REGRESI LINIER SEDERHANA KULIAH #2 ANALISIS REGRESI Usman Bustaman
Feb 24, 2016
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 LEBIH LANJUT…- Linier dalam paramater…- Persamaan Linier orde 1:- Persamaan Linier orde 2:- Dst… (orde pangkat tertinggi yang terdapat pada
variabel bebasnya)
SEDERHANARelasi 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
FUNGSI RATA-2 (Mean Function)
If you know something about X, this knowledge helps you predict something about Y.
REGRESI TERBAIK = MINIMISASI ERROR- Semua residual harus nol- Minimum Jumlah residual
- Minimum jumlah absolut residual
- Minimum versi Tshebysheff
- Minimum jumlah kuadrat residual OLS
ASSUMPTIONSLinear 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
KOEFISIEN REGRESIXbYb 10 =
21x
xy
xx
xy
SS
b
==
nX
X =nY
Y = 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
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
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
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 toVarianceXby explained Variance2 ==
SSTSSRR
Coefficient of Determinationto judge the adequacy of the regression model
Maknanya: …. ?
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
SSS
r
rRR
==
== 2
Correlationmeasures the strength of the linear association between two
variables.
Pearson Correlation…?
Buktikan…!
ASSUMPTIONSLinear 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 RLSLinear 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
Selang Kepercayaan koefisien regresi
xx
ekn
xx
ekn S
StbSStb
2
)1(,2/11
2
)1(,2/1 + b
Confidence Interval for b1