This is a powerful way of expressing how much observing something makes the probability of something else stronger or weaker. It has its roots in general medicine and epidemiology in diagnosis and the disease model which may be partly why I don’t see it a lot in our therapy quantitative research world. This may be partly because a lot of our research output (not mine I think!) likes to ride with the disease model but to ignore the realities and logic of this. However, it’s useful to know about it.
Details #
Likelihood ratios (LRs) may be positive: finding something happening may make the probability of the other thing being present or coming more likely, or they can be negative: finding the indicator making the other thing less likely.
Let’s put this into a therapy framework. Perhaps we are interested in whether being able to observe a particular kind of alliance rupture is associated with early termination of therapy defined as not returning after that session. Then formally, mathematically, the positive LR is:
$$\frac{(\text{rupture observed} \text{ | } \text{early termination})}{(\text{rupture observed} \text{ | } \text{no early termination)}}$$
In English that is “The probability, in our series of clients, that rupture was observed in sessions that turned out to be early termination sessions, divided by the probability that rupture was observed but that client came back for at least one more session, i.e. that the rupture was not in a termination session.”
Let’s say a very diligent research team looked at 100 therapies and rated sessions for the presence or absence of the defined rupture not knowing whether or not the session was a termination session and the results were that there were 21 of the 100 sessions in which the rupture was observed and of these 18 were termination sessions and 3 were not and of the remaining 79 sessions, none were termination sessions. Now our positive LR is this.
$$\frac{18 / 21}{3 / 79}\cong \text{22.6}$$
(That $\cong$ sign means “approximately equal to”.)
So what? Well now we go to odds ratios. Odds ratios are the bread and butter of gambling on things and they make it easy to handle probabilities with simple multiplication and division which, perhaps counterintuitively, we can’t with probabilities themselves. The odds of something are:
`$$\frac{\text{probability of it happening}}{1-\text{probability of it happening}}$$
So the odds of termination in a session were:
$$\frac{18}{100-18} \cong \text{.2195}$$
The nice thing about odds and LRs is that the “posterior odds” or something given that you have observed something else and have the LR are
$$\text{posterior odds} = \text{prior odds} * LR$$
So the odds of early termination if observing a rupture (i.e. posterior to that observation of the rupture) are:
$$\text{posterior odds} = \text{prior odds} * LR \cong \text{.2195} * \text{22.6} \cong \text{4.95}$$
And, given that we can work out the probability of an event from its odds as:
$$\frac{\text{odds}}{1 + \text{odds}} \cong \frac{4.95}{5.95} \cong \text{.832}$$
our probability of termination given such a rupture has now gone up from 21% to 83%, a pretty huge increase.
Thats a bit of cumbersome way to get to the increase in probability given the LR. A useful mnemoic is “1, 2, 3 = 15%, 30%, 45%) i.e. LRs of 1, 2 or 3 increase the probability of an event (if it was not very rare or overwhelmingly common) by those multiples of 15%: 15%, 30% and 45%. We can see that our LR of 23.6 leading to an increase of 62% fits with that.
This is a very artificial example as there are all sorts of issues arising from the fact that across the 100 therapies there will have been far more than 100 sessions but I hope by pulling the issue into our realm it is easier to follow. However, the LR remains something that you are most likely to see in the context of screening. The LR of a screening test is:
$$\frac{\text{sensitivity}}{1 – \text{specificity}}$$
Try also #
- Odds ratio
- Probability
- Screening
- Sensitivity
- Specificity
Chapters #
Not covered in the OMbook.
Online resources #
None currently.
Dates #
First created 22.vii.25, link to new entry for odds ratio added 24.vii.25.
