Senin, 12 November 2012

FIXED, RANDOM &

MIXED MODELS

Outline’s

� Introduction

� Single Factor Models

� Two Factor Models

� EMS (Expected Mean Square) RulesEMS (Expected Mean Square) Rules

� The Pseudo-F Test

Introduction

� Setiap peneliti sebelum me-runningeksperimen harus menentukan jenislevelnya.

� Jenis level yaitu Fixed, Random atau Mixed

Introduction

Fixed

� Level ditentukan oleh eksperimenter.

� Kesimpulan hanya berlaku untuk level yg telahditentukan.

� Tidak bisa digeneralisasi untuk populasi.� Tidak bisa digeneralisasi untuk populasi.

� Biasanya : level terdiri dari ekstrim bawah, tengah, atas

Introduction

RANDOM

� Experimenter tidak menentukan level

� Kesimpulan : bisa digeneralisasi untuk populasi

� Kelemahan : level yang terpilih tdkmencerminkan kondisi sebenarnya.mencerminkan kondisi sebenarnya.

� Untuk kasus single faktor, perbedaannya hanyaterkait dengan generalisasi kesimpulan.

Introduction

MIXED

� Kombinasi Fixed dan Random

� Menutup kelemahan masing-masing

� Lebih sesuai dengan kondisi nyataLebih sesuai dengan kondisi nyata

Single Factor Models

Model Matematika Completely Random Design

( Bab III , Hicks )

ijjijY ετµ ++= Fix atau Random

Asumsi pada

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 )

Kenapa levelnya termasuk jenis fixed ?

Single Factor Models

ijjijY ετµ ++= Fix atau Random

Level jenis Fixed :

� Asumsi pada

Fix 0)(11

=−= ∑∑==

µµτk

jj

k

jj

Single Factor Models

Level jenis Fixed :

� Cara Analisis ( Lihat BAB III , Hicks )

� Expected Mean Square ( EMS )

EMS df Source

� Hipotesis

j semuauntuk ,0:0 =jH τ

2

2

)1(

1

ε

τε

σε

φστ

−

+−

nk

nk

ij

j

Single Factor Models

� Level jenis Random :

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

Kenapa levelnya termasuk jenis random ?

Single Factor Models

ijjijY ετµ ++= Fix atau Random

� Level jenis Random :

Random ),0(NID 2τσ

Normal and Independent Disitribution

� Asumsi pada

Single Factor Models

� Level jenis Random :� Cara Analisis ( Lihat BAB III , Hicks )

� Expected Mean Square ( EMS )

EMS df Source

� Hipotesis

0: 20 =τσH

2

22

)1(

1

ε

τε

σε

σστ

−

+−

nk

nk

ij

j

Two Factor Models

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

ABBAY ijkijjiij )(++++= εµ

nkbjai ,...,2,1,....,2,1,..,2,1 ===

Two Factor Models

Two Factor Models

Two Factor Models

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

Langkah-Langkah Merumuskan EMS

1. Tulis sumber variasi pd kolom paling kiri.

2. Tulis indeks sbg judul kolom (i,j,k), diatas indeksditulis jenis levelnya ( F u / fixed, & R u / Random), diatasnya lagi tulis masing-masing jumlah levelnya (diatas I,j) dan jumlah replikasi (diatas K).diatasnya lagi tulis masing-masing jumlah levelnya (diatas I,j) dan jumlah replikasi (diatas K).

3. Tulis (kopi )jumlah level ke dalam tabel. Syarat : jumlah level tdk boleh muncul di baris yang adaindeks bersangkutan. Di kolom k, ditulis replikasinyasaja.

Langkah-Langkah Merumuskan EMS

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.

Langkah-Langkah Merumuskan EMS

6. Rumus EMS :

a. EMS dihitung baris demi baris.

b. Tutup kolom, jk indeks kolom muncul pd baris ygsedang dicari.

c. Kalikan angka-angka yg ada. Akan mjd koef. Pd rumusc. Kalikan angka-angka yg ada. Akan mjd koef. Pd rumusEMS.

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

Mesin ( Machine = M → Fixed

Contoh 1 ( Hicks, hal 163)

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

Contoh 1 ( Hicks, hal 163)

22 /102513

FEMS k j i df Source

R F R

2 5 4

σσ MSMST +

2)(

22

22

22

11120

/220112

/822044

/102513

ε

ε

ε

ε

σε

σσ

φσσ

σσ

ijk

errorTMTMij

TMMTMij

errorTTj

MSMSTM

MSMSMM

MSMST

+

++

+

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

Contoh 2

Contoh 2

Contoh 3 :

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),� Random Effects factor

The DataDriver Tire Mileage Driver Tire Mileage

1 A 39.6 3 A 33.91 A 38.6 3 A 43.21 A 41.9 3 A 41.31 B 18.1 3 B 17.81 B 20.4 3 B 21.31 B 19 3 B 22.31 C 31.1 3 C 31.31 C 29.8 3 C 28.71 C 26.6 3 C 29.71 C 26.6 3 C 29.72 A 38.1 4 A 36.92 A 35.4 4 A 30.32 A 38.8 4 A 352 B 18.2 4 B 17.82 B 14 4 B 21.22 B 15.6 4 B 24.32 C 30.2 4 C 27.42 C 27.9 4 C 26.62 C 27.2 4 C 21

Asking SPSS to perform Univariate ANOVA

Select the dependent variable, fixed factors, random factors

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

SourceHypothesis

Error

Intercept

Hypothesis

Error

TIRE

Type IIISum of

Squares dfMean

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 MSAB

This is contrary to the advice of some texts

The Anova table for the two factor model

(A – fixed, B - random)

( ) ijkijjiijky εαββαµ ++++=

Source SS df MS EMS F

A SSA a -1 MSA MSA/MSAB

B SSA b - 1 MSB MSB/MSError

( )∑=−++

a

iiAB a

nbn

1

222

1ασσ

22Bnaσσ +B SSA b - 1 MSB MSB/MSError

AB SSAB (a -1)(b -1) MSAB MSAB/MSError

Error SSError ab(n – 1) MSError 2σ

Bnaσσ +

22ABnσσ +

Note: The divisor for testing the main effects of A is no longer MSError but MSAB.

References Guenther, W. C. “Analysis of Variance” Prentice Hall, 1964

The Anova table for the two factor model

(A – fixed, B - random)

( ) ijkijjiijky εαββαµ ++++=

Source SS df MS EMS F

A SSA a -1 MSA MSA/MSAB

B SSA b - 1 MSB MSB/MSAB

( )∑=−++

a

iiAB a

nbn

1

222

1ασσ

222BAB nan σσσ ++B SSA b - 1 MSB MSB/MSAB

AB SSAB (a -1)(b -1) MSAB MSAB/MSError

Error SSError ab(n – 1) MSError 2σ

BAB nan σσσ ++

22ABnσσ +

Note: In this case the divisor for testing the main effects of A is MSAB . This is the approach used by SPSS.

References Searle “Linear Models” John Wiley, 1964

Pseudo – F Test

Pseudo – F Test

Pseudo – F Test

Pseudo – F Test

Pseudo – F Test

Pseudo – F Test

Pseudo – F Test

Inspirasi Hari Ini

http://www.stat.purdue.edu/~kuczek/stat514/