Statistics for the Social Sciences

1. Statistics for the Social Sciences Psychology 340 Spring 2005 Hypothesis testing with Correlation and Regression

2. Outline • Hypothesis testing with: • Correlation (effect sizes too) • See textbook Chapter 3 appendix (pgs 109-111) • Regression analyses • Bi-variate & Multiple regression

3. Y X Y 6 6 6 5 1 2 4 5 6 3 3 4 2 3 2 1 X 1 2 3 4 5 6 Hypothesis testing with Pearson’s r • Recall our previous example • Appears linear • Positive relationship • Fairly strong relationship • .89 is far from 0, near +1 • Fairly strong, but stronger than you’d expect by chance?

4. Hypothesis testing with Pearson’s r • Hypothesis testing • Core logic of hypothesis testing • Considers the probability that the result of a study could have come about if the experimental procedure had no effect • If this probability is low, scenario of no effect is rejected and the theory behind the experimental procedure is supported • A five step program • Step 1: State your hypotheses • Step 2: Set your decision criteria • Step 3: Collect your data • Step 4: Compute your test statistics • Step 5: Make a decision about your null hypothesis

5. Hypothesis testing with Pearson’s r • Step 1: State your hypotheses: as a research hypothesis and a null hypothesis about the populations • Null hypothesis (H0) • Research hypothesis (HA) • There are no correlation between the variables (they are independent) • Generally, the variables correlated (they are not independent)

6. Hypothesis testing with Pearson’s r • Step 1: State your hypotheses One -tailed • Our theory is that the variables are negatively correlated Note: sometimes the symbol (rho) is used H0: r≥  HA: r < 

7. Hypothesis testing with Pearson’s r • Step 1: State your hypotheses One -tailed Two -tailed • Our theory is that the variables are negatively correlated • Our theory is that the variables are not correlated H0: H0: r> r = HA: r <  HA: r ≠ 

8. Hypothesis testing with Pearson’s r • Step 2: Set your decision criteria • Your alpha () level will be your guide for when to reject or fail to reject the null hypothesis. • Based on the probability of making making an certain type of error

9. X Y 6 6 1 2 5 6 3 4 3 2 Hypothesis testing with Pearson’s r • Step 3: Collect your data r = 0.89 • Descriptive statistics (Pearson’s r) • Compute your degrees of freedom (df) df = n - 2 = 5 - 2 =3

10. Hypothesis testing with Pearson’s r • Step 4: Compute your test statistics r = 0.89 • Descriptive statistics (Pearson’s r) • Inferential statistics: 2 choices (really the same): • A t-test & the t-table • Use the Pearson’s r table (if available)

11. Reject H0 • Conclude that the correlation is ≠0 Hypothesis testing with Pearson’s r • Step 4: Compute your test statistics r = 0.89 • Descriptive statistics (Pearson’s r) • Inferential statistics: 2 choices (really the same): • A t-test & the t-table • Use the Pearson’s r table (if available) • Step 5: Make a decision about your null hypothesis • From table, with df = n - 2 = 3: tcrit = 3.18

12. Hypothesis testing with Pearson’s r • Step 4: Compute your test statistics r = 0.89 • Descriptive statistics (Pearson’s r) • Inferential statistics: 2 choices (really the same): • A t-test & the t-table • Use the Pearson’s r table (if available) • Step 5: Make a decision about your null hypothesis • From table • -level = 0.05 • Two-tailed • df = n - 2 = 3 • rcrit = 0.878 • Reject H0 • Conclude that the correlation is ≠0

13. Effect sizes with Pearson’s r • Small r = 0.10 • Medium r = 0.30 • Large r = 0.50

14. Y 6 5 4 3 2 Hypothesis testing on each of these 1 X 1 2 3 4 5 6 Hypothesis testing with Regression • A brief review of regression Y = (X)(slope) + (intercept)

15. Hypothesis testing with Regression • These t-tests test hypotheses • Both: • Standardized coefficients • Unstandardized coefficients • H0: Slope = 0 • H0: Intercept (constant) =0

16. “residual” “fit” Hypothesis testing with Regression • Multiple Regression • Typically researchers are interested in predicting with more than one explanatory variable • In multiple regression, an additional predictor variable (or set of variables) is used to predict the residuals left over from the first predictor.

17. First Explanatory Variable Second Explanatory Variable Third Explanatory Variable Fourth Explanatory Variable Hypothesis testing with Regression • Multiple Regression • We can test hypotheses about each of these explanatory hypotheses within a regression model • So it’ll tell us whether that variable is explaining a “significant”amount of the variance in the response variable

18. H0: Coefficient for var1 = 0 • p < 0.05, so reject H0, var1 is a significant predictor • H0: Coefficient for var2 = 0 • p > 0.05, so fail to reject H0, var2 is a not a significant predictor Multiple Regression in SPSS • Null Hypotheses

19. Hypothesis testing with Regression • Multiple Regression • We can test hypotheses about each of these explanatory hypotheses within a regression model • So it’ll tell us whether that variable is explaining a “significant”amount of the variance in the response variable • We can also use hypothesis testing to examine if the change in r2 is statistically significant

20. Second Predictor variable into the Independent Variable field • Click Statistics Hypothesis testing with Regression • Method 2 cont: • Enter:

21. Hypothesis testing with Regression • Click the ‘R squared change’ box

22. Hypothesis testing with Regression • Shows the results of two models • The variables in the first model (math SAT) • The variables in the second model (math and verbal SAT) • r2 for the first model • Model 1 • Coefficients for var1 (var name)

23. Coefficients for var1 (var name) • Coefficients for var2 (var name) Hypothesis testing with Regression • Shows the results of two models • The variables in the first model (math SAT) • The variables in the second model (math and verbal SAT) • r2 for the second model • Model 2

24. The 0.002 change in r2 is not statistically significant (p = 0.46) Hypothesis testing with Regression • Shows the results of two models • The variables in the first model (math SAT) • The variables in the second model (math and verbal SAT) • Change statistics: is the change in r2 from Model 1 to Model 2 statistically significant?

25. Next week • Review for the exam (Oct. 31) • I’ll put together two review labs

26. Relating Critical t’s and r’s • Inferential statistics: 2 choices (really the same): • A t-test & the t-table • Use the Pearson’s r table (if available)