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14.2.5 Correlation and Regression Analysis with SPSS During this tutorial you will learn how to use SPSS to investigate the association between two continuous variables and how to describe it graphically 2.5.1 In this practical session you will analyse some bivariate data. To enter the data: File / Open worksheet / Merlin4 (saved in the ANOVA Worksheet, 14.2.4) In order to see what this file contains: Analyse / Descriptive Statistics / Descriptives Select all the variables but not the filters. Descriptiv e Statistics N Minimum Maximum Mean Std. Deviation Ages of Employees 60 17 64 37.75 11.037 Sex of Employee 60 1 2 1.50 .504 Job category 60 1 4 2.27 1.039 Valid N (listwise) 60 We have three variables. Each is measured on 60 cases There are no missing values. We give each of the employees a salary and then see if it is associated with their age. In Variable View, in the first empty row name a new variable SALARY and label it Annual salary (£000). Type the following in Data View in one column: (Work down these columns one after the other) 38.1 38.9 23.2 22.9 19.8 19.7 15.6 31.7 17.3 37.8 18.7 42.8 19.6 47.5 31.3 8.5 28.5 14.1 33.5 32.9 42.3 60.1 15.5 15.8 59.3 15.9 37.3 20.3 13.7 9.8 25.9 60.7 20.7 35.9 33.8 39.3 32.9 19.8 6.4 15.2 53.6 75.2 40.2 25.3 24.5 14.5 23.9 28.2 35.3 8.7 37.6 10.9 28.5 63.2 32.0 10.2 8.6 25.3 12.5 38.4 The variables of interest in this practical session are the continuous variables SALARY and AGE. We shall investigate the relationship between the Salaries earned by the employees of Merlin and their ages. 2.5.2 Save revised datafile as Merlin5 2.5.3 Produce a scatter plot Graphs / Scatter / Define as Simple/ Select SALARY for Y and AGE for X. Give your graph a suitable title. 1 Salary v Age 80.0 Annual salary (£'000) 60.0 40.0 20.0 0.0 0 20 40 60 80 Age Examine the plot. You should find it does suggest a rather poor linear relationship. Does there appear to be a relationship? Yes but not strong Have a guess as to whether this is likely to be significant or not. . . . . . Yes. . . . . . . 2.5.4 Calculate the correlation coefficient. Analyse / Correlate / Bivariate Select SALARY and AGE as the variables. Correlations Annual salary (£'000) Age Annual salary (£'000) Pearson Correlation 1 .398** Sig. (2-tailed) .002 N 60 60 Age Pearson Correlation .398** 1 Sig. (2-tailed) .002 N 60 60 **. Correlation is significant at the 0.01 level (2-tailed). What is the value of the correlation coefficient?. . . . . . . . . . . . . . . 0.398 . . . . . . . . . What is the probability of it being zero? . . . . . . . . . . . . . . . .0.002 . .. . . . . . . . Is this significant at 5%? . . . . . . . . . . . . . . .Yes . . . . . . . . . . If the p value is less than 0.05 the correlation coefficient is significant at the 5% level of significance. 2 2.5.5 Find the regression equation: Analyse / Regression / Linear/ Select SALARY as Dependent, AGE as Independent. Use Method Enter. a Coe fficients Unstandardized Standardized Coefficients Coefficients Model B Std. Error Beta t Sig. 1 (Constant) 7.728 6.593 1.172 .246 Age .554 .168 .398 3.306 .002 a. Dependent Variable: Annual salary (£'000) The regression equation, as produced by SPSS, is not at all obvious. In the Coefficients table, under unstandardised coefficients and in the column under B you will find the constant, a, and the coefficient of Age, b. Write down the regression equation y = 7.73 + 0.554x. . . . . . . . . . . . . . 2.5.6 Produce the regression line on your scatterplot. Graphs / Scatter / Define as simple / Select SALARY for Y and AGE for X and OK Double click on the graph to get into editing mode. Click on the points to select them. Chart / Add chart element / Fit line at tot al Salary v Age 80.0 Annual salary (£'000) 60.0 40.0 20.0 R Sq Linear = 0.159 0.0 0 20 40 60 80 Age 3 2.5.7 Carry out residual analysis: (This could have been added at step 5) Analyse / Regression / Linear / Select SALARY as dependent and AGE as independent variables. Select Plots / Standardised residual plots / histogram. Select Save / Residuals / Unstandardised a Residuals Statistics Minimum Maximum Mean Std. Deviation N Predicted Value 17.154 43.216 28.660 6.1200 60 Residual -25.0975 37.5294 .0000 14.0992 60 Std. Predicted Value -1.880 2.378 .000 1.000 60 Std. Residual -1.765 2.639 .000 .991 60 a. Dependent Variable: Annual salary (£'000) Histogram Dependent Variable: Annual salary (£'000) 12 10 Frequency 8 6 4 2 Mean = -6.94E-18 0 Std. Dev. = 0.991 -2 -1 0 1 2 3 N = 60 Regression Standardized Residual You should see that your residuals appear reasonably normal on the histogram . To see if the mean is zero and the standard deviation low: Analyse / Descriptive Statistics / Descriptives / Select the SALARY and UNSTANDARDISED RESIDUALS. Descriptive Statistics N Minimum Maximum Mean Std. Deviation Annual salary (£'000) 60 6.4 75.2 28.660 15.3701 Unstandardized Residual 60 -25.09753 37.52944 .0000000 14.09917166 Valid N (listwise) 60 You should see that the mean of the residuals is zero. The standard deviation has been reduced (but not by much for these data). In the next three tasks we look at the males and females separately. 4 2.5.8 Produce the two regression lines on your scatterplot. Graphs / Scatter / Define as simple Select SALARY for Y and AGE for X and Set marker by SEX. OK Double click on the graph to get into editing mode. Click on the male points to select them (easiest to do in the legend). Chart / Add chart element / Fit line at total Repeat for the female points Salary v Age 80.0 Sex of Employee Male Female Annual salary (£'000) 60.0 40.0 20.0 R Sq Linear = 0.083 R Sq Linear = 0.233 0.0 0 20 40 60 80 Age . Describe the general differences between the male and the female salaries. . . . . . . . .Male salaries start higher and rise more quickly than do those for females . . . 2.5.9 In the Data editor: Data / Select cases / If condition is satisfied / If Sex = 1. Continue. Leave unselected cases as Filtered. OK Repeat tasks 2.5.4 and, if the correlation coefficient is significant, task 2.5.5 for the males only. 5 Correlations for M ale s only Annual salary (£'000) Age Annual salary (£'000) Pearson Correlation 1 .483** Sig. (2-tailed) .007 N 30 30 Age Pearson Correlation .483** 1 Sig. (2-tailed) .007 N 30 30 **. Correlation is significant at the 0.01 level (2-tailed). a Coe fficients Unstandardized Standardized Coefficients Coefficients Model B Std. Error Beta t Sig. 1 (Constant) 6.880 10.157 .677 .504 Age .734 .252 .483 2.917 .007 a. Dependent Variable: Annual salary (£'000) Write down the correlation coefficient and the regression equation, if appropriate.. . . . . . . . . . . . . . . 0.483. . . . . . . y = 6.88 + 0.734x . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.10In the Data editor: Data / Select cases / All cases. Then If condition is satisfied / If Sex = 2. Leave unselected cases as Filtered. Repeat tasks 2.5.4 and, if the correlation coefficient is significant, task 2.5.5 for the females only. Correlations for fe males only Annual salary (£'000) Age Annual salary (£'000) Pearson Correlation 1 .288 Sig. (2-tailed) .123 N 30 30 Age Pearson Correlation .288 1 Sig. (2-tailed) .123 N 30 30 Write down the correlation coefficient and the regression equation, if appropriate. . . . . . . . . .0.288 . . . . . . Not significant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

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