tl;dr - What follows is a partially organized set of thoughts and initial findings from starting to learn about and possibly calculating a receiver operator characteristic (ROC) curve using alert categorization data available via Opsweekly. The goal is to measure how effective our monitoring system is at identifying actionable failure events that require an operator's attention.
Given the current data set I have to work with at the moment, I'm unable to calculate a ROC curve. I have some ideas on how to complete the data to make this possible.
I have much more to learn and apply here.
Improving the Signal-to-Noise Ratio
Last year I posted some ideas on solving monitoring. Most relevant here is the notion of reducing noise and strengthening signal in our monitoring systems. Before we can improve it we must be able to measure it.
I'm fortunate to work with some amazing people, not least among them, John Allspaw. He and I got to talking about signal and noise in monitoring systems one day and he turned me on to a little bit of signal detection theory called the receiver operating characteristic or ROC.
In summary (and probably a terrible one at that), the idea behind the ROC is to help determine the efficacy of a test's ability to accurately detect the presence of whatever the test is supposed to detect. ROC came out of attempts after World War II to understand just how well radar operators could accurately detect the presence of enemy aircraft. After all, that little blip on the screen could be friendly aircraft; it could be a flock of geese; it could even be a defect in the screen itself.
ROC curves have most notably (I've found thus far) made their way into medicine where they're used to gauge the effectiveness of a given test's ability to identify the presence of disease in a set of patients. Luckily, there is a lot of literature to start digging into to understand ROC curves. Unfortunately, there is a lot of literature to start digging into to understand ROC curves.
Suffice it to say that in order to derive a ROC curve, one needs to have a measure of the true and false positive results and true and false negative results of a test.
Using ROC Curves
What can one do with a ROC curve? Once a ROC curve is created, one can calculate the detectability index (d'; read "dee prime") for the set. The higher the value, the better the test or system is at positively detecting what it's designed to detect. Related to d' (I think) is the area under the curve (AUC). This measurement helps determine how accurate a given test is.
Hypothesis: Using alert data gathered from Opsweekly, we can measure how well our monitoring system is at detecting actionable failure events. Using ROC curves, we can compare those data on at least a weekly basis to gauge how good or bad on-call periods have been. Sure, boiling down an on-call week to a single value is highly reductive, but I am curious what, if anything, the ROC can teach us.
NOTE: At Etsy, we use Opsweekly to categorize the alerts we've received from Nagios. Everything that follows from this point assumes some familiarity with Nagios, at least, and Opsweekly, hopefully.
Reviewing the alert categories we currently have defined in Opsweekly, I felt confident that we could calculate the sensitivity (true positive rate) of the alerts. 
Opsweekly Alert Categories
The categories feel pretty clear about what is a true positive (i.e. action taken) versus a false positive (i.e. no action taken due to downtime not being set). However, to calculate a ROC curve, one needs to also know the specificity (false positive rate) of a test's results. Unfortunately, we don't have such data for our alerts. We're not capturing data on true negatives (correct rejections), that is where Nagios did not alert because a given check/test's results were within "normal" limits/thresholds. Nor do we have data on false negatives (misses) where Nagios didn't alert but should have. 
Proving a Negative
To collect data for true negatives, we can probably calculate the expected count of correct rejections based on a given time period, the frequency of checks, and the known count of observed true/false positives. This may be brittle and not very explicit (i.e. we're not counting correct rejections as they occur) but it may suffice to get started.
Getting data on false negatives seems to be the biggest challenge to quantify because our checks' thresholds are defined by human operators and therefore need intervention by those operators when they're found to be inadequate/inappropriate for surfacing problems. If/when such a review were to occur (say, during a postmortem review), we'd also need to document somewhere (ideally Opsweekly) that Nagios failed to notify. This also feels brittle in that it depends on:
- Someone identifying a check failed to notify.
- Someone manually updating a data set to indicate what was a false negative.
That having been said, in the medical literature I read, it occurred to me that there had to have been some independent review and/or confirmation of a test's results (i.e. via some secondary test) before a ROC curve could be calculated. I found nothing clearly stating as much but it was implied because a ROC curve can't be accurately calculated without complete information.
In general, I imagine Opsweekly will need to be extended in some form to accommodate the collection of negative data.
I still have more to learn about ROC curves and their application. I have identified that I have an incomplete data set preventing me from generating a ROC curve (and related measurements) to understand how well our monitoring system is performing, in terms of identifying actionable failure events that require an operator's attention.
There is a very interesting paper by Sorkin and Woods titled "Systems with Human Monitors: A Signal Detection Analysis" (PDF) that describes the use of ROC curves in determining the effectiveness of monitoring systems. Notably they review what they term "two-stage detection systems" which basically boils down to computers and humans cooperating to monitor systems. I dig on computers augmenting humans as a functioning part of the team. I'll be revisting this paper soon.
 It turns out I was incorrect. Calculating the sensitivity of a test also
requires that one know the count of false negatives. Sensitivity is calculated
P(T+|E+) = TP/(TP+FN). In other words, the probability of the test
indicating an event occurred given that an event did occur is equal to the count
of true positive results divided by the sum of the counts of true positive and
false negative results. Return
 I considered the case where Nagios can't communicate with a host's NRPE daemon being possibly defined as a miss, but Nagios will still alert with a status of UNKNOWN in this case and feels a little more like a false positive. Return