*145*

When two variables have a linear relationship, you can often useÂ **simple linear regression**Â to quantify their relationship.

However, simple linear regression doesnâ€™t work well when two variables have a non-linear relationship. In these cases, you can try usingÂ **quadratic regression**.Â

This tutorial explains how to perform quadratic regression in SPSS.

**Example: Quadratic Regression in SPSS**

Suppose we are interested in understanding the relationship between number of hours worked and happiness. We have the following data on the number of hours worked per week and the reported happiness level (on a scale of 0-100) for 16 different people:

Use the following steps to perform a quadratic regression in SPSS.

**Step 1: Visualize the data.**

Before we perform quadratic regression, letâ€™s make a scatterplot to visualize the relationship between hours worked and happiness to verify that the two variables actually have a quadratic relationship.

Click theÂ **GraphsÂ **tab, thenÂ **Chart Builder**:

In the new window that pops up, chooseÂ **Scatter/DotÂ **in theÂ **Choose fromÂ **list. Then drag the chart titledÂ **Simple ScatterÂ **into the main editing window. Drag the variable **hoursÂ **onto the x-axis and **happinessÂ **onto the y-axis. Then clickÂ **OK**.

The following scatterplot will appear:

We can clearly see that a non-linear relationship exists between hours worked and happiness. This tells us that quadratic regression is an appropriate technique to use in this situation.

**Step 2: Create a new variable.**

Before we can perform quadratic regression, we need to create a predictor variable for hours^{2}.

Click theÂ **TransformÂ **tab, thenÂ **Compute variable**:

In the new window that pops up, name the target variableÂ **hours2Â **and define it asÂ **hours*hours**:

Once you clickÂ **OK**, the variable **hours2Â **will appear in a new column:

**Step 3: Perform quadratic regression.**

Next, we will perform quadratic regression. Click on theÂ **AnalyzeÂ **tab, thenÂ **Regression**, thenÂ **Linear**:

In the new window that pops up, dragÂ **happinessÂ **into the boxed labeled Dependent. DragÂ **hours** andÂ **hours2** into the box labeled Independent(s). Then clickÂ **OK**.

**Step 4: Interpret the results.**

Once you clickÂ **OK**, the results of the quadratic regression will appear in a new window.

The first table weâ€™re interested in is titledÂ **Model Summary**:

Here is how to interpret the most relevant numbers in this table:

**R Square:Â**This is the proportion of the variance in the response variable that can be explained by the explanatory variables. In this example,Â**90.9%**Â of the variation in happiness can be explained by the variablesÂ**hoursÂ**andÂ**hours**.^{2}**Std. Error of the Estimate:Â**TheÂ standard errorÂ is the average distance that the observed values fall from the regression line. In this example,Â the observed values fall an average of**9.519Â**units from the regression line.

The next table weâ€™re interested in is titledÂ **ANOVA**:

Here is how to interpret the most relevant numbers in this table:

**F:Â**This is the overall F statistic for the regression model, calculated as Mean Square Regression / Mean Square Residual.**Sig:Â**This is the p-value associated with the overall F statistic. It tells us whether or not the regression model as a whole is statistically significant. In this case the p-value is equal to 0.000, which indicates that the explanatory variables**hours**andÂ**hours**Â combined have a statistically significant association withÂ exam score.^{2}

The next table weâ€™re interested in is titledÂ **Coefficients**:

We can use the values in the columnÂ **Unstandardized BÂ **to form the estimated regression equation for this dataset:

Estimated happiness level =Â -30.253 + 7.173*(hours) â€“ .107*(hours^{2})

We can use this equation to find the estimated happiness level for an individual based on the number of hours they work per week. For example, an individual that works 60 hours per week is expected to have a happiness level of 14.97:

Estimated happiness level =Â -30.253 + 7.173*(60) â€“ .107*(60^{2}) = **14.97**.

Conversely, an individual that works 30 hours perkÂ week is predicted to have a happiness level of 88.65:

Estimated happiness level =Â -30.253 + 7.173*(30) â€“ .107*(30^{2}) = **88.65**.

**Step 5: Report the results.**

Lastly, we want to report the results of our quadratic regression. Here is an example of how to do so:

A quadratic regression was performed to quantify the relationship between the number of hours worked by an individual and their corresponding happiness level (measured from 0 to 100). A sample of 16 individuals was used in the analysis.

Â

Results showed that there was a statistically significant relationship between the explanatory variablesÂ

hoursÂandÂhoursand the response variableÂ^{2Â }happinessÂ (F(2, 13) = 65.095, pÂ

Combined, these two explanatory variables accounted for 90.9% of variability in happiness.Â

Â

The regression equation was found to be:

Â

Estimated happiness levelÂ =Â -30.253 + 7.173(hours) â€“ .107(hours

^{2})