Survival Analysis in SAS¶

In many mdeical studies, the main outcome variable is the time to the occurrence of a particular event. In a randomized controlled trial of treatment for cancer, for example, surgery, radiation, and chemotherapy might be compared with respect to time from randomization and the start of therapy until death. In this case the event of interest is death of a patient, but in other situations, it might be remission from a disease, relief from symptoms, or the recurrence of a particular condition. Such observations are generally referred to by the generic term survival data even when the endpoint or event considered is not death but something else. Such data generally requiree special techniques for their analysis for two main reasons:

Survival data are generally right skewed instead of normally distributed
AT the completion of the study, some patients may not have reached the endpoint of interest (death, relapse, etc.). Consequently, the exact survival times are not known. All that is known is that the survival time is longer thant he subject was in the study. This is known as censored data (right censored in this case).

In this lecture, we will examing the WHAS500 data set which contains data on the Worcester Heart Attach Study. This study examined several factors such as age, gender and BMI, that may influence survival time after a heart attack. Follow up time for all patients begins at the time of hospital admission after a heart attach and ends with death or loss to follow up (censoring). We will use the following variables

lenfol: length of follow up, terminated either by death or censoring
fstat: the censoring variable. 0 = loss to follow up, 1 = death
age: age at hospitalization
bmi: body mass index
hr: initial heart rate
gender: 0 = male, 1 = females

The Survival and Hazard Functions¶

Traditionally, important functions in statistics are the density function and the cumulative distribution function. Let T represent our survival time for uncensored patients, f represent the density, and $F(t)=Pr(T\leq t)$ be the cumualitive distribution function.

LIBNAME surv "H:\BiostatCourses\PublicHealthComputing\Lectures\Week11Survival\SAS";

ODS SELECT Histogram;
proc univariate data = surv.whas500(where=(fstat=1));
var lenfol;
histogram lenfol / kernel;
run;

The histogram with kernal density estimate of the (uncensored) survival time show that the highest risk of death is shortly after the heart attack and decreases as time passes. (Notice that it is right skewed which is very typical for survival data).

ODS SELECT cdfplot;
proc univariate data = surv.whas500(where=(fstat=1));
var lenfol;
cdfplot lenfol;
run;

The estimate cdf for these uncensored survival times show that after around 200 days a patients has accumulated quite a bit of the risk of death (around 50%), that is the probability of living dying before 200 days is 0.5. After this point the cdf increase more slowly. A faster increase in the cdf occurs in the time when there were more deaths (where the probability of death is more likely).

With censored data, we cannot estimate either of these functions. With (right) censeored data, we can estimate the survival and hazard rate functions. The survival function is

$$S(t)=1-F(t)=Pr(T>t)$$

which described the probability of living longer than time t. In the presence of censoring, the survival function is estimated using the nonparametric estimator known as the Kaplan-Meier estimator. To calculate the Kaplan-Meier estimator, let

$t_1
$d_j$ represent the number of deaths at time $t_j$
$r_j$ represent the number of patients still at risk at time $t_j$ (alive and not censored)

Then the Kaplan-Meier estimator is given by

$$\hat{S}(t)=\Pi_{t_j\leq t}\left(1-\dfrac{d_j}{r_j}\right)$$

The Greenwood estimator of the variance of the Kaplan-Meier estimatory is

$$V\left(\hat{S}(t)\right)=\hat{S}^2(t)\sum_{t_j\leq t}\dfrac{d_j}{r_j(r_j-d_j)}$$

To get the Kaplan-Meier estimator in SAS, you will use PROC LIFETEST.

ODS SELECT SURVIVALPLOT;
proc lifetest data=surv.whas500 plots=survival(atrisk cb);
ODS OUTPUT ProductLimitEstimates = ple;
time lenfol*fstat(0);
run; 

PROC PRINT DATA=ple(obs=25);
RUN;

The TIME statement is required with PROC LIFETEST and is read as *(values that correspond to right censoring). In this plot, the probability of survivng more than 1500 days following a heart attach is 0.6.

The hazard function is another important function in survival anlysis. The hazard function is defined as

$$h(t)=\lim_{\Delta t\downarrow 0}\dfrac{Pr(t\leq T\leq t+\Delta t|T\geq t)}{s}=\dfrac{f(t)}{S(t)}.$$

This function (approximately) describes the probability of dying at time t given that the patient has lived up to time t.

ODS SELECT HAZARDPLOT;
proc lifetest data=surv.whas500 plots=hazard(bw=200);
time lenfol*fstat(0);
run;

This type of "bathtub" shape for the hazard fucntion is common. The risk if death is typically highests immediately following hospitilization and then drops until eventually other factors such as age lead to the risk of death increasing again.

Comparing Survival Functions¶

The survival and hazard rate functions provide useful summaries of survival times for a single group of patients, but usually the interest is in comparing groups by comparing their survival curves (for example compare surivival times of males vs females or new vs old cancer treatmenat). In the following, we compare the survival times after a heart attack for men vs women;

PROC FORMAT;
VALUE gdr 0 = "male" 1 = "female";

ODS SELECT SurvivalPlot HomTests;
proc lifetest data=surv.whas500 atrisk plots=survival(atrisk cb) outs=outwhas500;
strata gender / test=(all);
time lenfol*fstat(0);
FORMAT gender gdr.;
run;

These test are of

$$H_0: S_1=S_2\;\text{ vs }\;H_1:S_1\neq S_2$$

The most common test is the log rank test, but SAS has other tests of these hypotheses as well. The log rank test is popular becuase under certain conditions it is the most powerful test. In this case, all of the tests conclud that the survival functions are significantly different between males and females. Females generally have a worse surival time.

Cox's Proportional Hazards Regression Model¶

In Cox's proportional hazards regression model, we model the hazard function with a baseline hazard function, $h_0(t)$, and a function of the covariates in the following way.

$$h(t|X)=h_0(t)\exp(\beta_1X_1+\cdots+\beta_pX_p)$$

$h_0(t)$ modeled nonparametrically, that is no assumptions are made about its functional form
the covariates are in an exponential term to restrict the values of the hazard function to remain positive

If we look at the ratio of the hazard functions at two different values of the covariates, then we have a quantity that is independent of time. This is the assumption that the hazards are proportional

$$\dfrac{h(t|x_2)}{h(t|x_1)}=\dfrac{h_0(t)\exp(\beta_1x_2)}{h_0(t)\exp(\beta_1x_1}=\exp(\beta_1(x_2-x_1)$$

This means that this model assums that if a subject has a risk of death twice as high as another subject at some time point then the risk of death is always twice as high at every time point.

In this model, $\beta_1$ is interpreted as "the risk of dying for group 2 is $\exp{\beta_1}$ times the risk of dying for group when, holding all other variables constant."

proc phreg data = surv.whas500 plots=survival;
class gender;
model lenfol*fstat(0) = gender age;
FORMAT gender gdr.;
run;

The resulting model is

$$h(t|gender,age}=h_0(t)\exp(-0.066*x_gender + 0.067*age)$$

The global test of the null hypothesis: BETA=0 is a test of all the BETAS in the model being equal to zero. In this case, the model is significant, so at least on predictor is useful at predicting hazard rates.
The estimated hazard rate ratio between females and males is $\exp{-0.06556}=0.937$ which is roughly a 6% decrease in risk. This coefficient is not significant though.
The estimated hazard rate increase by about 7% (hazard rate ratio = $\exp(0.06683)=1.069$) for each additional year.

Model Diagnostics¶

Cox's model is a semiparametric model meaning that we have made some parametric (functional form of the covariates and proportional hazards) assumptions and some nonparametric assumptions (no assumptions on the form of the baseline hazard). Just in multiple regression, we can use diagnostic plots to assess the these assumptions:

proportional hazards assumption
functional form of the covariates

To assess the whether or not the functional form of the covariates is correct we can examine residual plots. For Cox's model, there are different types of residuals such as Cox-Snell, diviance, martingale and Schoenfeld residuals. How all these residuals are calculated and the theory behind them is beyond the scope of this course, but we will discuss how to use them in diagnostic plots. To assess the funcitonal form, we can plot the martingal residuals vs each variables. Similar to multiple regression, we should the relationship modeling between these two should be a flat line through 0. We could for example use a loess curve to estimate this relationship.

/*full model with linear and quadratric term for bmi */
ODS SELECT NONE;
proc phreg data = surv.whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi hr;
outout out=residfull resmart=martingale;
run;

ODS SELECT ALL;
proc loess data = residfull plots=ResidualsBySmooth(smooth);
model martingale = bmi / smooth=0.2 0.4 0.6 0.8;
run;

In this case, the loess curves do vary some about the line y=0 but not too much. We may want to condiser a different functional form for a better fit though. Another way to assess the functional form is with the assess statement in proc phreg.

proc phreg data = surv.whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi hr;
assess var=(age bmi hr) / resample;
run;

What we want to see in these plots is the solid line doesn't extend beyond the region set by the many dashed lines. The solid line should stay well within this region if the form is correct. These plots are accompanied by simulated p-value. A small p-value indicates that the model should be modified. Most of these plots look fine, but the bmi plot does have a slighly high residual in the lower bmi range around 20-27.

proc phreg data = surv.whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr;
assess var=(age bmi bmi*bmi hr) / resample;
run;

The residuals for bmi are now smaller on the lower end of bmi. All of these plots look reasonable.

To assess the proportional hazards assumption, we can use several methods. If the predictor is categorical we can visually plot the two estimated Kaplan-Meier curves to see if this assumption is violated as we did for males and female above. We can also use the Schoenfel residuals. Plots of the Schoenfeld resisudal vs each predictor in the model should not show any estimated mean relationship (i.e a the line y=0). We can estimate this using a loess curve.

ODS SELECT NONE;
proc phreg data=surv.whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr;
output out=schoen ressch=schgender schage schgenderage
   schbmi schbmibmi schhr;
run;

ODS SELECT ALL;
proc loess data = schoen;
model schage=lenfol / smooth=(0.2 0.4 0.6 0.8);
run;

The smoothed loess curve appears roughly flat at 0 showing suggesting that the coefficient for age does not change over time and that the proportional hazards assumption holds for this covariate.

A third way we can check this assumption is by using the assess statement.

proc phreg data=surv.whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr;
assess var=(age bmi bmi*bmi hr) ph / resample ;
run;

Again, we are looking for solid lines that stay within the region defined by the dashed simulated lines. None of these lines looks particularly worrisome and all of the p-values are large suggesting that the proportional hazards assumption holds for each of these covariates.

Obs	STRATUM	LENFOL	Censor	Survival	Failure	StdErr	Failed	Left
1	1	0.00	.	1.0000	0	0	0	500
2	1	1.00	0	.	.	.	1	499
3	1	1.00	0	.	.	.	2	498
4	1	1.00	0	.	.	.	3	497
5	1	1.00	0	.	.	.	4	496
6	1	1.00	0	.	.	.	5	495
7	1	1.00	0	.	.	.	6	494
8	1	1.00	0	.	.	.	7	493
9	1	1.00	0	0.9840	0.0160	0.00561	8	492
10	1	2.00	0	.	.	.	9	491
11	1	2.00	0	.	.	.	10	490
12	1	2.00	0	.	.	.	11	489
13	1	2.00	0	.	.	.	12	488
14	1	2.00	0	.	.	.	13	487
15	1	2.00	0	.	.	.	14	486
16	1	2.00	0	.	.	.	15	485
17	1	2.00	0	0.9680	0.0320	0.00787	16	484
18	1	3.00	0	.	.	.	17	483
19	1	3.00	0	.	.	.	18	482
20	1	3.00	0	0.9620	0.0380	0.00855	19	481
21	1	4.00	0	.	.	.	20	480
22	1	4.00	0	0.9580	0.0420	0.00897	21	479
23	1	5.00	0	.	.	.	22	478
24	1	5.00	0	0.9540	0.0460	0.00937	23	477
25	1	6.00	0	.	.	.	24	476

Quartile Estimates
Percent	Point Estimate	95% Confidence Interval
Percent	Point Estimate	Transform	[Lower	Upper)
75	2353.00	LOGLOG	2350.00	2358.00
50	1317.00	LOGLOG	865.00	1579.00
25	174.00	LOGLOG	57.00	359.00

Quartile Estimates
Percent	Point Estimate	95% Confidence Interval
Percent	Point Estimate	Transform	[Lower	Upper)
75	.	LOGLOG	.	.
50	2160.00	LOGLOG	1624.00	.
25	368.00	LOGLOG	187.00	614.00

Summary of the Number of Censored and Uncensored Values
Stratum	GENDER	Total	Failed	Censored	Percent Censored
1	female	200	104	96	48.00
2	male	300	111	189	63.00
Total		500	215	285	57.00

Rank Statistics
GENDER	Log-Rank	Wilcoxon	Tarone	Peto	ModifiedPeto	Fleming
female	19.726	6271.0	345.2	14.516	14.474	14.570
male	-19.726	-6271.0	-345.2	-14.516	-14.474	-14.570

Product-Limit Survival Estimates
LENFOL		Number at Risk	Observed Events	Survival	Failure	Survival Standard Error	Number Failed	Number Left
0.00		200	0	1.0000	0	0	0	200
1.00		.	.	.	.	.	1	199
1.00		200	2	0.9900	0.0100	0.00704	2	198
2.00		.	.	.	.	.	3	197
2.00		.	.	.	.	.	4	196
2.00		.	.	.	.	.	5	195
2.00		.	.	.	.	.	6	194
2.00		198	5	0.9650	0.0350	0.0130	7	193
3.00		.	.	.	.	.	8	192
3.00		193	2	0.9550	0.0450	0.0147	9	191
4.00		191	1	0.9500	0.0500	0.0154	10	190
6.00		.	.	.	.	.	11	189
6.00		190	2	0.9400	0.0600	0.0168	12	188
7.00		.	.	.	.	.	13	187
7.00		188	2	0.9300	0.0700	0.0180	14	186
10.00		.	.	.	.	.	15	185
10.00		186	2	0.9200	0.0800	0.0192	16	184
11.00		.	.	.	.	.	17	183
11.00		.	.	.	.	.	18	182
11.00		.	.	.	.	.	19	181
11.00		184	4	0.9000	0.1000	0.0212	20	180
14.00		180	1	0.8950	0.1050	0.0217	21	179
16.00		179	1	0.8900	0.1100	0.0221	22	178
19.00		.	.	.	.	.	23	177
19.00		.	.	.	.	.	24	176
19.00		178	3	0.8750	0.1250	0.0234	25	175
22.00		.	.	.	.	.	26	174
22.00		175	2	0.8650	0.1350	0.0242	27	173
31.00		173	1	0.8600	0.1400	0.0245	28	172
32.00		172	1	0.8550	0.1450	0.0249	29	171
33.00		171	1	0.8500	0.1500	0.0252	30	170
34.00		170	1	0.8450	0.1550	0.0256	31	169
37.00		169	1	0.8400	0.1600	0.0259	32	168
42.00		168	1	0.8350	0.1650	0.0262	33	167
46.00		167	1	0.8300	0.1700	0.0266	34	166
49.00		166	1	0.8250	0.1750	0.0269	35	165
53.00		165	1	0.8200	0.1800	0.0272	36	164
57.00		.	.	.	.	.	37	163
57.00		164	2	0.8100	0.1900	0.0277	38	162
64.00		162	1	0.8050	0.1950	0.0280	39	161
83.00		161	1	0.8000	0.2000	0.0283	40	160
93.00		160	1	0.7950	0.2050	0.0285	41	159
95.00		159	1	0.7900	0.2100	0.0288	42	158
101.00		158	1	0.7850	0.2150	0.0290	43	157
132.00		157	1	0.7800	0.2200	0.0293	44	156
134.00		156	1	0.7750	0.2250	0.0295	45	155
135.00		155	1	0.7700	0.2300	0.0298	46	154
137.00		154	1	0.7650	0.2350	0.0300	47	153
145.00		153	1	0.7600	0.2400	0.0302	48	152
146.00		152	1	0.7550	0.2450	0.0304	49	151
151.00		151	1	0.7500	0.2500	0.0306	50	150
197.00		150	1	0.7450	0.2550	0.0308	51	149
200.00		149	1	0.7400	0.2600	0.0310	52	148
226.00		148	1	0.7350	0.2650	0.0312	53	147
235.00		147	1	0.7300	0.2700	0.0314	54	146
274.00		146	1	0.7250	0.2750	0.0316	55	145
287.00		145	1	0.7200	0.2800	0.0317	56	144
289.00		144	1	0.7150	0.2850	0.0319	57	143
321.00		143	1	0.7100	0.2900	0.0321	58	142
328.00		142	1	0.7050	0.2950	0.0322	59	141
358.00		141	1	0.7000	0.3000	0.0324	60	140
359.00		.	.	.	.	.	61	139
359.00		140	2	0.6900	0.3100	0.0327	62	138
363.00		138	1	0.6850	0.3150	0.0328	63	137
371.00	*	.	0	.	.	.	63	136
371.00	*	137	0	.	.	.	63	135
373.00	*	135	0	.	.	.	63	134
385.00		134	1	0.6799	0.3201	0.0330	64	133
386.00	*	133	0	.	.	.	64	132
392.00		132	1	0.6747	0.3253	0.0331	65	131
397.00	*	131	0	.	.	.	65	130
400.00	*	130	0	.	.	.	65	129
412.00	*	129	0	.	.	.	65	128
422.00		128	1	0.6695	0.3305	0.0333	66	127
424.00	*	127	0	.	.	.	66	126
442.00		126	1	0.6642	0.3358	0.0335	67	125
445.00	*	.	0	.	.	.	67	124
445.00	*	125	0	.	.	.	67	123
446.00		123	1	0.6588	0.3412	0.0336	68	122
449.00	*	122	0	.	.	.	68	121
451.00	*	121	0	.	.	.	68	120
458.00	*	120	0	.	.	.	68	119
465.00		119	1	0.6532	0.3468	0.0338	69	118
478.00	*	118	0	.	.	.	69	117
479.00		117	1	0.6476	0.3524	0.0340	70	116
497.00	*	116	0	.	.	.	70	115
506.00	*	115	0	.	.	.	70	114
516.00	*	114	0	.	.	.	70	113
521.00	*	113	0	.	.	.	70	112
522.00	*	112	0	.	.	.	70	111
524.00	*	111	0	.	.	.	70	110
535.00		110	1	0.6417	0.3583	0.0342	71	109
535.00	*	.	0	.	.	.	71	108
537.00		108	1	0.6358	0.3642	0.0344	72	107
542.00		107	1	0.6299	0.3701	0.0345	73	106
542.00	*	.	0	.	.	.	73	105
550.00	*	105	0	.	.	.	73	104
551.00	*	104	0	.	.	.	73	103
552.00		103	1	0.6237	0.3763	0.0347	74	102
578.00	*	102	0	.	.	.	74	101
589.00	*	101	0	.	.	.	74	100
606.00	*	100	0	.	.	.	74	99
631.00	*	99	0	.	.	.	74	98
632.00		98	1	0.6174	0.3826	0.0350	75	97
646.00		97	1	0.6110	0.3890	0.0352	76	96
649.00		96	1	0.6047	0.3953	0.0354	77	95
659.00	*	95	0	.	.	.	77	94
662.00	*	94	0	.	.	.	77	93
670.00		93	1	0.5982	0.4018	0.0356	78	92
673.00		92	1	0.5916	0.4084	0.0358	79	91
704.00		91	1	0.5851	0.4149	0.0360	80	90
714.00		90	1	0.5786	0.4214	0.0362	81	89
725.00	*	89	0	.	.	.	81	88
865.00		88	1	0.5721	0.4279	0.0364	82	87
905.00		87	1	0.5655	0.4345	0.0365	83	86
920.00		86	1	0.5589	0.4411	0.0367	84	85
1065.00		85	1	0.5523	0.4477	0.0368	85	84
1096.00		84	1	0.5458	0.4542	0.0370	86	83
1103.00	*	83	0	.	.	.	86	82
1117.00	*	82	0	.	.	.	86	81
1126.00	*	81	0	.	.	.	86	80
1140.00	*	80	0	.	.	.	86	79
1152.00		79	1	0.5389	0.4611	0.0372	87	78
1157.00	*	78	0	.	.	.	87	77
1160.00	*	77	0	.	.	.	87	76
1161.00	*	76	0	.	.	.	87	75
1162.00	*	75	0	.	.	.	87	74
1165.00		74	1	0.5316	0.4684	0.0374	88	73
1170.00	*	73	0	.	.	.	88	72
1174.00		72	1	0.5242	0.4758	0.0376	89	71
1174.00	*	.	0	.	.	.	89	70
1187.00	*	.	0	.	.	.	89	69
1187.00	*	70	0	.	.	.	89	68
1191.00	*	68	0	.	.	.	89	67
1199.00	*	67	0	.	.	.	89	66
1200.00		66	1	0.5163	0.4837	0.0378	90	65
1217.00		65	1	0.5083	0.4917	0.0381	91	64
1224.00	*	64	0	.	.	.	91	63
1232.00		63	1	0.5002	0.4998	0.0383	92	62
1244.00	*	62	0	.	.	.	92	61
1245.00	*	61	0	.	.	.	92	60
1251.00	*	60	0	.	.	.	92	59
1256.00	*	59	0	.	.	.	92	58
1257.00	*	58	0	.	.	.	92	57
1265.00	*	57	0	.	.	.	92	56
1274.00	*	56	0	.	.	.	92	55
1277.00	*	55	0	.	.	.	92	54
1279.00	*	54	0	.	.	.	92	53
1295.00	*	53	0	.	.	.	92	52
1302.00	*	52	0	.	.	.	92	51
1317.00		51	1	0.4904	0.5096	0.0388	93	50
1320.00	*	.	0	.	.	.	93	49
1320.00	*	50	0	.	.	.	93	48
1325.00	*	48	0	.	.	.	93	47
1332.00	*	47	0	.	.	.	93	46
1333.00	*	46	0	.	.	.	93	45
1338.00	*	45	0	.	.	.	93	44
1346.00	*	44	0	.	.	.	93	43
1359.00		43	1	0.4790	0.5210	0.0395	94	42
1363.00	*	42	0	.	.	.	94	41
1374.00	*	41	0	.	.	.	94	40
1377.00		40	1	0.4671	0.5329	0.0403	95	39
1385.00	*	39	0	.	.	.	95	38
1420.00	*	38	0	.	.	.	95	37
1451.00	*	37	0	.	.	.	95	36
1454.00	*	36	0	.	.	.	95	35
1536.00		35	1	0.4537	0.5463	0.0413	96	34
1576.00		34	1	0.4404	0.5596	0.0422	97	33
1577.00		33	1	0.4270	0.5730	0.0430	98	32
1579.00		32	1	0.4137	0.5863	0.0437	99	31
1627.00		31	1	0.4003	0.5997	0.0442	100	30
1836.00	*	30	0	.	.	.	100	29
1858.00	*	29	0	.	.	.	100	28
1885.00	*	28	0	.	.	.	100	27
1887.00	*	.	0	.	.	.	100	26
1887.00	*	27	0	.	.	.	100	25
1914.00	*	25	0	.	.	.	100	24
1919.00	*	24	0	.	.	.	100	23
1926.00		23	1	0.3829	0.6171	0.0456	101	22
1931.00	*	22	0	.	.	.	101	21
1933.00	*	21	0	.	.	.	101	20
1936.00	*	20	0	.	.	.	101	19
1941.00	*	19	0	.	.	.	101	18
1955.00	*	18	0	.	.	.	101	17
1964.00	*	17	0	.	.	.	101	16
1969.00	*	16	0	.	.	.	101	15
1979.00	*	15	0	.	.	.	101	14
2009.00	*	14	0	.	.	.	101	13
2057.00	*	13	0	.	.	.	101	12
2064.00	*	12	0	.	.	.	101	11
2108.00	*	11	0	.	.	.	101	10
2114.00	*	10	0	.	.	.	101	9
2123.00	*	9	0	.	.	.	101	8
2125.00	*	8	0	.	.	.	101	7
2132.00	*	7	0	.	.	.	101	6
2145.00	*	6	0	.	.	.	101	5
2156.00	*	5	0	.	.	.	101	4
2190.00	*	4	0	.	.	.	101	3
2350.00		3	1	0.2553	0.7447	0.1086	102	2
2353.00		2	1	0.1276	0.8724	0.1053	103	1
2358.00		1	1	0	1.0000	.	104	0

Test of Equality over Strata
Test	Chi-Square	DF	Pr > Chi-Square
Log-Rank	7.7911	1	0.0053
Wilcoxon	5.5370	1	0.0186
Tarone	6.6664	1	0.0098
Peto	6.7602	1	0.0093
Modified Peto	6.7611	1	0.0093
Fleming(1)	6.7309	1	0.0095

Model Information
Data Set	SURV.WHAS500
Dependent Variable	LENFOL
Censoring Variable	FSTAT
Censoring Value(s)	0
Ties Handling	BRESLOW

Model Fit Statistics
Criterion	Without Covariates	With Covariates
-2 LOG L	2455.158	2313.140
AIC	2455.158	2317.140
SBC	2455.158	2323.882

Testing Global Null Hypothesis: BETA=0
Test	Chi-Square	DF	Pr > ChiSq
Likelihood Ratio	142.0177	2	<.0001
Score	126.6381	2	<.0001
Wald	119.3806	2	<.0001

Type 3 Tests
Effect	DF	Wald Chi-Square	Pr > ChiSq
GENDER	1	0.2175	0.6410
AGE	1	116.3986	<.0001

Analysis of Maximum Likelihood Estimates
Parameter		DF	Parameter Estimate	Standard Error	Chi-Square	Pr > ChiSq	Hazard Ratio	Label
GENDER	female	1	-0.06556	0.14057	0.2175	0.6410	0.937	GENDER female
AGE		1	0.06683	0.00619	116.3986	<.0001	1.069

Independent Variable Scaling
Scaling applied: None
Statistic	BMI
Minimum Value	13.04546
Maximum Value	44.83886

Fit Summary
Fit Method	kd Tree
Blending	Linear
Number of Observations	500
Number of Fitting Points	33
kd Tree Bucket Size	20
Degree of Local Polynomials	1
Smoothing Parameter	0.20000
Points in Local Neighborhood	100
Residual Sum of Squares	205.38952

Supremum Test for Functional Form
Variable	Maximum Absolute Value	Replications	Seed	Pr > MaxAbsVal
AGE	11.2240	1000	177039786	0.1480
BMI	11.0212	1000	177039786	0.2460
HR	9.3459	1000	177039786	0.3760

Product-Limit Survival Estimates
LENFOL	Number at Risk	Observed Events	Survival	Failure	Survival Standard Error	Number Failed	Number Left
0.00	300	0	1.0000	0	0	0	300
1.00	.	.	.	.	.	1	299
1.00	.	.	.	.	.	2	298
1.00	.	.	.	.	.	3	297
1.00	.	.	.	.	.	4	296
1.00	.	.	.	.	.	5	295
1.00	300	6	0.9800	0.0200	0.00808	6	294
2.00	.	.	.	.	.	7	293
2.00	.	.	.	.	.	8	292
2.00	294	3	0.9700	0.0300	0.00985	9	291
3.00	291	1	0.9667	0.0333	0.0104	10	290
4.00	290	1	0.9633	0.0367	0.0109	11	289
5.00	.	.	.	.	.	12	288
5.00	289	2	0.9567	0.0433	0.0118	13	287
6.00	.	.	.	.	.	14	286
6.00	.	.	.	.	.	15	285
6.00	287	3	0.9467	0.0533	0.0130	16	284
7.00	.	.	.	.	.	17	283
7.00	.	.	.	.	.	18	282
7.00	.	.	.	.	.	19	281
7.00	284	4	0.9333	0.0667	0.0144	20	280
10.00	280	1	0.9300	0.0700	0.0147	21	279
14.00	279	1	0.9267	0.0733	0.0151	22	278
17.00	.	.	.	.	.	23	277
17.00	278	2	0.9200	0.0800	0.0157	24	276
18.00	.	.	.	.	.	25	275
18.00	.	.	.	.	.	26	274
18.00	276	3	0.9100	0.0900	0.0165	27	273
20.00	.	.	.	.	.	28	272
20.00	273	2	0.9033	0.0967	0.0171	29	271
26.00	271	1	0.9000	0.1000	0.0173	30	270
32.00	270	1	0.8967	0.1033	0.0176	31	269
33.00	.	.	.	.	.	32	268
33.00	269	2	0.8900	0.1100	0.0181	33	267
52.00	267	1	0.8867	0.1133	0.0183	34	266
55.00	266	1	0.8833	0.1167	0.0185	35	265
60.00	265	1	0.8800	0.1200	0.0188	36	264
61.00	264	1	0.8767	0.1233	0.0190	37	263
62.00	263	1	0.8733	0.1267	0.0192	38	262

Class Level Information
Class	Value	Design Variables
GENDER	female	1
	male	0

Class Level Information
Class	Value	Design Variables
GENDER	0	1
	1	0

Survival Analysis in SAS¶

The Survival and Hazard Functions¶

SAS Output

The UNIVARIATE Procedure

LENFOL

Histogram 1

Panel 1

SAS Output

The UNIVARIATE Procedure

LENFOL

CDF Plot 1

Panel 1

SAS Output

The LIFETEST Procedure

Stratum 1

Survival Curve

The PRINT Procedure

Data Set WORK.PLE

SAS Output

The LIFETEST Procedure

Stratum 1

Estimated Smoothed Hazard Curves

Comparing Survival Functions¶

SAS Output

The LIFETEST Procedure

Stratum 1

Product-Limit Estimates

Summary of LENFOL

Quartiles of the Survival Distribution

Mean

Stratum 2

Product-Limit Estimates

Summary of LENFOL

Quartiles of the Survival Distribution

Mean

Censored Summary

Strata Homogeneity

Rank Statistics

Log-Rank Covariance

Wilcoxon Covariance

Tarone Covariance

Peto Covariance

Modified Peto Covariance

Fleming Covariance

Homogeneity Tests

Survival Curves

Cox's Proportional Hazards Regression Model¶

SAS Output

The PHREG Procedure

Model Information

Number of Observations

Class Level Information

Summary of Event and Censored Observations

Convergence Status

Model Fit Statistics

Test of Global Null Hypothesis

Type 3 Tests

Maximum Likelihood Estimates of Model Parameters

Survivorship

Reference Set of Covariates

Model Diagnostics¶

SAS Output

The LOESS Procedure

Scaling Information

Smoothing Parameter: 0.2

Fit Summary

Fit Plot

Residual Plots

BMI

Diagnostic Plots

Fit Diagnostics

Smoothing Parameter: 0.4

Fit Summary

Fit Plot

Residual Plots

BMI

Diagnostic Plots

Fit Diagnostics

Smoothing Parameter: 0.6

Fit Summary