12.7 - Further Example

Example 12-5: Poverty and Teen Birth Rate Data Section

(Data source: The U.S. Census Bureau and Mind On Statistics, (3rd edition), Utts and Heckard). In this example, the observations are the 50 states of the United States (Poverty data - Note: remove data from the District of Columbia). The variables are y = percentage of each state’s population living in households with income below the federally defined poverty level in the year 2002, \(x_{1}\) = birth rate for females 15 to 17 years old in 2002, calculated as births per 1000 persons in the age group, and \(x_{2}\) = birth rate for females 18 to 19 years old in 2002, calculated as births per 1000 persons in the age group.

 

The two x-variables are correlated (so we have multicollinearity). The correlation is about 0.95. A plot of the two x-variables is given below.

scatterplot of birth 18 tp 19 vs birth 15 to 17

The figure below shows plots of y = poverty percentage versus each x-variable separately. Both x-variables are linear predictors of the poverty percentage.

matrix plot

Minitab results for the two possible simple regressions and the multiple regression are given below.

Regression Analysis: PovPct versus Brth15to17
Regression Equation

PovPct = 4.49 + 0.387 Brth15to17

Predictor Coef SE Coef T P
Constant 4.487 1.318 3.40 0.001
Brth15to17 0.38718 0.05720 6.77 0.000
S = 2.98209 R-Sq = 48.8% R-Sq(adj) = 47.8%
Regression Analysis: PovPct versus Brth18to19
Regression Equation

PovPct = 3.05 + 0.138 Brth18to19

Predictor Coef SE Coef T P
Constant 3.053 1.832 1.67 0.102
Brth18to19 0.13842 0.02482 5.58 0.000
S = 3.24777 R-Sq = 39.3% R-Sq(adj) = 38.0%
Regression Analysis: PovPct versus Brth15to17, Brth18to19
Regression Equation

PovPct = 6.44 + 0.632 Brth15to17 - 0.102 Brth18to19

Predictor Coef SE Coef T P
Constant 6.440 1.959 3.29 0.002
Brth15to17 0.6323 0.1918 3.30 0.002
Brth18to19 -0.10227 0.07642 -1.34 0.187
s = 2.95782 R-Sq = 50.7% R-Sq(adj) = 48.6%

We note the following:

  1. The value of the sample coefficient that multiplies a particular x-variable is not same in the multiple regression as it is in the relevant simple regression.
  2. The \(R^{2}\) for the multiple regression is not the sum of the \(R^{2}\) values for the simple regressions. An x-variable (either one) is not making an independent “add-on” in the multiple regression.
  3. The 18 to 19 year-old birth rate variable is significant in the simple regression, but is not in the multiple regression. This discrepancy is caused by the correlation between the two x-variables. The 15 to 17 year-old birth rate is the stronger of the two x-variables, and given it’s presence in the equation, the 18 to 19 year-old rate does not improve \(R^{2}\) enough to be significant. More specifically, the correlation between the two x-variables has increased the standard errors of the coefficients, so we have less precise estimates of the individual slopes.