4.2 Finding Patterns in Random Noise

Humans are so good at identifying patterns that we often see them even when it is really noise in masquerade. When we think we have seen a pattern, how do we quantify the level of confidence correctly? We describe common pitfalls that lead to an overconfidence in an apparent pattern, some that even prey on the inattentive scientist!

The Lesson in Context

This lesson continues 4.1 Signal and Noise by elaborating on ways in which random noise can emulate signals (produce apparent patterns) in many different contexts. We introduce the idea of [math]\displaystyle{ p }[/math]-values to quantify the statistical significance of patterns and describe various tempting statistical fallacies we tend to make as laypersons or scientists, such as gambler's fallacy and [math]\displaystyle{ p }[/math]-hacking. We play a game in which students try to produce a random string of coin tosses by thought, which reveals that a truly random string in fact contains more apparent patterns than one intuitively expects. Two other activities also illustrate how spurious patterns are in fact expected to arise from random noise.

Earlier Lessons

4.1 Signal and Noise

• A signal is a particular pattern of the data that we are seeking, while noise is something that introduces uncertainty or error into the measurement of that data, in a way that sometimes produces spurious signals.
• [math]\displaystyle{ p }[/math]-values are one way to quantify how statistically significant a measured signal is compared to the noise. It is defined as the probability that the measured signal is produced entirely by random noise alone even when the underlying cause of a signal is absent.

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Later Lessons

9.2 Biases

• Cognitive heuristics and biases such as confirmation bias may mislead us into seeing a pattern in random data where there is none.

11.1 Pathological Science

• [math]\displaystyle{ p }[/math]-hacking is one source of pathological science, where authors fail to disclose the measurements in which the supposed signal is not observed, thereby falsely inflating the statistical significance of the reported signal.

Takeaways

After this lesson, students should

1. Understand that people tend to see any regularity as a meaningful pattern (i.e., see more signal than there is), even when "patterns" occur by chance (i.e. are pure noise).
2. Recognize cases of the Look Elsewhere Effect in daily life when you hear phrases such as "what are the odds".
3. Recognize and explain the flaw in scenarios in which scientists and other people mistake noise for signal.
4. Resist the opposing temptations of both the Gambler's Fallacy (the expectation that a run of similar events will soon break and quickly balance out, because of the assumption that small samples resemble large samples) and the Hot-hand Fallacy (the expectation that a run will continue, because runs suggest non-randomness).
5. (Data Science) Describe the difference between the effect size (strength of pattern) and credence level (probability that the pattern is real), and identify the role each plays in decision making.

People underestimate the frequency of apparent patterns produced by randomness, leading to over-perception of spurious signal much more frequently than people account for. Events that are just coincidental are much more likely than most people expect.

Look Elsewhere Effect

If there is a low probability of obtaining a false positive in any given instance, the more times you try (the more questions you ask, measures you take, or studies you run without statistical correction), the more you increase the probability of getting some false positive.

Since humans are so good at identifying noise that looks like signals, it is easy to find and fall victim to this even if it doesn't seem like we're asking too many questions. The look elsewhere effect can be avoided by clearly stating the questions you're asking before seeing the data.

Things that Cause the Look Elsewhere Effect

• Asking too many questions of the same data set, reporting only statistically significant results.
• Asking the same question of multiple data sets, reporting only statistically significant results.
• Running a test or similar tests too many times, reporting only statistically significant results.
• The effect also occurs in everyday life, e.g. when one looks at a whole lot of phenomena and only takes note of the most surprising-looking patterns, not properly taking into account the larger number of unsurprising patterns/lack of pattern.

[math]\displaystyle{ p }[/math]-hacking

A subset of the Look Elsewhere Effect that occurs when people conduct multiple statistical tests and only report those with [math]\displaystyle{ p }[/math]-values under .05 (the traditional threshold for publication and statistical significance, which indicates a tolerance of 5% false positives).

A [math]\displaystyle{ p }[/math]-value cutoff of .05 thus indicates that, 1 out of 20 analyses of pure noise would discover a spurious signal. [math]\displaystyle{ p }[/math]-hacking is statistically problematic but more often a result of misunderstanding than deliberate fraudulence.

Common Techniques for [math]\displaystyle{ p }[/math]-hacking

• Running different statistical analyses on the same dataset and only reporting the statistically significant ones.
• Analyzing multiple DVs and only reporting the statistically significant ones.
• Gradually increasing the sample size until the [math]\displaystyle{ p }[/math]-value falls below .05.

[math]\displaystyle{ p }[/math]-hacking sounds malicious. But, it's easy to do inadvertently even as a professional researcher!

HARKing

Hypothesizing after the results are known (HARKing) is the act of finding a hypothesis that your data supports after collecting and looking through it.

Gambler's Fallacy

Expecting that streaks (e.g. Tails Tails Tails Tails) will be broken, such that future results will quickly "average out" earlier ones, even when all trials are independent.

Hot-hand Fallacy

Expecting that streaks (e.g. winning hands in Poker) will continue, even when all trials are independent.

These two fallacies lead in opposite directions, but are both a result of the misconception that small samples (or short runs) will resemble large samples (or long runs), forgetting about statistical uncertainty. The Gambler's Fallacy arises because people assume a small sample will look like a large sample, such that a run of e.g. Tails will quickly be balanced out by Heads. The Hot Hand Fallacy arises when a run (e.g. of Tails) makes people think the sequence isn't truly random, but an effect of skill or luck that will continue. Both fallacies arise because long runs are commoner in random sequences than people expect.

File-drawer Effect

Also called publication bias. The effect where researchers tend to publish results which confirm their hypothesis. Null results, which are less exciting, may be left in the "file-drawer."

Mr. Goxx

A hamster that actively manages a cryptocurrency portfolio by running in his "intention wheel" to determine what he's buying/selling and then goes through either a "BUY" or "SELL" decision-tunnel to decide what he's doing with it. As of october 2021, his portfolio was up nearly 30% from june when he started trading crypto. His decisions are streamed live on Twitch.

Mr. Goxx on Twitch

Live stream of Mr. Goxx's trading decisions.

Cold Reading in a Crowd

A medium shouts out to a crowd somewhat common names, such as William and Butler, and common ailments such as, "heart problem" or "passed in their sleep". It is very likely that a couple people in the crowd can match a majority of these features to someone in their life, making the claims seem impressively accurate (seemingly small [math]\displaystyle{ p }[/math]-value), but this ignores all the other members of the audience who could not match any of the claims. This is an illustration of [math]\displaystyle{ p }[/math]-hacking.

James Randi and a Psychic Artist

A demonstration of this in action.

[math]\displaystyle{ p }[/math]-hacking Through Incrementing Sample Size

"We conducted the study on 1000 participants, and our [math]\displaystyle{ p }[/math]-value is just slightly above 0.05. Let's recruit another 20 participants to see if our [math]\displaystyle{ p }[/math]-value can dip under 0.05." If they keep increasing their sample size by 20 each time, the [math]\displaystyle{ p }[/math]-value will fluctuate just from chance alone, possibly dipping below 0.05 even if there isn't a real signal. This is equivalent to only selecting a subsample of the data that would confirm a hypothesis, omitting that an even larger sample would have rejected the same hypothesis.

"What are the odds!"

Every time you hear this, be suspicious. The odds are probably higher than you'd think, if you take into account all the similar events that didn't include anything surprising.

Running Into Someone

You're on vacation in London and you run into an old friend that was also there on vacation. It may be unlikely that this specific friend came to this exact spot on vacation. But, you're bound to run into someone you've met before on some trip at some point, especially if you associate with people likely to visit similar places.

COVID Origin

What are the odds that COVID originated in a market where there just so happened to be a nearby major virology research institute and a worker went home sick just before the outbreak? Viral outbreaks are likely to occur in densely populated areas in major cities. Major cities tend to have major research institutes/hospitals that might have some virology component, especially in locations where there are higher risks of new viruses emerging like high population density and/or wet markets. Anything in the city could be considered nearby. Workers go home sick all the time. All in all, it is insufficient to conclude whether or not COVID originated from that specific lab based solely on the perceived unlikeliness of the circumstances.

Exemplary Quotes

“I know it seems super meaningful that we ran into each other in Australia, when neither of us live in Australia, but I guess the chances of running into someone you know at some point, if you travel a lot and know a lot of people, are pretty high.”

“There are many many cases of people making insanely correct predictions, so many that some people are convinced clairvoyance is real. But there are many more cases of people making totally wrong predictions. So it's probably just noise; with enough predictions, someone will be correct by luck.”

The Look Elsewhere Effect means that more data leads to more misinterpretations, or "too much data is bad."

In fact, more data reduces statistical uncertainty and thus strengthens inferences. What can lead to more misinterpretations through the Look Elsewhere Effect is when people ask too many statistical questions of the dataset without correcting for multiple comparisons.

More separate analyses someone runs, the better their analysis will be.

In fact, the more analyses someone runs, the more likely they are to hit upon a false positive, unless they correct for multiple comparisons (which can be done statistically).

Wow, that was such a striking coincidence! It must have some hidden significance.

Even very striking coincidences are bound to happen if you look across enough events. We forget how many events we have looked through, and remember only the coincidence. (See 10.1 Confirmation Bias.)

After this lesson, students should

1. Attitudes
   1. Be wary of our tendency to see patterns that do not exist (to see signal where there is in fact only noise).
2. Concept Acquisition
   1. People are (evolutionarily?) disposed to over-perceive signal (i.e., noise often gets misinterpreted as signal), perhaps because the cost of missing real signal (false negatives) is typically higher than the cost of mistaking noise for signal (false positives).
   2. People tend to see any regularity as a pattern (i.e., see more signal than there is), even when "patterns" occur by chance (i.e. are pure noise).
      1. e.g. People underestimate the frequency of apparent patterns produced by randomness, leading to overperception of spurious signal much more frequently than people account for. (Events that are just coincidental are much more likely than most people expect.)
   3. Gambler's Fallacy: Expecting that streaks will be broken, such that future results will "average out" earlier ones, even when all trials are independent.
   4. Hot-Hand Fallacy: Expecting that streaks will continue, even when all trials are independent.
   5. Look Elsewhere Effect: Even if there is a low probability of pure noise passing a given threshold for signal, if we look at enough noise some of it will pass that threshold by chance. That is, if there is a low probability of obtaining a false positive in any given instance, the more times you try (the more questions you ask, measures you take, or studies you run without statistical correction), the more you increase the probability of getting a false positive. This occurs when one:
      1. Asks too many questions of the same data set, reporting only statistically significant results.
      2. Asks the same question of multiple data sets, reporting only statistically significant results.
      3. Runs a test or similar tests too many times, reporting only statistically significant results.
      4. This also occurs in everyday life, e.g. when one looks at a whole lot of phenomena and only takes note of the most surprising-looking patterns, not properly taking into account the larger number of unsurprising patterns/lack of pattern.
   6. Statistical Significance: How unlikely a given set of results would be if the null hypothesis were true (i.e. if the hypothesized effect did not actually exist).
   7. [math]\displaystyle{ p }[/math]-values: The probability of getting a result as extreme or more if in fact the hypothesis is false, simply through random noise.
   8. Lack of statistical significance does NOT prove the null hypothesis.
3. Concept Application
   1. Describe how scientists guard against detecting a signal that does not exist.
   2. Recognize and explain the flaw in everyday scenarios in which people mistake noise for signal (e.g. Look Elsewhere Effect, gambler's fallacy, hot-hand effect).
   3. Recognize and explain the flaw in a scenario where scientists mistake noise for signal.
   4. Given a news article or other concrete example, correctly extract the effect size versus statistical significance of a causal factor, and explain how each affects the importance and usefulness of the results.

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