# Why the polls failed in the US Presidential Election of 2020

In the wake of the widespread failure of opinion polls to accurately predict the outcome of the 2020 US Presidential election, the American Association for Public Opinion Research (AAPOR) commissioned a study to investigate the causes and make recommendations. Their findings were recently released.

(This is the key question for 2020 opinion pollsters. The answer is yes, but they don't know why. Image source: Wikimedia)

# Summary of AAPOR's findings

I've read the report and I've dug through the findings. Here's my summary:

1. The polls overstated support for Democratic candidates.
2. We don't really know why.
3. Er... that's it.

Yes, I'm being harsh, but I'm underwhelmed by the report and I find some of the statements in it unconvincing. I'll present some of their main findings and talk through them. I encourage you to read the report for yourself and reach your own conclusions.

(We don't know why we didn't get the results right.)

# Factors they ruled out for 2020

• Late-breaking changes in favor of Republican candidates. This happened in 2016, but didn't happen in 2020. The polls were directionally consistent throughout the campaign.
• Weighting for education. In 2016, most polls didn't weight for education and education did seem to be a factor. In 2020, most polls did weigh for education. Educational weighting wasn't a factor.
• Pollsters got the demographics wrong. Pollsters don't use random sampling, they often use stratified sampling based on demographics. There's no evidence that errors in demographics led to widespread polling errors in 2020.
• People were afraid to say they voted for Trump. In races not involving Trump, the opinion polls were still wrong and still favored Democratic candidates. Trump wasn't the cause.
• Intention to vote vs. actually voting. The results can't be explained by voters saying they were going to vote but who didn't actually vote. For example, if Democratic voters said they were going to vote Democratic and didn't actually vote, this would explain the error, but it didn't happen.
• Proportion of early voters or election day voters. Early voting/election day voting didn't make a difference to the polling error.

# Factors they couldn't rule out

• Republican voters chose not to take part in surveys at a higher number than Democratic voters.
• The weighting model used to adjust sampling may have been wrong. Pollsters use models of the electorate to adjust their results. If these models are wrong, the results will be biased.
• Many more people voted in 2020 than in 2016 ("new voters" in the report) - maybe pollsters couldn't model these new voters very well.

Here's a paragraph from the report:

"Unfortunately, the ability to determine the cause or causes of polling error in 2020 is limited by the available data. Unless the composition of the overall electorate is known, looking only at who responded says nothing about who did not respond. Not knowing if the Republicans (or unaffiliated voters, or new voters) who responded to polls were more supportive of Biden than those who did not respond, for example, it is impossible to identify the primary source of polling error."

Let me put that paragraph another way: we don't have enough data to investigate the problem so we can't say what went wrong.

# Rinse and repeat - or just don't

I'm going to quote some sentences from the report's conclusions and comment:

• "Considering that the average margin of error among the state-level presidential polls in 2020 was 3.9 points, that means candidate margins smaller than 7.8 points would be difficult to statistically distinguish from zero using conventional levels of statistical significance. Furthermore, accounting for uncertainty of statistical adjustments and other factors, the total survey error would be even larger."
• "Most pre-election polls lack the precision necessary to predict the outcome of semi-close contests."
• "Our investigation reveals a systemic overstatement of the Democratic-Republican margin in nearly every contest, regardless of mode or proximity to the election. This overstatement is largest in states with more Republican supporters"

Some of the report's statements are extraordinary if you stop and think for a moment. I want you to ponder the key question: "what use are polls"?

The people paying for polls are mostly (but not completely) political campaigns and the media. The media want to report on an accurate snapshot of where the election is now and make an assessment of who will win. Political campaigns largely want the same thing.

In places like Alaska or Hawaii, polls aren't very useful because voters tend to vote strongly Democratic or Republican. For example, Wyoming is overwhelmingly a Republican stronghold and Washington D.C. a Democratic stronghold. My forecast for 2024 is simple: Wyoming will vote Republican and Washington D.C. Democratic.

Polls are useful where the race is close, or, in the words of the report "semi-close". But, according to the report, polls in semi-close states don't have sufficient accuracy to predict the result.

So, if polls aren't useful in strongly Democratic or Republican states, and they lack predictive power in "semi-close" races, what use are they? Why should anyone pay for them?

There's an even deadlier issue for the polling organizations. You can very clearly judge the accuracy of political opinion polls. Opinion poll companies run all kinds of polls on all kinds of topics, not just elections. How accurate are they in other areas where their success is harder to assess?

# Where to next?

The polling industry has an existential credibility crisis. It can't continue to sell a product that doesn't work. It's extraordinary that an industry that's been around for nearly 100 years doesn't have the data to diagnose its failures. The industry needs to come together to fix its problems as soon as possible - or face irrelevancy in the near future.

# What were the numbers?

Often in business, we're presented with charts where the y-axis is unlabeled because the presenter wants to conceal the numbers. Are there ways of reconstructing the labels and figuring out what the data is? Surprisingly, yes there are.

Given a chart like this:

you can often figure out what the chart values should be.

The great Evan Miller posted on this topic several years ago ("How To Read an Unlabeled Sales Chart"). He discussed two methods:

• Greatest common divisor (gcd)
• Poisson distribution

In this blog post, I'm going to take his gcd work a step further and present code and a process for reconstructing numbers under certain circumstances. In another blog post, I'll explain the Poisson method.

The process I'm going to describe here will only work:

• Where the underlying data is integers
• Where there's 'enough' range in the underlying data.
• Where the maximum underlying data is less than about 200.
• Where the y-axis includes zero.

# The results

I generated this chart without axes labels, the goal being to recreate the underlying data. I measured screen y-coordinates of the top and bottom plot borders (187 and 677) and I measured the y coordinates of the top of each of the bars. Using the process and code I describe below, I was able to correctly recreate the underlying data values, which were $$[33, 30, 32, 23, 32, 26, 18, 59, 47]$$.

# How plotting packages work

To understand the method, we need to understand how a plotting package will render a set of integers on a chart.

Let's take the list of numbers $$[1, 2, 3, 5, 7, 11, 13, 17, 19, 23]$$ and call them $$y_o$$.

When a plotting package renders $$y_o$$ on the screen, it will put them into a chart with screen x-y coordinates. It's helpful to think about the chart on the screen as a viewport with x and y screen dimensions. Because we only care about the y dimensions, that's what I'll talk about. On the screen, the viewport might go from 963 pixels to 30 pixels on the y-axis, a total range of 933 y-pixels.

Here's how the numbers $$y_o$$ might appear on the screen and how they map to the viewport y-coordinates. Note the origin is top left, not bottom right. I'll "correct" for the different origin.

The plotting package will translate the numbers $$y_o$$ to a set of screen coordinates I'll call $$y_s$$. Assuming our viewport starts from 0, we have:

$y_s = my_o$

Let's just look at the longest bar that corresponds to the number 23. My measurements of the start and end are 563 and 27, which gives a length of 536. $$m$$ in this case is 536/23, or 23.3.

There are three things to bear in mind:

• The set of numbers $$y_o$$ are integers
• The set of numbers $$y_s$$ are integers - we can't have half a pixel for example.
• The scalar $$m$$ is a real number

# Integer only solutions for $$m$$

In Evan Miller's original post, he only considered integer values of $$m$$. If we restrict ourselves to integers, then most of the time:

$m = gcd(y_s)$

where gcd is the greatest common divisor.

To see how this works, let's take:

$y_o = [1 , 2, 3]$

and

$m = 8$

These numbers give us:

$y_s = [8, 16, 24]$

To find the gcd in Python:

np.gcd.reduce([8, 16, 24])

which gives $$m = 8$$, which is correct.

If we could guarantee $$m$$ was an integer, we'd have an answer; we'd be able to reconstruct the original data just using the gcd function. But we can't do that in practice for three reasons:
1. $$m$$ isn't always an integer.
2. There are measurement errors which mean there will be some uncertainty in our $$y_s$$ values.
3. It's possible the original data set $$y_o$$ has a gcd which is not 1.

In practice, we gather screen coordinates using a manual process which will introduce errors. At most, we're likely to be off by a few pixels for each measurement, however, even the smallest error will mean the gcd method won't work. For example, if the value on the screen should be 500 but we might incorrectly measure it as 499, this small error means the method fails (there is a way around this failure that will work for small measurement errors.)

If our original data set has a gcd greater than 1, the method won't work. Let's say our data was:

$y_o = [2, 4, 6]$

and:

$m=8$

we would have:

$y_s = [16, 32, 48]$

which has a gcd of 16, which is an incorrect estimate of $$m$$. In practice, the odds of the original data set $$y_o$$ having a gcd > 1 are low.

The real killer for this approach is the fact that $$m$$ is highly likely in practice to be a real number.

# Real solutions for $$m$$

The only way I've found for solving for $$m$$ is to try different values for $$m$$ to see what succeeds. To get this to work, we have to constrain $$m$$ because otherwise there would be an infinite number of values to try. Here's how I constrain $$m$$:

• I limit the steps for different $$m$$ values to 0.01.
• I start my m values from just over 1 and I stop at a maximum $$m$$ value. My maximum $$m$$ value I get from assuming the smallest value I measure on the screen corresponds to a data value of 1, for example, if the smallest measurement is 24 pixels, the smallest possible original data is 1, so the maximum value for $$m$$ is 24.

Now we've constrained $$m$$, how do we evaluate $$y_s = my_o$$? First off, we define an error function. We want our estimates of the original data $$y_o$$ to be integers, so the further away we are from an integer, the worse the error. For the $$i$$th element of our estimate of $$y_o$$, the error estimate is:

$\frac{y_{si}}{m_{estimate}} - \frac{y_{si}}{m_{estimate}}$

we're choosing the least square error, which means minimizing:

$\frac{1}{n} \sum \left ( round \left ( \frac{y_{si}}{m_{estimate}} \right ) - \frac{y_{si}}{m_{estimate}} \right )^2$

in code, this comes out as:

sum([(round(_y/div) - _y/div)**2 for _y in y])/len(y)

Our goal is to try different values of $$m$$ and choose the solution that yields the lowest error estimate.

# The solution in practice

Before I show you how this works, there are two practicalities. The first is that $$m=1$$ is always a solution and will always give a zero error, but it's probably not the right solution, so we're going to ignore $$m=1$$. Secondly, there will be an error in our measurements due to human error. I'm going to assume the maximum error is 3 pixels for any measurement. To calculate a length, we take a measurement of the start and end of the bar (if it's a bar chart), which means our maximum uncertainty is 2*3. That's why I set my maximum $$m$$ to be min(y) + 2*MAX_ERROR.

To show you how this works, I'll talk you through an example.

The first step is measurement. We need to measure the screen y-coordinates of the plot borders and the top of the bars (or the position of the points on a scatter chart). If the plot doesn't have borders, just measure the position of the bottom of the bars and the coordinate of the highest bar. Here are some measurements I took.

Here are the measurements of the top of the bars (_y_measured): $$[482, 500, 489, 541, 489, 523, 571, 329, 399]$$

Here are the start and stop coordinates of the plot borders (_start, _stop):  $$677, 187$$

To convert these to lengths, the code is just: [_start - _y_m for _y_m in _y_measured]

The length of the screen from the top to the bottom is: _start - _stop = $$490$$

This gives us measured length (y_measured): $$[195, 177, 188, 136, 188, 154, 106, 348, 278]$$

Now we run this code:

MAX_ERROR = 3

STEP = 0.01

ERROR_THRESHOLD = 0.01

def mse(y, div):

"""Means square error calculation."""

return sum([(round(_y/div) - _y/div)**2 for _y in y])/len(y)

def find_divider(y):

"""Return the non-integer that minimizes the error function."""

error_list = []

for _div in np.arange(1 + STEP,

min(y) + 2*MAX_ERROR,

STEP):

error_list.append({"divider": _div,

"error":mse(y, _div)})

df_error = pd.DataFrame(error_list)

df_error.plot(x='divider', y='error', kind='scatter')

_slice = df_error[df_error['error'] == df_error['error'].min()]

divider = _slice['divider'].to_list()[0]

error = _slice['error'].to_list()[0]

if error > ERROR_THRESHOLD:

raise ValueError('The estimated error is {0} which is '

'too large for a reliable result.'.format(error))

return divider

def find_estimate(y, y_extent):

"""Make an estimate of the underlying data."""

if (max(y_measured) - min(y_measured))/y_extent < 0.1:

raise ValueError('Too little range in the data to make an estimate.')

m = find_divider(y)

return [round(_e/m) for _e in y_measured], m

estimate, m = find_estimate(y_measured, y_extent)

This gives us this output:

Original numbers: [33, 30, 32, 23, 32, 26, 18, 59, 47]

Measured y values: [195, 177, 188, 136, 188, 154, 106, 348, 278]

Divider (m) estimate: 5.900000000000004

Estimated original numbers: [33, 30, 32, 23, 32, 26, 18, 59, 47]

Which is correct.

# Limitations of this approach

Here's when it won't work:

• If there's little variation in the numbers on the chart, then measurement errors tend to overwhelm the calculations and the results aren't good.
• In a similar vein, if the numbers are all close to the top or the bottom of the chart, measurement errors lead to poor results.
• $$m < 1$$, which as the maximum y viewport range is usually in the range 500-900 pixels, it won't work for numbers greater than about 500.
• I've found in practice that if $$m < 3$$ the results can be unreliable. Arbitrarily, I call any error greater than 0.01 too high to protect against poor results. Maybe, I should limit the results to $$m > 3$$.

I'm not entirely convinced my error function is correct; I'd like an error function that better discriminates between values. I tried a couple of alternatives, but they didn't give good results. Perhaps you can do better.

Notice that the error function is 'denser' closer to 1, suggesting I should use a variable step size or a different algorithm. It might be that the closer you get to 1, the more errors and the effects of rounding overwhelm the calculation. I've played around with smaller step sizes and not had much luck.

# Future work

If the data is Poisson distributed, there's an easier approach you can take. In a future blog post, I'll talk you through it.

# Where to get the code

I've put the code on my Github page here: https://github.com/MikeWoodward/CodeExamples/blob/master/UnlabeledChart/approxrealgcd.py

# Why is education important?

High-paying jobs tend to be knowledge-intensive. If you can't get the qualified workers you want in the UK you could move your operations to Poland, Spain, or Czechia. This obviously applies to IT jobs, but also to specialized manufacturing jobs and jobs in other areas. Governments are in a race or a beauty contest to attract employers with high-paying jobs to set up in their country. High-paying jobs support an ecosystem of other employment, from transportation workers to baristas.

(In the modern economy, those who train well and invest will win. Image source: Wikimedia, Author:Ub-K0G76A. License: Creative Commons.)

That's why the UK's flat educational performance is so worrying and why it's puzzling there isn't more concern about it in the country. To summarize what I'm going to tell you: educational achievement in the UK has been stagnant for over a decade and later-in-life learning is flat or declining.

I'm going to focus on England because it's the largest of the UK countries, but the story is similar in Scotland, Wales, and Northern Ireland.

# A slice of PISA

The OECD has run PISA (Programme for International Student Assessment) every three years since 2000. It's an assessment of 15-year-old students' achievement in science, math, and reading across several countries. The idea is you can compare performance across nations across time using the same yardstick. Currently, 79 countries take part.

Of course, not every 15-year old in every country takes the test. In 2018 in the UK, around 13,000 students took the test - which means there was sampling. Sampling implies there will be some uncertainty in the results, so small year-to-year changes are insignificant. How statistical significance is calculated for a sample like this is well-known and I won't go into the theory here.

The press (and governments) give a lot of attention to the country league table of results, but I'm going to focus on the scores themselves.

# Standing still

The National Foundation for Education Research in the UK produces summary results and more detailed results for England, Northern Ireland, Scotland, and Wales. I'm just going to focus on the results for England.

Here are the results for math, science, and reading. Note the y-axis and the fact I've zoomed in on the scores.

With the exception of Math in 2009 and 2012, there have been no statistically significant changes in results from the 2018 results. This is so important I'm going to say it again in another way. The performance of English 15-year olds in math, science, and reading has not measurably changed from 2006 to 2018.

Let that sink in. Despite 12 years of government policy, despite 12 years of research, despite 12 years of the widespread adoption of computing technology in the classroom, English students' performance has not measurably changed when measured with a fair and consistent international test.

# Not learning later in life

We live in a world where change is a constant. New technology is re-making old industries all the time. Whatever qualifications you have, at some point you'll need retraining. Are people in the UK learning post-formal education?

The Learning and Work Institute tracks participation in learning post-formal education; they have estimates of the fraction of the UK adult population that are currently taking some form of training. Here are the results from 1996 to 2019. The blue line is the fraction of the population currently taking some form of training, and the red line is the fraction of the population who have never taken any form of training course since leaving full-time education.

The rates of participation in lifelong learning are at best steady-state and at worst declining. The Institute breaks the data down by social class (important in the UK) and age when someone left full-time education (a proxy for their level of education). Unsurprisingly, participation rates in lifelong learning are higher for those with more education and they're lower the further down the social class scale you go.

# Younger people have worse skills

In 2016, the OECD published a study on adult skills in England. The study was worrying. To quote from the report:

• "In most countries, but not in England, younger people have stronger basic skills than the generation of people approaching retirement."
• "In England, one-third of those aged 16-19 have low basic skills."
• "England has three times more low-skilled people among those aged 16-19 than the best-performing countries like Finland, Japan, Korea and the Netherlands. Much of this arises from weak numeracy (and to a lesser extent literacy) performance on average."
• "Around one in ten of all university students in England have numeracy or literacy levels below level 2."
• "Most low-skilled people of working age are in employment."

For context, people with level 2 skills or below "struggle to estimate how much petrol is left in the petrol tank from a sight of the gauge, or not be able to fully understand instructions on a bottle of aspirin".

# Education at a glance

The OECD summarized a lot of data for the UK as a whole in a 2020 summary. The results are a mixed bag, some good things, a lot of bad things. Here are a few cherry-picked quotes:

• "In United Kingdom, the proportion of adults employed in the private sector and participating in job-related non-formal education and training not sponsored by the employer is low compared to other OECD and partner countries. (3 %, rank 31/36 , 2016)"
• "In United Kingdom, the number of annual hours of participation of adults in formal education and training is comparatively low (169 %, rank 26/26 , 2016)"
• "In United Kingdom, the share of capital expediture on primary education is one of the smallest among OECD and partner countries with available data. (3.3 %, rank 27/32 , 2017)"

# Politics and the press

Education is a long-term process. If a government invests in education for 5 year-olds, it will be 10 years or more before the effects are apparent in exam results. Most education ministers only last a few years in the job and they want some kind of results quickly. This tends to focus policy on the short-term.

In the UK, the government has tinkered with the qualification system for 16 and 18-year olds. The net effect is to make it hard to compare results over time. For many years, average grades were going up and many commentators were convinced standards were slipping, but of course, it delivered the results politicians wanted.

The PISA results cut through all this and expose a system that's not improving. The politicians' response was to point at the country league tables and make vaguely positive comments about pupil achievements.

What about the decrease in adult education and training and the OECD report on skills? As far as I can tell, silence. I couldn't even find a decent discussion in the press.

# Is there a way forward?

I don't think there are any easy answers for 15-year olds, and certainly, none that are quick and cheap. What might help is a more mature discussion of what's going on, with some honesty and openness from politicians. Less tinkering with the exam system would be a good idea.

For adult learning, I'm very skeptical of a way forward without a significant cultural change. I can see a huge and widening educational divide in the UK. Graduates are training and retraining, while non-graduates are not. This is not good for society.

# $$\beta$$ is $$\alpha$$ if there's an effect

In hypothesis testing, there are two kinds of errors:

• Type I - we say there's an effect when there isn't. The threshold here is $$\alpha$$.
• Type II - we say there's no effect when there really is an effect. The threshold here is $$\beta$$.
This blog post is all about explaining and calculating $$\beta$$.

# The null hypothesis

Let's say we do an A/B test to measure the effect of a change to a website. Our control branch is the A branch and the treatment branch is the B branch. We're going to measure the conversion rate $$C$$ on both branches. Here are our null and alternative hypotheses:

• $$H_0: C_B - C_A = 0$$ there is no difference between the branches
• $$H_1: C_B - C_A \neq 0$$ there is a difference between the branches

Remember, we don't know if there really is an effect, we're using procedures to make our best guess about whether there is an effect or not, but we could be wrong. We can say there is an effect when there isn't (Type I error) or we can say there is no effect when there is (Type II error).

Mathematically, we're taking the mean of thousands of samples so the central limit theorem (CLT) applies and we expect the quantity $$C_B - C_A$$ to be normally distributed. If there is no effect, then $$C_B - C_A = 0$$, if there is an effect $$C_B - C_A \neq 0$$.

# $$\alpha$$ in a picture

Let's assume there is no effect. We can plot out our expected probability distribution and define an acceptance region (blue, 95% of the distribution) and two rejection regions (red, 5% of the distribution). If our measured $$C_B - C_A$$ result lands in the blue region, we will accept the null hypothesis and say there is no effect, If our result lands in the red region, we'll reject the null hypothesis and say there is an effect. The red region is defined by $$\alpha$$.

One way of looking at the blue area is to think of it as a confidence interval around the mean $$x_0$$:

$\bar x_0 + z_\frac{\alpha}{2} s \; and \; \bar x_0 + z_{1-\frac{\alpha}{2}} s$

In this equation, s is the standard error in our measurement. The probability of a measurement $$x$$ lying in this range is:

$0.95 = P \left [ \bar x_0 + z_\frac{\alpha}{2} s < x < \bar x_0 + z_{1-\frac{\alpha}{2}} s \right ]$

If we transform our measurement $$x$$ to the standard normal $$z$$, and we're using a 95% acceptance region (boundaries given by $$z$$ values of 1.96 and -1.96), then we have for the null hypothesis:

$0.95 = P[-1.96 < z < 1.96]$

# $$\beta$$ in a picture

Now let's assume there is an effect. How likely is it that we'll say there's no effect when there really is an effect? This is the threshold $$\beta$$.

To draw this in pictures, I want to take a step back. We have two hypotheses:

• $$H_0: C_B - C_A = 0$$ there is no difference between the branches
• $$H_1: C_B - C_A \neq 0$$ there is a difference between the branches

We can draw a distribution for each of these hypotheses. Only one distribution will apply, but we don't know which one.

If the null hypothesis is true, the blue region is where our true negatives lie and the red region is where the false positives lie. The boundaries of the red/blue regions are set by $$\alpha$$. The value of $$\alpha$$ gives us the probability of a false positive.

If the alternate hypothesis is true, the true positives will be in the green region and the false negatives will be in the orange region. The boundary of the green/orange regions is set by $$\beta$$. The value of $$\beta$$ gives us the probability of a false negative.

# Calculating $$\beta$$

Calculating $$\beta$$ is calculating the orange area of the alternative hypothesis chart. The boundaries are set by $$\alpha$$ from the null hypothesis. This is a bit twisty, so I'm going to say it again with more words to make it easier to understand.

$$\beta$$ is about false negatives. A false negative occurs when there is an effect, but we say there isn't. When we say there isn't an effect, we're saying the null hypothesis is true. For us to say there isn't an effect, the measured result must lie in the blue region of the null hypothesis distribution.

To calculate $$\beta$$, we need to know what fraction of the alternate hypothesis lies in the acceptance region of the null hypothesis distribution.

Let's take an example so I can show you the process step by step.

1. Assuming the null hypothesis, set up the boundaries of the acceptance and rejection region. Assuming a 95% acceptance region and an estimated mean of x, this gives the acceptance region as:
$P \left [ \bar x_0 + z_\frac{\alpha}{2} s < x < \bar x_0 + z_{1-\frac{\alpha}{2}} s \right ]$ which is the mean and 95% confidence interval for the null hypothesis. Our measurement $$x$$ must lie between these bounds.
2. Now assume the alternate hypothesis is true. If the alternate hypothesis is true, then our mean is $$\bar x_1$$.
3. We're still using this equation from before, but this time, our distribution is the alternate hypothesis.
$P \left [ \bar x_0 + z_\frac{\alpha}{2} s < x < \bar x_0 + z_{1-\frac{\alpha}{2}} s \right ] ]$
4. Transforming to the standard normal distribution using the formula $$z = \frac{x - \bar x_1}{\sigma}$$, we can write the probability $$\beta$$ as:
$\beta = P \left [ \frac{\bar x_0 + z_\frac{\alpha}{2} s - \bar x_1}{s} < z < \frac{ \bar x_0 + z_{1-\frac{\alpha}{2}} s - \bar x_1}{s} \right ]$

This time, let's put some numbers in.

• $$n = 200,000$$ (100,000 per branch)
• $$C_B = 0.062$$
• $$C_A = 0.06$$
• $$\bar x_0= 0$$ - the null hypothesis
• $$\bar x_1 = 0.002$$ - the alternate hypothesis
• $$s = 0.00107$$  - this comes from combining the standard errors of both branches, so $$s^2 = s_A^2 + s_B^2$$, and I'm using the usual formula for the standard error of a proportion, for example $$s_A = \sqrt{\frac{C_A(1-C_A)}{n} }$$
Plugging them all in, this gives:
$\beta = P[ -3.829 < z < 0.090]$
which gives $$\beta = 0.536$$

# This is too hard

This process is complex and involves lots of steps. In my view, it's too complex. It feels to me that there must be an easier way of constructing tests. Bayesian statistics holds out the hope for a simpler approach, but widespread adoption of Bayesian statistics is probably a generation or two away. We're stuck with an overly complex process using very difficult language.

# Scientific fraud and business manipulation

Sadly, there's a long history of scientific fraud and misrepresentation of data. Modern computing technology has provided better tools for those trying to mislead, but the fortunate flip side is, modern tools provide ways of exposing misrepresented data. It turns out, the right tools can indicate what's really going on.

(Author: Nick Youngson. License: Creative Commons. Source: Wikimedia)

In business, companies often say they can increase sales, or reduce costs, or do so some other desirable thing. The evidence is sometimes in the form of summary statistics like means and standard deviations. Do you think you could assess the credibility of evidence based on the mean and standard deviation summary data alone?

In this blog post, I'm going to talk about how you can use one tool to investigate the credibility of mean and standard deviation evidence.

# Discrete quantities

Discrete quantities are quantities that can only take discrete values. An example is a count, for example, a count of the number of sales. You can have 0, 1, 2, 3... sales, but you can't have -1 sales or 563.27 sales.

Some business quantities are measured on scales of 1 to 5 or 1 to 10, for example, net promoter scores or employee satisfaction scores. These scales are often called Likert scales.

For our example, let's imagine a company is selling a product on the internet and asks its customers how likely they are to recommend the product. The recommendation is on a scale of 0 to 10, where 0 is very unlikely to recommend and 10 is very likely to recommend. This is obviously based on the net promoter idea, but I'm simplifying things here.

Very unlikely to recommend                   Very likely to recommend
0 1 2 3 4 5 6 7 8 9 10

Imagine the salesperson for the company tells you the results of a 500 person study are a mean of 9 and a standard deviation of 2.5. They tell you that customers love the product, but obviously, there's some variation. The standard deviation shows you that not everyone's satisfied and that the numbers are therefore credible.

But are these numbers really credible?

Stop for a second and think about it. It's quite possible that their customers love the product. A mean of 9 on a scale of 10 isn't perfection, and the standard deviation of 2.5 suggests there is some variation, which you would expect. Would you believe these numbers?

# Investigating credibility

We have three numbers; a mean, a standard deviation, and a sample size. Lots of different distributions could have given rise to these numbers, how can we backtrack to the original data?

The answer is, we can't fully backtrack, but we can investigate possibilities.

In 2018, a group of academic researchers in The Netherlands and the US released software you can use to backtrack to possible distributions from mean and standard deviation data. Their goal was to provide a tool to help investigate academic fraud. They wrote up how their software works and published it online, you can read their writeup here. They called their software SPRITE (Sample Parameter Reconstruction via Iterative TEchniques) and made it open-source, even going so far as to make a version of it available online. The software will show you the possible distributions that could give rise to the summary statistics you have.

One of the online versions is here. Let's plug in the salesperson's numbers to see if they're credible.

If you go to the SPRITE site, you'll see a menu on the left-hand side. In my screenshot, I've plugged in the numbers we have:

• Our scale goes from 0 to 10,
• Our mean is 9,
• Our standard deviation is 2.5,
• The number of samples is 500.
• We'll choose 2 decimal places for now
• We'll just see the top 9 possible distributions.

Here are the top 9 results.

Something doesn't smell right.  I would expect the data to show some form of more even distribution about the mean. For a mean of 9, I would expect there to be a number of 10s and a number of 8s too. These estimated distributions suggest that almost everyone is deliriously happy, with just a small handful of people unhappy. Is this credible in the real world? Probably not.

I don't have outright evidence of wrong-doing, but I'm now suspicious of the data. A good next step would be to ask for the underlying data. At the very least, I should view any other data the salesperson provides with suspicion. To be fair to the salesperson, they were probably provided with the data by someone else.

What if the salesperson had given me different numbers, for example, a mean of 8.5, a standard deviation of 1.2, and 100 samples? Looking at the results from SPRITE, the possible distributions seem much more likely. Yes, misrepresentation is still possible, but on the face of it, the data is credible.

# Did you spot the other problem?

There's another, more obvious problem with the data. The scale is from 0 to 10, but the results are a mean of 9 and a standard deviation of 2.5, which imply a confidence interval of 6.5 to 11.5. To state the obvious, the maximum score is 10 but the upper range of the confidence interval is 11.5. This type of mistake is very common and doesn't of itself indicate fraud. I'll blog more about this type of mistake later.

# What does this mean?

Due diligence is about checking claims for veracity before spending money. If there's a lot of money involved, it behooves the person doing the due diligence to check the consistency of the numbers they've been given. Tools like SPRITE are very helpful for sniffing out areas to check in more detail. However, just because a tool like SPRITE flags something up it doesn't mean to say there's fraud; people make mistakes with statistics all the time. However, if something is flagged up, you need to get to the bottom of it.

# The echoes of history

Sometimes, there are weird connections between very different cultural areas and we see the echoes of history playing out. I'm going to tell you how pulsars, Nobel Prizes, an iconic album cover, Nazi atrocities, and software chart plotting all came to be connected.

# The discovery of pulsars

In 1967, Jocelyn Bell was working on her Ph.D. as a post-graduate researcher at the Mullard Radio Astronomy Observatory, near Cambridge in the UK. She had helped build a new radio telescope and now she was operating it. On November 28, 1967, she saw a strikingly unusual and regular signal, which the team nicknamed "little green men". The signal turned out to be a pulsar, a type of star new to science.

This was an outstanding discovery that shook up astronomy. The team published a paper in Nature, but that wasn't the end of it. In 1974, the Nobel committee awarded the Nobel Physics Prize to the team. To everyone on the team except Jocelyn Bell

Over the years, there's been a lot of controversy over the decision, with many people thinking she was robbed of her share of the prize, either because she was a Ph.D. student or because she was a woman. Bell herself has been very gracious about the whole thing; she is indeed a very classy lady.

# The pulsar and early computer graphics

In the late 1960s, a group of Ph.D. students from Cornell University were analyzing data from the pulsar Bell discovered. Among them was Harold Craft, who used early computer systems to visualize the results. Here's what he said to the Scientific American in 2015: "I found that it was just too confusing. So then, I wrote the program so that I would block out when a hill here was high enough, then the stuff behind it would stay hidden."

Here are three pages from Craft's Ph.D. thesis. Take a close look at the center plot. If Craft had made every line visible, it would have been very difficult to see what was going on. Craft re-imagined the data as if he were looking at it at an angle, for example, as if it were a mountain range ridgeline he was looking down on.  With a mountain ridgeline, the taller peaks hide what's behind them. It was a simple idea, but very effective.

(Credit: JEN CHRISTIANSEN/HAROLD D. CRAFT)

The center plot is very striking. So striking in fact, that it found its way into the Cambridge Encyclopaedia of Astronomy (1977 edition):

(Cambridge Encyclopedia of Astronomy, 1977 edition, via Tim O'Riley)

# Joy Division

England in the 1970s was not a happy place, especially in the de-industrialized north. Four young men in Manchester had formed a band and recorded an album. The story goes that one of them, Bernard Sumner, was working in central Manchester and took a break in the city library. He came across the pulsar image in the encyclopedia and liked it a lot.

The band needed an image for their debut album, so they selected this one. They gave it to a recently graduated designer called Peter Saville, with the instructions it was to be a black on white image. Saville felt the image would look better white on black, so he designed this cover.

This is the iconic Unknown Pleasures album from Joy Division.

The starkness of the cover, without the band's name or the album's name, set it apart. The album itself was critically acclaimed, but it never rose high in the charts at the time. However, over time, the iconic status of the band and the album cover grew. In 1980, the lead singer, Ian Curtis, committed suicide. The remaining band members formed a new band, New Order, that went on to massive international fame.

By the 21st century, versions of the album cover were on beach towels, shoes, and tattoos.

# Joy plots

In 2017, Claus Wilke created a new charting library for R, ggjoy.  His package enabled developers to create plots like the famous Unknown Pleasures album cover. In honor of the album cover, he called these plots joy plots.

# Ridgeline plots

This story has a final twist to it. Although joy plots sound great, there's a problem.

Joy Division took their name from a real Nazi atrocity fictionalized in a book called House of Dolls. In some of their concentration camps, the Nazis forced women into prostitution. The camp brothels were called Joy Division

The name joy plots was meant to be fun and a callback to an iconic data visualization, but there's little joy in evil. Given this history, Wilke renamed his package ggridges and the plots ridgeline plots.

Here's an example of the great visualizations you can produce with it.

If you search around online, you can find people who've re-created the pulsar image using ggridges.

It's not just R programmers who are playing with Unknown Pleasures, Python programmers have got into the act too. Nicolas P. Rougier created a great animation based on the pulsar data set using the venerable Matplotlib plotting package - you can see the animation here.

# If you liked this post

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# What is truth?

Statistical testing is ultimately all about probabilities and thresholds for believing an effect is there or not. These thresholds and associated ideas are crucial to the decision-making process but are widely misunderstood and misapplied. In this blog post, I'm going to talk about three testing concepts: confidence, significance, and p-values; I'll deal with the hugely important topic of statistical power in a later post.

(peas - not p-values. Author: Malyadri, Source: Wikimedia Commons, License: Creative Commons)

# Types of error

To simplify, there are two kinds of errors in statistical testing:

• Type I - false positive. You say there's an effect when there isn't. This is decided by a threshold $$\alpha$$, usually set to 5%. $$\alpha$$ is called significance.
• Type II - false negative. You say there isn't an effect but there is. This is decided by a threshold $$\beta$$ but is usually expressed as the statistical power which is $$1 - \beta$$.

In this blog post, I'm going to talk about the first kind of error, Type I.

# Distribution and significance

Let's imagine you're running an A/B test on a website and you're measuring conversion on the A branch ($$c_A$$) and on the B branch ($$c_B$$). The null hypothesis is that there's no effect, which we can write as:

$H_0: c_A - c_B = 0$

This next piece is a little technical but bear with me. Tests of conversion are usually large tests (mostly, > 10,000 samples in practice). The conversion rate is the mean conversion for all website visitors. Because there are a large number of samples, and we're measuring a mean, the Central Limit Theorem (CLT) applies, which means the mean conversion rates will be normally distributed. By extension from the CLT, the quantity $$c_A - c_B$$ will also be normally distributed. If we could take many measurements of $$c_A - c_B$$ and the null hypothesis was true, we would theoretically expect the results to look like something like this.

Look closely at the chart. Although I've cut off the x-axis, the values go off to $$\pm \infty$$. If all values of $$c_A - c_B$$ are possible, how can we reject the null hypothesis and say there's an effect?

# Significance - $$\alpha$$

To decide if there's an effect there, we use a threshold. This threshold is referred to as the level of significance and is called $$\alpha$$. It's usually set at the 0.05 or 5% level. Confusingly, sometimes people refer to confidence instead, which is 1 - significance, so a 5% significance level corresponds to a 95% confidence level.

In the chart below, I've colored the 95% region around the mean value blue and the 5% region (2.5% at each end) red. The blue region is called the acceptance region and the red region is called the rejection region.

What we do is compare the measurement we actually make with the chart. If our measurement lands in the red zone, we decide there's an effect there (reject the null), if our measurement lands in the blue zone, we'll decide there isn't an effect there (fail to reject the null or 'accept' the null).

# One-sided or two-sided tests

On the chart with the blue and the red region, there are two rejection (red) regions. This means we'll reject the null hypothesis if we get a value that's more than a threshold above or below our null value. In most tests, this is what we want; we're trying to detect if there's an effect there or not and the effect can be positive or negative. This is called a two-sided test because we can reject the null in two ways (too negative, too positive).

But sometimes, we only want to detect if the treatment effect is bigger than the control. This is called a one-sided test. Technically, the null hypothesis in this case is:

$H_0: c_A - c_B \leq 0$

Graphically, it looks like this:

So we'll reject the null hypothesis only if our measured value lands in the red region on the right. Because there's only one rejection region and it's on one side of the chart, we call this a one-sided test.

# p-values

I've very blithely talked about measured values landing in the rejection region or not. In practice, that's not what we do; in practice, we use p-values.

Let's say we measured some value x. What's the probability we would measure this value if the null hypothesis were true (in other words, if there were no effect)? Technically, zero because the distribution is continuous, but that isn't helpful. Let's try a more helpful form of words. Assuming the null hypothesis is true, what's the probability we would see a value of x or a more extreme value? Graphically, this looks something like the green area on the chart below.

Let me say again what the p-value is. If there's no effect at all, it's the probability we would see the result (or a more extreme result) that we measured, or how likely is it that our measurement could have been due to chance alone?

The chart below shows the probability the null hypothesis is true; it shows the acceptance region (blue), the rejection region (red), and the measurement p-value (green). The p-value is in the rejection region, so we'll reject the null hypothesis in this case. If the green overlapped with the blue region, we would accept (fail to reject) the null hypothesis.

# Misunderstandings

There are some common misunderstandings around testing that can have quite bad commercial effects.

• 95% confidence is too high a bar - we should drop the threshold to 90%. In effect, this means you'll accept a lot of changes that have no effect. This will reduce the overall effectiveness of your testing program (see this prior blog post for an explanation).
• One-sided tests are OK and give a smaller sample size, so we should use them. This is true, but it's often important to determine if a change is having a negative effect. In general, hypothesis testing tests a single hypothesis, but sadly, people try and read more into test results than they should and want to answer several questions with a single question.
• p-values represent the probability of an effect being present. This is just not true at all.
• A small p-value indicates a big effect. p-values do not make any indication about the size of an effect; a low p-value does not mean there's a big effect.

# Practical tensions

In practice, there can be considerable tension between business and technical people over statistical tests. A lot of statistical practices (e.g. 5% significance levels, two-sided testing) are based on experience built up over a long time. Unfortunately, this all sounds very academic to the business person who needs results now and wants to take shortcuts. Sadly, in the long run, shortcuts always catch you up. There's an old saying that's very true: "there ain't no such thing as a free lunch."