Correlation & Linear Regression in SPSS

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1 Petra Petrovics Correlation & Linear Regression in SPSS 4 th seminar

2 Types of dependence association between two nominal data mixed between a nominal and a ratio data correlation among ratio data

3 Correlation describes the strength of a relationship, the degree to which one variable is linearly related to another X (or X 1, X 2,, X p ): known variable(s) / independent variable(s) / predictor(s) Y: unknown variable / dependent variable causal relationship: X causes Y to change Regression shows us how to determine the nature of a relationship between two or more variables

4 Correlation Measures 1. Covariance 2. Coefficient of correlation 3. Coefficient of determination 4. Coefficient of rank correlation

5 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. C x, y x x y 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!!!

6 2. Coefficient of correlation r C s s x y = Σd d x 2 x d d y 2 y 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!

7 3. Coefficient of determination r 2 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. r 2 S S yˆ y =1- S S e y

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

9 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! + -

10 Output Mean Std. Deviation N Current Salary $34, $17, Beginning Salary $17, $7, Current Salary Beginning Salary Current Salary Beginning Salary Pearson Correlation 1,880(**) Sig. (2-tailed),000 Sum of Squares and Cross-products , ,73 Covariance , ,27 N Pearson Correlation,880(**) 1 Sig. (2-tailed),000 Sum of Squares and Cross-products , ,45 Covariance , ,96 N

11 Exercise 2 Multiple Correlation Is there any relation between the current salary previous experience (month) month since hire beginning salary? MULTIPLE CORRELATION

12 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! + -

13 Output View Inverse relationship & weak dependence Direct relationship & strong dependence Current Salary Previous Experience (months) Months since Hire Beginning Salary r C 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 *. Correlation is significant at the 0.05 level (2-tailed). Correlations **. Correlation is significant at the 0.01 level (2-tailed). Matrix Previous Experience Months Beginning Current Salary (months) since Hire Salary 1 -,097*,084,880**,034,067,000 1,379E , ,5 5,59E , , , ,097* 1,003,045,034,948, , , , , ,281 3, , ,084, ,020,067,948, , , , , ,823 3, , , ,880**,045 -,020 1,000,327, , ,5 2,93E , , , Inverse relationship Direct relationship

14 Linear regression y ŷ = b 0 + b 1 x b 1 : for every 1 unit increase in x we expect y to change by b 1 units on average b 0 : when x=0, y=b 0 x

15 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!

16 Transform / Compute Variable Create a new variable: age = this year date of birth (in year) This year

17 Regression Analyze / Regression / Linear

18 Model 1 Model Summary Adjusted Std. Error of R R Square R Square the Estimate,146 a,021,019 $16, a. Predictors: (Constant), age Multiple correlation coefficient Adjusted multiple determination coefficient R r 2 y1 r 2 y2 2r 1 r y r y2 r It expresses the combined effect of all the variables acting on the dependent variable Weak dependence 12 R 2 n 1 1 (1 R n p 1 Multiple determination coefficient It enables to compare the How many percent of the multiple determination variation of the dependent coefficient among variable can be explained by populations / samples the variation of all the with different size and independent variables different number of dependent variables as it The dependent variable s control for the number of (current salary) variation is sample / population size explained in 2,1% by the (n) and the number of regression model independent variables (p) 2 )

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

20 Model 1 b 0 b 1 (Constant) age Unstandardized Coefficients a. Dependent Variable: Current Salary The regression line: ŷ = b 0 + b 1 x Coefficients a b 0 : If the x variable is 0, how much is the y. Standardized Coefficients B Std. Error Beta t Sig , ,686 17,613, ,609 66,124 -,146-3,200,001 If the employees are 0-year-old, they earn $41543,805 (It doesn t mean anything.) b 1 : If the x increases by 1 unit, what is the difference in y. We can accept the parameters at every significance level. If the employees are 1 year older, they earn less money with $211,609 on average.

21 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?

22 Analyze / Regression / Curve Estimation Linear Compound Power To get a chart

23 Output View Linear Model Summary R R Square Adjusted R Square Std. Error of the Estimate,146,021, ,804 The independent variable is age. Model Summary Compound The highest R 2 Power R R Square Adjusted R Square Std. Error of the Estimate,215,046,044,389 The independent variable is age. Model Summary R R Square Adjusted R Square Std. Error of the Estimate,156,024,022,393 The independent variable is age.

24 Also in the Output View Faculty of Economics

25 Weak dependence. The age has 4,6% influence on the current salary s variation The model is significant.

26 b a ŷ = a b x = x a: no analyzation The parameters are significant. b: When an employee is 1 year older, the current salary will be times higher on average.

28 Thank You for Your Attention

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