Explaining Statistical Significance and Confidence in A/B tests

If you've been presenting or listening to A/B test results (from online or offline tests) for a while, you'll probably have been asked to explain what 'confidence' or 'statistical significance' is.

A simple way of describing the measure of confidence is:

The probability (or likelihood) that this result (win or lose) will continue.

100% means this result is certain to continue, 50% means it's 50-50 on if it will win or lose. Please note that this is just a SIMPLE way of describing confidence, it's not mathematically rigorous.

Statistical significance (or just 'significance') is achieved when the results reach a certain pre-agreed level, typically 75%, 80% or 90%.

It's worth mentioning that confidence doesn't give us the likelihood that the magnitude of the win will remain the same.  You can't say that a particular recipe will continue to win at +5.3% revenue per visitor (it might rise to 5.5%, or fall to 4.1%), but you can say that it will continue to outperform control.  As the sample size increases, the magnitude of the win will also start to settle down to a particular figure, and if you reach 100% confidence then you can also expect the level of the win to settle down to a specific figure too.

A note: noise and anomalous results in the early part of the test may lead you to see large wins with high confidence.  You need to consider the volume of orders (or successes) and traffic in your results, and observe the daily results for your test, until you can see that the effects of these early anomalies have been reduced.

Online testers frequently ask how long a test should run for - what measures should we look at, and when are we safe to assume that our test is complete (and the data is reliable).  I would say that looking at confidence and at daily trends should give you a good idea.

It's infuriating, but there are occasions when more time means less conclusive results: a test can start with a clear winner, but after time the result starts to flatten out (i.e. the winning lift decreases and confidence falls).  If you see this trend, then it's definitely time to switch the test off.

Conversely, you hope that you'll see flattish results initially, and then a clear winner begin to develop, with one recipe consistently outperforming the other(s).  Feeding more time, more traffic and more orders into the test gives you an increasingly clear picture of the test winner; the lifts will start to stabilise and the confidence will also start to grow.  So the question isn't "How long do I keep my test running?" but "How many days of consistent uplift do you look for?  And what level of confidence do I require to call a recipe a winner?"

What level of confidence do I need to call a test a winner?

Note that you may have different criteria for calling a winner compared to calling a loser.  I'm sure the mathematical purists will cry foul, and say that this sounds like cooking the books, or fiddling the results, but consider this:  if you're looking for a winner that you're going to implement through additional coding (and which may require an investment of time and money) then you'll probably want to be sure that you've got a definite winner that will provide a return on your money, so perhaps the win criteria would be 85% confidence with at least five days of consistent positive trending.

On the other hand, if your test is losing, then every day that you keep it running is going to cost you money (after all, you're funneling a fraction of your traffic through a sub-optimal experience).  So perhaps you'll call a loser with just 75% confidence and five days of consistent under-performing.  Here, the question becomes "How much is it going to cost me in immediate revenue to keep it running for another day?" and the answer is probably "Too much! Switch it off!!"  This is not a mathematical pursuit, along the lines of "How much money do we need to lose to achieve our agreed confidence levels?", this is real life profit-and-loss.

In a future blog post, I'll provide a more mathematical treatment of confidence, explaining how it's calculate from a statistical standpoint, so that you have a clear understanding of the foundations behind the final figures.

Geometry: Changing the steepness of a hill by zig-zagging

Even if a hill or a road is too steep to climb, there is still a way to make progress, and that's by zig-zagging.  Instead of going directly up the hill in the shortest route, it's possible to take an angled approach up the slope, increasing the path length, but making the climb angle less steep.

It is easier to outline this in a simplified diagram:

This triangular prism represents the face of a hill.
The angle directly up the hill is α and is shown in the pink triangle.
The angle of approach (i.e. the degree of zigzag, the deviation from the straight-up route) is ß, and is shown by the red and pink triangles combined.
The resultant angle (i.e. the actual angle of ascent) is δ and is shown by the blue triangle.

Each of the triangles is right-angled, so standard trigonometry functions can be applied (I haven't shown all the right angles in the diagram, but it is a regular triangular prism).

Considering each of these three angles in turn:  the way to get to a simplified expression for δ is to express the three angles in the fewest numbers of lines.  It's possible to express α, ß and δ in terms of the external dimensions of the prism (let's call them x, y and z) but this just leads to incompatible expressions that can't be simplified or combined.

α

  



ß


δ


The strategy here is to substitute for y and p in the expression for δ, and then to simplify.

Firstly, rearrange the expressions for α and ß to make y and p the subjects of those equations.


A very simple and elegant equation:  the angle of ascent depends on how steep the hill is, and the amount by which you zigzag, and is completely independent of the size of the hill (i.e. none of the lengths are relevant in the calculation).

A few sanity checks:

If ß is zero, or close to zero, then δ approaches α - i.e. if you don't zigzag, then you approach the hill at its actual angle.

If ß approaches 90 degrees, then  δ approaches zero - you hardly climb at all, but you'll need to travel much further to climb the hill.  In fact, as ß tends towards 90 degrees, path length p tends to infinity.

If α increases, then δ increases for constant ß (something that was worth checking).

An interesting note:

At first glance, you may think that a path (or zigzag) angle of 45 degrees would reduce the angle of ascent by half (e.g. from 60 degrees to 30 degrees), simply because 45 is half of 90.  However, this isn't the case.  In order to get a reduction of a half, cos ß needs to equal 0.5.  If cos ß = 0.5, then ß = 60 degrees.  A much larger deviation from the straight-up angle is needed.

In conclusion

This question was first put to me when I was in high school (a few years ago now) and it's been nagging at me ever since.  I'm pleased to have been able to solve it, and I'm pleased with how surprisingly simple the final expression is (previously, my 3-D geometry and logic weren't quite up to scratch, and I ended up going round in circles!).

Calculating the tetrahedral bond angle

Calculating the Tetrahedral Bond Angle

Every Chemistry textbook which covers molecular shapes will state with utmost authority that the bond angle in tetrahedral molecules is 109.5 degrees. Methane (CH4) is frequently quoted as the example, shown to be completely symmetrical and tetrahedral. And then the 109.5 degrees.  There's no proof given (after all, Chemistry textbooks aren't dealing with geometry, and there's no need to show something just for the sake of mathematical proof - rightly, the content is all about reactivity and structure).  However, the lack of proof has bugged me on-and-off for about 20 years, and recently I decided it was time to do something about it and prove it for myself.

There are various websites showing the geometry of a tetrahedron and how it relates to a cube, and those sites use the relationship between a cube and a tetrahedron in order to calculate the angle, but I'm going to demonstrate an alternative proof using solely the properties of a tetrahedron  - its symmetry and its equilateral triangular faces.

To start with, calculate the horizontal distance from one of the vertices to the centre of the opposite triangular face (the point directly below the central 'atom').  In this diagram, E is the top corner, D is the central "atom" (representing the centre of the tetrahedron) and C is the point directly below D, such that CDE is a straight line, and C is the centre of the shaded face (the base).

This gives a large right-angled triangle ACE, where the hypotenuse is one edge of the tetrahedron (length AE = l); one side is the line we'll be calculating (length AC, using the triangle ABC); and the third, CE, is the line extending from the top of the tetrahedron through the central atom down to the centre of the base.

In triangle ABC, length AB = l/2, angle A is 30 degrees, angle B is 90 degrees.  We need to calculate length AC:

cos 30 = l/2 / AC
AC = l /2 cos 30

Since we have two sides and an angle of a right-angled triangle, we can determine the other two angles; we're primarily interested in the angle at the top, labelled α.

sin α = AC / l

And as we know that AC = 1 / 2 cos 30 this simplifies to

sin α = 1 / (2 cos 30)

Evaluating:  1 / (2 cos 30) = 0.5773

sin α = 0.5773
α = 35.26 degrees.

Looking now at the triangle ADE which contains the tetrahedral bond angle at D:  the bond angle D can be calculating through symmetry, since ADE is an isosceles triangle.

D = 180 - (2*35.26) = 109.47 degrees, as we've been told all along.

QED

If you find this post helpful, you may also be interested in:

Calculating the packing density of closely-packed spheres
Calculating the packing density of closely-packed circles
How a Chemistry graduate got into online A/B testing

How did a Chemistry Graduate get into Online Testing?

When people examine my CV, they are often intrigued by how a graduate specialising in chemistry transferred into web analytics, and into online testing and optimisation.  Surely there's nothing in common between the two?

I am at a slight disadvantage - after all, I can't exactly say that I always wanted to go into website analysis when I was younger.  No; I was quite happy playing on my home computer, an Acorn Electron with its 32KB of RAM and 8-bit processor running at 1MHz, and the internet hadn't been invented yet.  You needed to buy an external interface just to connect it to a temperature gauge or control an electrical circuit - we certainly weren't talking about the 'internet of things'.  But at school, I was good at maths, and particularly good at science which was something I especially enjoyed.  I carried on my studies, specialising in maths, chemistry and physics, pursuing them further at university.  Along the way, I bought my first PC - a 286 with 640KB memory, then upgraded to a 486SX 25MHz with 2MB RAM, which was enough to support my scientific studies, and enabled me to start accessing the information superhighway.

Nearly twenty years later, I'm now an established web optimization professional, but I still have my interest in science, and in particular chemistry.  Earlier this week, I was reading through a chemistry textbook (yes, it's still that level of interest), and found this interesting passage on experimental method.  It may not seem immediately relevant, but substitute "online testing" or "online optimisation" for Chemistry, and read on.

Despite what some theoreticians would have us believe, chemistry is founded on experimental work.   An investigative sequence begins with a hypothesis which is tested by experiment and, on the basis of the observed results, is ratified, modified or discarded.   At every stage of this process, the accurate and unbiased recording of results is crucial to success.  However, whilst it is true that such rational analysis can lead the scientist towards his goal, this happy sequence of events occurs much less frequently than many would care to admit. 

I'm sure you can see how the practice and thought processes behind chemical experiments translates into care and planning for online testing.  I've been blogging about valid hypotheses and tests for years now - clearly the scientific thinking in me successfully made the journey from the lab to the website.  And the comment that the "happy sequence of experiment winners happen less frequently than many would care to admit" is particularly pertinent, and I would have to agree with it (although I wouldn't like to admit it).  Be honest, how many of your tests win?  After all, we're not doing experimental research purely for academic purposes - we're trying to make money, and our jobs are to get winners implemented and make money for our companies (while upholding our reputations as subject-matter experts).

The textbook continues...

Having made the all important experimental observations, transmitting this information clearly to other workers in the field is of equal importance.   The record of your observations must be made in such a manner that others as well as yourself can repeat the work at a later stage.   Omission of a small detail, such as the degree of purity of a particular reagent, can often render a procedure irreproducible, invalidating your claims and leaving you exposed to criticism.   The scientific community is rightly suspicious of results which can only be obtained in the hands of one particular worker!

The terminology is quite subject-specific here, but with a little translation, you can see how this also applies to online testing.  In the scientific world, there's a far greater emphasis on sharing results with peers - in industry, we tend to keep our major winners to ourselves, unless we're writing case studies (and ask yourself why do we read case studies anyway?) or presenting at conferences.  But when we do write or publish our results, it's important that we do explain exactly how we achieved that massive 197% lift in conversion - otherwise we'll end up  "invalidating our claims and leaving us exposed to criticism.  The scientific community [and the online community even moreso] is rightly suspicious of results which can only be obtained in the hands of one particular worker!"  Isn't that the truth?

Having faced rigorous scrutiny and peer review of my work in a laboratory, I know how to address questions about the performance of my online tests.   Working with online traffic is far safer than handling hazardous chemicals, but the effects of publishing spurious or inaccurate results are equally damaging to an online marketer or a laboratory-based chemist.  Online and offline scientists alike have to be thoughtful in their experimental practice, rigorous in their analysis and transparent in their methodology and calculations.  

Excerpts taken from Experimental Organic Chemistry: Principles and Practice by L M Harwood and C J Moody, published by Blackwell Scientific Publications in 1989 and reprinted in 1990.

Another day I haven't used Algebra

So, there's a meme floating around Facebook, which says, "Well, another day has passed, and I still haven't used algebra."  Really?  If it's true, it's not something to be especially proud of.  And the likelihood is that it's not true anyway.

For starters, there are many things that I learned at school that I don't use on a daily basis any more.  Foreign languages, for a start (although I probably do use them more than I realise).  Do I regularly apply the map-reading skills I learned at school? We have satnavs and apps for that.  And do I refer the Stuarts and the Tudors?  I suppose I should probably proudly announce that I haven't once consulted a history book this week, and rile all the historians I know.  Somehow though, Maths - probably due to its apparent difficulty or complexity - is seen as something that we should abandon, forget or even be proud of ignoring:

"Why do they make us learn math? It's not like I'll ever use it."
"Yeah, it's not like math teaches you how to work out complex problems logically."

However, Maths (and to some extent algebra) still permeates many areas of our life.  If you want to cook a meal (and you might), then you'll need to know when to start cooking it, in order to achieve a particular mealtime.  Or you might just start cooking as soon as you get home, and eat it as soon as it's ready.  But when will that be?  How long will it take you to get home if you drive at 30 mph?  40 mph?  Are you so sure that another day has passed and you really haven't used algebra?

And then there are those delightful puzzles on Facebook.  You know the sort - if three buckets are equal to 30, and two buckets and two spades are equal to 26, and a bucket and a spade and a flag are equal to 24, what's a flag worth?  I really don't think it's possible to solve that problem without using algebra (call it what you will).  How do you solve those problems?  Here's some help on BODMAS problems (or PEMDAS, if you're from the US).

Once you assign a numerical value (or a time, or a price) to an item (or a distance), and then start doing any sort of calculation on it, you are doing algebra.  Have you ever wondered which was better value in the sales?  The Black Friday sales?  The pre- or post-Christmas sales?  3 for 2 offers? Or buy-one-get-one-free? Or buy-one-get-one-half-price?

And if you have a £10 note in your pocket, and you want to know how many widgets you can buy without overspending... you used algebra.  I think it's fair to say that so far today, I have used algebra numerous times - you might even say X times.

Doctor Who: Sea Devils

Starring John Pertwee (1970-74 era)

I watched this story after the Peter Davison story I reviewed recently (Warriors of the Deep) - clearly I should have watched it first in order to fully grasp the chronology of the Sea Devils and other sub-aquatic life forms (even time travellers tell their stories in chronological order most of the time).  This story features tense atmospheric locations in contrast to the studio-driven episodes with Peter Davison, and, as the DVD notes point out, was filmed with considerable co-operation of the Royal Navy (who provided stock footage royalty-free, and whose staff provided many of the extras for the naval base scenes).  The range of footage of a submarine and helicopter included in the episodes lend the story a sense of realism and scale.

I selected this story from the DVDs on offer at my local charity shop as I've not previously seen the Master in his truly scary form.  Forgive me, but apart from one exception, I've never found John Simms' Master to be scary - he's always been too funny. Even Derek Jacobi managed more presence in his single episode than John Simms ever did - with the one exception during "The Sound of the Drums"/"The Last of the Time Lords".  The subtlety in the portrayal of his behind-the-scenes violence towards his wife and the Jones family, combined with his seemingly blasé approach to everything else was decidedly scary.  So, I was very interested to see how Roger Delgado played the role (I initially had him confused with a distant memory of Anthony Ainley as the Master, but I was still interested to see any previous 'classic' version of this complex character).

Anyway:  the story starts with the Doctor visiting the Master on an island prison.  I'll be honest - I very quickly guessed that the Master is in fact running the prison (the recent BBC Sherlock episode where Sherlock visits his sister in prison is a modern version of the same theme).  The Master is a charismatic, hypnotic character with a considerable degree of repressed anger - and he's not cracking jokes and twirling around like John Sims.  He comes across as a strategic thinker - being in prison isn't going to thwart his plans, he's thinking long-game, big picture.  The storyline is similar to the Master's approach - it too has a long developing time, moving the characters into position and building the tension gradually.  I like this approach - in contrast to the modern day "wrap it up in 40 minutes and then thrown in a bit of 'arc' at the end" which is now becoming frustratingly cliched.

I have to say that one of the most unfortunate parts of the episodes is the soundtrack.  It's loud, and it isn't very musical.  I guess the sound engineering and recording team were having fun trying out all the new sounds they could produce, but it's overpowering and intrusive, and it detracts significantly from the atmosphere.  One of the most tragic cases is during a fight scene between the Master and the Doctor.  The two characters duel with swords, in an old stone fortress on an isolated island; there's a sense of history and a clash of the titans.  And instead of drama and atmosphere, the soundtrack is an anachronism of burps and whistles which sound like an 8-bit computer struggling to run properly.  

As I said, the plot takes its time - there's a real sense that the Master is quietly and covertly carrying out his plot while the Doctor struggles to understand it, but pieces together clues from the other events going on.  Both are geniuses - the Master cobbles together a device for contacting the Sea Devils, while the Doctor demonstrates his ability to manufacture a radio transmitter from a few spare parts.  

The Master works with cunning and stealth to execute his plot; the Doctor has to negotiate his way past, through and around the Royal Navy - until he is imprisoned by the Master during one of his many visits to the prison.   The Doctor is released by his companion, Jo, and the two of them hurriedly escape towards the island's coastline.  The Master and the prison officers chase them down (there's an odd and almost comical scene where the prison guards use a Citroen 2CV across country), and as they reach the coastline, the Master uses his Sea-Devil-Summoning Device to call the Sea Devils onto the shore.  The sequence makes for the most dynamic action in the story, as the Doctor and Jo negotiate barbed wire (it may have been quicker to use the sonic screwdriver, setting 2428D?) and then detonate the mines in front of the advancing Sea Devils, with explosions galore.

The Sea Devils' initial invasion is unsuccessful, but the Doctor and Jo are forced to retreat to the naval base HMS Seaspite; true to his character the Doctor is determined to broker a peace deal between the humans and the Sea Devils (while the Master is proceeding to stir up trouble).  The Doctor's attempt at peaceful negotiations are thwarted, even though he's taken to the Sea Devils' base on a peaceful understanding.  A senior politician and obstinate military-minded man Robert Walker, orders a military strike on the Sea Devils (it's a recurring theme - mankind never seems to get past its own fears and reach out with truly peaceful intentions).  The Doctor flees from the base under the cover of the attack, his peaceful negotiations in tatters.  Subsequently recaptured, he again tries to persuade the military to seek a non-hostile settlement, and is again thwarted - this time by the Master and the Sea Devils who capture him and force him to help the Master complete his device to awaken all the Sea Devils' colonies globally.

When they return to the Sea Devil base, the Master completes his fiendish plot and successfully activates the device.  However, as they have now outlived their usefulness, the Sea Devils imprison both Time Lords.  In a cunning plan of his own, the Doctor has sabotaged the device, and it begins to overload.  The two Time Lords escape from the base using escape equipment from the captured submarine.  The massive power feedback from the sabotaged device destroys the Sea Devil colony before the planned military attack can begin. The Master once again evades capture - this time he fakes a heart attack and hijacks a rescue hovercraft and flees the scene to fight another day.

Overall, I enjoyed this series; I've mentioned the soundtrack and I'll say no more on that subject.  There's depth, there's slow and steady pace (which could be quickened), and there's plenty of under-the-surface tension (not just below the surface of the sea, but below the surface of the characters).  The Master's covert scheme is handled in such a way that it makes him look clever without making the Doctor look naive and simple, which is a potential pitfall in these kinds of stories.  I enjoyed this one, and moreso than the Warriors of the Deep (even despite the soundtrack). 

My other Doctor Who Reviews

Doctor Who: Asylum of the Daleks
Doctor Who: Sea Devils
Doctor Who: Warriors of the Deep
The Space Babies/The Devil's Chord

Other sci-fi TV reviews:

Star Trek Picard - Season 1 and Season 2
The Book of Boba Fett

Quantitative and Qualitative Testing - Just tell me why!

"And so, you see, we achieved a 197% uplift in conversions with Recipe B!"
"Yes, but why?"
"Well, the page exit rate was down 14% and the click-through-rate to cart was up 12%."
"Yes, but WHY?"

If you've ever been on the receiving end of one of these conversations, you'll probably recognise it immediately.  You're presenting test results, where your new design has won, and you're sharing the good news with the boss.  Or, worse still, the test lost, and you're having to defend your choice of test recipe.  You're showing slide after slide of test metrics - all the KPIs you could think of, and all the ones in every big book you've read - and still you're just not getting to the heart of the matter.  WHY did your test lose?

No amount of numerical data will fully answer the "why" questions, and this is the significant drawback of quantitative testing.  What you need is qualitative testing.

Quantitative testing - think of "quantity" - numbers - will tell you how many, how often, how much, how expensive, or how large.  It can give you ratios, fractions and percentages.

Qualitative testing - think of "qualities" - will tell you what shape, what colour, good, bad, opinions, views and things that can't be counted.  It will tell you the answer to the question you're asking, and if you're asking why, you'll get the answer why.  It won't, however, tell you what the profitability of having a green button instead of a red one will be - it'll just tell you that people prefer green because respondents said it was more calming compared to the angry red one.

Neither is easier than the other to implement well, and neither is less important than the other.  In fact, both can easily be done badly.  Online testing and research may have placed the emphasis may be on A/B testing, and its rigid, reliable, mathematical nature, in contrast to qualitative testing where it's harder to provide concise, precise summaries, but a good research facility will require practitioners of both types of testing.

In fact, there are cases where one form of testing is more beneficial than the other.  If you're building a business case to get a new design fully developed and implemented, then A/B testing will tell you how much profit it will generate (which can then be offset against full development costs).  User testing won't give you a revenue figure like that.

Going back to my introductory conversation - quantitative testing will tell you why your new design has failed.  Why didn't people click the big green button?  Was it because they didn't see it, or because the wording was unhelpful, or because they didn't have enough information to progress?  A click-through-rate of 5% may be low, but "5%" isn't going to tell you why.  Even if you segment your data, you'll still not get a decent answer to the either-or question.  

Let's suppose that 85% of people prefer green apples to red.  
Why?
There's a difference between men and women:  95% of men prefer green apples; compared to just 75% of women.
Great.  Why?  In fact, in the 30-40 year old age group, nearly 98% of men prefer green apples; compared to just 76% of women in the age range.

See?  All this segmentation is getting us no closer to understanding the difference - is it colour; flavour or texture??

However, quantitative testing will get you the answer pretty quickly - you could just ask people directly.

You could liken it to quantitative testing being like the black and white outline of a picture, (or, if you're really good, a grey-scale picture) with qualitative being the colours that fit into the picture.  One will give you a clear outline, one will set the hues. You need both to see the full picture.