1 / 46

Data: Crab mating patterns

Data: Challenger (Binomial with random effects). Data: Crab mating patterns. (Poisson Regression, ZIP model, Negative Binomial). Flu Data: (Binomial with random effects). Data: Typists (Poisson with random effects). D.A.D. Mixed (not generalized) Models:

elkan
Download Presentation

Data: Crab mating patterns

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. Data: Challenger (Binomial with random effects) Data: Crab mating patterns (Poisson Regression, ZIP model, Negative Binomial) Flu Data: (Binomial with random effects) Data: Typists (Poisson with random effects)

  2. D.A.D. Mixed (not generalized) Models: Fixed Effects and Random Effects

  3. D.A.D. SAS Global Forum 2010 “Generalized”  non normal distribution Binary for probabilities: Y=0 or 1 Mean p Pr{Y=j}= pj(1-p)(1-j) Link: L=ln(p/(1-p)) = “Logit” Range (over all L): 0<p<1 Poisson for counts: Y in {0,1,2,3,4, ….} Mean count l Pr{Y=j} = exp(- l )(lj)/(j!) Link: L = log(l) Range (over all L): l >0

  4. D.A.D. SAS Global Forum 2010 Generalized (not mixed) linear models. Use link L = g(E{Y}), e.g. ln(p/(1-p)) =ln(E{Y}/(1-E{Y}) Assume L is linear model in the inputs with fixed effects. Estimate model for L, e.g. L=g(E{Y})=bo + b1 X Use maximum likelihood Example: L = -1 + .18*dose Dose = 10, L=0.8, p=exp(0.8)/(1+exp(0.8))= inverse link = 0.86

  5. D.A.D. SAS Global Forum 2010 Challenger was mission 24 From 23 previous launches we have: 6 O-rings per mission Y=0 no damage, Y=1 erosion or blowby p = Pr {Y=1} = f{mission, launch temperature) Features: Random mission effects Logistic link for p procglimmix data=O_ring; class mission; model fail = temp/dist=binomial s; random mission; run; • Generalized • Mixed

  6. Demo O_rings.sas

  7. D.A.D. SAS Global Forum 2010 Estimated G matrix is not positive definite. Covariance Parameter Estimates Cov Standard Parm Estimate Error mission 2.25E-18 . Solutions for Fixed Effects Effect Estimate Error DF t Value Pr > |t| Intercept 5.0850 3.0525 21 1.67 0.1106 temp -0.1156 0.04702 115 -2.46 0.0154

  8. Just logistic regression – no mission variance component

  9. D.A.D. SAS Global Forum 2010 Flu Data CDC Active Flu Virus Weekly Data % positive data FLU; input fluseasn year t week pos specimens; pct_pos=100*pos/specimens; logit=log(pct_pos/100/(1+(pct_pos/100))); label pos = "# positive specimens"; label pct_pos="% positive specimens"; label t = "Week into flu season (first = week 40)"; label week = "Actual week of year"; label fluseasn = "Year flu season started"; logit pct. pos.

  10. Demo Get_Flu.sas

  11. “Sinusoids” S(j) = sin(2pjt/52) C(j)=cos(2pjt/52) PROCGLM DATA=FLU; class fluseasn; model logit = s1 c1 fluseasn*s1 fluseasn*c1 fluseasn*s2 fluseasn*c2 fluseasn*s3 fluseasn*c3 fluseasn*s4 fluseasn*c4; output out=out1 p=p; data out1; set out1; P_hat = exp(p)/(1+exp(p)); label P_hat = "Pr{pos. sample} (est.)"; run; (1) GLM all effects fixed (harmonic main effects insignificant) Logit scale

  12. Demo Flu_GLM.sas

  13. PROCMIXED DATA=FLU method=ml; ** reduced model; class fluseasn; model logit = s1 c1 /outp=outp outpm=outpm ddfm=kr; random intercept/subject=fluseasn; random s1 c1/subject=fluseasn type=toep(1); random s2 c2/subject=fluseasn type=toep(1); random s3 c3/subject=fluseasn type=toep(1); random s4 c4/subject=fluseasn type=toep(1); run; (2) MIXED analysis on logits Random harmonics. Normality assumed Logit scale Probability scale

  14. Demo Flu_MIXED.sas

  15. PROCGLIMMIX DATA=FLU; title2 "GLIMMIX Analysis"; class fluseasn; model pos/specimens = s1 c1 ; * s2 c2 s3 c3 s4 c4; random intercept/subject=fluseasn; random s1 c1/subject=fluseasn type=toep(1); random s2 c2/subject=fluseasn; ** Toep(1) - no converge; random s3 c3/subject=fluseasn type=toep(1); random s4 c4/subject=fluseasn type=toep(1); random _residual_; output out=out2 pred(ilink blup)=pblup pred(ilink noblup) overallpearson = p_resid; run; (3) GLIMMIX analysis Random harmonics. Binomial assumed (overdispersed – lab effects?) Mean – no BLUPs

  16. Demo Flu_GLIMMIX.sas

  17. D.A.D. SAS Global Forum 2010 Flu data Binomial random _residual_ does not affect the fit (just standard errors) Could try Beta distribution instead: PROCGLIMMIX DATA=FLU; title2 "GLIMMIX Analysis"; class fluseasn; model f = s1 c1 /dist=beta link=logit s; random intercept/subject=fluseasn; random s1 c1/subject=fluseasn type=toep(1); random s2 c2/subject=fluseasn type=toep(1); random s3 c3/subject=fluseasn type=toep(1); random s4 c4/subject=fluseasn type=toep(1); output out=out3 pred(ilink blup)=pblup pred(ilink noblup)=overall pearson=p_residbeta; run;

  18. D.A.D. SAS Global Forum 2010 Horseshoe Crab study (reference: SAS GLIMMIX course notes): Female nests have “satellite” males Count data – Poisson?Generalized Linear Features (predictors): Carapace Width, Weight, Color, Spine condition Random Effect: SiteMixed Model

  19. Demo Get_Horseshoe.sas

  20. procglimmix data=crab; class site; model satellites = weight width / dist=poi solution ddfm=kr; random int / subject=site; output out=overdisp pearson=pearson; run; procmeans data=overdisp n mean var; var pearson; run; procunivariate data=crab normal plot; var satellites; run; N Mean Variance --------------------------- 173 -0.0258264 2.6737114 --------------------------- Histogram # Boxplot 15.5+* 1 0 .* 1 0 . 12.5+* 1 | .* 1 | .** 3 | 9.5+** 3 | .*** 6 | .** 4 | 6.5+******* 13 | .******** 15 +-----+ .********** 19 | | 3.5+********** 19 | | .***** 9 *--+--* .******** 16 | | 0.5+******************************* 62 +-----+ ----+----+----+----+----+----+- * may represent up to 2 counts Fit Statistics Gener. Chi-Square / DF 2.77 Cov Parm Subject Estimate Intercept site 0.1625 Effect Estimate Pr > |t| Intercept -1.1019 0.2527 weight 0.5042 0.0035 width 0.0318 0.5229 Zero Inflated ?

  21. Demo Crabs_OVERDISP.sas

  22. D.A.D. SAS Global Forum 2010 Zero Inflated Poisson (ZIP)

  23. D.A.D. SAS Global Forum 2010 Zero Inflated Poisson (ZIP) procnlmixed data=crab; parms b0=0 bwidth=0 bweight=0 c0=-2 c1=0s2u1=1 s2u2=1; x=c0+c1*width+u1; p0 = exp(x)/(1+exp(x)); eta= b0+bwidth*width +bweight*weight +u2; lambda=exp(eta); if satellites=0 then loglike = log(p0 +(1-p0)*exp(-lambda)); else loglike = log(1-p0)+satellites*log(lambda)-lambda-lgamma(satellites+1); expected=(1-p0)*lambda; id p0 expected lambda; model satellites~general(loglike); Random U1 U2~N([0,0],[s2u1,0,s2u2]) subject=site; predict p0+(1-p0)*exp(-lambda) out=out1; run;

  24. D.A.D. SAS Global Forum 2010 Zero Inflated Poisson (ZIP) Parameter Estimates Parameter Estimate t Pr>|t| Lower Upper b0 2.7897 2.55 0.0268 0.3853 5.1942 bwidth -0.0944 -1.65 0.1267-0.2202 0.0314 bweight 0.4649 2.38 0.0366 0.0347 0.8952 c0 13.3739 4.42 0.0010 6.7078 20.0401 c1 -0.5447 -4.61 0.0008 -0.8049 -0.2844 s2u1 0.5114 1.12 0.2852 -0.4905 1.5133 s2u2 0.1054 1.67 0.1239 -0.0339 0.2447

  25. Demo Crabs_ZIP.sas

  26. D.A.D. SAS Global Forum 2010 From fixed part of model, compute Pr{count=j} and plot (3D) versus Weight, Carapace width

  27. D.A.D. SAS Global Forum 2010 Another possibility: Negative binomial Number of failures until kth success ( p=Prob{success} )

  28. D.A.D. SAS Global Forum 2010 3 trials before first success Negative Binomial Crab beer Crab beer Satellites

  29. D.A.D. Negative binomial: In SAS, k is our 1/k SAS Global Forum 2010 procglimmix data=crab; class site; model satellites = weight width / dist=nb solution ddfm=kr; random int / subject=site; run; Fit Statistics -2 Res Log Pseudo-Likelihood 539.06 Generalized Chi-Square 174.83 Gener. Chi-Square / DF 1.03 Covariance Parameter Estimates Cov Parm Subject Estimate Std. Error Intercept site 0.09527 0.07979 Scale 0.7659 0.1349 Standard Effect Estimate Error DF t Value Pr > |t| Intercept -1.2022 1.6883 168.5 -0.71 0.4774 weight 0.6759 0.3239 156.6 2.09 0.0386 width 0.01905 0.08943 166.2 0.21 0.8316 Num Den Effect DF DF F Value Pr > F weight 1 156.6 4.35 0.0386 width 1 166.2 0.05 0.8316

  30. Demo Crabs_NEGBIN.sas

  31. Population average model vs. Individual Specific Model 8 typists Y=Error counts (Poisson distributed) ln(li)= ln(mean of Poisson) = m+Ui for typist i. conditionally (individual specific) Distributions for Y, U~N(0,1) and m=1 l=em=e1=2.7183 = mean for “typical” typist

  32. D.A.D. SAS Global Forum 2010 Population average model Expectation ||||| | | of individual distributions averaged across population of all typists. Run same simulation for 8000 typists, compute mean of conditional population means, exp(m+U). The MEANS Procedure Variable N Mean Std Dev Std Error ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ lambda 8000 4.4280478 6.0083831 0.067175 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Z=(4.428-2.7183)/0.06718 = 25.46 !! Population mean is not em Conditional means, m+U, are lognormal. Log(Y)~N(1,1)  E{Y}=exp(m+0.5s2) = e1.5 = 4.4817

  33. Demo Typists.sas

  34. D.A.D. SAS Global Forum 2010 • Main points: • Generalized linear models with random effects are subject specific models. • Subject specific models have fixed effects that represent an individual with random effects 0 (individual at the random effect distributional means). • Subject specific models when averaged over the subjects do not give the model fixed effects. • Models with only fixed effects do give the fixed effect part of the model when averaged over subjects and are thus called population average models.

More Related