FIXED, RANDOM & MIXED MODELS. Senin, 12 November 2012

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Senin, 12 November 2012

FIXED, RANDOM & MIXED MODELS

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Outline’s

Introduction

Single Factor Models Two Factor Models

EMS (Expected Mean Square) Rules EMS (Expected Mean Square) Rules The Pseudo-F Test

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Introduction

Setiap peneliti sebelum me-running

eksperimen harus menentukan jenis

levelnya.

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Introduction Fixed

Level ditentukan oleh eksperimenter.

Kesimpulan hanya berlaku untuk level yg telah ditentukan.

Tidak bisa digeneralisasi untuk populasi. Tidak bisa digeneralisasi untuk populasi. Biasanya : level terdiri dari ekstrim bawah, tengah, atas

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Introduction

RANDOM

Experimenter tidak menentukan level

Kesimpulan : bisa digeneralisasi untuk populasi Kelemahan : level yang terpilih tdk

mencerminkan kondisi sebenarnya. mencerminkan kondisi sebenarnya.

Untuk kasus single faktor, perbedaannya hanya terkait dengan generalisasi kesimpulan.

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Introduction MIXED

Kombinasi Fixed dan Random

Menutup kelemahan masing-masing Lebih sesuai dengan kondisi nyata Lebih sesuai dengan kondisi nyata

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Single Factor Models

Model Matematika Completely Random Design ( Bab III , Hicks )

ij j

ij

Y

=

µ

+

τ

+

ε

Fix atau Random

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Single Factor Models Level jenis Fixed :

Masih INGAT CONTOH 3.1 di BAB III , Hicks halaman 50-51 ? ( resistance to abrasion )

halaman 50-51 ? ( resistance to abrasion )

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ij j

ij

Y

=

µ

+

τ

+

ε

Fix atau Random

Level jenis Fixed :

Asumsi pada Fix ( ) 0 1 1 = − =

_∑

∑

= =

µ

τ

k j j k j j

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Level jenis Fixed :

Cara Analisis ( Lihat BAB III , Hicks ) Expected Mean Square ( EMS )

EMS df Source Hipotesis j semua untuk , 0 : 0 j = H τ 2 2 ) 1 ( 1 ε τ ε σ ε φ σ τ − + − n k n k ij j

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Single Factor Models Level jenis Random :

Masih INGAT CONTOH 3.2 di BAB III , Hicks halaman 65-66 ? ( quality of the incoming material )

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ij j

ij

Y

=

µ

+

τ

+

ε

Fix atau Random

Level jenis Random :

Random

NID

(

0 ,

σ

2τ

)

Normal and Independent Disitribution

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Single Factor Models Level jenis Random :

Cara Analisis ( Lihat BAB III , Hicks ) Expected Mean Square ( EMS )

EMS df Source Hipotesis 0 : 2 0 στ = H 2 2 2 ) 1 ( 1 ε τ ε σ ε σ σ τ − + − n k n k ij j

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Two Factor Models

Model Matematika ( Bab VI , Hicks , hal. 159-160 )

AB

B

A

Y

_ij

=

µ

+

_i

+

_j

+

_ij

+

ε

₍_k₎_ij

n

k

b

j

a

i

=

1 ,

2 ,..,

=

1 ,

2 ,....,

=

1 ,

2 ,...,

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Expected Mean Square (EMS)

Penting untuk eksperimen yg kompleks (ex : random, mixed).

EMS untuk menguji signifikasi suatu faktor (melakukan uji pseudo-F ).

(melakukan uji pseudo-F ).

EMS berfungsi sebagai denominator (pembagi) dalam uji pseudo-F

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Langkah-Langkah Merumuskan EMS

1. Tulis sumber variasi pd kolom paling kiri.

2. Tulis indeks sbg judul kolom (i,j,k), diatas indeks

ditulis jenis levelnya ( F u / fixed, & R u / Random), diatasnya lagi tulis masing-masing jumlah levelnya (di atas I,j) dan jumlah replikasi (diatas K).

diatasnya lagi tulis masing-masing jumlah levelnya (di atas I,j) dan jumlah replikasi (diatas K).

3. Tulis (kopi )jumlah level ke dalam tabel. Syarat :

jumlah level tdk boleh muncul di baris yang ada

indeks bersangkutan. Di kolom k, ditulis replikasinya saja.

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4.Tulis angka 1 pada baris dimana indeks ditulis dalam tanda “ ( )”.

5. Isi sel yg lain dgn angka 0 & 1. Angka 0 u/ jika level pd kolom tsb fixed, dan angka 1 jika levelnya random.

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6. Rumus EMS :

a. EMS dihitung baris demi baris.

b. Tutup kolom, jk indeks kolom muncul pd baris yg

sedang dicari.

c. Kalikan angka-angka yg ada. Akan mjd koef. Pd rumus c. Kalikan angka-angka yg ada. Akan mjd koef. Pd rumus

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Contoh 1 ( Hicks, hal 163)

Viskositas sebuah slurry diuji oleh 4 laboran yang dipilih secara random. Material yang diuji oleh laboran dimasukan dalam botol dan diuji viskositasnya dengan 5 mesin yang ada. Masing-masing laboran menguji sebanyak 2 kali.

masing laboran menguji sebanyak 2 kali. Laboran ( Technician = T ) → Random

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Deciding What to Use as the Denominator of Your F-test

For an all fixed model the Error MS is the denominator of all F-tests.

For an all random or mix model,

1. Ignore the last component of the expected mean square. 2. Look for the expected mean square that now looks this

expected mean square. expected mean square.

3. The mean square associated with this expected mean square

will be the denominator of the F-test.

4. If you can’t find an expected mean square that matches the

one mentioned above, then you need to develop a Synthetic Error Term

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Contoh 1 ( Hicks, hal 163) 2 2 / 10 2 5 1 3 F EMS k j i df Source R F R 2 5 4 σ σ MS MS T + 2 ) ( 2 2 2 2 2 2 1 1 1 20 / 2 2 0 1 12 / 8 2 2 0 4 4 / 10 2 5 1 3 ε ε ε ε σ ε σ σ φ σ σ σ σ ij k error TM TM ij TM M TM ij error T T j MS MS TM MS MS M M MS MS T + + + + Perhatikan :

Uji F ( Uji pseudo F ) untuk M bukan dibagi dengan MS error

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Example 4 : ( Stat485 Lecture)

In this Example a Taxi company is interested in comparing the effects of three brands of tires (A, B and C) on

mileage (mpg). Mileage will also be effected by driver. The company selects at random b = 4 drivers at random from its collection of drivers. Each driver has n = 3 opportunities to use each brand of tire in which mileage is measured.

to use each brand of tire in which mileage is measured. Dependent

Mileage

Independent

Tire brand (A, B, C),

Fixed Effect Factor

Driver (1, 2, 3, 4),

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The Data

Driver Tire Mileage Driver Tire Mileage 1 A 39.6 3 A 33.9 1 A 38.6 3 A 43.2 1 A 41.9 3 A 41.3 1 B 18.1 3 B 17.8 1 B 20.4 3 B 21.3 1 B 19 3 B 22.3 1 C 31.1 3 C 31.3 1 C 29.8 3 C 28.7 1 C 26.6 3 C 29.7 1 C 26.6 3 C 29.7 2 A 38.1 4 A 36.9 2 A 35.4 4 A 30.3 2 A 38.8 4 A 35 2 B 18.2 4 B 17.8 2 B 14 4 B 21.2 2 B 15.6 4 B 24.3 2 C 30.2 4 C 27.4 2 C 27.9 4 C 26.6 2 C 27.2 4 C 21

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The Output

Tests of Between-Subjects Effects

Dependent Variable: MILEAGE

28928.340 1 28928.340 1270.836 .000 68.290 3 22.763a 2072.931 2 1036.465 71.374 .000 87.129 6 14.522b Source Hypothesis Error Intercept Hypothesis Error TIRE Type III Sum of Squares df Mean Square F Sig. 87.129 6 14.522 68.290 3 22.763 1.568 .292 87.129 6 14.522b 87.129 6 14.522 2.039 .099 170.940 24 7.123c Error Hypothesis Error DRIVER Hypothesis Error TIRE * DRIVER MS(DRIVER) a. MS(TIRE * DRIVER) b. MS(Error) c.

The divisor for both the fixed and the random main effect is MS_AB This is contrary to the advice of some texts

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The Anova table for the two factor model (A – fixed, B - random)

( )

ij ijk j i ijk y = µ +α + β + αβ + ε Source SS df MS EMS F A SS_A a -1 MS_A MS_A/MS_AB B SS_A b - 1 MS_B MS_B/MS_Error ( − )∑= + + a i i AB a nb n 1 2 2 2 1 α σ σ 2 2 B naσ σ + B SS_A b - 1 MS_B MS_B/MS_Error AB SS_AB (a -1)(b -1) MS_AB MS_AB/MS_Error

Error SS_Error ab(n – 1) MS_Error σ2

B naσ σ + 2 2 AB nσ σ +

Note: The divisor for testing the main effects of A is no longer

MS_Error but MS_AB.

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The Anova table for the two factor model (A – fixed, B - random)

( )

ij ijk j i ijk y = µ +α + β + αβ + ε Source SS df MS EMS F A SS_A a -1 MS_A MS_A/MS_AB B SS_A b - 1 MS_B MS_B/MS_AB ( − )∑= + + a i i AB a nb n 1 2 2 2 1 α σ σ 2 2 2 B AB na nσ σ σ + + B SS_A b - 1 MS_B MS_B/MS_AB AB SS_AB (a -1)(b -1) MS_AB MS_AB/MS_Error

Error SS_Error ab(n – 1) MS_Error σ2

B AB na nσ σ σ + + 2 2 AB nσ σ +

Note: In this case the divisor for testing the main effects of A is

MS_AB. This is the approach used by SPSS.

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