Log · Entry 05
What is the cost of β?
To recap. Eudaemon turns a person’s records into evidence strong enough to act on, and turns the goals a person brings into questions that evidence can answer. Entry 02 showed that correlations only warrant predictions, and that advice needs a directional cause (e.g. late coffee causes a later bedtime). These directional causes can be established by repeatedly trialling an intervention, with a coin deciding each time whether said intervention happens. Entry 03 introduced β, the size of the effect an intervention has (e.g. how many minutes does a late coffee delay bedtime?), and entry 04 showed how frequently one could trial an intervention per day, and therefore how much sooner a trial can establish β. This entry looks to bring those variables together in an equation such that we can define the time taken to establish β in days. Put simply, this equation answers the question: what is the cost of β?
Cost vs. benefit
The blueprint for Eudaemon refers to an “experiment engine”. Its function is to prompt the user to run N-of-1 trials on themselves, letting a coin toss dictate a discrete action that day: hot shower or cold shower? normal coffee or decaf? to gym or not to gym? These trials are what is required to establish directional causes; however, they are a burden on the user, as a caffeine-free morning followed by the gym and a cold shower may not be all that palatable on a dark February morning. The user may be asking themselves: how much longer do I have to do this? Hence, before building the experiment engine, it would seem wise to do some cost-benefit analysis. The benefit, how much better the advice received from Eudaemon on a goal or decision becomes, will be the focus of future entries; the cost, in terms of time, is defined below. If the cost is low and/or the benefit is high, the experiment engine becomes central, and building it the priority.
The equation for cost
Tap or click any of the terms below to find their definition.
- Tβ, the days to β
- The days a trial must run to establish which direction a cause runs and how big its effect is: does a late coffee cause a later bedtime and, if so, by how many minutes?
- β, the size of the effect
- How much a change in one variable moves another. For example, the difference, in minutes, between the average bedtime after a true coffee and the average bedtime after a decaf. Entry 03, The moving target
- σ, the standard deviation
- How far one night’s reading typically sits from its average, in minutes. The more an outcome varies by itself, the longer a given β takes to show. Standard deviation, explained
- s, the split
- People may find the distribution of interventions dictated by a fair coin intolerable (e.g. too many cold showers or decaf coffees), and therefore want an unfair one. The price of that luxury is a longer Tβ, because the rarer average has to catch up. s is the share of occasions on which the coin chooses the more favourable intervention (i.e. its bias towards true coffees and warm showers); ½ for a fair coin.
- ρ, how much one day repeats the last
- From 0, days varying freely, to 1, each day repeating the one before. The more the days repeat, the fewer independent comparisons a record holds. When a coin decides, this term becomes 1. Entry 02, Watch or try
- f, clean comparisons a day
- How often the coin can be tossed in a day before one toss’s effect spills into the next reading. At 3 a day an answer arrives 3 times sooner than at 1 a day. Entry 04, How fast can a trial reach an answer?
- 7.85, the convention
- The convention most medical trials follow. If the coffee did nothing, there’d be only a 5% (1 in 20) chance of the trial wrongly finding an effect; if it really does move bedtime by the amount being looked for, there’s an 80% chance the trial detects it. 7.85 is (1.96 + 0.84)², the two numbers that set those odds. Lehr turned it into a rule of thumb for sizing trials in 1992. Lehr, Statistics in Medicine, 1992
- M, the next entry
- The subject of the next entry.
NB: M will be discussed in the next entry.
How much is known
On any one person, very little. ρ has been measured in a small number of studies, with values as high as .87 depending on the person and what was measured, and most medical N-of-1 trials ignore it (83.8% of 115 reviewed). β against σ has been measured for single interventions people actually trialled, but nobody has measured how it spreads across one person’s many questions. f has never been measured: a single trial’s protocol (HeartSteps) set 5 a day, of which about 4 were usable; one’s life may give higher numbers. M hasn’t been measured either. s is a choice, not a measurement, and 7.85 is a convention. Nobody has put these together on one person and turned them into a number of days or a cost. Even typical values, measured on one person across many questions, would do that: they’d put a figure on the cost half of the cost-benefit analysis, and with the benefit, decide whether the experiment engine becomes Eudaemon’s focus.
Next. M.