Day 14: Assumptions of Regression Analysis in SPSS – Ensuring Valid Models

Day 14: Assumptions of Regression Analysis in SPSS – Ensuring Valid Models

Welcome to Day 14 of your 50-day SPSS learning journey! Today, we’ll focus on the assumptions of regression analysis, which are critical for ensuring that your models produce valid and reliable results. Ignoring these assumptions can lead to misleading conclusions, so let’s dive into what they are and how to test them in SPSS.

What Are the Assumptions of Regression?

Regression analysis relies on several key assumptions:

  1. Linearity: The relationship between the independent and dependent variables is linear.
  2. Normality of Residuals: The residuals (errors) are normally distributed.
  3. Homoscedasticity: The variance of residuals is constant across all levels of the independent variable(s).
  4. No Multicollinearity: Independent variables are not highly correlated with each other.
  5. Independence of Residuals: Residuals are independent of each other (for time-series or repeated measures data).

Why Are These Assumptions Important?

Violating these assumptions can lead to:

  • Biased estimates of regression coefficients.
  • Misleading significance tests.
  • Poor predictive performance of the model.

By testing and addressing these assumptions, you can ensure your model is valid and interpretable.

How to Test Assumptions in SPSS

1. Linearity

Linearity assumes that the relationship between the predictors and the outcome is linear.

Test: Create scatterplots of the dependent variable against each independent variable.

  1. Go to Graphs > Chart Builder.
  2. Drag Scatter/Dot to the preview pane.
  3. Add the independent variable to the x-axis and the dependent variable to the y-axis.
  4. Look for a linear pattern in the scatterplot (a straight-line trend).

2. Normality of Residuals

Residuals (the differences between observed and predicted values) should follow a normal distribution.

Test:

  1. After running your regression model, save the residuals:
    • Go to Analyze > Regression > Linear.
    • Click Save and select Unstandardized Residuals.
  2. Check normality:
    • Go to Analyze > Descriptive Statistics > Explore.
    • Add the saved residuals to the Dependent List.
    • Click Plots and select Normality Plots with Tests.
    • Check the histogram and the Shapiro-Wilk Test in the output.
    • If p > 0.05, residuals are normally distributed.

3. Homoscedasticity

Homoscedasticity means that residuals have constant variance across all levels of the independent variable(s).

Test:

  1. After running your regression model, save the residuals and predicted values.
  2. Create a scatterplot:
    • Go to Graphs > Chart Builder.
    • Plot the residuals on the y-axis and predicted values on the x-axis.
    • Look for random scatter (no cone-shaped pattern).

4. No Multicollinearity

Multicollinearity occurs when independent variables are highly correlated, making it difficult to estimate their individual effects.

Test:

  1. Run the regression model:
    • Go to Analyze > Regression > Linear.
    • Click Statistics and check Collinearity Diagnostics.
  2. Check the VIF (Variance Inflation Factor) in the Coefficients table:
    • If VIF > 10, multicollinearity may be a concern.

5. Independence of Residuals

Residuals should be independent of each other, especially in time-series data.

Test:

  1. Use the Durbin-Watson Test:
    • After running your regression model, check the Durbin-Watson statistic in the Model Summary table.
    • A value between 1.5 and 2.5 indicates independence.

Example: Testing Assumptions

Use the following dataset:

ID Hours_Studied Attendance Test_Score
1 2 60 50
2 4 70 60
3 6 80 70
4 8 90 80
5 10 95 90
  1. Run a Multiple Regression with Test_Score as the dependent variable and Hours_Studied and Attendance as the predictors.
  2. Test for each assumption:
    • Create scatterplots for linearity.
    • Test normality of residuals with the Shapiro-Wilk Test.
    • Check homoscedasticity using residual vs. predicted value plots.
    • Verify multicollinearity with VIF values.
    • Use the Durbin-Watson statistic to test independence of residuals.

How to Handle Violations of Assumptions

If any assumptions are violated:

  1. Linearity: Transform variables (e.g., log, square root) or use polynomial regression.
  2. Normality of Residuals: Use non-parametric tests (e.g., Spearman’s correlation or rank regression).
  3. Homoscedasticity: Transform the dependent variable (e.g., log transformation).
  4. Multicollinearity: Remove highly correlated predictors or use techniques like Principal Component Analysis (PCA).
  5. Independence of Residuals: Use time-series models (e.g., ARIMA) for dependent data.

Common Mistakes to Avoid

  1. Ignoring Assumptions: Always test assumptions before interpreting results.
  2. Relying Solely on Visuals: Combine visual inspection with statistical tests for accuracy.
  3. Overlooking Multicollinearity: Ignoring multicollinearity can lead to unreliable coefficient estimates.

Key Takeaways

  • Valid regression analysis depends on meeting key assumptions.
  • Use scatterplots, normality tests, and diagnostic statistics to test assumptions.
  • Address violations with transformations, alternate models, or removing problematic predictors.

What’s Next?

In Day 15 of your 50-day SPSS learning journey, we’ll explore Data Transformation Techniques in SPSS. You’ll learn how to create new variables, recode data, and apply transformations to improve your analysis. Stay tuned for more hands-on learning!


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