Definitions for maths#

work in progress

All of the quantities we will define are only valid in the time range covered by our data, which is \(0\leq t \leq t_{\mathrm{ne}}\) for the no-effect time \(t_{\mathrm{ne}}\).

Probability#

Firstly we will define some probability distributions in terms of \(x\), which is a given modified Rankin Scale (mRS), and \(t\), the time from onset to treatment.

  • \(P(\mathrm{mRS}=x\ |\ t)\), the probability distribution of mRS.

  • \(P(\mathrm{mRS}\leq x\ |\ t)\), the cumulative probability.

In all cases, we know that:

  • The probabilities of a condition being met and it not being met sum to 1: \( P(\mathrm{mRS}>x\ |\ t) = 1 - P(\mathrm{mRS}\leq x\ |\ t) \)

  • The probability of mRS\(\leq6\) is always equal to 1: \(P(\mathrm{mRS}\leq6\ |\ t)=1\) at all \(t\).

Odds#

The odds are an alternative way of expressing the likeliness of an event occurring. Odds are expressed in terms of how much more (or less) likely it is that a given outcome will be attained rather than any other outcomes.

  • \(\mathrm{odds} = \frac{\mathrm{probability\ of\ this\ outcome}}{\mathrm{probability\ of\ other\ outcomes}} \phantom{g} \mathrm{(single\ patient)} \phantom{gap} = \frac{\mathrm{number\ of\ this\ outcome}}{\mathrm{number\ of\ other\ outcomes}} \phantom{g} \mathrm{(group\ of\ patients)}\)

  • \(O(\mathrm{mRS}\leq x\ |\ t) = \frac{P(\mathrm{mRS}\leq x\ |\ t)}{P(\mathrm{mRS}>x\ |\ t)}\)

Special cases:

  • When only half of the outcomes are good, i.e. probability=0.5, then odds=1 and log(odds)=0.

  • Odds and log(odds) for \(\mathrm{mRS}\leq6\) are not defined for any \(t\). This is because \(P(\mathrm{mRS}\leq6)=1.0\) at all times, and so: $\(O(\mathrm{mRS}\leq6) = \frac{P(\mathrm{mRS}\leq6)}{1-P(\mathrm{mRS}\leq6)} = \frac{1}{0}\)$

Here we will calculate odds directly from probabilities, and so the odds will be given as non-integer positive numbers rather than fractions.

We will also consider the natural log of odds, i.e. \(\log_{e}(\mathrm{odds})\). Whenever this notebook says “log”, it means natural log (base \(e\)).