2. Digression - Hypotheses Many research designs involve statistical tests – involve accepting or rejecting a hypothesis Null (statistical) hypotheses assume no relationship between two or more variables. Statistics are used to test null hypotheses –E.g. We assume that there is no relationship between weight and fast food consumption until we find statistical evidence that there is 1 2 3

3. Probability Probability is the odds that a certain event will occur In research, we deal with the odds that patterns in data have emerged by chance vs. they are representative of a real relationship Remember – inference is the key…samples and populations Alpha (α) is the probability level (or significance level) set, in advance, by the researcher as the odds that something occurs by chance 1 2 3 4

4. Probability Alpha levels (cont.) –E.g.  =.05 means that there will be a 5% chance that significant findings are due to chance rather than a relationship in the data 1 2

5. Probability Most statistical tests produce a p-value that is then compared to the  -level to accept or reject the null hypothesis E.g. Researcher sets significance level at.05 a priori; test results show p =.02. Researcher can then reject the null hypothesis and conclude the result was not due to chance but to there being a real relationship in the data How about p =.051, when  -level =.05? 1

6. Error Significance levels (e.g.  =.05) are set in order to avoid error –Type I error = rejection of the null hypothesis when it was actually true Conclusion = relationship; there wasn’t one (false positive) (=  ) –Type II error = acceptance of the null hypothesis when it was actually false Conclusion = no relationship; there was one 1 2 3 4

7. Error – Truth Table Null TrueNull False Accept Type II error RejectType I error 1 2 3 4

8. Back to Our Example Conclusion: No relationship exists between weight and fast food consumption with this group of respondents 1

9. 2 Really? Conclusion: We have found no evidence that a relationship exists between weight and fast food consumption with this group of subjects –Do you believe this? Can you critique it? Construct validity? External validity? –Thinking in this fashion will help you adopt a critical stance when reading research 1

10. Another Example Now let’s see if a relationship exists between weight and the number of piercings a person has –What’s your guess (hypothesis) about how the results of this test will turn out? –It’s fine to guess, but remember that our null hypothesis is that no relationship exists, until the data shows otherwise 1

11. Another Example (continued) What can we conclude from this test? Does this mean that  weight causes  piercings, or vice versa, or what? 1 2

12. Correlations and causality Correlations only describe the relationship, they do not prove cause and effect Correlation is a necessary, but not sufficient condition for determining causality There are Three Requirements to Infer a Causal Relationship 1

13. Causality…  A statistically significant relationship between the variables  The causal variable occurred prior to the other variable  There are no other factors that could account for the cause  Correlation studies do not meet the last requirement and may not meet the second requirement (go back to internal validity – 497) 1

14.  If there is a relationship between weight and # piercings it could be because  weight  # piercings  weight  # piercings  weight  some other factor  # piercings  Which do you think is most likely here? 4 Correlations and causality 1 2 3

15. Other Types of Correlations Other measures of correlation between two variables: –Point-biserial correlation=use when you have a dichotomous variable The formula for computing a PBC is actually just a mathematical simplification of the formula used to compute Pearson’s r, so to compute a PBC in SPSS, just compute r and the result is the same 1

16. Other Types of Correlations Other measures of correlation between two variables: (cont.) –Spearman rho correlation; use with ordinal (rank) data Computed in SPSS the same way as Pearson’s r…simply toggle the Spearman button on the Bivariate Correlations window 1

17. Coefficient of Determination  Correlation Coefficient Squared  Percentage of the variability among scores on one variable that can be attributed to differences in the scores on the other variable  The coefficient of determination is useful because it gives the proportion of the variance of one variable that is predictable from the other variable  Next week we will discuss regression, which builds upon correlation and utilizes this coefficient of determination 1 2 3

18. Correlation in excel Use the function “correl” The “arguments” (components) of the function are the two arrays 1

19. http://www.stat.uiuc.edu/courses/stat100/java/GCApplet/GCAppletFrame.html http:// www.stat.tamu.edu /~west/applets/clicktest.htmlhttp:// www.stat.tamu.edu /~west/applets/clicktest.html http://www.stat.tamu.edu/~west/applets/rplot.html 2 Applets (see applets page) 1