Statistics for Everyone Workshop Fall 2010

1. Statistics for Everyone Workshop Fall 2010 Part 6A Assessing the Relationship Between 2 Numerical Variables Using Correlation Workshop presented by Linda Henkel and Laura McSweeney of Fairfield University Funded by the Core Integration Initiative and the Center for Academic Excellence at Fairfield University

2. Assessing the Relationship Between 2 Numerical Variables With Correlation What if the research question we want to ask is whether or how strongly two variables are related to each other, but we do not have experimental control over those variables and we rely on already existing conditions? • TV viewing and obesity • Age and severity of flu symptoms • Tire pressure and gas mileage • Amount of pressure and compression of insulation

3. Assessing the Relationship Between 2 Variables With Correlation The nature of the research question is about the association between two variables, not about whether one causes the other Correlation describes and quantifies the systematic, linear relation between variables Correlation  Causation

4. Statistics as a Tool in Scientific Research Types of Research Questions • Descriptive (What does X look like?) • Correlational (Is there an association between X and Y? As X increases, what does Y do?) • Experimental (Do changes in X cause changes in Y?) Different statistical procedures allow us to answer the different kinds of research questions

5. Correlation: type and strength of linear relationship between X and Y Linear Regression: making predictions of Variable Y based on knowing the value of Variable X

6. Linear Correlation Test Statistical test for Pearson correlation coefficient (r) Used for: Analyzing the strength of the linear relationship between two numerical variables Use when: Both variables are numerical (interval or ratio) and the research question is about the type and strength of relation (not about causality)

7. Other Correlation Coefficients There are many different types of correlation coefficients – used for different types of variables -- but we focus only on Pearson r Point biserial (rPB): Use when one variable is nominal and has two levels (e.g., gender [male/female], type of car [gas-powered, hybrid]) and one variable is numerical (e.g., reaction time; miles per gallon) Spearman rank order(rS): Use for ordinal data or numerical data that are not normally distributed or linear Calculators for these are available at: http://faculty.vassar.edu/lowry/VassarStats.html

8. The Essence of the Correlation Coefficient The correlation coefficient indicates whether or not a relationship exists (association, co-occurrence, covariation) The value of the correlation coefficient tells you about the type and strength of the relationship • Positive, negative • Strong, moderately strong, weak, no relation

9. Different Types of Correlations Positive: As X increased, Y increased and as X decreased, Y decreased Negative: As X increased, Y decreased and as X decreased, Y increased

10. Different Types of Correlations • No systematic relation between X and Y • High values of X are associated with both high & low values of Y; • Low values of X are associated with both high & low values of Y

11. Different Types of Correlations Curvilinear: Not straight U: As X increased, Y decreased up to a point then increased Inverted U: As X increased, Y increased up to a point, then decreased

12. Describing Linear Correlation Pearson correlation coefficient (r) Direction/type: Positive, Negative, or No correlation Magnitude/strength: r =  1  perfect correlation r closer to 1  strong correlation r closer to 0  weak or no correlation • r is unitless; r is NOT a proportion or percentage • It doesn’t matter which variable you call X or Y when calculating r

13. Issues in Interpreting Correlation Coefficient Magnitude of the Effect: |r| Conclusion About Relationship 0 to .20 negligible to weak .20 to .40 weak to moderate .40 to .60 moderate to strong .60 to 1.0 strong

14. Example 1: What can we say?

15. What can we say? There is little or no correlation between oxide thickness and deposit time. In fact, r = .002.

16. Example 2: What can we say?

17. What can we say? There was a strong, positive correlation between surface area to volume and the drug release rate. As the surface area to volume increased, the drug release rate increased The smaller the surface area to volume, the lower the drug release rate

18. What More Does the Correlation Coefficient Tell Us? r = .99

19. Interpreting Correlation Coefficient A strong linear association was found between the surface area to volume and the drug release rate, r = .99 There was a significant positive correlation between the surface area to volume and the drug release rate, r = .99 As the surface area to volume increased the drug release rate increased, and this was a strong linear relationship, r = .99

20. Testing for Linear Correlation Correlations are descriptive We can describe the type and strength of the linear relationship between two variables by examining the r value r values are associated with p values that reflect the probability that the obtained relationship is real or is more likely just due to chance

21. Running and Interpreting a Correlation The key research question in a correlational design is: Is there a real linear relationship between the two variables or is the obtained pattern no different what we would expect just by chance? i.e. H0: The population correlation coefficient is zero HA: The population correlation coefficient is not zero Teaching tip:In order to understand what we mean by “a real relationship,” students must understand probability

22. What Do We Mean by “Just Due to Chance”? p value = probability of results being due to chance When the p value is high (p > .05), the obtained relationship is probably due to chance .99 .75 .55 .25 .15 .10 .07 When the p value is low (p < .05), the obtained relationship is probably NOT due to chance and more likely reflects a real relationship .04 .03 .02 .01 .001

23. What Do We Mean by “Just Due to Chance”? p value = probability of results being due to chance [Probability of observing your data (or more severe) if H0 were true] When the p value is high (p > .05), the obtained relationship is probably due to chance [Data likely if H0 were true] .99 .75 .55 .25 .15 .10 .07 When the p value is low (p < .05), the obtained relationship is probably NOT due to chance and more likely reflects a real relationship [Data unlikely if H0 were true, so data support HA] .04 .03 .02 .01 .001

24. What Do We Mean by “Just Due to Chance”? In science, a p value of .05is a conventionally accepted cutoff point for saying when a result is more likely due to chance or more likely due to a real effect Not significant = the obtained relationship is probably due to chance; the relationship observed does not appear to really differ from what would be expected based on chance; p > .05 Statistically significant = the obtained relationship is probably NOT due to chance and is likely a real linear relationship between the two variables; p < .05

25. Finding the Linear Correlation Using Excel Step 1: Make a scatterplot of the data If there is a linear trend, go to Step 2 Step 2: Get the Pearson Correlation Coefficient (r) Using the = correl(X, Y) function [Refer to handouts for more detail]

26. Running a Test for Linear Correlation Using Excel Pearson’s r:Both variables are numerical (interval or ratio) and the research question is about the type and strength of relation (not about causality) Need: • Pearson correlation coefficient (r) • Number of pairs of observations (N) To run: Open Excel file “SFE Statistical Tests” and go to page called Linear Correlation Test Enter the correlation coefficient (r) and number of pairs of observations (N) Output: Computer calculates the p value

27. Running a Test for Linear Correlation Using SPSS When to Calculate Pearson’s r:Both variables are numerical (interval or ratio) and the research question is about the type and strength of relation (not about causality) To run: Analyze  Correlate  Bivariate Move the X and Y variables over. Check “Pearson Correlation” and “Two-tailed test of significance”. Then click Ok. Output: Computer calculates r and the p value

28. Reporting Results of Correlation If the relationship was significant (p < .05) (a) Say: A [say something about size: weak, moderate, strong] [say direction: positive/negative] correlation was found between [X] and [Y] that was statistically significant, r = .xx, p = .xx (b) Describe the relation: Thus as X increased, Y [increased/decreased] Note: If the r value is positive, then as X increased, Y increased; if r is negative, then as X increased, Y decreased e.g., A strong positive correlation was found between surface area to volume ratio and the drug release rate that was statistically significant, r = .99, p = .0001. As the surface area to volume ratio increased, the drug release rate increased.

29. Reporting Results of Correlation If the relationship was not significant (p > .05) Say: No statistically significant correlation between [X] and [Y] was found, r = .xx, p = .xx. Thus as X increased, Y neither increased nor decreased systematically. e.g., No statistically significant correlation between oxide thickness and deposit time was found, r = .002, p = .99. Thus as oxide thickness increased, deposit time neither increased nor decreased systematically.

30. Teaching Tip Impress upon your students that an association does not imply causation e.g., Average life expectancy and the average number of TVs per household are highly correlated. But you can’t increase life expectancy by increasing the number of TVs. They are related, but it isn’t a cause-and-effect relationship.

31. More Teaching Tips You can ask your students to report either: • the exact p value (p = .03, p = .45) • the cutoff: say either p < .05 (significant) or p > .05 (not significant) You should specify which style you expect. Ambiguity confuses them! Tell students they can only use the word “significant” only when they mean it (i.e., the probability the results are due to chance is less than 5%) and to not use it with adjectives (i.e., they often mistakenly think one test can be “more significant” or “less significant” than another). Emphasize that “significant” is a cutoff that is either met or not met -- Just like you are either found guilty or not guilty, pregnant or not pregnant. There are no gradients. Lower p values = less likelihood results are due to chance, not “more significant”

32. Correlations are Predictive You can predict one variable from the other if there is a strong correlation. If so, linear regression can be used to find a linear model which can be used to make predictions. But, remember to make a scatterplot of the data to see if a linear model is appropriate and/or the best model!

33. Correlation: type and strength of linear relationship between X and Y Linear Regression: making predictions of Variable Y based on knowing the value of Variable X

34. Example 1 A linear regression seems appropriate since the data have an overall linear trend.

35. Example 2 Even though r = -.92 a linear regression model is not appropriate here. A different model should be used.

36. Terminology X variable: predictor (independent variable, explanatory variable, what we know) Y variable: criterion (dependent variable, response variable, what we want to predict) Stronger correlation = better prediction

37. Making Predictions

38. Making Predictions

39. Using Regression to Make Predictions Low or no correlation(e.g., Oxide thickness and Deposit time) • Best prediction of Y is the mean of Y (knowing X doesn’t add anything) High correlation (e.g., Surface area to volume ratio vs. Drug release rate ) • Best prediction of Y is based on knowing X

40. Sample Scatterplots & Regression Lines If r = +1, Y’ = Y If r = 0, Y’ = MY

41. Terminology Regression line: • Best fitting straight line that summarizes a linear relation • Comprised of the predicted values of Y (denoted by Y’ )

42. What Do We Mean by the “Best Fit” Line?

43. What Do We Mean by the “Best Fit” Line?

44. What Do We Mean by the “Best Fit” Line? It is the line that minimizes the vertical distances from the observed points to the line

45. Least Squares Line Error or residual = observed Y – predicted Y = Y – Y’ The least-squares line is the “best fit” line that minimizes the sum of the square errors/residuals i.e: It minimizes

46. Best Fit Line Also known as linear regression model or least squares line Used to predict Y using a single predictor X and a linear model Y’ = bX + a Y’ = predicted y-value X = known x-value b = slope a = y-intercept or the point where line crosses y-axis [We can use more advanced techniques for multiple predictors or nonlinear models]

47. Interpreting the Model Y’ = bX + a What is b? • Slope of the regression line • The ratio of how much Y changes relative to a change in one unit of X • Same sign as the correlation What is a? • Y-intercept or where the line crosses the Y axis (where X = 0)

48. Getting the Least Squares Line While Excel will get the least squares line, we give the following formulas for completeness.

49. Getting the Least Squares Line in Excel Linear Regression: When you want to predict a numerical variable Y from another numerical variable X Need: • Enter X and Y data in Excel • Be sure that X column is directly to the left of the Y column To get linear model: Make a scatterplot of Y vs. X. Click on the Chart and choose Chart/Add Trendline Choose the Option tab and select Display equation Output: Least squares line

50. Getting the Least Squares Line in SPSS Linear Regression: When you want to predict a numerical variable Y from another numerical variable X Need:Enter X and Y data in SPSS To get linear model: Analyze  Regression  Linear; Move the Y variable to Dependent and the X variable to Independent; Click Ok. Output:Least squares line