1 MISSING DATA TECHNIQUES WITH SAS IDRE Statistical Consulting Group
2 ROAD MAP FOR TODAY To discuss: 1. Commonly used techniques for handling missing data, focusing on multiple imputation 2. Issues that could arise when these techniques are used 3. Implementation of SAS Proc MI procedure Assuming MVN Assuming FCS 4. Imputation Diagnostics
3 GOALS OF STATISTICAL ANALYSIS WITH MISSING DATA Minimize bias Maximize use of available information Obtain appropriate estimates of uncertainty
4 THE MISSING DATA MECHANISM DESCRIBES THE PROCESS THAT IS BELIEVED TO HAVE GENERATED THE MISSING VALUES. 1. Missing completely at random (MCAR) Neither the unobserved values of the variable with missing nor the other variables in the dataset predict whether a value will be missing. Example: Planned missingness 2. Missing at random (MAR) Other variables (but not the variable with missing itself) in the dataset can be used to predict missingness. Example: Men may be more likely to decline to answer some questions than women 3. Missing not at random (MNAR) The unobserved value of the variable with missing predicts missingness. Example: Individuals with very high incomes are more likely to decline to answer questions about their own income
5 OUR DATA High School and Beyond N= Variables Student Demographics and Achievement including test scores
6 ANALYSIS OF FULL DATA
7 COMMON TECHNIQUES FOR DEALING WITH MISSING DATA 1. Complete case analysis (listwise deletion) 2. Mean Imputation 3. Single Imputation 4. Stochastic Imputation
8 Method: Drop cases with missing data on any variable of interest Appeal: Nothing to implement default method Drawbacks: COMPLETE CASE ANALYSIS (LISTWISE DELETION) Loss of cases/data Biased estimates unless MCAR
9 COMPLETE CASE ANALYSIS (LISTWISE DELETION) proc means data = ats.hsb_mar nmiss N min max mean std; var _numeric_ ; run;
10 LISTWISE DELETION ANALYSIS DROPS OBSERVATIONS WITH MISSING VALUES
11 UNCONDITIONAL MEAN IMPUTATION Method: Replace missing values for a variable with its overall estimated mean Appeal: Simple and easily implemented Drawbacks: Artificial reduction in variability b/c imputing values at the mean. Changes the magnitude of correlations between the imputed variables and other variables.
12 MEAN, SD AND CORRELATION MATRIX OF 5 VARIABLES BEFORE & AFTER MEAN IMPUTATION
13 SINGLE OR DETERMINISTIC (REGRESSION) IMPUTATION Method: Replace missing values with predicted scores from a regression equation. Appeal: Uses complete information to impute values. Drawback: All predicted values fall directly on the regression line, decreasing variability. Also known as regression imputation
14 SINGLE OR DETERMINISTIC (REGRESSION) IMPUTATION p.46, Applied Missing Data Analysis, Craig Enders (2010)
15 SINGLE OR DETERMINISTIC (REGRESSION) IMPUTATION Imputing values directly on the regression line: Underestimates uncertainty Inflates associations between variables because it imputes perfectly correlated values Upwardly biases R-squared statistics, even under the assumption of MCAR
16 STOCHASTIC IMPUTATION Stochastic imputation addresses these problems with regression imputation by incorporating or "adding back" lost variability. Method: Add randomly drawn residual to imputed value from regression imputation. Distribution of residuals based on residual variance from regression model.
18 STOCHASTIC IMPUTATION Appeals: Restores some lost variability. Superior to the previous methods as it will produce unbiased coefficient estimates under MAR. Drawback: SE s produced during stochastic estimation, while less biased, will still be attenuated.
19 WHAT IS MULTIPLE IMPUTATION? Iterative form of stochastic imputation. Multiple values are imputed rather than a single value to reflect the uncertainty around the true value. Each imputed value includes a random component whose magnitude reflects the extent to which other variables in the model cannot predict it's true value Common misconception: imputed values should represent "real" values. Purpose: To correctly reproduce the full data variance/covariance matrix
20 ISN'T MULTIPLE IMPUTATION JUST MAKING UP DATA? No. This is argument applies to single imputation methods MI analysis methods account for the uncertainty/error associated with the imputed values. Estimated parameters never depend on a single value.
21 THREE PHASES 1. Imputation or Fill-in Phase: Missing values are imputed, forming a complete data set. This process is repeated m times. 2. Analysis Phase: Each of the m complete data sets is then analyzed using a statistical model (e.g linear regression). 3. Pooling Phase: The parameter estimates (e.g. coefficients and standard errors) obtained from each analyzed data set are then combined for inference.
22 THE IMPORTANCE OF BEING COMPATIBLE The imputation model should be "congenial to or consistent with your analytic model: Includes, at the very least, the same variables as the analytic model. Includes any transformations to variables in the analytic model E.g. logarithmic and squaring transformations, interaction terms All relationships between variables should be represented and estimated simultaneously. Otherwise, you are imputing values assuming they are uncorrelated with the variables you did not include.
23 PREPARING FOR MULTIPLE IMPUTATION 1. Examine the number and proportion of missing values among your variables of interest. 2. Examine Missing Data Patterns among your variables of interest. 3. If necessary, identify potential auxiliary variables 4. Determine imputation method
24 EXAMINE MISSING VALUES: PROC MEANS NMISS OPTION
25 EXAMINE MISSING VALUES: NOTE VARIABLE(S ) WITH HIGH PROPORTION OF MISSING -- THEY WILL IMPACT MODEL CONVERGENCE THE MOST
26 EXAMINE MISSING DATA PATTERNS: SYNTAX proc mi data=hsb_mar nimpute=0 ; var write read female math prog ; ods select misspattern; run;
27 EXAMINE MISSING DATA PATTERNS
28 IDENTIFY POTENTIAL AUXILIARY VARIABLES Characteristics: Correlated with missing variable (rule of thumb: r> 0.4) Predictor of missingness Not of analytic interest, so only used in imputation model Why? Including auxiliary variables in the imputation model can: Improve the quality of imputed values Increase power, especially with high fraction of missing information (FMI >25%) Be especially important when imputing DV Increase plausibility of MAR
29 HOW DO YOU IDENTIFY AUXILIARY VARIABLES? A priori knowledge Previous literature Identify associations in data
30 AUXILIARY VARIABLES ARE CORRELATED WITH MISSING VARIABLE
31 AUXILIARY VARIABLES ARE PREDICTORS OF MISSINGNESS
32 IMPUTATION MODEL EXAMPLE 1: MI USING MULTIVARIATE NORMAL DISTRIBUTION (MVN)
33 ASSUMING A JOINT MULTIVARIATE NORMAL DISTRIBUTION Probably the most common parametric approach for multiple imputation. Assumes variables are individually and jointly normally distributed Assuming a MVN distribution is robust to violations of normality given a large enough N. Uses the data augmentation (DA) algorithm to impute. Biased estimates may result when N is relatively small and the fraction of missing information is high.
34 IMPUTATION PHASE proc mi data= new nimpute=10 out=mi_mvn seed=54321; var socst science write read female math progcat1 progcat2 ; run;
35 MULTIPLY IMPUTED DATASET
36 ANALYSIS PHASE: ESTIMATE MODEL FOR EACH IMPUTED DATASET proc glm data = mi_mvn ; model read = write female math progcat1 progcat2; by _imputation_; ods output ParameterEstimates=a_mvn; run; quit;
37 PARAMETER ESTIMATE DATASET
38 proc mianalyze parms=a_mvn; modeleffects intercept write female math run; POOLING PHASE- COMBINING PARAMETER ESTIMATES ACROSS DATASETS progcat1 progcat2;
40 COMPARE MIANALYZE ESTIMATES TO ANALYSIS WITH FULL DATA FULL DATA ANALYSIS MIANALYZE OUPUT
41 HOW DOES PROC MIANALYZE WORK PROC MIANALYZE combines results across imputations Regression coefficients are averaged across imputations Standard errors incorporate uncertainty from 2 sources: "within imputation" - variability in estimate expected with no missing data The usual uncertainty regarding a regression coefficient "between imputation" - variability due to missing information. The uncertainty surrounding missing values
42 OUTPUT FROM MIANALYZE
43 VARIANCE WITHIN Sampling variability expected with no missing data. Average of variability of coefficients within an imputation Equal to arithmetic mean of sampling variances (SE 2 ) Example: Add together 10 estimated SE 2 for write and divide by 10 V w =
44 VARIANCE BETWEEN Variability in estimates across imputations i.e. the variance of the m coefficients Estimates the additional variation (uncertainty) that results from missing data. Example: take all 10 of the parameter estimates (β) for write and calculate the variance V B =
45 TOTAL VARIANCE The total variance is sum of 3 sources of variance. Within, Between Additional source of sampling variance. What is the sampling variance? Variance Between divided by number of imputations Represents sampling error associated with the overall coefficient estimates. Serves as a correction factor for using a specific number of imputations.
46 DEGREES OF FREEDOM DF for combined results are determined by the number of imputations. *By default DF = infinity, typically not a problem with large N but can be with smaller samples The standard formula to estimate DF can yield estimates that are fractional or that far exceed the DF for complete data. Correction to adjust for the problem of inflated DF has been implemented Use the EDF option on the proc mianalyze line to indicate to SAS what is the proper adjusted DF.
47 RELATIVE INCREASES IN VARIANCE (RVI) Proportional increase in total sampling variance due to missing information [V B + V B /m] V w For example, the RVI for write coefficient is 0.048, meaning that the sampling variance is 4.8% larger than it would have been with complete data.
48 FRACTION OF MISSING INFORMATION (FMI) Directly related to RVI. Proportion of total sampling variance that is due to missing data [V B + V B /m] V T For a given variable, FMI based on percentage missing and correlations with other imputation model variables. Interpretation similar to an R 2. Example: FMI=.046 for write means that 4.6% of sampling variance is attributable to missing data.
49 RELATIVE EFFICIENCY: IS 5 IMPUTATIONS ENOUGH? Captures how well true population parameters are estimated Related to both the FMI and m Low FMI + few m = high efficiency As FMI increase so should m: Better statistical power and more stable estimate
50 DIAGN0STICS: HOW DO I KNOW IF IT WORKED? Compare means and frequencies of observed and imputed values. Use boxplots to compare distributions Look at Variance Information tables from the proc mianalyze output Plots - Assess convergence of DA algorithm
51 TRACE PLOTS: DID MY IMPUTATION MODEL CONVERGE? Convergence for each imputed variable can be assessed using trace plots. Examine for each imputed variables Special attention to variables with a high FMI proc mi data= ats.hsb_mar nimpute=10 out=mi_mvn; mcmc plots=trace plots=acf ; var socst write read female math; run;
53 EXAMPLE OF A POOR TRACE PLOT
54 AUTOCORRELATION PLOTS: DID MY IMPUTATION MODEL CONVERGE? Assess possible auto correlation of parameter values between iterations. Assess the magnitude of the observed dependency of imputed values across iterations. proc mi data= ats.hsb_mar nimpute=10 out=mi_mvn; mcmc plots=trace plots=acf ; var socst write read female math; run;
56 IMPUTATION MODEL EXAMPLE 2: MI USING FULLY CONDITIONAL SPECIFICATION (FCS)
57 WHAT IF I DON T WANT TO ASSUME A MULTIVARIATE NORMAL DISTRIBUTION? Alternative method for imputation is Fully Conditional Method (FCS) FCS does not assume a joint distribution and allows the use of different distributions across variables. Each variable with missing is allowed its own type of regression (linear, logistic, etc) for imputation Example uses: Logistic model for binary outcome Poisson model for count variable Other bounded values
58 AVAILABLE DISTRIBUTIONS FCS methods available: Discriminant function or logistic regression for binary/categorical variables Linear regression and predictive mean matching for continuous variables. Properties to Note: 1. Discriminant function only continuous vars as covariates (default). 2. Logistic regression assumes ordering of class variables if more then two levels (default). 3. Regression is default imputation method for continuous vars. 4. PMM will provide plausible values. 1. For an observation missing on X, finds cases in data with similar values on other covariates, then randomly selects an X value from those cases
59 IMPUTATION PHASE proc mi data= ats.hsb_mar nimpute=20 out=mi_fcs ; class female prog; fcs plots=trace(mean std); var socst write read female math science prog; fcs discrim(female prog /classeffects=include) nbiter =100 ; run;
60 ALTERNATE EXAMPLE proc mi data= ats.hsb_mar nimpute=20 out=mi_new1; class female prog; var socst write read female math science prog; fcs logistic (female= socst science) ; fcs logistic (prog =math socst /link=glogit) regpmm(math read write); run;
61 ANALYSIS PHASE: ESTIMATE GLM MODEL USING EACH IMPUTED DATASET proc genmod data=mi_fcs; class female prog; model read=write female math prog/dist=normal; by _imputation_; ods output ParameterEstimates=gm_fcs; run;
63 POOLING PHASE- COMBINING PARAMETER ESTIMATES ACROSS DATASETS PROC MIANALYZE parms(classvar=level)=gm_fcs; class female prog; MODELEFFECTS INTERCEPT write female math prog; RUN;
64 TRACE PLOTS: DID MY IMPUTATION MODEL CONVERGE?
65 FCS HAS SEVERAL PROPERTIES THAT MAKE IT AN ATTRACTIVE ALTERNATIVE 1. FCS allows each variable to be imputed using its own conditional distribution 2. Different imputation models can be specified for different variables. However, this can also cause estimation problems. Beware: Convergence Issues such as complete and quasi-complete separation (e.g. zero cells) when imputing categorical variables.
66 BOTTOM LINE MI improves over single imputation methods because: Single value never used Appropriate estimates of uncertainty The nature of your data and model will determine if you choose MVN or FCS Both are state of the art methods for handling missing data Produce unbiased estimates assuming MAR
ABSTRACT Paper 3295-2015 Imputing Missing Data using SAS Christopher Yim, California Polytechnic State University, San Luis Obispo Missing data is an unfortunate reality of statistics. However, there are
Missing Data: Part 1 What to Do? Carol B. Thompson Johns Hopkins Biostatistics Center SON Brown Bag 3/20/13 Overview Missingness and impact on statistical analysis Missing data assumptions/mechanisms Conventional
ABSTRACT Paper SD-08 Dealing with Missing Data for Credit Scoring Steve Fleming, Clarity Services Inc. How well can statistical adjustments account for missing data in the development of credit scores?
APPLIED MISSING DATA ANALYSIS Craig K. Enders Series Editor's Note by Todd D. little THE GUILFORD PRESS New York London Contents 1 An Introduction to Missing Data 1 1.1 Introduction 1 1.2 Chapter Overview
John Fox Sociology 740 Winter 2014 Outline Why Missing Data Arise Why Missing Data Arise Global or unit non-response. In a survey, certain respondents may be unreachable or may refuse to participate. Item
VASA Mission of VA Statisticians Association (VASA) Promote & disseminate statistical methodological research relevant to VA studies; Facilitate communication & collaboration among VA-affiliated statisticians;
Handling missing data in Stata a whirlwind tour 2012 Italian Stata Users Group Meeting Jonathan Bartlett www.missingdata.org.uk 20th September 2012 1/55 Outline The problem of missing data and a principled
Analyzing Structural Equation Models With Missing Data Craig Enders* Arizona State University firstname.lastname@example.org based on Enders, C. K. (006). Analyzing structural equation models with missing data. In G.
Modern Methods for Missing Data Paul D. Allison, Ph.D. Statistical Horizons LLC www.statisticalhorizons.com 1 Introduction Missing data problems are nearly universal in statistical practice. Last 25 years
IBM SPSS Missing Values 22 Note Before using this information and the product it supports, read the information in Notices on page 23. Product Information This edition applies to version 22, release 0,
Paper SAS270-2014 Sensitivity Analysis in Multiple Imputation for Missing Data Yang Yuan, SAS Institute Inc. ABSTRACT Multiple imputation, a popular strategy for dealing with missing values, usually assumes
Res. Lett. Inf. Math. Sci. (2002) 3, 153-160 Available online at http://www.massey.ac.nz/~wwiims/research/letters/ Dealing with Missing Data Judi Scheffer I.I.M.S. Quad A, Massey University, P.O. Box 102904
Multiple Imputation for Missing Data: A Cautionary Tale Paul D. Allison University of Pennsylvania Address correspondence to Paul D. Allison, Sociology Department, University of Pennsylvania, 3718 Locust
Int. Journal of Math. Analysis, Vol. 5, 2011, no. 1, 1-13 Review of the Methods for Handling Missing Data in Longitudinal Data Analysis Michikazu Nakai and Weiming Ke Department of Mathematics and Statistics
4 Missing Data Paul D. Allison INTRODUCTION Missing data are ubiquitous in psychological research. By missing data, I mean data that are missing for some (but not all) variables and for some (but not all)
Missing Data Katyn & Elena What to do with Missing Data Standard is complete case analysis/listwise dele;on ie. Delete cases with missing data so only complete cases are le> Two other popular op;ons: Mul;ple
00 Jeeshim and KUCC65. (003-05-09) Multicollinearity.doc Introduction Multicollinearity in Regression Models Multicollinearity is a high degree of correlation (linear dependency) among several independent
1 Statistical Methods Principles Missing Data Dr Eleni Matechou email@example.com References: R.J.A. Little and D.B. Rubin 2nd edition Statistical Analysis with Missing Data J.L. Schafer and J.W.
Multiple regression Introduction Multiple regression is a logical extension of the principles of simple linear regression to situations in which there are several predictor variables. For instance if we
Longitudinal and Life Course Studies 2009 Volume 1 Issue 1 Pp 63-72 Handling attrition and non-response in longitudinal data Harvey Goldstein University of Bristol Correspondence. Professor H. Goldstein
IBM SPSS Missing Values 20 Note: Before using this information and the product it supports, read the general information under Notices on p. 87. This edition applies to IBM SPSS Statistics 20 and to all
Introduction to Multivariate Models: Modeling Multivariate Outcomes with Mixed Model Repeated Measures Analyses Applied Multilevel Models for Cross Sectional Data Lecture 11 ICPSR Summer Workshop University
Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates
A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 4: September
5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4
Mgt 540 Research Methods Data Analysis 1 Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm http://web.utk.edu/~dap/random/order/start.htm
Journal of Abnormal Psychology Copyright 2003 by the American Psychological Association, Inc. 2003, Vol. 112, No. 4, 545 557 0021-843X/03/$12.00 DOI: 10.1037/0021-843X.112.4.545 Missing Data Techniques
123 Kwantitatieve Methoden (1999), 62, 123-138. A REVIEW OF CURRENT SOFTWARE FOR HANDLING MISSING DATA Joop J. Hox 1 ABSTRACT. When we deal with a large data set with missing data, we have to undertake
SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling
The simple linear model Regression Analysis: Basic Concepts Allin Cottrell Represents the dependent variable, y i, as a linear function of one independent variable, x i, subject to a random disturbance
Name Financial Econometrics Jeffrey R. Russell Final Exam You have 3 hours to complete the exam. Use can use a calculator. Try to fit all your work in the space provided. If you find you need more space
How To Use A Monte Carlo Study To Decide On Sample Size and Determine Power Linda K. Muthén Muthén & Muthén 11965 Venice Blvd., Suite 407 Los Angeles, CA 90066 Telephone: (310) 391-9971 Fax: (310) 391-8971
Craig K. Enders Arizona State University Department of Psychology firstname.lastname@example.org Topic Page Missing Data Patterns And Missing Data Mechanisms 1 Traditional Missing Data Techniques 7 Maximum Likelihood
Running head: ASSUMPTIONS IN MULTIPLE REGRESSION 1 Assumptions in Multiple Regression: A Tutorial Dianne L. Ballance ID#00939966 University of Calgary APSY 607 ASSUMPTIONS IN MULTIPLE REGRESSION 2 Assumptions
SPSS TRAINING SESSION 3 ADVANCED TOPICS (PASW STATISTICS 17.0) Sun Li Centre for Academic Computing email@example.com IN SPSS SESSION 2, WE HAVE LEARNT: Elementary Data Analysis Group Comparison & One-way
HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate
Stats - Moderation Moderation A moderator is a variable that specifies conditions under which a given predictor is related to an outcome. The moderator explains when a DV and IV are related. Moderation
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
Auxiliary Variables in Mixture Modeling: 3-Step Approaches Using Mplus Tihomir Asparouhov and Bengt Muthén Mplus Web Notes: No. 15 Version 8, August 5, 2014 1 Abstract This paper discusses alternatives
Data Mining and Data Warehousing Henryk Maciejewski Data Mining Predictive modelling: regression Algorithms for Predictive Modelling Contents Regression Classification Auxiliary topics: Estimation of prediction
Brockmeier, Kromrey, & Hogarty Nonrandomly Missing Data in Multiple Regression Analysis: An Empirical Comparison of Ten s Lantry L. Brockmeier Jeffrey D. Kromrey Kristine Y. Hogarty Florida A & M University
Regression Analysis Pekka Tolonen Outline of Topics Simple linear regression: the form and estimation Hypothesis testing and statistical significance Empirical application: the capital asset pricing model
Multiple Linear Regression A regression with two or more explanatory variables is called a multiple regression. Rather than modeling the mean response as a straight line, as in simple regression, it is
Lecture 16: Generalized Additive Models Regression III: Advanced Methods Bill Jacoby Michigan State University http://polisci.msu.edu/jacoby/icpsr/regress3 Goals of the Lecture Introduce Additive Models
Topic 4 - Analysis of Variance Approach to Regression Outline Partitioning sums of squares Degrees of freedom Expected mean squares General linear test - Fall 2013 R 2 and the coefficient of correlation
Regression Analysis Using JMP Yiming Peng, Department of Statistics February 12, 2013 2 Presentation and Data http://www.lisa.stat.vt.edu Short Courses Regression Analysis Using JMP Download Data to Desktop
Journal of School Psychology 48 (2010) 5 37 An introduction to modern missing data analyses Amanda N. Baraldi, Craig K. Enders Arizona State University, United States Received 19 October 2009; accepted
Bivariate Regression Analysis The beginning of many types of regression TOPICS Beyond Correlation Forecasting Two points to estimate the slope Meeting the BLUE criterion The OLS method Purpose of Regression
[Leeuw, Edith D. de, and Joop Hox. (2008). Missing Data. Encyclopedia of Survey Research Methods. Retrieved from http://sage-ereference.com/survey/article_n298.html] Missing Data An important indicator
MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level of Significance
DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 Analysis of covariance and multiple regression So far in this course,
Multiple Regression in SPSS This example shows you how to perform multiple regression. The basic command is regression : linear. In the main dialog box, input the dependent variable and several predictors.
DISCRIMINANT FUNCTION ANALYSIS (DA) John Poulsen and Aaron French Key words: assumptions, further reading, computations, standardized coefficents, structure matrix, tests of signficance Introduction Discriminant
Statistical modelling with missing data using multiple imputation Session 4: Sensitivity Analysis after Multiple Imputation James Carpenter London School of Hygiene & Tropical Medicine Email: firstname.lastname@example.org
EDUCATION AND VOCABULARY MULTIPLE REGRESSION IN ACTION EDUCATION AND VOCABULARY 5-10 hours of input weekly is enough to pick up a new language (Schiff & Myers, 1988). Dutch children spend 5.5 hours/day
136 Poisson Regression Analysis 13. Poisson Regression Analysis We have so far considered situations where the outcome variable is numeric and Normally distributed, or binary. In clinical work one often
Regression in ANOVA James H. Steiger Department of Psychology and Human Development Vanderbilt University James H. Steiger (Vanderbilt University) 1 / 30 Regression in ANOVA 1 Introduction 2 Basic Linear
Journal of Official Statistics, Vol. 23, No. 4, 2007, pp. 467 475 Discussion Seppo Laaksonen 1 1. Introduction Bjørnstad s article is a welcome contribution to the discussion on multiple imputation (MI)
Linear Regression Analysis using PROC GLM Regression analysis is a statistical method of obtaining an equation that represents a linear relationship between two variables (simple linear regression), or
21 Multivariate analysis of variance In previous chapters, we explored the use of analysis of variance to compare groups on a single dependent variable. In many research situations, however, we are interested
2. Linear regression with multiple regressors Aim of this section: Introduction of the multiple regression model OLS estimation in multiple regression Measures-of-fit in multiple regression Assumptions
Chapter 3 Student Lecture Notes 3- Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing
4. Simple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/4 Outline The simple linear model Least squares estimation Forecasting with regression Non-linear functional forms Regression
IAPRI Quantitative Analysis Capacity Building Series Multiple regression analysis & interpreting results How important is R-squared? R-squared Published in Agricultural Economics 0.45 Best article of the
Data analysis process Data collection and preparation Collect data Prepare codebook Set up structure of data Enter data Screen data for errors Exploration of data Descriptive Statistics Graphs Analysis
Introduction to Multilevel Modeling Using HLM 6 By ATS Statistical Consulting Group Multilevel data structure Students nested within schools Children nested within families Respondents nested within interviewers
University of California, Berkeley Prof. Ken Chay Department of Economics Fall Semester, 005 ECON 14 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE # Question 1: a. Below are the scatter plots of hourly wages
SPSS Guide: Regression Analysis I put this together to give you a step-by-step guide for replicating what we did in the computer lab. It should help you run the tests we covered. The best way to get familiar
New SAS Procedures for Analysis of Sample Survey Data Anthony An and Donna Watts, SAS Institute Inc, Cary, NC Abstract Researchers use sample surveys to obtain information on a wide variety of issues Many
What s Missing? Handling Missing Data with SAS Timothy B. Gravelle Principal Scientist & Director, Insights Lab September 13, 2013 2000-2013 PriceMetrix Inc. Patents granted and pending. Missing data Survey
PharmaSUG2011 - Paper SP04 Implementation of Pattern-Mixture Models Using Standard SAS/STAT Procedures Bohdana Ratitch, Quintiles, Montreal, Quebec, Canada Michael O Kelly, Quintiles, Dublin, Ireland ABSTRACT
PSY 512: Advanced Statistics for Psychological and Behavioral Research 2 Understand linear regression with a single predictor Understand how we assess the fit of a regression model Total Sum of Squares
Simple Linear (OLS) Regression Regression is a method for studying the relationship of a dependent variable and one or more independent variables. Simple Linear Regression tells you the amount of variance
1 Multivariate Logistic Regression As in univariate logistic regression, let π(x) represent the probability of an event that depends on p covariates or independent variables. Then, using an inv.logit formulation
Multiple Linear Regression in Data Mining Contents 2.1. A Review of Multiple Linear Regression 2.2. Illustration of the Regression Process 2.3. Subset Selection in Linear Regression 1 2 Chap. 2 Multiple
Lecture 9 1. Continuous and categorical predictors 2. Indicator variables 3. General linear model: child s IQ example 4. Proc GLM and Proc Reg 5. ANOVA example: Rat diets 6. Factorial design: 2 2 7. Interactions
Chapter 311 Introduction Often, theory and experience give only general direction as to which of a pool of candidate variables (including transformed variables) should be included in the regression model.
Combining Multiple Imputation and Inverse Probability Weighting Shaun Seaman 1, Ian White 1, Andrew Copas 2,3, Leah Li 4 1 MRC Biostatistics Unit, Cambridge 2 MRC Clinical Trials Unit, London 3 UCL Research
Paper 264-26 Model Fitting in PROC GENMOD Jean G. Orelien, Analytical Sciences, Inc. Abstract: There are several procedures in the SAS System for statistical modeling. Most statisticians who use the SAS
Multivariate hypothesis tests for fixed effects Testing homogeneity of level-1 variances In the following sections, we use the model displayed in the figure below to illustrate the hypothesis tests. Partial
Multiple Regression Simple or total correlation: relationship between one dependent and one independent variable, Y versus X Coefficient of simple determination: r (or r, r ) YX YX XX Partial correlation:
Correlational research is used to describe the relationship between two or more naturally occurring variables. Is age related to political conservativism? Are highly extraverted people less afraid of rejection
PSY 512: Advanced Statistics for Psychological and Behavioral Research 2 Understand when to use multiple Understand the multiple equation and what the coefficients represent Understand different methods
Multiple Imputation for Missing Data: Concepts and New Development (Version 9.0) Yang C. Yuan, SAS Institute Inc., Rockville, MD Abstract Multiple imputation provides a useful strategy for dealing with
STA63/CBB540: Statistical methods in computational biology Logistic Regression (/24/3) Lecturer: Barbara Engelhardt Scribe: Dinesh Manandhar Introduction Logistic regression is model for regression used
Logistic Regression (a type of Generalized Linear Model) 1/36 Today Review of GLMs Logistic Regression 2/36 How do we find patterns in data? We begin with a model of how the world works We use our knowledge