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Simple and Multiple Regression - Cognitive Psychology - Lecture Notes, Study notes of Cognitive Psychology

English and Foreign Languages UniversityCognitive Psychology

Simple and Multiple Regression, Simple Linear Regression, Variance, Mean, Ordinary Least Squares, Standardization of Variables, Original Regression, Colinearity, Nonlinearity are points from this lecture notes.

Typology: Study notes

2011/2012

Uploaded on 11/21/2012

ashakiran
ashakiran 🇮🇳

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Ch. 2: Simple and Multiple Regression
I. Simple Linear Regression
A. ,
ii bXaY +=
ˆ
iiiii ebXaeYY ++=+= ˆ
B. Mean of Y
E(Y) = E(a + bX + e) = a + bE(X) + E(e)
If E(e) = 0, then E(Y) = a + bE(X)
C. Variance of Y
Var(Y) = Var(Y) + Var(e) + 2Cov(Ye)
ˆ ˆ
= Var(Y) + Var(e) + 2bCov(Xe)
ˆ
D. If we force the line to pass through the mean of X, the mean of Y, and the center of
the cluster, then we are forcing the covariance between X and e to be near zero.
Thus, Var(Y) = Var(Y) + Var(e).
ˆ
Since Var(Y) = b
ˆ2Var(X), Var(Y) = b2Var(X) + Var(e).
E. Ordinary Least Squares
The regression can be established by the following formulae.
b = )(
)(
XVar
XYCov , and
a = E(Y) - bE(X)
II. Standardization of variables
A. If we standardize both X and Y, then a = 0 – 0 = 0, and
xxy ZZZ
β
β
=
+
=0
ˆ.
B. Original regression
Y = Y + e
ˆ
= a + bX + e
If manipulate the equation a bit (subtracting y
μ
from both side and adding
xx bb
μ
μ
−+ into the right side), then
Y - y
μ
= a + b x
μ
- y
μ
+ bX - b x
μ
+ e
= (a + b x
μ
- y
μ
) + b(X - x
μ
) + e * y
μ
= a + b x
μ
= b(X - x
μ
) + e (intercept is zero)
C. If we divide both sides by y
σ
and insert xx
σ
σ
/ into the right side, then
y
y
Y
σ
μ
− =
yxy
xx e
Xb
σσσ
μσ
+
−)(
y
Z = yxyx eZb
σ
σ
σ
/)/(
+
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Download Simple and Multiple Regression - Cognitive Psychology - Lecture Notes and more Study notes Cognitive Psychology in PDF only on Docsity!

Ch. 2: Simple and Multiple Regression I. Simple Linear Regression A. Y ˆ i = a + bXi , Yi = Yi + ei = a + bXi + e i

B. Mean of Y E(Y) = E(a + bX + e) = a + bE(X) + E(e) If E(e) = 0, then E(Y) = a + bE(X) C. Variance of Y Var(Y) = Var( Y ˆ^ ) + Var(e) + 2Cov( Y ˆ e) = Var( Y ˆ ) + Var(e) + 2bCov(Xe) D. If we force the line to pass through the mean of X, the mean of Y, and the center of the cluster, then we are forcing the covariance between X and e to be near zero. Thus, Var(Y) = Var( Y ˆ ) + Var(e). Since Var( Y ˆ^ ) = b^2 Var(X), Var(Y) = b^2 Var(X) + Var(e). E. Ordinary Least Squares The regression can be established by the following formulae. b = ( )

Var X Cov XY , and a = E(Y) - bE(X) II. Standardization of variables A. If we standardize both X and Y, then a = 0 – 0 = 0, and

Zy ˆ = 0 + β Z x = β Z x.

B. Original regression Y = Y ˆ + e = a + bX + e

If manipulate the equation a bit (subtracting μ y from both side and adding

  • b μ (^) x − b μ x into the right side), then

Y - μ y = a + b μ x - μ y + bX - b μ x + e

= (a + b μ x - μ y ) + b(X - μ x ) + e * μ y = a + b μ x

= b(X - μ^ x ) + e (intercept is zero)

C. If we divide both sides by σ y and insert σ x / σ x into the right side, then

y Y y

y x y b x X x e

Z y = ( b σ x / σ y ) Zx + e / σ y

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Thus, Z y ˆ = ( b σ x / σ y ) Zx and β = b σ^ x /^ σ^ y.

D. If both X and Y are standardized, b = β = r = Cov(XY).

III. Multiple Regression A. Y ˆ^ = a + b 1 X 1 + b 2 X 2 +... + bkXk B. Y = Y ˆ^ + e = a + b 1 X 1 + b 2 X 2 +... + bkXk + e C. Dummy variables for testing two regression lines Y = a + b 1 X 1 + b 2 X 2 + b 3 Z 1 + e where Z 1 can take the value of 1 or 0 for different groups. D. Colinearity

  1. Assume X 1 and X 2 are perfectly correlated. Y ˆ^ = a + b 1 X 1 + b 2 X 2 can be rewritten as Y ˆ^ = a + (b 1 + b 2 )X. It will give us a common slope value for X, which is b 1 + b 2.
  2. Thus, there is no unique solution for the values of b 1 and b 2 if X 1 and X 2 are perfectly correlated.
  3. If two variables are highly correlated, it will be hard to accurately estimate both b 1 and b 2. E. Interaction
  4. The magnitude of the effect of one variable on another is different depending on the third variable value.
  5. Y ˆ^ = a + b 1 X 1 + b 2 X 2 + b 3 X 1 X 2 F. Nonlinearity (Curve-Linearity)
  6. Y ˆ^ = a + b 1 X 1 + b 2 X 1 2
  7. Nonlinear transformation can be done as a form of logX, 1/X, or (^) X.

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