either the numerator or the deminator of s = 0 set s = 1

  • Hotelling's Trace Criterion

    Roy's Largest Latent Root

    Pillai's Trace Criterion

  • Which of these is "best? Schatzoff (1966) Roy's largest-latent root was the most sensitive when population centroids differed along a single dimension, but was otherwise least sensative. Under most conditions it was a toss-up between Wilks' and Hotelling's criteria.

    Olson (1976) Pillai's criteria was the most robust to violations of assumptions concerning homogeneity of the covariance matrix. Under diffuse noncentrality the ordering was Pillai, Wilks, Hotelling and Roy. Under concentrated noncentrality the ordering is Roy, Hotelling, Wilks and Pillai.

    Final "Best" When sample sizes are very large the Wilks, Hotelling and Pillai become asymptotically equivalent

  • Tabel Manova

    Sumber Variasi

    Matriks Jumlah Kuadrat

    dan Hasil Kali Silang

    Derajat Bebas











  • Uji hipotesa

    menyangkut generalized variance.

    ditolak bila generalized variance



    ditemukan oleh Wilks).

    Distribusi yang eksak untuk

    diberikan dalam tabel






  • Tabel Distribusi Wilks Lamda

    Jumlah Variabel

    Jumlah Grup

    Distribusi sampling data multivariat













  • Bila

    benar dan


    berdistribusi mendekati Khi kuadrat dengan derajat bebas


    Jadi, untuk


    ditolak pada tingkat signifikansi bila:








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