Correlation & Linear Regression in SPSS

• Faculty of Economics •

Gazdaságelméleti és Módszertani Intézet

Correlation & Linear Regression in SPSS 4th seminar

Petra Petrovics



Types of dependence • association – between two nominal data

• mixed – between a nominal and a ratio data • correlation – among ratio data



Correlation describes the strength of a relationship, the degree to which one variable is linearly related to another

Regression shows us how to determine the nature of a relationship between two or more variables

• X (or X1, X2, … , Xp): known variable(s) / independent variable(s) / predictor(s) • Y: unknown variable / dependent variable • causal relationship: X „causes” Y to change



Correlation Measures 1. Covariance 2. Coefficient of correlation

3. Coefficient of determination 4. Coefficient of rank correlation



1. Covariance • A measure of the joint variation of the two variables; • An average value of the product of the deviations of observations on 2 random variables from their sample means.

x  x    y  y   Cx, y   n 1

– – – –

ranges from -  to +; C = 0, when X and Y are uncorrelated; its sign shows the direction of correlation it doesn’t measure the degree of relationship!!!



2. Coefficient of correlation Σd x d y C r = s xs y d 2xd 2y

• • • • •

Pearson correlation A measure of how closely related two data series are. Its sign shows the direction of correlation It measures the strength of correlation 0 < r  < 1  statistical dependence r = 0  X and Y are uncorrelated  r = -1  negative ☻ r = 1  positive ☺ • You can use only in case of linear relationship!



3. Coefficient of determination • r2 • The square of the sample correlation coefficient between the outcomes and their predicted values. • Measures the degree of correlation in percentage (%) • It provides a measure of how well future outcomes are likely to be predicted by the model. • Vary from 0 to 1. 2

r 

S yˆ Sy

=1-

Se Sy



Exercise 1 - Correlation File / Open / Employee data.sav Is there any relation between - current salary & - beginning salary? CORRELATION



Analyze / Correlate / Bivariate… 0  I r I  0,3 weak dependence 0,3  I r I  0,7 medium-strong dependence

0,7  I r I 1  strong dependence

r

Shows direction and strength

C Just direction! +

-



Output Current Salary Beginning Salary

Mean Std. Deviation $34,419.57 $17,075.661 $17,016.09 $7,870.638

N 474 474 Beginning Salary

Current Salary Current Salary

Beginning Salary

Pearson Correlation

Sig. (2-tailed) Sum of Squares and Cross-products Covariance N Pearson Correlation Sig. (2-tailed) Sum of Squares and Cross-products Covariance N

1

,880(**) ,000 55948605047,73 118284577,27 474 1

137916495436,340 291578214,45 474 ,880(**) ,000 55948605047,73 29300904965,45 118284577,27 474

61946944,96 474



Exercise 2 – Multiple Correlation Is there any relation between

• the current salary • previous experience (month)

• month since hire • beginning salary? MULTIPLE CORRELATION



Analyze / Correlate / Bivariate… 0  I r I  0,3 weak dependence 0,3  I r I  0,7 medium-strong dependence 0,7  I r I 1  strong dependence

C

Just direction!

r

Shows direction and strength

+

-



Output View Correlations

Current Salary

r

C Inverse relationship & weak dependence

Direct relationship & strong dependence

Previous Experience (months)

Months since Hire

Beginning Salary

Pearson Correlation Sig. (2-tailed) Sum of Squares and Cross-products Covariance N Pearson Correlation Sig. (2-tailed) Sum of Squares and Cross-products Covariance N Pearson Correlation Sig. (2-tailed) Sum of Squares and Cross-products Covariance N Pearson Correlation Sig. (2-tailed) Sum of Squares and Cross-products Covariance N

Matrix Previous Experience (months) -,097* ,034

Months since Hire ,084 ,067

Beginning Salary ,880** ,000

1,379E+011

-82332343,5

6833347,5

5,59E+010

291578214,5 474 -,097* ,034

-174064,151 474 1

14446,823 474 ,003 ,948

118284577 474 ,045 ,327

-82332343,54

5173806,810

1482,241

17573777

-174064,151 474 ,084 ,067

10938,281 474 ,003 ,948

3,134 474 1

37153,862 474 -,020 ,668

6833347,489

1482,241

47878,295

-739866,50

3,134 474 ,045 ,327

101,223 474 -,020 ,668

-1564,200 474 1

55948605048

17573776,7

-739866,5

2,93E+010

118284577,3 474

37153,862 474

-1564,200 474

61946945 474

Current Salary 1

14446,823 474 ,880** ,000

*. Correlation is significant at the 0.05 level (2-tailed). **. Correlation is significant at the 0.01 level (2-tailed).

Inverse relationship Direct relationship



Linear regression y ŷ = b0 + b 1 x

b1: for every 1 unit increase in x we expect y to change by b1 units on average b0: when x=0, y=b0 x



Exercise 3 – Linear Regression File / Open / Employee data.sav Determine a linear relationship between the salary and the age of the employees!

 Create a new variable!



Transform / Compute Variable…

Create a new variable: age = this year – date of birth (in year)

This year



Regression

Analyze / Regression / Linear…



Model Summary Model 1

R ,146a

R Square ,021

Adjusted R Square ,019

Std. Error of the Estimate $16,928.804

a. Predictors: (Constant), age

Multiple correlation coefficient R

Adjusted multiple determination coefficient

ry21  ry22  2ry1  ry 2  r12 1  r122

R 2  1

Multiple determination coefficient

It expresses the combined effect of all the variables acting on the dependent variable

How many percent of the variation of the dependent variable can be explained by the variation of all the independent variables

Weak dependence

The dependent variable’s (current salary) variation is explained in 2,1% by the regression model

n 1  (1  R 2 ) n  p 1

It enables to compare the multiple determination coefficient among populations / samples with different size and different number of dependent variables as it control for the number of sample / population size (n) and the number of independent variables (p)



F-test: for model testing

We can accept the model in every significance level. The F ratio (in the Analysis of Variance Table) is 10.241 and significant at p=.001. This provides evidence of existence of a linear relationship between the variables


Gazdaságelméleti és Módszertani Intézet Coefficientsa

Model b0 (Constant) 1 b1 age

Unstandardized Coefficients B Std. Error 41543,805 2358,686 -211,609 66,124

Standardized Coefficients Beta -,146

t 17,613 -3,200

Sig. ,000 ,001

a. Dependent Variable: Current Salary

The regression line: ŷ = b0 + b1x

We can accept the parameters at every significance level.

b0: If the x variable is 0, how much is the y.

 If the employees are 0-year-old, they earn $41543,805 (It doesn’t mean anything.) b1: If the x increases by 1 unit, what is the difference in y.  If the employees are 1 year older, they earn less money with $211,609 on average.



Exercise 4 – Curve Estimation File / Open / Employee data.sav Determine the relationship between the salary and the age of the employees! Which regression model fit the most?



Analyze / Regression / Curve Estimation…

• Linear • Compound • Power

To get a chart



Output View Model Summary

Linear

R ,146

R Square ,021


Std. Error of the Estimate 16928,804

The independent variable is age. Model Summary

Compound

R ,215

R Square ,046


Std. Error of the Estimate ,389

The independent variable is age.

The highest R2

Power

Model Summary R ,156

R Square ,024


The independent variable is age.

Std. Error of the Estimate ,393

• Faculty of Economics • Also in the Output View…




• Weak dependence. • The age has 4,6% influence on the current salary’s variation

The model is significant.



b a

ŷ = a  bx = 40482.362  0.993x a: no analyzation

The parameters are significant.

b: When an employee is 1 year older, the current salary will be 0.993 times higher on average.





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Correlation & Linear Regression in SPSS

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