ࡱ> ^`]q_ Pbjbjzyzy .|6b6bHTTT8\ ppppKKKC E E E E E E $!r$i KKi pp~ ppC C _pKf6.+/  0 3,$$_$_KhETKKKi i KKK $KKKKKKKKKTB : CORRELATION and REGRESSION Earlier we discussed the relationship between two or more variables that were categorical. Here we discuss the relationship between two or more variables that are quantitative. (This is an introductory course so we will stick to two variables.) Correlation means to study the relationship between variables, while regression involves making predictions of one variable based on other variables. Response variable: a variable that measures an outcome Explanatory variable: a variable the may explain or influence a response variable There can be many response and explanatory variables and in fact this happens quite often. However, we will only worry about one of each and call the response variable y and the explanatory variable x. Sometimes we are just interested in studying the relationship between variables in this case no distinction needs to be made between the two. As an example a person interested in health might look for relationships between many measures of health such as BMI, blood pressure, cholesterol level, etc. In this case no distinction needs to be made. If you wish to predict one (or more) variable from another (or several others) the one(s) that are to be predicted are response variable(s) and the one(s) the prediction is based on are the explanatory variable(s). As an example you might want to use such things as BMI, blood pressure, cholesterol level and other variables to predict the life expectancy of a person. In this case the BMI, blood pressure, cholesterol level and so forth are explanatory variables and the life expectancy is the response variable. Scatterplot: A scatterplot shows the relationship between two quantitative variables measured on the same individuals. Each individual is one dot. If there is an explanatory variable put it on the x-axis. Example: What do you think of ranking state school systems by average SAT scores? Is a 559 state SAT math average better than a 548? At first you may think so but there is another variable that might affect SAT math average other than the states school system. This other variable is the percent of students that actually take the SAT. Below is a scatterplot of the 50 states and DC. On the x-axis is the percent taking the SAT and on the y-axis is the SAT math average. We think that the percent taking might explain the average so it goes on the x-axis. In examining a scatterplot you should look for the overall pattern (direction, form, strength) and any outliers. What do you see in the scatterplot with SAT math averages?  This scatterplot has negative relationship. That means that as x increases, y tends to decrease. A positive relationship will have the property that as x increases, y tends to also increase. Scatterplots do not have to have a positive or a negative relationship, they can certainly be neither. In our SAT example you could actually see which states are doing the best. It is not the ones with the highest SAT math averages, but rather the states that have their dots towards the upper part of the general pattern. (The states that have their dots towards the bottom are not doing as well.) The data for the scatterplot was for 2002. Colorado had 28% take the SAT and a SAT math average of 548. Alabama had 9% take the SAT and a math average of 559. Which state do you think is better? Recall the Palm Beach scatterplot:  This scatterplot shows a positive relationship that is basically linear with an extreme outlier! Recall two-way tables. We saw this one before. InstructorAssistant ProfessorAssociate ProfessorFull ProfessorTotalFemale615655024761604Male3961868313832723Total1001183118518594327 Notice that it is like a scatterplot but instead of putting 4327 dots which can only be in 8 places we tell how many dots are in each of the 8 places. So a two-way table is almost like the categorical data version of a scatterplot. Here are a couple of scatterplots that show a strong relationship but it is not linear.   Here is an example of a relationship that at places is positive and at others is negative and is quite strong. This graph is not a scatterplot, but was obtained from a scatterplot. (It shows the curve of best fit.)  Sometimes we want to add categorical variables to a scatterplot. For example in the SAT math example, perhaps you might want to color the southern states a different color. If you could obtain data from say 60 years ago guess where the southern states would be. Where do you think they are in 2002? MEASURING THE STRENGTH OF A LINEAR RELATIONSHIP The population number is  EMBED Equation.3  (rho), which we wont calculate, the sample number is r.  EMBED Equation.3  LINE OF BEST FIT Sometimes it makes sense to find the Prediction Line = Regression Line = Line of Best Fit = Least Squares Line = Least Squares Regression Line. (They are all the same thing.) This is a line through the scatterplot that minimizes the sum of the squares of how far vertically the points are from the line. The following give the equation for the line of best fit for sample data:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  FACTS AND CAUTIONS ABOUT CORRELATION AND REGRESSION If you switch x and y the correlation stays the same. You can tell by looking at the formula for r. r does not change if we change the units of measurement on x and/or y. This is because r uses standardized values. Standardized means how many standard deviations from the mean. If you measure your weight in both pounds and find that you are 1.43 standard deviations above the mean and then convert to kilograms you will still be 1.43 standard deviations above the mean. If r is positive there is a positive association between x and y. If r is negative there is a negative association between x and y. You can tell by looking at the formula for r. If there is a positive relationship then bigger than average xs will tend to have bigger than average ys and smaller than average xs will tend to have smaller than average ys. This will tend to make  EMBED Equation.3  either the product of two positives or the product of two negatives, which are both positive. In the formula for r these are added up. If you add up numbers that tend to be positive you will get a positive. On the other hand, if there is a negative relationship then bigger than average xs will tend to have smaller than average ys and smaller than average xs will tend to have larger than average ys. This will tend to make  EMBED Equation.3  the product of a negative and a positive which is negative. In the formula for r these are added up. If you add up numbers that tend to be negative you will get a negative. r is always between -1 and 1(not obvious, but can be proven). r near 0 indicates a very weak linear relationship. As r moves away from 0 the linear relationship is stronger. At r=1 and r=-1 you have perfect linear relationships. We can see some examples of scatterplots with different rs by following the link at the end of these slides. r measures only the strength of a linear relationship. It will not describe how strong other relationships are. You should plot your data before finding r. You may have a case with r=.5, a somewhat weak linear relationship, but the relationship is extremely strong, just not linear. Again we can see examples with the link at the end of these slides. Like the mean and standard deviation r is not resistant to outliers. These are all strongly affected by a few extreme observations. Again we can see examples with the link at the end of these slides. If you switch x and y you get different regression lines. You can see why this is so by recalling that the regression line minimizes the sum of the squares of the vertical errors. If you switch x and y then instead of vertical errors it will be horizontal errors. Because  EMBED Equation.3 , a change in 1 standard deviation of x results in a change of r standard deviations of y. Recall that slope is rise over run and so it tells us the general change in y relative to x. Also recall that how many standard deviations from the mean (standardized scores) allows for universal comparisons. If we had x=height and y=weight measured in inches and pounds then the slope would give us the how many pounds per inch on average. But we could also conclude that an increase in one standard deviation in height would result in r standard deviations increase in weight.  EMBED Equation.3  tells the percent of differences in y attributed to the regression line on x. There may be many factors that explain the differences in y, the one we are studying is x. If you attempt to find other factors that explain the differences in y, you may very well have your percents greatly exceed 100%. This is because most likely the factors will overlap. y varies for two reasons. There is scatter and because x changes.  EMBED Equation.3 measures the variation due to x changing divided by the total variation. This fact is not obvious but can be proven. You should not find the line of best fit unless the data shows a linear pattern. You can do the calculations no matter, but it may be nonsense. Again we can see examples with the link at the end of these slides. The regression line, just like r, is not resistant to outliers. Again we can see examples with the link at the end of these slides. Extrapolation is the use of the regression line far outside the range of values obtained for x. These predictions are almost never trustworthy. As an example, suppose you collect data from an elementary school for students. You record the ages and the heights and find the line of best fit. Then you try to predict the height of someone that you know is 80 years old. What would happen? The thing to remember is that predictions are only somewhat reliable near where your data was collected. Association does not imply causation. A high correlation does not mean that x causes y, rather that x explains y. Recall the example we mentioned about ice cream sales and sunburns. Suppose we measured ice cream sales and sunburns each day in some mountain resort. What would we find? Does this mean that the ice cream is causing the sunburns? Many times people try to claim a high correlation means causation. As you become more statistically literate, you will not be fooled by such people. Often even when there is causation it may not be the entire explanation for the correlation. As an example suppose someone is interested in genetic causation of BMIs in mothers and daughters. Data is collected and it is found that for the correlation of BMIs for mothers and daughters that r=0.55. This suggests a genetic causation. However, there is also an environmental factor. Daughters learn many habits from their mothers that will affect their BMI, such as exercise and eating habits. Even though there is some genetic causation the r=0.55 not entirely due to genetics, rather it is a mix of genetics and environment. It is extremely difficult to find out how much genetics is contributing to the value of r. Note that it is even possible (although unlikely) that the environment factor is actually negative and the genetic factor contributes more than 0.55 to the value of r. This would happen for example if the daughters saw bad habits and their consequences of their mothers and learned to do the opposite. This may happen from time to time but these are surely anecdotes. We could find the true genetic causation by removing the daughters from their mothers, but people would complain! It is usually quite difficult to determine how much one variable directly affects another one when there are many other variables lurking in the background without doing an experiment. Experiments divide subjects up randomly and by doing so eliminate lurking variables. Recall experiments defeated the lurking variable of the womens importance of their health from the studies that suggested that hormone replacement therapy would cause a reduction in heartattacks in women at menopause. However, experiments are not always possible. Can we use statistics to determine some of the questions below? What is the genetic effect of mothers BMI on daughters BMI? Would stricter gun control laws reduce or increase violent crime? Has increased free trade widened the gap between the incomes of the less educated and more educated? In all these cases experiments are not practical. Statistics can try to answer these questions, but it can very difficult or even impossible. Trying to pinpoint cause and effect in settings involving complex relations between many variables is nearly impossible if experiments cant be done. Still, people that have specific points of view will try to use statistics to further their agenda. Do not fall prey to their shaky arguments. Next we will give two examples in which no experiment have been done and give potential lurking variables that could explain the association and hence mean there is no causation. Does smoking cause lung cancer? No experiments have been done on humans. There is a strong association between smoking and lung cancer. That does not mean that smoking causes lung cancer, recall there is a strong association between ice cream sales and sunburns, and ice cream sales do not cause sunburns! Perhaps there is a lurking variable such as a gene that does two things. It causes people to tend to be addicted to nicotine and it tends to cause tumors in the lungs of people. If this were the case, it would explain why there is such a strong association between smoking and lung cancer. It would be the gene that caused cancer and not the smoking itself. Tobacco executives not too long ago claimed in front of congress that there was no evidence that smoking caused lung cancer. People have pretty much ruled the possibility of such a lurking variable. People who smoke more are more likely to get lung cancer. People who stop smoking reduce their risk. Lung cancer increased dramatically as smoking became more popular (with a lag of about 30 years, since lung cancer develops after years of smoking). Lung cancer was rare among women until smoking became popular among women (again with a lag of about 30 years). Experiments on animals show that the tars from smoking do cause cancer. With all these facts it is unlikely such a genetic factor exists. If it did, this gene started showing up in humans at the same time that smoking became popular! Perhaps there is another lurking variable we havent thought of. Medical authorities dont hesitate to say that smoking causes lung cancer. It is written on packs of cigarettes! They have very good evidence of this. But the evidence would be better if experiments were done on people, but this almost surely will never happen. Does being educated cause people to be wealthier? There is a strong association between education level and wealth. But no experiments have been done. We would have to divide people up at random and educate some and not others! Perhaps there is a lurking variable such as how motivated a person is. Being motivated would cause people to be wealthy and also get an education. If a motivated person did not have the chance for an education perhaps they would put their effort towards becoming wealthy? Do you think that this lurking variable (motivation level) explains all the association between education and wealth? Do you think that it explains part of it? We have seen that lurking variables can make it appear that x has a strong direct affect on y when it doesnt. As another example consider TV sets per 1000 people and life expectancy of the countries of the world.  From the graph it may look like we could cause an increase in life expectancies of poorer countries by shipping them TVs. Do you think this would work? However, lurking variables can also hide an effect or at least make a causation appear not as strong as it really is. Consider data on a household. Degree days per month and amount of electricity used in a month are given (assume the house is heating with electricity and is in the high mountains of Colorado so no AC is needed in the summer). A degree day is the number of degrees that the mean temperature for that day is below 65 F. Suppose that data is keep for several years but that just over a year ago the household installed a few solar panels to produce electricity. Suppose the solar panels were not mentioned and whether or not there were solar panels was a lurking variable. Here is the scatterplot.  The relationship is apparent and not surprising. However if we use different dots to show the months in which the solar panels were in use we get the following:  Notice the association is much stronger within each category than when looked at overall. Here is another reason to include categories in a scatterplot (recall the southern states example earlier). Below is a scatterplot where each dot represents a state. We are comparing SAT Math averages with SAT Verbal averages. Notice a very strong relationship. What do you think would happen if instead of 50 states we had 50 individual students picked at random? Do you think the relationship would be stronger or weaker or the same? Another way of asking this is, do you think there would be more or less or the same amount of scatter?  It is important to remember that if your data is averages you should not apply correlation and regression to individuals with your data. The correlation for averages is much higher than individuals and any prediction for individuals based on averages will not be as trustworthy. There are many other types of regression and correlation besides linear. Examples are quadratic, square root, cubic, exponential, logarithmic, and logistic, just to name a few. There are hundreds of possible forms. Below are the graphs of some common forms. Quadratic . Exponential . Logarithmic . 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