6  Parametric Survival Regression

mindmap
  root((Regression 
  Analysis)
    Continuous <br/>Outcome Y
      {{Unbounded <br/>Outcome Y}}
        )Chapter 3: <br/>Ordinary <br/>Least Squares <br/>Regression(
          (Normal <br/>Outcome Y)
      {{Nonnegative <br/>Outcome Y}}
        )Chapter 4: <br/>Gamma Regression(
          (Gamma <br/>Outcome Y)
      {{Bounded <br/>Outcome Y <br/> between 0 and 1}}
        )Chapter 5: Beta <br/>Regression(
          (Beta <br/>Outcome Y)
      {{Nonnegative <br/>Survival <br/>Time Y}}
        )Chapter 6: <br/>Parametric <br/> Survival <br/>Regression(
          (Exponential <br/>Outcome Y)
          (Weibull <br/>Outcome Y)
          (Lognormal <br/>Outcome Y)
    Discrete <br/>Outcome Y

Figure 6.1

Heads-up on the properties of the cumulative distribution function!

It is important to clarify that a valid CDF \(F_Y(y)\) fulfils the following properties:

1. \(F_Y(y)\) must never be a decreasing function.
2. Given that \(F_Y(y) : \mathbb{R} \rightarrow [0, 1]\), it must never evaluate to be \(< 0\) or \(> 1\). The output of a CDF is a cumulative probability, hence the previous bounds.
3. When \(y \rightarrow -\infty\), if follows that \(F_Y(y) \rightarrow 0\).
4. When \(y \rightarrow \infty\), if follows that \(F_Y(y) \rightarrow 1\).

Now, in the case of a CDF corresponding to a continuous random variable \(Y\), there is an additional handy property that relates the CDF \(F_Y(y)\) to the PDF \(f_Y(y)\):

\[ f_Y(y) = \frac{\mathrm{d}}{\mathrm{d}y} F_Y(y). \tag{6.1}\]

Equation 6.1 indicates that the PDF of \(Y\) can be obtained by taking the first derivative of the CDF with respect to \(y\).