Linear Regression #

Linear regression is a statistical method used to study the relationship between two or more variables. In simple linear regression, we explore the linear relationship between an independent variable (X) and a dependent variable (Y). The model estimates the regression coefficients using the least squares method, which are then used to predict the value of the dependent variable.

Simple Linear Regression Analysis #

The core of simple linear regression analysis is modeling through the linear equation ( Y = \beta_0 + \beta_1 X + \epsilon ), where:

• ( Y ) is the dependent variable (the variable to be predicted)
• ( X ) is the independent variable (the variable used to predict the dependent variable)
• ( \beta_0 ) is the intercept, representing the value of ( Y ) when ( X = 0 )
• ( \beta_1 ) is the regression coefficient, representing the change in ( Y ) when ( X ) increases by one unit
• ( \epsilon ) is the error term

Practical Operation: Steps in SPSS #

Suppose we have a dataset containing the independent variable Hours_Studied (study hours) and the dependent variable Test_Score (test scores). We wish to build a linear regression model to predict the test scores.

Steps:

Step 1: Data Preparation

1. Open SPSS and go to Data View.
2. Enter the data, with variables as follows:
   • Hours_Studied (independent variable): Study hours
   • Test_Score (dependent variable): Test scores Example data:

   Hours_Studied	Test_Score
   1	55
   2	60
   3	65
   4	70
   5	75
   6	80
   7	85

Step 2: Conduct Regression Analysis

1. Click on the Analyze menu, select Regression, and then click Linear.
2. In the pop-up dialog:
   • Move Test_Score (dependent variable) to the Dependent box.
   • Move Hours_Studied (independent variable) to the Independent(s) box.
3. If desired, click the Statistics button to select options such as Descriptive Statistics and Confidence Intervals to obtain more analysis information.
4. Click OK to run the regression analysis in SPSS, and the output results will be generated.

Step 3: View and Interpret the Output Results

The key results in the SPSS output include:

• Model Summary: This table provides the goodness of fit for the regression model. Important is R², which tells us the percentage of variance in the dependent variable explained by the independent variable.

  Example output:

  Model	R	R²	Adjusted R²	Std. Error of the Estimate
  1	0.98	0.96	0.95	2.00

  Here, R² = 0.96 means that 96% of the variance in the test scores can be explained by study hours.

• ANOVA Table: This table tests the overall significance of the regression model. If the Sig. value is less than 0.05, the model is statistically significant.

  Example output:

  Model	Sum of Squares	df	Mean Square	F	Sig.
  1	300.00	1	300.00	50.00	0.000

  Sig. = 0.000 indicates that the model is statistically significant.

• Coefficients Table: This table provides the coefficients of the regression equation, including the intercept and slope.

  Example output:

  Predictor	B	Std. Error	Beta	t	Sig.
  (Constant)	50.00	2.00		25.00	0.000
  Hours Studied	5.00	0.50	0.98	10.00	0.000

  • Intercept: 50.00, meaning that if a student studies for 0 hours, the predicted test score is 50.
  • Slope: 5.00, meaning that for every additional hour of study, the test score increases by 5 points.

Step 4: Make Predictions Using the Regression Equation

Based on the regression equation:

[ \text{Test Score} = 50.00 + 5.00 \times (\text{Hours Studied}) ]

we can predict test scores for different study hours.

For example, if a student studies for 8 hours:

[ \text{Test Score} = 50.00 + 5.00 \times 8 = 90 ]

The predicted test score for this student is 90.

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Summary

1. To perform simple linear regression in SPSS, first prepare the data.
2. Use Analyze → Regression → Linear to run the regression analysis.
3. Focus on the R², ANOVA Table, and Coefficients Table in the output.
4. Use the regression equation to make predictions.

Through these steps, we can perform simple linear regression analysis in SPSS and use the model for predictions.

#SPSS

Last modified on 2024-01-06