Advanced Biostatistics Chapter 7: Pharmacokinetics of Theophylline 17 April 2006

Notes:
“By groups”

proc sort data=theoph; by subject; run;

proc nlmixed data=theoph;

parms th1=2 th2=0.2 kap=0.1 sig2=2;

ex1=exp(-th1*time); ex2=exp(-th2*time); diff=ex2-ex1; rec=1/(th1-th2);

phi=1/kap; tmax=rec*log(th1/th2); cmax=th1*phi*exp(-th1*tmax);

mean=dose*th1*th2*phi*rec*diff;

model conc~normal(mean,sig2);

by subject;

run;

th1hat th2hat kaphat

1.7774 0.05395 0.01992

1.9427 0.10170 0.04470

2.4536 0.08142 0.03956

1.1715 0.08747 0.03740

1.4715 0.08844 0.04360

1.1637 0.09953 0.05114

0.6797 0.10220 0.05159

1.3755 0.09196 0.04646

8.8656 0.08663 0.03269

0.6955 0.07397 0.03244

3.8490 0.09812 0.05725

0.8329 0.10560 0.04200

First NLMixed

proc nlmixed data=theoph;

parms th1=2 th2=0.2 kap=0.1 saa=0.3 sbb=0.4 see=0.5;

th1i=th1+a; kapi=kap+b;

ex1=exp(-th1i*time); ex2=exp(-th2*time); diff=ex2-ex1; rec=1/(th1-th2);

phii=1/kapi;

mean=dose*th1i*th2*phii*rec*diff;

model conc~normal(mean,see);

random a b ~ normal([0,0],[saa,0,sbb]) subject=subject;

run;

Fit Statistics

-2 Log Likelihood 399.4

AIC (smaller is better) 411.4

Parameter Estimates

Standard

Parameter Estimate Error DF t Value Pr > |t| Alpha Lower Upper

th1 1.7707 0.2861 10 6.19 0.0001 0.05 1.1332 2.4081

th2 0.08425 0.004596 10 18.33 <.0001 0.05 0.07401 0.09449

kap 0.03967 0.008676 10 4.57 0.0010 0.05 0.02034 0.05900

saa 0.8092 0.4585 10 1.76 0.1080 0.05 -0.2124 1.8308

sbb 0.000425 0.000258 10 1.65 0.1296 0.05 -0.00015 0.000999

see 0.5695 0.08587 10 6.63 <.0001 0.05 0.3781 0.7608

Second NLMixed

proc nlmixed data=theoph;

parms th1=2 th2=0.2 kap=0.1 saa=0.3 sab=0 sbb=0.4 see=0.5;

th1i=th1*exp(ap); kapi=kap*exp(bp);

ex1=exp(-th1i*time); ex2=exp(-th2*time); diff=ex2-ex1; rec=1/(th1-th2);

phii=1/kapi;

mean=dose*th1i*th2*phii*rec*diff;

model conc~normal(mean,see);

random ap bp ~ normal([0,0],[saa,sab,sbb]) subject=subject;

run;

Fit Statistics

-2 Log Likelihood 356.1

AIC (smaller is better) 370.1

Parameter Estimates

Standard

Parameter Estimate Error DF t Value Pr > |t| Alpha Lower Upper

th1 1.6142 0.3172 10 5.09 0.0005 0.05 0.9075 2.3209

th2 0.08540 0.004370 10 19.54 <.0001 0.05 0.07566 0.09514

kap 0.03920 0.002337 10 16.77 <.0001 0.05 0.03399 0.04440

saa 0.4228 0.1932 10 2.19 0.0535 0.05 -0.00778 0.8533

sab 0.4438 0.2054 10 2.16 0.0560 0.05 -0.01377 0.9015

sbb 0.4949 0.2248 10 2.20 0.0523 0.05 -0.00595 0.9958

see 0.5017 0.06838 10 7.34 <.0001 0.05 0.3493 0.6540

Third NLMixed

proc nlmixed data=theoph;

parms th1=2 th2=0.2 kap=0.1 saa=0.3 sbb=0.4 see=0.5;

th1i=th1*exp(ap); kapi=kap*exp(bp);

ex1=exp(-th1i*time); ex2=exp(-th2*time); diff=ex2-ex1; rec=1/(th1-th2);

phii=1/kapi;

mean=dose*th1i*th2*phii*rec*diff;

model conc~normal(mean,see);

random ap bp ~ normal([0,0],[saa,0,sbb]) subject=subject;

run;

Fit Statistics

-2 Log Likelihood 388.3

AIC (smaller is better) 400.3

Parameter Estimates

Standard

Parameter Estimate Error DF t Value Pr > |t| Alpha Lower Upper

th1 1.5957 0.2818 10 5.66 0.0002 0.05 0.9678 2.2235

th2 0.08482 0.004398 10 19.29 <.0001 0.05 0.07502 0.09462

kap 0.03920 0.01004 10 3.90 0.0029 0.05 0.01682 0.06157

saa 0.3365 0.1545 10 2.18 0.0544 0.05 -0.00775 0.6808

sbb 0.3979 0.1833 10 2.17 0.0551 0.05 -0.01048 0.8062

see 0.5152 0.07218 10 7.14 <.0001 0.05 0.3544 0.6760