8. The Spearman correlation

"The rapid growth of video game popularity has generated concern among practitioners, parents, scholars, and politicians," wrote researchers Hope M. Cummings and Elizabeth A. Vandewater. In their study, Cummings and Vandewater measured the time adolescents spent playing video games as well as time spent doing other activities, such as interacting with family and friends, reading or doing homework, or playing sports. [Source: Cummings, H., & Vandewater, E. (2007). Relation of adolescent video game play to time spent in other activities. Archives of Pediatrics & Adolescent Medicine, 161(7), 684-689]

After reading about the study conducted by Cummings and Vandewater, you decide to conduct a similar study among a sample of 10 teenage girls. You ask the girls to keep a log of their activities for a day. You want to test whether the amount of time girls spend playing video games is correlated with the amount of time they read for pleasure. You realize that because some of the girls in your sample do not play video games at all, your data will have many zeros for the amount of time they play video games.

You decide to use the Spearman correlation for the data. The first thing you do is convert the data to ranks. You convert the amount of time each girl plays video games by assigning the lowest rank to the six girls who don't play video games and the highest rank to those who play video games the most. The rank you assign to each of the six girls who don't play video games is options: 1, 6, 5.75, 3.5

The following table consists of the data for the 10 girls in your study. The data are sorted in order of the amount of time the girls spent playing video games. Fill in the missing ranks for the time spent playing video games and the ranks for the time spent reading for pleasure. Note that the time spent reading for pleasure is notsorted in order.

Time Spent Playing Video Games (Minutes/Day)	Time Spent Reading for Pleasure (Minutes/Day)	Rank of Time Spent Playing Video Games	Rank of Time Spent Reading for Pleasure
0	10	{your answer above}	4
0	60	{your answer above}	9
0	50	{your answer above}	options: 8, 3.5, 5.75, 10
0	0	{your answer above}	options: 1.5, 1.25, 5, 6
0	22	{your answer above}	options: 6, 1, 4.33, 7
0	11	{your answer above}	options: 5, 3, 3.25, 11
20	0	options: 7, 2, 7.25, 8.5	options: 1.5, 5, 5.25, 8
33	45	options: 8, 2.75, 4, 8.5	options: 7, 2.75, 4, 8
82	65	options: 9, 4.5, 9.25, 10	options: 10, 2.33, 3, 8
126	5	options: 10, 6.5, 7, 7.25	options: 3, 3.33, 4, 10

Using your completed table, you can calculate the Spearman correlation in two different ways. One way is to use the Pearson correlation formula on the ranks of the data. When you do this, you get rss= 0.02. The second way to calculate the Spearman correlation is to use the following formula:

rss

=

1- 6D/n(n-1)

In this formula, D represents the difference between ranks for each individual. Using this formula, you get rss= 0.12. These two estimates of the Spearman correlation are different for the following reason:

Choose the right answers

1) You do not use the Pearson correlation formula when calculating the Spearman correlation.

2) The Pearson correlation formula applied to the ranks loses accuracy when there are many ties.

3) The formula that uses the differences (D) is always only a rough estimate of rss.

4) The formula that uses the differences (D) loses accuracy when there are many ties.