Intro to Stats

1. Intro to Stats Linear Regression

2. Linear Regression • Uses correlations • Predicts value of one variable from the value of another • ***computes UKNOWN outcomes from present, known outcomes • If we know correlation between two variables and one value, we can predict other value • In other words, what value on Y would be predicted by a score on X?

3. When to use it: • You are examining a relationship between continuous variables • You wish to predict scores on one variable from scores on the other

4. Line of best fit – regression equation • Fit a line between the two variables that best captures the scores • Minimal distance between each data point and the line • Allows for the best guess at a score on the second variable given some data point on the first • Error in prediction: Distance from each point to the regression line • If the correlation were perfect, data points would be at a 45-degree angle.

5. Standard Error

6. Formula for a Line Y’ = bX + a Y’ = predicted score of Y based on X b = slope of the line a = point where line crosses the y-axis X = score used as the predictor

7. Parts of the linear equation • b • The value of b is the slope • From this we can tell how much the Y variable will change when X increases by 1 point • a • The Y-intercept • This tells us what Y would be if X = 0 • This is where the line crosses the Y axis

8. First, b b = ΣXY – (ΣXΣY / n) ΣX2 – [(ΣX)2 / n]

9. Second, a a = ΣY - bΣX n

10. Standard Error • Can examine how closely the actual Y values approximate the predicted Y values • If averaged across all data points, this is the standard error of the estimate • Estimates the imprecision of the line

11. Example 1 • 1. State hypotheses • Null hypothesis: no relationship between years of education and income • H0: β = 0 • Research hypothesis: years of education predicts income • H1: β ≠ 0

12. Hypothesis Test • We’ll use SPSS output to test if the x significantly predicts changes in y • Partitions variance into variance accounted for by predictors • And variance unaccounted for by predictors (the residual) • The output will include a significance test of whether the variance accounted for significantly differs from zero (an F-statistic)

13. Example 1 • 5. Use SPSS output

14. Example 1 • 5. Use SPSS output for the standardized beta and the test statistic

15. Example 1 • 6. The output indicates that b = 3.54 and β = .95, with a p < .05 (actually p < .01) • So it does exceed the critical value • 7. If over the critical value, reject the null • & conclude that years of education significantly predicts income

16. Example 1 • In results • Years of education significantly predicted income, b = 3.54, t = 6.11, p < .05, such that more years of education predicted greater income. • Could further say that: for every additional year of education, participants made an additional $35,400 per year (3.54 x10,000 dollars).

17. Multiple Regression • Predict an outcome Y-value with multiple predictor X-values • **This is the real advantage over a correlation coefficient • Determine whether each predictor makes a unique improvement to the prediction of Y

18. Example 1