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multiple regression analysis
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REGRESSION ANALYSIS
-P H SHAMEER
• An introduction to regression model
• Performing it on SPSS
INTRODUCTION
• What is regression model?An explanatory methodForecast expressed as a function of a
certain no. of variables that influences its outcome
2 types of variables
1. DEPENDENT
-which we want to forecast
2. INDEPENDENT
-or predictor variables
• Eg:
• Predict how much an individual enjoys his/her job
• Dependent variable: job satisfaction
• Independent variables:
salary, academic qualification, age, sex,
no. of years, socio-economic status….
assumptions
1. LINEAR RELATIONSHIP exists
2. HOMOSCEDASTICITY exists
3. Residuals are INDEPENDENT of one another
4. MULTICOLLINEARITY doesn’t exist
Analysis for Linearity
Not Linear Linear
x x
Y
x
Y
x
Residual Analysis for Homoscedasticity
Non-constant variance Constant variance
x x
Y
x x
Y
resi
dua
ls
resi
dua
ls
SCATTER PLOTS• -helps to visualize, graphically the
relationship between pairs of variables
Regression Equation
where
a is y intercept
&
b1, b2,..bi are regression coefficients
1 1 2 2' i iy a b x b x b x
How a & b can be calculated?
• Method of least squares
this method determines the values in such a way that the sum of squared deviations (errors) is minimized
and hence the name least squares
b=(∑x*y/n) ─ (x * y)
( ∑x2 / n) ─ (x)2
a = y- bx where y = ∑y/n
x= ∑x/n
n is the no. of observations
forecasting
• Once the relationship is determined , it can be used to make any no. of forecasts simply by inserting the values of X’s
• y = a+b1x1+b2x2+…+bixi
• Caution: the basic relationship should be assessed periodically
terminology
b - standard regression coefficient: Measure of how strongly each predictor
variable influences the dependent variableE.g.: if b=2.5
change of one standard deviation in the predictor will change 2.5 standard deviations in the forecasting variable
terminology
RMeasure of correlation between observed
& predicted value of the dependent variable
R -1 t0 1R= n*∑xi*yi-∑xi*∑yi
√(n∑xi2- (∑xi)2) √(n∑yi
2- (∑yi)2)
Scatter Plots of Data with Various Correlation Coefficients
Y
X
Y
X
Y
X
Y
X
Y
X
r = -1 r = -.6 r = 0
r = +.3r = +1
Y
Xr = 0
Slide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall
terminology…..
R2
variation in Y accounted for by the set of predictors
Measure of how good a forecasting of dep. variable by knowing the independent variables.
When applied to reality, R2 over estimate the success
terminology…Adjusted R2
The adjustment takes into account the size of the sample and number of predictors
Gives most useful measure of success of our model ( goodness of fit)
R2 range:0 to 1.If R2=0.75, success will be 75%
Is each X contributing to the prediction of Y?
• Test if each regression coefficient is significantly different than zero given the variables standard error.
– T-test for each regression coefficient
Performing regression in spss
Eg:importance of several psycholinguistic variables on spelling performance
variables
Independent:
standardized spelling score(spellsc), chronological age(age), reading age(readage), standardized reading score(standsc)
Dependent variable:
percentage correct spelling(spelperc)
Performing regression in spss
• SPPS=Statistical Packages in Social Sciences
Enter the data
Cont..
>Analyze>regression> lineardialogue box appears
now enter dependent and independent variables
Selection methods:on relative contribution of independent
variables
1. simultaneous/ enter method
2. Hierarchical method
3. Statistical methods
a. Forward
b. Backward
c. Stepwise
d. Remove
Now click the statistics button
Now click ‘continue’> then ‘ok’
Output:
Cont…
• Here reading age is not a significant predictor
result:
percentage correct spelling=
-232+.406*chronological age
+.394*standardized reading score
+.786*standardized spelling score
references
Forecasting methods for management
by Spyros Makridas & Steven C Wheelwright
SPSS for psychologists
by Nicola Brace, Richard Kemp & Rosemary Snelger
Research Methods for M.Com
by L.R Potti
THANKYOU…