DOCSLIB.ORG

Derivation of the Mean and Variance of a Linear Function

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Derivation of the Mean and Variance of a Linear Function APPENDIX A Derivation of the Mean and Variance of a Linear Function Regression analysis is based on the use of linear functions. Indeed, the predicted value of the dependent variable in a regression model is a linear function of the values of the independent variables. As a result, it is possible to compute the mean and variance of the predicted value of the dependent variable directly from the means and variances of the independent variables in a regression equation. In vector nota- tion, the equation for the predicted value of the dependent variable, in the case of a multiple regression equation with two independent vari- ables, is given by: ~, = ua + Xlb I + x2b 2 Using this equation, it is a relatively simple matter to derive the mean of the predicted values of the dependent variable from the means of the independent variables, the regression coefficients for these vari- ables, and the intercept. To begin with, it will be recalled that the mean of a variable, using vector algebra, is given by: 1 Mean(~,) = -~- (u':~) The product of these two vectors in this equation can be simplified as follows: 192 APPENDtXA u'~ = u'(ua + xlb I + x2b 2) = u'ua + U'Xlb I + u'x2b 2 = Na + ~xub I + ~x2ib 2 = Na + bl~Xli + b2~x2i = Na + blN~ 1 + b2N~ 2 This quantity can be substituted into the equation for the mean of a variable such that: 1 1 Mean(~)) = ~-(u'~)= ~ (Na + blN~ 1 + b2Nx2) = a + bl~ 1 + b2~ 2 Therefore, the mean of the predicted value of the dependent variable is a function of the means of the independent variables, their regres- sion coefficients, and the intercept. The variance of the predicted value of the dependent variable can be derived in a similar manner. However, we can greatly simplify this derivation by introducing the assumption that both the dependent variable and the independent variables have means that are equal to zero. This simplifying assumption is of no real consequence because the variance of a variable, which is the average squared deviation from the mean, is unaffected by the value of the mean itself. Moreover, because of the manner in which the intercept is computed, the mean of the predicted value of the dependent variable will be equal to zero whenever the mean of the dependent variable is equal to zero. Consequently, the sum of squares of the predicted values of the dependent variable is given by: ss(y) = - - y) = = (Xlb I + x2b2)'(xlb I + x2b 2) = blX~(Xlb I + x2b 2) + b2x~(Xlb I + x2b 2) THE MEAN AND VARIANCE OF A LINEAR FUNCHON 193 = blX]Xlb 1 + blX~X2b 2 + b2x~xlb I + b2x~x2b 2 This equation can be simplified even further by noting the following equivalencies: b,x;xlb, = b x;xl = b2x[x2b x = b2x[x2 = b2y.x2 i blxlx2b 2 = b2X[Xlb I = bib2xlx 2 = blb2Y.xlix2i As a result, the sum of squares of the predicted values of the depen- dent variable is given by: SS(~) = b 2 Y~x2 + b 2y.x2i + 2bib 2 ~XliX2i Therefore, we can derive the variance of the predicted value of the dependent variable as follows: 1 2 2 Var(~) = (b2y.x 2) + ~(b2~'.x2i ) + 1 ~" 2(bib2 EXliX2i) -- 2i + 2 bl b21 E Xli x2i Of course, this equation can be reduced to a series of more familiar quantities as follows: Var(~) = b2Var(xl) + b2Var(x2) + 2blb2Cov(xl,x2) In other words, the variance of the predicted value of the dependent variable is a function of the regression coefficients of the independent variables, the variances of the independent variables, and the covari- 19 4 APPENDIXA ance between these independent variables. This equation can readily be expanded to obtain the variance of the predicted value of the dependent variable whenever there are more than two independent variables. For example, in the case of three independent variables, there are three terms involving the variances and regression coeffi- cients of each of these independent variables and three terms involv- ing the regression coefficients of these independent variables and the covariances between each pair of independent variables. Finally, it must be noted that this equation has a special interpreta- tion whenever all of the variables are expressed in standard form such that both the dependent and independent variables have variances that are equal to one. Specifically, in this case, the variance of the predicted values of the dependent variable is given by: Var(~y) = b~ 2 Var(zl) + b2 2 Var(z 2) + 2b~ b~Cov(zl,z2) = b~ 2 + b2 2 + 2b~b2r12 Moreover, it will be recalled that the coefficient of determination is equal to the ratio of the variance of the predicted value of the depen- dent variable to the variance of the observed value of that variable such that: R2.12 _ Var(~) _ Var(~.y) = Var(~y) Var(y) Var(Zy) = 2 + 2 + 2blb2 2 In short, the coefficient of determination can be computed directly from the standardized regression coefficients of the independent vari- ables and the correlations among these independent variables. APPENDIX B Derivation of the Least-Squares Regression Coefficient The derivation of the least-squares regression coefficient requires some familiarity with calculus. From a mathematical point of view, the main problem posed by regression analysis involves the estimation of a regression coefficient and an intercept that minimize some func- tion of the errors of prediction. The least-squares estimation proce- dure adopts a solution to this problem that is based on minimizing the sum of the squared errors of prediction. This criterion can be stated algebraically as follows: Minimize SS(e) = Ee~ = ~-~(Yi- ~i)2 where Yi = a + b x i Obviously, the values of the predicted scores and, therefore, the values of the errors of prediction are determined by the choice of a particu- lar intercept and regression coefficient. If we substitute this linear function for the predicted value of the dependent variable, the problem becomes one of finding the regres- sion coefficient, b, and the intercept, a, that minimize the following function: S(a,b) = ~(Yi - a - bxi) 2 19 6 APPENDIXB In order to minimize this function, we calculate the partial derivatives of both b and a using the chain rule of differentiation and the rule that the derivative of a sum is equal to the sum of the derivatives as follows: 3S o~---b = ~ 2 (Yi - a - bx i ) (-x i ) ~9S o3"-a = ~2(Yi- a - bxi)(-1) Performing the indicated multiplication, dividing both partial deriva- tives by -2, and setting them both equal to zero, we obtain: ~(--xiY i + bx? + axi) = 0 ~(-yi+bxi +a) = 0 Rearranging the terms in these equations, we obtain: b ~,x? - X~xiYi + a ~,x i = 0 aN - ~-.dYi + b ~,xi = 0 These equations can be greatly simplified if we assume, for the sake of simplicity and without loss of generality, that the independent variable, x, is expressed in mean deviation form such that: ~,x i = 0 Specifically, under this assumption, these equations reduce to: ~, xi Yi b= Xx? a = = N THE LEAST-SQUARESREGRESSION COEFFICIENT 19 7 If we divide both the numerator and the denominator of the equa- tion for b by N, we obtain the more familiar equation for the least- squares regression coefficient as the covariance of two variables divided by the variance of the independent variable: Cov(x,y) b = Var(x) Moreover, the equation for a, whenever the independent variable is expressed in mean deviation form, is simply a special case of the more general equation for the intercept such that: a=y-b~ where ~=0 This result obtains because the mean of any variable expressed in mean deviation form is zero. The least-squares intercept insures that the predicted value of the dependent variable is equal to the mean of the dependent variable whenever the independent variable is equal to its mean. This result is obvious if we simply rearrange the equation for the intercept as follows: y = a+ b~ These results were obtained using the simplifying assumption that the independent variable is expressed in mean deviation form. However, it can be demonstrated that this assumption is not necessary to obtain these equations. Indeed, the covariance between two variables and the variance of the independent variable are the same whether or not the independent variable is expressed in mean deviation form. APPENDIX C Derivation of the Standard Error of the Simple Regression Coefficient In order to test for the significance of a regression coefficient, we must be able to estimate its standard error. We begin with the generic equation for the simple regression coefficient as a ratio of the covari- ance between two variables to the variance of the independent variable such that: Cov(x,y) b - Var(x) For the sake of simplicity and without loss of generality, we shall assume that both the dependent variable, y, and the independent vari- able, x, are expressed in mean deviation form. First, we multiply both the numerator and the denominator of this equation by N, the number of cases, and simplify as follows: N C xy X xi Yi b = N s 2x Xx? This equation can be expressed somewhat differently such that: THE STANDARD ERROR OF THE SIMPLE REGRESSION COEFFICIENT 199 Consequently, the regression coefficient can be expressed as a sum of the weighted values of the dependent variable, y, as follows: b= Ewiy where xi w/= In other words, the simple regression coefficient is equal to a linear function of the values of the dependent variable in which each value of the dependent variable, y, is multiplied by the ratio of the value of the corresponding independent variable, x, to its sum of squares.
Load more

Recommended publications
  • A Generalized Linear Model for Principal Component Analysis of Binary Data
    A Generalized Linear Model for Principal Component Analysis of Binary Data Andrew I. Schein Lawrence K. Saul Lyle H. Ungar Department of Computer and Information Science University of Pennsylvania Moore School Building 200 South 33rd Street Philadelphia, PA 19104-6389 {ais,lsaul,ungar}@cis.upenn.edu Abstract they are not generally appropriate for other data types. Recently, Collins et al.[5] derived generalized criteria We investigate a generalized linear model for for dimensionality reduction by appealing to proper- dimensionality reduction of binary data. The ties of distributions in the exponential family. In their model is related to principal component anal- framework, the conventional PCA of real-valued data ysis (PCA) in the same way that logistic re- emerges naturally from assuming a Gaussian distribu- gression is related to linear regression. Thus tion over a set of observations, while generalized ver- we refer to the model as logistic PCA. In this sions of PCA for binary and nonnegative data emerge paper, we derive an alternating least squares respectively by substituting the Bernoulli and Pois- method to estimate the basis vectors and gen- son distributions for the Gaussian. For binary data, eralized linear coefficients of the logistic PCA the generalized model's relationship to PCA is anal- model. The resulting updates have a simple ogous to the relationship between logistic and linear closed form and are guaranteed at each iter- regression[12]. In particular, the model exploits the ation to improve the model's likelihood. We log-odds as the natural parameter of the Bernoulli dis- evaluate the performance of logistic PCA|as tribution and the logistic function as its canonical link.
    [Show full text]
  • Linear, Ridge Regression, and Principal Component Analysis
    Linear, Ridge Regression, and Principal Component Analysis Linear, Ridge Regression, and Principal Component Analysis Jia Li Department of Statistics The Pennsylvania State University Email: [email protected] http://www.stat.psu.edu/∼jiali Jia Li http://www.stat.psu.edu/∼jiali Linear, Ridge Regression, and Principal Component Analysis Introduction to Regression I Input vector: X = (X1, X2, ..., Xp). I Output Y is real-valued. I Predict Y from X by f (X ) so that the expected loss function E(L(Y , f (X ))) is minimized. I Square loss: L(Y , f (X )) = (Y − f (X ))2 . I The optimal predictor ∗ 2 f (X ) = argminf (X )E(Y − f (X )) = E(Y | X ) . I The function E(Y | X ) is the regression function. Jia Li http://www.stat.psu.edu/∼jiali Linear, Ridge Regression, and Principal Component Analysis Example The number of active physicians in a Standard Metropolitan Statistical Area (SMSA), denoted by Y , is expected to be related to total population (X1, measured in thousands), land area (X2, measured in square miles), and total personal income (X3, measured in millions of dollars). Data are collected for 141 SMSAs, as shown in the following table. i : 1 2 3 ... 139 140 141 X1 9387 7031 7017 ... 233 232 231 X2 1348 4069 3719 ... 1011 813 654 X3 72100 52737 54542 ... 1337 1589 1148 Y 25627 15389 13326 ... 264 371 140 Goal: Predict Y from X1, X2, and X3. Jia Li http://www.stat.psu.edu/∼jiali Linear, Ridge Regression, and Principal Component Analysis Linear Methods I The linear regression model p X f (X ) = β0 + Xj βj .
    [Show full text]
  • Principal Component Analysis (PCA) As a Statistical Tool for Identifying Key Indicators of Nuclear Power Plant Cable Insulation
    Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2017 Principal component analysis (PCA) as a statistical tool for identifying key indicators of nuclear power plant cable insulation degradation Chamila Chandima De Silva Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Materials Science and Engineering Commons, Mechanics of Materials Commons, and the Statistics and Probability Commons Recommended Citation De Silva, Chamila Chandima, "Principal component analysis (PCA) as a statistical tool for identifying key indicators of nuclear power plant cable insulation degradation" (2017). Graduate Theses and Dissertations. 16120. https://lib.dr.iastate.edu/etd/16120 This Thesis is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Principal component analysis (PCA) as a statistical tool for identifying key indicators of nuclear power plant cable insulation degradation by Chamila C. De Silva A thesis submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Materials Science and Engineering Program of Study Committee: Nicola Bowler, Major Professor Richard LeSar Steve Martin The student author and the program of study committee are solely responsible for the content of this thesis. The Graduate College will ensure this thesis is globally accessible and will not permit alterations after a degree is conferred.
    [Show full text]
  • Generalized Linear Models
    CHAPTER 6 Generalized linear models 6.1 Introduction Generalized linear modeling is a framework for statistical analysis that includes linear and logistic regression as special cases. Linear regression directly predicts continuous data y from a linear predictor Xβ = β0 + X1β1 + + Xkβk.Logistic regression predicts Pr(y =1)forbinarydatafromalinearpredictorwithaninverse-··· logit transformation. A generalized linear model involves: 1. A data vector y =(y1,...,yn) 2. Predictors X and coefficients β,formingalinearpredictorXβ 1 3. A link function g,yieldingavectoroftransformeddataˆy = g− (Xβ)thatare used to model the data 4. A data distribution, p(y yˆ) | 5. Possibly other parameters, such as variances, overdispersions, and cutpoints, involved in the predictors, link function, and data distribution. The options in a generalized linear model are the transformation g and the data distribution p. In linear regression,thetransformationistheidentity(thatis,g(u) u)and • the data distribution is normal, with standard deviation σ estimated from≡ data. 1 1 In logistic regression,thetransformationistheinverse-logit,g− (u)=logit− (u) • (see Figure 5.2a on page 80) and the data distribution is defined by the proba- bility for binary data: Pr(y =1)=y ˆ. This chapter discusses several other classes of generalized linear model, which we list here for convenience: The Poisson model (Section 6.2) is used for count data; that is, where each • data point yi can equal 0, 1, 2, ....Theusualtransformationg used here is the logarithmic, so that g(u)=exp(u)transformsacontinuouslinearpredictorXiβ to a positivey ˆi.ThedatadistributionisPoisson. It is usually a good idea to add a parameter to this model to capture overdis- persion,thatis,variationinthedatabeyondwhatwouldbepredictedfromthe Poisson distribution alone.
    [Show full text]
  • Time-Series Regression and Generalized Least Squares in R*
    Time-Series Regression and Generalized Least Squares in R* An Appendix to An R Companion to Applied Regression, third edition John Fox & Sanford Weisberg last revision: 2018-09-26 Abstract Generalized least-squares (GLS) regression extends ordinary least-squares (OLS) estimation of the normal linear model by providing for possibly unequal error variances and for correlations between different errors. A common application of GLS estimation is to time-series regression, in which it is generally implausible to assume that errors are independent. This appendix to Fox and Weisberg (2019) briefly reviews GLS estimation and demonstrates its application to time-series data using the gls() function in the nlme package, which is part of the standard R distribution. 1 Generalized Least Squares In the standard linear model (for example, in Chapter 4 of the R Companion), E(yjX) = Xβ or, equivalently y = Xβ + " where y is the n×1 response vector; X is an n×k +1 model matrix, typically with an initial column of 1s for the regression constant; β is a k + 1 ×1 vector of regression coefficients to estimate; and " is 2 an n×1 vector of errors. Assuming that " ∼ Nn(0; σ In), or at least that the errors are uncorrelated and equally variable, leads to the familiar ordinary-least-squares (OLS) estimator of β, 0 −1 0 bOLS = (X X) X y with covariance matrix 2 0 −1 Var(bOLS) = σ (X X) More generally, we can assume that " ∼ Nn(0; Σ), where the error covariance matrix Σ is sym- metric and positive-definite. Different diagonal entries in Σ error variances that are not necessarily all equal, while nonzero off-diagonal entries correspond to correlated errors.
    [Show full text]
  • Simple Linear Regression: Straight Line Regression Between an Outcome Variable (Y ) and a Single Explanatory Or Predictor Variable (X)
    1 Introduction to Regression \Regression" is a generic term for statistical methods that attempt to fit a model to data, in order to quantify the relationship between the dependent (outcome) variable and the predictor (independent) variable(s). Assuming it fits the data reasonable well, the estimated model may then be used either to merely describe the relationship between the two groups of variables (explanatory), or to predict new values (prediction). There are many types of regression models, here are a few most common to epidemiology: Simple Linear Regression: Straight line regression between an outcome variable (Y ) and a single explanatory or predictor variable (X). E(Y ) = α + β × X Multiple Linear Regression: Same as Simple Linear Regression, but now with possibly multiple explanatory or predictor variables. E(Y ) = α + β1 × X1 + β2 × X2 + β3 × X3 + ::: A special case is polynomial regression. 2 3 E(Y ) = α + β1 × X + β2 × X + β3 × X + ::: Generalized Linear Model: Same as Multiple Linear Regression, but with a possibly transformed Y variable. This introduces considerable flexibil- ity, as non-linear and non-normal situations can be easily handled. G(E(Y )) = α + β1 × X1 + β2 × X2 + β3 × X3 + ::: In general, the transformation function G(Y ) can take any form, but a few forms are especially common: • Taking G(Y ) = logit(Y ) describes a logistic regression model: E(Y ) log( ) = α + β × X + β × X + β × X + ::: 1 − E(Y ) 1 1 2 2 3 3 2 • Taking G(Y ) = log(Y ) is also very common, leading to Poisson regression for count data, and other so called \log-linear" models.
    [Show full text]
  • Chapter 2 Simple Linear Regression Analysis the Simple
    Chapter 2 Simple Linear Regression Analysis The simple linear regression model We consider the modelling between the dependent and one independent variable. When there is only one independent variable in the linear regression model, the model is generally termed as a simple linear regression model. When there are more than one independent variables in the model, then the linear model is termed as the multiple linear regression model. The linear model Consider a simple linear regression model yX01 where y is termed as the dependent or study variable and X is termed as the independent or explanatory variable. The terms 0 and 1 are the parameters of the model. The parameter 0 is termed as an intercept term, and the parameter 1 is termed as the slope parameter. These parameters are usually called as regression coefficients. The unobservable error component accounts for the failure of data to lie on the straight line and represents the difference between the true and observed realization of y . There can be several reasons for such difference, e.g., the effect of all deleted variables in the model, variables may be qualitative, inherent randomness in the observations etc. We assume that is observed as independent and identically distributed random variable with mean zero and constant variance 2 . Later, we will additionally assume that is normally distributed. The independent variables are viewed as controlled by the experimenter, so it is considered as non-stochastic whereas y is viewed as a random variable with Ey()01 X and Var() y 2 . Sometimes X can also be a random variable.
    [Show full text]
  • Week 7: Multiple Regression
    Week 7: Multiple Regression Brandon Stewart1 Princeton October 24, 26, 2016 1These slides are heavily influenced by Matt Blackwell, Adam Glynn, Jens Hainmueller and Danny Hidalgo. Stewart (Princeton) Week 7: Multiple Regression October 24, 26, 2016 1 / 145 Where We've Been and Where We're Going... Last Week I regression with two variables I omitted variables, multicollinearity, interactions This Week I Monday: F matrix form of linear regression I Wednesday: F hypothesis tests Next Week I break! I then ::: regression in social science Long Run I probability ! inference ! regression Questions? Stewart (Princeton) Week 7: Multiple Regression October 24, 26, 2016 2 / 145 1 Matrix Algebra Refresher 2 OLS in matrix form 3 OLS inference in matrix form 4 Inference via the Bootstrap 5 Some Technical Details 6 Fun With Weights 7 Appendix 8 Testing Hypotheses about Individual Coefficients 9 Testing Linear Hypotheses: A Simple Case 10 Testing Joint Significance 11 Testing Linear Hypotheses: The General Case 12 Fun With(out) Weights Stewart (Princeton) Week 7: Multiple Regression October 24, 26, 2016 3 / 145 Why Matrices and Vectors? Here's one way to write the full multiple regression model: yi = β0 + xi1β1 + xi2β2 + ··· + xiK βK + ui Notation is going to get needlessly messy as we add variables Matrices are clean, but they are like a foreign language You need to build intuitions over a long period of time (and they will return in Soc504) Reminder of Parameter Interpretation: β1 is the effect of a one-unit change in xi1 conditional on all other xik . We are going to review the key points quite quickly just to refresh the basics.
    [Show full text]
  • Linear Regression in Matrix Form
    Statistics 512: Applied Linear Models Topic 3 Topic Overview This topic will cover • thinking in terms of matrices • regression on multiple predictor variables • case study: CS majors • Text Example (KNNL 236) Chapter 5: Linear Regression in Matrix Form The SLR Model in Scalar Form iid 2 Yi = β0 + β1Xi + i where i ∼ N(0,σ ) Consider now writing an equation for each observation: Y1 = β0 + β1X1 + 1 Y2 = β0 + β1X2 + 2 . Yn = β0 + β1Xn + n TheSLRModelinMatrixForm Y1 β0 + β1X1 1 Y2 β0 + β1X2 2 . = . + . . . . Yn β0 + β1Xn n Y X 1 1 1 1 Y X 2 1 2 β0 2 . = . + . . . β1 . Yn 1 Xn n (I will try to use bold symbols for matrices. At first, I will also indicate the dimensions as a subscript to the symbol.) 1 • X is called the design matrix. • β is the vector of parameters • is the error vector • Y is the response vector The Design Matrix 1 X1 1 X2 Xn×2 = . . 1 Xn Vector of Parameters β0 β2×1 = β1 Vector of Error Terms 1 2 n×1 = . . n Vector of Responses Y1 Y2 Yn×1 = . . Yn Thus, Y = Xβ + Yn×1 = Xn×2β2×1 + n×1 2 Variance-Covariance Matrix In general, for any set of variables U1,U2,... ,Un,theirvariance-covariance matrix is defined to be 2 σ {U1} σ{U1,U2} ··· σ{U1,Un} . σ{U ,U } σ2{U } ... σ2{ } 2 1 2 . U = . . .. .. σ{Un−1,Un} 2 σ{Un,U1} ··· σ{Un,Un−1} σ {Un} 2 where σ {Ui} is the variance of Ui,andσ{Ui,Uj} is the covariance of Ui and Uj.
    [Show full text]
  • AP Statistics Chapter 7 Linear Regression Objectives
    AP Statistics Chapter 7 Linear Regression Objectives: • Linear model • Predicted value • Residuals • Least squares • Regression to the mean • Regression line • Line of best fit • Slope intercept • se 2 • R 2 Fat Versus Protein: An Example • The following is a scatterplot of total fat versus protein for 30 items on the Burger King menu: The Linear Model • The correlation in this example is 0.83. It says “There seems to be a linear association between these two variables,” but it doesn’t tell what that association is. • We can say more about the linear relationship between two quantitative variables with a model. • A model simplifies reality to help us understand underlying patterns and relationships. The Linear Model • The linear model is just an equation of a straight line through the data. – The points in the scatterplot don’t all line up, but a straight line can summarize the general pattern with only a couple of parameters. – The linear model can help us understand how the values are associated. The Linear Model • Unlike correlation, the linear model requires that there be an explanatory variable and a response variable. • Linear Model 1. A line that describes how a response variable y changes as an explanatory variable x changes. 2. Used to predict the value of y for a given value of x. 3. Linear model of the form: 6 The Linear Model • The model won’t be perfect, regardless of the line we draw. • Some points will be above the line and some will be below. • The estimate made from a model is the predicted value (denoted as yˆ ).
    [Show full text]
  • Logistic Regression
    CHAPTER 5 Logistic regression Logistic regression is the standard way to model binary outcomes (that is, data yi that take on the values 0 or 1). Section 5.1 introduces logistic regression in a simple example with one predictor, then for most of the rest of the chapter we work through an extended example with multiple predictors and interactions. 5.1 Logistic regression with a single predictor Example: modeling political preference given income Conservative parties generally receive more support among voters with higher in- comes. We illustrate classical logistic regression with a simple analysis of this pat- tern from the National Election Study in 1992. For each respondent i in this poll, we label yi =1ifheorshepreferredGeorgeBush(theRepublicancandidatefor president) or 0 if he or she preferred Bill Clinton (the Democratic candidate), for now excluding respondents who preferred Ross Perot or other candidates, or had no opinion. We predict preferences given the respondent’s income level, which is characterized on a five-point scale.1 The data are shown as (jittered) dots in Figure 5.1, along with the fitted logistic regression line, a curve that is constrained to lie between 0 and 1. We interpret the line as the probability that y =1givenx—in mathematical notation, Pr(y =1x). We fit and display the logistic regression using the following R function calls:| fit.1 <- glm (vote ~ income, family=binomial(link="logit")) Rcode display (fit.1) to yield coef.est coef.se Routput (Intercept) -1.40 0.19 income 0.33 0.06 n=1179,k=2 residual deviance = 1556.9, null deviance = 1591.2 (difference = 34.3) 1 The fitted model is Pr(yi =1)=logit− ( 1.40 + 0.33 income).
    [Show full text]
  • Chapter 12: Logistic Regression
    Chapter 12: Logistic Regression So far we used linear models to predict a continuous reponse variable y using a set of continuous or discrete predictors . Now we turn our attention to predicting a discrete reponse variable using both continuous and discrete predictors. There are many applications where the response variable is discrete, such as predicting 1.) if a team will win a sporting event, 2.) if a patient will have a heart attack, or 3.) if a medical device will fail. Let's assume the response variable y has two possible outcomes, 0 and 1. (This is not really a limitation, as response variable with multiple outcomes can be modeled as a series of binary outcomes.) Linear models are unbounded, real-valued functions. The responses predicted by linear models can therefore range from negative to positive infinity. There is no way to use only field algebra or linear algebra to fit a function that predicts only two events. Instead, we use a link function to translate the continuous output of a linear model into a binary prediction. Rather than predict the binary response, we could try predicting the probability that the response is equal to 1. Probabilities are continous, but they are bounded to the interval [0,1]. We don't have a method to force a linear models to only make predictions in this range. We could instead predict the odds of the response variable, which is the probability that the response equals 1 divided by the probability that the response equals 0. If the , then the probability that y equals 1 is twice as large as the probability that y equals 0.
    [Show full text]