Regression line. Regression. Regression line. Slope intercept form review 9/16/09

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1 Regression FPP 10 kind of Regression line Correlation coefficient a nice numerical summary of two quantitative variables It indicates direction and strength of association But does it quantify the association? It would be of interest to do this for Predictions Understanding phenomena Regression line Correlation measures the direction and strength of the straight-line (linear) relationship between two quantitative variables Slope intercept form review If a scatter plot shows a linear relationship, we would like to summarize this overall pattern by drawing a line on the scatter plot This line represents a mathematical model. Later we will make the mathematical model a statistical one. 1

2 Regression line Slope intercept form notation y = mx + b Regression Price of Homes Based on Square Feet Price = SQFT r = Regression form notation ˆ y = a + bx Which line is best Which model to use Different people might draw different lines by eye on a scatterplot Price = SQFT (red) Price = SQFT (blue) Price = SQFT (green) What are some ways we can determine which model(line) out of all the possible models(lines) is the best one? What are some ways that we can numerically rank the different models? (i.e. the different lines) This will come later in the course 2

3 Slope interpretation ˆ y = a + bx The slope, b, of a regression line is almost always important for interpreting the data. The slope is the rate of change, the mean amount of change in y-hat when x increases by 1 Slope interpretation Price of Homes Based on Square Feet Price = SQFT r = For every 1 sqft increase in size of home on average the house price increases by $159.8 dollars Intercept interpretation ˆ y = a + bx The intercept, a, of the regression line is the value of y-hat when x = 0. Although we need the value of the intercept to draw the line, it is statistically meaningful only when x can actually take values close to zero. Intercept interpretation Price of Homes Based on Square Feet Price = SQFT r = If the sqft of a home was 0 on average the house price will be -$90, dollars This doesn t make much sense here because x (sqft) doesn t take on values close to zero. 3

4 Prediction Price of Homes Based on Square Feet Price = SQFT r = For a 3500 sqft home we would predict the selling price to be price = *3500 price = $469,054.2 OECD data: Income and unemployment in the U.S. What is the relationship between households disposable income and the nation s unemployment rate? Data from the U.S to 1998 (data provided by the economics department at Duke) Disposable income vs unemployment rates Disposable income and unemployment rates regression output 4

5 Does regression fit data well? A regression line is reasonable if Association between two variables is indeed linear When points are randomly scattered around line Income/unemployment rate data well-described by regression line. Regression of AIDS rates per 1000 people of GNP per capita Line is too low for GDP values near zero and too high for big GDP values. We shouldn t use line for predictions Birth and death rates in 74 countries Changing the response variable When the regression line fits the data badly, sometimes you can transform variables to obtain a better fitting line. With monetary variables, typically this can be accomplished by taking logarithms. 5

6 Regression of log(aids) on log(gnp) Much better fit Predict log(aids) from log(gnp). Exponentiate to estimate AIDS Facts about regression The distinction between explanatory and response variable is essential in regression If you have a slope computed using x as the explanatory and y as the response variable you can t back solve to get predictions of x given y If you want to predict x given a y then you must find the intercept and slope with y being the explanatory variable and x being the resopnse Facts about regression There is a close relationship between the correlation coefficient and the slope of a regression line b = r SD y SD x They have the same sign They are proportional to each other The intercept has no relationship with the correlation coefficient but here is the formula a = y bx Warnings about regression Predicting y at values of x beyond the range of x in the data is called extrapolation This is risky, because we have no evidence to believe that the association between x and y remains linear for unseen x values Extrapolated predictions can be absolutely wrong 6

7 Extrapolation Diamond price and carat Explanatory variable is measured by carats and response variable is dollars Extrapolation The relationship between diamond carat and price doesn t remain linear after a carat size of about 0.4 Predict price of hope diamond ˆ y = (45.52) = $110, Extrapolation Green line is linear fit with only diamonds less then 0.4 carats Blue line is linear fit with all carat sizes Red curve a quadratic fit Lurking variable A variable not being considered could be driving the relationship In practice this is a difficult issue to tackle. Especially when everything seems OK 7

8 Influential point An outlier in either the X or Y direction which, if removed, would markedly change the value of the slope and y-interept. applet Causality On its own, regression only quantifies an association between x and y It does not prove causality Under a carefully designed experiment (or in some cases observational studies) regression can be used to show causality. 8