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Linear Regression
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Slide 2
Aims
Understand linear regression with one predictor
Understand how we assess the fit of a regressionmodel
Total Sum of Squares Model Sum of Squares
Residual Sum of Squares
F
R
2
Know how to do Regression on PASW/SPSS
Interpret a regression model
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Slide 3
What is Regression?
A way of predicting the value of one variable
from another.
It is a hypothetical model of the relationship
between two variables.
The model used is a linear one.
Therefore, we describe the relationship using the
equation of a straight line.
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Slide 4
Describing a Straight Line
bi
Regression coefficient for the predictor Gradient (slope) of the regression line
Direction/Strength of Relationship
b0
Intercept (value of Y when X = 0)
Point at which the regression line crosses the Y-
axis (ordinate)
iii XbbY 10
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Intercepts and Gradients
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The Method of Least Squares
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How Good is the Model?
The regression line is only a model
based on the data.
This model might not reflect reality.
We need some way of testing how well
the model fits the observed data.
How?
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Slide 8
Sums of Squares
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Summary
SST Total variability (variability between scores and the mean).
SSR Residual/Error variability (variability between the
regression model and the actual data).
SSM
Model variability (difference in variability between the
model and the mean).
2
T iSS y y
2
pR i iSS y y
2
pM iSS y y
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Testing the Model: ANOVA
If the model results in better predictionthan using the mean, then we expect SSMto
be much greater than SSR
SSR
Error in Model
SSM
Improvement Due to the Model
SST
Total Variance In The Data
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Slide 11
Testing the Model: ANOVA
Mean Squared Error
Sums of Squares are total values.
They can be expressed as averages.
These are called Mean Squares, MS
R
MMS
MS
F
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Slide 12
Testing the Model: R2
R2
The proportion of variance accounted for by the
regression model.
The Pearson Correlation Coefficient Squared
T
M
SSSSR 2
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Regression: An Example
A record company boss was interested in
predicting record sales from advertising.
Data
200 different album releases
Outcome variable:
Sales (CDs and Downloads) in the week after release
Predictor variable:
The amount (in s) spent promoting the record before
release.
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Step One: Graph the Data
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Regression Using PASW/SPSS
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Output: Model Summary
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Output: ANOVA
SSM
SSR
SST
MSR
MSM
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Slide 18
SPSS Output: Model Parameters
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Using The Model
iii bb
BudgetgAdvertisin09612.014.134
BudgetgAdvertisinSalesRecord 10
75.143
10009612.014.134
BudgetgAdvertisin09612.014.134SalesRecord
ii