## The Digital Cell: Statistical tests

Statistical hypothesis testing, commonly referred to as “statistics”, is a topic of consternation among cell biologists.

This is a short practical guide I put together for my lab. Hopefully it will be useful to others. Note that statistical hypothesis testing is a huge topic and one post cannot hope to cover everything that you need to know.

What statistical test should I do?

To figure out what statistical test you need to do, look at the table below. But before that, you need to ask yourself a few things.

• What are you comparing?
• What is n?
• What will the test tell you? What is your hypothesis?
• What will the p value (or other summary statistic) mean?

If you are not sure about any of these things, whichever test you do is unlikely to tell you much.

The most important question is: what type of data do you have? This will help you pick the right test.

• Measurement – most data you analyse in cell biology will be in this category. Examples are: number of spots per cell, mean GFP intensity per cell, diameter of nucleus, speed of cell migration…
• Normally-distributed – this means it follows a “bell-shaped curve” otherwise called “Gaussian distribution”.
• Not normally-distributed – data that doesn’t fit a normal distribution: skewed data, or better described by other types of curve.
• Binomial – this is data where there are two possible outcomes. A good example here in cell biology would be a mitotic index measurement (the proportion of cells in mitosis). A cell is either in mitosis or it is not.
• Other – maybe you have ranked or scored data. This is not very common in cell biology. A typical example here would be a scoring chart for a behavioural effect with agreed criteria (0 = normal, 5 = epileptic seizures). For a cell biology experiment, you might have a scoring system for a phenotype, e.g. fragmented Golgi (0 = is not fragmented, 5 = is totally dispersed). These arbitrary systems are a not a good idea. Especially, if the person scoring is unblinded to the experimental procedure. Try to come up with an unbiased measurement procedure.

 What do you want to do? Measurement (Normal) Measurement (not Normal) Binomial Describe one group Mean, SD Median, IQR Proportion Compare one group to a value One-sample t-test Wilcoxon test Chi-square Compare two unpaired groups Unpaired t-test Wilcoxon-Mann-Whitney two-sample rank test Fisher’s exact test or Chi-square Compare two paired groups Paired t-test Wilcoxon signed rank test McNemar’s test Compare three or more unmatched groups One-way ANOVA Kruskal-Wallis test Chi-square test Compare three or more matched groups Repeated-measures ANOVA Friedman test Cochran’s Q test Quantify association between two variables Pearson correlation Spearman correlation Predict value from another measured variable Simple linear regression Nonparametric regression Simple logistic regression Predict value from several measured or binomial variables Multiple linear (or nonlinear) regression Multiple logistic regression

Modified from Table 37.1 (p. 298) in Intuitive Biostatistics by Harvey Motulsky, 1995 OUP.

What do “paired/unpaired” and “matched/unmatched” mean?

Most of the data you will get in cell biology is unpaired or unmatched. Individual cells are measured and you have say, 20 cells in the control group and 18 different cells in the test group. These are unpaired (or unmatched in the case of more than one test group) because the cells are different in each group. If you had the same cell in two (or more) groups, the data would be paired (or matched). An example of a paired dataset would be where you have 10 cells that you treat with a drug. You take a measurement from each of them before treatment and a measurement after. So you have paired measurements: one for cell A before treatment, one after; one for cell B before and after, and so on.

How to do some of these tests in IgorPRO

The examples below assume that you have values in waves called data0, data1, data2,… substitute the wavenames for your actual wave names.

Is it normally distributed?

The simplest way is to plot them and see. You can plot out your data using Analysis>Histogram… or Analysis>Packages>Percentiles and BoxPlot… Another possibility is to look at skewness or kurtosis of the dataset (you can do this with WaveStats, see below)

However, if you only have a small number of measurements, or you want to be sure, you can do a test. There are several tests you can do (Kolmogorov-Smirnoff, Jarque-Bera, Shapiro-Wilk). The easiest to do and most intuitive (in Igor) is Shapiro-Wilk.


StatsShapiroWilkTest data0



If p < 0.05 then the data are not normally distributed. Statistical tests on normally distributed data are called parametric, while those on non-normally distributed data are non-parametric.

Describe one group

To get the mean and SD (and lots of other statistics from your data):


Wavestats data0



To get the median and IQR:


StatsQuantiles/ALL data0



The mean and sd are also stored as variables (V_avg, V_sdev). StatsQuantiles calculates V_median, V_Q25, V_Q75, V_IQR, etc. Note that you can just get the median by typing Print StatsMedian(data0) or – in Igor7 – Print median(data0). There is often more than one way to do something in Igor.

Compare one group to a value

It is unlikely that you will need to do this. In cell biology, most of the time we do not have hypothetical values for comparison, we have experimental values from appropriate controls. If you need to do this:


StatsTTest/CI/T=1 data0



Compare two unpaired groups

Use this for normally distributed data where you have test versus control, with no other groups. For paired data, use the additional flag /PAIR.


StatsTTest/CI/T=1 data0,data1



For the non-parametric equivalent, if n is large computation takes a long time. Use additional flag /APRX=2. If the data are paired, use the additional flag /WSRT.


StatsWilcoxonRankTest/T=1/TAIL=4 data0,data1



For binomial data, your waves will have 2 points. Where point 0 corresponds to one outcome and point 1, the other. Note that you can compare to expected values here, for example a genetic cross experiment can be compared to expected Mendelian frequencies. To do Fisher’s exact test, you need a 2D wave representing a contingency table. McNemar’s test for paired binomial data is not available in Igor

StatsChiTest/S/T=1 data0,data1


If you have more than two groups, do not do multiple versions of these tests, use the correct method from the table.

Compare three or more unmatched groups

For normally-distributed data, you need to do a 1-way ANOVA followed by a post-hoc test. The ANOVA will tell you if there are any differences among the groups and if it is possible to investigate further with a post-hoc test. You can discern which groups are different using a post-hoc test. There are several tests available, e.g. Dunnet’s is useful where you have one control value and a bunch of test conditions. We tend to use Tukey’s post-hoc comparison (the /NK flag also does Newman-Keuls test).


StatsAnova1Test/T=1/Q/W/BF data0,data1,data2,data3
StatsTukeyTest/T=1/Q/NK data0,data1,data2,data3



The non-parametric equivalent is Kruskal-Wallis followed by a multiple comparison test. Dunn-Holland-Wolfe method is used.


StatsKSTest/T=1/Q data0,data1,data2,data3
StatsNPMCTest/T=1/DHW/Q data0,data1,data2,data3



Compare three or more matched groups

It’s unlikely that this kind of data will be obtained in a typical cell biology experiment.

StatsANOVA2RMTest/T=1 data0,data1,data2,data3


There are also operations for StatsFriedmanTest and StatsCochranTest.

Correlation

Straightforward command for two waves or one 2D wave. Waves (or columns) must be of the same length


StatsCorrelation data0



At this point, you probably want to plot out the data and use Igor’s fitting functions. The best way to get started is with the example experiment, or just display your data and Analysis>Curve Fitting…

Hazard and survival data

In the lab we have, in the past, done survival/hazard analysis. This is a bit more complex and we used SPSS and would do so again as Igor does not provide these functions.

Notes for use

The good news is that all of this is a lot more intuitive in Igor 7! There is a new Menu item called Statistics, where most of these functions have a dialog with more information. In Igor 6.3 you are stuck with the command line. Igor 7 will be out soon (July 2016).

• Note that there are further options to most of these commands, if you need to see them
• check the manual or Igor Help
• or type ShowHelpTopic “StatsMedian” in the Command Window (put whatever command you want help with between the quotes).
• Extra options are specified by “flags”, these are things like “/Q” that come after the command. For example, /Q means “quiet” i.e. don’t print the output into the history window.
• You should always either print the results to the history or put them into a table so that we can check them. Note that the table gets over written if you do the same test with different data, so printing in this case is a good idea.
• The defaults in Igor are setup OK for our needs. For example, Igor does two-tailed comparison, alpha = 0.05, Welch’s correction, etc.
• Most operations can handle waves of different length (or have flags set to handle this case).
• If you are used to doing statistical tests in Excel, you might be wondering about tails and equal variances. The flags are set in the examples to do two-tailed analysis and unequal variances are handled by Welch’s correction.
• There’s a school of thought that says that using non-parametric tests is best to be cautious. These tests are not as powerful and so it is best to use parametric tests (t test, ANOVA) when you can.

Part of a series on the future of cell biology in quantitative terms.

## The Digital Cell: Getting started with IgorPRO

This post follows on from “Getting Started“.

In the lab we use IgorPRO for pretty much everything. We have many analysis routines that run in Igor, we have scripts for processing microscope metadata etc, and we use it for generating all figures for our papers. Even so, people in the lab engage with it to varying extents. The main battle is that the use of Excel is pretty ubiquitous.

I am currently working on getting more people in the lab started with using Igor. I’ve found that everyone is keen to learn. The approach so far has been workshops to go through the basics. This post accompanies the first workshop, which is coupled to the first few pages of the Manual. If you’re interested in using Igor read on… otherwise you can skip to the part where I explain why I don’t want people in the lab to use Excel.

IgorPro is very powerful and the learning curve is steep, but the investment is worth it.

These are some of the things that Igor can do: Publication-quality graphics, High-speed data display, Ability to handle large data sets, Curve-fitting, Fourier transforms, smoothing, statistics, and other data analysis, Waveform arithmetic, Matrix math, Image display and processing, Combination graphical and command-line user interface, Automation and data processing via a built-in programming environment, Extensibility through modules written in the C and C++ languages. You can even play games in it!

The basics

The first thing to learn is about the objects in the Igor environment and how they work.There are four basic objects that all Igor users will encounter straight away.

• Waves
• Graphs
• Tables
• Layouts

All data is stored as waveforms (or waves for short). Waves can be displayed in graphs or tables. Graphs and tables can be placed in a Layout. This is basically how you make a figure.

The next things to check out are the command window (which displays the history), the data browser and the procedure window.

Essential IgorPro

• Tables are not spreadsheets! Most important thing to understand. Tables are just a way of displaying a wave. They may look like a spreadsheet, but they are not.
• Igor is case insensitive.
• Spaces. Igor can handle spaces in names of objects, but IMO are best avoided.
• Igor is 0-based not 1-based
• Logical naming and logical thought – beginners struggle with this and it’s difficult to get this right when you are working on a project, but consistent naming of objects makes life easier.
• Programming versus not programming – you can get a long way without programming but at some point it will be necessary and it will save you a lot of time.

Pretty soon, you will go beyond the four basic objects and encounter other things. These include: Numeric and string variables, Data folders, Notebooks, Control panels, 3D plots – a.k.a. gizmo, Procedures.

Why don’t we use Excel?

• Excel can’t make high quality graphics for publication.
• We do that in Igor.
• So any effort in Excel is a waste of time.
• Excel is error-prone.
• Too easy for mistakes to be introduced.
• Not auditable. Tough/impossible to find mistakes.
• Igor has a history window that allows us to see what has happened.
• Most people don’t know how to use it properly.
• Not good for biological data – Transcription factor Oct4 gets converted to a date.
• Limited to 1048576 rows and 16384 columns.

But we do use Excel a lot

• Excel is useful for quick calculations and for preparing simple charts to show at lab meeting.
• Same way that Powerpoint is OK to do rough figures for lab meeting.
• But neither are publication-quality.
• We do use Excel for Tracking Tables, Databases(!) etc.

The transition is tough, but worth it

Writing formulae in Excel is straightforward, and the first thing you will find is that to achieve the same thing in Igor is more complicated. For example, working out the mean for each row in an array (a1:y20) in Excel would mean typing =AVERAGE(A1:y1) in cell z1 and copying this cell down to z20. Done. In Igor there are several ways to do this, which itself can be unnerving. One way is to use the Waves Average panel. You need to know how this works to get it to do what you want.

But before you turn back, thinking I’ll just do this in Excel and then import it… imagine you now want to subtract a baseline value from the data, scale it and then average. Imagine that your data are sampled at different intervals. How would you do that? Dealing with those simple cases in Excel is difficult-to-impossible. In Igor, it’s straightforward.

Resources for learning more Igor:

• Igor Help – fantastic resource containing the manual and more. Access via Help or by typing ShowHelpTopic “thing I want to search for”.
• Igor Manual – This PDF is available online or in Applications/Igor Pro/Manual. This used to be a distributed as a hard copy… it is now ~3000 pages.
• Guided Tour of IgorPro – this is a great way to start and will form the basis of the workshops.
• Demos – Igor comes packed with Demos for most things from simple to advanced applications.
• IgorExchange – Lots of code snippets and a forum to ask for advice or search for past answers.
• Igor Tips – I’ve honestly never used these, you can turn on tips in Igor which reveal help on mouse over.
• Igor mailing list – topics discussed here are pretty advanced.
• Introduction to IgorPRO from Payam Minoofar is good. A faster start to learning to program that reading the manual.
• Hands-on experience!

Part of a series on the future of cell biology in quantitative terms.

## Everything In Its Right Place

Something that has driven me nuts for a while is the bug in FIJI/ImageJ when making montages of image stacks. This post is about a solution to this problem.

What’s a montage?

You have a stack of images and you want to array them in m rows by n columns. This is useful for showing a gallery of each frame in a movie or to separate the channels in a multichannel image.

What’s the bug/feature in ImageJ?

If you select Image>Stacks>Make Montage… you can specify how you want to layout your montage. You can specify a “border” for this. Let’s say we have a stack of 12 images that are 300 x 300 pixels. Let’s arrange them into 3 rows and 4 columns with 0 border.

So far so good. We have an image that is 1200 x 900. But it looks a bit rubbish, we need some grouting (white pixel space between the images). We don’t need a border, but let’s ignore that for the moment. So the only way to do this in ImageJ is to specify a border of 8 pixels.

Looks a lot better. Ok there’s a border around the outside, which is no use, but it looks good. But wait a minute! Check out the size of the image (1204 x 904). This is only 4 pixels bigger in x and y, yet we added all that grouting, what’s going on?

The montage is not pixel perfect.

So the first image is not 300 x 300 any more. It is 288 x 288. Hmmm, maybe we can live with losing some data… but what’s this?

The next image in the row is not even square! It’s 292 x 288. How much this annoys you will depend on how much you like things being correct… The way I see it, this is science, if we don’t look after the details, who will? If I start with 300 x 300 images, it’s not too much to ask to end up with 300 x 300 images, is it? I needed to fix this.

Solutions

I searched for a while for a solution. It had clearly bothered other people in the past, but I guess people just found their own workaround.

ImageJ solution for multichannel array

So for a multichannel image, where the grayscale images are arrayed next to the merge, I wrote something in ImageJ to handle this. These macros are available here. There is a macro for doing the separation and arraying. Then there is a macro to combine these into a bigger figure.

Igor solution

For the exact case described above, where large stacks need to be tiled out into and m x n array, I have to admit I struggled to write something for ImageJ and instead wrote something for IgorPRO. Specifying 3 rows, 4 columns and a grout of 8 pixels gives the correct TIFF 1224 x 916, with each frame showing in full and square. The code is available here, it works for 8 bit greyscale and RGB images.

I might update the code at some point to make sure it can handle all data types and to allow labelling and adding of a scale bar etc.

The post title is taken from “Everything In Its Right Place” by Radiohead from album Kid A.

## The Digital Cell: Getting Started

More on the theme of “The Digital Cell“: using quantitative, computational approaches in cell biology.

So you want to get started? Well, the short version of this post is:

Find something that you need to automate and get going!

Programming

I make no claim to be a computer wizard. My first taste of programming was the same as anyone who went to school in the UK in the 1980s: BBC Basic. Although my programming only went as far as copying a few examples from the book (right), this experience definitely reduced the “fear of the command line”. My next encounter with coding was to learn HTML when I was an undergraduate. It was not until I was a postdoc that I realised that I needed to write scripts in order get computers to do what I wanted them to do for my research.

Image analysis

I work in cell biology. My work involves a lot of microscopy. From the start, I used computer-based methods to quantify images. My first paper mentions quantifying images, but it wasn’t until I was a PhD student that I first used NIH Image (as it was called then) to extract quantitative information from confocal micrographs. I was also introduced to IgorPRO (version 3!) as a PhD student, but did no programming. That came later. As a postdoc, we used Scanalytics’ IPLab and Igor (as well as a bit of ImageJ as it had become). IPLab had an easy scripting language and it was in this program that I learned to write macros for analysis. At this time there were people in the lab who were writing software in IgorPro and MATLAB. While I didn’t pick up programming in IgorPRO or MATLAB then, it made me realise what was possible.

When I started my own group I discovered that IPLab had been acquired by BD Biosciences and then stripped out. I had hundreds of useless scripts and needed a new solution. ImageJ had improved enormously by this time and so this became our default image analysis program. The first data analysis package I bought was IgorPro (version 6) and I have stuck with it since then. In a future post, I will probably return to whether or not this was a good path.

Getting started with programming

Around 2009, I was still unable to program properly. I needed a macro for baseline subtraction – something really simple – and realised I didn’t know how to do it. We didn’t have just one or two traces to modify, we had hundreds. This was simply not possible by hand. It was this situation that made me realise I needed to learn to program.

…having a concrete problem that is impossible to crack any other way is the best motivator for learning to program.

This might seem obvious, but having a concrete problem that is impossible to crack any other way is the best motivator for learning to program. I know many people who have decided they “want to learn to code” or they are “going to learn to use R”. This approach rarely works. Sitting down and learning this stuff without sufficient motivation is really tough. So I would advise someone wanting to learn programming to find something that needs automation and just get going. Just get something to work!

Don’t worry (initially) about any of the following:

• What program/language to use – as long as it is possible, just pick something and do it
• If your code is ugly or embarrassing to show to an expert – as long as it runs, it doesn’t matter
• About copy-and-pasting from examples – it’s OK as long as you take time to understand what you are doing, this is a quick way to make progress. Resources such as stackoverflow are excellent for this
• Bugs – you can squish them, they will frustrate you, but you might need some…
• Help – ask for help. Online forums are great, experts love showing off their knowledge. If you have local expertise, even better!

Once you have written something (and it works)… congratulations, you are a computer programmer!

Seriously, that is all there is to it. OK, it’s a long way to being a good programmer or even a competent one, but you have made a start. Like Obi Wan Kenobi says: you’ve taken your first step into a larger world.

So how do you get started with an environment like IgorPro? This will be the topic for next time.

Part of a series on the future of cell biology in quantitative terms.

I needed to generate a uniform random distribution of points inside a circle and, later, a sphere. This is part of a bigger project, but the code to do this is kind of interesting. There were no solutions available for IgorPro, but stackexchange had plenty of examples in python and mathematica. There are many ways to do this. The most popular seems to be to generate a uniform random set of points in a square or cube and then discard those that are greater than the radius away from the origin. I didn’t like this idea, because I needed to extend it to spheroids eventually, and as I saw it the computation time saved was minimal.

Here is the version for points in a circle (radius = 1, centred on the origin).

This gives a nice set of points, 1000 shown here.

And here is the version inside a sphere. This code has variable radius for the sphere.

The three waves (xw,yw,zw) can be concatenated and displayed in a Gizmo. The code just plots out the three views.

My code uses var + enoise(var) to get a random variable from 0,var. This is because enoise goes from -var to +var. There is an interesting discussion about whether this is a truly flat PDF here.

This is part of a bigger project where I’ve had invaluable help from Tom Honnor from Statistics.

This post is part of a series on esoterica in computer programming.

An occasional series in esoteric programming issues.

As part of a larger analysis project I needed to implement a short program to determine the closest distance of two line segments in 3D space. This will be used to sort out which segments to compare… like I say, part of a bigger project. The best method to do this is to find the closest distance one segment is to the other when the other one is represented as an infinite line. You can then check if that point is beyond the segment if it is you use the limits of the segment to calculate the distance. There’s a discussion on stackoverflow here. The solutions point to one in C++ and one in MATLAB. The C++ version is easiest to port to Igor due to the similarity of languages, but the explanation of the MATLAB code was more approachable. So I ported that to Igor to figure out how it works.

The MATLAB version is:

>> P = [-0.43256      -1.6656      0.12533]
P =
-0.4326   -1.6656    0.1253
>> Q = [0.28768      -1.1465       1.1909]
Q =
0.2877   -1.1465    1.1909
>> R = [1.1892    -0.037633      0.32729]
R =
1.1892   -0.0376    0.3273
>> S = [0.17464     -0.18671      0.72579]
S =
0.1746   -0.1867    0.7258
>> N = null(P-Q)
N =
-0.3743   -0.7683
0.9078   -0.1893
-0.1893    0.6115
>> r = (R-P)*N
r =
0.8327   -1.4306
>> s = (S-P)*N
s =
1.0016   -0.3792
>> n = (s - r)*[0 -1;1 0];
>> n = n/norm(n);
>> n
n =
0.9873   -0.1587
>> d = dot(n,r)
d =
1.0491
>> d = dot(n,s)
d =
1.0491
>> v = dot(s-r,d*n-r)/dot(s-r,s-r)
v =
1.2024
>> u = (Q-P)'$$(S - (S*N)*N') - P)' u = 0.9590 >> P + u*(Q-P) ans = 0.2582 -1.1678 1.1473 >> norm(P + u*(Q-P) - S) ans = 1.0710  and in IgorPro: Function MakeVectors() Make/O/D/N=(1,3) matP={{-0.43256},{-1.6656},{0.12533}} Make/O/D/N=(1,3) matQ={{0.28768},{-1.1465},{1.1909}} Make/O/D/N=(1,3) matR={{1.1892},{-0.037633},{0.32729}} Make/O/D/N=(1,3) matS={{0.17464},{-0.18671},{0.72579}} End Function MakeVectors() Make/O/D/N=(1,3) matP={{-0.43256},{-1.6656},{0.12533}} Make/O/D/N=(1,3) matQ={{0.28768},{-1.1465},{1.1909}} Make/O/D/N=(1,3) matR={{1.1892},{-0.037633},{0.32729}} Make/O/D/N=(1,3) matS={{0.17464},{-0.18671},{0.72579}} End Function DoCalcs() WAVE matP,matQ,matR,matS MatrixOp/O tempMat = matP - matQ MatrixSVD tempMat Make/O/D/N=(3,2) matN Wave M_VT matN = M_VT[p][q+1] MatrixOp/O tempMat2 = (matR - matP) MatrixMultiply tempMat2, matN Wave M_product Duplicate/O M_product, mat_r MatrixOp/O tempMat2 = (matS - matP) MatrixMultiply tempMat2, matN Duplicate/O M_product, mat_s Make/O/D/N=(2,2) MatUnit matUnit = {{0,1},{-1,0}} MatrixOp/O tempMat2 = (mat_s - mat_r) MatrixMultiply tempMat2,MatUnit Duplicate/O M_Product, Mat_n Variable nn nn = norm(mat_n) MatrixOP/O new_n = mat_n / nn //new_n is now a vector with unit length Variable dd dd = MatrixDot(new_n,mat_r) //print dd //dd = MatrixDot(new_n,mat_s) //print dd dd = abs(dd) // now find v // v = dot(s-r,d*n-r)/dot(s-r,s-r) variable vv MatrixOp/O mat_s_r = mat_s - mat_r MatrixOp/O tempmat2 = dd * mat_n - mat_r vv = MatrixDot(mat_s_r,tempmat2) / MatrixDot(mat_s_r,mat_s_r) //print vv //because vv &amp;gt; 1 then closest post is s (because rs(1) = s) and therefore closest point on RS to infinite line PQ is S //what about the point on PQ is this also outside the segment? // u = (Q-P)'\((S - (S*N)*N') - P)' variable uu MatrixOp/O matQ_P = matQ - matP MatrixTranspose matQ_P //MatrixOP/O tempMat2 = ((matS - (matS * matN) * MatrixTranspose(MatN)) - MatrixTranspose(matP)) Duplicate/O MatN, matNprime MatrixTranspose matNprime MatrixMultiply matS, matN Duplicate/O M_Product, matSN MatrixMultiply M_Product, matNprime MatrixOP/O tempMat2 = ((matS - M_product) - matP) MatrixTranspose tempMat2 MatrixLLS matQ_P tempMat2 Wave M_B uu = M_B[0] // find point on PQ that is closest to RS // P + u*(Q-P) MatrixOp/O matQ_P = matQ - matP MatrixOp/O matPoint = MatP + (uu * MatQ_P) MatrixOP/O distpoint = matPoint - matS Variable dist dist = norm(distpoint) Print dist End  The sticking points were finding the Igor equivalents of • null() • norm() • dot() • \ otherwise known as mldivide Which are: • MatrixSVD (answer is in the final two columns of wave M_VT) • norm() • MatrixDot() • MatrixLLS MatrixLLS wouldn’t accept a mix of single-precision and double-precision waves, so this needed to be factored into the code. As you can see, the Igor code is much longer. Overall, I think MATLAB handles Matrix Math better than Igor. It is certainly easier to write. I suspect that there are a series of Igor operations that can do what I am trying to do here, but this was an exercise in direct porting. More work is needed to condense this down and also deal with every possible case. Then it needs to be incorporated into the bigger program. SIMPLE! Anyway, hope this helps somebody. The post title is taken from the band Adventures In Stereo. ## Weak Superhero: how to win and lose at Marvel Top Trumps Top Trumps is a card game for children. The mind can wander when playing such games with kids… typically, I start thinking: what is the best strategy for this game? But also, as the game drags on: what is the quickest way to lose? Since Top Trumps is based on numerical values with simple outcomes, it seemed straightforward to analyse the cards and to simulate different scenarios to look at these questions. Many Top Trumps variants exist, but the pack I’ll focus on is Marvel Universe “Who’s Your Hero?” made by Winning Moves (cat. No.: 3399736). Note though that the approach can probably be adapted to handle any other Top Trumps set. There are 30 cards featuring Marvel characters. Each card has six categories: 1. Strength 2. Skill 3. Size 4. Wisecracks 5. Mystique 6. Top Trumps Rating. What is the best card and which one is the worst? In order to determine this I pulled in all the data and compared each value to every other card’s value, and repeated this per category (code is here, the data are here). The scaling is different between category, but that’s OK, because the game only uses within field comparisons. This technique allowed me to add up how many cards have a lower value for a certain field for a given card, i.e. how many cards would that card beat. These victories could then be summed across all six fields to determine the “winningest card”. The cumulative victories can be used to rank the cards and a category plot illustrates how “winningness” is distributed throughout the deck. As an aside: looking at the way the scores for each category are distributed is interesting too. Understanding these distributions and the way that each are scaled gives a better feel for whether a score of say 2 in Wisecracks is any good (it is). The best card in the deck is Iron Man. What is interesting is that Spider-Man has the designation Top Trump (see card), but he’s actually second in terms of wins over all other cards. Head-to-head, Spider-Man beats Iron Man in Skill and Mystique. They draw on Top Trumps Rating. But Iron Man beats Spider-Man on the three remaining fields. So if Iron Man comes up in your hand, you are most likely to defeat your opponent. At the other end of the “winningest card” plot, the worst card, is Wasp. Followed by Ant Man and Bucky Barnes. There needs to be a terrible card in every Top Trump deck, and Wasp is it. She has pitiful scores in most fields. And can collectively only win 9 out of (6 * 29) = 174 contests. If this card comes up, you are pretty much screwed. What about draws? It’s true that a draw doesn’t mean losing and the active player gets another turn, so a draw does have some value. To make sure I wasn’t overlooking this with my system of counting victories, I recalculated the values using a Football League points system (3 points for a win, 1 point for a draw and 0 for a loss). The result is the same, with only some minor changes in the ranking. I went with the first evaluation system in order to simulate the games. I wrote a first version of the code that would printout what was happening so I could check that the simulation ran OK. Once that was done, it was possible to call the function that runs the game, do this multiple (1 x 10^6) times and record who won (player 1 or player 2) and for how many rounds each game lasted. A typical printout of a game (first 9 rounds) is shown here. So now I could test out different strategies: What is the best way to win and what is the best way to lose? Strategy 1: pick your best category and play If you knew which category was the most likely to win, you could pick that one and just win every game? Well, not quite. If both players take this strategy, then the player that goes first has a slight advantage and wins 57.8% of the time. The games can go on and on, the longest is over 500 rounds. I timed a few rounds and it worked out around 15 s per round. So the longest game would take just over 2 hours. Strategy 2: pick one category and stick with it This one requires very little brainpower and suits the disengaged adult: just keep picking the same category. In this scenario, Player 1 just picks strength every time while Player 2 picks their best category. This is a great way to lose. Just 0.02% of games are won using this strategy. Strategy 3: pick categories at random The next scenario was to just pick random categories. I set up Player 1 to do this and play against Player 2 picking their best category. This means 0.2% of wins for Player 1. The games are over fairly quickly with the longest of 1 x 10^6 games stretching to 200 rounds. If both players take this strategy, it results in much longer games (almost 2000 rounds for the longest). The player-goes-first advantage disappears and the wins are split 49.9 to 50.1%. Strategy 4: pick your worst category How does all of this compare with selecting the worst category? To look at this I made Player 2 take this strategy, while Player 1 picked the best category. The result was definitive, it is simply not possible for Player 2 to win. Player 1 wins 100% of all 1 x 10^6 games. The games are over in less than 60 rounds, with most being wrapped up in less than 35 rounds. Of course this would require almost as much knowledge of the deck as the winning strategy, but if you are determined to lose then it is the best strategy. The hand you’re dealt Head-to-head, the best strategy is to pick your best category (no surprise there), but whether you win or lose depends on the cards you are dealt. I looked at which player is dealt the worst card Wasp and at the outcome. The split of wins for player 1 (58% of games) are with 54% of those, Player 2 stated with Wasp. Being dealt this card is a disadvantage but it is not the kiss of death. This analysis could be extended to look at the outcome if the n worst cards end up in your hand. I’d predict that this would influence the outcome further than just having Wasp. So there you have it: every last drop of fun squeezed out of a children’s game by computational analysis. At quantixed, we aim to please. The post title is taken from “Weak Superhero” by Rocket From The Crypt off their debut LP “Paint As A Fragrance” on Headhunter Records ## Pledging My Time II 2016 was the 400 year anniversary of William Shakespeare’s death. Stratford-upon-Avon Rotary Club held the Shakespeare Marathon on the same weekend. Runners had an option of half or full marathon. There were apparently 3.5 K runners. Only 700 of whom were doing the full marathon. The chip results were uploaded last night and can be found here. Similar to my post on the Coventry Half Marathon, I thought I’d quickly analyse the data. The breakdown of runners by category. M and F are male and female runners under 35 years of age. M35 is 35-45, F55 is 55-65 etc. Only a single runner in the F65 category! The best time was 02:34:51 by Adam Holland of Notfast. Fastest female runner was 3:14:39 by Josie Hinton of London Heathside. Congrats to everyone who ran and thanks to the organisers and all the supporters out on the course. The post title is taken from “Pledging My Time” a track from Blonde on Blonde by Bob Dylan ## Wrote for Luck Fans of probability love random processes. And lotteries are a great example of random number generation. The UK National Lottery ran in one format from 19/11/1994 until 7/10/2015. I was talking to somebody who had played the same set of numbers in all of these lottery draws and I wondered what the net gain or loss has been for them over this period. The basic format is that people buy a line of numbers (6 numbers, from 1-49) and try to match the six numbers (from 49 balls numbered 1-49) drawn from a machine. The aim is to match all six balls and win the jackpot. The odds of this are fantastically small (1 in ~14 million), but if they are the only person matching these numbers they can take away £3-5 million. There are prizes for matching three numbers (1 in ~56 chance), four numbers (1 in ~1,032), five numbers (1 in ~55,491) or five numbers plus a seventh “bonus ball” (1 in ~2,330,636). Typical prizes are £10, £100, £1,500, or £50,000, respectively. The data for all draws are available here. I pulled all draws regardless of machine that was used or what set of balls was used. This is what the data look like. The rows are the seven balls (colour coded 1-49) that came out of the machine over 2065 draws. I wrote a quick bit of code which generated all possible combinations of lottery numbers and compared all of these combinations to the real-life draws. The 1 in 14 million that I referred to earlier is actually \(^{n}C_r = ^{49}C_6$$

$$\frac{49!}{\left ( 6! \left (49-6 \right )! \right )} = 13,983,816$$

This gives us the following.

Crunching these combinations against the real-life draw outcomes tells us what would have happened if every possible ticket had been bought for all draws. If we assume a £1 stake for each draw and ~14 million people each buying a unique combination line. Each person has staked £2065 for the draws we are considering.

• The unluckiest line is 6, 7, 10, 21, 26, 36. This would’ve only won 12 lots of three balls, i.e. £120 – a net loss of £1945
• The luckiest line is 3, 6, 13, 23, 27, 49. These numbers won 41 x three ball, 2 x four ball, 1 x jackpot, 1 x 5 balls + bonus.
• Out of all possible combinations, 13728621 of them are in the red by anything from £5 to £1945. This is 98.2% of combinations.

Pretty terrible odds all-in-all. Note that I used the typical payout values for this calculation. If all possible tickets had been purchased the payouts would be much higher. So this calculation tells us what an individual could expect if they played the same numbers for every draw.

Note that the unluckiest line and the luckiest line have an equal probability of success in the 2066th draw. There is nothing intrinsically unlucky or lucky about these numbers!

I played the lottery a few times when it started with a specified set of numbers. I matched 3 balls twice and 4 balls once. I’ve not played since 1998 or so. Using another function in my code, I could check what would’ve happened if I’d kept playing all those intervening years. Fortunately, I would’ve looked forward to a net loss with 43 x three balls and 2 x four balls. Since I actually had a ticket for some of those wins and hardly any for the 2020 losing draws, I feel OK about that. Discovering that my line had actually matched the jackpot would’ve been weird, so I’m glad that wasn’t the case.

There’s lots of fun to be had with this dataset and a quick google suggests that there are plenty of sites on the web doing just that.

Here’s a quick plot for fun. The frequency of balls drawn in the dataset:

• The ball drawn the least is 13
• The one drawn the most is 38
• Expected number of appearances is 295 (14455/49).
• 14455 is 7 balls from 2065 draws

Since October 2015, the Lottery changed to 1-59 balls and so the dataset used here is effectively complete unless they revert to the old format.

The title of this post comes from “Wrote for Luck” by The Happy Mondays from their 1988 LP Bummed. The Manic Street Preachers recorded a great cover version which was on the B-Side of Roses in The Hospital single.

## Pledging My Time

The end of the month sees the Coventry Half Marathon. I looked at what constitutes a good time over this course, based on 2015 results. I thought I’d post this here in case any one is interested.

The breakdown of runners by category for the 2015 event. Male Senior (MSEN) category has the most runners, constituting a wide age grouping. There were 3565 runners in total, 5 in an undetermined category and 9 DNFs. These 14 were not included in the analysis.

The best time last year was 01:10:21!

Good luck to everyone running this (or any other event) this year.

Edit: The 2016 Coventry Half Marathon happened today. I’m updating this post with the new data.

The width of violins has no special significance compared to 2015. Fastest time this year was 1:08:40 in the MSEN category.

There were more runners this year than last (4212 finishers), across all categories. Also this year there was a wheelchair category, which is not included here as there were only four competitors. FWIW, I placed somewhere in the first violin, in the lower whisker :-).

Congrats to everyone who ran and thanks to the all the supporters out on the course.

The post title is taken from “Pledging My Time” a track from Blonde on Blonde by Bob Dylan