1 / 25

Lecture 4, part 1 : Linear Regression Analysis: Two Advanced Topics

Lecture 4, part 1 : Linear Regression Analysis: Two Advanced Topics. Karen Bandeen -Roche, PhD Department of Biostatistics Johns Hopkins University. July 14, 2011. Introduction to Statistical Measurement and Modeling. Data examples. Boxing and neurological injury

pink
Download Presentation

Lecture 4, part 1 : Linear Regression Analysis: Two Advanced Topics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Lecture 4, part 1: Linear RegressionAnalysis: Two Advanced Topics Karen Bandeen-Roche, PhD Department of Biostatistics Johns Hopkins University July 14, 2011 Introduction to Statistical Measurement and Modeling

  2. Data examples • Boxing and neurological injury • Scientific question: Does amateur boxing lead to decline in neurological performance? • Some related statistical questions: • Is there a dose-response increase in the rate of cognitive decline with increased boxing exposure? • Is boxing-associated decline independent of initial cognition and age? • Is there a threshold of boxing that initiates harm?

  3. Boxing data

  4. Outline • Topic #1: Confounding • Handling this is crucial if we are to draw correct conclusions about risk factors • Topic #2: Signal / noise decomposition • Signal: Regression model predictions • Noise: Residual variation • Another way of approaching inference, precision of prediction

  5. Topic # 1: Confounding • Confound means to “confuse” • When the comparison is between groups that are otherwise not similar in ways that affect the outcome • Lurking variables,….

  6. Confounding Example: Drowning and Eating Ice Cream * * * * * * * Drowning rate * * * * * * * * * * * * * * * * * * * Ice Cream eaten

  7. Confounding Epidemiology definition: A characteristic “C” is a confounder if it is associated (related) with both the outcome (Y: drowning) and the risk factor (X: ice cream) and is not causally in between Ice Cream Consumption Drowning rate ?? JHU Intro to Clinical Research

  8. Confounding Statistical definition: A characteristic “C” is a confounder if the strength of relationship between the outcome (Y: drowning) and the risk factor (X: ice cream) differs with, versus without, adjustment for C Ice Cream Eaten Drowning rate Outdoor Temperature

  9. Confounding Example: Drowning and Eating Ice Cream * * * * * * * Drowning rate * * * * * * * * * Warm temperature * * * * * * * * * * Cool temperature Ice Cream eaten

  10. Effect modification A characteristic “E” is an effect modifier if the strength of relationship between the outcome (Y: drowning) and the risk factor (X: ice cream) differs within levels of E Ice Cream Consumption Drowning rate Outdoor temperature JHU Intro to Clinical Research

  11. Effect Modification: Drowning and Eating Ice Cream * * * * * * * * * * Drowning rate * * * * * * Warm temperature * * * * * * * * * * Cool temperature Ice Cream eaten

  12. Topic #2: Signal/Noise Decomposition • Lovely due to geometry of least squares • Facilitates testing involving multiple parameters at once • Provides insight into R-squared

  13. Signal/Noise Decomposition • First step: decomposition of variance • “Regression” part: Variance of s • “Error” or “Residual” part: Variance of e • Together: These determine “total” variance of Ys • “Sums of Squares” (SS) rather than variance per se • Regression SS (SSR): • Error SS (SSE): • Total SS (SST):

  14. Signal/Noise Decomposition • Properties • SST = SSR + SSE • SSR/SST = “proportion of variance explained” by regression = R-squared • Follows from geometry • SSR and SSE are independent (assuming A1-A5) and have easily characterized probability distributions • Provides convenient testing methods • Follows from geometry plus assumptions

  15. Signal/Noise Decomposition • SSR and SSE are independent • Define M = span(X) and take “Y” as centered at • It is possible to orthogonally rotate the coordinate axes so that first p axes ε M; remaining n-p-1 axes ε M⊥ • Gram-Schmidt orthogonalization • Doing this transforms Y into TY :=Z, for some orthonormal matrix T with columns:= {e1,...,en-1} • Distribution of Z = N(TE[Y|X],σ2I)

  16. Signal/Noise Decomposition • SSR and SSE are independent - continued • TY=Z Y = T’Z • SSE = squared length of = • SSR = squared length of = • Claim now follows: SSR & SSE are independent because (Z1,…,Zp) and (Zp+1,…,Zn-1) are independent

  17. Signal/Noise Decomposition • Under A1-A5 SSE, SSR and their scaled ratio have convenient distributions • Under A1-A2: E[Y|X] ε M, E[Zj|X] =0, all j>p • Recall {Z1,...,Zn-1} are mutually independent normal with variance=σ2 • Thus SSE = = ~ σ2χ2n-p-1 under A1-A5 (a sum of k independent squared N(0,1) is )

  18. Signal/Noise Decomposition • Under A1-A5 SSE, SSR and their scaled ratio have convenient distributions • For j ≤ p E[Zj|X] ≠ 0 in general • Exception: H0: β1=…=βp = 0 • Then SSR = ~ σ2χ2p under A1-A5 and ~ Fp,n-p-1 ~ with numerator and denominator independent.

  19. Signal/Noise Decomposition • An organizational tool: The analysis of variance (ANOVA) table F = MSR/MSE

  20. “Global” hypothesis tests • These involve sets of parameters • Hypotheses of the form H0: βj = 0 for all j in a defined subset of {j=1,...,p} vs. H1: βj ≠ 0 for at least one of the j Example 1: H0: βLATITUDE = 0 and βLONGITUDE = 0 Example 2: H0: all polynomial or spline coefficients involving a given variable = 0. Example 3: H0: all coefficients involving a variable = 0.

  21. “Global” hypothesis tests • Testing method: Sequential decomposition of sums of squares • Hypothesis to be tested is H0: βj1=...=βjk = 0 in full model • Fit model excluding xj1,...,xjpj: Save SSE = SSEs • Fit “full” (or larger) model adding xj1,...,xjpj to smaller model. Save SSE=SSEL, often=overall SSE • Test statistic S = [(SSES-SSEL)/pj]/[SSEL(n-p-1)] • Distribution under null: F(pj,n-p-1) • Define rejection region based on this distribution • Compute S • Reject or not as S is in rejection region or not

  22. Signal/Noise Decomposition • An augmented version for global testing F = MSR(2|1)/MSE

  23. R-squared – Another view • From last lecture: ECDF Corr(Y, ) squared • More conventional: R2 = SSR/SST • Geometry justifies why they are the same • Cov(Y, ) = Cov(Y- + , ) = Cov(e, ) + Var( ) • Covariance = inner product first term = 0 • A measure of precision with which regression model describes individual responses

  24. Outline: A few more topics • Colinearity • Overfitting • Influence • Mediation • Multiple comparisons

  25. Main points • Confounding occurs when an apparent association between a predictor and outcome reflects the association of each with a third variable • A primary goal of regression is to “adjust” for confounding • Least squares decomposition of Y into fit and residual provides an appealing statistical testing framework • An association of an outcome with predictors is evidenced if SS due to regression is large relative to SSE • Geometry: orthogonal decomposition provides convenient sampling distribution, view of R2 • ANOVA

More Related