Statistics of Contingency Tables - Extension to I x J. stat 557 Heike Hofmann

Transcription

1 Statistics of Contingency Tables - Extension to I x J stat 557 Heike Hofmann

2 Outline Testing Independence Local Odds Ratios Concordance & Discordance Intro to GLMs

3 Simpson s paradox Simpson s paradox: marginal association between X and Y is opposite to conditional associations between X and Y for each level of Z due to: very strong marginal association between X and Z or Y and Z

4 Conditional Odds Ratios X, Y are conditionally independent for level k of Z, if the conditional log odds ratio is 0 X,Y are conditionally independent given Z, if all conditional odds ratios are 0. (Does not imply marginal independence) X,Y have homogenous association, if all conditional odds ratios given Z are constant.

5 Testing Independence Odds ratio of 1 indicates independence, confidence interval helps to determine deviation from independence, but CI is approximation. Alternative solution: table tests

6 Testing independence null hypothesis: Score Test (Pearson, 1900): X 2 = i,j Likelihood-Ratio Test: π ij = π i. π.j (n ij µ ˆ ij ) 2 ˆ µ ij G 2 =2 i,j i, j log nij µ ˆ ij both X 2 and G 2 have the same limiting distribution of chi 2 (I-1)(J-1)

7 Example: Cholesterol/Heart Disease 1329 patients of same age/sex Coronary Disease present absent mg/l Cholesterol 220 > 220 y11= 20 y12= 553 y21= 72 y22= 684

8 Cholesterol/Heart Disease Expected Values under independence Coronary Disease present absent Total mg/l Cholesterol 220 > 220 Total

9 Cholesterol/Heart Disease loglikelihood ratio test G 2 = 19.8 Pearson score test X 2 = 18.4 with df = (2-1)*(2-1) = 1 independence seems to be violated

10 Extensions to I x J Contingency Tables

11 Local Odds Ratios Each set of four cells forming a rectangle yields one odds ratio Local Odds Ratio: Use only neighboring cells local odds ratios form a minimal sufficient set a c b d

12 Example: Marijuana Use Study on Marijuana use (based on parental use) student never occasional regular parent neither one both evidence of association?

13 Example: Marijuana Use Student by Parent Use student prodplot(mj, count~student+parent, c("vspine","hspine"), subset=level==2) positive association? parent neither one both

14 Summaries of I x J Tables (ordinal variables) Concordance/Discordance: For each pair of subjects count #concordant/discordant pairs, where Y=1 Y=2... Y=J X=1 X=2 X=i π11 π12 πii π21 π22 π2i... πj1 πj2... πji a pair is concordant, if subject 2 is ranked higher on X, it is also ranked higher on Y a pair is discordant, if subject 2 is ranked higher on X, but ranked lower on Y

15 Concordance/Discordance concordance to (i,j): <i, <j or >i, >j i,j discordance to (i,j): >i, <j or >i, <j Π c =2 I J π ij π c ij i=1 j=1 h>i k>j π hk = i,j

16 Gamma Statistic let C, D be the probabilities for concordance and discordance, resp. γ = Π C Π D Π C + Π D approx. normal with σ 2 (ˆγ) = 16 n(π C + Π D ) 4 I i=1 J j=1 π ij Π C π d ij Π D π c ij 2

17 Marijuana Example: C=2 (141 ( )+ 68 (11+19)+ 54 (51+19) )=48562 D=2 (17 ( )+ 68 (54+40)+ 11 (40+51) )=24732 γ = (C D)/(C + D) = = sd(γ) = fairly strong indication of a positive association

18 Generalized Linear Models Three components: random, systematic, link Random part: distribution of observations y Systematic component: linear combination of explanatory variables link function: link between random and systematic component

19 Exponential Family Two parameter family, φ > 0 density f(yi; θi, φ) = exp ((yiθi b(θi))/a(φ) + c(yi, φ)) θi is the natural parameter, φ is called dispersion parameter. E[Y] = b (θ) Var[Y] = b (θ) a(φ)

20 Normal Distribution f(x; µ, σ) = 1 exp 2πσ 2 (x µ)2 2σ 2 θx (θ/2) 2 f(x; θ, φ) = exp x 2 /φ 1/2 log(φπ) φ define θ = 2µ, φ = 2σ 2 ( )

21 Binomial Distribution = exp P (X = x) = n π x (1 π) n x x π P (X = x; θ = log xθ n log(1 + e θ ) + log 1 π, φ = 1) = n x

22 Poisson Distribution λ λx P (X = x; λ) =e x! P (X = x; θ = log λ, φ = 1) = exp xθ e θ log(x!)

23 Systematic Component Let X1, X2,..., Xp be explanatory variables systematic component: η i = p β j x ij j=1 with parameters ß1, ß2,..., ßp

24 Link Function g() is called the link function of the GLM, if g differentiable and monotonic, g(e[y]) = Xjßj g is called the natural link, if g(e[y]) = θ