CHAPTER 23 Survival Regression 333

Here are some important points about Figure 23-2:»

» ^{Figure 23-2a shows a typical survival curve. It’s not defined by any algebraic}

formula. It just graphs the table of values obtained by a life-table or Kaplan-

Meier calculation.»

» ^{Figure 23-2b shows how the baseline survival curve is flexed by raising every}

baseline survival value to a power. You get the lower curve by setting h = 2

and squaring every baseline survival value. You get the upper curve by setting

h = 0.05 and taking the square root of every baseline survival value. Notice

that the two flexed curves keep all the distinctive zigs and zags of the baseline

curve, in that every step occurs at the same time value as it occurs in the

baseline curve.

• ^{The lower curve represents a group of participants who had a worse}

survival outcome than those making up the baseline group. This means

that at any instant in time, they were somewhat more likely to die than a

baseline participant at that same moment. Another way of saying this is

that the participants in the lower curve have a higher hazard rate than the

baseline participants.

• ^{The upper curve represents participants who had better survival than}

a baseline person at any given moment — meaning they had a lower

hazard rate.

Obviously, there is a mathematical relationship between the chance of dying at

any instant in time, which is called hazard, and the chance of surviving up to some

point in time, which we call survival. It turns out that raising the survival curve to

the h power is exactly equivalent to multiplying the hazard curve by the natural

logarithm of h. Because every point in the hazard curve is being multiplied by the

same amount — by Log(h) — raising a survival curve to a power is referred to as

a proportional hazards transformation.

But what should the value of h be? The h value varies from one individual to

another. Keep in mind that the baseline curve describes the survival of a perfectly

average participant, but no individual is completely average. You can think of

every participant in the data as having her very own personalized survival curve,

based on her very own h value, that provides the best estimate of that partici-

pant’s chance of survival over time.

Seeing how predictor variables influence h

The final piece of the PH regression puzzle is to figure out how the predictor vari-

ables influence h, which influences survival. As you likely know, all regression

procedures estimate the values of the coefficients that make the predicted values

agree as much as possible with the observed values. For PH regression, the