# The QUANTLIFE Procedure

### Example 94.1 Primary Biliary Cirrhosis Study

This example illustrates how to use quantile regression analysis to detect varying covariate effects on survival time. Consider a study of primary biliary cirrhosis, a rare but fatal chronic liver disease, discussed by Fleming and Harrington (1991). Researchers followed 418 patients who had this disease, 161 of whom died during the study.

The data set contains the following variables:

• Time, follow-up time, in years

• Status, event indicator, with value 1 for death time and value 0 for censored time

• Age, age from birth to study registration, in years

• Albumin, serum albumin level, in g/dl

• Bilirubin, serum bilirubin level, in mg/dl

• Edema, edema presence

• Protime, prothrombin time, in seconds

The following statements create the data set PBC, which is used in this example:

data pbc;
input Time Status Age Albumin Bilirubin Edema Protime @@;
label Time="Follow-Up Time in Days";
logAlbumin   = log(Albumin);
logBilirubin = log(Bilirubin);
logProtime   = log(Protime);
datalines;
400 1 58.7652 2.60 14.5 1.0 12.2 4500 0 56.4463 4.14  1.1 0.0 10.6
1012 1 70.0726 3.48  1.4 0.5 12.0 1925 1 54.7406 2.54  1.8 0.5 10.3
1504 0 38.1054 3.53  3.4 0.0 10.9 2503 1 66.2587 3.98  0.8 0.0 11.0
1832 0 55.5346 4.09  1.0 0.0  9.7 2466 1 53.0568 4.00  0.3 0.0 11.0
2400 1 42.5079 3.08  3.2 0.0 11.0   51 1 70.5599 2.74 12.6 1.0 11.5
3762 1 53.7139 4.16  1.4 0.0 12.0  304 1 59.1376 3.52  3.6 0.0 13.6

... more lines ...

989 0 35.0000 3.23  0.7 0.0 10.8  681 1 67.0000 2.96  1.2 0.0 10.9
1103 0 39.0000 3.83  0.9 0.0 11.2 1055 0 57.0000 3.42  1.6 0.0  9.9
691 0 58.0000 3.75  0.8 0.0 10.4  976 0 53.0000 3.29  0.7 0.0 10.6
;

The next statements fit a linear model for the log of survival time of the PBC patients with the covariates logBilirubin, logProtime, logAlbumin, Age, and Edema:

ods graphics on;
proc quantlife data=pbc log method=na plot=(quantplot survival) seed=1268;
model Time*Status(0)=logBilirubin logProtime logAlbumin Age Edema
/quantile=(.1 .2 .3 .4 .5 .6 .75);
run;

You use the QUANTILE= option to specify a set of quantiles of interest for comparing quantile-specific covariate effects. The METHOD= option specifies the Nelson-Aalen method for estimating the regression parameters.

The QUANTLIFE procedure provides resampling methods for computing confidence limits for the parameters; for more information, see the section Confidence Interval. By default, the repetition number is 200. You can request a different number of repetitions by specifying the NREP= option. You can also use the SEED= option to specify the seed for generating random numbers so that you can later reproduce the results.

Output 94.1.1 displays model information and information about censoring in the data. Out of 418 observations, 257 are censored.

Output 94.1.1: Model Information

The QUANTLIFE Procedure

Model Information
Data Set WORK.PBC
Dependent Variable Log(Time)
Censoring Variable Status
Censoring Value(s) 0
Number of Observations 418
Method Nelson-Aalen
Replications 200
Seed for Random Number Generator 1268

Summary of the Number of Event and Censored
Values
Total Event Censored Percent
Censored
418 161 257 61.48

Output 94.1.2 provides the parameter estimates. Each quantile level has a set of parameter estimates and confidence limits.

Output 94.1.2: Parameter Estimates at Different Quantiles

Parameter Estimates
Quantile Parameter DF Estimate Standard
Error
95% Confidence Limits t Value Pr > |t|
0.1000 Intercept 1 14.8030 4.0967 6.7736 22.8325 3.61 0.0003
logBilirubin 1 -0.4488 0.1485 -0.7398 -0.1578 -3.02 0.0027
logProtime 1 -3.6378 1.4560 -6.4915 -0.7841 -2.50 0.0129
logAlbumin 1 1.9286 0.9756 0.0165 3.8408 1.98 0.0487
Age 1 -0.0244 0.0107 -0.0455 -0.00334 -2.27 0.0237
Edema 1 -1.0712 0.6688 -2.3820 0.2396 -1.60 0.1100
0.2000 Intercept 1 15.1800 2.6664 9.9540 20.4060 5.69 <.0001
logBilirubin 1 -0.6532 0.0886 -0.8268 -0.4796 -7.37 <.0001
logProtime 1 -3.3273 0.9401 -5.1699 -1.4847 -3.54 0.0004
logAlbumin 1 1.6842 0.6888 0.3343 3.0342 2.45 0.0149
Age 1 -0.0291 0.00687 -0.0425 -0.0156 -4.23 <.0001
Edema 1 -0.7265 0.3179 -1.3497 -0.1034 -2.29 0.0228
0.3000 Intercept 1 13.2382 2.5296 8.2804 18.1961 5.23 <.0001
logBilirubin 1 -0.6013 0.0762 -0.7506 -0.4521 -7.90 <.0001
logProtime 1 -2.5816 0.8907 -4.3273 -0.8359 -2.90 0.0039
logAlbumin 1 1.7246 0.7142 0.3248 3.1245 2.41 0.0162
Age 1 -0.0244 0.00716 -0.0385 -0.0104 -3.41 0.0007
Edema 1 -0.8577 0.2763 -1.3992 -0.3163 -3.10 0.0020
0.4000 Intercept 1 13.4716 3.0874 7.4204 19.5228 4.36 <.0001
logBilirubin 1 -0.6047 0.0846 -0.7705 -0.4389 -7.15 <.0001
logProtime 1 -2.1632 1.1726 -4.4615 0.1351 -1.84 0.0658
logAlbumin 1 0.9819 0.7191 -0.4274 2.3912 1.37 0.1728
Age 1 -0.0255 0.00681 -0.0389 -0.0122 -3.74 0.0002
Edema 1 -1.0589 0.3104 -1.6672 -0.4506 -3.41 0.0007
0.5000 Intercept 1 10.9205 2.8047 5.4235 16.4175 3.89 0.0001
logBilirubin 1 -0.5315 0.0904 -0.7087 -0.3543 -5.88 <.0001
logProtime 1 -1.2222 1.2142 -3.6020 1.1577 -1.01 0.3148
logAlbumin 1 1.5700 0.6284 0.3383 2.8016 2.50 0.0129
Age 1 -0.0318 0.00883 -0.0491 -0.0145 -3.60 0.0004
Edema 1 -0.7316 0.3743 -1.4653 0.00202 -1.95 0.0513
0.6000 Intercept 1 11.2381 2.6294 6.0846 16.3917 4.27 <.0001
logBilirubin 1 -0.5701 0.0852 -0.7370 -0.4031 -6.69 <.0001
logProtime 1 -1.3508 1.1402 -3.5856 0.8840 -1.18 0.2368
logAlbumin 1 1.3704 0.5091 0.3726 2.3682 2.69 0.0074
Age 1 -0.0226 0.0109 -0.0440 -0.00111 -2.06 0.0399
Edema 1 -0.5141 0.3088 -1.1193 0.0912 -1.66 0.0968
0.7500 Intercept 1 10.0954 3.1893 3.8445 16.3463 3.17 0.0017
logBilirubin 1 -0.6366 0.1071 -0.8466 -0.4267 -5.94 <.0001
logProtime 1 -0.9670 1.2343 -3.3862 1.4521 -0.78 0.4338
logAlbumin 1 1.8148 0.5883 0.6618 2.9678 3.08 0.0022
Age 1 -0.0203 0.0156 -0.0509 0.0102 -1.30 0.1931
Edema 1 -0.3529 0.3120 -0.9644 0.2586 -1.13 0.2587

For comparison, the following statements use the LIFEREG procedure to fit a Weibull distribution to the data. The LIFEREG procedure fits an accelerated failure time model, which assumes that the independent variables have a multiplicative effect on the event time.

proc lifereg data=pbc;
model Time*Status(0)=logBilirubin logProtime logAlbumin Age Edema;
run;

Output 94.1.3 provides the parameter estimates that are computed by PROC LIFEREG.

Output 94.1.3: Parameter Estimates from PROC LIFEREG

The LIFEREG Procedure

Analysis of Maximum Likelihood Parameter Estimates
Parameter DF Estimate Standard
Error
95% Confidence Limits Chi-Square Pr > ChiSq
Intercept 1 12.2155 1.4539 9.3658 15.0651 70.59 <.0001
logBilirubin 1 -0.5770 0.0556 -0.6861 -0.4680 107.55 <.0001
logProtime 1 -1.7565 0.5248 -2.7850 -0.7280 11.20 0.0008
logAlbumin 1 1.6694 0.4276 0.8313 2.5074 15.24 <.0001
Age 1 -0.0265 0.0053 -0.0368 -0.0162 25.35 <.0001
Edema 1 -0.6303 0.1805 -0.9842 -0.2764 12.19 0.0005
Scale 1 0.6807 0.0430 0.6014 0.7704
Weibull Shape 1 1.4691 0.0928 1.2980 1.6628

The p-value for logProtime is very small. For this same variable, the p-values that result from the quantile regression analysis are 0.3148 for the 0.5th quantile and 0.4338 for the 0.75th quantile, and the p-values are much smaller for the lower quantiles. Apparently, the effect of this covariate depends on which side of the response distribution is being modeled.

The PLOT=QUANTPLOT option in the PROC QUANTLIFE statement requests the quantile process plots in Output 94.1.4 and Output 94.1.5, which plot the estimated regression parameter against the quantile level. You can use these plots to compare quantile-specific covariate effects. If the curve is not constant, it can indicate heterogeneity in the data. The interpretation of the regression coefficients at a given quantile is similar to that of classical regression analysis. That is, the coefficient from a given covariate indicates the effect on log(Time) of a unit change in that covariate, assuming that the other covariates are fixed.

Output 94.1.4: Quantile Processes with 95% Confidence Bands

Output 94.1.5: Quantile Processes with 95% Confidence Bands

In Output 94.1.4, you can see that the effect of logProtime has a negative effect over the lower quantiles, which diminishes in magnitude at the median and upper quantiles. This insight would be missed if you were using the accelerated failure model.