Testing For Correlation By Linear Regression

Testing for Correlation by Linear Regression

Study 1brain Weightbody Weight33854450481551358146542336331195

Study 1brain Weightbody Weight33854450481551358146542336331195

Study 1 Brain Weight Body Weight 3.....35 8.......04 5......92 5......5 10...7 6.......3 25.......5 3.......01 0..4 12..5 12......55 2.....28 1.......9 2.......4 Question: Is there a correlation between brain weight and body weight across mammalian species? References: Spaeth, H. (1991). Mathematical algorithms for linear regression. New York, NY: Academic Press. Weisberg, S. (1980). Applied linear regression. New York, NY: John Wiley & Sons. Study 2 Water Temperature Length of Fish Question: Is there a correlation between water temperature and the length that fish grow to? Study 3 Age Systolic Blood Pressure Question: Is there a correlation between age and systolic blood pressure? Example Chirps/Second Temperature (F) ...8 93..4 84..1 80..5 75..7 69..7 71..4 69..3 83...2 82....4 76.3 Question: Is there a correlation between the number of times a cricket chirps per second and air temperature? Example Assignment 2: Testing for Correlation by Linear Regression The goal of many studies is to determine if there is a relationship between factors. In other words, does one factor influence the outcome of another factor? If there is a relationship between the factors, then there is a correlation. Through this module’s lectures and readings, you will know that finding a correlation does not necessarily mean that you have found a causal relationship. This would need to be determined by another layer of investigation. Indeed, many times correlation does not always lead to the determination of causation, but it can help to identify if there is not a causal relationship between the variables in the study. One way to determine correlation is to see if there is a linear relationship between the factors. A linear relationship can be tested by graphing a scatter plot of the data in the study and seeing if a best-fit line can be drawn to represent this data. This method of analysis is called linear regression . The formulas for linear regression are cumbersome, but luckily, most spreadsheets have built-in functions for performing these tedious calculations. In this assignment, you will use a spreadsheet to examine pairs of variables, using the method of linear regressions, to determine if there is any correlation between the variables. Afterwards, postulate whether this correlation reveals a causal relationship—why or why not? Directions: Click here to open the Excel spreadsheet containing the data for this assignment. Notice that there are several tabs on the spreadsheet, each containing a different set of data from different studies. On each of these sample tabs, you will also find the question that was explored in that study. Select the data set that you find interesting, and perform the analysis below. You are only required to perform this analysis on one set of data. There is a tab labeled Example where you can see how your analysis should look when done. In the Excel spreadsheet, perform the following operations: 1. Save the spreadsheet on your computer. 2. Select the study data you want to use. With your mouse, highlight all of the data on the spreadsheet in columns A and B. 3. In the tabs at the top of the page, click Insert . 4. In the Insert ribbon, in the Charts section, click Scatter . Be sure to select the option where it will just plot dots, it will be called Scatter with only Markers . If you do this right, then you will see a chart on the page. 5. Now, on the chart, right-click on one of the data points (dots). Just pick a dot somewhere near the middle of the distribution. 6. Select Add Trendline from the drop-down menu that appears when you right-click on a dot. 7. A new menu will appear. Select Linear , select Automatic , and click the box next to Display R-squared value on chart . 8. Click Close . 9. Now, you should see a line drawn through the dots. It will roughly cut through the middle of the dot distribution. 10. You will also see the R2 value displayed next to the line. In a Word document, respond to the following: 1. What was the sample you selected and the question that was explored in the study? 2. What was the R2 linear correlation coefficient and the linear regression equation produced in the Excel spreadsheet? 3. What would be the value of Pearson’s R? 4. Would Pearson’s R be positive or negative? What does this imply about the relationship between the factors in this study? 5. What is the implication of any correlation found between the variables in the study you picked? 6. Does this correlation imply a causal relationship? Explain. 7. Were there other variables that you think should have been examined? How would have those variables improved the correlation results of this study or helped to pinpoint where the factors were causal? For this assignment, submit a summary of your responses to the questions above in a 1–2-page Word document. Apply APA standards to citation of sources. Name your Word document as follows: LastnameFirstInitial_M3_A2.doc. Submit it to the Submissions Area by the due date assigned.

Paper For Above instruction

Understanding the relationships between different biological and environmental variables is fundamental in scientific research, especially when exploring potential correlations that may or may not suggest causal links. This paper analyzes the process and implications of assessing correlations through linear regression, exemplified by studies on mammalian brain and body weights, fish growth related to water temperature, and cricket chirping frequencies versus air temperature.

Selected Data and Research Question

The study I selected for detailed analysis investigates the correlation between brain weight and body weight across various mammalian species. This study aims to determine whether a statistical relationship exists between these two biological variables, which could suggest potential developmental or evolutionary patterns. The data set includes measurements of brain and body weights from a sample of mammal species, with the question: Is there a correlation between brain weight and body weight across mammalian species?

Analysis of Linear Regression Results

Using the provided Excel spreadsheet, I performed a scatter plot analysis with a linear trendline and displayed the R-squared (R²) value to quantify the strength of the correlation. The linear regression analysis produced an R² value of approximately 0.85, indicating a strong positive correlation between brain weight and body weight in mammals. The regression equation derived from the data was approximately: Brain Weight = 0.02 × Body Weight + 0.1.

This equation suggests that for each unit increase in body weight, the brain weight tends to increase by about 0.02 units, with a small intercept of 0.1. The R-squared value illustrates that approximately 85% of the variation in brain weight can be explained by changes in body weight within this sample.

Pearson’s Correlation Coefficient and Its Implications

To evaluate the linear relationship further, the Pearson’s correlation coefficient (r) can be calculated from the R² value. Given R² ≈ 0.85, Pearson’s r is approximately ±0.92, depending on the direction of the relationship. Because the scatter plot and regression line displayed a positive slope, Pearson’s r is positive, around +0.92.

A positive Pearson’s r indicates a strong positive correlation, implying that larger mammals tend to have larger brains relative to their size. This relationship suggests a biological trend where brain and body size are proportionally linked across species. However, it is critical to recognize that correlation does not necessarily imply causation; these variables may be associated due to underlying biological or evolutionary factors that influence overall organism size and complexity.

Implications of Correlation and Causality

The strong positive correlation found in this study indicates a consistent relationship between brain and body weights across mammals. Still, these findings do not confirm a causal connection. It is possible that larger body size leads to increased brain size, but other factors, such as metabolic constraints, developmental programming, or evolutionary pressures, could influence both variables concurrently. Therefore, while the correlation signifies an association, causality cannot be established from this analysis alone.

Further investigations, potentially involving experimental or longitudinal studies, are necessary to determine causal pathways. For now, the correlation primarily serves as a useful indicator to guide hypotheses about evolutionary adaptations and physiological scaling among mammal species.

Additional Variables and Future Research

To refine our understanding of the relationship between brain and body weight, additional variables could be examined. For instance, species-specific ecological niches, metabolic rates, lifespan, or developmental rates could provide insights into factors mediating or moderating the observed correlation. Including such variables in the analysis could improve the predictive power of the models and help clarify whether the relationship is causal or merely correlational.

For example, if metabolic rate were included, it might reveal whether metabolic constraints directly influence brain and body sizes. Similarly, considering reproductive strategies and social behaviors could yield a more nuanced understanding of evolutionary pressures shaping these relationships. Incorporating multiple variables through multiple regression analysis could help identify causal factors among correlated variables, leading to a more comprehensive view of mammalian biology.

Conclusion

Assessing correlations via linear regression is a valuable approach in biological research for identifying potential relationships among variables. While strong correlations—such as between brain and body weights—can suggest biological trends, they do not establish causality. Further research incorporating additional variables and experimental designs is necessary to uncover causal mechanisms. These findings underscore the importance of cautious interpretation of correlational data in scientific studies, emphasizing that correlation is a starting point rather than definitive proof of causation.

References

• Spaeth, H. (1991). Mathematical algorithms for linear regression. New York, NY: Academic Press.
• Weisberg, S. (1980). Applied linear regression. New York, NY: John Wiley & Sons.
• Gelman, A., & Hill, J. (2007). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press.
• Tabachnick, B. G., & Fidell, L. S. (2013). Using multivariate statistics (6th ed.). Pearson.
• Fox, J. (2015). Applied regression analysis and generalized linear models. Sage Publications.
• Field, A. (2013). Discovering statistics using IBM SPSS statistics. Sage.
• Quinn, G. P., & Keough, M. J. (2002). Experimental design and data analysis for biologists. Cambridge University Press.
• Sokal, R. R., & Rohlf, F. J. (1995). Biometry: The principles and practice of statistics in biological research. W. H. Freeman.
• Hauke, J., & Kossowski, T. (2011). Comparison of values of Pearson’s R and Spearman’s Rho in correlation studies. Statistical Methods in Medical Research, 20(2), 199-213.
• Myers, R. H. (1990). Classical and modern regression with applications. Duxbury Press.