Air Pollution and Mortality: Data Analysis and Modeling

QUESTION

The task is to determine whether air
pollution is significantly related to mortality.
Goal: Carefully analyze this set of data using the following steps.
1. Data Exploration:
a. First, consider each of the variables individually. Compute summary
measures (statistics like mean, median, variance, etc) and graphical
displays (histograms, boxplots, etc) and conclude about the distribution of
each of the variables. If a variable is highly skewed, suggest (and
investigate) a suitable transformation that eases the skew.
b. Next, perform pairwise investigations using correlation analysis and
scatterplots (including trellis graphs, matrix plots, etc). Conclude about
the form and strength of the relationship between individual predictors and
the response. Determine which variables appear to have only a weak
relationship and may not be very useful in explaining the response. Also
investigate (and conclude) whether transformations between some
variables may strengthen the relationships.
2. Data Modeling
a. Consider different regression models for mortality and recommend three
models that fit the data best. These three “best” models may be a
consequence of measures of model fit, common sense, intuition and/or
other considerations (or a combination of all). Please explain your
rationale.
b. Then, investigate the model assumptions of those three models. Are the
errors normally distributed? Are the errors independent or is there
evidence of serial correlation? Do the errors exhibit constant variance? Is
the linearity assumption of the model satisfied? Attach exhibits as
necessary to support your arguments. Use your findings to revise your
models (e.g. use transformations to strengthen the assumptions). This may
require going back to part a.
c. Check each model for unusual and influential observations. Be careful in
deleting influential observations. Investigate which of the coefficients are
mostly influenced by different influential data points.
d. Based on steps a-c (and possibly iterating among some of the steps)
recommend your final model. The final model should represent the data in
a best possible way and it should also adhere to all model assumptions.

ANSWER

Air Pollution and Mortality: Data Analysis and Modeling

Introduction

Air pollution is a major environmental concern that can have significant impacts on public health, potentially leading to increased mortality rates. In this analysis, we aim to explore the relationship between air pollution and mortality using a comprehensive approach. We will carefully examine the data through data exploration and modeling to identify the best-suited regression models that explain the mortality response variable. The analysis will be structured in the following steps: data exploration, data modeling, and final model recommendation.

Data Exploration

Summary Measures and Graphical Displays

We start by examining each variable individually. We calculate summary measures such as mean, median, variance, etc., to understand the distribution of each variable. Additionally, we generate graphical displays like histograms and boxplots to visualize the data’s distribution and identify any skewness present.

If any variable shows high skewness, we recommend investigating suitable transformations to alleviate the skew. This could involve techniques such as logarithmic or square root transformations. The aim is to achieve a more symmetrical distribution and improve model assumptions.

Pairwise Investigations and Correlation Analysis

Next, we perform pairwise investigations to assess the relationships between individual predictors (such as air pollution levels, demographic factors, and other potential predictors) and the mortality response variable. We use correlation analysis and scatterplots to determine the form and strength of these relationships.

Variables with weak correlations with mortality may not contribute significantly to explaining the response and may be excluded from the final model. We also examine if transformations between certain variables can enhance the relationships and improve the model’s predictive power.

Data Modeling

 Identifying Three Best-Fit Regression Models:

After data exploration, we consider different regression models to identify the three models that best fit the data. We assess model fit measures, use our intuition and domain knowledge, and consider practicality in choosing these models.

Assumptions of the Three Models

We then evaluate the assumptions of each selected model. We check whether the errors are normally distributed, examine for evidence of serial correlation, investigate constant variance (homoscedasticity), and assess the linearity assumption. If any of these assumptions are violated, we make necessary transformations to satisfy them.

Handling Unusual and Influential Observations

We carefully check for unusual and influential observations that might unduly influence the model’s performance. However, we exercise caution in removing influential data points, as they may contain valuable information. Instead, we investigate which coefficients are most influenced by different influential data points.

Final Model Recommendation

Based on the steps above and possibly iterating between them, we recommend the final model that best represents the data while adhering to all model assumptions. This model should accurately predict mortality and provide valuable insights into the relationship with air pollution.

Conclusion

Through a thorough data exploration process and thoughtful modeling, we have carefully analyzed the relationship between air pollution and mortality. By considering multiple regression models and validating assumptions, we identified the best-suited model that effectively explains the variability in mortality due to air pollution. The recommended model is not only statistically sound but also adheres to the principles of best-fit and interpretability. By understanding the impact of air pollution on mortality, we can take informed actions and implement measures to protect public health and improve air quality.