Section 9.2: Time-Dependent Covariates
Table 9.1 on page 273 based on data set bone_marrow. This table is done with three different models. We first have to create centered variables in a data step. The time-varying covariates are created on-the-fly in proc phreg.
data bone_marrow1;
set bone_marrow;
z11 = z1 - 28;
z22 = z2 - 28;
z1xz2 = z11 * z22;
g1 = ( g = 1 );
g2 = ( g = 2 );
g3 = ( g = 3 );
run;
proc phreg data= bone_marrow1;
model t2*dfree(0) = za g2 g3 ;
if ( t2 >= ta &a=1) then za = 1;
else za = 0;
run;
proc phreg data= bone_marrow1;
model t2*dfree(0)= zc g2 g3;
if ( t2 >= tc & c = 1) then zc = 1;
else zc = 0;
run;
proc phreg data= bone_marrow1;
model t2*dfree(0)= zp g2 g3;
if (t2 >= tp & p=1) then zp = 1;
else zp = 0;
run;
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
za 1 0.31836 0.28514 1.2466 0.2642 1.375
g2 1 -0.55164 0.28799 3.6690 0.0554 0.576
g3 1 0.43381 0.27222 2.5396 0.1110 1.543
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
zc 1 -0.19478 0.28757 0.4587 0.4982 0.823
g2 1 -0.62251 0.29622 4.4163 0.0356 0.537
g3 1 0.36567 0.26850 1.8548 0.1732 1.441
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
zp 1 -1.12986 0.32800 11.8658 0.0006 0.323
g2 1 -0.49624 0.28924 2.9435 0.0862 0.609
g3 1 0.38134 0.26761 2.0306 0.1542 1.464
Table 9.2 on page 274 based on three models.
/*Table 9.2 Fixed Factors Only*/ proc phreg data= bone_marrow1; model t2*dfree(0) = g2 g3 z8 z11 z22 z1xz2; run; /*Table 9.2 Time-Dependent Factor*/ proc phreg data= bone_marrow1; model t2*dfree(0)= zp g2 g3; if (t2 >= tp & p=1) then zp = 1; else zp = 0; run; /*Table 9.2 All Factor*/ proc phreg data= bone_marrow1; model t2*dfree(0) = g2 g3 z8 z11 z22 z1xz2 zp; if (t2 >= tp & p=1) then zp = 1; else zp = 0; run;
First model: Model Fit Statistics
Without With
Criterion Covariates Covariates
-2 LOG L 746.719 713.982
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
g2 1 -1.09058 0.35430 9.4752 0.0021 0.336
g3 1 -0.40439 0.36278 1.2425 0.2650 0.667
z8 1 0.83687 0.27853 9.0276 0.0027 2.309
z11 1 0.00686 0.01968 0.1214 0.7276 1.007
z22 1 0.00391 0.01826 0.0458 0.8305 1.004
z1xz2 1 0.00315 0.0009498 11.0074 0.0009 1.003
Second model: Model Fit Statistics
Without With
Criterion Covariates Covariates
-2 LOG L 746.719 723.647
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
zp 1 -1.12986 0.32800 11.8658 0.0006 0.323
g2 1 -0.49624 0.28924 2.9435 0.0862 0.609
g3 1 0.38134 0.26761 2.0306 0.1542 1.464
Third model: Model Fit Statistics
Without With
Criterion Covariates Covariates
-2 LOG L 746.719 706.611
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
g2 1 -1.03234 0.35317 8.5444 0.0035 0.356
g3 1 -0.41538 0.36526 1.2932 0.2555 0.660
z8 1 0.81302 0.28339 8.2309 0.0041 2.255
z11 1 0.00950 0.01947 0.2379 0.6258 1.010
z22 1 0.00448 0.01795 0.0623 0.8029 1.004
z1xz2 1 0.00286 0.0009346 9.3528 0.0022 1.003
zp 1 -0.99618 0.33651 8.7638 0.0031 0.369
Table 9.3 on page 275. The log likelihood statistics is contained in each of the SAS output. We omit them here to save some space. We will only show the output from the test statement. SAS performs Wald test on each one and it gives very similar results.
proc phreg data=bone_marrow1;
model t2*dfree(0) = zp g2 g3 z11 z22 z1xz2 z8 zpg2 zpg3;
if ( t2 >= tp & p = 1) then zp = 1;
else zp = 0;
zpg2 = zp*g2; zpg3 =zp*g3;
model1: test zpg2, zpg3;
run;
proc phreg data=bone_marrow1;
model t2*dfree(0) = zp g2 g3 z11 z22 z1xz2 z8 z8zp;
if ( t2 >= tp & p = 1) then zp = 1;
else zp = 0;
z8zp = z8*zp;
model2: test z8zp;
run;
proc phreg data=bone_marrow1;
model t2*dfree(0) = zp g2 g3 z11 z22 z1xz2 z8 z1zp z2zp zxzp;
if ( t2 >= tp & p = 1) then zp = 1;
else zp = 0;
z1zp = z11*zp; z2zp = z22*zp; zxzp = z1xz2*zp;
model3: test z1zp, z2zp, zxzp;
run;
proc phreg data=bone_marrow1;
model t2*dfree(0) = zp g2 g3 z11 z22 z1xz2 z8 zpg2 zpg3 z8zp;
if ( t2 >= tp & p = 1) then zp = 1;
else zp = 0;
zpg2 = zp*g2; zpg3 =zp*g3;
z8zp = z8*zp;
model4: test z8zp;
run;
proc phreg data=bone_marrow1;
model t2*dfree(0) = zp g2 g3 z11 z22 z1xz2 z8 zpg2 zpg3 z1zp z2zp zxzp;
if ( t2 >= tp & p = 1) then zp = 1;
else zp = 0;
zpg2 = zp*g2; zpg3 =zp*g3;
z1zp = z11*zp; z2zp = z22*zp; zxzp = z1xz2*zp;
model5: test z1zp, z2zp, zxzp;
run;
proc phreg data=bone_marrow1;
model t2*dfree(0) = zp g2 g3 z11 z22 z1xz2 z8 zpg2 zpg3 z1zp z2zp zxzp z8zp;
if ( t2 >= tp & p = 1) then zp = 1;
else zp = 0;
zpg2 = zp*g2; zpg3 =zp*g3;
z1zp = z11*zp; z2zp = z22*zp; zxzp = z1xz2*zp;
z8zp = z8*zp;
model6: test z8zp;
run;
Linear Hypotheses Testing Results
Wald
Label Chi-Square DF Pr > ChiSq
model1 7.4997 2 0.0235
model2 3.1518 1 0.0758
model3 7.8877 3 0.0484
model4 4.1729 1 0.0411
model5 12.5410 3 0.0057
model6 4.4670 1 0.0346
Table 9.4 on page 276.
proc phreg data=bone_marrow1;
model t2*dfree(0) = zp g2 g3 z11 z22 z1xz2 z8 zpg2 zpg3 z1zp z2zp zxzp z8zp;
if ( t2 >= tp & p = 1) then zp = 1;
else zp = 0;
zpg2 = zp*g2; zpg3 =zp*g3;
z1zp = z11*zp; z2zp = z22*zp; zxzp = z1xz2*zp;
z8zp = z8*zp;
run;
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
zp 1 -0.30621 0.69362 0.1949 0.6589 0.736
g2 1 1.30734 0.81864 2.5503 0.1103 3.696
g3 1 1.10712 1.22422 0.8178 0.3658 3.026
z11 1 -0.15375 0.05454 7.9486 0.0048 0.857
z22 1 0.11660 0.04337 7.2286 0.0072 1.124
z1xz2 1 0.00260 0.00195 1.7863 0.1814 1.003
z8 1 -1.23479 1.11389 1.2288 0.2676 0.291
zpg2 1 -3.03742 0.92574 10.7653 0.0010 0.048
zpg3 1 -1.86753 1.29076 2.0934 0.1479 0.155
z1zp 1 0.19335 0.05878 10.8208 0.0010 1.213
z2zp 1 -0.14705 0.04800 9.3831 0.0022 0.863
zxzp 1 0.0001265 0.00230 0.0030 0.9561 1.000
z8zp 1 2.45353 1.16086 4.4670 0.0346 11.629
Example 9.2 on page 277 based on data set kidney.
proc phreg data=kidney; model time*infect(0)= z1 z2; z1 = placement-1; z2 = log(time)*z1; run;
The PHREG Procedure
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 14.6737 2 0.0007
Score 12.7062 2 0.0017
Wald 8.0705 2 0.0177
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
z1 1 1.42235 1.03145 1.9016 0.1679 4.147
z2 1 -1.46217 0.58748 6.1944 0.0128 0.232
Table 9.5 on page 278 based on data set bone_marrow1 that is created in the previous Section 9.2. We have included the code to create the table and will only show the output from each of the test statement.
/*Factor of group*/ proc phreg data = bone_marrow1; model t2*dfree(0) = g1 g2 g1p g2p; g1p = log(t2) * g1; g2p = log(t2) * g2; group: test g1p=g2p=0; run; /*Effect of Waiting time*/ proc phreg data = bone_marrow1; model t2*dfree(0) = z7 z7p; z7p = log(t2)*z7; Waiting_time: test z7p; run; /*Factor of Fab status*/ proc phreg data = bone_marrow1; model t2*dfree(0) = z8 z8p; z8p = log(t2)*z8; Fab: test z8p; run; /*Factor of MTX*/ proc phreg data = bone_marrow1; model t2*dfree(0) = z10 z10p; z10p = log(t2)*z10; MTX: test z10p; run; /*Factor of Sex*/ proc phreg data = bone_marrow1; model t2*dfree(0) = z3 z4 z3z4 z3p z4p z3z4p; z3z4 = z3*z4; z3p = log(t2)*z3; z4p = log(t2)*z4; z3z4p = log(t2)*z3z4; Sex: test z3p, z4p, z3z4p; run; /*Factor of CMV status*/ proc phreg data = bone_marrow1; model t2*dfree(0) = z5 z6 z5z6 z5p z6p z5z6p; z5z6 = z5*z6; z5p = log(t2)*z5; z6p = log(t2)*z6; z5z6p = log(t2)*z5z6; CMV: test z5p, z6p, z5z6p; run; /*Factor of Age*/ proc phreg data = bone_marrow1; model t2*dfree(0) = z11 z22 z1xz2 z1p z2p z1z2p; z1p = log(t2)*z11; z2p = log(t2)*z22; z1z2p = log(t2)*z1xz2; Age: test z1p, z2p, z1z2p; run;
Linear Hypotheses Testing Results
Wald
Label Chi-Square DF Pr > ChiSq
group 1.7349 2 0.4200
Waiting_time 0.0049 1 0.9442
Fab 0.4441 1 0.5051
MTX 4.3226 1 0.0376
Sex 0.2203 3 0.9742
CMV 1.6870 3 0.6398
Age 4.7593 3 0.1903
Table 9.6 on page 280 based on the data set kidney.
/*to extract the event time*/
proc sql noprint;
select distinct time into :event_time separated by ' '
from kidney
where infect = 1;
quit;
%put &event_time;
%macro event_lpl(data, time, censor, var, delta_list);
%let k=1;
%let whole =;
%let dep = %scan(&delta_list, &k, ' ');
%do %while(&dep NE );
ods listing close;
proc phreg data=&data;
model &time*&censor(0) = &var z2;
if &time > &dep then z2 = &var;
else z2 = 0;
ods output FitStatistics = _temp&k;
run;
ods output close;
ods listing;
%let whole= &whole%str( _temp&k);
%let k = %eval(&k + 1);
%let dep = %scan(&delta_list, &k, ' ');
%end;
data whole;
set &whole;
if Criterion = "-2 LOG L" ;
run;
data whole;
set whole;
e_time = scan("&delta_list", _n_, ' ');
logp = - withcovariates/2;
run;
proc print data = whole noobs;
var e_time logp;
run;
%mend;
data kidney1;
set kidney;
z1 = placement - 1;
run;
%event_lpl(kidney1, time, infect, z1, &event_time)
e_time logp
0.5 -97.878
1.5 -100.224
2.5 -97.630
3.5 -97.501
4.5 -99.683
5.5 -100.493
6.5 -98.856
8.5 -100.428
9.5 -101.084
10.5 -101.668
11.5 -102.168
15.5 -100.829
16.5 -101.477
18.5 -102.059
23.5 -102.620
26.5 -103.229
The anova table on page 280.
proc phreg data = kidney1; model time*infect(0) = z2 z3 ; if time > 3.5 then do z2=z1; z3=0; end; else do z2=0; z3=z1; end; run;
The PHREG Procedure
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 13.9049 2 0.0010
Score 12.8330 2 0.0016
Wald 9.4677 2 0.0088
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
z2 1 -2.08891 0.75973 7.5600 0.0060 0.124
z3 1 1.08175 0.78320 1.9077 0.1672 2.950
Section 9.3: Stratified Proportional Hazards Models
Table 9.7 on page 283.
proc phreg data=bone_marrow1;
model t2*dfree(0) = zp g2 g3 z11 z22 z1xz2 z8;
if ( t2 >= tp & p = 1) then zp = 1;
else zp = 0;
strata z10;
run;
The PHREG Procedure
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 38.2728 7 <.0001
Score 44.6457 7 <.0001
Wald 40.3675 7 <.0001
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
zp 1 -1.00334 0.34453 8.4810 0.0036 0.367
g2 1 -0.99030 0.36658 7.2980 0.0069 0.371
g3 1 -0.36327 0.37138 0.9568 0.3280 0.695
z11 1 0.00954 0.01984 0.2314 0.6305 1.010
z22 1 -0.00141 0.01791 0.0062 0.9373 0.999
z1xz2 1 0.00255 0.0009364 7.4245 0.0064 1.003
z8 1 0.89203 0.28347 9.9024 0.0017 2.440
Table 9.8 on page 284.
proc phreg data=bone_marrow1;
by z10;
model t2*dfree(0) = zp g2 g3 z11 z22 z1xz2 z8;
if ( t2 >= tp & p = 1) then zp = 1;
else zp = 0;
ods output ParameterEstimates=table9_8;
run;
proc sort data=table9_8;
by variable;
run;
proc transpose data = table9_8 prefix=g
out=table9_8a (where=(_name_="Estimate" or _name_="StdErr")) ;
by variable;
id z10;
run;
data table9_8b;
set table9_8a;
num = lag(gno)-lag(gyes);
if _name_="StdErr" then do;
chisq = num**2/(gno**2 + gyes**2);
pvalue=1 - probchi(chisq, 1);
end;
drop num;
run;
proc print data = table9_8b noobs;
run;
Variable _NAME_ gNo gYes chisq pvalue
g2 Estimate -1.19819 -0.56257 . .
g2 StdErr 0.45851 0.63852 0.65380 0.41876
g3 Estimate -0.29627 -0.85960 . .
g3 StdErr 0.44541 0.91752 0.30506 0.58073
z11 Estimate 0.02765 0.01141 . .
z11 StdErr 0.02587 0.03909 0.12005 0.72898
z1xz2 Estimate 0.00225 0.00139 . .
z1xz2 StdErr 0.00140 0.00227 0.10247 0.74889
z22 Estimate -0.02027 0.03433 . .
z22 StdErr 0.02533 0.03104 1.85815 0.17284
z8 Estimate 1.08877 0.34588 . .
z8 StdErr 0.33846 0.65111 1.02484 0.31137
zp Estimate -0.88294 -1.00895 . .
zp StdErr 0.47591 0.55113 0.02995 0.86261
Section 9.4: Left Truncation
Table 9.9 on page 288.
proc phreg data=bone_marrow1; model t2*dfree(0) = g2 g3 z11 z22 z1xz2 z8 /entry=tp; strata z10; run;
The PHREG Procedure
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 37.1792 6 <.0001
Score 36.0545 6 <.0001
Wald 32.1483 6 <.0001
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
g2 1 -1.75212 0.43761 16.0307 <.0001 0.173
g3 1 -0.75036 0.40769 3.3874 0.0657 0.472
z11 1 0.04174 0.02229 3.5066 0.0611 1.043
z22 1 -0.03460 0.02068 2.7986 0.0943 0.966
z1xz2 1 0.00228 0.00122 3.4931 0.0616 1.002
z8 1 1.27753 0.32494 15.4571 <.0001 3.588
Section 9.5: Synthesis of Time-varying Effects (Multistate Modeling)
Table 9.10 on page 291 based on data bone_marrow1 created in Section 9.1.
proc phreg data=bone_marrow1;
model tp*p(0) = z11 z22 z1xz2 z10;
run;
The PHREG Procedure
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 30.7549 4 <.0001
Score 27.4464 4 <.0001
Wald 25.9884 4 <.0001
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
z11 1 0.03606 0.01626 4.9175 0.0266 1.037
z22 1 -0.02616 0.01477 3.1367 0.0766 0.974
z1xz2 1 -0.00272 0.0009715 7.8217 0.0052 0.997
z10 1 -1.04226 0.22329 21.7882 <.0001 0.353
The first model in Table 9.11 on page 292 based on the models specified on page 291.
code here.
The left-truncated model before platelet recovery. We need to modify our data set by censoring any individual whose platelets recover prior to death or relapse at their platelet recovery time. This is done in the data step below.
data bone_marrow2;
set bone_marrow1;
m2 = dfree;
myt = t2;
if tp <= t2 and p = 1 then do;
m2= 0;
myt = tp;
end;
run;
proc phreg data=bone_marrow2;
model myt*m2(0) = g2 g3 z8 z11 z22 z1xz2 ;
run;
The PHREG Procedure
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 14.2991 6 0.0265
Score 13.4770 6 0.0361
Wald 12.1243 6 0.0593
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
g2 1 1.46620 0.91746 2.5539 0.1100 4.333
g3 1 1.44752 1.33334 1.1786 0.2776 4.253
z8 1 -1.75629 1.32144 1.7664 0.1838 0.173
z11 1 -0.16154 0.06198 6.7936 0.0091 0.851
z22 1 0.12580 0.04755 7.0003 0.0081 1.134
z1xz2 1 0.00318 0.00210 2.2880 0.1304 1.003
The left-truncated model after platelet recovery. This is done using the entry= option in proc phreg.
proc phreg data=bone_marrow2; model t2*dfree(0) = g2 g3 z8 z11 z22 z1xz2 /entry = tp; run;
The PHREG Procedure
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 37.7176 6 <.0001
Score 36.9965 6 <.0001
Wald 32.9733 6 <.0001
Analysis of Maximum Likelihood Estimates
Parameter Standard Hazard
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio
g2 1 -1.71600 0.42554 16.2616 <.0001 0.180
g3 1 -0.75650 0.40752 3.4460 0.0634 0.469
z8 1 1.21153 0.32231 14.1291 0.0002 3.359
z11 1 0.03872 0.02177 3.1625 0.0753 1.039
z22 1 -0.02923 0.02051 2.0321 0.1540 0.971
z1xz2 1 0.00266 0.00123 4.6829 0.0305 1.003