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CRIM 483. Chapter 5: Correlation Coefficients. Correlation Coefficients. Correlation coefficient=numerical index that reflects the linear relationship between two variables for in the dataset Range -1.00 to +1.00 Known as a bivariate correlation

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## CRIM 483

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**CRIM 483**Chapter 5: Correlation Coefficients**Correlation Coefficients**• Correlation coefficient=numerical index that reflects the linear relationship between two variables for in the dataset • Range -1.00 to +1.00 • Known as a bivariate correlation • Statistic often used to measure correlations=Pearson r correlation (rxy) • Use with continuous variables**Descriptions Continued**• Correlations can indicate two types of relationships: • Direct/positive correlation: both variables change in the same direction • Indirect/negative: variables change in different directions • Ultimately, the correlation coefficient represents the amount of variability shared between two variables**Things to Remember**• The absolute value of the correlation coefficient indicates strength: • .70 and -.70 are equal in strength, but the relationship is in a different direction • .50 is a weaker correlation than -.70 • There will be no correlation in the following cases • When two variables do not share variance • Examining the relationship between education and age when all subjects are the same age (no variance in age) • When the range of one variable is constrained • Examining reading comprehension and grades among high-achieving children**Coefficient of Determination**• Coefficient of determination (CD)=the percentage of variance in one variable that is accounted for by the variance in the other variable • The more two variables share in common, the more related they will be—they share variability • CD=rstudying*GPA2 • rstudying*GPA=.7 rstudying*GPA2 =.49 or 49% of GPA variance is explained by studying time • Conversely, 51% of GPA is not explained by studying time=coefficient of alienation or coefficient of nondetermination…amount of x not explained by y • CD helps to determine the meaningfulness of the relationship**Association v. Causation**• Be careful when interpreting correlations • Bivariate relationships can lead to spurious conclusions • For example, ice cream sales are correlated highly with crime • Does this mean that increased ice cream consumption causes crime? • Correlations do not account for other variables that may be related to both factors examined • Pearson’s r only one type of correlation statistic—others are found in Table 5.3**CRIM 483**Chapter 13: Correlation Coefficients and Statistical Significance**Example**• You want to test the relationship between the quality of marriage and the quality of parent-child relationships • Once you have selected the test statistic, follow these steps: • State the null hypothesis and research hypothesis • What is the null? • What is the research hypothesis? • Set the level of risk for statistical significance:__% • Select the appropriate test statistic**Testing Differences/Relationships**• Compute the test statistic value using the formula on page 81 • What is the computed correlation coefficient? The coefficient IS your test statistic. • To determine significance, you will need the Degrees of Freedom, which is DF = n-2 • Degrees of freedom represents a measure of the number of independent observations in the sample that can be used to estimate the standard deviation of the parent population • NOTE: A t-test distribution (similar to a z-score) is usually computed—in this case, the text makes it a little easier for you • Determine the critical value—the value needed to reject the null hypothesis • Turn to Table B4 in the appendix • What is the critical value in the this table for .05 • Since it is non-directional, you must use the two-tailed figures**Testing, Continued**• Compare the obtained value to the critical value • What is the comparison? • Which is a better reflection of this comparison, #7 or #8? • If obtained value > critical value, reject the null • Observed differences/relationships are not due to chance • If obtained value < critical value, do not reject the null Observed differences/relationships are due to chance What is the final answer to your research question using a correlation coefficient?**Interpretation: Always Remember**• Cause v. Associations • Correlation coefficients are only bivariate • They do not control for any other variables nor do they determine which variable came first • Thus, they are limited in their ability to signify cause • Significance v. Meaningfulness • A test statistic can be significant but it may not be very meaningful • For instance, .393 was significant in this example, but the coefficient of determination shows that only 15.4% of the variance is shared • Thus, the correlation leaves a lot of room for doubt and speculation for what other factors are more important**Example #2 Using SPSS: Ch. 13 Data Set 1**Figure 13.2. Chapter 13 Data Set 1**Figure 13.4. SPSS Output for testing the Significance of the**Correlation Coefficient

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