REGRESI LINIER SEDERHANA

KULIAH #2 ANALISIS REGRESI

Usman Bustaman

APA ITU?• Regresi

• Linier

• Sederhana

REGRESI (Buku 5: Kutner, Et All P. 5)

Sir Francis Galton (latter part of the 19th century):

- studied the relation between heights of parents and children

- noted that the heights of children of both tall and short parents appeared to "revert" or "regress" to the mean of the group.

- developed a mathematical description of this regression tendency,

- today's regression models (to describe statistical relations between variables).

LINIER

Masih ingat Y=mX+B? Slope? Konstanta?

B

m

X

Y

LINIER LEBIH LANJUT…- Linier dalam paramater…

- Persamaan Linier orde 1:

- Persamaan Linier orde 2:

- Dst… (orde pangkat tertinggi yang terdapat pada variabel bebasnya)

SEDERHANA

Relasi antar 2 variabel:

1 variabel bebas (independent variable)

1 variabel tak bebas (dependent variable)

Y=mX+B?

Mana variabel bebas?

Mana variabel tak bebas?

B

m

X

Y

BAGAIMANA MEMBANGUN MODEL REGRESI LINIER SEDERHANA?

Analisis/Comment Grafik-2 Berikut:

Analisis/Comment Grafik-2 Berikut:

A B

C D

FUNGSI RATA-2 (Mean Function)

If you know something about X, this knowledge helps you predict something about Y.

PREDIKSI TERBAIK…

Bagaimana mengestimasi parameter dengan cara terbaik…

Regresi Linier

Regresi Linier

Koefisien regresi

Populasi

Sampel

Y = b0 + b1Xi

Y =𝛽0+𝛽1 𝑋

Regresi Linier Model

ie

X

Y

Y X b b0 1+=Yi

Xi

? (the actual value of Yi)

REGRESI TERBAIK = MINIMISASI ERROR- Semua residual harus nol

- Minimum Jumlah residual

- Minimum jumlah absolut residual

- Minimum versi Tshebysheff

- Minimum jumlah kuadrat residual OLS

ORDINARY LEAST SQUARE (OLS)

ASSUMPTIONS

Linear regression assumes that… • 1. The relationship between X and Y is linear• 2. Y is distributed normally at each value of X• 3. The variance of Y at every value of X is the same

(homogeneity of variances)• 4. The observations are independent

ASUMSI LEBIH LANJUT…

Alexander Von Eye & Christof Schuster (1998) Regression Analysis

for Social Sciences

ASUMSI LEBIH LANJUT…

Alexander Von Eye & Christof Schuster (1998) Regression Analysis

for Social Sciences

PROSES ESTIMASI PARAMETER (Drapper & Smith)

KOEFISIEN REGRESI

XbYb 10 21x

xy

xx

xy

S

Sb

n

XX

n

YY observasi jumlah n

1

)( 1

2

n

YYYVar

n

i

1)( 1

2

n

XXXVar

n

i

xxS

)(SSTS yy

xyS

1),(Covar 1

n

YYXXYX

n

i

SIMBOL-2 (Weisberg p. 22)

MAKNA KOEFISIEN REGRESI

b0 ≈ …..

b1 ≈ …..

?x = 0

- Tinggi vs berat badan- Nilai math vs stat

- Lama sekolah vs pendptn- Lama training vs jml produksi

…….

C A

B

A

yi

 

x

yyi

 

C

B

ii xy

y

A2 B2 C2

SST Total squared distance of observations from naïve mean of y Total variation

SSR Distance from regression line to naïve mean of y

 Variability due to x (regression)   

SSEVariance around the regression line  Additional variability not explained by x—what least squares method aims to minimize

n

iii

n

i

n

iii yyyyyy

1

2

1 1

22 )ˆ()ˆ()(

REGRESSION PICTURE

Y

Variance to beexplained by predictors

(SST)

SST (SUM SQUARE TOTAL)

Y

X

Variance NOT explained by X

(SSE)

Variance explained by X

(SSR)

SSE & SSR

Y

X

Variance NOT explained by X

(SSE)

Variance explained by X

(SSR)

SST = SSR + SSE Variance to beexplained by predictors

(SST)

Koefisien Determinasi

orsby Predict explained be toVariance

Xby explained Variance2 SST

SSRR

Coefficient of Determinationto judge the adequacy of the regression model

Maknanya: …. ?

Koefisien Determinasi

SALAH PAHAM TTG R2

1. R2 tinggi prediksi semakin baik ….

2. R2 tinggi model regresi cocok dgn datanya …

3. R2 rendah (mendekati nol) tidak ada hubungan antara variabel X dan Y …

Korelasi

yx

xy

yyxx

xyxy

xy

SS

Sr

rRR

2

Correlationmeasures the strength of the linear association between two

variables.

Pearson Correlation…?

Buktikan…!

KORELASI & REGRESI

21x

xy

xx

xy

S

Sb

yx

xy

yyxx

xyxy

SS

Sr

𝑺𝒀=√𝑺𝒀𝒀

𝑺𝑿=√𝑺𝑿𝑿

ASSUMPTIONS

Linear regression assumes that… • 1. The relationship between X and Y is linear• 2. Y is distributed normally at each value of X• 3. The variance of Y at every value of X is the same

(homogeneity of variances)• 4. The observations are independent

UJI PARAMETER RLS

Linear regression assumes that… • 1. The relationship between X and Y is linear• 2. Y is distributed normally at each value of X• 3. The variance of Y at every value of X is the same

(homogeneity of variances)• 4. The observations are independent

DISTRIBUSI SAMPLING B1

b1 ~ Normal ~ Normal

Uji koefisien regresi

ib

iikn S

bt

)1(

0:

0:

1

0

i

i

H

H

Uji koefisien regresi

xx

eekn

SS

b

bS

bt

2

11

1

11)1( )(

0:

0:

1

10

AH

H

Selang Kepercayaan koefisien regresi

xx

ekn

xx

ekn S

Stb

S

Stb

2

)1(,2/11

2

)1(,2/1

Confidence Interval for b1

Uji koefisien regresi

xxe

ekn

SX

nS

b

bS

bt

22

00

0

00)1(

1)(

0:

0:

0

00

AH

H

xxekn

xxekn S

X

nStb

S

X

nStb

22

)1(,2/00

22

)1(,2/0

11

Confidence Interval for the intercept

Selang Kepercayaan koefisien regresi