A Tale of Three Numbers

Statistical Significance,Effect Size, and

Sample Size

BRIEF REVIEW

Causation vs. Correlation

When two variables A and B are correlated, there are four possibilities:

1. A causes B2. B causes A3. A common cause C causes both A and B4. The correlation is accidental

So, discovering that countries with democratic elections get in fewer wars, we might conclude:

1. Democracy causes peace.2. Peace causes democracy.3. Christianity causes both democracy and peace.4. Democracy and peace are only accidentally

correlated.

Observational Studies

Importantly, if we just observe the facts and collect data on how things are, we cannot tell which hypothesis is true.

Observational studies find correlations, not the causal structure of the world. (This is what HW4 was about.)

The Best Evidence

So far, we’ve learned that a good experiment or clinical trial is:

• Randomized• Double-blind• Controlled

This is often abbreviated ‘RCT’: Randomized Controlled Trial.

Controls

An experiment with no controls is useless.

It tells us what happens when we do X, but not what happens when we don’t do X (control).

Maybe the same results would happen from not doing X. Maybe X does nothing. Or a lot. Or a little. With no controls, it is impossible to tell.

Randomization

An experiment or trial is randomized when each person who is participating in the experiment/ trial has a fair and equal chance of ending up either in the control group or the experimental group.

Benefits of Randomization

Proper randomization:

Minimizes experimenter bias– the experimenter can’t bias who goes into which group.

Minimizes allocation bias– lowers the chance that the control group and experimental group differ in important ways.

Selection Bias

Randomization cannot get rid of all selection bias.

For example, many psychology experiments are just performed on American undergraduates by their professors.

This means both groups over-represent young Westerners. (“Sampling bias”)

Allocation Bias

Randomization also guards against allocation bias, where the control group and experimental group are different in important ways.

For example, if you assign the first 20 people to enroll in the experiment to the control and the next 20 to the experimental group, there may be allocation bias: the first to enroll may be more eager to take part, because they are sicker.

The Importance of Randomization

Previously we saw that improper randomization procedures on average exaggerated effects by 41%.

This is an average result, so improper randomization often leads to exaggerations that are even larger than 41%.

Why RCTs?

The importance of the experimental method (as opposed to scientific observation) is that it allows us to discern the causal structure of the world.

Causal Structure

If we find a correlation between our experimental treatment T and our desired outcome O, we can rule out:• O caused T in the experiment.• A common cause C caused both O and T in the

experiment.

Causal Structure

But can we determine whether the correlation between T and O is real in the first place and not accidental?

Yes!

STATISTICAL SIGNIFICANCE

Statistical Significance

We say that an experimental correlation is statistically significant if it’s unlikely to be accidental.

How can we tell when it’s unlikely to be accidental?

Null Hypothesis

We give a name to the claim that there is no causal connection between the variables being studied.

It is called the null hypothesis.

Our goal is to reject the null hypothesis when it is false, and to accept it when it is true.

Rejecting the Null Hypothesis

All experimental data is consistent with the null hypothesis. Any correlation can always be due entirely to chance.

But sometimes the null hypothesis doesn’t fit the data very well. When the null hypothesis suggests that our actual observations are very unlikely, we reject the null hypothesis.

P-Values

One way to characterize the significance of an observed correlation is with a p-value.

The p-value is the probability that we would observe our data on the assumption that the null hypothesis is true.

p = P(observations/ null hypothesis = true)

P-Values

Obviously lower p-values are better, that means your observed correlation is more likely to be true.

In science we have an arbitrary cut-off point, 5%. We say that an experimental result with p < .05 is statistically significant.

Statistical Significance

What does p < .05 mean?

It means that the probability that our experimental results would happen if the null hypothesis is true is less than 5%.

According to the null hypothesis, there is less than a 1 in 20 chance that we would obtain these results.

Note

Importantly, p-values are not measures of how likely the null hypothesis is, given the data. They are measures of how likely the data is, given the null hypothesis.

p = P(data/ null hypothesis = true) ≠

P(null hypothesis = true/ data)

Example

Suppose I have a coin, and I hypothesize that the coin is biased toward heads.

The null hypothesis might be “this is a fair coin, it is equally likely to land heads or tails”.

Suppose I then flip it 5 times and it lands HHHHH– heads 5 times in a row.

Example

We know that the probability of this happening if the coin is fair is 1/25 = 1/32 = 0.03125 or about 3%.

P(HHHHH/ the coin is fair) =P(HHHHH/ null hypothesis = true) =p = 3%

Example

So p = .03 < .05, and we can reject the null hypothesis. The bias toward heads is statistically significant.

Importance

Just because the results of an experiment (or observational study) are “statistically significant” does not mean the revealed correlations are important.

The effect size also matters, that is the strength of the correlation.

EFFECT SIZES

Effect Size

One NAEP analysis of 100,000 American students found that science test scores for men were higher than the test scores for women, and this effect was statistically significant

These results are unlikely if the null hypothesis, that gender plays no role in science scores, were true.

Effect Size

However, the average difference between men and women on the test was just 4 points out of 300, or 1.3% of the total score.

Yes, there was a real (statistically significant) difference. It was just a very, very small difference.

Effect Size

One way to put the point might be: “p-values tell you when to reject the null hypothesis. But they do not tell you when to care about the results.”

Measures of Effect Size

There are lots of measures of effect size:

Pearson’s r, Cohen’s f, Cohen’s d, Hedges’ g, Cramér’s V,…

Here we will just talk about two measures that are commonly reported: odds ratios and relative risks.

Odds Ratio

First, let’s introduce the idea of a binary variable. A binary variable is a variable that can have only two values.

“height” is not a binary variable, because there are more than two heights people can have.“got an A” is a binary variable, because either you got an A or you didn’t.

Odds

Whenever you have a binary variable, you can ask about the odds of that variable– what are the odds of getting an A?

If 10 students got A’s out of 50 students, then 10 students passed and 40 failed. The odds of getting an A are 10:40 or 1:4 or 25%.

Odds vs. Probabilities

Odds are not probabilities. There are 50 students and 10 of them got A’s.

The probability of getting an A: 10/50 = 20%

The odds of getting an A: 10/40 = 25%

Odds Ratios

Suppose I have another binary variable “studied”– students either studied for the exam or they didn’t.

I can ask about the odds that a student who studied got an A, and the odds that a student who didn’t study got an A.

In Table Format

Got an A = yes

Got an A = no

Totals

Study = yes 6 15 21Study = no 4 25 29Totals 10 40 50

Odds Ratio

So the odds of getting an A among studiers are 6:15 or 40%.

And the odds of getting an A among non-studiers are 4/25 or 16%.

Odds Ratio

The odds ratio is the ratio of these odds, or 40%:16% ≈ 2.5

This means that (in our example) studying raises the odds that someone will get an A by 150%.

Alternatively: a student who studies has two and a half times better odds of getting an A.

Relative Risk

While odds ratios are appropriate when we have two correlated binary variables in an observational study (as when I observe the effects of studying on getting an A), the effect sizes in RCTs are usually reported by relative risks, which are also called risk ratios.

Relative Risk

Relative risks are just like odds ratios except they compare probabilities and not odds.

The odds that a studying student passes are 6:15 = 40%

The probability is 6/(6 + 15) = 6/21 ≈ 29%

Example

The odds that a non-studying student passes are 4:25 = 16%.

The probability is 4/(4 + 25) = 4/29 ≈ 14%.

Example

Whereas the odds ratio was 40:16 = 250%, we get a relative risk of:

29%:14% = 29:14 = 2.07 = 207%

These numbers are similar, but obviously not the same. The risk ratio tells you that a student who studies is twice as likely to get an A.

Relation

As the probabilities of events get smaller the odds approach the probabilities, and odds ratios and relative risks are similar.

However, as the probabilities of the events get higher, the odds and risk ratios get very different.

Here’s our table again…

Got an A = yes

Got an A = no

Totals

Study = yes 6 15 21Study = no 4 25 29

Totals 10 40 50

Odds Ratio for High Probability Events

The probability of not getting an A is much higher than the probability of getting an A: 40/50 >> 10/50. The odds of study = no, A = no: 25/4 = 6.25The odds of study = yes, A = no: 15/6 = 2.5Odds ratio: 6.25/2.5 = 250%.Not studying increases odds of A = no by one and a half times.

Relative Risk for High Probability Events

What about probabilities?

P(A = no/ study = no) = 86%P(A = no/ study = yes) = 71%Relative risk = 86/71 = 121%

So not studying increases your risk of not getting an A by 21%.

What This Means

What this means is that if you see an effect size reported in the news you must know whether it is an odds ratio or a risk ratio.

Otherwise a seemingly very big difference might actually be a very small difference.

Real Life Case

Here’s a real headline from the NY Times:

“Doctors are only 60% as likely to order cardiac catheterization for women and blacks as for men and whites.”

This sounds like a risk ratio. Doctors refer white men n% of the time and blacks and women 60% of n% of the time. Right?

Large Difference in Risk!

The study found that doctors referred white men to heart specialists 90.6% of the time.

If the “60%” figure is a risk ratio, then they referred blacks and women 60% x 90.6% = 54.4% of the time.

That’s a big difference!

Actually… No

But people who write newspaper articles don’t understand odds ratios and risk ratios.

The probability of a doctor referring a black man or a woman to a heart specialist was 84.7%, not 54.4%.

The article was confusing an odds ratio with a risk ratio.

What’s Going On?

If 90.6% of white males were referred, then 9.4% were not referred, and so a white male's odds of being referred were 90.6/9.4 ≈ 9.6.

Since 84.7% of blacks and women were referred, 15.3% were not referred, and so for them, the odds of referral were 84.7/15.3 ≈ 5.5.

The odds ratio was therefore 5.5/9.6 ≈ 60%. The odds of a referral if you were black or a woman were about 60% of the odds of referral if you were a white man.

But the risk ratio was much higher. If you were black or a woman, the probability that you would be referred was 93% of the probability that a white man would be referred.

This Happens All the Time

This is from “Childhood Asthma Gene Identified by scientists” from the Science Editor of The Independent:

“Inheriting the gene raises the risk of developing asthma by between 60 and 70 per cent– enough for researchers to believe that the discovery may eventually open the way to new treatments for the condition.”

Complete Misrepresentation

The quote talks about “raising the risk”.

Is that what the scientists found? No. The number reported wasn’t an increased risk, it was an odds ratio.

The gene only raised the risk of asthma by 19%.

More “Science Editors”

This quote is from the Science Editor of the London Times in “Genetic breakthrough offers MS sufferers new hopes for treatment”:

“Research has identified two genetic variants that each raises a person's risk of developing MS by about 30 per cent, shedding new light on the origins of the autoimmune disease that could ultimately lead to better therapies”

Complete Misrepresentation

The quote talks about genes “raising the risk” of Multiple Sclerosis.

Is that what the researchers found? No!

The first gene raised the risk of MS by only 3%, the second by only 4%!

SAMPLE SIZE

Sample

In statistics, the people who we are studying are called the sample.

Our question is then: what sample size is needed for a result that generalizes to the population?

Non-Random Samples

The first thing we should realize is that it’s not going to do us any good to ask a non-random group of people.

Suppose everyone who goes to ILoveMitt.com is voting for Mitt. If I ask them, it will seem like 100% of the population will vote for Mitt, even if only 3% will really vote for him.

Internet Polls

Internet polls are not trustworthy. They are biased toward people who have the internet, people who visit the site that the poll is on, and people who care enough to vote on a useless internet poll.

Representative Samples

The opposite of a biased sample is a representative sample.

A perfectly representative sample is one where if n% of the population is X, then n% of the sample is X, for every X.

For example, if 10% of the population smokes, 10% of the sample smokes.

Random Sampling

One way to get a representative sample is to randomly select people from the population, so that each has a fair and equal chance of ending up in the sample.

Confidence Interval

Suppose I poll a sample of some population and find out that 50% of the sample will vote for candidate C. I might be:• 90% certain that 48-52% of the population will

vote for C• 95% certain that 45-55% of the population will

vote for C• 99% certain that 40-50% of the population will

vote for C

Margin of Error

The margin error is half of some confidence interval (usually 95%).

So if I’m 95% certain that between 45 and 55% of people will vote for C, then the margin of error is ±5%.

Error Bars

Often, confidence intervals/ margins of error are presented graphically as error bars.