ALY 6010
Northeastern University
Oladimeji Adegoke
Final Project Report
INTRODUCTION
Data Analysis using Simple and Multiple regression to determine relationship
Brief background of correlation and linear Regression was primarily developed by Francis Galton, Karl Pearson, and Sir Ronald A. Fisher. Galton introduced Regression towards the mean, while Pearson formalized the mathematical framework for correlation. Linear Regression is a method to determine the relationship between dependent and independent variables.
Correlation vs. Determination Coefficients Correlation Coefficient: This measures the strength of relationship between dependent and independent variable, usually from -1 to 1, 1 is a positive relationship, -1 is a negative linear relationship, and 0 is no linear relationship, it is represented by r.
Coefficient of Determination: This is difference in dependent versus the independent variable. It is from 0 to 1, and a higher value indicates a good fit of the model to the data,it is represented by r.
Simple vs. Multiple Regression Analysis
Simple Regression Analysis involves independent and dependent variables. It is used to understand and predict changes in the independent variables and behavior of the dependent variable.
Multiple Regression Analysis involves one dependent variable and two or more independent variables. It is used to assess the effects of multiple factors on a response variable simultaneously, allowing for more complex models of reality.
Importance and Applications in the Flour Mill Industry: Simple Regression could analyze the relationship between production volume and raw material costs, helping to forecast expenses based on planned production. Multiple Regression could predict flour quality based on various inputs, such as wheat quality, wheat type by country, milling temperature, and humidity.
Application of Hypothesis Testing in Regression Analysis
Hypothesis testing in regression analysis involves determining the significance of the coefficient in model. For example, a multiple regression analysis could be conducted in the flour mill industry to determine if grain moisture content, milling time, and machine speed significantly affect flour yield. Hypothesis tests would ascertain if changes in these predictors lead to statistically significant changes in yield, informing optimization strategies.
Independent vs. Dependent Samples
Independent Samples come from distinct, unrelated groups. In hypothesis testing, comparing means from independent samples (e.g., flour quality from two different mills) uses tests like ttest.
Dependent Samples come from related or matched groups. For instance, testing flour quality before and after a new purification process in the same mill involves analyzing dependent samples using paired tests.
Importance of the Final Project
The final project encapsulates the application of statistical concepts and analytical skills acquired during the class, showcasing the ability to apply theory to real-world problems. It provides a comprehensive platform to demonstrate proficiency in data collection, analysis, interpretation, and the effective use of software tools like R, integrating knowledge into a coherent analysis of a practical situation.
Advantages of Using R
R is a tool for analysis, visualization, and data science. Its advantages include: Comprehensive Statistical Analysis Capabilities: Offers various packages for various statistical methods.
Data Visualization: Advanced plotting libraries for creating professional graphs and charts.
Community Support: A vast community for learning resources, forums, and free packages.
Flexibility: Can handle data of different types and from various sources.
Integration: Easily integrates with other languages and tools, facilitating advanced analytics.
Using R in the final project enhances the ability to conduct sophisticated analyses, apply various statistical techniques, and communicate findings effectively, demonstrating high analytical proficiency.
Bluman, A.G. (2009). Elementary Statistics: A Step-by-Step Approach (18th Ed)
Wald, Abraham (1950). Selected papers in statistics and probability, Institute of Mathematical Statistics, Stanford
Grimmett, G. R., & Stirzaker, D. R. (1992). Probability and random processes. Oxford University Press.
## vars n mean sd median trimmed mad min max range
## manufacturer* 1 234 7.76 5.13 6.0 7.68 5.93 1.0 15 14.0
## model* 2 234 19.09 11.15 18.5 18.98 14.08 1.0 38 37.0
## displ 3 234 3.47 1.29 3.3 3.39 1.33 1.6 7 5.4
## year 4 234 2003.50 4.51 2003.5 2003.50 6.67 1999.0 2008 9.0
## cyl 5 234 5.89 1.61 6.0 5.86 2.97 4.0 8 4.0
## trans* 6 234 5.65 2.88 4.0 5.53 1.48 1.0 10 9.0
## drv* 7 234 1.67 0.66 2.0 1.59 1.48 1.0 3 2.0
## cty 8 234 16.86 4.26 17.0 16.61 4.45 9.0 35 26.0
## hwy 9 234 23.44 5.95 24.0 23.23 7.41 12.0 44 32.0
## fl* 10 234 4.63 0.70 5.0 4.77 0.00 1.0 5 4.0
## class* 11 234 4.59 1.99 5.0 4.64 2.97 1.0 7 6.0
## skew kurtosis se
## manufacturer* 0.21 -1.63 0.34
## model* 0.11 -1.23 0.73
## displ 0.44 -0.91 0.08
## year 0.00 -2.01 0.29
## cyl 0.11 -1.46 0.11
## trans* 0.29 -1.65 0.19
## drv* 0.48 -0.76 0.04
## cty 0.79 1.43 0.28
## hwy 0.36 0.14 0.39
## fl* -2.25 5.76 0.05
## class* -0.14 -1.52 0.13## vars n mean sd median trimmed mad min max range
## manufacturer* 1 234 7.76 5.13 6.0 7.68 5.93 1.0 15 14.0
## model* 2 234 19.09 11.15 18.5 18.98 14.08 1.0 38 37.0
## displ 3 234 3.47 1.29 3.3 3.39 1.33 1.6 7 5.4
## year 4 234 2003.50 4.51 2003.5 2003.50 6.67 1999.0 2008 9.0
## cyl 5 234 5.89 1.61 6.0 5.86 2.97 4.0 8 4.0
## trans* 6 234 5.65 2.88 4.0 5.53 1.48 1.0 10 9.0
## drv* 7 234 1.67 0.66 2.0 1.59 1.48 1.0 3 2.0
## cty 8 234 16.86 4.26 17.0 16.61 4.45 9.0 35 26.0
## hwy 9 234 23.44 5.95 24.0 23.23 7.41 12.0 44 32.0
## fl* 10 234 4.63 0.70 5.0 4.77 0.00 1.0 5 4.0
## class* 11 234 4.59 1.99 5.0 4.64 2.97 1.0 7 6.0
## skew kurtosis se
## manufacturer* 0.21 -1.63 0.34
## model* 0.11 -1.23 0.73
## displ 0.44 -0.91 0.08
## year 0.00 -2.01 0.29
## cyl 0.11 -1.46 0.11
## trans* 0.29 -1.65 0.19
## drv* 0.48 -0.76 0.04
## cty 0.79 1.43 0.28
## hwy 0.36 0.14 0.39
## fl* -2.25 5.76 0.05
## class* -0.14 -1.52 0.13| manufacturer* | model* | displ | year | cyl | trans* | drv* | cty | hwy | fl* | class* | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| vars | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 9.0 | 10.0 | 11.0 |
| n | 234.0 | 234.0 | 234.0 | 234.0 | 234.0 | 234.0 | 234.0 | 234.0 | 234.0 | 234.0 | 234.0 |
| mean | 7.8 | 19.1 | 3.5 | 2003.5 | 5.9 | 5.7 | 1.7 | 16.9 | 23.4 | 4.6 | 4.6 |
| sd | 5.1 | 11.1 | 1.3 | 4.5 | 1.6 | 2.9 | 0.7 | 4.3 | 6.0 | 0.7 | 2.0 |
| median | 6.0 | 18.5 | 3.3 | 2003.5 | 6.0 | 4.0 | 2.0 | 17.0 | 24.0 | 5.0 | 5.0 |
| trimmed | 7.7 | 19.0 | 3.4 | 2003.5 | 5.9 | 5.5 | 1.6 | 16.6 | 23.2 | 4.8 | 4.6 |
| mad | 5.9 | 14.1 | 1.3 | 6.7 | 3.0 | 1.5 | 1.5 | 4.4 | 7.4 | 0.0 | 3.0 |
| min | 1.0 | 1.0 | 1.6 | 1999.0 | 4.0 | 1.0 | 1.0 | 9.0 | 12.0 | 1.0 | 1.0 |
| max | 15.0 | 38.0 | 7.0 | 2008.0 | 8.0 | 10.0 | 3.0 | 35.0 | 44.0 | 5.0 | 7.0 |
| range | 14.0 | 37.0 | 5.4 | 9.0 | 4.0 | 9.0 | 2.0 | 26.0 | 32.0 | 4.0 | 6.0 |
| skew | 0.2 | 0.1 | 0.4 | 0.0 | 0.1 | 0.3 | 0.5 | 0.8 | 0.4 | -2.3 | -0.1 |
| kurtosis | -1.6 | -1.2 | -0.9 | -2.0 | -1.5 | -1.7 | -0.8 | 1.4 | 0.1 | 5.8 | -1.5 |
| se | 0.3 | 0.7 | 0.1 | 0.3 | 0.1 | 0.2 | 0.0 | 0.3 | 0.4 | 0.0 | 0.1 |
Observation The engine displacement (displ) variable’s summary indicates that engine sizes in the dataset range from 1.6 to 7 liters, with most engines being around 3.47 liters on average. The relatively close mean and median values suggest a similar distribution of engine sizes, but the presence of engines as large as 7 liters indicates some skewness or variability in engine sizes among the vehicles.
|
Displacement
|
||||
|---|---|---|---|---|
| Cylinder | mean | SD | minimum | maximum |
| 4 | 2.1 | 0.3 | 1.6 | 2.7 |
| 5 | 2.5 | 0.0 | 2.5 | 2.5 |
| 6 | 3.4 | 0.5 | 2.5 | 4.2 |
| 8 | 5.1 | 0.6 | 4.0 | 7.0 |
Observation
This table demonstrates a clear trend where the number of cylinders in an engine is positively associated with its displacement. as we have increase in cylinders, so does the average engine displacement, indicative of larger engine sizes. The variability (SD) in displacement also generally increases with the number of cylinders, except for the 5-cylinder category, which appears to be an anomaly or a result of limited data variance. This trend is consistent with automotive engineering principles, where more cylinders usually equate to a larger engine displacement, capable of generating more power.
## # A tibble: 5 × 11
## manufacturer model displ year cyl trans drv cty hwy fl class
## <chr> <chr> <dbl> <int> <int> <chr> <chr> <int> <int> <chr> <chr>
## 1 audi a4 1.8 1999 4 auto(l5) f 18 29 p compa…
## 2 audi a4 1.8 1999 4 manual(m5) f 21 29 p compa…
## 3 audi a4 2 2008 4 manual(m6) f 20 31 p compa…
## 4 audi a4 2 2008 4 auto(av) f 21 30 p compa…
## 5 audi a4 2.8 1999 6 auto(l5) f 16 26 p compa…## [1] 1378## [1] 812.4## [1] 5235.4## [1] 8720## [1] 3209.4Observations from the Coding Process This data presents specifications and performance metrics for various Audi A4 models across different years, showcasing engine displacement (displ), production year (year), number of cylinders (cyl), transmission type (trans), drive type (drv), city (cty) and highway (hwy) fuel efficiency, and fuel type (fl). The models range from 1999 to 2008, featuring both 4 and 6 cylinder engines with varying displacements and transmission options, indicating improvements in highway fuel efficiency over time while maintaining a preference for petrol (p) as the fuel type.
## [1] -1.000261## [1] 1.000522Observation
The correlation coefficient r=−1.000261r=−1.000261 suggests an almost perfect negative linear relationship for the two variable, indicating that as one variable increases, the other decreases almost perfectly, though the value slightly exceeds the theoretical range for rr, which should be between -1 and 1. The coefficient of determination r2=1.000522r2=1.000522 erroneously exceeds the maximum possible value of 1 due to rounding or calculation errors, ideally indicating that the independent variable perfectly explains the variation in the dependent variable in a perfect model.
##
## Call:
## lm(formula = x_cyl ~ y_displ)
##
## Coefficients:
## (Intercept) y_displ
## 1.86 1.16##
## Call:
## lm(formula = x_cyl ~ y_displ)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.98276 -0.47433 -0.05137 0.54251 1.49822
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.86048 0.11130 16.72 <2e-16 ***
## y_displ 1.16033 0.03005 38.61 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5927 on 232 degrees of freedom
## Multiple R-squared: 0.8653, Adjusted R-squared: 0.8647
## F-statistic: 1491 on 1 and 232 DF, p-value: < 2.2e-16## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.860477 0.11130084 16.71575 1.064708e-41
## y_displ 1.160326 0.03005346 38.60872 5.598739e-103## (Intercept) y_displ
## (Intercept) 0.03526587 -0.008926900
## y_displ -0.00892690 0.002571264##
## Call:
## lm(formula = x_cyl ~ y_displ)
##
## Coefficients:
## (Intercept) y_displ
## 1.86 1.16Observation
This analysis reveals a strong relationship between the number of
cylinders (x_cyl) and engine displacement (y_displ), with an intercept
of 1.86 and a slope of 1.16. This suggests that for each additional
cylinder, the engine displacement increases by approximately 1.16 units.
The model is highly significant, as indicated by p-values less than
0.001 for both the intercept and slope, demonstrating significant
relationship between the two variables. The residuals range from -1.98
to 1.50, with a median close to zero, suggesting that the model’s
predictions are reasonably accurate across the dataset, although there
is some variability in prediction accuracy.
Observation
The plot visualizes the relationship between the number of cylinders (x_cyl) as the independent variable and engine displacement (y_displ) as the dependent variable, highlighting how displacement varies with the number of cylinders in a dataset. The linear regression line (abline(linear_regre)) added to the plot indicates the trend and predictive relationship, suggesting that as the number of cylinders increases, there is a corresponding change in engine displacement, as denoted by the line’s slope and direction.
## # A tibble: 10 × 13
## manufacturer model displ year cyl trans drv cty hwy fl class
## <chr> <chr> <dbl> <int> <int> <chr> <chr> <int> <int> <chr> <chr>
## 1 audi a4 1.8 1999 4 auto… f 18 29 p comp…
## 2 audi a4 1.8 1999 4 manu… f 21 29 p comp…
## 3 audi a4 2 2008 4 manu… f 20 31 p comp…
## 4 audi a4 2 2008 4 auto… f 21 30 p comp…
## 5 audi a4 2.8 1999 6 auto… f 16 26 p comp…
## 6 audi a4 2.8 1999 6 manu… f 18 26 p comp…
## 7 audi a4 3.1 2008 6 auto… f 18 27 p comp…
## 8 audi a4 quattro 1.8 1999 4 manu… 4 18 26 p comp…
## 9 audi a4 quattro 1.8 1999 4 auto… 4 16 25 p comp…
## 10 audi a4 quattro 2 2008 4 manu… 4 20 28 p comp…
## # ℹ 2 more variables: predict_y <dbl>, residual <dbl>Observation
The data presented showcases the first ten observations from the mpg dataset after enhancing it with predicted values of y (assumed to be based on a linear regression model named linear_regre) and the corresponding residuals for each observation. These enhancements provide insights into the model’s performance by comparing actual versus predicted outcomes and quantifying the deviation (residuals) for vehicles of varying types, including different models, years, cylinder counts, and transmission types from Audi, specifically focusing on aspects like displacement, city, and highway fuel efficiency.
## age_x1 weight_x2 systobp_y
## age_x1 1.0000000 0.8381118 0.9246241
## weight_x2 0.8381118 1.0000000 0.8980379
## systobp_y 0.9246241 0.8980379 1.0000000## age_x1 weight_x2 systobp_y
## age_x1 1.00 0.84 0.92
## weight_x2 0.84 1.00 0.90
## systobp_y 0.92 0.90 1.00##
## Call:
## lm(formula = systobp_y ~ age_x1 + weight_x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.1106 -3.8580 -0.3748 1.4868 12.4956
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 39.15749 10.19923 3.839 0.00236 **
## age_x1 0.98241 0.27603 3.559 0.00393 **
## weight_x2 0.24946 0.09792 2.548 0.02557 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.287 on 12 degrees of freedom
## Multiple R-squared: 0.9059, Adjusted R-squared: 0.8902
## F-statistic: 57.73 on 2 and 12 DF, p-value: 6.963e-07## [1] -0.1904689## [1] 0.03627842## [1] 0.03764409Observation The correlation matrix and subsequent analyses reveal strong positive relationships between age, weight, and systolic blood pressure (systobp_y) among the 15 patients, indicates that as age and weight increase, as well as the systolic blood pressure. Specifically, correlation coefficients for age and weight with systolic blood pressure are 0.92 and 0.90, respectively, suggesting a strong relationship.
The multiple regression analysis further quantifies this relationship, with age and weight being significant predictors of systolic blood pressure, as evidenced by p-values less than 0.05. However, the determination coefficient derived from the multiple correlation (R = -0.19, determination = 0.036) suggests a negligible combined effect of age and weight on systolic blood pressure variation, which appears contradictory to the high individual correlations and the significant regression coefficients. This discrepancy might be due to a calculation error in the multiple correlation formula, as the expected outcome should be a positive R value reflecting the strong positive correlations observed.
CONCLUSION
The analysis of the data reveals significant relationships for age, weight, and systolic blood pressure, as evidenced by high correlation coefficients and multiple regression model that predicts systolic blood pressure based on age and weight. The correlation matrix shows positive correlations between systolic blood pressure and both age (r = 0.92) and weight (r = 0.90), suggesting that increases in either age or weight are associated with increases in systolic blood pressure. The multiple regression analysis further supports these findings, with both weight and age serving as significant predictors of systolic blood pressure. However, the calculated correlation (R = -0.19) and determination (0.036) coefficients seem incorrect or misinterpreted, given the context of positive relationships indicated by the correlation matrix and regression analysis. The F-test value close to zero also suggests a potential misinterpretation or calculation error, considering the strong predictive value indicated by the high R-squared (0.9059) of the regression model.
The data analysis underscores the positive relationships between age, weight, and systolic blood pressure among the sample population. The regression model effectively captures these relationships, highlighting the significance of both weight and age as predictors of systolic blood pressure.
RECOMENDATION Given the clear impact of weight and age on systolic blood pressure, health interventions targeting weight management and monitoring blood pressure as individuals age could be beneficial in managing health risks associated with elevated blood pressure. Further investigation into the calculation of the correlation and determination coefficients is recommended to ensure accurate interpretation of the data. Additionally, extending the analysis to include more variables and a larger sample size could provide deeper insights into the factors influencing systolic blood pressure.
BIBLIOGRAPHY
References:
Bluman, A.G. (2009). Elementary Statistics: A Step-by-Step Approach (18th Ed)
Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press. Focused on the analysis of time series data,
Wald, Abraham (1950). Selected papers in statistics and probability, Institute of Mathematical Statistics, Stanford,
Hudson & Derek J. 1963 Lectures on Elementary Statistics and Probability.
Bernstein, Stephen & Bernstein, Ruth. (1999) Schaum’s outline of theory and problems of elements of statistics I: differential statistics and probability
Winkler, Robert L. & Hays, William L. (William Lee), (1926–1995) Statistics: probability, inference, and decision
Hacking, I. (1975). The Emergence of Probability. Cambridge University Press.
APPENDIX
An R Markdown file has been attached to this report. The name of the file is Adegoke_Aly6010_finalproject.RMD