Confidence intervals

The frequentist interpretation of probability

Before we start taking about confidence intervals, it is important to recall the frequentist interpretation of probability. Under this interpretation, the probability \(P(A)\) represents a long-run frequency of event \(A\) over a large number of identical repetitions of an experiment. In our calculation of confidence intervals and in performance of null hypothesis significance tests, we will directly apply this interpretation of probability again and again, using our computers to “repeat” experiments many times and tally the frequencies of what we see.

Confidence intervals

Consider the following question. If I were to do the experiment again, what value might I expect for my plug-in estimate for my functional? What if I did it again and again and again? These are reasonable questions because the plug-in estimate is something that can vary meaningfully from experiment to experiment. Remember, with the frequentist interpretation of probability, we cannot assign a probability to a parameter value. A parameter has one value, and that’s that. We can describe the long-term frequency of observing results about random variables. Because a plug-in estimate for a statistical functional does vary from experiment to experiment, it is a random variable. So, we can define a 95% confidence interval as follows.

If an experiment is repeated over and over again, the estimate I compute for a parameter, \(\hat{\theta}\), will lie between the bounds of the 95% confidence interval for 95% of the experiments.

While this is a correct definition of a confidence interval, some statisticians prefer another. To quote Larry Wasserman from his book, All of Statistics,

[The above definition] is correct but useless since we rarely repeat the same experiment over and over. A better interpretation is this: On day 1, you collect data and construct a 95 percent confidence interval for a parameter \(\theta_1\). On day 2, you collect new data and construct a 95 percent confidence interval for an unrelated parameter \(\theta_2\). On day 3, you collect new data and construct a 95 percent confidence interval for an unrelated parameter \(\theta_3\). You continue this way constructing confidence intervals for a sequence of unrelated parameters \(\theta_1, \theta_2, \ldots\). Then 95 percent of your intervals will trap the true parameter value. There us no need to introduce the idea of repeating the same experiment over and over.

In other words, the confidence interval describes the construction of the confidence interval itself. 95% of the time, it will contain the true (unknown) parameter value. Wasserman’s description contains a reference to the true parameter value, so if you are going to talk about the true parameter value, this description is useful. However, the first definition of the confidence interval is quite useful if you want to think about how repeated experiments will end up.

We will use the first definition in thinking about how to construct a confidence interval. To construct the confidence interval, then, we will repeat the experiment over and over again, each time computing \(\hat{\theta}\). We will then generate an ECDF of our \(\hat{\theta}\) values, and report the 2.5th and 97.5th percentile to get our 95% confidence interval. But wait, how will we repeat the experiment so many times?

Bootstrap confidence intervals

Remember that the data come from a generative distribution with CDF \(F(x)\). Doing an experiment where we make \(n\) measurements amounts to drawing \(n\) numbers out of \(F(x)\) 1. So, we could draw out of \(F(x)\) over and over again. The problem is, we do not know what \(F(x)\) is. However, we do have an empirical approximation for \(F(x)\), namely \(\hat{F}(x)\). So, we could draw \(n\) samples out of \(\hat{F}(x)\), compute \(\hat{\theta}\) from these samples, and repeat. This procedure is called bootstrapping.

To get the terminology down, a bootstrap sample, \(\mathbf{x}^*\), is a set of \(n\) \(x\) values drawn from \(\hat{F}(x)\). A bootstrap replicate is the estimate \(\hat{\theta}^*\) obtained from the bootstrap sample \(\mathbf{x}^*\). To generate a bootstrap sample, consider an array of measured values \(\mathbf{x}\). We draw \(n\) values out of this array with replacement to give us \(\mathbf{x}^*\). This is equivalent to sampling out of \(\hat{F}(x)\).

So, the recipe for generating a bootstrap confidence interval is as follows.

  1. Generate \(B\) independent bootstrap samples. Each one is generated by drawing \(n\) values out of the data array with replacement.

  2. Compute \(\hat{\theta}^*\) for each bootstrap sample to get the bootstrap replicates.

  3. The central \(100 (1-\alpha)\) percent confidence interval consists of the percentiles \(100\alpha/2\) and \(100(1-\alpha/2)\) of the bootstrap replicates.

This procedure works for any estimate \(\hat{\theta}\), be it the mean, median, variance, skewness, kurtosis, or any other thing you can think of. Note that we use the empirical distribution, so there is never any assumption of an underlying “true” distribution. We are employing the plug-in principle for repeating experiments. Instead of sampling out of the generative distribution (which is what performing an experiment is), we plug-in the empirical distribution and sample out of it, instead. Thus, we are doing nonparametric inference on what we would expect for parameters coming out of unknown distributions; we only know the data.

There are plenty of subtleties and improvements to this procedure, but this is most of the story. We will discuss the mechanics of how to programmatically generate bootstrap replicates in forthcoming lessons, but we have already covered the main idea.

1

We’re being loose with language here. We’re drawing out of the distribution that has CDF \(F(x)\), but we’re saying “draw out of F” for short.