MATH 829: Introduction to Data Mining and Analysis Linear Regression: statistical tests
|
|
- Charity Jones
- 6 years ago
- Views:
Transcription
1 1/16 MATH 829: Introduction to Data Mining and Analysis Linear Regression: statistical tests Dominique Guillot Departments of Mathematical Sciences University of Delaware February 17, 2016
2 Statistical hypothesis testing 2/16 Suppose we have a linear model for some data: Y = β 0 + β 1 X β p X p + ɛ.
3 Statistical hypothesis testing 2/16 Suppose we have a linear model for some data: Y = β 0 + β 1 X β p X p + ɛ. An important problem is to identify which variables are really useful in predicting Y.
4 Statistical hypothesis testing 2/16 Suppose we have a linear model for some data: Y = β 0 + β 1 X β p X p + ɛ. An important problem is to identify which variables are really useful in predicting Y. We want to decide if β i = 0 or not with some level of condence. Also want to test if groups of coecients {β ik : k = 1,..., l} are zero.
5 Statistical hypothesis testing (cont.) 3/16 Recall: to do a statistical test: 1 State a null hypothesis H 0 and an alternative hypothesis H 1. 2 Construct an appropriate test statistics. 3 Derive the distribution of the test statistic under the null hypothesis. 4 Select a signicance level α (typically 5% or 1%). 5 Compute the test statistics and decide if the null hypothesis is rejected at the given signicance level.
6 Example 4/16 Example: Suppose X N(µ, 1) with µ unknown. We want to test: H 0 : µ = 0 H 1 : µ 0. We have an iid sample X 1,..., X n N(µ, 1).
7 Example 4/16 Example: Suppose X N(µ, 1) with µ unknown. We want to test: H 0 : µ = 0 H 1 : µ 0. We have an iid sample X 1,..., X n N(µ, 1). Recall: if X N(µ 1, σ1 2) and Y N(µ 2, σ2 2 ) are independent, then X + Y N(µ 1 + µ 2, σ1 2 + σ2 2 ).
8 Example 4/16 Example: Suppose X N(µ, 1) with µ unknown. We want to test: H 0 : µ = 0 H 1 : µ 0. We have an iid sample X 1,..., X n N(µ, 1). Recall: if X N(µ 1, σ1 2) and Y N(µ 2, σ2 2 ) are independent, then X + Y N(µ 1 + µ 2, σ1 2 + σ2 2 ). Therefore, under H 0, we have ˆµ = 1 n X i N(0, 1 n n ). Test statistics: n ˆµ N(0, 1).
9 Example 4/16 Example: Suppose X N(µ, 1) with µ unknown. We want to test: H 0 : µ = 0 H 1 : µ 0. We have an iid sample X 1,..., X n N(µ, 1). Recall: if X N(µ 1, σ1 2) and Y N(µ 2, σ2 2 ) are independent, then X + Y N(µ 1 + µ 2, σ1 2 + σ2 2 ). Therefore, under H 0, we have ˆµ = 1 n X i N(0, 1 n n ). Test statistics: n ˆµ N(0, 1). Suppose we observed: nˆµ = k.
10 Example 4/16 Example: Suppose X N(µ, 1) with µ unknown. We want to test: H 0 : µ = 0 H 1 : µ 0. We have an iid sample X 1,..., X n N(µ, 1). Recall: if X N(µ 1, σ1 2) and Y N(µ 2, σ2 2 ) are independent, then X + Y N(µ 1 + µ 2, σ1 2 + σ2 2 ). Therefore, under H 0, we have ˆµ = 1 n X i N(0, 1 n n ). Test statistics: n ˆµ N(0, 1). Suppose we observed: nˆµ = k. We compute: P ( z α nˆµ z α ) = P ( z α N(0, 1) z α ) = 1 α.
11 Example Example: Suppose X N(µ, 1) with µ unknown. We want to test: H 0 : µ = 0 H 1 : µ 0. We have an iid sample X 1,..., X n N(µ, 1). Recall: if X N(µ 1, σ1 2) and Y N(µ 2, σ2 2 ) are independent, then X + Y N(µ 1 + µ 2, σ1 2 + σ2 2 ). Therefore, under H 0, we have ˆµ = 1 n X i N(0, 1 n n ). Test statistics: n ˆµ N(0, 1). Suppose we observed: nˆµ = k. We compute: P ( z α nˆµ z α ) = P ( z α N(0, 1) z α ) = 1 α. Reject the null hypothesis if k [ z α, z α ]. 4/16
12 Example (cont.) 5/16 For example, suppose: α = 0.05, nˆµ = 2.2. If Z N(0, 1), then P ( 1.96 Z 1.96) = 0.95
13 Example (cont.) 5/16 For example, suppose: α = 0.05, nˆµ = 2.2. If Z N(0, 1), then P ( 1.96 Z 1.96) = 0.95 Therefore, it is very unlikely to observe nˆµ = 2.2 if µ = 0. We reject the null hypothesis.
14 Example (cont.) 5/16 For example, suppose: α = 0.05, nˆµ = 2.2. If Z N(0, 1), then P ( 1.96 Z 1.96) = 0.95 Therefore, it is very unlikely to observe nˆµ = 2.2 if µ = 0. We reject the null hypothesis. In fact, P ( 2.2 Z 2.2) So our p-value is
15 Example (cont.) 5/16 For example, suppose: α = 0.05, nˆµ = 2.2. If Z N(0, 1), then P ( 1.96 Z 1.96) = 0.95 Therefore, it is very unlikely to observe nˆµ = 2.2 if µ = 0. We reject the null hypothesis. In fact, P ( 2.2 Z 2.2) So our p-value is Type I error: H 0 true, but rejected False positive. (Controlled by the level α).
16 Example (cont.) For example, suppose: α = 0.05, nˆµ = 2.2. If Z N(0, 1), then P ( 1.96 Z 1.96) = 0.95 Therefore, it is very unlikely to observe nˆµ = 2.2 if µ = 0. We reject the null hypothesis. In fact, P ( 2.2 Z 2.2) So our p-value is Type I error: H 0 true, but rejected False positive. (Controlled by the level α). Type II error: H 0 false, but not rejected False negative. (Power of the test). 5/16
17 Testing if coecients are zero 6/16 In practice, we often include unnecessary variables in linear models.
18 Testing if coecients are zero 6/16 In practice, we often include unnecessary variables in linear models. These variables bias our estimator, and can lead to poor performance.
19 Testing if coecients are zero 6/16 In practice, we often include unnecessary variables in linear models. These variables bias our estimator, and can lead to poor performance. Need ways of identifying a good set of predictors. We now discuss a classical approach that uses statistical tests.
20 Testing if coecients are zero 6/16 In practice, we often include unnecessary variables in linear models. These variables bias our estimator, and can lead to poor performance. Need ways of identifying a good set of predictors. We now discuss a classical approach that uses statistical tests. Before, we tested if the mean of a N(µ, 1) is zero: H 0 : µ = 0 H 1 : µ 0. assuming σ 2 = 1 is known. What if the variance is unknown? Sample variance: s 2 = 1 n 1 n (x i x) 2, x = 1 n n x i.
21 7/16 In general, suppose X N(µ, σ 2 ) with σ 2 known and we want to test H 0 : µ = µ 0 H 1 : µ µ 0.
22 7/16 In general, suppose X N(µ, σ 2 ) with σ 2 known and we want to test H 0 : µ = µ 0 H 1 : µ µ 0. Under the H 0 hypothesis, we have X := 1 n X i N n ) (µ 0, σ2 n
23 7/16 In general, suppose X N(µ, σ 2 ) with σ 2 known and we want to test H 0 : µ = µ 0 H 1 : µ µ 0. Under the H 0 hypothesis, we have X := 1 n X i N n Therefore, we use the test statistic ) (µ 0, σ2 n Z = X µ 0 σ/ n N(0, 1).
24 7/16 In general, suppose X N(µ, σ 2 ) with σ 2 known and we want to test H 0 : µ = µ 0 H 1 : µ µ 0. Under the H 0 hypothesis, we have X := 1 n X i N n Therefore, we use the test statistic Z = X µ 0 σ/ n ) (µ 0, σ2 n N(0, 1). If the variance is unknown, we replace σ by its sample version s T = X µ 0 s/ n t n 1.
25 Review: the student distribution The student t ν distribution with ν degrees of freedom: f ν (t) = Γ ( ) ν+1 ( ) ν+1 2 ( νπ Γ ν ) 1 + t2 2, ν 2 where Γ is the Gamma function. When X 1,..., X n are iid N(µ, σ 2 ), then X µ s/ n t n 1. 8/16
26 9/16 Back to testing regression coecients: suppose y i = x i,1 β 1 + x i,2 β x i,p β p + ɛ i, where (x ij ) is a xed matrix, and ɛ i are iid N(0, σ 2 ). We saw that this implies ˆβ N(β, (X T X) 1 σ 2 ).
27 9/16 Back to testing regression coecients: suppose y i = x i,1 β 1 + x i,2 β x i,p β p + ɛ i, where (x ij ) is a xed matrix, and ɛ i are iid N(0, σ 2 ). We saw that this implies ˆβ N(β, (X T X) 1 σ 2 ). In particular, ˆβ i N(β i, v i σ 2 ), where v i is the i-th diagonal element of (X T X) 1.
28 9/16 Back to testing regression coecients: suppose y i = x i,1 β 1 + x i,2 β x i,p β p + ɛ i, where (x ij ) is a xed matrix, and ɛ i are iid N(0, σ 2 ). We saw that this implies ˆβ N(β, (X T X) 1 σ 2 ). In particular, ˆβ i N(β i, v i σ 2 ), where v i is the i-th diagonal element of (X T X) 1. We want to test: H 0 : β i = 0 H 1 : β i 0.
29 Back to testing regression coecients: suppose y i = x i,1 β 1 + x i,2 β x i,p β p + ɛ i, where (x ij ) is a xed matrix, and ɛ i are iid N(0, σ 2 ). We saw that this implies ˆβ N(β, (X T X) 1 σ 2 ). In particular, ˆβ i N(β i, v i σ 2 ), where v i is the i-th diagonal element of (X T X) 1. We want to test: H 0 : β i = 0 H 1 : β i 0. Note: v i is known, but σ is unknown. 9/16
30 10/16 Recall: y i = x i,1 β 1 + x i,2 β x i,p β p + ɛ i, Problem: How do we estimate σ?
31 10/16 Recall: y i = x i,1 β 1 + x i,2 β x i,p β p + ɛ i, Problem: How do we estimate σ? Let ˆɛ i = y i (x i,1 ˆβ1 + x i,2 ˆβ2 + + x i,p ˆβp ).
32 10/16 Recall: y i = x i,1 β 1 + x i,2 β x i,p β p + ɛ i, Problem: How do we estimate σ? Let ˆɛ i = y i (x i,1 ˆβ1 + x i,2 ˆβ2 + + x i,p ˆβp ). What about: ˆσ 2 = 1 n ˆɛ 2 i? n What is E(ˆσ 2 )?
33 Recall: y i = x i,1 β 1 + x i,2 β x i,p β p + ɛ i, Problem: How do we estimate σ? Let ˆɛ i = y i (x i,1 ˆβ1 + x i,2 ˆβ2 + + x i,p ˆβp ). What about: ˆσ 2 = 1 n ˆɛ 2 i? n What is E(ˆσ 2 )? We have y = Xβ + ɛ and ˆβ = (X T X) 1 X T y. Thus, ˆɛ = y X ˆβ = y X(X T X) 1 X T y = (I X(X T X) 1 X T )y = (I X(X T X) 1 X T )(Xβ + ɛ) = (I X(X T X) 1 X T )ɛ = Mɛ. 10/16
34 11/16 We showed ˆɛ = Mɛ where M := I X(X T X) 1 X T. Now
35 11/16 We showed ˆɛ = Mɛ where M := I X(X T X) 1 X T. Now n ˆɛ 2 i = ˆɛ T ˆɛ = ɛ T M T Mɛ.
36 11/16 We showed ˆɛ = Mɛ where M := I X(X T X) 1 X T. Now n ˆɛ 2 i = ˆɛ T ˆɛ = ɛ T M T Mɛ. Note: M T = M and M T M = M 2 = (I X(X T X) 1 X T )(I X(X T X) 1 X T ) = I X(X T X) 1 X T = M. (In other words, M is idempotent.)
37 11/16 We showed ˆɛ = Mɛ where M := I X(X T X) 1 X T. Now n ˆɛ 2 i = ˆɛ T ˆɛ = ɛ T M T Mɛ. Note: M T = M and M T M = M 2 = (I X(X T X) 1 X T )(I X(X T X) 1 X T ) = I X(X T X) 1 X T = M. (In other words, M is idempotent.) Therefore, n ˆɛ 2 i = ɛ T Mɛ.
38 We showed ˆɛ = Mɛ where M := I X(X T X) 1 X T. Now n ˆɛ 2 i = ˆɛ T ˆɛ = ɛ T M T Mɛ. Note: M T = M and M T M = M 2 = (I X(X T X) 1 X T )(I X(X T X) 1 X T ) = I X(X T X) 1 X T = M. (In other words, M is idempotent.) Therefore, n ˆɛ 2 i = ɛ T Mɛ. Now, E(ɛ T Mɛ) = E(tr(Mɛɛ T )) = tr E(Mɛɛ T ) = tr ME(ɛɛ T ) = tr Mσ 2 I = σ 2 tr M. 11/16
39 12/16 We proved: E( n ˆɛ 2 i ) = σ 2 tr M, where M = I X(X T X) 1 X T. (Here I = I n, the n n identity matrix.) What is tr M?
40 12/16 We proved: E( n ˆɛ 2 i ) = σ 2 tr M, where M = I X(X T X) 1 X T. (Here I = I n, the n n identity matrix.) What is tr M? Recall tr(ab) = tr(ba). Thus,
41 12/16 We proved: E( n ˆɛ 2 i ) = σ 2 tr M, where M = I X(X T X) 1 X T. (Here I = I n, the n n identity matrix.) What is tr M? Recall tr(ab) = tr(ba). Thus, tr M = tr(i X(X T X) 1 X T ) = n tr(x(x T X) 1 X T ) = n tr(x T X(X T X) 1 ) = n tr(i p ) = n p.
42 We proved: E( n ˆɛ 2 i ) = σ 2 tr M, where M = I X(X T X) 1 X T. (Here I = I n, the n n identity matrix.) What is tr M? Recall tr(ab) = tr(ba). Thus, Therefore, tr M = tr(i X(X T X) 1 X T ) = n tr(x(x T X) 1 X T ) = n tr(x T X(X T X) 1 ) = n tr(i p ) = n p. 1 n n p E( ˆɛ 2 i ) = σ 2. 12/16
43 13/16 As a result of the previous calculation, our estimator of the variance σ 2 in the regression model will be s 2 = 1 n (y i ŷ i ) 2, n p where ŷ i := x i,1 ˆβ1 + x i,2 ˆβ2 + + x i,p ˆβp is our prediction of y i.
44 13/16 As a result of the previous calculation, our estimator of the variance σ 2 in the regression model will be s 2 = 1 n (y i ŷ i ) 2, n p where ŷ i := x i,1 ˆβ1 + x i,2 ˆβ2 + + x i,p ˆβp is our prediction of y i. Our test statistic is T = ˆβ i s, v i = ((X T X) 1 ) ii. v i
45 13/16 As a result of the previous calculation, our estimator of the variance σ 2 in the regression model will be s 2 = 1 n (y i ŷ i ) 2, n p where ŷ i := x i,1 ˆβ1 + x i,2 ˆβ2 + + x i,p ˆβp is our prediction of y i. Our test statistic is T = ˆβ i s, v i = ((X T X) 1 ) ii. v i Under the null hypothesis H 0 : β i = 0, one can show that the above T statistic has a student distribution with n p degrees of freedom: T t n p.
46 13/16 As a result of the previous calculation, our estimator of the variance σ 2 in the regression model will be s 2 = 1 n (y i ŷ i ) 2, n p where ŷ i := x i,1 ˆβ1 + x i,2 ˆβ2 + + x i,p ˆβp is our prediction of y i. Our test statistic is T = ˆβ i s, v i = ((X T X) 1 ) ii. v i Under the null hypothesis H 0 : β i = 0, one can show that the above T statistic has a student distribution with n p degrees of freedom: T t n p. Thus, to test if β i = 0, we compute the value of the T satistic, say T = ˆT and reject the null hypothesis (at the α = 5% level) if P ( t n p ˆT ) 0.05.
47 As a result of the previous calculation, our estimator of the variance σ 2 in the regression model will be s 2 = 1 n (y i ŷ i ) 2, n p where ŷ i := x i,1 ˆβ1 + x i,2 ˆβ2 + + x i,p ˆβp is our prediction of y i. Our test statistic is T = ˆβ i s, v i = ((X T X) 1 ) ii. v i Under the null hypothesis H 0 : β i = 0, one can show that the above T statistic has a student distribution with n p degrees of freedom: T t n p. Thus, to test if β i = 0, we compute the value of the T satistic, say T = ˆT and reject the null hypothesis (at the α = 5% level) if P ( t n p ˆT ) Important: This procedure cannot be iterated to remove multiple coecients. We will see how this is done later. 13/16
48 Condence intervals for the regression coecients 14/16 Recall that ˆβ i N(β i, v i σ 2 ). Using our esimate s 2 for σ 2, we can construct a 1 2α condence interval for β i : ( ˆβi z (1 α) v i s, ˆβ i + z (1 α) v i s). Here z (1 α) is the (1 α)-th percentile of the N(0, 1) distribution, i.e., P (Z z 1 α ) = 1 α.
49 Python 15/16 Unfortunately, scikit-learn doesn't compute t-statistics and condence intervals. However, the module statsmodels provides exactly what we need. import numpy as np import statsmodels.api as sm import statsmodels.formula.api as smf # Load data dat = sm.datasets.get_rdataset("guerry", "HistData").data # Fit regression model (using the natural log # of one of the regressors) results = smf.ols('lottery ~ Literacy + np.log(pop1831)', data=dat).fit() # Inspect the results print results.summary()
50 Python (cont.) 16/16
MATH 829: Introduction to Data Mining and Analysis Consistency of Linear Regression
1/9 MATH 829: Introduction to Data Mining and Analysis Consistency of Linear Regression Dominique Guillot Deartments of Mathematical Sciences University of Delaware February 15, 2016 Distribution of regression
More informationBANA 7046 Data Mining I Lecture 2. Linear Regression, Model Assessment, and Cross-validation 1
BANA 7046 Data Mining I Lecture 2. Linear Regression, Model Assessment, and Cross-validation 1 Shaobo Li University of Cincinnati 1 Partially based on Hastie, et al. (2009) ESL, and James, et al. (2013)
More informationSTAT 135 Lab 13 (Review) Linear Regression, Multivariate Random Variables, Prediction, Logistic Regression and the δ-method.
STAT 135 Lab 13 (Review) Linear Regression, Multivariate Random Variables, Prediction, Logistic Regression and the δ-method. Rebecca Barter May 5, 2015 Linear Regression Review Linear Regression Review
More informationSimple and Multiple Linear Regression
Sta. 113 Chapter 12 and 13 of Devore March 12, 2010 Table of contents 1 Simple Linear Regression 2 Model Simple Linear Regression A simple linear regression model is given by Y = β 0 + β 1 x + ɛ where
More informationCh 2: Simple Linear Regression
Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component
More informationLinear Regression. In this problem sheet, we consider the problem of linear regression with p predictors and one intercept,
Linear Regression In this problem sheet, we consider the problem of linear regression with p predictors and one intercept, y = Xβ + ɛ, where y t = (y 1,..., y n ) is the column vector of target values,
More informationBusiness Statistics. Tommaso Proietti. Linear Regression. DEF - Università di Roma 'Tor Vergata'
Business Statistics Tommaso Proietti DEF - Università di Roma 'Tor Vergata' Linear Regression Specication Let Y be a univariate quantitative response variable. We model Y as follows: Y = f(x) + ε where
More informationProbability and Statistics Notes
Probability and Statistics Notes Chapter Seven Jesse Crawford Department of Mathematics Tarleton State University Spring 2011 (Tarleton State University) Chapter Seven Notes Spring 2011 1 / 42 Outline
More information18.S096 Problem Set 3 Fall 2013 Regression Analysis Due Date: 10/8/2013
18.S096 Problem Set 3 Fall 013 Regression Analysis Due Date: 10/8/013 he Projection( Hat ) Matrix and Case Influence/Leverage Recall the setup for a linear regression model y = Xβ + ɛ where y and ɛ are
More informationCh 3: Multiple Linear Regression
Ch 3: Multiple Linear Regression 1. Multiple Linear Regression Model Multiple regression model has more than one regressor. For example, we have one response variable and two regressor variables: 1. delivery
More informationReview: General Approach to Hypothesis Testing. 1. Define the research question and formulate the appropriate null and alternative hypotheses.
1 Review: Let X 1, X,..., X n denote n independent random variables sampled from some distribution might not be normal!) with mean µ) and standard deviation σ). Then X µ σ n In other words, X is approximately
More information3 Multiple Linear Regression
3 Multiple Linear Regression 3.1 The Model Essentially, all models are wrong, but some are useful. Quote by George E.P. Box. Models are supposed to be exact descriptions of the population, but that is
More informationMaximum Likelihood Estimation
Maximum Likelihood Estimation Merlise Clyde STA721 Linear Models Duke University August 31, 2017 Outline Topics Likelihood Function Projections Maximum Likelihood Estimates Readings: Christensen Chapter
More informationLecture 15. Hypothesis testing in the linear model
14. Lecture 15. Hypothesis testing in the linear model Lecture 15. Hypothesis testing in the linear model 1 (1 1) Preliminary lemma 15. Hypothesis testing in the linear model 15.1. Preliminary lemma Lemma
More informationMath 423/533: The Main Theoretical Topics
Math 423/533: The Main Theoretical Topics Notation sample size n, data index i number of predictors, p (p = 2 for simple linear regression) y i : response for individual i x i = (x i1,..., x ip ) (1 p)
More informationMATH 829: Introduction to Data Mining and Analysis Least angle regression
1/14 MATH 829: Introduction to Data Mining and Analysis Least angle regression Dominique Guillot Departments of Mathematical Sciences University of Delaware February 29, 2016 Least angle regression (LARS)
More informationMultiple Linear Regression
Multiple Linear Regression Simple linear regression tries to fit a simple line between two variables Y and X. If X is linearly related to Y this explains some of the variability in Y. In most cases, there
More informationMATH 829: Introduction to Data Mining and Analysis Linear Regression: old and new
1/15 MATH 829: Introduction to Data Mining and Analysis Linear Regression: old and new Dominique Guillot Departments of Mathematical Sciences University of Delaware February 10, 2016 Linear Regression:
More informationSimple Linear Regression
Simple Linear Regression ST 430/514 Recall: A regression model describes how a dependent variable (or response) Y is affected, on average, by one or more independent variables (or factors, or covariates)
More informationData Mining Stat 588
Data Mining Stat 588 Lecture 02: Linear Methods for Regression Department of Statistics & Biostatistics Rutgers University September 13 2011 Regression Problem Quantitative generic output variable Y. Generic
More information1 Least Squares Estimation - multiple regression.
Introduction to multiple regression. Fall 2010 1 Least Squares Estimation - multiple regression. Let y = {y 1,, y n } be a n 1 vector of dependent variable observations. Let β = {β 0, β 1 } be the 2 1
More informationLinear models and their mathematical foundations: Simple linear regression
Linear models and their mathematical foundations: Simple linear regression Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/21 Introduction
More informationSTA442/2101: Assignment 5
STA442/2101: Assignment 5 Craig Burkett Quiz on: Oct 23 rd, 2015 The questions are practice for the quiz next week, and are not to be handed in. I would like you to bring in all of the code you used to
More informationMAT2377. Rafa l Kulik. Version 2015/November/26. Rafa l Kulik
MAT2377 Rafa l Kulik Version 2015/November/26 Rafa l Kulik Bivariate data and scatterplot Data: Hydrocarbon level (x) and Oxygen level (y): x: 0.99, 1.02, 1.15, 1.29, 1.46, 1.36, 0.87, 1.23, 1.55, 1.40,
More informationvariability of the model, represented by σ 2 and not accounted for by Xβ
Posterior Predictive Distribution Suppose we have observed a new set of explanatory variables X and we want to predict the outcomes ỹ using the regression model. Components of uncertainty in p(ỹ y) variability
More informationLecture 6 Multiple Linear Regression, cont.
Lecture 6 Multiple Linear Regression, cont. BIOST 515 January 22, 2004 BIOST 515, Lecture 6 Testing general linear hypotheses Suppose we are interested in testing linear combinations of the regression
More informationINTERVAL ESTIMATION AND HYPOTHESES TESTING
INTERVAL ESTIMATION AND HYPOTHESES TESTING 1. IDEA An interval rather than a point estimate is often of interest. Confidence intervals are thus important in empirical work. To construct interval estimates,
More information3. For a given dataset and linear model, what do you think is true about least squares estimates? Is Ŷ always unique? Yes. Is ˆβ always unique? No.
7. LEAST SQUARES ESTIMATION 1 EXERCISE: Least-Squares Estimation and Uniqueness of Estimates 1. For n real numbers a 1,...,a n, what value of a minimizes the sum of squared distances from a to each of
More informationLinear Regression. September 27, Chapter 3. Chapter 3 September 27, / 77
Linear Regression Chapter 3 September 27, 2016 Chapter 3 September 27, 2016 1 / 77 1 3.1. Simple linear regression 2 3.2 Multiple linear regression 3 3.3. The least squares estimation 4 3.4. The statistical
More informationRegression and Statistical Inference
Regression and Statistical Inference Walid Mnif wmnif@uwo.ca Department of Applied Mathematics The University of Western Ontario, London, Canada 1 Elements of Probability 2 Elements of Probability CDF&PDF
More informationProperties of the least squares estimates
Properties of the least squares estimates 2019-01-18 Warmup Let a and b be scalar constants, and X be a scalar random variable. Fill in the blanks E ax + b) = Var ax + b) = Goal Recall that the least squares
More information20.1. Balanced One-Way Classification Cell means parametrization: ε 1. ε I. + ˆɛ 2 ij =
20. ONE-WAY ANALYSIS OF VARIANCE 1 20.1. Balanced One-Way Classification Cell means parametrization: Y ij = µ i + ε ij, i = 1,..., I; j = 1,..., J, ε ij N(0, σ 2 ), In matrix form, Y = Xβ + ε, or 1 Y J
More informationStatistical Inference
Statistical Inference Bernhard Klingenberg Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Outline Estimation: Review of concepts
More informationWeighted Least Squares
Weighted Least Squares The standard linear model assumes that Var(ε i ) = σ 2 for i = 1,..., n. As we have seen, however, there are instances where Var(Y X = x i ) = Var(ε i ) = σ2 w i. Here w 1,..., w
More informationMatrix Approach to Simple Linear Regression: An Overview
Matrix Approach to Simple Linear Regression: An Overview Aspects of matrices that you should know: Definition of a matrix Addition/subtraction/multiplication of matrices Symmetric/diagonal/identity matrix
More informationMATH 644: Regression Analysis Methods
MATH 644: Regression Analysis Methods FINAL EXAM Fall, 2012 INSTRUCTIONS TO STUDENTS: 1. This test contains SIX questions. It comprises ELEVEN printed pages. 2. Answer ALL questions for a total of 100
More informationEconometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018
Econometrics I KS Module 2: Multivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: April 16, 2018 Alexander Ahammer (JKU) Module 2: Multivariate
More informationBias Variance Trade-off
Bias Variance Trade-off The mean squared error of an estimator MSE(ˆθ) = E([ˆθ θ] 2 ) Can be re-expressed MSE(ˆθ) = Var(ˆθ) + (B(ˆθ) 2 ) MSE = VAR + BIAS 2 Proof MSE(ˆθ) = E((ˆθ θ) 2 ) = E(([ˆθ E(ˆθ)]
More informationMATH 567: Mathematical Techniques in Data Science Linear Regression: old and new
1/14 MATH 567: Mathematical Techniques in Data Science Linear Regression: old and new Dominique Guillot Departments of Mathematical Sciences University of Delaware February 13, 2017 Linear Regression:
More information4 Multiple Linear Regression
4 Multiple Linear Regression 4. The Model Definition 4.. random variable Y fits a Multiple Linear Regression Model, iff there exist β, β,..., β k R so that for all (x, x 2,..., x k ) R k where ε N (, σ
More informationLecture 2. The Simple Linear Regression Model: Matrix Approach
Lecture 2 The Simple Linear Regression Model: Matrix Approach Matrix algebra Matrix representation of simple linear regression model 1 Vectors and Matrices Where it is necessary to consider a distribution
More informationHypothesis testing Goodness of fit Multicollinearity Prediction. Applied Statistics. Lecturer: Serena Arima
Applied Statistics Lecturer: Serena Arima Hypothesis testing for the linear model Under the Gauss-Markov assumptions and the normality of the error terms, we saw that β N(β, σ 2 (X X ) 1 ) and hence s
More informationStatement: With my signature I confirm that the solutions are the product of my own work. Name: Signature:.
MATHEMATICAL STATISTICS Homework assignment Instructions Please turn in the homework with this cover page. You do not need to edit the solutions. Just make sure the handwriting is legible. You may discuss
More informationSTA 2101/442 Assignment Four 1
STA 2101/442 Assignment Four 1 One version of the general linear model with fixed effects is y = Xβ + ɛ, where X is an n p matrix of known constants with n > p and the columns of X linearly independent.
More informationThis model of the conditional expectation is linear in the parameters. A more practical and relaxed attitude towards linear regression is to say that
Linear Regression For (X, Y ) a pair of random variables with values in R p R we assume that E(Y X) = β 0 + with β R p+1. p X j β j = (1, X T )β j=1 This model of the conditional expectation is linear
More informationSTAT5044: Regression and Anova. Inyoung Kim
STAT5044: Regression and Anova Inyoung Kim 2 / 51 Outline 1 Matrix Expression 2 Linear and quadratic forms 3 Properties of quadratic form 4 Properties of estimates 5 Distributional properties 3 / 51 Matrix
More informationChapter 12: Multiple Linear Regression
Chapter 12: Multiple Linear Regression Seungchul Baek Department of Statistics, University of South Carolina STAT 509: Statistics for Engineers 1 / 55 Introduction A regression model can be expressed as
More information[y i α βx i ] 2 (2) Q = i=1
Least squares fits This section has no probability in it. There are no random variables. We are given n points (x i, y i ) and want to find the equation of the line that best fits them. We take the equation
More informationRegression #3: Properties of OLS Estimator
Regression #3: Properties of OLS Estimator Econ 671 Purdue University Justin L. Tobias (Purdue) Regression #3 1 / 20 Introduction In this lecture, we establish some desirable properties associated with
More informationUNIVERSITY OF MASSACHUSETTS. Department of Mathematics and Statistics. Basic Exam - Applied Statistics. Tuesday, January 17, 2017
UNIVERSITY OF MASSACHUSETTS Department of Mathematics and Statistics Basic Exam - Applied Statistics Tuesday, January 17, 2017 Work all problems 60 points are needed to pass at the Masters Level and 75
More informationRegression Review. Statistics 149. Spring Copyright c 2006 by Mark E. Irwin
Regression Review Statistics 149 Spring 2006 Copyright c 2006 by Mark E. Irwin Matrix Approach to Regression Linear Model: Y i = β 0 + β 1 X i1 +... + β p X ip + ɛ i ; ɛ i iid N(0, σ 2 ), i = 1,..., n
More informationOutline. Remedial Measures) Extra Sums of Squares Standardized Version of the Multiple Regression Model
Outline 1 Multiple Linear Regression (Estimation, Inference, Diagnostics and Remedial Measures) 2 Special Topics for Multiple Regression Extra Sums of Squares Standardized Version of the Multiple Regression
More information11 Hypothesis Testing
28 11 Hypothesis Testing 111 Introduction Suppose we want to test the hypothesis: H : A q p β p 1 q 1 In terms of the rows of A this can be written as a 1 a q β, ie a i β for each row of A (here a i denotes
More informationMATH 829: Introduction to Data Mining and Analysis Graphical Models I
MATH 829: Introduction to Data Mining and Analysis Graphical Models I Dominique Guillot Departments of Mathematical Sciences University of Delaware May 2, 2016 1/12 Independence and conditional independence:
More informationSimple Linear Regression
Simple Linear Regression ST 370 Regression models are used to study the relationship of a response variable and one or more predictors. The response is also called the dependent variable, and the predictors
More informationMATH 829: Introduction to Data Mining and Analysis Graphical Models III - Gaussian Graphical Models (cont.)
1/12 MATH 829: Introduction to Data Mining and Analysis Graphical Models III - Gaussian Graphical Models (cont.) Dominique Guillot Departments of Mathematical Sciences University of Delaware May 6, 2016
More informationScatter plot of data from the study. Linear Regression
1 2 Linear Regression Scatter plot of data from the study. Consider a study to relate birthweight to the estriol level of pregnant women. The data is below. i Weight (g / 100) i Weight (g / 100) 1 7 25
More informationPart IB Statistics. Theorems with proof. Based on lectures by D. Spiegelhalter Notes taken by Dexter Chua. Lent 2015
Part IB Statistics Theorems with proof Based on lectures by D. Spiegelhalter Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationWeighted Least Squares
Weighted Least Squares The standard linear model assumes that Var(ε i ) = σ 2 for i = 1,..., n. As we have seen, however, there are instances where Var(Y X = x i ) = Var(ε i ) = σ2 w i. Here w 1,..., w
More informationREGRESSION WITH SPATIALLY MISALIGNED DATA. Lisa Madsen Oregon State University David Ruppert Cornell University
REGRESSION ITH SPATIALL MISALIGNED DATA Lisa Madsen Oregon State University David Ruppert Cornell University SPATIALL MISALIGNED DATA 10 X X X X X X X X 5 X X X X X 0 X 0 5 10 OUTLINE 1. Introduction 2.
More informationMultivariate Linear Regression Models
Multivariate Linear Regression Models Regression analysis is used to predict the value of one or more responses from a set of predictors. It can also be used to estimate the linear association between
More informationSimple Linear Regression
Simple Linear Regression In simple linear regression we are concerned about the relationship between two variables, X and Y. There are two components to such a relationship. 1. The strength of the relationship.
More informationMatrices and vectors A matrix is a rectangular array of numbers. Here s an example: A =
Matrices and vectors A matrix is a rectangular array of numbers Here s an example: 23 14 17 A = 225 0 2 This matrix has dimensions 2 3 The number of rows is first, then the number of columns We can write
More informationLeast Squares Estimation
Least Squares Estimation Using the least squares estimator for β we can obtain predicted values and compute residuals: Ŷ = Z ˆβ = Z(Z Z) 1 Z Y ˆɛ = Y Ŷ = Y Z(Z Z) 1 Z Y = [I Z(Z Z) 1 Z ]Y. The usual decomposition
More informationScatter plot of data from the study. Linear Regression
1 2 Linear Regression Scatter plot of data from the study. Consider a study to relate birthweight to the estriol level of pregnant women. The data is below. i Weight (g / 100) i Weight (g / 100) 1 7 25
More informationMath 494: Mathematical Statistics
Math 494: Mathematical Statistics Instructor: Jimin Ding jmding@wustl.edu Department of Mathematics Washington University in St. Louis Class materials are available on course website (www.math.wustl.edu/
More informationHypothesis Testing One Sample Tests
STATISTICS Lecture no. 13 Department of Econometrics FEM UO Brno office 69a, tel. 973 442029 email:jiri.neubauer@unob.cz 12. 1. 2010 Tests on Mean of a Normal distribution Tests on Variance of a Normal
More informationChapter 12. Analysis of variance
Serik Sagitov, Chalmers and GU, January 9, 016 Chapter 1. Analysis of variance Chapter 11: I = samples independent samples paired samples Chapter 1: I 3 samples of equal size J one-way layout two-way layout
More informationChapter 1. Linear Regression with One Predictor Variable
Chapter 1. Linear Regression with One Predictor Variable 1.1 Statistical Relation Between Two Variables To motivate statistical relationships, let us consider a mathematical relation between two mathematical
More informationNeed for Several Predictor Variables
Multiple regression One of the most widely used tools in statistical analysis Matrix expressions for multiple regression are the same as for simple linear regression Need for Several Predictor Variables
More informationChapter 14. Linear least squares
Serik Sagitov, Chalmers and GU, March 5, 2018 Chapter 14 Linear least squares 1 Simple linear regression model A linear model for the random response Y = Y (x) to an independent variable X = x For a given
More informationManaging Uncertainty
Managing Uncertainty Bayesian Linear Regression and Kalman Filter December 4, 2017 Objectives The goal of this lab is multiple: 1. First it is a reminder of some central elementary notions of Bayesian
More informationST430 Exam 1 with Answers
ST430 Exam 1 with Answers Date: October 5, 2015 Name: Guideline: You may use one-page (front and back of a standard A4 paper) of notes. No laptop or textook are permitted but you may use a calculator.
More informationHigh-dimensional regression
High-dimensional regression Advanced Methods for Data Analysis 36-402/36-608) Spring 2014 1 Back to linear regression 1.1 Shortcomings Suppose that we are given outcome measurements y 1,... y n R, and
More informationLecture 11: Regression Methods I (Linear Regression)
Lecture 11: Regression Methods I (Linear Regression) Fall, 2017 1 / 40 Outline Linear Model Introduction 1 Regression: Supervised Learning with Continuous Responses 2 Linear Models and Multiple Linear
More informationThe outline for Unit 3
The outline for Unit 3 Unit 1. Introduction: The regression model. Unit 2. Estimation principles. Unit 3: Hypothesis testing principles. 3.1 Wald test. 3.2 Lagrange Multiplier. 3.3 Likelihood Ratio Test.
More informationHypothesis Testing The basic ingredients of a hypothesis test are
Hypothesis Testing The basic ingredients of a hypothesis test are 1 the null hypothesis, denoted as H o 2 the alternative hypothesis, denoted as H a 3 the test statistic 4 the data 5 the conclusion. The
More information14 Multiple Linear Regression
B.Sc./Cert./M.Sc. Qualif. - Statistics: Theory and Practice 14 Multiple Linear Regression 14.1 The multiple linear regression model In simple linear regression, the response variable y is expressed in
More informationSummary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1)
Summary of Chapter 7 (Sections 7.2-7.5) and Chapter 8 (Section 8.1) Chapter 7. Tests of Statistical Hypotheses 7.2. Tests about One Mean (1) Test about One Mean Case 1: σ is known. Assume that X N(µ, σ
More informationM(t) = 1 t. (1 t), 6 M (0) = 20 P (95. X i 110) i=1
Math 66/566 - Midterm Solutions NOTE: These solutions are for both the 66 and 566 exam. The problems are the same until questions and 5. 1. The moment generating function of a random variable X is M(t)
More informationMLES & Multivariate Normal Theory
Merlise Clyde September 6, 2016 Outline Expectations of Quadratic Forms Distribution Linear Transformations Distribution of estimates under normality Properties of MLE s Recap Ŷ = ˆµ is an unbiased estimate
More informationChapter 2 Multiple Regression I (Part 1)
Chapter 2 Multiple Regression I (Part 1) 1 Regression several predictor variables The response Y depends on several predictor variables X 1,, X p response {}}{ Y predictor variables {}}{ X 1, X 2,, X p
More informationApplied Regression Analysis
Applied Regression Analysis Chapter 3 Multiple Linear Regression Hongcheng Li April, 6, 2013 Recall simple linear regression 1 Recall simple linear regression 2 Parameter Estimation 3 Interpretations of
More informationSTAT763: Applied Regression Analysis. Multiple linear regression. 4.4 Hypothesis testing
STAT763: Applied Regression Analysis Multiple linear regression 4.4 Hypothesis testing Chunsheng Ma E-mail: cma@math.wichita.edu 4.4.1 Significance of regression Null hypothesis (Test whether all β j =
More informationSo far our focus has been on estimation of the parameter vector β in the. y = Xβ + u
Interval estimation and hypothesis tests So far our focus has been on estimation of the parameter vector β in the linear model y i = β 1 x 1i + β 2 x 2i +... + β K x Ki + u i = x iβ + u i for i = 1, 2,...,
More informationST 740: Linear Models and Multivariate Normal Inference
ST 740: Linear Models and Multivariate Normal Inference Alyson Wilson Department of Statistics North Carolina State University November 4, 2013 A. Wilson (NCSU STAT) Linear Models November 4, 2013 1 /
More informationProblem Selected Scores
Statistics Ph.D. Qualifying Exam: Part II November 20, 2010 Student Name: 1. Answer 8 out of 12 problems. Mark the problems you selected in the following table. Problem 1 2 3 4 5 6 7 8 9 10 11 12 Selected
More informationSimple Regression Model Setup Estimation Inference Prediction. Model Diagnostic. Multiple Regression. Model Setup and Estimation.
Statistical Computation Math 475 Jimin Ding Department of Mathematics Washington University in St. Louis www.math.wustl.edu/ jmding/math475/index.html October 10, 2013 Ridge Part IV October 10, 2013 1
More informationLecture 14 Simple Linear Regression
Lecture 4 Simple Linear Regression Ordinary Least Squares (OLS) Consider the following simple linear regression model where, for each unit i, Y i is the dependent variable (response). X i is the independent
More informationMultiple Linear Regression
Multiple Linear Regression ST 430/514 Recall: a regression model describes how a dependent variable (or response) Y is affected, on average, by one or more independent variables (or factors, or covariates).
More informationEconometrics. Week 11. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 11 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 30 Recommended Reading For the today Advanced Time Series Topics Selected topics
More informationMath 5305 Notes. Diagnostics and Remedial Measures. Jesse Crawford. Department of Mathematics Tarleton State University
Math 5305 Notes Diagnostics and Remedial Measures Jesse Crawford Department of Mathematics Tarleton State University (Tarleton State University) Diagnostics and Remedial Measures 1 / 44 Model Assumptions
More informationANALYSIS OF VARIANCE AND QUADRATIC FORMS
4 ANALYSIS OF VARIANCE AND QUADRATIC FORMS The previous chapter developed the regression results involving linear functions of the dependent variable, β, Ŷ, and e. All were shown to be normally distributed
More informationLecture 10 Multiple Linear Regression
Lecture 10 Multiple Linear Regression STAT 512 Spring 2011 Background Reading KNNL: 6.1-6.5 10-1 Topic Overview Multiple Linear Regression Model 10-2 Data for Multiple Regression Y i is the response variable
More informationBayesian Linear Regression
Bayesian Linear Regression Sudipto Banerjee 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. September 15, 2010 1 Linear regression models: a Bayesian perspective
More informationChapter 17: Undirected Graphical Models
Chapter 17: Undirected Graphical Models The Elements of Statistical Learning Biaobin Jiang Department of Biological Sciences Purdue University bjiang@purdue.edu October 30, 2014 Biaobin Jiang (Purdue)
More informationLecture 4 Multiple linear regression
Lecture 4 Multiple linear regression BIOST 515 January 15, 2004 Outline 1 Motivation for the multiple regression model Multiple regression in matrix notation Least squares estimation of model parameters
More informationThe linear model is the most fundamental of all serious statistical models encompassing:
Linear Regression Models: A Bayesian perspective Ingredients of a linear model include an n 1 response vector y = (y 1,..., y n ) T and an n p design matrix (e.g. including regressors) X = [x 1,..., x
More informationAnswers to Problem Set #4
Answers to Problem Set #4 Problems. Suppose that, from a sample of 63 observations, the least squares estimates and the corresponding estimated variance covariance matrix are given by: bβ bβ 2 bβ 3 = 2
More informationThe assumptions are needed to give us... valid standard errors valid confidence intervals valid hypothesis tests and p-values
Statistical Consulting Topics The Bootstrap... The bootstrap is a computer-based method for assigning measures of accuracy to statistical estimates. (Efron and Tibshrani, 1998.) What do we do when our
More information