Quant: Linear Regression in SPSS

The aim of this analysis is to look at the relationship between a father’s education level (dependent variable) when you know the mother’s education level (independent variable). The variable names are “paeduc” and “maeduc.” Thus, the hope is to determine the linear regression equation for predicting the father’s education level from the mother’s education.

Introduction

The aim of this analysis is to look at the relationship between a father’s education level (dependent variable) when you know the mother’s education level (independent variable). The variable names are “paeduc” and “maeduc.” Thus, the hope is to determine the linear regression equation for predicting the father’s education level from the mother’s education.

From the SPSS outputs the following questions will be addressed:

  • How much of the total variance have you accounted for with the equation?
  • Based upon your equation, what level of education would you predict for the father when the mother has 16 years of education?

Methodology

For this project, the gss.sav file is loaded into SPSS (GSS, n.d.).  The goal is to look at the relationships between the following variables: paeduc (HIGHEST YEAR SCHOOL COMPLETED, FATHER) and maeduc (HIGHEST YEAR SCHOOL COMPLETED, MOTHER). To conduct a linear regression analysis navigate through Analyze > Regression > Linear Regression.  The variable paeduc was placed in the “Dependent List” box, and maeduc was placed under “Independent(s)” box.  The procedures for this analysis are provided in video tutorial form by Miller (n.d.).  The following output was observed in the next four tables.

The relationship between paeduc and maeduc are plotted in a scatterplot by using the chart builder.  Code to run the chart builder code is shown in the code section, and the resulting image is shown in the results section.

Results

Table 1: Variables Entered/Removed

Model Variables Entered Variables Removed Method
1 HIGHEST YEAR SCHOOL COMPLETED, MOTHERb . Enter
a. Dependent Variable: HIGHEST YEAR SCHOOL COMPLETED, FATHER
b. All requested variables entered.

Table 1, reports that for the linear regression analysis the dependent variable is the highest years of school completed for the father and the independent variable is the highest year of school completed by the mother.  No variables were removed.

Table 2: Model Summary

Model R R Square Adjusted R Square Std. Error of the Estimate
1 .639a .408 .407 3.162
a. Predictors: (Constant), HIGHEST YEAR SCHOOL COMPLETED, MOTHER
b. Dependent Variable: HIGHEST YEAR SCHOOL COMPLETED, FATHER

For a linear regression trying to predict the father’s highest year of school completed based on his wife’s highest year of school completed, the correlation is positive with a value of 0.639, which can only 0.408 of the variance explained (Table 2) and 0.582 of the variance is unexplained.  The linear regression formula or line of best fit (Table 4) is: y = 0.76 x + (2.572 years) + e.  The line of best fit essentially explains in equation form the mathematical relationship between two variables and in this case the father’s and mother’s highest education level.  Thus, if the mother has completed her bachelors’ degree (16th year), then this equation would yield (y = 2.572 years + 0.76 (16 years) + e = 14.732 years + e).  The e is the error in this prediction formula, and it exists because of the r2 value is not exactly -1.0 or +1.0.  The ANOVA table (Table 3) describes that this relationship between these two variables is statistically significant at the 0.05 level.

Table 3: ANOVA Table

Model Sum of Squares df Mean Square F Sig.
1 Regression 6231.521 1 6231.521 623.457 .000b
Residual 9045.579 905 9.995
Total 15277.100 906
a. Dependent Variable: HIGHEST YEAR SCHOOL COMPLETED, FATHER
b. Predictors: (Constant), HIGHEST YEAR SCHOOL COMPLETED, MOTHER

Table 4: Coefficients

Model Unstandardized Coefficients Standardized Coefficients t Sig.
B Std. Error Beta
1 (Constant) 2.572 .367 7.009 .000
HIGHEST YEAR SCHOOL COMPLETED, MOTHER .760 .030 .639 24.969 .000
a. Dependent Variable: HIGHEST YEAR SCHOOL COMPLETED, FATHER

The image below (Figure 1), is a scatter plot, which is plotting the highest year of school completed by the mother vs. the father along with the linear regression line (Table 4) and box plot images of each respective distribution.  There are more outliers in the husband’s education level compared to those of the wife’s education level, and the spread of the education level is more concentrated about the median for the husband’s education level.

u4db1f1.png

Figure 1: Highest year of school completed by the mother vs the father scatter plot with regression line and box plot images of each respective distribution.

Conclusion

There is a statistically significant relation between the husband’s and wife’s highest year of education completed.  The line of best-fit formula shows a moderately positive correlation and is defined as y = 0.76 x + (2.572 years) + e; which can only explain 40.8% of the variance, while 58.2% of the variance is unexplained.

SPSS Code

DATASET NAME DataSet1 WINDOW=FRONT.

REGRESSION

  /MISSING LISTWISE

  /STATISTICS COEFF OUTS R ANOVA

  /CRITERIA=PIN(.05) POUT(.10)

  /NOORIGIN

  /DEPENDENT paeduc

  /METHOD=ENTER maeduc

  /CASEWISE PLOT(ZRESID) OUTLIERS(3).

STATS REGRESS PLOT YVARS=paeduc XVARS=maeduc

/OPTIONS CATEGORICAL=BARS GROUP=1 BOXPLOTS INDENT=15 YSCALE=75

/FITLINES LINEAR APPLYTO=TOTAL.

References:

Quant: Regression and Correlations

Top management of a large company has told you that they really would like to be able to determine what the impact of years of service at their company has on workers’ productivity levels, and they would like to be able to predict potential productivity based upon years of service. The company has data on all of its employees and has been using a valid productivity measure that assesses each employee’s productivity. You have told management that there is a possible way to do that.

Through a regression analysis, it should be possible to predict the potential productivity based upon years of service, depending on two factors: (1) that the productivity assessment tool is valid and reliable (Creswell, 2014) and (2) we have a large enough sample size to conduct our analysis and be able to draw statistical inference of the population based on the sample data which has been collected (Huck, 2011). Assuming these two conditions are met, then regression analysis could be made on the data to create a prediction formula. Regression formulas are useful for summarizing the relationship between the variables in question (Huck, 2011). There are multiple types of regression all of them are tests of prediction: Linear, Multiple, Log-Linear, Quadratic, Cubic, etc. (Huck, 2011; Schumacker, 2014).  The linear regression is the most well-known because it uses basic algebra, a straight line, and the Pearson correlation coefficient to aid in stating the regression’s prediction strength (Huck, 2011; Schumacker, 2014).  The linear regression formula is: y = a + bx + e, where y is the dependent variable (in this case the productivity measure), x is the independent variable (years of service), a (the intercept) and b (the regression weight) are a constants that are to be defined through the regression analysis, and e is the regression prediction error (Field, 2013; Schumacker, 2014).  The sum of the errors should be equal to zero (Schumacker, 2014).

Linear regression models try to describe the relationship between one dependent and one independent variable, which are measured at the ratios or interval level (Schumacker, 2014).  However, other regression models are tested to find the best regression fit over the data.  Even though these are different regression tests, the goal for each regression model is to try to describe the current relationship between the dependent variable and the independent variable(s) and for predicting.  Multiple regression is used when there are multiple independent variables (Huck, 2011; Schumacker, 2014). Log-Linear Regression is using a categorical or continuously independent variable (Schumacker, 2014). Quadratic and Cubic regressions use a quadratic and cubic formula to help predict trends that are quadratic or cubic in nature respectively (Field, 2013).  When modeling predict potential productivity based upon years of service the regression with the strongest correlation will be used as it is that regression formula that explains the variance between the variables the best.   However, just because the regression formula can predict some or most of the variance between the variables, it will never imply causation (Field, 2013).

Correlations help define the strength of the regression formula in defining the relationships between the variables, and can vary in value from -1 to +1.  The closer the correlation coefficient is to -1 or +1; it informs the researcher that the regression formula is a good predictor of the variance between the variables.  The closer the correlation coefficient is to zero, indicates that there is hardly any relationship between the variable (Field, 2013; Huck, 2011; Schumacker, 2014).  A negative correlation could show that as the years of service increases the productivity measured is decreased, which could be caused by apathy or some other factor that has yet to be measured.  A positive correlation could show that as the years of service increases the productivity also measured increases, which could also be influenced by other factors that are not directly related to the years of service.  Thus, correlation doesn’t imply causation, but can help determine the percentage of the variances between the variables by the regression formula result, when the correlation value is squared (r2) (Field, 2013).

References

  • Creswell, J. W. (2014) Research design: Qualitative, quantitative and mixed method approaches (4th ed.). California, SAGE Publications, Inc. VitalBook file.
  • Field, A. (2013) Discovering Statistics Using IBM SPSS Statistics (4th ed.). UK: Sage Publications Ltd. VitalBook file.
  • Huck, S. W. (2011) Reading Statistics and Research (6th ed.). Pearson Learning Solutions. VitalBook file.
  • Schumacker, R. E. (2014) Learning statistics using R. California, SAGE Publications, Inc, VitalBook file.