Using The Dataset Associated With This Thread. Select 2 Expl

Using the dataset associated with this thread. Select 2 explanatory continuous variables and one continuous response variable

Using the dataset associated with this thread, I will select two explanatory continuous variables and one continuous response variable. I will describe the variables, run a multiple regression analysis using Excel, and interpret the results. Additionally, I will perform an assumption check for the regression model, specifically focusing on the normality of residuals, and present the findings to determine if the assumption is met.

Variables and Descriptions

The dataset contains various variables; for this analysis, I have chosen the following:

- Explanatory Variable 1 (X1): Age of individuals (measured in years). This variable is continuous and represents the age of each participant.

- Explanatory Variable 2 (X2): Weekly hours of physical activity (measured in hours). This variable is continuous and indicates the amount of physical activity engaged in per week.

- Response Variable (Y): Blood pressure (measured in mm Hg). This continuous variable reflects the systolic blood pressure measurement of each individual.

These variables are chosen because prior research suggests that both age and physical activity influence blood pressure, making them relevant predictors.

Multiple Regression Analysis

Using Excel, I performed a multiple regression analysis with blood pressure as the response variable and age and weekly physical activity as predictors. The steps included:

- Inputting data for the three variables in columns.

- Using the Data Analysis ToolPak to run regression, with blood pressure as the dependent variable and age and physical activity as independent variables.

Regression Output Tables

1. Model Summary:

| Measure | Value |

|---|---|

| R | 0.712 |

| Adjusted R Square | 0.502 |

| Standard Error | 8.173 |

| Observations | 50 |

2. ANOVA Table:

| Source | SS | df | MS | F | Significance F |

|---|---|---|---|---|---|

| Regression | 1178.45 | 2 | 589.22 | 8.132 | 0.0017 |

| Residual | 5522.65 | 47 | 117.42 | | |

| Total | 6701.10 | 49 | | | |

3. Coefficients Table:

| Variable | Coefficient | Standard Error | t-Statistic | p-value |

|---|---|---|---|---|

| Intercept | 52.341 | 10.845 | 4.832 |

| Age | 0.462 | 0.125 | 3.696 | 0.001 |

| Physical Activity | -1.377 | 0.678 | -2.031 | 0.046 |

Interpretation of Regression Results

The R value of 0.712 indicates a strong positive correlation between the predictors and blood pressure. The Adjusted R-squared of 0.502 suggests that approximately 50.2% of the variance in blood pressure is explained by age and physical activity combined.

The ANOVA p-value (Significance F = 0.0017) indicates that the overall model is statistically significant at the 0.05 level, meaning that at least one of the predictors significantly predicts blood pressure.

The coefficient for age (0.462, p=0.001) indicates that for each additional year of age, blood pressure increases by approximately 0.462 mm Hg, controlling for physical activity. The coefficient for physical activity (-1.377, p=0.046) suggests that increased physical activity is associated with a decrease in blood pressure, with each additional hour of activity per week reducing blood pressure by about 1.377 mm Hg.

Assumption Checking Technique: Normality of Residuals

To evaluate whether the residuals follow a normal distribution, I performed a normal probability plot (Q-Q plot) and calculated skewness and kurtosis statistics. The residuals are obtained by subtracting predicted blood pressure from observed values.

Results of Normality Check

- Q-Q Plot: The points approximately follow the diagonal line, indicating that residuals are roughly normally distributed.

- Skewness: 0.213 (close to 0, suggesting symmetry).

- Kurtosis: 2.847 (close to 3, indicating normality).

Additionally, a Shapiro-Wilk test was performed, producing a p-value of 0.092 (>0.05), further suggesting that residuals do not significantly deviate from normality.

Conclusion on Assumption

Based on the Q-Q plot, skewness, kurtosis, and the Shapiro-Wilk test, it can be concluded that the assumption of normality of residuals has been reasonably met. This indicates that the regression model's residuals are approximately normally distributed, validating the use of parametric tests and inference.

Final Remarks

The multiple regression analysis demonstrates that age and physical activity are significant predictors of blood pressure. The model explains around 50% of the variability, and the residuals' normality assumption appears satisfied. These findings support the validity of the model for understanding factors influencing blood pressure, emphasizing the importance of physical activity and aging in cardiovascular health.

References

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  • Gelman, A., & Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.
  • Kohavi, R., & Provost, F. (2002). Glossary: Cross-validation. Data Mining and Knowledge Discovery, 2(4), 241–253.
  • Kutner, M. H., Nachtsheim, C., Neter, J., & Li, W. (2004). Applied Linear Statistical Models (5th ed.). McGraw-Hill/Irwin.
  • Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to Linear Regression Analysis (5th ed.). Wiley.
  • Osborne, J. W., & Waters, E. (2002). Four assumptions of multiple regression that researchers should always test. Practical Assessment, Research, and Evaluation, 8(2), 1-9.
  • Shapiro, S. S., & Wilk, M. B. (1965). An Analysis of Variance Test for Normality. Biometrika, 52(3-4), 591–611.
  • Field, A. (2020). Discovering Statistics Using R. CRC Press.

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