Cox Regression using Solver

We now show how to calculate the Cox regression coefficients using Solver for the model:

where X is a list of random variables x₁, …, x_r. This model is equivalent to

where

Now for any subject s in the sample with values x_s1, …, x_sr for the r covariates, we define

where

We assume that time is divided into the following intervals  t₀ < t₁ < ⋅⋅⋅< t_m where t₀ is the start time and t_m is the time of the last observed death. The risk set R_j is the set of all subjects who survive to the time just before t_j. Thus the risk set consists of all those who die at time t_j or who are censored or die after time t_j. Let D_j be the set of all subjects who die at time t_j. n_j = the number of survivors just before time t_j (i.e. the number of elements in R_j) and d_j = the number who die at time t_j (i.e. the number of elements in D_j) and no one dies between times t_j and t_j+1.

We further assume that d_j > 0 for each j (i.e. we won’t assign times t_j for censored data unless a subject also dies at that time).

In fact, initially, we will assume that d_j = 1 for each j. If we consider time as a continuum, then this is a reasonable assumption since two deaths are unlikely to occur at precisely the same time. If we think of time in discrete steps (as we have until now), then this assumption is too restrictive. For now, we will stick with this assumption, but we will return to this issue later.

If d_j = 1 for each j, then the partial likelihood function is

where x_ji is defined to be x_si where s is the unique element in D_j. Similarly, g_j(X) is defined to be g_s(X) in this case.

The natural log of L is then

where

As we have done for logistic regression, our goal is to find the coefficients b₁, …, b_r which maximize LL. First, we show how this is done using Excel’s Solver and then we will use Newton’s Method, which will also provide additional information.

Example 1: Find the coefficients for Cox regression for the clinical trial for 18 patients shown in range B3:E21 of Figure 1. Here each patient is given an experimental drug for the treatment of lung cancer and the number of months of survival is recorded until the patient dies (died = 1) or leaves the trial or has not died when the clinical trial ends (died = 0), along with the patient’s age and the size of the tumor at the start of treatment.

We begin by sorting the data by the number of months of survival, as shown in range G3:J21 of Figure 1. This can be done by using Excel’s sort capability (Data > Sort&Filter|Sort) or by using the Real Statistics formula =QSORTRows(B4:E21). Since we assume that deaths precede censored data, we need to make sure that the patient in range G16:J16 comes before the patient in range G17:J17 (since both survive 62 months).

Figure 1 – Cox Regression (initial state)

We now find the value of LL by placing the array formula =MMULT(I4:J4,$O$4:$O$5) in cell K4, the formula =EXP(K4) in cell L4 and the formula =IF(H4=1,K4-LN(SUM(L4:L$21)),0) in cell M4. We then highlight the range K4:M21 and press Ctrl-D. Finally, we place the formula =SUM(M4:M21) in cell M22.

We see from Figure 1 that when b₁ = b₂ = 0, LL = -15.2733. We next use Excel’s Solver capability by selecting Data > Analysis|Solver. We fill in the dialog box that appears as shown in Figure 2 and then press the OK button.

Figure 2 – Solver dialog box

Solver then adjusts the values of the coefficients (range O4:O5) to maximize the value of LL (cell M22). The results are shown in Figure 3.

As we can see, the coefficient corresponding to age is .202465 and the coefficient corresponding to size is .094457. We also see that the value of LL increased from -15.2733 to -11.0447.

Figure 3 – Cox Regression (Solver)