survival data analysis model development · pdf filesurvival data analysis model development...

Survival Data Analysis

Model Development

Sandra Gardner, PhD

Dalla Lana School of Public Health

University of Toronto March 4, 2015

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Agenda

Model development ▫ Purposeful selection of covariates method

Reference: Hosmer, Lemeshow & May 2008

▫ Model development tips

▫ SAS coding examples

March 4, 2015 CHL5209H

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Model development

• Select model looking at overall K-M survival plot and other diagnostic plots

• Which model?

▫ non-parametric

▫ Cox model

▫ parametric models (e.g. Weibull )


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Model development

• Choose covariates for the model

▫ Time varying covariates?

▫ Variable selection methods

Forward, backward, stepwise, best score

Not currently available for Proc Lifereg

▫ Purposeful selection of covariates (Reference: Hosmer, Lemeshow and May, 2008)


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Purposeful selection of covariates (1)

• Step 1 ▫ Model each covariate separately (univariate

analysis) ▫ Fit multivariate model including all variables

where p<.25

• Step 2 ▫ Identify covariates to be removed from

multivariate models

• Step 3 ▫ Check for confounding (important changes in beta

values).


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• Step 4

▫ Add variables previously excluded step 1 to check for confounding

• Step 5

▫ Examine scale of continuous covariates

Linearity, transformation of covariates

• Step 6

▫ Check for interactions


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• Step 7

▫ Model evaluation

▫ Goodness of fit


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Step 1: explore data

Variables in Creation Order

# Variable Type Len Format Informat

1 treat Num 8 BEST12. BEST32.

2 resect75 Num 8 BEST12. BEST32.

3 age Num 8 BEST12. BEST32.

4 interval Num 8 BEST12. BEST32.

5 karn Num 8 BEST12. BEST32.

6 race Num 8 BEST12. BEST32.

7 local Num 8 BEST12. BEST32.

8 male Num 8 BEST12. BEST32.

9 nitro Num 8 BEST12. BEST32.

10 weeks Num 8 BEST12. BEST32.

11 event Num 8 BEST12. BEST32.

12 path Num 8 BEST12. BEST32.

13 grade Num 8 BEST12. BEST32.

14 lweeks Num 8

15 age50 Num 8


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Estimated median=27.4 and mean=44.5

Overall Survival

Step 1: explore data (examples)

male Frequency Percent

Cumulative

Frequency

Cumulative

Percent

0 79 35.59 79 35.59

1 143 64.41 222 100.00


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path Frequency Percent

Cumulative

Frequency

Cumulative

Percent

1 149 67.12 149 67.12

2 30 13.51 179 80.63

3 35 15.77 214 96.40

4 8 3.60 222 100.00

Step 1: data manipulation

• Do you understand how the data was collected?

▫ What is the quality of the data?

• Calculating outcome and censoring variables

• Data linkage

• Double check the results of any data manipulation


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Step 1: missing data

• Check patterns of missing data • Strategies to consider ▫ Delete observations ▫ Add missing category ▫ Missing data imputation

• Strategy will depend on amount of missing data and why the data is missing

• Are data missing at random? • Missing a covariate? ▫ Add random effects to model?


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Step 1: other model development tips

• Clean and label data before analysis • Explore the distribution of covariates ▫ Assess need to recode or rescale covariates at this step

or at univariate modeling (step 1) ▫ Check for highly correlated (collinear) relationships

amongst covariates

• Consult ▫ Subject matter specialists ▫ Statistical and medical literature

• Present ▫ Use graphics to supplement tabular results


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Step 1/2 – remove variables?

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Analysis of Maximum Likelihood Parameter Estimates

Parameter DF Estimate

Standard

Error

95%

Confidence

Limits Chi-Square Pr > ChiSq

Intercept 1 2.7478 0.1227 2.5073 2.9882 501.57 <.0001

path 1 0.3974 0.0692 0.2617 0.5331 32.94 <.0001

Scale 1 0.8934 0.0447 0.8100 0.9855

Parameter DF Estimate Std Err

95% Lower

Confidence

Limit

95% Upper

Confidence

Limit ChiSquare Pr>Chi treat 1 0.1775 0.1282 -0.0737 0.4288 1.92 0.1660

age50 1 -0.4161 0.1260 -0.6631 -0.1691 10.90 0.0010

age 1 -0.0176 0.0048 -0.0270 -0.0081 13.14 0.0003

male 1 0.1109 0.1342 -0.1522 0.3740 0.68 0.4088

race 1 -0.7266 0.2276 -1.1727 -0.2805 10.19 0.0014

karn 1 0.5376 0.1237 0.2951 0.7801 18.88 <.0001

local 1 0.1919 0.1516 -0.1052 0.4890 1.60 0.2055

grade 1 0.3380 0.2423 -0.1370 0.8130 1.95 0.1631

path 1 0.3974 0.0692 0.2617 0.5331 32.94 <.0001

resect75 1 0.4085 0.1437 0.1269 0.6902 8.08 0.0045

nitro 1 -0.4702 0.1250 -0.7151 -0.2252 14.15 0.0002

interval 1 0.1768 0.0347 0.1089 0.2447 26.02 <.0001

Example univariate model

Table of univariate results

Step 1: multivariate model (1)

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Standard

Error

95% Confidence


Intercept 1 3.0454 0.3345 2.3897 3.7010 82.87 <.0001

treat 1 0.2208 0.1045 0.0159 0.4257 4.46 0.0347

age 1 -0.0097 0.0044 -0.0183 -0.0011 4.91 0.0267

race 1 -0.3972 0.1930 -0.7753 -0.0190 4.24 0.0396

karn 1 0.3324 0.1102 0.1165 0.5483 9.11 0.0025

local 1 0.2652 0.1247 0.0207 0.5096 4.52 0.0335

grade 1 0.4349 0.1993 0.0443 0.8256 4.76 0.0291

path 1 0.2582 0.0633 0.1341 0.3823 16.62 <.0001

resect75 1 0.2397 0.1215 0.0016 0.4778 3.89 0.0484

nitro 1 -0.3022 0.1085 -0.5148 -0.0896 7.76 0.0053

interval 1 0.1054 0.0325 0.0418 0.1690 10.53 0.0012

Scale 1 0.7667 0.0384 0.6950 0.8458

Step 3/4: multivariate model (2)

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Standard

Error

95% Confidence


Intercept 1 3.0177 0.3368 2.3576 3.6778 80.28 <.0001

treat 1 0.2164 0.1047 0.0113 0.4216 4.28 0.0387

age 1 -0.0099 0.0044 -0.0185 -0.0013 5.12 0.0236

male 1 0.0773 0.1112 -0.1407 0.2952 0.48 0.4873

race 1 -0.3998 0.1930 -0.7781 -0.0216 4.29 0.0383

karn 1 0.3184 0.1119 0.0990 0.5378 8.09 0.0044

local 1 0.2605 0.1248 0.0158 0.5051 4.35 0.0369

grade 1 0.4259 0.1997 0.0346 0.8173 4.55 0.0329

path 1 0.2597 0.0633 0.1355 0.3838 16.80 <.0001

resect75 1 0.2480 0.1220 0.0088 0.4871 4.13 0.0421

nitro 1 -0.3067 0.1086 -0.5196 -0.0938 7.97 0.0048

interval 1 0.1050 0.0325 0.0414 0.1687 10.46 0.0012

Scale 1 0.7663 0.0384 0.6946 0.8453

Step 5: quadratic?

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Standard

Error

95% Confidence


Intercept 1 3.1532 0.6884 1.8039 4.5025 20.98 <.0001

treat 1 0.2218 0.1052 0.0156 0.4279 4.45 0.0350

age 1 -0.0150 0.0276 -0.0691 0.0390 0.30 0.5850

age*age 1 0.0001 0.0003 -0.0005 0.0006 0.04 0.8421

race 1 -0.3983 0.1931 -0.7768 -0.0198 4.25 0.0392

karn 1 0.3319 0.1102 0.1160 0.5478 9.08 0.0026

local 1 0.2656 0.1247 0.0211 0.5100 4.53 0.0332

grade 1 0.4351 0.1994 0.0442 0.8259 4.76 0.0291

path 1 0.2581 0.0633 0.1340 0.3823 16.62 <.0001

resect75 1 0.2417 0.1220 0.0025 0.4809 3.92 0.0477

nitro 1 -0.3011 0.1096 -0.5159 -0.0864 7.55 0.0060

interval 1 0.1162 0.1013 -0.0824 0.3147 1.32 0.2514

interval*interval 1 -0.0015 0.0131 -0.0272 0.0243 0.01 0.9106

Scale 1 0.7666 0.0384 0.6949 0.8457

Step 5: continuous or categorical?

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Step 6: interactions

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Standard

Error

95% Confidence


Intercept 1 2.4132 0.4041 1.6212 3.2051 35.67 <.0001

treat 1 0.2373 0.1030 0.0354 0.4391 5.31 0.0212

age 1 0.0028 0.0063 -0.0096 0.0151 0.19 0.6628

race 1 -0.4147 0.1897 -0.7865 -0.0428 4.78 0.0289

karn 1 1.4432 0.4093 0.6409 2.2455 12.43 0.0004

local 1 0.2703 0.1227 0.0298 0.5108 4.85 0.0276

grade 1 0.3340 0.8534 -1.3385 2.0066 0.15 0.6955

path 1 0.2563 0.0625 0.1338 0.3788 16.81 <.0001

resect75 1 0.2207 0.1196 -0.0137 0.4552 3.40 0.0650

nitro 1 -0.2995 0.1067 -0.5086 -0.0905 7.89 0.0050

interval 1 0.1166 0.0322 0.0534 0.1798 13.06 0.0003

age*karn 1 -0.0230 0.0082 -0.0390 -0.0070 7.92 0.0049

age*grade 1 0.0022 0.0157 -0.0287 0.0330 0.02 0.8896

Scale 1 0.7535 0.0377 0.6831 0.8312

Step 3/6: interaction example

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Table of age50 by karn

age50 karn

Frequency

Percent

Row Pct

Col Pct 0 1 Total

0 43

19.37

37.07

40.95

73

32.88

62.93

62.39

116

52.25

1 62

27.93

58.49

59.05

44

19.82

41.51

37.61

106

47.75

Total 105

47.30

117

52.70

222

100.00

P=0.0014

Step 6

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Unadjusted/adjusted:

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Standard

Error

95% Confidence


Intercept 1 4.2052 0.2404 3.7340 4.6765 305.87 <.0001

age 1 -0.0176 0.0048 -0.0270 -0.0081 13.14 0.0003

Scale 1 0.9284 0.0465 0.8415 1.0243



Standard

Error

95% Confidence


Intercept 1 3.7787 0.2593 3.2705 4.2870 212.34 <.0001

age 1 -0.0137 0.0048 -0.0231 -0.0043 8.18 0.0042

karn 1 0.4619 0.1244 0.2182 0.7056 13.80 0.0002

Scale 1 0.9000 0.0451 0.8158 0.9929

Stratifying:

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karn=1

karn=0



Standard

Error

95% Confidence


Intercept 1 3.2973 0.3389 2.6330 3.9616 94.64 <.0001

age 1 -0.0043 0.0065 -0.0170 0.0084 0.44 0.5091

Scale 1 0.8359 0.0590 0.7278 0.9600



Standard

Error

95% Confidence


Intercept 1 4.6371 0.3289 3.9924 5.2818 198.72 <.0001

age 1 -0.0224 0.0070 -0.0361 -0.0087 10.30 0.0013

Scale 1 0.9478 0.0673 0.8246 1.0893

Interaction:

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Standard

Error

95% Confidence


Intercept 1 3.2994 0.3622 2.5894 4.0093 82.97 <.0001

age 1 -0.0043 0.0069 -0.0179 0.0093 0.39 0.5347

karn 1 1.3256 0.4765 0.3916 2.2596 7.74 0.0054

age*karn 1 -0.0179 0.0095 -0.0366 0.0008 3.52 0.0605

Scale 1 0.8933 0.0448 0.8097 0.9854

Estimate

Label Estimate

Standard

Error z Value Pr > |z| Exponentiated

age, karn=0 -0.00430 0.006921 -0.62 0.5347 0.9957

Estimate

Label Estimate

Standard


age, karn=1 -0.02222 0.006576 -3.38 0.0007 0.9780

Step 6

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Step 3/6: confounding example

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p=0.0088

Table of resect75 by karn

resect75 karn

Frequency

Percent

Row Pct

Col Pct 0 1 Total

0 36

16.22

62.07

34.29

22

9.91

37.93

18.80

58

26.13

1 69

31.08

42.07

65.71

95

42.79

57.93

81.20

164

73.87

Total 105

47.30

117

52.70

222

100.00

Unadjusted/adjusted:

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Standard

Error

95%

Confidence


Intercept 1 3.0637 0.1233 2.8221 3.3054 617.50 <.0001

resect75 1 0.4085 0.1437 0.1269 0.6902 8.08 0.0045

Scale 1 0.9390 0.0471 0.8511 1.0359



Standard

Error

95%

Confidence


Intercept 1 2.8780 0.1281 2.6270 3.1290 504.96 <.0001

resect75 1 0.3101 0.1409 0.0338 0.5863 4.84 0.0278

karn 1 0.4896 0.1244 0.2458 0.7334 15.50 <.0001

Scale 1 0.9068 0.0455 0.8220 1.0004

Confounding:

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58.3173.31

3101.0

4896.0)3793.05793.0(100

3101.0

3101.04085.0100

)(100100%ˆ

1

221

1

111

aa

Stratifying:

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Standard

Error

95%

Confidence


Intercept 1 2.9096 0.1381 2.6388 3.1803 443.60 <.0001

resect75 1 0.2598 0.1706 -0.0745 0.5941 2.32 0.1277

Scale 1 0.8289 0.0585 0.7217 0.9519



Standard

Error

95%

Confidence


Intercept 1 3.3160 0.2088 2.9068 3.7252 252.28 <.0001

resect75 1 0.3801 0.2322 -0.0750 0.8352 2.68 0.1017

Scale 1 0.9792 0.0696 0.8519 1.1256

karn=1

karn=0

No interaction:

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Standard

Error

95% Confidence


Intercept 1 2.9096 0.1511 2.6134 3.2057 370.77 <.0001

resect75 1 0.2620 0.1866 -0.1038 0.6277 1.97 0.1603

karn 1 0.4064 0.2453 -0.0744 0.8873 2.74 0.0976

resect75*karn 1 0.1119 0.2846 -0.4459 0.6697 0.15 0.6942

Scale 1 0.9066 0.0454 0.8218 1.0002

Estimate

Label Estimate

Standard


resection, karn=0 0.2620 0.1866 1.40 0.1603 1.2995

Estimate

Label Estimate

Standard


resection, karn=1 0.3739 0.2149 1.74 0.0820 1.4533

Step 7: final model?

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Standard

Error

95% Confidence


Intercept 1 2.4051 0.4000 1.6212 3.1891 36.15 <.0001

treat 1 0.2377 0.1030 0.0359 0.4395 5.33 0.0210

age 1 0.0029 0.0062 -0.0092 0.0151 0.22 0.6385

race 1 -0.4151 0.1897 -0.7870 -0.0432 4.79 0.0287

karn 1 1.4431 0.4094 0.6407 2.2456 12.42 0.0004

local 1 0.2697 0.1226 0.0293 0.5101 4.84 0.0279

grade 1 0.4494 0.1961 0.0650 0.8337 5.25 0.0219

path 1 0.2570 0.0623 0.1349 0.3791 17.02 <.0001

resect75 1 0.2203 0.1196 -0.0142 0.4547 3.39 0.0656

nitro 1 -0.2997 0.1067 -0.5088 -0.0906 7.89 0.0050

interval 1 0.1167 0.0322 0.0535 0.1799 13.11 0.0003

age*karn 1 -0.0230 0.0082 -0.0390 -0.0070 7.92 0.0049

Scale 1 0.7537 0.0377 0.6832 0.8313

Step 7: Cox-Snell residuals

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Additional SAS code

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proc sort data=sda.brain out=sbrain;

by descending treat;

run;

proc lifereg data=sbrain order=data;

class treat;

model weeks*event(0)=treat age/d=lnormal;

effectplot/noobs;

title 'LifeReg: effect plot';

run;


class age50 karn;

model weeks*event(0)=age50 karn

age50*karn/d=lnormal;

slice age50*karn/sliceby=karn diff cl exp;

effectplot interaction(x=age50 sliceby=karn) / noobs link;

title 'LifeReg: Age groups and Karnofsky score groups -

LogNormal model';

run;


class treat path;

model weeks*event(0)=treat path/d=lnormal;

/* joint test 3 higher path categories different from 1 */

lsmeans path / diff exp;

lsmestimate path 'path 2,3,4 vs 1' .33 .33 .33 -1;

title 'LifeReg: Treatment & Pathology - LogNormal

model';

run;

Effect plots

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Standard

Error

95% Confidence


Intercept 1 4.1190 0.2456 3.6376 4.6004 281.26 <.0001

treat 1 1 0.1883 0.1245 -0.0558 0.4323 2.29 0.1306

treat 0 0 0.0000 . . . . .

age 1 -0.0177 0.0048 -0.0271 -0.0083 13.52 0.0002

Scale 1 0.9230 0.0463 0.8366 1.0183

age xbeta1 xbeta0

20 3.9533 4.4189

40 3.5993 4.0649

60 3.2453 3.7109

80 2.8913 3.3569

Interaction plots

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Standard

Error

95% Confidence


Intercept 1 3.1154 0.1371 2.8468 3.3840 516.70 <.0001

age50 1 1 -0.0579 0.1782 -0.4072 0.2915 0.11 0.7454

age50 0 0 0.0000 . . . . .

karn 1 1 0.7133 0.1734 0.3734 1.0531 16.92 <.0001

karn 0 0 0.0000 . . . . .

age50*karn 1 1 1 -0.4958 0.2477 -0.9813 -0.0103 4.01 0.0453

age50*karn 1 0 0 0.0000 . . . . .

age50*karn 0 1 0 0.0000 . . . . .

age50*karn 0 0 0 0.0000 . . . . .

Scale 1 0.8962 0.0449 0.8123 0.9886

Hypothesis tests

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Standard

Error

95% Confidence


Intercept 1 3.0473 0.0952 2.8607 3.2340 1023.77 <.0001

treat 1 1 0.1951 0.1203 -0.0407 0.4309 2.63 0.1048

treat 0 0 0.0000 . . . . .

path 4 1 1.1837 0.3307 0.5354 1.8319 12.81 0.0003

path 3 1 0.8068 0.1702 0.4732 1.1405 22.47 <.0001

path 2 1 0.3911 0.1784 0.0415 0.7408 4.81 0.0283

path 1 0 0.0000 . . . . .

Scale 1 0.8875 0.0444 0.8046 0.9790

Least Squares Means Estimate

Effect Label Estimate

Standard

Error z Value Pr > |z|

path path 2,3,4 vs 1 0.7545 0.1486 5.08 <.0001

Reference

• Applied Survival Analysis, D.W. Hosmer, S. Lemeshow, S. May, Wiley 2008


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