Chapter 16 Multiple Regression

Chapter 16 Multiple Regression

16.1 Getting Started

• This chapter expands regression analysis to include multiple independent variables in a model.
• Review of simple regression interpretation is provided to enhance communication on results before transitioning to multiple regression.
• For practical implementation in R, load the countries2 data set and attach the DescTools and stargazer libraries.

16.2 Organizing the Regression Output

• Presentation of results is crucial in research as it aids the reader's understanding (professors, clients, supervisors).
• The output from R can be complex, creating barriers for comprehension especially for non-R users.
• Stargazer package in R can produce publication-quality tables that enhance readability and convey important information from regression outputs.

Analysis of Impact of Fertility Rates on Life Expectancy

• Previous findings from Chapter 14 indicated that fertility rate and mean level of education significantly relate to life expectancy, while population size does not.
• In the following regression analysis, results will be organized into a table using stargazer instead of the typical R output.

Table Example

• The resultant table will have a clear structure:
  • Dependent variable: Life Expectancy
  • Fertility Rate coefficient: -4.911 (standard error: 0.231)
  • Constant: 85.946 (standard error: 0.700)
  • Observations: 185
  • R²: 0.711
  • Adjusted R²: 0.710
  • Residual Std. Error: 4.032 (degrees of freedom = 183)
  • F Statistic: 450.410 (degrees of freedom = 1; 183)
• Note significance levels: *p < 0.1; **p < 0.05; ***p < 0.01

Running the Model in R

• The R code snippet to run the fertility model is:

  fertility <- (lm(countries2$lifexp ~ countries2$fert1520))
  stargazer(fertility, type="text", dep.var.labels=c("Life Expectancy"), covariate.labels=c("Fertility Rate"))

16.2.1 Summarizing Life Expectancy Models

• Key interpretations of previous models focus on the determinants of life expectancy:

  • Fertility Rate Impact (Model 1): Strong negative relationship.

  • Formula for prediction: $\hat{y} = 85.946 - 4.911(fert1520)$

  • Interpretation: Each unit increase in fertility rate decreases life expectancy by 4.911 years, accounting for approximately 71% variation in life expectancy.

  • Education Level Impact (Model 2): Strong positive relation.

  • Formula for prediction: $\hat{y} = 56.663 + 1.839(mnschool)$

  • Interpretation: Each additional year in education increases life expectancy by 1.839 years.

  • Educational attainment alone explains about 59% of variation in life expectancy across nations.

  • Population Size Impact (Model 3): No statistically significant effect (p = 0.43).

  • Notable observation: Constant value in population model (73.214) near mean life expectancy indicates poor predictive capacity of population size alone.

16.3 Multiple Regression

• Move from individual regression models to multiple regression allows for consideration of the impact of multiple independent variables concurrently, thus controlling for inter-variable relationships.
• This methodology addresses overlapping relationships and improves predictive ability, using correlation and regression principles.
  • General formula for a multiple regression model with two independent variables:
    $y_i = a + b_1x_{1i} + b_2x_{2i} + … + b_kx_{ki} + e_i$
• Here, each partial slope indicates the impact of an independent variable on the dependent variable while controlling for others.

Working with Multiple Regression in R

• To run a multiple regression in R, specify all independent variables in the lm() function.
• Example R code:

  fit <- lm(countries2$lifexp ~ countries2$fert15 + countries2$mnschool + log10(countries2$pop), na.action=na.exclude)
  stargazer(fit, type="text", dep.var.labels=c("Life Expectancy"), covariate.labels=c("Fertility Rate", "Mean Years of Education", "Log10 Population"))