UNDERSTANDING CORRELATION

Transcription

1 Supporting Military Families Through Research and Outreach UNDERSTANDING CORRELATION A HOW-TO GUIDE Military REACH University of Arizona Norton School of Family and Consumer Sciences

2 Introduc on Understanding Correla on: A How To Guide Perhaps one of the most basic and founda onal sta s cal analysis techniques is the correla on. The proper name for correla on is the Pearson Product Moment Correla on; however, it is rarely described as such and instead more commonly referred to as a correla on, Pearson correla on, or Pearson r. In this How To Guide we will cover the basics of correla on as well as provide examples of how correla on is used in academic research. What is a Correla on? A correla on can be defined as the associa on between two variables¹. It is the degree to which two variables share a common rela onship. Consider for example two variables, one called X and one called Y. For each variable there is a distribu on of scores, some scores are high and some scores are low and others are in between. For each variable there is a range of scores that increases from a low score to a high score. If Y increases at a similar rate as X, these two variables would be described as posi vely correlated. The image below on the le provides an example of a posi ve correla on. Alterna vely if Y decreases as X increases then the two variables would be described as nega vely correlated. The image below on the right provides an example of a nega ve correla on. Posi ve Correla on Nega ve Correla on As another example, imagine that instructor quality was assessed in several youth programs. Some instructors are more effec ve than others. A correla on could provide useful informa on about what is associated with students percep on that the instruc on is be er. Many variables could be measured, but an obvious one would be how much me the instructor spent preparing. One would expect that those teachers who prepare more thoroughly are perceived more favorably. A posi ve correla on would confirm that these a ributes of teaching are related. ¹ A variable refers to something that varies. An example might be the age of the participants in a study. The participants may differ in their age and thus will vary between one another. Another example of a variable is something that is measured. For example, in a study on youth programs, instructor quality might be measured. Because some instructors are more effective than others, scores for instructor quality will vary between instructors. 2

3 Proper es of a Correla on Correla on will o en be reported as a number denoted by an r. The coefficient r can range between 1.0 and +1.0.The plus and minus signs indicate the direc on of the rela onship. Minus means for example that lots of prepara on leads to poorer teaching outcomes. In the social sciences a value of.10 is considered small whether it is a posi ve or a nega ve rela onship,.30 is considered medium, and.50 is considered large. The size of the correla on coefficient is important because it communicates the degree to which the two variables are associated. Correla on and Causa on Correla ons do not imply causa on. Two variables can be related without one causing the other. In other words, a correla on between two variables does not suggest that one variable causes another. A correla on can actually be interpreted in three different ways. Take the example of a correla on between instructor quality and instructor prepara on. The first way that this correla on could be interpreted is that greater prepara on leads to greater instructor quality. This interpreta on suggests that instructors that prepare more provide be er quality instruc on. The second way that this correla on could be interpreted is that high quality instructors spend more me preparing. This interpreta on suggests that it may not be the case that greater prepara on me leads to be er instruc on, but rather, that higher quality instructors spend more me preparing. The third interpreta on is that a third possibly unknown variable influences both instructor quality and instructor prepara on. For example, teaching fewer ac vi es may result in be er instructor quality as well as provide more prepara on me for the instructor. All three interpreta ons of the correla on are possible. Repor ng Correla ons in Research Ar cles There are two primary ways that correla ons are reported in research ar cles: (1) in the text and (2) as a correla on matrix². Below is an example of how a correla on might be reported in the text of a research ar cle. Pearson correla ons were conducted to determine the associa on between three variables: instructor quality, instructor prepara on, and number of ac vi es taught. All correla ons were significant at the p <.05 level. The correla on between instructor quality and instructor prepara on was moderate in size (r =.36) while the correla on between instructor quality and number of teaching ac vi es was somewhat small (r =.24). The correla on between instructor prepara on and number of teaching ac vi es emerged as large in size (r =.47). From this example we learn that the correla ons ranged from small to large, that they were all posi ve, and that all were sta s cally significant at the p <.05 level (see the How To Guide on Sta s cal Language for more informa on on p values and significance tes ng). This informa on provides a basis for making judgments about the rela ve importance of each of the variables. In this example, prepara on me had the largest correla on with instructor quality. ² A correlation matrix is a simple and visual way to present several correlations at once. A correlation matrix will be presented as a table in research articles. 3

4 Generally, when correla ons are conducted between three or more variables, the correla ons will be presented in a table as a correla on matrix. Below is an example of what a correla on matrix might look like: Variable Correla on Instructor Quality 2. Instructor Prepara on.36 * 3. Ac vi es Taught.24 *.47 * Note: * p <.05 A correla on matrix is a useful way to summarize a set of correla ons and is more commonly used as the number of correla ons conducted increases. From a correla on matrix we can observe the size, direc on, and significance level of several correla ons at once. Each variable is numbered and named in the le column. In the top row, only the number of the variable appears. To determine how highly related instructor prepara on (variable 2) is to instructor quality (variable 1), look for the number at the intersec on of row 2 (for prepara on) and column 1 (for quality). At this spot, one finds the size of the correla on, here it is.36. The hyphens or dash marks indicate no correla on because that is the space deno ng variable 1 s rela onship to itself. Summary This How To Guide illustrates how correla ons might be presented in research ar cles. The goal of this guide is to improve skills to cri cally read scien fic research.. Readers are encouraged to review our other How To guides on the website that provide informa on about Regression, ANOVA, t Tests, and Sta s cal Language. 4

5 The Arizona Center for Research and Outreach (REACH) Lynne M. Borden, Principal Inves gator Extension Specialist and Professor The Military REACH Team Lynne M. Borden, Ph.D. Principal Inves gator Leslie Bosch, M.S. Noel A. Card, Ph.D. Deborah M. Casper, M.S. Sandra Duque, M.S. Stacy Ann Hawkins, Ph.D. Bryna Koch, M.P.H Mary Koss, Ph.D. Leslie Langbert, M.S.W. Ann Mastergeorge, Ph.D. Stephen Russell, Ph.D. Amy Schaller, M.A. Ashley Jones, B.S. Gabriel Schlomer, Ph.D. (Primary Author) Chris ne Bracamonte Wiggs, M.S. (Co Principal Inves gator) Supporting Military Families Through Research and Outreach The Arizona Center for Research and Outreach The University of Arizona Tucson, Arizona Phone: Website: h p://reachmilitaryfamilies.arizona.edu/ John & Doris Norton School of Family and Consumer Sciences The University of Arizona Tucson, Arizona Phone: Website: h p://ag.arizona.edu/fcs/ 5