LESSON 4.1. MULTIPLE LINEAR REGRESSION

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LESSON 4.1.MULTIPLE LINEAR REGRESSION
Design and Data Analysis in Psychology
II Salvador Chacón Moscoso Susana Sanduvete
Chaves
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1. INTRODUCTION
x1
x2
Y
x3
xK
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1. INTRODUCTION

• Model components
• More than one independent variable (X)
• Qualitative.
• Quantitative.
• A quantitative dependent variable (Y).
• Example
• X1 educative level.
• X2 economic level.
• X3 personality characteristics.
• X4 gender.
• Y drug dependence level.

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1. INTRODUCTION

• Reasons why it is interesting to increase the
  simple linear regression model
• Human behavior is complex (multiple regression is
  more realistic).
• It increases statistical power (probability of
  rejecting null hypothesis and taking a good
  decision).

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1. INTRODUCTION

• Regression equation
• Raw scores

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1. INTRODUCTION

• Regression equation
• Deviation scores
• Standard scores

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2. ASSUMPTIONS

1. Linearity.
2. Independence of errors
3. Homoscedasticity the variances are constant.
4. Normality the punctuations are distributed in a
   normal way.
5. The predictor variables cannot correlate
   perfectly between them.

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3. PROPERTIES
The errors do not correlate with the predictor
variables or the predicted scores.
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4. INTERPRETATION
Example 1 quantitative variables
X1
0.48
Maternal stimulation
0.01
Y
X2
3-year-old development level
6-year-old development level
0.62
X3
b020.8
Paternal stimulation
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4. INTERPRETATION
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4. INTERPRETATION
Example 2 quantitative and qualitative variables
X1
0.157
Emotional tiredness
-0.7
Y
X2
Gender 0woman 1man
Stress symptoms
b01.987
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4. INTERPRETATION
The same slope, different constant parallel
lines
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4. INTERPRETATION
Example 3 two qualitative variables
X1
-0.915
Gender 0woman 1man
-0.096
Y
X2
Work 0public 1 private
Stress symptoms
b05.206
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4. INTERPRETATION

• Women, public organization
• Women, private organization
• Men, public organization
• Men, private organization

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5. COMPONENTS OF VARIATION

• SSTOTAL SSEXPLAINED SSRESIDUAL

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6. GOODNESS OF FITCOEFFICIENT OF DETERMINATION

• 2 possibilities
• r12 0
• b) r12 ? 0

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6. GOODNESS OF FITCOEFFICIENT OF DETERMINATION

• r12 0

Y
b
a
X1
X2
X2
X1
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6. GOODNESS OF FITCOEFFICIENT OF DETERMINATION

1. r12 ? 0

Y
b
a
c
X1
X2
X2
X1
(the area c would be summed twice)
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6. GOODNESS OF FITCOEFFICIENT OF DETERMINATION
Semi partial correlation coefficient square
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7. MODEL VALIDATION
Sources of variation Sums of squares df Variances F
Regression or explained k
Residual or unexplained N-k-1
Total N-1
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7. MODEL VALIDATION

• ? Null hypothesis
  is rejected. The variables are related. The model
  is valid.
• ? Null hypothesis
  is accepted. The variables are not related. The
  model is not valid.
• (k number of independent variables)

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7. MODEL VALIDATION EXAMPLE

• A linear regression equation was estimated in
  order to study the possible relationship between
  the level of familiar cohesion (Y) and the
  variables gender (X1) and time working outside,
  instead at home (X2). Some of the most relevant
  results were the following

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7. MODEL VALIDATION EXAMPLE
Sources of variation Sums of squares df Variances F Sig.
Regression or explained 436.580 2 218.290 14.898 0.000
Residual or unexplained 1142.852 78 14.652 14.898 0.000
Total 1579.432 80
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7. MODEL VALIDATION EXAMPLE

1. Which is the proportion of unexplained
   variability by the model?
2. Can the model be considered valid? Justify your
   answer (a0.05).

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7. MODEL VALIDATION EXAMPLE

• Which is the proportion of unexplained
  variability by the model?
• 2. Can the model be considered valid? Justify
  your answer (a0.05).
• Yes, because the significance (sig.) is lower to
  a0.05.

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8. SIGNIFICANCE OF REGRESSION PARAMETERS

• Statistic

In SPSS it is called standard error (error
típico)
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8. SIGNIFICANCE OF REGRESSION PARAMETERS

• 3. Comparison and conclusions (for each
  independent variable)
• ? Null hypothesis is
  rejected. The slope is statistically different to
  0. As a conclusion, there is relationship between
  variables. It is recommended to maintain the
  variable as part of the model.
• ? Null hypothesis is
  accepted. The slope is statistically equal to 0.
  As a conclusion, there is not relationship
  between variables. It is recommended to remove
  the variable from the model.

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8. SIGNIFICANCE OF REGRESSION PARAMETERS EXAMPLE

• We studied the relationship between the variables
  nationality (0 Moroccan, 1 Filipino) and gender
  (0man, 1woman) with the variable depression in
  a 148-participant sample. We know that F is equal
  to 8.889, and the values obtained in the
  following table

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8. SIGNIFICANCE OF REGRESSION PARAMETERS EXAMPLE
Non-standard coefficients Non-standard coefficients Standard c.
B Stand. error Beta t Sig.
(Constant) ? 1.194 15.391 0.000
Gender ? 1.519 0.242 2.944 0.004
Nationality ? 1.495 0.31 -3.768 0.000

1. Calculate R2.
2. Calculate the regression equation in raw scores.
3. Would you remove any variable from the model?
   Justify your answer (a0.05).

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8. SIGNIFICANCE OF REGRESSION PARAMETERS EXAMPLE

• Calculate R2.

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8. SIGNIFICANCE OF REGRESSION PARAMETERS EXAMPLE

• 2. Calculate the regression equation in raw
  scores.

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8. SIGNIFICANCE OF REGRESSION PARAMETERS EXAMPLE

• 3. Would you remove any variable from the model?
  Justify your answer (a0.05).
• No, because the t of the three parameters present
  a significance (sig.) lower than a0.05