Download Linear Regression Two - Political Science - Lecture Slides and more Slides Political Science in PDF only on Docsity!
Linear Regression II: Making Sense of
Regression Results
- Interpreting SPSS regression output
- Coefficients for independent variables
- Fit of the regression: R Square
- Statistical significance
- How to reject the null hypothesis
- Multivariate regressions
- College graduation rates
- Ethnicity and voting
Linear Regression: Review
- Want to draw a line that best represents the relationship between the IV (X) and DV (Y). - Y = a + b*X - Allows us to predict DV given value of IV
- Regression finds the values for a and b that minimizes the distance between the points and the line.
- Technically, a and b are population parameters. We only get to calculate sample statistics, a-hat and b-hat.
Interpreting SPSS regression output
- An SPSS regression output includes two key
tables for interpreting your results:
- A “Coefficients” table that contains the y-intercept (or “constant”) of the regression, a coefficient for every independent variable, and the standard error of that coefficient.
- A “Model Summary” table that gives you information on the fit of your regression.
Interpreting SPSS regression output:
Coefficients
Coefficientsa
4.236 7.048 .601. 5.88E-02 .007 .588 8.778.
(Constant) Average SAT Score
Model 1 B Error^ Std.
Unstandardized Coefficients Beta
Standardized Coefficients t Sig.
a. Dependent Variable: Graduation Rate In this class, we will ONLY LOOK AT UN STANDARDIZED COEFFICIENTS!
- The y-intercept is 4.2% with a standard error of 7.0%
- The coefficient for SAT Scores is 0.059%, with a standard error of 0.007%.
Interpreting SPSS regression output:
Coefficients
- The y-intercept or constant is the predicted
value of the dependent variable when the
independent variable takes on the value of
zero.
- This basic model predicts that when a college admits a class of students who averaged zero on their SAT, 4.2% of them will graduate.
- The constant is not the most helpful statistic.
Interpreting SPSS regression output:
Coefficients
- The coefficient of an independent variable is
the predicted change in the dependent
variable that results from a one unit increase in
the independent variable.
- A college with students whose SAT scores are one point higher on average will have a graduation rate that is 0.059% higher.
- Increasing SAT scores by 200 points leads to a (200)(0.059%) = 11.8% rise in graduation rates
R Square Examples
Statistical Significance
- What would the null hypothesis look like in a
scatterplot?
- If the independent variable has no effect on the dependent variable, the scatterplot should look random, the regression line should be flat, and its slope should be zero.
- Null hypothesis: The regression coefficient (b) for an independent variable equals zero.
- Can we reject null b=0 based on our estimate of b- hat?
Statistical Significance
- So, if a coefficient is more than twice the size
of its standard error, we reject the null
hypothesis with 95% confidence.
- This works whether the coefficient is negative or positive.
- The coefficient/standard error ratio is called the “test statistic” or “t-stat.”
- A t-stat bigger than 2 or less than -2 indicates at statistically significant correlation.
Interpreting SPSS regression output: T-
Stats Coefficientsa
4.236 7.048 .601. 5.88E-02 .007 .588 8.778.
(Constant) Average SAT Score
Model 1 B Error^ Std.
Unstandardized Coefficients Beta
Standardized Coefficients t Sig.
a. Dependent Variable: Graduation Rate
Multivariate Regressions
Year of
Founding
SAT Scores
Graduation
Tuition Rates
Student/Faculty
Ratio
Multivariate Regressions
- Again, want to estimate coefficients:
Est. Grad. Rate = a + b 1 *SAT Score + b 2 *Year Founded+ b 3 *Tuition + b 4 *Faculty Ratio
Multivariate Regressions
- Holding all other factors constant, a 200 point increase in SAT scores leads to a predicted (200)(0.042) = 8.4% increase in the graduation rate, and this effect is statistically significant.
- Controlling for other factors, a college that is 100 years younger should have a graduation rate that is (100)(-0.021) = 2.1% lower, but this effect is not significantly different from zero.
