Topic 10 - Linear Regression Least squares principle - pages 301 – – 309 Hypothesis tests/confidence intervals/prediction intervals for regression - pages

Regression How much should you pay for a house? Would you consider the median or mean sales price in your area over the past year as a reasonable price? What factors are important in determining a reasonable price? –Amenities –Location – Square footage To determine a price, you might consider a model of the form: Price = f(square footage) + 

Scatter plots To determine the proper functional relationship between two variables, construct a scatter plot. For the home sales data below, what sort of functional relationship exists between Price and SQFT (square footage)?home sales

Simple linear regression The simplest model form to consider is Y i =  0 +  1 X i +  i Y i is called the dependent variable or response. X i is called the independent variable or predictor.  i is the random error term which is typically assumed to have a Normal distribution with mean 0 and variance  2. We also assume that error terms are independent of each other.

Least squares criterion If the simple linear model is appropriate then we need to estimate the values  0 and  1. To determine the line that best fits our data, we choose the line that minimizes the sum of squared vertical deviations from our observed points to the line. In other words, we minimize

Least squares estimators

Home sales example For the home sales data, what are least squares estimates for the line of best fit for Price as a function of SQFT?home sales

Inference Often times, inference for the slope parameter,  1, is most important.  1 tells us the expected change in Y per unit change in X. If we conclude that  1 equals 0, then we are concluding that there is no linear relationship between Y and X. If we conclude that  1 equals 0, then it makes no sense to use our linear model with X to predict Y. has a Normal distribution with a mean of  1 and a variance of.

Hypothesis test for  1 To test H 0 :  1 =  0, use the test statistic HAHA Reject H 0 if  1 <  0 T < - t , n-2  1 >  0 T > t , n-2  1 ≠  0 | T | > t  /2, n-2

Home sales example For the home sales data, is the linear relationship between Price and SQFT significant?home sales

Confidence interval for  1 A (1-  )100% confidence interval for  1 is For the home sales data, what is a 95% confidence interval for the expected increase in price for each additional square foot?home sales

Confidence interval for mean response Sometimes we want a confidence interval for the average (expected) value of Y at a given value of X = x *. With the home sales data, suppose a realtor says the average sales price of a 2000 square foot home is $120,000. Do you believe her?home sales has a Normal distribution with a mean of  0 +  1 x * and a variance of

Confidence interval for mean response A (1-  )100% confidence interval for  0 +  1 x * is With the home sales data, do you believe the realtor’s claim?home sales

Prediction interval for a new response Sometimes we want a prediction interval for a new value of Y at a given value of X = x *. A (1-  )100% prediction interval for Y when X = x * is With the home sales data, what is a 95% prediction interval for the amount you will pay for a 2000 square foot home?home sales

Extrapolation Prediction outside the range of the data is risky and not appropriate as these predictions can be grossly inaccurate. This is called extrapolation. For our home sales example, the prediction formula was developed for homes that were less than 3750 square feet, is it appropriate to use the regression model to predict the price of a home that is 5000 square feet?

Correlation The correlation coefficient, r, describes the direction and strength of the straight-line association between two variables. We will use StatCrunch to calculate r and focus on interpretation. If r is negative, then the association is negative. (A car’s value vs. its age) If r is positive, then the association is positive. (Height vs. weight) r is always between –1 and 1 (-1 < r < 1). –At –1 or 1, there is a perfect straight line relationship. –The closer to –1 or 1, the stronger the relationship. –The closer to 0, the weaker the relationship. Understanding Correlation Correlation by eye

Home sales example For the home sales data, consider the correlation between the variables.home sales

Correlation and regression The square of the correlation, r 2, is the proportion of variation in the value of Y that is explained by the regression model with X. 0  r 2  1 always. The closer r 2 is to 1, the better our model fits the data and the more confident we are in our prediction from the regression model. For the home sales example, r 2 = between price and square footage, so about 71% of the variation in price is due to square footage. Other factors are responsible for the remaining variation.

Association and causation A strong relationship between two variables does not always mean a change in one variable causes changes in the other. The relationship between two variables is often due to both variables being influenced by other variables lurking in the background. The best evidence for causation comes from properly designed randomized comparative experiments.

Does smoking cause lung cancer? Unethical to investigate this relationship with a randomized comparative experiment. Observational studies show strong association between smoking and lung cancer. The evidence from several studies show consistent association between smoking and lung cancer. More and longer cigarettes smoked, the more often lung cancer occurs. Smokers with lung cancer usually began smoking before they developed lung cancer. It is plausible that smoking causes lung cancer Serves as evidence that smoking causes lung cancer, but not as strong as evidence from an experiment.