Regression Models - Introduction

Transcription

1 Regression Models - Introduction In regression models there are two types of variables that are studied: A dependent variable, Y, also called response variable. It is modeled as random. An independent variable, X, also called predictor variable or explanatory variable. It is sometimes modeled as random and sometimes it has fixed value for each observation. In regression models we are fitting a statistical model to data. We generally use regression to be able to predict the value of one variable given the value of others. STA61 week 13 1

2 Simple Linear Regression - Introduction Simple linear regression studies the relationship between a quantitative response variable Y, and a single explanatory variable X. Idea of statistical model: Actual observed value of Y = Box (a well know statistician) claim: All models are wrong, some are useful. Useful means that they describe the data well and can be used for predictions and inferences. Recall: parameters are constants in a statistical model which we usually don t know but will use data to estimate. STA61 week 13

3 Simple Linear Regression Models The statistical model for simple linear regression is a straight line model of the form X where Y 0 1 For particular points, Yi 0 1X i i, i 1,..., n We expect that different values of X will produce different mean response. In particular we have that for each value of X, the possible values of Y follow a distribution whose mean is... Formally it means that. STA61 week 13 3

4 Estimation Least Square Method Estimates of the unknown parameters β 0 and β 1 based on our observed data are usually denoted by b 0 and b 1. For each observed value x i of X the fitted value of Y is This is an equation of a straight line. yˆ i b0 b1 xi. The deviations from the line in vertical direction are the errors in prediction of Y and are called residuals. They are defined as e i y i y ˆi. The estimates b 0 and b 1 are found by the Method of Lease Squares which is based on minimizing sum of squares of residuals. Note, the least-squares estimates are found without making any statistical assumptions about the data. STA61 week 13 4

5 Derivation of Least-Squares Estimates Let S n yi b0 b1 x i i1 We want to find b 0 and b 1 that minimize S. Use calculus. STA61 week 13 5

6 Statistical Assumptions for SLR Recall, the simple linear regression model is Y i = β 0 + β 1 X i + ε i where i = 1,, n. The assumptions for the simple linear regression model are: 1) E(ε i )=0 ) Var(ε i ) = σ 3) ε i s are uncorrelated. These assumptions are also called Gauss-Markov conditions. The above assumptions can be stated in terms of Y s STA61 week 13 6

7 Possible Violations of Assumptions Straight line model is inappropriate Var(Y i ) increase with X i. Linear model is not appropriate for all the data STA61 week 13 7

8 Properties of Least Squares Estimates The least-square estimates b 0 and b 1 are linear in Y s. That it, there exists constants c i, d i such that, b 0 Proof: Exercise.. c i Y i, b 1 The least squares estimates are unbiased estimators for β 0 and β 1. Proof: d i Yi STA61 week 13 8

9 Gauss-Markov Theorem The least-squares estimates are BLUE (Best Linear, Unbiased Estimators). Of all the possible linear, unbiased estimators of β 0 and β 1 the least squares estimates have the smallest variance. The variance of the least-squares estimates is STA61 week 13 9

10 Estimation of Error Term Variance σ The variance σ of the error terms ε i s needs to be estimated to obtain indication of the variability of the probability distribution of Y. Further, a variety of inferences concerning the regression function and the prediction of Y require an estimate of σ. Recall, for random variable Z the estimates of the mean and variance of Z based on n realization of Z are. Similarly, the estimate of σ is 1 s n n i1 e i S is called the MSE (Mean Square Error) it is an unbiased estimator of σ. STA61 week 13 10

11 Normal Error Regression Model In order to make inference we need one more assumption about ε i s. We assume that ε i s have a Normal distribution, that is ε i ~ N(0, σ ). The Normality assumption implies that the errors ε i s are independent (since they are uncorrelated). Under the Normality assumption of the errors, the least squares estimates of β 0 and β 1 are equivalent to their maximum likelihood estimators. This results in additional nice properties of MLE s: they are consistent, sufficient and MVUE. STA61 week 13 11

12 Inference about the Slope and Intercept Recall, we have established that the least square estimates b 0 and b 1 are linear combinations of the Y i s. Further, we have showed that they are unbiased and have the following variances Var 1 X 0 Var n S b and b 1 XX S XX In order to make inference we assume that ε i s have a Normal distribution, that is ε i ~ N(0, σ ). This in turn means that the Y i s are normally distributed. Since both b 0 and b 1 are linear combination of the Y i s they also have a Normal distribution. STA61 week 13 1

13 Inference for β 1 in Normal Error Regression Model The least square estimate of β 1 is b 1, because it is a linear combination of normally distributed random variables (Y i s) we have the following result: b 1 ~ N 1, S XX We estimate the variance of b 1 by S /S XX where S is the MSE which has n- df. Claim: The distribution 1 1 of is t with n- df. Proof: b S S XX STA61 week 13 13

14 Tests and CIs for β 1 The hypothesis of interest about the slope in a Normal linear regression model is H 0 : β 1 = 0. The test statistic for this hypothesis is t stat S b 1 S XX We compare the above test statistic to a t with n- df distribution to obtain the P-value. b 1 S. E b 1 Further, 100(1-α)% CI for β1 is: S b t n ; S b1 tn; S Eb1 1. XX STA61 week 13 14

15 Important Comment Similar results can be obtained about the intercept in a Normal linear regression model. However, in many cases the intercept does not have any practical meaning and therefore it is not necessary to make inference about it. STA61 week 13 15