Week 8 Responses QR Part 1

Please respond to my colleague post separately:

Colleague post #1: Listed below are my responses to this weeks discussion questions. I look forward to your responses and questions.

A polynomial regression analysis is a way to describe a curved relationship instead of a straight line relationship (Pati & Pati, 2021). In research, this could help explain the patterns when the line is not straight (Pati & Pati, 2021). The benefit of a polynomial regression analysis is that it helps the researcher to understand the relationship between the independent and dependent variables when the pattern does not follow a straight line (Pati & Pati, 2021). An example of this could be stress. While some stress has been shown to be beneficial in a persons life, too much stress can have an adverse effect. A polynomial regression analysis could catch this effect, while a linear regression analysis would miss it.

What is the difference between a linear, quadratic, and cubic regression analysis?

A linear regression analysis shows the relationship in a straight line (Pati & Pati, 2021). An example would be getting from point A to point B. If a person wants to get a higher test grade in a subject they are not good at, they have to spend more time studying to understand the material better.

A quadratic regression analysis shows the relationship in a u shape curve (Faith, 2024, para.1). The u can be up or down. An example would be a student who did not study for an exam and fails the exam. The student studies for the next exam and passes. In an attempt to not fail any more exams, the student starts studying excessively, becomes tired, oversleeps, and misses the exam.

A cubic regression analysis shows the relationship between the independent and the dependent variables in a up and down motion (Statology, 2025). It can go up, down, and back up again (Wisdom Library, 2025). An example would be a student wanting to do well on a midterm exam. The person studies and memorizes the material to pass the exam. After the exam, the student forgets the information. Because the final is cumulative in the students class, the student has to study the information again to pass the exam.

When looking at the SPSS output, how do you know what is the best fitting model?

From what I have learned in this class, it is the one that tells the most based on the data in the simplest fashion.

What values do you need from the SPSS output in order to report the findings in the results section in APA style?

The f value tells you if the model mattered. The p-value for f tells you if the f value was significant. The R2 value tells you the percentage. The B tells you the direction of the effect (positive or negative). The t value helps to determine whether the effect is meaningful by measuring how far away the effect is from 0, and the p value here tells the researcher whether the effect was unlikely to have occurred due to chance.

Course Reflection

Let me start with, holy cow, was this class intense! Please do not misunderstand me, I did learn, but wow. On that note, there were two parts of this class that I enjoyed. The first one was learning about mediators and moderators. That was beyond interesting and showed me that statistics can show the relationship between the independent and dependent variables and how something occurred. The second was using SPSS on a regular basis. The more I used this software, the better I got, and the more equations I had to create, the more comfortable I became interpreting them.

References

Faith, C. (2024). Quadratic Regression Model. Retrieved March 3, 2026, from

https://www.researchgate.net/publication/385492279_Quadratic_Regression_Model

Patil, S., & Patil, S. (2021). Linear with polynomial regression: Overview. International Journal of Applied Research, 7(8),

273-275.

. (2025). Cubic Regression in Excel (Step-by-Step). Retrieved March 3, 2026, from

Wisdom Library. org. (2025). Significance of Cubic Regression. Retrieved March 3, 2026, from

https://www.wisdomlib.org/concept/cubic-regression#:~:text=Cubic%20regression%2C%20as%20

defined%20in,in%20a%20non%2Dlinear%20fashion.

Colleague post 2: Polynomial regression analysis is a statistical technique used when the relationship between variables follows a curved pattern rather than a straight-line relationship. In many research situations, a simple linear regression is insufficient because the association between variables varies across levels of the independent variable. McDonald (2014) explains that curvilinear regression is appropriate when the plotted data suggest that the relationship between the predictor and outcome variables is curvilinear rather than linear. In these situations, polynomial regression allows researchers to include higher-order terms such as squared (X2) or cubed (X3) versions of the independent variable so that the model can capture nonlinear patterns in the data.

One of the major benefits of polynomial regression models is their ability to more accurately represent complex relationships that cannot be explained by linear models alone. When the true relationship between variables is nonlinear, forcing a linear model onto the data can lead to inaccurate interpretations and weaker predictive power. By incorporating polynomial terms into the regression equation, researchers can better approximate the curvature of the data and improve model fit. McDonald (2014) notes that the goodness of fit for a regression model is typically evaluated using the coefficient of determination, R2, which represents the proportion of variance in the dependent variable explained by the model. As additional polynomial terms are introduced, the model often captures additional variance because the equation becomes more flexible in describing the relationship between variables.

The primary distinction between linear, quadratic, and cubic regression models lies in the number of polynomial terms included and the shape of the relationship they represent. A linear regression model includes only the first-order term (X), producing a straight-line relationship between variables. A quadratic regression model includes an additional squared term (X2), which produces a parabolic curve in which the slope changes direction once. In contrast, a cubic regression model includes both squared and cubed terms (X2 and X3), allowing the slope to change direction twice and creating an S-shaped pattern in the data. Wuensch (2015) demonstrates that these higher-order models are evaluated sequentially, beginning with a linear model and then adding quadratic and cubic terms to determine whether each additional component significantly improves the models explanatory power.

When evaluating SPSS output to determine the best-fitting model, researchers typically examine the change in R2 when higher-order terms are added to the regression equation. Wuensch (2015) explains that although adding polynomial terms will almost always increase R2, the key question is whether the increase in explained variance is statistically significant and theoretically meaningful. In practice, the preferred model is the simplest one that provides a statistically significant improvement in fit while avoiding unnecessary complexity.

To report polynomial regression results in APA style, several values must be extracted from the SPSS output. These typically include the overall R2 value, the F statistic with its associated degrees of freedom, the significance level (p-value), and the regression coefficients for each term included in the final model. Reporting these statistics allows researchers to clearly communicate the strength of the relationship, the statistical significance of the model, and the contribution of each polynomial component to the overall equation. Together, these statistics provide the evidence needed to justify the selection of the final regression model.

Colleague post #3: Regression analysis using polynomials (or polynomial regression) is an appropriate method of analysis when there exists a non-linear relationship between two variables. Most real world data, especially in the field of Psychology, do not follow a straight-line relationship between variables. The relationship could be curvilinear or even reverse direction depending on the level of the predictor variable. By allowing for polynomial terms, specifically second-order and third-order, of the predictor variable, polynomial regression provides the ability to accurately represent the curvature of the relationship in the data rather than forcing the relationship into a straight-line model (McDonald, 2014).

There are three basic types of polynomial regression: Linear, Quadratic and Cubic. The type of polynomial regression used depends on the number of polynomial terms that are included in the equation. For example, a linear regression equation will only include the original predictor variable and assume a straight-line relationship. A quadratic regression equation will include both the original predictor variable and its square, thus allowing the regression equation to create one bend (such as a U-shape or an upside-down U-shape). A cubic regression equation will include the original predictor variable and its first and second-order terms, allowing for two bends or reversals in the relationship. To determine the most appropriate polynomial regression equation, researchers will compare the model fit of the different equations based upon the addition of the new terms. Model fit is often evaluated by determining if the additional terms add to the overall R-squared value of the model (i.e., the percentage of variance explained) and whether the additional terms are statistically significant, as indicated by an increase in R-squared of the model and/or a low probability value associated with the F-statistic (Wuensch, 2015).

When reporting polynomial regression findings in APA format, researchers need to include various statistics obtained from their SPSS output. Statistics typically required include the regression coefficient (B) and the standard error of each predictor term; the t-value and probability value (p-value) for each predictor term; and the overall model fit statistics, including the F-statistic and the degrees of freedom; the probability value associated with the F-statistic; the R-squared value (indicating the proportion of variance in the outcome variable that is explained by the predictors); and the degrees of freedom for the R-squared value. Overall, I found this course to have been very challenging but very helpful as it provided me with a greater appreciation for the relationship between statistical concepts and actual data analysis. Analyzing the SPSS output and interpreting the polynomial regression models has increased my confidence in understanding regression results and how they are presented in research studies.

References:

McDonald, J. H. (2014). Handbook of biological statistics (3rd ed.). Sparky House Publishing.

Wuensch, K. L. (2015). Polynomial regression with SPSS. East Carolina University.