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Additive Hazards Models in SAS

Sabuj Sarker

Saskatchewan Cancer Agency

May 10, 2012

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Background

Survival Analysis – Time to event analysis

Multiplicative Cox hazards model is a semi-parametric, multiplicative hazards model. Widely used model.

There may be times when a measure of the additive effect of a covariate is preferred over a relative measure.

Additive Hazards models are not commonly used like Cox model

Aalen’s and Lin & Ying’s additive models are used to consider additive effects

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Methods: Notations

x: time to some event at time t

Tj: the time to event for the jth patient; j=1,2, …., n

δj: the event indicator for the jth patient, where δj =1 if event has occurred, 0 if the life time is right-censored.

Zj(t)=(Zj1(t), Zj2(t), ……,Zjp(t))’ is a p-vector of, possibly, time-dependent covariates.

β= ( β1, ………, βp)’ is a parameter vector

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Cox Hazards Model

1. Cox Hazard Model (1972):

h(t/Z)=h0(t)exp (β1z1+ …… + βpzp)

Cox model assumes that covariates act in an

multiplicative manner on an unknown baseline hazard

(h0).

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Additive Hazard Model

1. Aalen’s Model (1989):

h(t/Z)=h0(t)+ β1(t)zj1(t)+ …..+ βp(t)zjp (t)

- β(t) is dependent on time t

- Covariates act in additive manner on unknown baseline hazard (h0)

2. Lin and Ying’s (L-Y) Model (1994):

h(t/Z)=h0(t)+β1zj1(t)+ …..+ βpzjp (t)

- Coefficient β is constant

Aalen’s additive hazards model in SAS

Data on 90 males with larynx cancer

The variables represented in the dataset are as follows:

Stage of disease (1=stage 1, 2=stage2, 3=stage 3, 4=stage 4)

Time to death or on-study time, months

Age at diagnosis of larynx cancer

Year of diagnosis of larynx cancer

Death indicator (0=alive, 1=dead)

Reference: Kardaun Stat. Nederlandica 37 (1983), 103-126.

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Cont…

data larynx; /* Just the Sample of Data */

input stage time age year_diag death;

label stage = "Stage of Disease"

age = "Age at Diagnosis"

year_diag = "Year of Diagnosis";

cards;

1 0.6 77 76 1

1 1.3 53 71 1

1 3.2 51 77 0

1 3.3 76 74 1

1 3.3 63 77 0

1 3.5 43 71 1

1 3.5 60 73 1

2 7.5 50 73 0

3 4.8 54 76 0

4 3.6 71 75 1

;

run;

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Cont..

/* Format */

proc format;

value stage 1 = "stage 1"

2 = "stage 2"

3 = "stage 3"

4 = "stage 4";

value dead 0 = "Alive"

1 = "Dead";

run;

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Cont..

/* Preparation for Analysis */

data larynx1;

length time death z1-z3 age 8.;

set larynx;

z1=(stage=2);

z2=(stage=3);

z3=(stage=4);

age=age-64.11;

keep time death z1 z2 z3 age;

run;

proc sort data = larynx1;

by time;

run;

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Cont..

%include 'E:\SUCCESS\additivemacro.sas';

/* Proc IML */

proc iml;

option={y, n, y, n, n};

contrast={0 1 -1 0 0, 0 0 1 -1 0};

effects={'z1', 'z2' ,'z3', 'age'};

timeunit={'Years'};

%additive (larynx1, 0.05, timeunit, effects, option, contrast, beta, test);

quit;

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Cont…

The data set beta created above has many columns. It has the following pattern.

Col1: time

Col2-6: beta_0-beta_4

col7-11: standard errors for beta_0-beta_4

col2-16: lower confidence limit for beta_0-beta_4

col7-21: upper confidence limit for beta_0-beta_4

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Cont.. Obs COL1 COL2 COL3 COL4 COL5 COL6 COL7 COL8

1 0.1 0.000015 0.000130 -0.00007 0.07751 -.000202225 0.000015 0.000130

2 0.2 -0.000134 0.057630 0.00068 0.07118 0.001860612 0.000149 0.057500

3 0.3 -0.000105 0.057359 0.07461 0.15573 0.001464296 0.000194 0.057512

4 0.3 -0.000105 0.057359 0.07461 0.15573 0.001464296 0.000194 0.057512

5 0.3 -0.000105 0.057359 0.07461 0.15573 0.001464296 0.000194 0.057512

6 0.4 -0.000174 0.058012 0.07467 0.24403 0.002421188 0.000206 0.057516

7 0.5 -0.000120 0.057496 0.11462 0.24540 0.001664510 0.000213 0.057518

8 0.6 0.030085 0.028127 0.08400 0.21261 0.003034450 0.030205 0.064582

9 0.7 0.030607 0.028573 0.12466 0.20909 0.004621786 0.030210 0.064584

10 0.8 0.031678 0.029490 0.16813 0.40186 0.007880698 0.030221 0.064587

Obs COL9 COL10 COL11 COL12 COL13 COL14 COL15

1 0.000074 0.07751 .000202225 -0.000014 -0.000125 -0.000220 -0.074405

2 0.000761 0.07777 .002072725 -0.000425 -0.055068 -0.000808 -0.081240

3 0.052185 0.11324 .002696663 -0.000485 -0.055362 -0.027669 -0.066225

4 0.052185 0.11324 .002696663 -0.000485 -0.055362 -0.027669 -0.066225

5 0.052185 0.11324 .002696663 -0.000485 -0.055362 -0.027669 -0.066225

6 0.052185 0.14360 .002861404 -0.000577 -0.054717 -0.027610 -0.037423

7 0.065723 0.14361 .002959762 -0.000536 -0.055237 -0.014191 -0.036060

8 0.072508 0.14730 .003261431 -0.029116 -0.098452 -0.058114 -0.076099

9 0.083129 0.14735 .003627198 -0.028603 -0.098008 -0.038273 -0.079704

10 0.093812 0.20297 .004369147 -0.027553 -0.097099 -0.015736 0.004050

Obs COL16 COL17 COL18 COL19 COL20 COL21

1 -.000598578 0.000043 0.00038 0.00007 0.22942 0.000194

2 -.002201855 0.000158 0.17033 0.00217 0.22360 0.005923

3 -.003821066 0.000274 0.17008 0.17689 0.37768 0.006750

4 -.003821066 0.000274 0.17008 0.17689 0.37768 0.006750

5 -.003821066 0.000274 0.17008 0.17689 0.37768 0.006750

6 -.003187060 0.000229 0.17074 0.17695 0.52548 0.008029

7 -.004136518 0.000297 0.17023 0.24344 0.52687 0.007466

8 -.003357836 0.089287 0.15471 0.22611 0.50132 0.009427

9 -.002487391 0.089817 0.15516 0.28759 0.49788 0.011731

10 -.000682672 0.090909 0.15608 0.35200 0.79966 0.01644

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Figure 1: Estimated Baseline Cumulative

Hazard Rate for Stage I Patients

title "Figure 1";

axis1 label=(j=c 'Years') minor=none;

axis2 label=(a=90 j=c 'Estimated Baseline Cumulative Hazard Rate for Stage I Patients')

minor = none;

proc gplot data=beta;

plot col2*col1 col12*col1 col17*col1/overlay haxis=axis1 vaxis=axis2;

run;

quit;

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Cont..

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Survival Time in Year

Estimated

Baseline

Cumulative

Hazards

Rates for

Stage I

Patients

Figure 1: Estimate cumulative baseline hazards with 95% confidence interval

Figure 2: Estimated Cumulative Excess

Risk of Stage II Patients

title "Figure 2";

axis1 label=(j=c 'Years') minor=none;

axis2 label=(a=90 j=c 'Estimated Cumulative Excess Risk of Stage II Patients')

order=(-0.4 to 0.6 by 0.2) minor = none;

proc gplot data=beta;

plot col3*col1 col13*col1 col18*col1/overlay haxis=axis1 vaxis=axis2;

run;

quit;

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Cont..

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Estimated

Cumulative

Excess Risk

of Stage II

as

Compared

to Stage I

Patients

Survival Time in Year

Figure 2: Estimate cumulative excess risk of hazards with 95% confidence interval

Testing in the Nonparametric Additive

Hazard Model

options nocenter;

proc iml;

option={ n, y, n, n, y};

contrast={0 1 -1 0 0, 0 0 1 -1 0};

effects={'z1', 'z2' ,'z3', 'age'};

timeunit={'Years'};

%additive (larynx1, 0.05, timeunit, effects, option, contrast, beta, test);

quit;

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Cont...

Global Test

Chi-Square d.f p-value

10.9613 4 0.0270

Analysis of Variance

Effect Chi-Square d.f p-value

z1 0.1456 1 0.7027

z2 3.0062 1 0.0829

z3 8.4655 1 0.0036

age 0.2333 1 0.6291

Test of Linear Combinations Contrast Matrix 0 1 -1 0 0

0 0 1 -1 0

Chi-Square d.f p-value

6.8131 2 0.0332

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For testing H0: beta_1=beta_2

title "beta_1=beta_2";

proc iml;

option={ n, y, n, n, y};

contrast={0 1 -1 0 0};

effects={'z1', 'z2' ,'z3', 'age'};

timeunit={'Years'};

%additive (larynx1, 0.05, timeunit, effects, option, contrast, beta, test);

quit;

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Cont..

Test of Linear Combinations Contrast Matrix 0 1 -1 0 0

Chi-Square d.f p-value

1.5884 1 0.2076

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For Testing Ho: beta_1=beta_3

title "beta_1=beta_3";

proc iml;

option={ n, y, n, n, y};

contrast={0 1 0 -1 0};

effects={'z1', 'z2' ,'z3', 'age'};

timeunit={'Years'};

%additive (larynx1, 0.05, timeunit, effects, option, contrast, beta, test);

quit;

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Cont..

Test of Linear Combinations Contrast Matrix 0 1 0 -1 0

Chi-Square d.f p-value

6.9390 1 0.0084

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Thank you

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