Explanatory and Response Variables. Chapter 4. Question. Scatterplot

Transcription

1 Explanatory and Response Variables Chapter 4 Scatterplots and Correlation Interested in studying the relationship between two variables by measuring both variables on the same individuals. a response variable measures an outcome of a study an explanatory variable explains or influences changes in a response variable sometimes there is no distinction 1 2 Question Consider a study to determine whether surgery or chemotherapy results in higher survival rates for a certain type of cancer What is the explanatory variable? the response variable? Scatterplot Graphs the relationship between two quantitative (numerical) variables measured on the same individuals. If a distinction exists, plot the explanatory variable on the horizontal (x) axis and plot the response variable on the vertical (y) axis. 3 4

2 Scatterplot Scatterplot Relationship between mean SAT verbal score and percent of high school grads taking SAT To add a categorical variable, use a different plot color or symbol for each category Southern states highlighted 5 6 Scatterplot Look for overall pattern and deviations from this pattern Describe pattern by form, direction, and strength of the relationship Look for outliers Linear Relationship Some relationships are such that the points of a scatterplot tend to fall along a straight line -- linear relationship 7 8

3 Direction Positive association above-average values of one variable tend to accompany above-average values of the other variable, and below-average values tend to occur together Positive Association Negative association above-average values of one variable tend to accompany below-average values of the other variable, and vice versa 9 10 Negative Association Examples From a scatterplot of college students, will the association between verbal SAT score and GPA be negative or positive? For used cars, is the association between the age of the car and the selling price positive or negative? 11 12

4 Examples of Relationships Measuring Strength & Direction of a Linear Relationship How closely does a non-horizontal straight line fit the points of a scatterplot? The correlation coefficient r (often referred to as just correlation): measure of the strength of the relationship: the stronger the relationship, the larger the magnitude of r. measure of the direction of the relationship: positive r indicates a positive relationship negative r indicates a negative relationship Correlation Coefficient special values for r : a perfect positive linear relationship would have r = +1 a perfect negative linear relationship would have r = -1 if there is no linear relationship, or if the scatterplot points are best fit by a horizontal line, then r = 0 Note: -1 r +1 both variables must be quantitative; no distinction between response and explanatory variables Examples of Correlations Husband s versus Wife s ages r =.94 Husband s versus Wife s heights r =.36 Professional Golfer s Putting Success: Distance of putt in feet versus percent success r = -.94 r has no units; does not change when measurement units are changed (ex: ft. or in.) 15 16

5 Linear Miles per Gallon versus Speed Linear Miles per Gallon versus Speed Linear relationship? Curved relationship. Correlation is close to zero. Correlation is misleading Problems with Correlations Outliers can inflate or deflate correlations (see next slide) A Outliers and Correlation B Groups combined inappropriately may mask relationships (a third variable) groups may have different relationships when separated For each scatterplot above, how does the outlier affect the correlation? A: outlier decreases the correlation B: outlier increases the correlation 19 20

6 Correlation Calculation Suppose we have data on variables X and Y for n individuals: x 1, x 2,, x n and y 1, y 2,, y n Each variable has a mean and std dev: ( x, s x ) and ( y, s y ) (see ch. 2 for s) r = 1 n - 1 n i =1 # % $ x i x s x &# ( y i y & % ' $ s ( y ' Case Study Per Capita Gross Domestic Product and Average Life Expectancy for Countries in Western Europe Case Study Country Per Capita GDP (x) Life Expectancy (y) Austria Belgium Finland France Germany Ireland Italy Netherlands Switzerland United Kingdom x y Case Study ( x i x )/s x = = s x =1.532 s y =0.795 sum =

7 Case Study Features of the Correlation Coefficient Suppose you measure the correlation between the temperature in New York and that in Boston during a month. Do you expect to get the same correlation if you measure it in Celsius than if you measure it in Fahrenheit? Recall the process of obtaining the correlation. The first step is to convert the samples of both variables to standard units. This implies that the correlation does not depend on units. Thus, no matter if you use Celsius of Fahrenheit, you will get the same correlation. Furthermore, since the correlation depends on the product of the two variables, it does not matter whether you consider the correlation the temperature in NY and that in Boston or the correlation between the temperature in Boston and that in NY Features of the Correlation Coefficient The correlation coefficient has the following properties The correlation is not affected when the two variables are interchanged. The correlation is not changed if the same number is added to all the values of one of the variables. The correlation is not changed if all the values of one of the variables is multiplied by the same positive number. It will change sign if the number is negative. Correlation If women always married men who were five years older, what would the correlation between their ages be? (Hint: It always helps to visualize the scatterplot.) True or False: If the correlation coefficient is then below-average values of the dependent variable tend to be associated with below-average values of the independent variable

8 Correlation A teaching assistant gives a quiz to her section. There are ten questions on the quiz, and no part credit is given. After grading the papers the TA writes down for each student the number of questions the student got right and the number wrong. The average number of wrong answers is 3.6, with the same SD of 2.0. The correlation between number right and number wrong is: can t tell without the data 29