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The contents and notes are from Coursera course "AI for Medical Prognosis".

What is Medical Prognosis

• prognosis = Predicting the risk of a future event
  • risk of illness
  • survival probability with illness
• examples
  • use clinical history, physical examinations, labs & imaging, etc. to predict the risk score/survival probability

Build & Evaluate Risk Models

Risk model types

• linear risk model
  • risk score = a linear sum of the coefficient x feature
• tree-based model
  • be cautious about missing data
    • identifying missing data types
      • missing completely at random = missingness not dependent on anything → no bias
      • missing at random = missingness dependent only on the available information (for somewhat feature criteria)
      • missing not at random = missingness dependent on unavailable information
    • use data imputation to handle missing data (Data preprocessing#^594d05)

Risk model evaluation

• Since the risk score can be any value, it needs to be compared in pairs. In a group of predicted patients,

  • concordant pair: if patient A has a higher risk outcome than patient B, and A’s risk score is higher than B’s risk score
  • risk ties: patient's risk scores are the same
  • permissible pair: patients’ outcomes are not the same

• Thus:

  • +1 for a permissible pair that is concordant
  • +0.5 for a permissible pair for risk tie

• C-index (max 1, 0.5 for any random constant risk)

  $$\frac{\mathrm{\#}concordantpairs+0.5\times \mathrm{\#}riskties}{\mathrm{\#}permissiblepairs}$$

Build & Evaluate Survival Model

What does survival model tell

• What is the probability of survival past any time t?
• With survival functions: S(t) = Pr(T>t)
  • always decreasing from 1 to 0: longer the t, harder to survive

Survival data structure

• in survival data, the labels are amounts of time to event
• censoring observations: no observations of events happening in the specified time period:
  • end-of-study censoring (no event)
  • loss-to-follow-up censoring (patients withdraw)
• right censoring = the time to events is only known to exceed a certain value (e.g. 12 months → 12m +)

Estimate survival

Let $i$ = 1, ..., $n$ be the cases, and let $T_i$ be the time when $i$ was censored or an event happened. Let $e_i=1$ if an event was observed for $i$ and 0 otherwise. Then let $X_t=\{i:T_i>t\}$, and let $M_t=\{i:e_i=1\text{ or }{T}_i>t\}$. The estimator will be: $$

\hat{S}(t) = \frac{|X_t|}{|M_t|} $$

• Kaplan Meier estimate: the probability of survival past t months with censored observations $$ S(t) = \prod 1- Pr(T=i | T >= i ) = \prod_{t_i \leq t} (1 - \frac{d_i}{n_i}) $$
• $t_i$ are the events observed in the dataset
• $d_i$ is the number of deaths at time $t_i$
• $n_i$ is the number of people who we know have survived up to time $t_i$.

Survival model types

• Hazard functions

  • Hazard: what’s a patient’s immediate risk of death if they make it to time t (risk of death if aged t)

    $$\lambda (t)=\prod Pr(T=t|T>=t)$$

  • cumulative hazard $$ \Lambda (t) = \int _0
    { #t}
    \lambda(u) du $$

  • individual hazard $$ \lambda_{individual} (t) = \lambda_{0} (t) exp (\sum_i {B_i X_i}) $$ where ${\lambda }_0(t)$ is a baseline hazard, B_i is coefficient of factor X_i

  • relation between survival and hazard: $$ S(t) = exp(- \int _0
    { #t}
    \lambda(u) du) $$

  $$\lambda (t)=-\frac{S^{\prime }(t)}{S(t)}$$
• Survival trees
  • Nelson Aalen Estimator: estimate the cumulative hazard of the population
  $$$$

{ #t}
\frac{d_i}{n_i}
$$

- Mortality score = a single score value of cumulative hazard at the event times

Survival model evaluation

• as a variation of evaluating prognostic models, with a slightly different definition of a concordant pair, a risk tie, and a permissible pair with survival data
  • here the risk outcome is the time to the event
  • concordant pairs: the patients with worse risk outcome (earlier event) have higher risk scores
  • permissible pairs: patients’ event times are not the same
    
  • Harrell’s C-Index: same as the C-index formula