Chapter 6: Exploring Data: Relationships Chi-Kwong Li Displaying Relationships: Scatterplots Regression Lines Correlation Least-Squares Regression Interpreting Correlation and Regression
Displaying Relationships 2 Relationship Between Two Variables Examine data for two variables to see if there is a relationship between the variables. Does one influence the other? Study both variables on the same individual. If a relationship exists between variables, typically one variable influences or causes a change in another variable. Response variable – A variable that measures an outcome or result of a study (observed outcome). Explanatory variable – A variable that explains or causes change in the response variable.
Displaying Relationships: Scatterplots A scatterplot is a graph that shows the relationship between two numerical variables, measured on the same individual. Explanatory variable and response variable are plotted as (x, y) pairs on the graph. 3 Example: A study done to see how the number of beers that a young adult drinks predicts his/her blood alcohol content (BAC). Results of 16 people: Young Adult12345678 Beers52983735 BAC0.100.030.190.120.040.0950.070.06 Young Adult910111213141516 Beers35465714 BAC0.020.050.070.100.850.090.010.05 Explanatory variable, x = beers drunk Response variable, y = BAC level
Displaying Relationships: Scatterplots Scatterplot of the data from the previous table are plotted as points on the graph at the right side. 4 About This Scatterplot… 1.What is the overall pattern (form, direction, and strength)? Form – Roughly a straight-line pattern. Direction – Positive association (both increase). Strength – Moderately strong (mostly on line). 2.Any striking deviations (outliers)? Not here. BAC vs. number of beers consumed Outliers – A deviation in a distribution of a data point falling outside the overall pattern.
Regression Lines They describe how response variables y change as explanatory variables x change. They are used to predict the value of y for a given x. 5 Beer Example continued: A regression line has been added to be able to predict blood alcohol content from the number of beers a young adult drinks. Graphically, you can predict that if x = 6 beers, then y = 0.95 BAC. (Legal limit for driving in most states is BAC = 0.08.) BAC vs. number of beers consumed
Regression Lines (continued) It is easier to use the equation of the line for predicting the value of y, given the value of x. Using the equation of the line for the previous example: predicted BAC = −0.0127 + (0.01796)(beers) y = −0.127 + 0.01796 (x) For a young adult drinking 6 beers (x = 6): predicted BAC = −0.0127 + 0.01796 (6) = 0.095 A straight line equation has the form: predicted y = a + b x In this equation, b is the slope, the amount by which y changes when x increases by 1 unit. The number a is the intercept, the value of y when x =0. 6
Correlation 7 Measures the direction and strength of the straight-line relationship between two numerical variables. Correlation is usually written as r, a number between −1 and 1. It has the same sign as the slope of a regression line. r > 0 for positive association. r < 0 for negative association. Perfect correlation r = 1 or r = −1 occurs only when all points lie exactly on a straight line. The correlation moves away from 1 or −1 (toward zero) as the straight-line relationship gets weaker. Correlation r = 0 indicates no straight-line relationship.
Equation of the Correlation 8 To calculate the correlation, suppose you have data on variable x and y for n individuals. From the data, you have the values calculated for the means and standard deviations x and y. The correlation r between x and y is: _ _ r = [ sum ( x_i - x )( y_i – y )] / [(n -1) s_x s_y] Correlation is strongly affected by a few outlying observations. (Also, the mean and standard deviation are affected by outliers.)
Correlation (continued) The scatterplots below show examples of how the correlation r measures the direction and strength of a straight-line association. 9
Least Squares Regression Least-Squares Regression Line A line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible. Equation of the Least-Squares Regression Line From the data for an explanatory variable x and a response variable y for n individuals, we have calculated the means ¯, ¯, and standard deviations s x, s y, as well as their correlation r. 10 This equation was used to calculate the line for predicting BAC for number of beers drunk. Predicted y = −0.0127 + 0.01796 x The least-squares regression line is the line: Predicted y = a + bx With slope … b = r s y /s x And intercept … a = ¯ − b ¯ xy xy
Interpreting Correlation and Regression Both the correlation r and least-squares regression line can be strongly influenced by a few outlying points. Always make a scatterplot before doing any calculations. Often the relationship between two variables is strongly influenced by other variables. Before conclusions are drawn based on correlation and regression, other possible effects of other variables should be considered. A strong association between two variables is not enough to draw conclusions about cause and effect. Sometimes an observed association really does reflect cause and effect (such as drinking beer causes increased BAC). Sometimes a strong association is explained by other variables that influence both x and y. Remember, association does not imply causation. 11
Discussion Examples of positive / negative correlations in daily life. Other comments? Thank you !