Chapter 17: Advanced Regression Topics

Chapter 17: Advanced Regression Topics

17.1 Get Started

• Purpose of the Chapter:

  • Analyze life expectancy considering other health-related factors.

  • Address evaluation issues related to multiple regression models.

• Data Set and Tools:

  • Load the countries2 data set.

  • Import the DescTools and stargazer libraries in R to follow along.

17.2 Incorporating Access to Health Care

• New Variables for Analysis:

  • Include variables that measure access to health care in life expectancy analysis:

  • Percent of population living in urban areas.

  • Number of doctors per 10,000 persons.

• Expectations:

  • Higher availability of healthcare resources generally correlates with higher life expectancy.

  • Urban areas typically have better healthcare availability than rural areas.

Regression Model Expansion

• Updated regression model:

  fit<-lm(countries2$lifexp ~ countries2$fert1520 + countries2$mnschool + log10(countries2$pop19_M) + countries2$urban + countries2$docs10k)

• Creating Table of Results:

  stargazer(fit, type="text", dep.var.labels=c("Life Expectancy"), 
            covariate.labels = c("Fertility Rate", "Mean Years of Education", "Log10 Population", "% Urban", "Doctors per 10,000"))

17.3 Regression Results Overview

• Regression Output:

  • Life Expectancy Model Output:

  • Fertility Rate: -3.237*** (0.341)

  • Mean Years of Education: 0.413** (0.163)

  • Log10 Population: 0.082 (0.331)

  • % Urban: 0.050*** (0.015)

  • Doctors per 10,000: 0.049* (0.027)

  • Constant: 73.923*** (2.069)

  • Statistical significance levels:

  • *p < 0.1; **p < 0.05; ***p < 0.01

• Model Performance:

  • Observations: 179

  • R²: 0.768 (explains 76.8% of variation in life expectancy)

  • Adjusted R²: 0.761 (improvement over previous model)

  • Residual Standard Error: 3.606 (df = 173)

  • F Statistic: 114.590*** (df = 5; 173)

Interpretations of Individual Variables

1. Fertility Rate:

   • Negative significant correlation: for every unit increase, life expectancy expected to decline by about 3.24 years, controlling for others.

2. Mean Years of Education:

   • Positive correlation: expectation of increase in life expectancy by about 0.413 years for each unit increase in education level.

3. % Urban Population:

   • Positive relationship: life expectancy increases by 0.05 years with each unit increase in urban population percentage.

4. Doctors per 10,000 Population:

   • Positive correlation: life expectancy increases by about 0.05 years for every additional doctor per 10,000, though p-value is borderline for statistical significance.

   • Clarification on Significance Testing:

     • Using a one-tailed test defines the p-value as less than 0.05.

Considerations on Missing Data

• Missing Cases:

  • Observation of 16 missing cases in the data set of 195 countries.

  • Importance of monitoring as listwise deletion occurs in multiple regression models, affecting data use.

17.4 Multicollinearity

• Definition and Implications:

  • Multicollinearity affects statistical significance when independent variables are highly correlated.

  • It complicates the interpretation of coefficients.

T-Score Influence

• T-Score Calculation:

  • For a regression coefficient b, the t-score is calculated as:
    $t = \frac{b}{S_b}$

  • Standard error is influenced by the correlation with other independent variables.

Diagnostic Assessment of Collinearity

• Correlation Matrix for Collinearity:

  • Example correlations: docs10k has:

  • Strong bivariate correlation with life expectancy (r = 0.7031).

  • Correlation with fert1520 (r = -0.6777), mnschool (r = 0.7666), and urban (r = 0.5543) suggests potential collinearity issues.

Measuring Collinearity

• Tolerance and VIF (Variance Inflation Factor):

  • Tolerance: Proportion of variance that isn’t accounted for by the other independent variables.

  • VIF Calculation:

  • VIF is derived from tolerance:
    $\text{VIF} = \frac{1}{\text{Tolerance}}$

  • Example for docs10k:

    • Fit regression with docs10k as DV:
      r fit_tol <- lm(countries2$docs10k ~ countries2$mnschool + log10(countries2$pop19_M) + countries2$urban)

    • Resulting VIF calculation suggests it allows identification of high multicollinearity.

  • Standard Values: VIF > 10 indicates serious concerns, but generally, values under 5 or 10 can be acceptable.

17.5 Addressing Linearity

• Scatterplot Analysis:

  • Initial model predictions don’t seem to fit the observed data; suggests exploring visual relationships can help detect nonlinear patterns.

  • Visual Confirmation:

  • A linear model does not fit the data well. A curvilinear model may suit better.

Non-Linear Model Transformations

• Log Transformations:

  • Use of logarithmic transformations to mitigate extreme values and explore curvilinear relationships.

  • Example model:

  • Transform docs10k:
    $\log_{10}(docs10k)$

Comparison of Model Fits

• Analysis of simple regression outputs for docs10k compared to its logged version, indicating improved fit:

  • Original Model:

  • R²: 0.487, RMSE: 5.290

  • Log-Transformed Model:

  • R²: 0.673, RMSE: 4.230

• The log-transformed model significantly enhances fit.

17.6 Statistical vs. Substantive Significance

• Model Comparisons:

  • Five-Variable Model:

  • R²: 0.772

  • Explains 77% of life expectancy variations with fertility rates and doctors per capita.

  • Two-Variable Alternative Model:

  • R²: 0.989

  • Complete explanation with fewer variables (male life expectancy, infant mortality).

• Caution on Simplicity:

  • The more simplified model lacks substantive explanatory capacity.

17.7 Next Steps

• Moving Forward:

  • Explore assumptions underlying OLS regression in Chapter 18 to enhance model confidence.

17.8 Assignments

• Complete regression analysis using states20 data set with specified independent variables.

• Produce publication-ready table, summarize results, check for multicollinearity, plot predicted vs. observed values.