Example for Fit Cox Model with Fixed Predictors only

Medical researchers want to determine whether the stage of larynx cancer affects the chance of death. The researchers plan to adjust the analysis for a patient's age. The researchers record the stage and age of 90 male cancer patients. Then, the researchers record the number of years between the first treatment and either the patient's death or the end of the study. Finally, the researchers record whether the patient died.

The medical researchers perform Cox regression to evaluate the relationship between death, age, and the stage of the cancer. The researchers also want to estimate the survival probability for a 60-year-old man for each stage.

Note

These data were adapted based on a public data set from Kardaun that can be found in Klein and Moeschberger (2003)¹. However, the results in this example do not match the textbook because the textbook uses the Breslow method for handling ties and this example uses the Efron method.

Open the sample data, LarynxCancer.MTW.
Choose Stat > Reliability/Survival > Cox Regression > Fit Cox Model with Fixed Predictors only.
In Response, enter Time.
In Censoring column (optional), enter Death.
In Continuous predictors, enter Age.
In Categorical predictors, enter Stage.
Select Graphs and check Display survival plot for predictor values.
From the drop down, select Enter individual values. Enter the following values in the table.

Age Stage

60 I

60 II

60 III

60 IV
Select OK in each dialog box.

Age	Stage
60	I
60	II
60	III
60	IV

Interpret the results

First, the researchers use the goodness-of-fit tests to evaluate the overall fit of the model. The p-values for all 3 tests are below 0.05, so the researchers conclude that the model fits the data well. Then the researchers use the ANOVA table to evaluate the effect of individual terms. The p-value for stage is significant at an α-level of 0.05. Therefore, the medical researchers conclude that the stage of the cancer has a statistically significant effect on the survival of the patient. However, the p-value for age is 0.182, so the effect of age is not significant at an α-level of 0.05. The coefficients for the predictors define an equation that describes the relationship between the stage, the age of the patient, and the survival time.

The researchers use the Relative Risks for Categorical Predictors table to assess the risk between different stages of cancer. For example, the risk of death among patients at stage IV is 5.5 times higher than the risk for patients in Stage I. Moreover, the confidence interval shows that the true risk of death for patients in Stage IV could be as little as 2.4 times or as much as 12.6 times more than the risk for patients at Stage I, at the 95 percent level of confidence. The confidence interval does not contain 1, so the difference between the risk of death for Stage I and Stage IV is statistically significant.

The survival plot displays the survival probability for a 60-year-old man for each stage of cancer over multiple years. A diagnosis of stage IV cancer has the greatest affect on survival probability. The plot shows that after 1 year a 60 year old with stage IV cancer has only a 64% chance of surviving. The other three stages have a probability of 85% or higher. After 2 years, the probability drops to 42% for a patient with stage IV, but remains at 74% or higher for the other three stages.

Method

Cox model type	Fixed predictors only
Categorical predictor coding	(1, 0)
Tie adjustment	Efron

Censoring Information

Uncensored Units	Censored Units	Total	Percent Censored
50	40	90	44.44%

Regression Equation

Stage
I	Risk Score	=	0.0 + 0.01903 Age

II	Risk Score	=	0.1400 + 0.01903 Age

III	Risk Score	=	0.6424 + 0.01903 Age

IV	Risk Score	=	1.706 + 0.01903 Age

Coefficients

Term	Coef	SE Coef	Z-Value	P-Value
Age	0.0190	0.0143	1.33	0.182
Stage
II	0.140	0.462	0.30	0.762
III	0.642	0.356	1.80	0.071
IV	1.706	0.422	4.04	0.000

Relative Risks for Continuous Predictors

	Unit of Change	Relative Risk	95% CI
Age	1	1.0192	(0.9911, 1.0481)

Relative Risks for Categorical Predictors

Level A	Level B	Relative Risk	95% CI
Stage
II	I	1.1503	(0.4647, 2.8477)
III	I	1.9010	(0.9459, 3.8204)
IV	I	5.5068	(2.4086, 12.5901)
III	II	1.6526	(0.6819, 4.0049)
IV	II	4.7872	(1.7825, 12.8566)
IV	III	2.8968	(1.2952, 6.4788)

Model Summary

Model	Log-Likelihood	R-sq	AIC	AICc	BIC
Without terms	-196.86	—	393.73	393.73	393.73
With terms	-187.71	18.65%	383.41	384.30	391.06

Goodness-of-Fit Tests

Test	DF	Chi-Square	P-Value
Likelihood Ratio	4	18.31	0.001
Wald	4	21.15	0.000
Score	4	24.78	0.000

Analysis of Variance

		Wald Test
Source	DF	Chi-Square	P-Value
Age	1	1.78	0.182
Stage	3	17.92	0.000

¹ Klein, J.P. & Moeschberger, M.L. (2003). Examples of survival data: Death times of male laryngeal cancer patients. Survival Analysis: Techniques for Censored and Truncated Data (2nd ed., pp. 9-10). Springer.