6  Crunchified Parametric Survival Regression

Fun fact!

Crunchified! Extra crunchy, borderline noisy; could probably shatter glass.

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

Definition of cumulative distribution function

Let \(Y\) be a random variable either discrete or continuous. Its cumulative distribution function (CDF) \(F_Y(y) : \mathbb{R} \rightarrow [0, 1]\) refers to the probability that \(Y\) is less or equal than an observed value \(y\):

\[ F_Y(y) = P(Y \leq y). \tag{6.1}\]

Then, we have the following by type of random variable:

  • When \(Y\) is discrete, whose support is \(\mathcal{Y}\), suppose it has a PMF \(P_Y(Y = y)\). Then, the CDF is mathematically represented as:

\[ F_Y(y) = \sum_{\substack{t \in \mathcal{Y} \\ t \leq y}} P_Y(Y = t). \tag{6.2}\]

  • When \(Y\) is continuous, whose support is \(\mathcal{Y}\), suppose it has a PDF \(f_Y(y)\). Then, the CDF is mathematically represented as:

\[ F_Y(y) = \int_{-\infty}^y f_Y(t) \mathrm{d}t. \tag{6.3}\]

Note that in Equation 6.2 and Equation 6.3, we use the auxiliary variable \(t\) since we do not compute the summation or integral over the observed \(y\) given its role on either the PMF or PDF. Therefore, we use this auxiliary variable \(t\).

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 \(f_Y(y)\):

\[ f_Y(y) = \frac{\mathrm{d}}{\mathrm{d}y} F_Y(y). \tag{6.4}\]

Equation 6.4 indicates that the PDF of \(Y\) can be obtained by taking the first derivative of the CDF with respect to \(y\).