DOCSLIB.ORG

A Absolute Standardized Mean Difference (ASMD), 121–122 ACEAIPW Precision Known Propensity Score Model Arbitrary Function, 77

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Absolute Standardized Mean Difference (ASMD), 121–122 ACEAIPW Precision Known Propensity Score Model Arbitrary Function, 77 Index A B Absolute standardized mean difference Bayesian approach, 218 (ASMD), 121–122 BlackBoost, 214 ACEAIPW precision Boosting model, 119–120 known propensity score model arbitrary function, 77 disadvantage, 79 C Monte Carlo computations, 79 cART. See Combined antiretroviral therapies quadratic function minimization, 78 (cART) simulated 100 datasets, 79, 80 Causal inference variance, 78 counterfactual outcome weighted mean squared error, 79 mediation, treatment effect, 8, 9 known response regression model, 80–81 post-treatment confounders, RCT, 7–8 Acquired immunodeficiency syndrome potential outcome, 5–6 (AIDS), 203 randomization, 5 AdaBoost, 213 selection bias, observational studies, 7 AIDS Clinical Trials Group (ACTG) Study epidemiology and clinical trials, 4 A5095, 204 ITT, 4 ASMD. See Absolute standardized mean statistical models difference (ASMD) case-control designs, 10 ATE. See Average treatment effect (ATE) causal mediation, 19–20 ATT. See Average treatment effect among the MAR mechanism, 9 treated (ATT) matching and propensity score Augmented inverse probability weighted matching, 10–11 (AIPW) estimator missing data, 9 construction, 75 MSMs, 12–13 correct PM, 74 post-treatment confounders, RCT, correct RRM, 74 13–19 double robustness property, 76 sequential ignorability (SI) and model HT estimator, 74, 75 identification, 20, 21 Average causal mediation effect (ACME), Causal mediation models 20 ACME, 20, 23 Average treatment effect (ATE), 112, 121 causal diagram, 244, 254 Average treatment effect among the treated CDF, random variable, 21 (ATT), 112, 121 direct effect/natural direct effect, 19 © Springer International Publishing Switzerland 2016 315 H. He et al. (eds.), Statistical Causal Inferences and Their Applications in Public Health Research, ICSA Book Series in Statistics, DOI 10.1007/978-3-319-41259-7 316 Index Causal mediation models (cont.) CDE. See Controlled direct effect (CDE) equivalence of different choices, 260 CFI. See Comparative fit index (CFI) estimation of parameters Child Resilience Project (CRP), 219, 234–236 general treatment X; 252–253 Chronic thromboembolic pulmonary maximum likelihood estimation, 250 hypertension (CTEPH), 104 moment estimation, 250 Cognitive behavioral smoking cessation three-value treatment, 251–252 therapy (CBT), 187 GMM, 243 Combined antiretroviral therapies (cART) identifiability of parameters ACTG A5095, 204, 209–211 continuous mediator M; 248–249 AdaBoost, 213 discrete variable M; 248 BlackBoost, 214 general conditions, 246–248 HIV-1 infected patients, 203 linear model of M; 249–250 methods indirect and direct effects, 241 missing data, 205–207 logistic regression equation, 258 two-stage designs, 207–209 LSEM, 21, 22 variance estimate, 209 matrix Geff, 261 non-parametric estimator, 212 means of estimates, 255 simulation studies, 211–212 mediator–outcome relationship, 242 The Commit to Quit (CTQ) study moderated-mediation model, 255–257 compliance model, 196–197 necessity and sufficiency theorem, 258–259 compliance regions, 200 notation and definitions, 243–246 data, 195–196 OLS estimates, 254 estimated causal effects, 199 OLS regression, 243 maximum likelihood estimates, 198 pure indirect effect, 20 principal effects, 196–197 three linear models, 242 two-stage ML approach, 200 total effect of treatment, 20 The Commit to Quit (CTQ) trials, 187 treatment–outcome relationship, 242 Comparative fit index (CFI), 303 unobserved pre-treatment confounder, 242 Compliance behavior, 13 Causal models Complier average causal effect (CACE), 14 CBT, 187 Controlled direct effect (CDE), 19, 269 continuous measures, 187 Cox regression, 104 CTQ study CRP. See Child Resilience Project (CRP) compliance model, 196–197 CTQ trials. See The Commit to Quit (CTQ) compliance regions, 200 trials data, 195–196 Cumulative distribution function (CDF), 21 estimated causal effects, 199 maximum likelihood estimates, 198 principal effects, 196–197 D two-stage ML approach, 200 Data-adaptive matching score, 117–118 likelihood and inference methods DomEXT Baseline, 234–235 compliance regions, 193–195 Donsker classes, 158–161 contribution, 192 two-stage approach, 192–193 placebo-controlled trials, 188 E principal stratification approach, 188 Empirical processes structural principal effects model average treatment effect, 161–163 compliance distributions, 191–192 Donsker classes, 158–161 ITT effects, 190–191 estimating equation, 157 notation and assumptions, 189–190 motivation and setup, 157–158 Causal relative risk, 176, 177 Estimated propensity variable (EPV), 62, CBT. See Cognitive behavioral smoking 63 cessation therapy (CBT) Estimating equation (EE), 37 CD4 cell, 210–211 Exposure to agents, 7 Index 317 F J Face-value average causal effect (FACE), 52 Jackknife method, 182–183 Fisher’s linear discriminant (LD), 61 Functional response models (FRM), 221 L Latent growth modeling (LGM), 296 G Likelihood and inference methods Generalized boosted model (GBM), 116–117 compliance regions, 193–195 Generalized Linear Structural Equation Models contribution, 192 (GLSEM), 21 two-stage approach, 192–193 Generalized method of moments (GMM), 243 Linear predictor (LP), 60 Genetic Epidemiology Network of Salt Linear SEM (LSEM), 21, 22 Sensitivity (GenSalt) Study LISREL formulation, 299–300 covariate adjustment, 43–44 Logistic regression covariates, 40 model construction, 70–71 outcomes, 39 propensity analysis, custodial sanctions parameter estimations, 40, 41 study, 71–73 pre vs. post score matching, 41–43 propensity score weighting approach, 43 treatment conditions, 39–40 M GMM. See Generalized method of moments Mahalanobis distance, 33 (GMM) Mahalanobis metric matching, 33 Greedy algorithm, 33 Mann-Whitney-Wilcoxon rank sum test, 222 MAR. See Missing at random (MAR) H Marginal structural models (MSMs), 12–13, Heteroscedasticity 274 balancing property of PS/PV, 67 mboost package, 207, 212 covariance matrices, 66 MCAR. See Missing Complete at Random linear discriminant, 67 (MCAR) QD, 67 Mean squared error (MSE), 278 simulations, 68–70 Missing at random (MAR), 9, 31, 302 High-Risk Youth Demonstration Grant Missing Complete at Random (MCAR), 235, Programs, 95 296, 308 Homoscedasticity MMDP. See Monotone missing data patterns asymptotic variance analysis (MMDP) EPV, 62, 63 Moderated-mediation model, 255–257 propensity variable, 62, 63 Monotone missing data patterns (MMDP), 227 sample size, 63 Monte Carlo (MC) cross-validation criteria, variance multiplier of coefficient, 63–64 122–123 model construction, 60–61 Monte Carlo (MC) mean, 278 precision, propensity analysis, 62 Monte Carlo (MC) replications, 212 simulations, 65, 66 MSE. See Mean squared error (MSE) Horvitz-Thompson (HT) estimator, 74, 75 MSMs. See Marginal structural models (MSMs) Multinomial logistic regression (MLR), 115 I Important variables stratification (IVS), 128 Intention to treat (ITT) approach, 4, 218, 219 N Inverse probability of treatment weights Natural direct effects (NDEs), 268–269 (IPTW), 102 Natural indirect effects (NIEs), 268–269 Inverse probability weighiting (IPW), 36, 106, Newton’s method, 170 273, 309 Nonparametric black-box algorithms, 112 ITT approach. See Intention to treat (ITT) Nonparametric curve regression methods, 37 approach Nonparametric density estimation, 119 318 Index Nonparametric models, 147 controlled effects, 274–275 Non-randomized controlled trials (non-RCTs), natural effects, 274 91 principal strata effects, 273–274 Nucleoside reverse-transciptase inhibitor identification, 289 (NRTI), 210–211 controlled effects, 272–274 natural effects, 271–272 principal strata effects, 270–271 O limitations, 289 Observational data, 91 RPM G-estimator, 275, 287 OLS regression. See Ordinary least squares simulation study (OLS) regression conditions, 275 Optimal pair matching (OPM) method data generation, 276–278 advantages, 126 IPWCDE, 286 control-Philadelphia creation MC SD, 286 covariate balance before and after no confounders, 279–282 matching, 130, 132–133 population values, 278–279 PSS illustration, 130, 131 post-T confounder, M and Y, 284 standardized differences, 133 pre-T confounder, M and Y, 281–284 stratification tree, 130, 132 pre-T confounder, T, M, and Y, 285–286 stratification variables and stratification squared MC SD, 278, 280 intervals, 129–130 TSLS IV estimator, 275, 285, 287 tolerance number of subclasses, 129 Principal stratification (PST), 15–16, 218 tolerance size of distance matrix, 129 PROC SYSLIN, 175 covariate balance, 126 Propensity analysis massive obstetric unit closures in balancing score, 58 Philadelphia, 127, 134 logistic regression, 70–73 rank-based Mahalanobis distances, 126, normal linear model (see 127 Heteroscedasticity; R package, 134 Homoscedasticity) stratification tree construction propensity variable, 59 checking matching feasibility, 127–128 PS, 58, 59 checking statistical criteria, 127 Propensity score (PS) estimation checking the number of strata after assessment steps, 97–98 propensity, 128 definition, 92 flowchart, 128, 129 empirical example IVS, 128 approximated Type IV Pearson PSS, 128 distribution, 95, 96 structure of data, 127 composite score, 30-day substance use, Ordinary least squares (OLS) regression, 243 95 empirical distribution of robustness, 95, 97 P empirical distribution of sensitivity Pearl’s causal framework, 50 indices, 95, 96 PHQ-9, 304 High-Risk Youth Demonstration Grant Potential outcome approaches Programs, 95 causal inference, 265–266 in literature, 105 define mediation effects, 287–288 missing confounder data CDE, 269 balance assessment, 107–108 controlled effects, 266 IPTW, 107 identification
Load more

Recommended publications
  • Auditor: an R Package for Model-Agnostic Visual Validation and Diagnostics
    auditor: an R Package for Model-Agnostic Visual Validation and Diagnostics APREPRINT Alicja Gosiewska Przemysław Biecek Faculty of Mathematics and Information Science Faculty of Mathematics and Information Science Warsaw University of Technology Warsaw University of Technology Poland Faculty of Mathematics, Informatics and Mechanics [email protected] University of Warsaw Poland [email protected] May 27, 2020 ABSTRACT Machine learning models have spread to almost every area of life. They are successfully applied in biology, medicine, finance, physics, and other fields. With modern software it is easy to train even a complex model that fits the training data and results in high accuracy on test set. The problem arises when models fail confronted with the real-world data. This paper describes methodology and tools for model-agnostic audit. Introduced tech- niques facilitate assessing and comparing the goodness of fit and performance of models. In addition, they may be used for analysis of the similarity of residuals and for identification of outliers and influential observations. The examination is carried out by diagnostic scores and visual verification. Presented methods were implemented in the auditor package for R. Due to flexible and con- sistent grammar, it is simple to validate models of any classes. K eywords machine learning, R, diagnostics, visualization, modeling 1 Introduction Predictive modeling is a process that uses mathematical and computational methods to forecast outcomes. arXiv:1809.07763v4 [stat.CO] 26 May 2020 Lots of algorithms in this area have been developed and are still being develop. Therefore, there are countless possible models to choose from and a lot of ways to train a new new complex model.
    [Show full text]
  • ROC Curve Analysis and Medical Decision Making
    ROC curve analysis and medical decision making: what’s the evidence that matters for evidence-based diagnosis ? Piergiorgio Duca Biometria e Statistica Medica – Dipartimento di Scienze Cliniche Luigi Sacco – Università degli Studi – Via GB Grassi 74 – 20157 MILANO (ITALY) [email protected] 1) The ROC (Receiver Operating Characteristic) curve and the Area Under the Curve (AUC) The ROC curve is the statistical tool used to analyse the accuracy of a diagnostic test with multiple cut-off points. The test could be based on a continuous diagnostic indicant, such as a serum enzyme level, or just on an ordinal one, such as a classification based on radiological imaging. The ROC curve is based on the probability density distributions, in actually diseased and non diseased patients, of the diagnostician’s confidence in a positive diagnosis, and upon a set of cut-off points to separate “positive” and “negative” test results (Egan, 1968; Bamber, 1975; Swets, Pickett, 1982; Metz, 1986; Zhou et al, 2002). The Sensitivity (SE) – the proportion of diseased turned out to be test positive – and the Specificity (SP) – the proportion of non diseased turned out to be test negative – will depend on the particular confidence threshold the observer applies to partition the continuously distributed perceptions of evidence into positive and negative test results. The ROC curve is the plot of all the pairs of True Positive Rates (TPR = SE), as ordinate, and False Positive Rates (FPR = (1 – SP)), as abscissa, related to all the possible cut-off points. An ROC curve represents all of the compromises between SE and SP can be achieved, changing the confidence threshold.
    [Show full text]
  • Estimating the Variance of a Propensity Score Matching Estimator for the Average Treatment Effect
    Observational Studies 4 (2018) 71-96 Submitted 5/15; Published 3/18 Estimating the variance of a propensity score matching estimator for the average treatment effect Ronnie Pingel [email protected] Department of Statistics Uppsala University Uppsala, Sweden Abstract This study considers variance estimation when estimating the asymptotic variance of a propensity score matching estimator for the average treatment effect. We investigate the role of smoothing parameters in a variance estimator based on matching. We also study the properties of estimators using local linear estimation. Simulations demonstrate that large gains can be made in terms of mean squared error, bias and coverage rate by prop- erly selecting smoothing parameters. Alternatively, a residual-based local linear estimator could be used as an estimator of the asymptotic variance. The variance estimators are implemented in analysis to evaluate the effect of right heart catheterisation. Keywords: Average Causal Effect, Causal Inference, Kernel estimator 1. Introduction Matching estimators belong to a class of estimators of average treatment effects (ATEs) in observational studies that seek to balance the distributions of covariates in a treatment and control group (Stuart, 2010). In this study we consider one frequently used matching method, the simple nearest-neighbour matching with replacement (Rubin, 1973; Abadie and Imbens, 2006). Rosenbaum and Rubin (1983) show that instead of matching on covariates directly to remove confounding, it is sufficient to match on the propensity score. In observational studies the propensity score must almost always be estimated. An important contribution is therefore Abadie and Imbens (2016), who derive the large sample distribution of a nearest- neighbour propensity score matching estimator when using the estimated propensity score.
    [Show full text]
  • An Introduction to Logistic Regression: from Basic Concepts to Interpretation with Particular Attention to Nursing Domain
    J Korean Acad Nurs Vol.43 No.2, 154 -164 J Korean Acad Nurs Vol.43 No.2 April 2013 http://dx.doi.org/10.4040/jkan.2013.43.2.154 An Introduction to Logistic Regression: From Basic Concepts to Interpretation with Particular Attention to Nursing Domain Park, Hyeoun-Ae College of Nursing and System Biomedical Informatics National Core Research Center, Seoul National University, Seoul, Korea Purpose: The purpose of this article is twofold: 1) introducing logistic regression (LR), a multivariable method for modeling the relationship between multiple independent variables and a categorical dependent variable, and 2) examining use and reporting of LR in the nursing literature. Methods: Text books on LR and research articles employing LR as main statistical analysis were reviewed. Twenty-three articles published between 2010 and 2011 in the Journal of Korean Academy of Nursing were analyzed for proper use and reporting of LR models. Results: Logistic regression from basic concepts such as odds, odds ratio, logit transformation and logistic curve, assumption, fitting, reporting and interpreting to cautions were presented. Substantial short- comings were found in both use of LR and reporting of results. For many studies, sample size was not sufficiently large to call into question the accuracy of the regression model. Additionally, only one study reported validation analysis. Conclusion: Nurs- ing researchers need to pay greater attention to guidelines concerning the use and reporting of LR models. Key words: Logit function, Maximum likelihood estimation, Odds, Odds ratio, Wald test INTRODUCTION The model serves two purposes: (1) it can predict the value of the depen- dent variable for new values of the independent variables, and (2) it can Multivariable methods of statistical analysis commonly appear in help describe the relative contribution of each independent variable to general health science literature (Bagley, White, & Golomb, 2001).
    [Show full text]
  • Njit-Etd2007-041
    Copyright Warning & Restrictions The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or reproduction is not to be “used for any purpose other than private study, scholarship, or research.” If a, user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of “fair use” that user may be liable for copyright infringement, This institution reserves the right to refuse to accept a copying order if, in its judgment, fulfillment of the order would involve violation of copyright law. Please Note: The author retains the copyright while the New Jersey Institute of Technology reserves the right to distribute this thesis or dissertation Printing note: If you do not wish to print this page, then select “Pages from: first page # to: last page #” on the print dialog screen The Van Houten library has removed some of the personal information and all signatures from the approval page and biographical sketches of theses and dissertations in order to protect the identity of NJIT graduates and faculty. ABSTRACT PROBLEMS RELATED TO EFFICACY MEASUREMENT AND ANALYSES by Sibabrata Banerjee In clinical research it is very common to compare two treatments on the basis of an efficacy vrbl Mr pfll f Χ nd Υ dnt th rpn f ptnt n th t trtnt A nd rptvl th
    [Show full text]
  • Propensity Score Analysis with Hierarchical Data
    Section on Statistics in Epidemiology Propensity score analysis with hierarchical data Fan Li, Alan M. Zaslavsky, Mary Beth Landrum Department of Health Care Policy, Harvard Medical School 180 Longwood Avenue, Boston, MA 02115 October 29, 2007 Abstract nors et al., 1996; D’Agostino, 1998, and references therein). This approach, which involves comparing subjects weighted Propensity score (Rosenbaum and Rubin, 1983) methods are (or stratified, matched) according to their propensity to re- being increasingly used as a less parametric alternative to tra- ceive treatment (i.e., propensity score), attempts to balance ditional regression methods in medical care and health policy subjects in treatment groups in terms of observed character- research. Data collected in these disciplines are often clus- istics as would occur in a randomized experiment. Propensity tered or hierarchically structured, in the sense that subjects are score methods permit control of all observed confounding fac- grouped together in one or more ways that may be relevant tors that might influence both choice of treatment and outcome to the analysis. However, propensity score was developed using a single composite measure, without requiring specifi- and has been applied in settings with unstructured data. In cation of the relationships between the control variables and this report, we present and compare several propensity-score- outcome. weighted estimators of treatment effect in the context of hier- Propensity score methods were developed and have been archically structured
    [Show full text]
  • Logistic Regression, Part I: Problems with the Linear Probability Model
    Logistic Regression, Part I: Problems with the Linear Probability Model (LPM) Richard Williams, University of Notre Dame, https://www3.nd.edu/~rwilliam/ Last revised February 22, 2015 This handout steals heavily from Linear probability, logit, and probit models, by John Aldrich and Forrest Nelson, paper # 45 in the Sage series on Quantitative Applications in the Social Sciences. INTRODUCTION. We are often interested in qualitative dependent variables: • Voting (does or does not vote) • Marital status (married or not) • Fertility (have children or not) • Immigration attitudes (opposes immigration or supports it) In the next few handouts, we will examine different techniques for analyzing qualitative dependent variables; in particular, dichotomous dependent variables. We will first examine the problems with using OLS, and then present logistic regression as a more desirable alternative. OLS AND DICHOTOMOUS DEPENDENT VARIABLES. While estimates derived from regression analysis may be robust against violations of some assumptions, other assumptions are crucial, and violations of them can lead to unreasonable estimates. Such is often the case when the dependent variable is a qualitative measure rather than a continuous, interval measure. If OLS Regression is done with a qualitative dependent variable • it may seriously misestimate the magnitude of the effects of IVs • all of the standard statistical inferences (e.g. hypothesis tests, construction of confidence intervals) are unjustified • regression estimates will be highly sensitive to the range of particular values observed (thus making extrapolations or forecasts beyond the range of the data especially unjustified) OLS REGRESSION AND THE LINEAR PROBABILITY MODEL (LPM). The regression model places no restrictions on the values that the independent variables take on.
    [Show full text]
  • Homoskedasticity
    Homoskedasticity How big is the difference between the OLS estimator and the true parameter? To answer this question, we make an additional assumption called homoskedasticity: Var (ujX) = σ2: (23) This means that the variance of the error term u is the same, regardless of the predictor variable X. If assumption (23) is violated, e.g. if Var (ujX) = σ2h(X), then we say the error term is heteroskedastic. Sylvia Fruhwirth-Schnatter¨ Econometrics I WS 2012/13 1-55 Homoskedasticity • Assumption (23) certainly holds, if u and X are assumed to be independent. However, (23) is a weaker assumption. • Assumption (23) implies that σ2 is also the unconditional variance of u, referred to as error variance: Var (u) = E(u2) − (E(u))2 = σ2: Its square root σ is the standard deviation of the error. • It follows that Var (Y jX) = σ2. Sylvia Fruhwirth-Schnatter¨ Econometrics I WS 2012/13 1-56 Variance of the OLS estimator How large is the variation of the OLS estimator around the true parameter? ^ • Difference β1 − β1 is 0 on average • Measure the variation of the OLS estimator around the true parameter through the expected squared difference, i.e. the variance: ( ) ^ ^ 2 Var β1 = E((β1 − β1) ) (24) ( ) ^ ^ ^ 2 • Similarly for β0: Var β0 = E((β0 − β0) ). Sylvia Fruhwirth-Schnatter¨ Econometrics I WS 2012/13 1-57 Variance of the OLS estimator ^ Variance of the slope estimator β1 follows from (22): ( ) N 1 X Var β^ = (x − x)2Var (u ) 1 N 2(s2)2 i i x i=1 σ2 XN σ2 = (x − x)2 = : (25) N 2(s2)2 i Ns2 x i=1 x • The variance of the slope estimator is the larger, the smaller the number of observations N (or the smaller, the larger N).
    [Show full text]
  • HOMOSCEDASTICITY PLOT Graphics Commands
    HOMOSCEDASTICITY PLOT Graphics Commands HOMOSCEDASTICITY PLOT PURPOSE Generates a homoscedasticity plot. DESCRIPTION A homoscedasticity plot is a graphical data analysis technique for assessing the assumption of constant variance across subsets of the data. The first variable is a response variable and the second variable identifies subsets of the data. The mean and standard deviation are calculated for each of these subsets. The following plot is generated: Vertical axis = subset standard deviations; Horizontal axis = subset means. The interpertation of this plot is that the greater the spread on the vertical axis, the less valid is the assumption of constant variance. A common pattern is for the spread (i.e., the standard deviation) to increase as the location (i.e., the mean) increases. This indicates the need for some type of transformation such as a log or square root. SYNTAX HOMOSCEDASTICITY PLOT <y> <tag> <SUBSET/EXCEPT/FOR qualification> where <y> is a response variable; <tag> identifies the subsets; and where the <SUBSET/EXCEPT/FOR qualification> is optional. EXAMPLES HOMOSCEDASTICITY PLOT Y1 TAG HOMOSCEDASTICITY PLOT Y1 TAG SUBSET TAG > 2 NOTE 1 One limitation of the homoscedasticity plot is that it does not give a convenient way to label the groups on the plot. This can be done by using the SUBSET command as in this example (assume Y is the response variable, X the group-id variable): X1LABEL MEANS Y1LABEL STANDARD DEVIATIONS CHARACTER X; LINE BLANK XLIMITS 0 5; YLIMITS 0 4 CHARACTER NORM; TITLE HOMOSCEDASTICITY PLOT HOMOSCEDASTICITY PLOT Y X SUBSET X = 1 PRE-ERASE OFF CHARACTER T HOMOSCEDASTICITY PLOT Y X SUBSET X = 2 CHARACTER CHIS HOMOSCEDASTICITY PLOT Y X SUBSET X = 3 CHARACTER UNIF HOMOSCEDASTICITY PLOT Y X SUBSET X = 4 CHARACTER F HOMOSCEDASTICITY PLOT Y X SUBSET X = 5 NOTE 2 Bartlett’s test is an analytic test for the assumption of constant variance.
    [Show full text]
  • Chapter 10 Heteroskedasticity
    Chapter 10 Heteroskedasticity In the multiple regression model yX , it is assumed that VI() 2 , i.e., 22 Var()i , Cov(ij ) 0, i j 1, 2,..., n . In this case, the diagonal elements of the covariance matrix of are the same indicating that the variance of each i is same and off-diagonal elements of the covariance matrix of are zero indicating that all disturbances are pairwise uncorrelated. This property of constancy of variance is termed as homoskedasticity and disturbances are called as homoskedastic disturbances. In many situations, this assumption may not be plausible, and the variances may not remain the same. The disturbances whose variances are not constant across the observations are called heteroskedastic disturbance, and this property is termed as heteroskedasticity. In this case 2 Var()ii , i 1,2,..., n and disturbances are pairwise uncorrelated. The covariance matrix of disturbances is 2 00 1 00 2 Vdiag( ) (22 , ,..., 2 )2 . 12 n 2 00 n Regression Analysis | Chapter 10 | Heteroskedasticity | Shalabh, IIT Kanpur 1 Graphically, the following pictures depict homoskedasticity and heteroskedasticity. Homoskedasticity Heteroskedasticity (Var(y) increases with x) Heteroskedasticity (Var(y) decreases with x) Examples: Suppose in a simple linear regression model, x denote the income and y denotes the expenditure on food. It is observed that as the income increases, the expenditure on food increases because of the choice and varieties in food increase, in general, up to a certain extent. So the variance of observations on y will not remain constant as income changes. The assumption of homoscedasticity implies that the consumption pattern of food will remain the same irrespective of the income of the person.
    [Show full text]
  • Assumptions of Multiple Linear Regression
    Statistics Solutions Advancement Through Clarity http://www.statisticssolutions.com Assumptions of Multiple Linear Regression Multiple linear regression analysis makes several key assumptions: Linear relationship Multivariate normality No or little multicollinearity No auto-correlation Homoscedasticity Multiple linear regression needs at least 3 variables of metric (ratio or interval) scale. A rule of thumb for the sample size is that regression analysis requires at least 20 cases per independent variable in the analysis, in the simplest case of having just two independent variables that requires n > 40. G*Power can also be used to calculate a more exact, appropriate sample size. Firstly, multiple linear regression needs the relationship between the independent and dependent variables to be linear. It is also important to check for outliers since multiple linear regression is sensitive to outlier effects. The linearity assumption can best be tested with scatter plots, the following two examples depict two cases, where no and little linearity is present. 1 / 5 Statistics Solutions Advancement Through Clarity http://www.statisticssolutions.com Secondly, the multiple linear regression analysis requires all variables to be normal. This assumption can best be checked with a histogram and a fitted normal curve or a Q-Q-Plot. Normality can be checked with a goodness of fit test, e.g., the Kolmogorov-Smirnof test. When the data is not normally distributed a non-linear transformation, e.g., log-transformation might fix this issue. However it can introduce effects of multicollinearity. Thirdly, multiple linear regression assumes that there is little or no multicollinearity in the data. Multicollinearity occurs when the independent variables are not independent from each other.
    [Show full text]
  • Simple Linear Regression 80 60 Rating 40 20
    Simple Linear 12 Regression Material from Devore’s book (Ed 8), and Cengagebrain.com Simple Linear Regression 80 60 Rating 40 20 0 5 10 15 Sugar 2 Simple Linear Regression 80 60 Rating 40 20 0 5 10 15 Sugar 3 Simple Linear Regression 80 60 Rating 40 xx 20 0 5 10 15 Sugar 4 The Simple Linear Regression Model The simplest deterministic mathematical relationship between two variables x and y is a linear relationship: y = β0 + β1x. The objective of this section is to develop an equivalent linear probabilistic model. If the two (random) variables are probabilistically related, then for a fixed value of x, there is uncertainty in the value of the second variable. So we assume Y = β0 + β1x + ε, where ε is a random variable. 2 variables are related linearly “on average” if for fixed x the actual value of Y differs from its expected value by a random amount (i.e. there is random error). 5 A Linear Probabilistic Model Definition The Simple Linear Regression Model 2 There are parameters β0, β1, and σ , such that for any fixed value of the independent variable x, the dependent variable is a random variable related to x through the model equation Y = β0 + β1x + ε The quantity ε in the model equation is the “error” -- a random variable, assumed to be symmetrically distributed with 2 2 E(ε) = 0 and V(ε) = σ ε = σ (no assumption made about the distribution of ε, yet) 6 A Linear Probabilistic Model X: the independent, predictor, or explanatory variable (usually known).
    [Show full text]