4/21/2012 10:07:00 PM Lecture 6-Chapter 4 part 2 Regression Analysis: Exploring Association Between Numeric Variables Review of Last Time:  a scatterplot is a graphical display of 2 numeric variables  correlation is the numeric summary for 2 numeric variables Modeling Statistical Trends  By using 2 variables( such as x and y) a tool for graphical examination (such as a scatterplot) and a measure of association (such as a correlation) we can ask whether one variable can be used to predict another. Models are equations that inform us about the validity of the data The Regression Line  the regression line is used to make predictions  the regression line is a way to summarize linear relationships  by fitting a line to the data we can use this line to make predictions  the simplest form is a straight line with an intercept and a slope - the equations similar to equations seen in algebra ( y=mx +b) in which y represents the outcome/predicted variable, m is the slope, and b is the intercept. However in statistics we will rewrite this equation to (y=a + bx). Y is the predicted variable, a is the intercept, and b is the slope. o Y is called the predicted variable because the line generates PREDICTIONS about its values not the actual values  These predictions come close to real world numbers but are not necessarily exact, there is a percentage of error. The Least Squares Line  the regression line is the “best fit” for the data - the best fit meaning minimizing the average squared vertical distances  this is only useful when the data is a linear model Finding the Best Fit  this is usually done by the computer  find the slope and the intercept- to find the slope (b, which is the ratio of the two variables standard deviations by the correlation coefficient) b=r sy/ sx - b is the slope. r is the correlation coefficient and sy and sx are the standard deviations of both variables. o If the correlation coefficient is near 0 then the slope will most likely be near 0 - Next find the intercept (a) with the value that you have calculated for the slope (b) - a = y – b x - a is the intercept, y bar is the mean of the y value, b is the slope, and x bar is the mean of x - next we take the values we have calculated and put them input them in to a final equation to calculate the predicted value predicted y = a -bx Interpreting the slope if the x variable is increased then y is also predicted to increased by an average of the value of the slope i.e. for each unit of increase in the x variable we predict our y variable to increase by an average value of the slope  if the x variable decreases then the slope will also decrease slope can only be used if the data is a linear model ‘ Interpreting the intercept  the y intercept is the value of y when x is 0 the y intercept is used to interpret data when: - it makes sense to have a value of 0 for x - the y-intercept value is meaningful (for example, if talking about age and the value is negative than the value is obviously not meaningful) - the data include values equal to or close to 0 More Regression the equation will change if x and y are switched (because the standard deviations are different, if they are the same then there is no effect of the equation) statistics language: - the x variable is known as the “explainatory”, “predictor”, “treatment” or “independent variable” - The y variable is the “predicted” “outcome variable” of “dependent variable” o It is important to know these terms because in statistics it is important to understand phrasing More Language In statistics ”please predict a womans BMI from her waist hip ratio (WHR)” - By using statistics language we know that the y variable is the BMI and the x variable is WHR Evaluating the Model General rule: not for non-linear models  if there is not a clear line don’t try to find one Outliers outliers have a strong effect on both correlation and the rquation of the regression line - This strong effect on the regression line is known as the influential point - If there is an influential point preform regression analysis with and without this point Aggreate Datanot aggreate data aggreate data  aggregate data for regression means that each point is represents the mean of all of the y values - This eliminates a lot of variability. (which is not necessarily bad, however it is important to acknowledge that by summarizing or averaging data in this way some varability is lost) BEWARE of extrapolation  only use the regression line to predict y values for x values that are with in the linear range of the dta -  the range is the maximum minus the minimum value, data must be within this range - if the value is out of this range, it could result in nonsensical results - Coefficient of Determination r ^2 (r squared)  r squared measures how much of the variation in y (the response variable) can be accounted for when using x (the explanatory variable).  r squared helps determine which explanatory variable (x) would be best at making predictions about y  we can only account for SOME of the