Topic 10 - Linear Regression

Transcription

1 Topic 10 - Linear Regression Least squares principle Hypothesis tests/confidence intervals/prediction intervals for regression 1

2 Linear 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) + 2

3 Scatter plots To determine the proper functional relationship between two variables, construct a scatter plot. For example, for the home sales data from Topic 8, what sort of functional relationship exists between Price and SQFT (square footage)? 3

4 Simple linear regression The simplest model form to consider is Y i = 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. 4

5 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 n Q ( Yi 0 1Xi) i 1 2 5

6 Scatterplot w/regression line and errors 6

7 7

8 Formulas for Least Squares Estimates ˆ S [( x x)( y y)] [( x x)( y y)] S ( x x) [( x x)(x x)] xy i i i i 1 2 xx i i i 2 ( x x) is the numerator in sample variance; ( x x) 2 s n 1 2, so it is the squared variation of the variable X. 2 ( x x) as [( x x)(x x)] Since you can rewrite i i, then the ˆ estimated slope 1 is the ratio of the joint variation of X and Y, to the variation of X alone. ˆ y ˆ x. Once the slope is identified, it s simply a matter of 0 0 positioning that line within the graph to minimize the squared variability. The intercept is a mathematical fallout position on the slope. 8

9 9

10 10

11 11

12 12

13 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? StatCrunch steps: Load the data set STAT > REGRESSION > SIMPLE LINEAR Set your X and Y values Click Calculate you ll get a result similar to that shown on the next slide. 13

14 Typical Regression Output Simple linear regression results: Dependent Variable: PRICE Independent Variable: SQFT PRICE = SQFT Sample size: 117 R (correlation coefficient) = R-sq = Estimate of error standard deviation: Parameter estimates: Parameter Estimate Std. Err. Alternative DF T-Stat P-Value Intercept Slope <

15 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. ˆ 1 has a Normal distribution with a mean of 1 and a variance of 2 n / ( X ) 2 i X i 1 15

16 Hypothesis test for 1 To test H 0 : 1 = 0, where 0 is the assumed slope for the hypothesis (frequently 0), use the test statistic; T n i 1 ˆ s 1 0 ( X X) i 2 H A 1 < 0 1 > Reject H 0 if T < -t,n-2 T > t,n-2 T > t /2,n-2 16

17 Home sales example For the home sales data, is the linear relationship between Price and SQFT significant or is there a significant linear relationship between the two variables? Simple linear regression results: Dependent Variable: PRICE Independent Variable: SQFT PRICE = SQFT Sample size: 117 R (correlation coefficient) = R-sq = Estimate of error standard deviation: Parameter estimates: Parameter Estimate Std. Err. Alternative DF T-Stat P-Value Intercept Slope <

18 Just to link some concepts together The denominator in the T calculation for the slope estimator is;, the standard error of the slope n i 1 s ( X X) i 2 s ˆ 1 s s ˆ 1 is the is the and the denominator can be solved for algebraically. It s the square root of the numerator in sample variance; 18

19 Confidence interval for 1 A (1-)100% confidence interval for 1 is ˆ t s / ( X X) 1 /2, n2 n i 1 i 2 For the home sales data, what is a 95% confidence interval for the expected increase in price for each additional square foot? Note that StatCrunch doesn t do the CI for you perse, but you do have all the terms you need.the standard error for the slope IS given to you by StatCrunch. 19

20 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? ˆ ˆ x * has a Normal distribution with a mean of x * and 0 1 a variance of * ( x X) n n 2 ( Xi X) i 1 20

21 Confidence interval for mean response A (1-)100% confidence interval for x * is ˆ * 2 ˆ 1 ( x X) x t s n n ( X X) * 0 1 /2, n 2 With the home sales data, do you believe the realtor s claim? i 1 i 2 Note that StatCrunch DOES do this for you. Additional steps once you re in Regression would be to click next, choose Predict Y for X, where X is the X* you re given in the problem and then calculate. You ll get all the basic regression outputs as well as a 95% CI and a 95% Prediction Interval (explained later). 21

22 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 * 2 ˆ ˆ 1 ( x X) x t s 1 n n ( X X) * 0 1 /2, n 2 i 1 i 2 With the home sales data, what is a 95% prediction interval for the amount you will pay for a 2000 square foot home? See slide on Confidence Interval for StatCrunch procedures. 22

23 Extrapolation Prediction outside the range of the data is risky and not appropriate as these predictions can be grossly inaccurate. This is called extrapolation. This is supposed to illustrate why extrapolation is bad. If the confidence intervals get wider the farther that you get from the mean X value, think about what they d look like if you picked some X value that was way out of range.the resulting CI would be so wide, it could be useless. 23

24 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. Run the correlation by eye applet in StatCrunch. 24

25 Home sales example For the home sales data, consider the correlation between the variables. 25

26 Correlation and regression As correlation (r) approaches either -1 (negative relationship) or +1 (positive relationship), it is visually more apparent that the relationship exists and the variation around the line is less. 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. If correlation increases (the data is closer to the line), there is less variation (error) that is unexplained by our model. Therefore, there is a direct relationship between r and r 2, as correlation increases, so does r 2. 26

27 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. 27

28 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. The additional file for Topic 10 contains discussion as well as calculations of key components in regression. 28