SAS PROC NLMIXED Mike Patefield The University of Reading 12 May

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1 SAS PROC NLMIXED Mike Patefield The University of Reading 1 May w.m.patefield@reading.ac.uk non-linear mixed models maximum likelihood repeated measurements on each subject (i) response vector yi random effect vector u i any distribution p( y ) i u,β i 1 u q( u ) i multivariate normal i β likelihood ( β ) = ( β ) ( β ) p y u, q u du i i 1 i i i u i multidimensional integral evaluated by Gaussian quadrature (default) likelihood maximised numerically; default Quasi-Newton 1

2 multi-level models e.g i = batch ( w ) i ~ N 0,σ w j = sample ( ) ij u k = sub-sample response vector y ( ) ij :p yij w i,uij likelihood ; i = 1,...,b u ~ N 0,σ ; j = 1,...,s = i wi j u ij ( ) ( ) p y w,u q u du q ( w ) dw ij i ij u ij ij w i i effectively a -D integral

3 SAS NLMIXED does not (yet) take advantage of this factorisation; it allows only one RANDOM statement Specifying, subject = batch(i) T random effect v = ( w,u,...,u ) i i i1 is y = ( y,..., y ) T T T i i1 is likelihood = ( ) ( ) an (s + 1) D integral i p y v qv dv vi T i i i i feasible for s reasonably low. 3

4 3 illustrations of NLMIXED Normal Bi-exponential model Binary 3-level non-linear model Repeated ordinal data proportional odds model with subject effects 4

5 Indomethicin Kenetics; Pinheiro & Bates (P&B) Indomethicin concentration (mcg/ml).0 Subject 1.0 Subject.0 Subject Subject 4 Subject Subject Time since drug administration (hr) 5

6 Normal Bi-exponential model; random coefficients ( ) ( ) y = β + b exp exp β + b t ij 1 1i i j ( ) ( ) ( β >β 4) +β+ b exp exp β+ b t +ε 3 3i 4 4i j ij i = 1,..., 6 subjects t j ; 11 times for each subject Normal theory: ( ) εij ~N 0, σ ( b,b,b,b ) T ~ N( 0,ψ) 1i i 3i 4i 6

7 Fitting: maximum likelihood P&B fit NLME models using linear approximations NLMIXED uses adaptive Gaussian quadrature - more computationally intensive - by using a lot of quadrature points can get accurate approximation to m.l.e. 7

8 P&B select model with SAS NLMIXED code s 1 ρ1ss 1 s Ψ= 0 0 s proc nlmixed gconv=1e-13 data=a ; parms beta1=.8 beta=0.89 beta3=0.61 beta4=-1.1 s1=1 s=1 s3=1 rho1=0 sigma=0.; bounds s1>0, s>0, s3>0, rho1>-1, rho1<1, sigma>0; model conc~normal((beta1+b1)*exp(-exp(beta+b)*time) +(beta3+b3)*exp(-exp(beta4)*time), sigma**); random b1 b b3 ~ normal( [0,0,0], [s1**,rho1*s1*s,s**,0,0,s3**]) subject=subject; run; 8

9 Results P & B NLMIXED (1 points) log L ˆ ˆβ ˆβ ˆβ ˆβ ŝ ŝ ŝ ˆρ ˆσ

10 ρ = Corr b,b = 1 set ( ) 1 1i i b = b s /s i 1i 1 s 1 ss 1 s Ψ= 0 0 s proc nlmixed gconv=1e-13 data=a ; parms beta1=.8100 beta= beta3= beta4= s1= s= s3=0.7 sigma= ; bounds s1>0, s>0, s3>0, sigma>0; model conc~normal((beta1+b1)*exp(-exp(beta+b1*s/s1)*time) +(beta3+b3)*exp(-exp(beta4)*time), sigma**); random b1 b3 ~ normal( [0,0], [s1**,0,s3**]) subject=subject; run; 10

11 Quadrature Points 5 Parameter Estimates Standard Parameter Estimate Error beta beta beta beta s s s sigma

12 Hyper pigmentation 3 levels : 33 panellists (k) armpits (j) 15 visits (i) BINARY MULTI-LEVEL MODEL response variable: y ijk = presence of hyperpigmentation ~ Binary ( p ) ijk Explanatory variables Group : at panellist level, coded 0,1 Armpit : Left/Right, coded 0/1 Product : at armpit level, coded 0,1 Visit :

13 Model ( ) ijk logit p = fixed effects + wk + ujk ( ) ( σ σ ) w ~ N 0,,u ~ N 0, ; j=1, k w jk u 13

14 MLwiN: RIGLS, 1 st order MQL 14

15 NLMIXED: Subject panellist (k) Random effects w,u k 1k,u k ( ) ~N0,ψ OR - require 3-D integrals σw 0 0 ψ= 0 σu σ reduce to -D vjk = wk + ujk, j = 1, v1k 0 σ w +σu σ w ~N, v k 0 σw σ w +σu 15 u

16 proc nlmixed gconv=1e-13 data=hyper; ods output ParameterEstimates=p1; parms beta1= beta= beta3= beta4= beta5= varw=10 varu=1; eta=beta1+beta*vgroup+beta3*vr+beta4*vprod+beta5*visit +vl*left+vr*right; model y~binary(1/(1+exp(-eta))); random left right ~ normal([0,0], [varw+varu, varw, varw+varu]) subject=panel; run; Quadrature Points 9 - Log Likelihood 37.3 Parameter Estimates NLMIXED MLwiN Standard Standard Parameter Estimate Error Estimate Error beta beta beta beta beta varw varu

17 proc nlmixed qpoints=41 gconv=1e-13 data=hyper; ods output ParameterEstimates=p; parms /data=p1; eta=beta1+beta*vgroup+beta3*vr+beta4*vprod+beta5*visit +vl*left+vr*right; model y~binary(1/(1+exp(-eta))); random left right ~ normal([0,0], [varw+varu, varw, varw+varu]) subject=panel; run; Quadrature Points Log Likelihood Parameter Estimates qpoints=9 qpoints=41 qpoints=81 qpoints=9 qpoints=41 qpoints=81 Standard Standard Standard Parameter Estimate Estimate Estimate Error Error Error beta beta beta beta beta varw varu

18 fitted logits and probabilities at visit 15 group arm product logit( ˆp ) ˆp 0 L L R R L L R R

19 visit p 1 model visit as non-linear logistic ( ) β6 visit ijk 5 logit p = linear effects + β e + random effects 19

20 proc nlmixed qmax=51 gconv=1e-13 data=hyper; parms beta1=. beta=4. beta3=1.5 beta4=-0.1 beta5=-0.5 beta6=0.1 varw=15 varu=0.76; eta=beta1+beta*vgroup+beta3*vr+beta4*vprod+beta5*exp(-beta6*visit) +vl*left+vr*right; model y~binary(1/(1+exp(-eta))); random left right ~ normal([0,0], [varw+varu, varw, varw+varu]) subject=panel; run; Quadrature Points Log Likelihood Parameter Estimates qpoints=31 qpoints=81 qpoints=31 qpoints=81 Standard Standard Parameter Estimate Estimate Error Error beta beta beta beta beta beta varw varu

21 fitted logits and probabilities linear logistic at visit 15 exponential logistic as visit group arm product logit( pˆ ) pˆ logit( ˆp ) ˆp 0 L L R R L L R R

22 Proportional odds model - for ordinal response at time t for subject i Pr (response is category j) ( ) p = P Y = jz,s j it it i zit covariates at time t for subject i s i subject effect (random) j = 1,...,k (categories)

23 j ( ) Q = P Y j it = p1+ p pj (j = 1,...,k 1) p = Q, p = Q Q,..., p = 1 Q k k 1 Proportional Odds ( ) T logit Q j =α j+ zitβ+ si (j = 1,...,k 1) ( ) s ~ N 0, σ ; σ = e δ i s s α j 's are cut-points. 3

24 ( ) nitj i = i it i = j P Y y z,s p t j where n itj = 1 if response is in category j for subject i at time t 0 otherwise likelihood ( ) nitj βαδ =,, p q(s )ds i si t j j i i 4

25 Trial of Eliprodil in Severe Head Injury (Bolland, 003) 440 patients 3 times (day 1, 90, 180) 3 ordinal responses on Glasgow outcome scale (GOS) j = 1 good recovery j = moderate disability j = 3 severe disability or worse covariates: - days = log(log(day)) gcsind01 = 0, Glasgow coma score 5 1, Glasgow coma score > 5 proportional odds model: logit Q =α +β days +β gcsind01+s ( ) j j 1 i 5

26 initial values; fit proportional odds model ignoring subject effects using GENMOD proc genmod data=hidata7 ; model gos3cat= lldays gcsind01/ d=multinomial ; run; Analysis Of Parameter Estimates Standard Wald 95% Parameter DF Estimate Error Confidence Limits Intercept Intercept lldays gcsind Scale NOTE: The scale parameter was held fixed. 6

27 Proportional odds subject effect model using general log-likelihood proc nlmixed qpoints=63 gconv=1e-14 data=hidata7; ods output ParameterEstimates=parm1; parms a1= a= b1= b= delta=; p1=1/(1+exp(-a1-b1*lldays-b*gcsind01+s)); q=1/(1+exp(-a-b1*lldays-b*gcsind01+s)); p=q-p1; p3=1-q; logl=n1*log(p1)+n*log(p)+n3*log(p3); model gos3cat~general(logl); random s~normal(0,exp(*delta)) subject=patid; run; Quadrature Points Log Likelihood Parameter Estimates qpoints=63 qpoints=17 qpoints=63 qpoints=17 Standard Standard Parameter Estimate Estimate Error Error a a b b delta

28 Further Applications of NLMIXED non-linear structural relationships (Patefield, 00) non-linear factor analysis multilevel Poisson modelling Checking ensure identifiability by examining eigenvalues of Hessian matrix 8