Understanding & Interpreting Regression Analysis

1 Understanding & InterpretingRegression AnalysisOCTRI BERD Program28 November 2018 Interpreting Regression analysis1 Workshop overview Welcome What this workshop is not a first course in statistics for those who desire a fundamental under-standing of what to do, or not do. We assume Regression Analysis isthe appropriate tool for your problems and you ve seen it before a detailed review, development or extension of what is typically seen ina standard course on Regression Analysis What this workshop is an adjuvant or corrective therapy for the interpretation of key scientificquantities (estimators) obtained from Regression analyses we meanmeans(viewed through the lens of Regression coefficients) is narrow in scope.

2 Providing the opportunity for much needed insightto clearly communicate research findingsInterpreting Regression analysis2 Workshop info: about the instructors Kyle HartBiostatistician, OHSU, Department of Obstetrics and Gynecology, Biostatis-tics & Design Program (BDP)14 years experience in biomedical research, 4th year at the OHSU & BDPP reviously worked as a data manager at the VA Portland Health Care Sys-tem and at a private medical device companyDegrees in Biostatistics (MS), English & Technical Writing (BS) Broad practitioner across many types of methods, including lots of sim-ple methods, some more exciting stuff, and lots of time mentoring juniorinvestigators.

3 And I like to play with synthesizersInterpreting Regression analysis3 Workshop info: about the instructors David YanezProfessor of Biostatistics, OHSU/PSU School of Public Health4th year at OHSU, Co-Director of BDPP rior to that, was Professor at the Department of Biostatistics, UW Collaboratively, worked extensively in CVD research & on projects in anes-thesia, emergency medicine, nephrology, nursing & pediatrics Statistical interests include: clinical trials, observational studies, longitudi-nal data Analysis , robust methods, measurement error models Taught a lot: medical biometry, Regression , survival, longitudinal data anal-ysis, mathematical statistics, measurement error models, biostatistical con-sulting & technical writing.

4 Tries not to be boringInterpreting Regression analysis4 Workshop info: fair warningPlease note: Some material may conflict with textbooks, wikipedia, and other easily ac-cessible resources Statistical methods ( , t-test) can be motivated/interpreted in more thanone way Not everyone writing about statistics knows this, or admits it In this workshop we will make/use motivations and interpretations thatmake fewer assumptions Why this approach? It should free you to think about what is relevant to your science andnot whether potentially relevant statistical assumptions are satisfied orviolatedInterpreting Regression analysis5 Workshop info: fair warningBackground knowledge tends to varya lot.

5 If issues arise, consider how youmight solve them The material was covered too fast Which parts/topics were confusing? The workshop lectures (slides, scribbles & audio) are being recorded;you can watch them repeatedly I ve seen this all What was different in these presentations, why present it like that tothis audience? Of course, you are welcome/encouraged to ask questions It doesn t feel like a math course It shouldn t, but that will depend on your background. Statistics usesmath but is not mathInterpreting Regression analysis6 Workshop info: fair warning There was no cookbook , flowchart or template of what I should do Such courses tend to cover a lot of methods but with little depth In some simple scenarios may be okay For more complex studies (involving humans), maybe not We ll keep the scope narrow with more depth Memorizing many formulas (recipes) tend to forget them Understanding concepts, fewer formulas tend to stick Interpreting Regression analysis7 Disclaimer Much of these materials have been acquired through courses taught anddiscussions had over the past 25+ years.

6 The lion s share of the creditgoes to Scott Emerson & Kenneth Rice, first-rate statisticians, consum-mate pedagogues. Through their excellent expositions and discussions onstatistical methods, in general, and general Regression methods, in partic-ular, these works flow Any/all errors contained herein and presented forthwith are the sole re-sponsibility of this presenter To Regression analysis8 Homily Everything is Regression . -Scott Emerson, Emeritus Professor of Biostatistics, UW While there are no absolutes, this ismostlytrue Statistical methods used to answer scientific questions almost exclusivelyfall into the following of individual clusters of observations or the distribution of some the distributions of some variable across groups We use Regression to address items 1, 3, 4 We will focus solely upon item 4 If you remember one thing, please remember Regression is an all-purpose tool for comparing groups Interpreting Regression analysis9 Preliminaries.

7 Notation For Regression , it is common (99 out of 100 statisticians ) to usethe following notation: Norndenote the number of subjects It is also called the sample size Ydenotes the outcome (or response) variable ( , FEV1, weight) Xdenotes the grouping (or predictor, independent) variable ( , treat-ment group, exposure group, age) When appropriate, we will use mnemonic variable names for the char-acteristics being studied ( , FEV1, AGE, HEIGHT, TRT, etc.) Interpreting Regression analysis10 Preliminaries: motivationInterpreting Regression analysis11 Preliminaries: motivation Q: Why Regression ? A: It is our best all-purpose tool for comparing a response variable acrosspopulations defined by some grouping variable Let s start with the Regression analysis12 Preliminaries: Anatomy of a Line In mathematics, we model a straight line of two variables,XandY, asY=a+bX, where Ydenotes thedependent(akaoutput) variable Xdenotes theindependent(akainput) variable We generally considerYto be a function ofX, ,(Y|X) =a+bX ais the value ofYwhenX= 0(Y|X= 0) =a+b(0)=a.

8 Bis thedifferenceinYfor a unit difference inX(Y|X= 1) (Y|X= 0) =a+b(1) [a+b(0)]=a+b a= Regression analysis13 Preliminaries: Anatomy of a LineX axisY axis 101201231 unitb = (Y | X=1) (Y | X=0)a = (Y | X=0)(Y | X)=a+bX Interpreting Regression analysis14 Preliminaries: simple linear Regression For Regression , we model theaverageorexpected valueofYasE(Y|X) = 0+ 1X E[ ]denotes themeanorexpected value 0is themeanvalue ofYwhenX= 0E(Y|X= 0) = 0+ 1(0)= 0. 1is themeandifferenceinYfor a unit difference inXE(Y|X= 1) E(Y|X= 0) = 0+ 1(1) [ 0+ 1(0)]= 0+ 1 0= 1. simple here means only one independent variable and an interceptInterpreting Regression analysis15 Preliminaries: simple linear RegressionGroup Variable (X)Response Variable (Y) 101201231 unit 1=E(Y|X=0) E(Y|X=1) 0=E(Y|X=0)E(Y|X)= 0+ 1X Interpreting Regression analysis16 Preliminaries: Anatomy of a LineX axisY axis 101201231 unitb = (Y | X=1) (Y | X=0)a = (Y | X=0)(Y | X)=a+bX Interpreting Regression analysis17 Preliminaries: Differences Between theLinear Equation & simple linear RegressionRelationships betweenYandX: The mathematical linear equation.

9 (Y|X) =a+bX Deterministic relationship betweenYandX We know whatYisexactly, givenX Typically knowaandb The simple linear Regression model:E(Y|X) = 0+ 1X Non-deterministic (stochastic) relationship betweenYandX We know whatYtends to, givenX theaveragedYat a given value ofX Typically don t know 0and 1 need to estimate themInterpreting Regression analysis18 Preliminaries: axisY axis 10120123(Y | X)=a+bXGroup Variable (X)Response Variable (Y) 10120123E(Y | X)= 0+ 1 XInterpreting Regression analysis19 Example 1: Is Studying Helpful? Please interpret the Regression coefficient, 1= 1, belowTime Spent Studying (hrs)Score Achieved (pts)lllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllDoes Studying Look Helpful?

10 Estimated Slope1 hour ^1= 1 Interpreting Regression analysis20 Example 1: Is Studying Helpful? Letters denote measurements for 20 different subjectsTime Spent Studying (hrs)Score Achieved (pts)aaabbbcccdddeeefffggghhhiiijjjkkkll lmmmnnnooopppqqqrrrssstttDoes Studying Look Helpful? Interpreting Regression analysis21 Example 1: Is Studying Helpful? Does your interpretation change now?Time Spent Studying (hrs)Score Achieved (pts)aaabbbcccdddeeefffggghhhiiijjjkkkll lmmmnnnooopppqqqrrrssstttDoes Studying Look Helpful?Change > 0 Cohort < 0 Interpreting Regression analysis22 Example 1: SummaryGoing back to our definition 1=E(Y|X= 1) E(Y|X= 0) 1is thedifferencein means of the outcome,Y, comparing two groups thatdifferby one unit inX not thechangein the mean (ofY) obtained byincreasingXby oneunit As this example illustrates, the effects of change may differ from the ob-serve association averaged over the populationInterpreting Regression analysis23 Example 2.