Correlation

1. Correlation Chapter 9

2. Overview • One of the major goals in Criminology is the etiology (cause) of crime, because of this criminologists studies are concerned with the relationship between two or more variables. • Instead of looking for differences between the groups as in ANOVA and the F-test, we want to see if they are related, and if so how strongly they are associated, one method of analysis that serves that purpose is correlation.

3. Defining Correlation • Correlation describes the relationship between two variables, the independent variable (X), and the dependent variable (Y). Correlation simply shows the degree to which these variables change together, the extent to which the variation in X and the variation in Y are related to one another. • There are several ways in which the variables can be relates to each other

4. Defining Correlation • Those ways are • Inversely (negatively) related • Positively related • No relationship • Spurious

5. Inversely Related • Consider the relationship between the size of the jail population (X) and the crime rate (Y). IN the first case, the size of the jail population may affect the crime rate by deterring and/or incapacitating criminals. • A deterrent effect or incapacitative effect would cause the crime rate to go down when the size of the jail population increases, we would assume that X and Y are inversely (or negatively) related. The values of the two variables change in opposite directions simultaneously

6. Perfect Inverse Correlation 10 9 8 7 6 5 4 3 2 1 0 Y 0 1 2 3 4 5 6 7 8 9 10 X

7. Positively Related • A second possibility in our example is that as the size of the jail population increases so does the crime rate. The other possibility for a positive relation is that as the size of the jail population (X) decreases so does the crime rate (Y). Both types are considered positive because X and Y change in the SAME direction, for the relationship to be positive the variables must both change in the SAME direction, either increase or decrease

8. Positive Correlation • As X increases, the value of Y also increases, hence they increase together.

9. Perfect Positive Correlation 10 9 8 7 6 5 4 3 2 1 0 Y 0 1 2 3 4 5 6 7 8 9 10 X

10. Strong Positive Correlation 10 9 8 7 6 5 4 3 2 1 0 Y 0 1 2 3 4 5 6 7 8 9 10 X

11. No relationship • The third possibility is that there is no relationship at all between the variables. The size of the jail population and the crime rate are unrelated, there is no correlation

12. Zero Correlation • When the points on the scatter plot do not follow any pattern, there is a zero correlation. The points on the graph do not follow any pattern, they are scattered through out the graph.

13. Zero Correlation 10 9 8 7 6 5 4 3 2 1 0 Y 0 1 2 3 4 5 6 7 8 9 10 X

14. Spurious Relationship • The final type of relationship could be spurious. The relationship between the jail population (X) and the crime rate (Y) could be associated with a third variable. • The size of the jail population (X1) could be related to the unemployment rate (X2), which may be strongly associated with the crime rate (Y).

15. Spurious Relationship • To isolate the effect of (X1) it is necessary to examine the correlation coefficient between both (X1) and (X2) upon Y. It may be that the original correlation between the size of the jail population and the crime rate was due to the relationship with unemployment rate. If when the unemployment rate is held constant, the effect of the size of the jail population on the crime rate disappears then the original correlation is spurious.

16. Spurious Relationship • If the size of the jail population has an independent effect upon the crime rate, the unemployment rate should have little or no effect upon the crime rate • Another possibility is that the unemployment rate has an indirect effect on the crime rate due to its relationship with the size of the jail population.

17. Spurious Relationship • Such a relationship could result when the unemployment rate (X2) as well as the jail population (X1) increases --- a positive correlation. • But the crime rate could also decrease as both (X1) and (X2) increase --- a negative or inverse correlation

18. Interpreting Correlation • With Correlation both the direction and the magnitude of the relationship should be considered. • Direction indicates the pattern of the relationship between X and Y. • The magnitude indicates the strength of the relationship between X and Y

19. Direction • To interpret the direction of the correlation coefficient you would use a scatter plot- a graph that indicates the pattern of the relationship between the two variables. The dots are paired values of X and Y. • The scatter plot shows that when X increases, the value of Y decreases.

20. Magnitude • Another issue in the interpretation of the correlation is the strength of the relationship. • The researcher must determine if the correlation coefficient is statistically significant. • If the probability level of the correlation coefficient is less than or equal to .05, the value of r is statistically significant.

21. Magnitude • Correlation coefficients (r) range in value from -1 to 1. A r value of -1 or 1 is a perfect correlation. 0 indicates no relationship. • The sign of r has nothing to do with its strength. • Negative correlations are not weaker than positive ones. • A negative sign is a indicator of an inverse relationship between X and Y. • The closer the r value is to 0 the weaker the relationship.

22. Percentage of Variance • Correlation is a measure if the relationship between the variance in the independent variable (X) and the dependent variable (Y), it determines to what extent they vary together. • The variance is the average squared deviations from the mean for each score in a distribution. • Here r2 is an important indicator of the relationship between X and Y. • r2 is also known as the coefficient of determination- the proportion by the variation in the dependant variable (Y) that is explained by the variation in the dependent variable (X).

23. Percentage of Variance • The coefficient of determination measures the proportion of the total variation in the dependent variable that is explained or accounted for by the variation in the independent variable. • The higher the r2 the greater the percentage of the variance in the dependant variable that is explained by the variance in the independent variable. • The higher the r2 the stronger the nature of the relationship between X and Y.

24. Considering Causation • Correlation is actually a measure of how the variance in the independent variable is related to that of the dependent variable. • Causation can be assumed, if some measures are followed and practiced

25. Five Criteria for a Causal Relationship • 1. Consistency of the association: Have different studies resulted in similar findings. If so, then the present analysis can further confirm the findings by finding additional support for them.

26. Five Criteria for a Causal Relationship • 2. Strength of the association: A weak association would offer little or no chance for a causal relationship. Never assume that because the correlation is strong the relationship is causal

27. Five Criteria for a Causal Relationship • 3. Specificity and coherence of the association: To what extent does one independent variable relate to one specific type of crime?

28. Five Criteria for a Causal Relationship • 4. Temporal relationship of the association: Causal Time order must be present. Exposure to the independent variable (X) must precede the dependent variable (Y) in time.

29. Five Criteria for a Causal Relationship • 5. Consideration of “rival causal hypotheses”: Were other independent variables considered? Effects of other independent variables were not considered and could account for the relationship and even make it spurious

30. Bivariate Correlation using SPSS • Using SPSS we examine one method of computing a correlation coefficient, the Pearson product-moment correlation coefficient. • This statistic known as “r” shows the degree of relationship between X and Y providing that they are both measured at the interval or ratio level of measurement, also the two variables are linearly related and the come from a normally distributed population.

31. Bivariate Correlation using SPSS • As with all statistical tests it is important not to violate the assumptions. Some statistical test can not withstand violations of assumptions very well, leading to a Type II error. • The consequences of stating that no relationship exists when in fact one does exist can be just as vexing as making a Type I error, stating that a relationship does exist when the null hypothesis should have been retained.

32. Bivariate Correlation using SPSS • Using the state data set we examine the relationship between two ratio level variables in each state: the percentage of people living in poverty, average 1996-1997 (X the independent variable) and the rate of burglary per 100,000 (Y the dependent variable).

33. Conclusion • This chapter reviews a measure of association between independent (X) and dependent (Y) variables- the correlation coefficient (r). • It considers the degree of relationship between X and Y. It requires that both X and Y be measured at the interval or ratio level of measurement, that they are linearly related, and that they come from normally distributed populations.

34. Conclusion • The direction “r” tells you if X and Y are positively or negatively (inversely) related . The best indication of the nature of the direction is the scatter diagram. • The closer the value of r is to 1 the stronger relationship. The value of r2 tells you that the percentage of the variance in the dependent variable (Y) is explained by the variance in the independent variable (X)