Likert scale analysis¶

Here’s a question from the Reddit statistics forum.

I have collected data regarding how individuals feel about a particular program. They reported their feelings on a scale of 1-5, with 1 being Strongly Disagree, 2 being Disagree, 3 being Neutral, 4 being Agree, and 5 being Strongly Agree.

I am looking to analyze the data for averages responses, but I see that a basic mean will not do the trick. I am looking for very simple statistical analysis on the data. Could someone help out regarding what I would do?

It sounds like OP has heard the advice that you should not compute the mean of values on a Likert scale. The Likert scale is ordinal, which means that the values are ordered, but it is not an interval scale, because the distances between successive points are not necessarily equal.

For example, if we imagine that “Neutral” maps to 0 on a number line and “Agree” maps to 1, it’s not clear where we should place “Strongly agree”. And an appropriate mapping might not be symmetric — for example, maybe the people who choose “Strongly agree” are content, but the people who choose “Strongly disagree” are angry. In an arithmetic mean, they would cancel each other out — but that might erase an important distinction.

Nevertheless, I think an absolute prohibition on computing means is too strong. I’ll show some examples where I think it’s a reasonable thing to do — but I’ll also suggest alternatives that might be better.

Political views¶

As an example, I’ll use data from the General Social Survey (GSS), which I have resampled to correct for stratified sampling.

The variable we’ll start with is polviews, which contains responses to this question:

We hear a lot of talk these days about liberals and conservatives. I’m going to show you a seven-point scale on which the political views that people might hold are arranged from extremely liberal–point 1–to extremely conservative–point 7. Where would you place yourself on this scale?

This is not a Likert scale, specifically, but it is ordinal. Here is the distribution of responses:

Here’s the mapping from numerical values to the options that were shown to respondents.

As always, it’s a good idea to visualize the distribution before we compute any summary statistics. I’ll use Pmf from empiricaldist to compute the PMF of the values.

Here’s what it looks like.

The modal value is “Moderate” and the distribution is roughly symmetric, so I think it’s reasonable to compute a mean. For example, suppose we want to know how self-reported political alignment has changed over time. We can compute the mean in each year like this:

And plot it like this.

I used LOWESS to plot a local regression line, which makes long-term trends easier to see. It looks like the center of mass trended toward conservative from the 1970s into the 1990s and trended toward liberal since then.

When you compute any summary statistic, you lose information. To see what we might be missing, we can use a normalized cross-tabulation to compute the distribution of responses in each year.

And we can use a heat map to visualize the results.

Based on the heat map, it seems like the general shape of the distribution has not changed much — so the mean is probably a good way to make comparisons over time.

However, it is hard to interpret the mean in absolute terms. For example, in the most recent data, the mean is about 4.1.

Since 4.0 maps to “Moderate”, we can say that the center of mass is slightly on the conservative side of moderate, but it’s hard to say what a difference of 0.1 means on this scale.

As an alternative, we could add up the percentage who identify as conservative or liberal, with or without an adverb.

And plot those percentages over time.

Or we could plot the difference in percentage points.

This figure shows the same trends we saw by plotting the mean, but the y-axis is more interpretable — for example, we could report that, at the peak of the Reagan era, conservatives outnumbered liberals by 10-15 percentage points.

Standard deviation¶

Suppose we are interested in polarization, so we want to see if the spread of the distribution has changed over time. Would it be OK to compute the standard deviation of the responses? As with the mean, my answer is yes and no.

First, let’s see what it looks like.

The standard deviation is easy to compute, and it makes it easy to see the long-term trend. If we interpret the spread of the distribution as a measure of polarization, it looks like it has increased in the last 30 years.

But it is not easy to interpret this result in context. If it increased from about 1.35 to 1.5, is that a lot? It’s hard to say.

As an alternative, let’s compute the mean absolute deviation (MAD), which we can think of like this: if we choose two people at random, how much will they differ on this scale, on average?

A quick way to estimate MAD is to draw two samples from the responses and compute the mean pairwise distance.

The result is about 1.5 points, which is bigger than the distance from moderate to slightly conservative, and smaller than the distance from moderate to conservative.

Rather than sampling, we can compute MAD deterministically by forming the joint distribution of response pairs and computing the expected value of the distances. For this computation, it is convenient to use NumPy functions for outer product and outer difference.

Again, the result is about 1.5 points.

Now we can see how this value has changed over time.

Here’s the result, along with the standard deviation.

The two figures tell the same story — polarization is increasing. But the MAD is easier to interpret. In the 1970s, if you chose two people at random, they would differ by less than 1.5 points on average. Now the difference would be almost 1.7 points. Considering that the difference between a moderate and a conservative is 2 points, it seems like we should still be able to get along.

I think MAD is more interpretable than standard deviation, but it is based on the same assumption that the points on the scale are equally spaced. In most cases, that’s not an assumption we can easily check, but in this example, maybe we can.

For another project, I selected 15 questions in the GSS where conservatives and liberals are most likely to disagree, and used them to estimate the number of conservative responses from each respondent. The following figure shows the average number of conservative responses to the 15 questions for each point on the self-reported scale.

The result is close to a straight line, which suggests that the assumption of equal spacing is not bad in this case.

When is the mean bad?¶

In the examples so far, computing the mean and standard deviation of a scale variable is not necessarily the best choice, but it could be a reasonable choice. Now we’ll see an example where it is probably a bad choice.

The variable homosex contains responses to this question:

What about sexual relations between two adults of the same sex–do you think it is always wrong, almost always wrong, wrong only sometimes, or not wrong at all?

If the wording of the question seems loaded, remember that many of the core questions in the GSS were written in the 1970s. Here is the encoding of the responses.

And here are the value counts.

Before we do anything else, let’s look at the distribution of responses.

There are several reasons it’s a bad idea to summarize this distribution by computing the mean. First, one of the responses is not ordered. If we include “Other” in the mean, the result is meaningless.

If we exclude “Other”, the remaining responses are ordered, but arguably not evenly spaced on a spectrum of opinion.

If we compute a mean and report that the average response is somewhere between “Sometimes wrong” and “Almost always wrong”, that is not an effective summary of the distribution.

The distribution of results is strongly bimodal — most people are either accepting of homosexuality or not. And that suggests a better way to summarize the distribution: we can simply report the fraction of respondents who choose one extreme or the other.

I’ll start by creating a binary variable that is 1 for respondents who chose “Not wrong at all”, 0 for the other responses, and NaN for people who were not asked the question, did not respond, or chose “Other”.

The mean of this variable is the fraction of respondents who chose “Not wrong at all”.

Now we can see how this fraction has changed over time.

The percentage of people who accept homosexuality was almost unchanged in the 1970s and 1980s, and began to increase quickly around 1990. For a discussion of this trend, and similar trends related to racism and sexism, you might be interested in Chapter 11 of Probably Overthinking It.

Discussion¶

Computing the mean of an ordinal variable can be a quick way to make comparisons between groups or show trends over time. The computation implicitly assumes that the points on the scale are equally spaced, which is not true in general, but in many cases it is close enough.

However, even when the mean (or standard deviation) is a reasonable choice, there is often an alternative that is easier to interpret in context.

It’s always good to look at the distribution before choosing a summary statistic. If it’s bimodal, the mean is probably not the best choice.

Data Q&A: Answering the real questions with Python

Copyright 2024 Allen B. Downey

License: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International

Appendix: Quantifying discord¶

We’ve seen that mean absolute deviation (MAD) can quantify the average level of disagreement between two people in a population. As a generalization, it can also quantify the discord between two groups.

I might come back and expand this example later.

Testing percentiles¶

Here’s a question from the Reddit statistics forum.

I have two different samples (about 100 observations per sample) drawn from the same population (or that’s what I hypothesize; the populations may in fact be different). The samples and population are approximately normal in distribution.

I want to estimate the 85th percentile value for both samples, and then see if there is a statistically significant difference between these two values. I cannot use a normal z- or t-test for this, can I? It’s my current understanding that those tests would only work if I were comparing the means of the samples.

As an extension of this, say I wanted to compare one of these 85th percentile values to a fixed value; again, if I was looking at the mean, I would just construct a confidence interval and see if the fixed value fell within it…but the percentile stuff is throwing me for a loop.

This is […] related to a research project I’m working on (in my job).

There are two questions here. The first is about testing a difference in percentiles between two groups. The second is about the difference between a percentile from an observed sample and an expected value.

We’ll answer the first question with a permutation test, and we’ll answer the second in two ways: bootstrap resampling and a Gaussian model.

Data¶

Since OP didn’t provide a dataset, we have to generate one. I’ll draw two samples from Gaussian distributions with the same standard deviation and different means.

Here’s what the distributions of the groups look like.

If we compute the 85th percentile in both groups, we see a difference, as expected.

Now let’s see if a difference of that size would be likely if the two samples were actually drawn from the same distribution.

Testing the Difference¶

When we test a difference between two groups, the usual model of the null hypothesis is that the groups are actually identical. If that’s true, the two samples came from the same distribution, so we can combine them into a single sample.

Now we can simulate the null hypothesis by permutation — that is, by shuffling the pooled data and splitting it into two groups with the same sizes as the originals. The following function generates two samples under this assumption, and returns the difference in their 85th percentiles.

If we call it many times, the result is a sample from the distribution of differences under the null hypothesis.

Here’s what it looks like, with a vertical line at the observed difference. I’m plotting it with cut=0 so the estimated density doesn’t extend past the minimum and maximum of the data.

The distribution is multimodal, which is the result of selecting a moderately high percentile from a moderately small dataset — the diversity of the results is limited. However, in this example we are interested in the tails of the distribution, so multimodality is not a problem.

To estimate a one-sided p-value, we can compute the fraction of the sample that exceeds the actual difference.

Or, for a two-sided p-value, we can compute the fraction of the sample that exceeds the actual difference in absolute value.

In this example, the result of the one-sided test would be considered significant at the 5% significance level, but the two-sided test would not. So which is it?

I think it’s not worth worrying about. My interpretation of the results is the same either way: they are inconclusive. Under the null hypothesis, a difference as big as the one we saw would be unlikely, but we can’t rule out the possibility that the groups are identical — or nearly so — and the apparent difference is due to random variation.

Testing a fixed value¶

Now let’s turn to the second question. Suppose we have reason to think that the actual value of the 85th percentile is 12.3, and we would like to know whether the data contradict this hypothesis.

We’ll test group1 first. Here’s the 85th percentile of group1 and its difference from the expected value.

Let’s see if a difference of this magnitude is likely to happen under the null hypothesis. One way to model the null hypothesis is to create a dataset that is similar to the observed data, but where the 85th percentile is exactly as expected. We can do that by shifting the observed data by the observed difference.

The 85th percentile of the shifted data is the expected value, exactly.

Now, to generate samples under the null hypothesis, we can use the following function, which takes a sample, shifts it to have the expected value of the 85th percentile, generates a bootstrap resample of the shifted values, and returns the difference between the 85th percentile of the sample and the expected value.

If we call this function many times, we get a sample of the differences we expect under the null hypothesis.

The following function shows the distribution of the sample with a vertical line at the observed value.

Without computing a p-value, we can see that a difference as big as actual_diff1 is entirely plausible under the null hypothesis. We can confirm that by computing a one-sided p-value.

So the observed difference in the first group is not statistically significant. Now let’s do the same thing for the second group.

Here’s what the distribution of differences looks like under the null hypothesis, with a vertical line at the observed value.

There are no differences in the sample that exceed the observed value.

We can conclude that a difference as big as that is very unlikely under the null hypothesis. There’s not much point in computing a p-value more precisely than that, but if it’s required, we can estimate it if we assume that the tail of the sampling distribution is roughly Gaussian. In that case, we can fit a KDE to the sampling distribution like this.

And use a Pmf object to approximate the estimated density.

Then we can use the corresponding CDF to compute the probability of a value that exceeds the observed difference.

So the p-value is quite small.

Model-based resampling¶

The bootstrap method in the previous section is a good choice if we are unsure about the distribution of the data, or if there are outliers. But multiple modes in the sampling distribution suggest that there might not be enough diversity in the data for bootstrapping to be reliable.

Fortunately, there is another way we might model the null hypothesis: using a Gaussian distribution. If we generate data from a continuous distribution, we expect the sampling distribution to be unimodal.

But there is a problem we have to solve first — we have to make an assumption about the standard deviation of the hypothetical Gaussian distribution. One option is to use the standard deviation of the data.

Now we need to find a Gaussian distribution with a given standard deviation that has the expected 85th percentile. We can do that by starting with a distribution centered at 0, computing it’s 85th percentile and then shifting it.

ppf stands for “percentile point function”, which is another name for the quantile function, which is the inverse of the CDF — it takes a cumulative probability and returns the corresponding quantity.

The following function takes one of the groups, fits a hypothetical model to it, generates a sample from the model, and returns the difference between the 85th percentile of the sample and the expected value.

If we call this function many times, we get the sampling distribution of the test statistic under the null hypothesis.

Here’s what the distribution looks like, compared to the corresponding distribution from the bootstrapped model.

The shapes of the distributions are different, but their ranges are comparable. And the conclusion is the same: a difference as big as actual_diff1 is entirely plausible under the null hypothesis.

Now let’s try the same test with Group 2.

Here’s the result, along with the result from the bootstrap model.

Again, the shapes of the distributions are different, but the conclusion is the same. A difference as big as actual_diff2 is unlikely under the null hypothesis.

As usual, the two-sided p-value is bigger by a factor of two, roughly, but the difference never matters in practice.

Under this model of the null hypothesis, the probability is small that the 85th percentile of the data would exceed the expected value by so much.

Discussion¶

This example demonstrates a kind of inconsistency in hypothesis testing. We found that Group 1 is not significantly different from the expected value — in the technical sense of significantly — but Group 2 is. So that suggests that Group 1 and Group 2 are different from each other, but when we test that hypothesis, the difference is not statistically significant.

People who are new to hypothesis testing find results like this surprising, but they are not rare. Generally, they are a consequence of the logic of null hypothesis testing and the arbitrariness of the significance threshold.

I think it helps to interpret p-values qualitatively.

• A p-value greater than 10% means that the observed effect is plausible under the null hypothesis, and could happen by chance.

• A p-value less than 1% means that an observed effect is unlikely under the null hypothesis — so it is unlikely to have happened by chance.

• Anything in between is inconclusive.

There is nothing special about 5%.

Data Q&A: Answering the real questions with Python

Copyright 2024 Allen B. Downey

License: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International

Bootstrapping percentiles¶

Here’s a question from the Reddit statistics forum.

I’m trying to figure out how to determine the confidence interval for the .2 percentile temperature for specific set of observed temperatures (all hourly temperatures during January, February, and December since 2000). I have recordings for 53128 of the 53424 possible hourly recordings.

How would I go about saying that I am X% sure that the actual .2 percentile value is between two numbers? Could anyone provide any insight on how to accomplish this. Thank you.

OP provided a link to the data, so this is a question we can answer! For computing confidence intervals, my first choice is bootstrap resampling, but as it turns out, it does not work well for this problem. I’ll show what goes wrong and how to fix it. Then we’ll answer a follow-up question about quantifying the effect of missing data.

Data¶

I downloaded the data as a CSV file, which we can read into a Pandas DataFrame.

There are 53424 measurements, of which 306 are missing.

The range of temperatures is from -9.9 degF to 89 degF.

And the 0.2 percentile is 1 degF.

Basic bootstrap¶

The following function takes the cleaned data, resamples it, and computes the 0.2 percentile of the bootstrapped sample.

If we call this function 1001 times, we get a sample from the sampling distribution of the percentile.

Here’s what that sample looks like.

Immediately we can see that something has gone wrong. The resampling process produces only 8 unique values.

If we try to compute a CI by pulling percentiles from the sample, the results are not credible.

This example demonstrates a limitation of bootstrap resampling — it does not work well when there are a small number of unique values.

However, because the data are temperature measurements, they are actually continuous quantities. So one option is to replace bootstrapping with a model that generates continuous quantities. We’ll try that with a normal model, see that it does not work, and they try again with KDE.

Resampling from a normal model¶

If we look at the CDF of the data, it resembles the characteristic sigma of the normal distribution.

So let’s see how it compares to a normal model. I’ll estimate the parameters by computing the mean and standard deviation of the data.

And compute the normal CDF within 4 standard deviations of the mean.

Here’s what the model looks like compared to the data.

It looks pretty good, but there are places where the data clearly deviate from the model. That’s enough to make me worry, but let’s proceed and see how it goes.

The following function takes the cleaned data, generates a random sample from the normal model, and returns the 0.2 percentile of the sample.

If we call it 1001 times, we hope the result is a sample from the sampling distribution of the percentile.

And at first glance it looks good.

But notice that range of the sampling distribution does not include the 0.2 percentile of the data, which is 1. We can compute a 90% CI, but again, it is not credible.

To see what went wrong, let’s look at the normal model and the data again, this time with the y axis on a log scale. The log scale is like a microscope that lets us see more clearly what is happening in the tail of the distribution.

On a linear scale, it seemed like the normal model might be good enough; on a log scale, it is clear that the data deviate from the model in the left tail.

In retrospect, it is not a surprise if a simple two-parameter model fails to capture every detail of the distribution — the world is a complicated place. So let’s try a nonparametric approach.

Resampling with KDE¶

We can use kernel density estimation (KDE) to model the distribution of the data, then use the model to resample. Here’s how we estimate the distribution.

To see what the result looks like, we can approximate the density of the model with a discrete PMF.

And then compare the CDF of the model with the CDF of the data.

The result shows that KDE is doing what it is meant to do — fitting a continuous distribution to the data with minimal assumptions.

The following function takes the cleaned data, uses the KDE model to generate a random sample, and returns the 0.2 percentile of the sample.

If we call it 1001 times, we hope once again that the result is a sample from the sampling distribution of the percentile.

And this time we get a better result. The sampling distribution looks good, and it contains the actual percentile of the data.

And the width of the 90% CI is plausible.

So with a couple of false starts, we have answered the original question. But it turns out there’s more.

Fill missing values¶

In a follow-up message, OP wrote:

Just in case it helps any, here’s what I’m ultimately trying to accomplish with this endeavor… I am trying to come up with a plausible way of demonstrating that the .2 percentile value (1 degF) that is derived from this data set is sufficiently representative of what the value would be if there were no missing data points (hourly readings) from the dataset.

OK, that’s a different question! However, the resampling framework can be extended naturally to estimate the effect of missing data. Here’s a function that takes the original data — including NaNs — and fills the missing values with a random selection of valid values. For historical reasons, this way of filling missing values is called “hot deck imputation”.

To test it, we can check that the result has no NaNs.

Now we can include fill_missing as part of the resampling pipeline. The following function takes the original data, fills missing values, generates a sample from a KDE model, and returns the 0.2 percentile of the sample.

If we call it many times, we get a sample from a distribution that represents the uncertainty of the estimate due to a combination of missing data and random sampling.

Here’s what the result looks like, compared to the sampling distribution from the previous section, which represents only uncertainty due to random sampling.

The difference does not seem substantial, and the CIs are similar.

We can conclude that missing data does not have much effect on the CI.

To estimate the effect more precisely, we could run this again with a sample size of 10,001 rather than 1001. But I won’t bother because with only 306 missing values out of 53,424, I did not expect the missing data to affect the results by much, and this result confirms it. Rather than estimate the CI more precisely, I would conclude that missing data is not a problem, and drop it.

Discussion¶

Normally I am quick to recommend bootstrap resampling because “it just works”. It makes almost no assumptions about the distribution of the data, and it is easy to extend to almost any statistic. But as this example shows, it is not infallible — the kryptonite of bootstrapping is lack of diversity in the data.

To diagnose this problem, it is a good idea to explore the sampling distribution. If bootstrapping goes well, the sampling distribution should have many unique values, and the range should contain the estimate computed directly from the data, usually close to the middle of the CI.

If the results from bootstrapping fail these tests, think about other ways to model the data-generating process. If a parametric model fits the data well, you can use the data to estimate parameters and then use the model to generate simulated samples. Otherwise, consider a non-parametric approach like KDE.

For filling missing data, hot deck imputation ignores serial correlation and other statistical structure in the data, so the imputed values are likely to be unrealistic. But in this case that’s probably a feature, because the results overestimate the effect of missing data. As a result, we can make the argument, “Even if we assume that the missing data is highly variable, it has no substantial effect on the estimated percentile or the computed CI.”

If it were necessary to fill missing data with more realistic values, we could use a time series method like ARIMA or a Gaussian process.

Data Q&A: Answering the real questions with Python

Copyright 2024 Allen B. Downey

License: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International

What does “strength” mean?¶

Here’s a question from the Reddit statistics forum.

I am currently doing a uni assignment and one of my tasks is analysing the correlation between two variables. When I use the correlation function in Excel, it returns a correlation of -0.0377. When I use the same data to create a scatter plot, the trend line is positive. I need to identify the correlation strength and direction and thereby, I am confused by these opposing outcomes. Can somebody please explain why the correlation is showing as negative but the trend line is positive? What does this indicate in terms of the strength and direction of the relationship between the two variables?

To answer the immediate question, correlation and the slope of a linear regression line always have the same sign. Mathematically, they are both related to the dot product of the x and y variables.

So there is something strange going on. It might be a simple error — for example, maybe the correlation and regression were based on different data. Or it might be that the trend computed by Excel is something other than linear regression. For example, a line that minimizes mean absolute error (MAE) rather than mean squared error (MSE) can have a slope with the opposite sign of the correlation.

Without more information it’s hard to be sure what’s going on, but for this example it might not matter. The computed correlation is negative but very small. If we fit a line (other than a regression line) to the same data and the slope is positive but similarly small, that is not necessarily inconsistent. Within statistical uncertainty, both are indistinguishable from zero.

OP also asks, “What does this indicate in terms of the strength and direction of the relationship between the two variables?” So let’s answer that question, too.

Interpreting correlation and slope¶

When people talk about the strength of a relationship, they might mean correlation or they might mean the slope of a fitted line. But these measures of “strength” are not always consistent.

For example, suppose we are concerned about the health effects of weight gain, so we plot weight versus age from 20 to 50 years old. I’ll generate two fake datasets to demonstrate the point.

I used the same random seed to generate both, so they look similar, as we can see in these scatter plots.

Nevertheless, they have substantially different correlations.

In the first dataset, the correlation is close to 0.75. In the second, it is close to 0.5. So we might think the first relationship is stronger.

But let’s look at the slopes of the regression lines. For the first dataset, the estimated slope is about 0.019 kilograms per year or about 0.56 kilograms over the 30-year range.

For the second dataset, the estimated slope is almost 10 times higher — about 0.18 kilograms per year or 5.3 kilograms per 30 years.

According to the correlations, the first relationship is stronger. According to the slopes, the second relationship is stronger. So which is it? The answer depends on context.

In this example, the slope of the regression line indicates the magnitude of weight gain. If we are concerned about the health effects of weight gain, the second relationship is probably more important.

On the other hand, correlation indicates how well we can predict one value based on the other. If, for some reason, we are trying to guess someone’s weight, based on their age, the first relationship would be more important.

Here are all the results in the same plot.

Because of the way the plots are scaled, the slope looks smaller in the second figure, but that’s misleading. So this example is a reminder to look at the labels of the y axis — which is where the effect size often hides.

Minimizing MAE¶

Earlier I said a line that minimizes mean absolute error (MAE) rather than mean squared error (MSE) can have a slope with the opposite sign of the correlation. To demonstrate, I’ll use the following function to minimize MAE.

Now I’ll generate a dataset where xs and ys are actually uncorrelated.

In this dataset, the correlation is slightly negative and the slope of the fitted line is slightly positive.

Here’s what the scatter plot looks like with the minimum MAE line.

To find this example, I generated datasets with different random number seeds. Out of the first 100 attempts, 19 yield correlation and slope with opposite signs.

So examples like this are not rare, if the actual correlation is close to zero.

Data Q&A: Answering the real questions with Python

Copyright 2024 Allen B. Downey

License: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International

What does a confidence interval mean?¶

Here’s a question from the Reddit statistics forum (with an edit for clarity):

Why does a confidence interval not tell you that 90% of the time, [the true value of the population parameter] will be in the interval, or something along those lines?

I understand that the interpretation of confidence intervals is that with repeated samples from the population, 90% of the time the interval would contain the true value of whatever it is you’re estimating. What I don’t understand is why this method doesn’t really tell you anything about what that parameter value is.

This is, to put it mildly, a common source of confusion. And here is one of the responses:

From a frequentist perspective, the true value of the parameter is fixed. Thus, once you have calculated your confidence interval, one if two things are true: either the true parameter value is inside the interval, or it is outside it. So the probability that the interval contains the true value is either 0 or 1, but you can never know which.

This response is the conventional answer to this question — it is what you find in most textbooks and what is taught in most classes. And, in my opinion, it is wrong. To explain why, I’ll start with a story.

Suppose Frank and Betsy visit a factory where 90% of the widgets are good and 10% are defective. Frank chooses a part at random and asks Betsy, “What is the probability that this part is good?”

Betsy says, “If 90% of the parts are good, and you choose one at random, the probability is 90% that it is good.”

“Wrong!” says Frank. “Since the part has already been manufactured, one of two things must be true: either it is good or it is defective. So the probability is either 100% or 0%, but we don’t know which.”

Frank’s argument is based on a strict interpretation of frequentism, which is a particular philosophy of probability. But it is not the only interpretation, and it is not a particularly good one. In fact, it suffers from several flaws. This example shows one of them — in many real-world scenarios where it would be meaningful and useful to assign a probability to a proposition, frequentism simply refuses to do so.

Fortunately, Betsy is under no obligation to adopt Frank’s interpretation of probability. She is free to adopt any of several alternatives that are consistent with her commonsense claim that a randomly-chosen part has a 90% probability of being functional.

Now let’s see how this story relates to confidence intervals.

Click here to run this notebook on Colab

Generating a confidence interval¶

Suppose that Frank is a statistics teacher and Betsy is one of his students. One day Frank teaches the class a process for computing confidence intervals that goes like this:

1. Collect a sample of size $n$.

2. Compute the sample mean, $m$, and the sample standard deviation, $s$.

3. If those estimates are correct, the sampling distribution of the mean is a normal distribution with mean $m$ and standard deviation $s / \sqrt{n}$.

4. Compute the 5th and 95th percentiles of this sampling distribution. The result is a 90% confidence interval.

As an example, Frank generates a sample with size 100 from a normal distribution with known parameters mean $\mu=10$ and standard deviation $\sigma=3$.

Then Betsy uses the following function to compute a 90% CI.

In this example, we know that the actual population mean is 10 so we can see that this CI contains the population mean. But if we draw another sample, we might get a sample mean that is substantially higher or lower than $\mu$, and the CI we compute might not contain $\mu$.

To see how often that happens, we’ll use this function, which generates a sample, computes a 90% CI, and checks whether the CI contains $\mu$.

If we run this function 1000 times, we can count how often the CI contains $\mu$.

The answer is close to 90% — that is, if we run this process many times, 90% of the CIs it generates contain $\mu$ and 10% don’t. So the CI-computing process is like a factory where 90% of the widgets are good and 10% are defective.

Now suppose Frank chooses a different value of $\mu$ and does not tell Betsy what it is. To simulate that scenario, I’ll choose a value from a random number generator with a specific seed.

And just for good measure, I’ll generate a random value for $\sigma$, too.

Next Frank generates a sample from a normal distribution with those parameters, and gives the sample to Betsy.

And Betsy uses the data to compute a CI.

Now suppose Frank asks, “What is the probability that this CI contains the actual value of $\mu$ that I chose?”

Betsy says, “We have established that 90% of the CIs generated by this process contain $\mu$, so the probability that this CI contains $\mu$ is 90%.”

And of course Frank says “Wrong! Now that we have computed the CI, it is unknown whether it contains the true parameter, but it is not random. The probability that it contains $\mu$ is either 100% or 0%. We can’t say it has a 90% chance of containing $\mu$.”

Once again, Frank is asserting a particular interpretation of probability — one that has the regrettable property of rendering probability nearly useless. Fortunately, Betsy is under no obligation to join Frank’s cult.

Under most reasonable interpretations of probability, you can say that a specific 90% CI has a 90% chance of containing the true parameter. There is no real philosophical problem with that.

But there might be practical problems.

Practical problems¶

The process we use to construct a CI takes into account variability due to random sampling, but it does not take into account other problems, like measurement error and non-representative sampling. To see why that matters, let’s consider a more realistic example.

Suppose we want to estimate the average height of adult male residents of the United States. If we define terms like “height”, “adult”, “male”, and “resident of the United States” precisely enough, we have defined a population that has a true, unknown average height. If we collect a representative sample from the population and measure their heights, we can use the sample mean to estimate the population mean and compute a confidence interval.

To demonstrate, I’ll use data from the Behavioral Risk Factor Surveillance System (BRFSS). Here’s an extract I prepared for Elements of Data Science, based on BRFSS data from 2021.

It includes data from 203,760 male respondents.

For 193,701 of them, we have their self-reported height recorded in centimeters.

We can use this data to compute a sample mean and 90% confidence interval.

Because the sample size is so large, the confidence interval is quite small — its width is only 0.03% of the estimate.

So there is very little variability in this estimate due to random sampling. That means the estimate is precise, but that doesn’t mean it’s accurate.

For one thing, the measurements in this dataset are self-reported. If people tend to round up — and they do — that would make the estimated mean too high.

For another thing, it is difficult to construct a representative sample of a population as large as the United States. The BRFSS is run by people who know what they are doing, but nothing is perfect — it is likely that some groups are systematically overrepresented or underrepresented. And that could make the estimated mean too high or too low.

Given that there is almost certainly some measurement error and some sampling bias, it is unlikely that the actual population falls in the very small confidence interval we computed.

And that’s true in general — when the sample size is large, variability due to random sampling is small, which means that other sources of error are likely to be bigger. So as sample size increases, the probability decreases that the CI contains the true value.

Summary¶

The way confidence intervals are taught in most statistics class is based on the frequentist interpretation of probability. But you are not obligated to adopt that interpretation, and there are good reasons you should not.

Some people will say that confidence intervals are a frequentist method that is inextricable from the frequentist interpretation. I don’t think that’s true — there is nothing about the computation of a confidence interval that depends on the frequentist interpretation. So you are free to interpret the CI under any philosophy of probability you like.

If you want to say that a 90% CI has a 90% chance of containing the true value, there is nothing wrong with that, philosophically. I think it is a meaningful and useful probabilistic claim.

However, it is only true if other sources of error — like sampling bias and measurement error — are small compared to variability due to random sampling.

For that reason, I think the best interpretation of a confidence interval, for practical purposes, is that it quantifies the precision of the estimate but says nothing about its accuracy.

Credit: I borrowed Frank and Betsy from my friend Ted Bunn. They first appeared in his blog post Who knows what evil lurks in the hearts of men? The Bayesian doesn’t care..

Is taking the SD of a count variable helpful?¶

Here’s a question from the Reddit statistics forum.

A student brought this up to me in class this week and I had no idea how to answer. For some context they are doing in experiment that involves a count variable and then have a choice of what inferential stat test they want to run. If they pick a t-test they need to show SD error bars on their graph but one student kept telling me it isn’t possible. I’ve spent time looking around forums and asked my PI who also shrugged since we’re chemistry people. I was just wondering if someone can explain if finding the SD of a count variable is possible, and if it is, does it tell you anything important or is it just a waste of time?

In a follow-up, OP provided more context:

To be specific students were counting the amount of eggs laid on two different types of substrate in petri dishes (something like wood and grass). They were then given the option to choose whatever inferential stat test they thought best fits the data and a number of them chose unpaired t-tests. They used [statistical software] to do the actual test (plugging in the count numbers from 2 variables with 3 replicates, so 6 numbers in total) .

It was my understanding that because the data probably doesn’t follow a normal distribution, SD error bars wouldn’t really be helpful.

There are several questions going on here, so let’s start with the headline — is taking the SD of a count variable helpful?

Click here to run this notebook on Colab

Generating a dataset¶

Count data are often well modeled by a negative binomial distribution, so that’s what I’ll use to generate a dataset.

Here’s what the distribution of values looks like. I’m using a Pmf object, which shows all of the values, rather than a histogram, which puts the values into bins.

We can compute the mean and standard deviation of the data in the usual way.