4. Measures of association - Spearman's rho How is unemployment related to immigration? Is a higher rate of unemployment in the United States associated with a higher immigration rate? The following table shows U.S. unemployment rates (seasonally adjusted), along with immigration rates in the state of California (percentage of people obtaining legal permanent resident status per 1,000 people) from 2003 to 2012. (Sources: "Yearbook of Immigration Statistics: 2012 Legal Permanent Residents," U.S. Dept. of Homeland Security, July 2013, and "Labor Force Statistics," Bureau of Labor and Statistics Economic Releases, accessed May 21, 2014, http://www.bls.gov/data). Suppose you want to find the strength and direction of the relationship between U.S. unemployment rate and immigration rate in the state of California. Since these are continuous ordinal-level variables, you decide to use Spearman's rho ( rs). Recall that you compute rs using the ranks of the scores, not the scores themselves. You first rank the scores (in decreasing order, so that the largest score has the smallest rank), and then find the squared difference (D2 ) between each pair of ranks. Complete the following computing table to calculate Spearman's rho for these data. (Note: Use a minus sign (-) to enter a negative value.) Using the completed computing table, now calculate Spearman's rho: rs= This result indicates a relationship between these variables. According to these data, as the U.S. unemployment rate increases, the number of people obtaining permanent legal residence in California tends to . This indicates relationship between the variables. According to the proportional reduction in error, or PRE, interpretation, taking the national unemployment rate into account would improve predictions of immigration rate in California by