Simple Linear Regression (OLS). Types of Correlation Positive correlationNegative correlationNo...

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Transcript of Simple Linear Regression (OLS). Types of Correlation Positive correlationNegative correlationNo...
Simple linear regression describes the linear relationship between an independentvariable, plotted on the xaxis, and a dependent variable, plotted on the yaxis
Independent Variable (X)
depe
nden
t Var
iabl
e (Y
)
Least Squares Regression
Residual (ε) =
Sum of squares of residuals =
Model line:
• we must find values of and that minimise o 1
XY 10
YY
2)( YY
2)(min YY
Descriptive Statistics
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2
n
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xxS
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Regression Statistics
SST
SSRR 2
Coefficient of Determinationto judge the adequacy of the regression model
Regression Statistics
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xy
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xy
SS
SR
RR
2
Correlation
measures the strength of the linear association between two variables.
Standard Error for the regression model
MSES
n
SSES
SS
e
e
ee
2
2
22
2
Regression Statistics
2)( YYSSE
ANOVA
df SS MS F Pvalue
Regression 1 SSR SSR / df MSR / MSE P(F)
Residual n2 SSE SSE / df
Total n1 SST
If P(F)< then we know that we get significantly better prediction of Y from the regression model than by just predicting mean of Y.
ANOVA to test significance of regression
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1
10
AH
H
Hypotheses Test the Correlation Coefficient
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AH
H
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2
R
nRT
We would reject the null hypothesis if 2,2/0 ntt
Diagnostic Tests For Regressions
i
Expected distribution of residuals for a linear model with normal distribution or residuals (errors).
iY
Regression – important points
1. Ensure that the range of valuessampled for the predictor variableis large enough to capture the fullrange to responses by the responsevariable.
Regression – important points
2. Ensure that the distribution ofpredictor values is approximatelyuniform within the sampled range.
Assumptions of Regression
1. The linear model correctly describes the functional relationship between X and Y.
Assumptions of Regression
1. The linear model correctly describes the functional relationship between X and Y.
Y
X
Assumptions of Regression
3. For any given value of X, the sampled Y values are independent
4. Residuals (errors) are normally distributed.
5. Variances are constant along the regression line.
The linear model with a singlepredictor variable X can easily be extended to two or more predictor variables.
1 1 2 2 ...o p pY X X X
Y
X1
Variance NOT explained by X1 and X2
Unique variance explained by X1
Unique variance explained by X2
X2
Common variance explained by X1 and X2
Partial Regression Coefficients (slopes): Regression coefficient of X after controlling for (holding all other predictors constant) influence of other variables from both X and Y.
1 1 2 2 ...o p pY X X X
Partial Regression Coefficients
intercept residuals
The matrix algebra of
Ordinary Least Square
1( ' ) 'X X X Y Predicted Values:
Residuals:
Intercept and Slopes:
XY
YY
Regression Statistics
SST
SSRR 2
Coefficient of Determinationto judge the adequacy of the regression model
Adjusted R2 are not biased!
n = sample sizek = number of independent variables
)1(1
11 22 R
kn
nRadj
Regression Statistics
Standard Error for the regression model
MSES
kn
SSES
SS
e
e
ee
2
2
22
1
Regression Statistics
2)( YYSSE
ANOVA
df SS MS F Pvalue
Regression k SSR SSR / df MSR / MSE P(F)
Residual nk1 SSE SSE / df
Total n1 SST
If P(F)< then we know that we get significantly better prediction of Y from the regression model than by just predicting mean of Y.
ANOVA to test significance of regression
0:
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iA
k
H
H
at least one!
Hypotheses Tests for Regression Coefficients
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ikn
CS
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bS
bt
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i
H
H
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S
S 2
Diagnostic Tests For Regressions
i
Expected distribution of residuals for a linear model with normal distribution or residuals (errors).
iX
X Residual Plot
5
0
5
10
0 2 4 6 8
XR
esid
uals
 Forward selectionThe ‘best’ predictor variables are entered, one by one.
 Backward eliminationThe ‘worst’ predictor variables are eliminated, one by one.
Model Selection
Model Selection: The General Case
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Reject H0 if :
The degree of correlation between Xs.
A high degree of multicolinearity produces unacceptable uncertainty (large variance) in regression coefficient estimates (i.e., large sampling variation)
Imprecise estimates of slopes and even the signs of the coefficients may be misleading.
ttests which fail to reveal significant factors.
Multicolinearity
Multicolinearity
If the Ftest for significance of regression is significant, but tests on the individual regression coefficients are not, multicolinearity may be present.
Variance Inflation Factors (VIFs) are very useful measures of multicolinearity. If any VIF exceed 5, multicolinearity is a problem.
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VIF
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