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2002 Prentice-Hall, Inc. Chap 11-1

Analisis Regresi

Noorlaily Fitdiarini, SE., MBA.

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Tujuan Analisis Regresi

X2 tests of independence digunakan untukmenentukan adanya suatu hubungan statistikantara 2 variabel, tetapi tidak menyebutkanbagaimana hubungannya.

Analisis Regresi & Korelasi menunjukkan baik sifatmaupun kuatnya suatu hubungan antara keduavariabel tersebut.

Setiap regresi pasti ada korelasinya, tetapi korelasibelum tentu dilanjutkan oleh regresi.

Korelasi yang tidak dilanjutkan dengan regresiadalah korelasi antara 2 variabel yang tidakmempunyai hubungan kausal/sebab-akibat, atauhubungan fungsional (berdasarkan teori/konsep)

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Macam-macam Model Regresi

Positive Linear Relationship

Negative Linear Relationship

Relationship NOT Linear

No Relationship

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Model Regresi Linear Sederhana

Hubungan di antara 2 variabeldigambarkan dengan fungsi linear

Perubahan pada satu variabelmenyebabkan perubahan pada variabelyang lain

Ada ketergantungan satu variabelterhadap yang lain

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Regresi Linear

Y interceptSlopeCoefficient

Random

Error

Dependent(Response)Variable

Independent(Explanatory)Variable

ii iY X

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Regresi Linear(continued)

ii iY X

= Random Error

Y

X

(Observed Value of Y) =

Observed Value of Y

i

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Interpretasi dariSlope dan Intercept

adalah rata-rata dari nilai Y

jika nilai X sama dengan 0.

mengukur perubahan pada

rata-rata nilai Y sebagai akibat dari perubahansatu unit X.

| 0E Y X

1

|E Y X

X

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adalah rata-rata dari nilai Y

yang diestimasi jika nilai X sama dengan 0.

perubahan yang diestimasi

pada rata-rata nilai Y sebagai akibat dari perubahan

satu unit X.

(continued)

| 0b E Y X

1

|E Y Xb

X

Interpretation of theSlope and the Intercept

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Regresi Linear Sederhana:Contoh Kasus

Anda ingin mengujiketergantungan linear

dari penjualan tahunandari suatu tokoterhadap ukuran/luastoko. Sampel data

untuk 7 toko diperoleh.Bagaimana persamaangaris lurus yang palingsesuai dengan datatersebut?

AnnualStore Square Sales

Feet ($1000)

1 1,726 3,6812 1,542 3,395

3 2,816 6,653

4 5,555 9,543

5 1,292 3,318

6 2,208 5,563

7 1,313 3,760

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Scatter Diagram: Contoh Kasus

0

2 0 0 0

4 0 0 0

6 0 0 0

8 0 0 0

1 0 0 0 0

1 2 0 0 0

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0

S q u a re F e e t

A

nnualSales

($000)

Excel Output

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Persamaan Garis Regresi:Contoh Kasus

0 1

1636.415 1.487

i i

i

Y b b X

X

From Excel Printout:

Coef f ic ients

I n t e r c e p t 1 6 3 6 . 4 1 4 7 2 6

X V a r i a b l e 1 . 4 8 6 6 3 3 6 5 7

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Graph of the SampleRegression Line: Example

0

2 0 0 0

4 0 0 0

6 0 0 0

8 0 0 0

1 0 0 0 0

1 2 0 0 0

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0

S q u a r e F e e t

A

nnualSales

($000)

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Interpretation of Results:Contoh Kasus

Slope = 1.487 berarti bahwa untuk setiappeningkatan satu unit X, kita memprediksikan rata-rata Y akan meningkat kira-kira sebesar 1,487 unit

Jadi setiap peningkatan 1 sq foot ukuran toko,diperkirana penjual tahunan akan meningkatsebesar $1487.

1636.415 1.487i i

Y X

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Measure of Variation:The Sum of Squares

(continued)

Xi

Y

X

Y

SST=(Yi-Y)2

SSE=(Yi-Yi)2

SSR=(Yi-Y)2

_

_

_

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The ANOVA Table in Excel

ANOVA

df SS MS FSignificance

F

Regression p SSRMSR

=SSR/pMSR/MSE

P-value of

the F Test

Residuals n-p-1 SSEMSE

=SSE/(n-p-1)

Total n-1 SST

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Measures of VariationThe Sum of Squares: Example

ANOVAdf SS MS F Significance F

Regression 1 30380456.12 30380456 81.17909 0.000281201

Residual 5 1871199.595 374239.92

Total 6 32251655.71

Excel Output for Produce Stores

SSR

SSERegression (explained) df

Degrees of freedom

Error (residual) df

Total df

SST

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The Coefficient of Determination &Standard Error of Estimate

Coefficient of Determination

Mengukur proporsi dari variasi Y yang diterangkanoleh variabel independen X pada model regresi

Standard Error of Estimate:

2 Regression Sum of Squares

Total Sum of Squares

SSRr

SST

2

1

2 2

n

i

i

YX

Y YSSE

Sn n

Standar deviasi dari variasi observasi sekitar garis

regresi

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Measures of Variation:Produce Store Example

R eg ressio n S ta tistics

M u l t ip le R 0 . 9 7 0 5 5 7 2

R S q u a re 0 . 9 4 1 9 8 1 2 9

A d j u s t e d R S q u a r e 0 . 9 3 0 3 7 7 5 4

S t a n d a rd E rro r 6 1 1 .7 5 1 5 1 7

O b s e rva t io n s 7

Excel Output for Produce Stores

r2 = .94

94% dari variasi penjualan tahunandijelaskan oleh variabilitas ukuran toko (yang

diukur dengan sq foot)

Syx

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Example: Produce Store

Data for Seven Stores:EstimatedRegressionEquation:

The slope of this

model is 1.487.Is square footage ofthe store affecting itsannual sales?

AnnualStore Square Sales

Feet ($000)

1 1,726 3,681

2 1,542 3,395

3 2,816 6,653

4 5,555 9,5435 1,292 3,318

6 2,208 5,563

7 1,313 3,760

Yi= 1636.415 +1.487Xi

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Inferences about the Slope:t Test Example

H0: 1 = 0

H1: 1 0

.05df7 - 2 = 5Critical Value(s):

Test Statistic:

Keputusan:

Kesimpulan:Terdapat bukti bahwaukuran toko (sq foot)

mempengaruhi penjualantahunan.

t0 2.5706-2.5706

.025

Reject Reject

.025

From Excel Printout

Reject H0

Coefficients Standard Error t Stat P-value

Intercept 1636.4147 451.4953 3.6244 0.01515

Footage 1.4866 0.1650 9.0099 0.00028

1b 1bS t

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Inferences about the Slope:Confidence Interval Example

Confidence Interval Estimate of the Slope:

11 2n bb t S

Excel Printout for Produce Stores

Pada 95% level of confidence, the confidenceinterval untuk slope adalah (1.062, 1.911). Di atas 0.

Kesimpulan: Terdapat suatu ketergantungan linear

yang signifikan dari penjualan tahunan terhadapukuran toko.

Lowe r 9 5% Upper 9 5%

I n t e r c e p t 4 7 5 . 81 0 9 26 2 7 97 .0 1 85 3

X V a r ia b l e 1 .0 6 24 9 0 37 1 .9 1 07 7 69 4

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ANOVA

df SS MS F Significance F

Regression 1 30380456.12 30380456.12 81.179 0.000281Residual 5 1871199.595 374239.919

Total 6 32251655.71

Inferences about the Slope:F Test Example

Test Statistic:

Decision:

Conclusion:

H0:1= 0H1:1 0.05numeratordf = 1denominatordf7 - 2 = 5

There is evidence thatsquare footage affects

annual sales.

From Excel Printout

Reject H0

0 6.61

Reject

= .05

1, 2nF

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Tujuan dari Analisis Korelasi

Digunakan untuk mengukur kuatnya hubungan(linear) antara 2 variabel. Tidak ada hubungan kausal

Population correlation coefficient (Rho) digunakanuntuk mengukur kuatnya hubungan (linear) antara 2variabel.

Sample correlation coefficient r adalah suatuestimasi dari dan digunakan untuk mengukurkuatnya hubungan (linear) pada sampel observasi

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Features ofand r

Unit free

Range between -1 and 1

The closer to -1, the stronger the negativelinear relationship

The closer to 1, the stronger the positivelinear relationship

The closer to 0, the weaker the linearrelationship

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Test for a Linear Relationship

Hypotheses

H0:= 0 (tidak ada korelasi)

H1: 0 (korelasi)

Test statistic

2

2 1

2 2

1 1

where

2

n

i i

i

n n

i ii i

rt

r

n

X X Y Y

r r

X X Y Y

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Example: Produce Stores

R eg ressio n S tat istics

M u l t ip le R 0 . 9 7 0 5 5 7 2

R S q u a re 0 . 9 4 1 9 8 1 2 9A d j u s t e d R S q u a r e 0 . 9 3 0 3 7 7 5 4

S t a n d a rd E rro r 6 1 1 .7 5 1 5 1 7

O b s e rva t io n s 7

From Excel Printout r

Is there anyevidence of a linear

relationship betweenthe annual sales of astore and its square

footage at .05 levelof significance? H0:= 0 (No association)

H1: 0 (Association).05df7 - 2 = 5

E l

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Example:Produce Store Solution

0 2.5706-2.5706

.025

Reject Reject

.025

Critical Value(s):

Conclusion:There is evidence of alinear relationship at 5%level of significance

Decision:Reject H0

2

.97069.0099

1 .9420

52

rt

r

n

The value of the t statistic isexactly the same as the tstatistic value for test on theslope coefficient

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0 1 1 2 2i i i k ki iY b b X b X b X e

Population

Y-intercept

Population slopes Random

Error

The Multiple Regression ModelRelationship between 1 dependent & 2 or more

independent variables is a linear function

Dependent (Response)

variable for sample

Independent (Explanatory)

variables for sample model

1 2i i i k ki iY X X X

Residual

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Oil (Gal) Temp Insulation Di

275.30 40 3 0.0094

363.80 27 3 0.0098

164.30 40 10 0.0496

40.80 73 6 0.0041

94.30 64 6 0.0001230.90 34 6 0.0295

366.70 9 6 0.1342

300.60 8 10 0.1328

237.80 23 10 0.0001

121.40 63 3 0.3083

31.40 65 10 0.1342203.50 41 6 0.0094

441.10 21 3 0.4941

323.00 38 3 0.0824

52.50 58 10 0.0062

Interpretation of Coefficient of

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Interpretation of Coefficient ofMultiple Determination

96.56% of the total variation in heating oil can be

explained by different temperature and amount ofinsulation

95.99% of the total fluctuation in heating oil canbe explained by different temperature and amountof insulation after adjusting for the number ofexplanatory variables and sample size

2

,12 .9656YSSR

rSST

2

adj .9599r

C ffi i t f M lti l

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Coefficient of MultipleDetermination

Regression Statistics

M u lt ip le R 0 .9 8 2 6 5 4 7 5 7

R S q u a re 0 .9 6 5 6 1 0 3 7 1

A d ju s t e d R S q u a re 0 . 9 5 9 8 7 8 7 6 6

S ta n d a rd E rro r 2 6 . 0 1 3 7 8 3 2 3

O b s e rva t io n s 1 5

Excel Output

SST

SSRr ,Y 2

12

Adjusted r2

reflects the number

of explanatory

variables and sample

size

is smaller than r2

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Testing for Overall Significance

Shows if there is a linear relationship betweenall of the X variables together and Y

Use F test statistic Hypotheses:

H0: k=0 (no linear relationship)

H1: at least one i ( at least one independent

variable affects Y)

The null hypothesis is a very strong statement

Almost always reject the null hypothesis

T t f O ll Si ifi

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Test for Overall SignificanceExample Solution

F0 3.89

H0:1=2= =p=0

H1: At least one i0 = .05

df = 2 and 12Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Reject at = 0.05

There is evidence that atleast one independentvariable affects Y

= 0.05

F 168.47(Excel Output)

T t f Si ifi

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Test for Significance:Individual Variables

Shows if there is a linear relationship betweenthe variable Xiand Y

Use t test statistic Hypotheses:

H0: i 0 (no linear relationship)

H1: i 0 (linear relationship between Xiand Y)

t T t St ti ti

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t Test StatisticExcel Output: Example

Coefficients Standard Error t StatIntercept 562.1510092 21.09310433 26.65093769

X Variable 1 -5.436580588 0.336216167 -16.16989642

X Variable 2 -20.01232067 2.342505227 -8.543127434

tTest Statistic for X1(Temperature)

tTest Statistic for X2(Insulation)

i

i

b

bt

S

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t Test : Example Solution

H0: 1 = 0

H1: 1 0

df = 12

Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Reject H0 at = 0.05

There is evidence of asignificant effect oftemperature on oilconsumption.

t0 2.1788-2.1788

.025

Reject H0

Reject H0

.025

Does temperature have a significant effect on monthly

consumption of heating oil? Test at = 0.05.

tTest Statistic = -16.1699

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