Example for Fit Cox Model in a Counting Process Form

Medical researchers want to determine the success rate of recovery from a bone marrow transplant as a treatment for acute leukemia. Recovery depends on factors such as the patient's Risk Category at the time of transplantation, their Disease Stage, and whether their platelet count returned to normal levels. Risk Category and Disease Stage are fixed predictors because they do not change throughout the study. However, a patient's platelet count is a time-dependent predictor because the count can change during the recovery process.

The medical researchers study 137 patients after they receive the transplant and record the number of days they are disease-free. A patient is not disease-free if they die before their platelet count returns to normal or if their leukemia returns after their platelet count returns to normal. A value of Yes indicates a disease-free patient and is a censored observation. A censored observation is when the event does not occur by the end of the observation time.

The data are in the counting process form, which means that multiple rows represent each patient. Each row describes a time interval when the values of all the variables are constant. Time-dependent predictors change between rows. The intervals begin just after the start time and include the end time.

For example, the following table contains the data for the patient with an ID of 1. The observed values of Risk Category and Disease Stage are the same in every row because those predictors are fixed. Because a normal platelet count can change during the study, each patient requires a new row of data whenever this predictor changes. The first row shows that the patient did not have a normal platelet count in the interval of the first 13 days after the transplant. The second row shows that the patient had a normal platelet count from after day 13 until the end of the study on day 2,081.

ID	Risk Category	Start Time	End Time	Disease Free	Normal Platelets	Disease Stage
1	1	0	13	Yes	No	Normal
1	1	13	2081	Yes	Yes	Normal

Note

These data were adapted based on a public data set from Copelan that is in Klein and Moeschberger (2003)¹.

Open the sample data, BoneMarrow.MTW.
Choose Stat > Reliability/Survival > Cox Regression > Fit Cox Model in a Counting Process Form.
In Start Time, enter Start Time. In End Time, enter End Time.
In Censoring column (optional), enter Disease Free. In Censoring value, select Yes.
In Categorical predictors, enter Risk Category, Normal Platelets, and Disease Stage.
Select Options. In Case identification (for subject residuals), enter ID.
Select OK in each dialog box.

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-values for all 3 terms are significant at an α-level of 0.05. Therefore, the medical researchers conclude that the patient's risk category at the time of transplantation, their disease stage, and whether their platelet count is at normal levels all have a statistically significant effect on whether a patient recovers from a bone marrow transplant.

The researchers use the Relative Risks for Categorical Predictors table to assess the risk between different levels of the predictors. For example, the risk of death or recurrence of leukemia among patients with normal platelets is 0.37 times lower than a patient without normal platelets. Moreover, the confidence interval shows that the true risk of death or recurrence for patients with normal platelets could be as little as 0.19 times or as much as 0.7 times less than the risk for patients without normal platelets, at the 95 percent level of confidence. The confidence interval does not contain 1, so the difference between the risk of death or recurrence for patients with and without normal platelets is statistically significant.

Cox Regression: (Start Time, End Time] versus Risk Categor, Normal Plate, ...

Method Cox model type Counting process form Categorical predictor coding (1, 0) Tie adjustment Efron Rows unused 1

Censoring Information Uncensored Censored Percent Units Units Total Censored 83 173 256 67.58% Censoring value: Disease Free = Yes

Regression Equation Risk Score = 0.0 Risk Category_1 - 0.793 Risk Category_2 - 0.033 Risk Category_3 + 0.0 Normal Platelets_No - 1.004 Normal Platelets_Yes + 0.0 Disease Stage_High Risk - 0.696 Disease Stage_Normal

Coefficients Term Coef SE Coef Z-Value P-Value Risk Category 2 -0.793 0.321 -2.47 0.014 3 -0.033 0.325 -0.10 0.919 Normal Platelets Yes -1.004 0.332 -3.02 0.003 Disease Stage Normal -0.696 0.275 -2.53 0.011

Relative Risks for Categorical Predictors Relative Level A Level B Risk 95% CI Risk Category 2 1 0.4524 (0.2409, 0.8495) 3 1 0.9673 (0.5116, 1.8290) 3 2 2.1383 (1.2487, 3.6616) Normal Platelets Yes No 0.3666 (0.1912, 0.7029) Disease Stage Normal High Risk 0.4986 (0.2909, 0.8547) Risk for level A relative to level B

Model Summary Model Log-Likelihood R-sq AIC AICc BIC Without terms -373.30 — 746.59 746.59 746.59 With terms -358.60 11.47% 725.20 725.71 734.88

Goodness-of-Fit Tests Test DF Chi-Square P-Value Likelihood Ratio 4 29.39 0.000 Wald 4 32.47 0.000 Score 4 35.22 0.000

Analysis of Variance Wald Test Source DF Chi-Square P-Value Risk Category 2 9.77 0.008 Normal Platelets 1 9.13 0.003 Disease Stage 1 6.41 0.011

¹ Klein, J.P. & Moeschberger, M.L. (2003). Semiparametric proportional hazards regression with fixed covariates. Survival Analysis: Techniques for Censored and Truncated Data (2nd ed., pp. 243-293). Springer.