# Lesson 4: Measures of Disease Frequency *Companion-podcast transcript • Sarah & Kiffer* *Office Hours episode to listen to after working through the lesson* --- **Sarah:** Welcome back to Office Hours. I'm Sarah. **Kiffer:** And I'm Kiffer. This episode goes with Lesson four, Measures of Disease Frequency. **Sarah:** It's meant for after you've worked through the lesson. We'll add some perspective and critique, work through the questions students tend to find thorny with this material, and do some extra worked examples, including a few that are harder than the ones on the lesson page. **Kiffer:** There are a few places where we ask you to work something out before we give the answer. When we do, you'll hear a few seconds of quiet. Pause the audio if you'd like more time. **Sarah:** Here are the four questions. Can a rate be bigger than one? How can a disease become more common while fewer people catch it? Why is the case fatality figure reported during an outbreak so unreliable? And what does a standardized rate actually tell you? We'll finish with DALYs, because Kiffer and I don't entirely agree about them. **Kiffer:** Let's start with the first one. **Sarah:** I'll admit this one confused me when I first learned it. I pictured a rate as something like a percentage, twelve percent of people or something like that. So a rate of two point five sounded impossible to me. **Kiffer:** That's a very common picture, and it comes from the way the word rate is used in the news. Here's an example. The American Lung Association says that adults get an average of two to three colds a year. Suppose we followed a group of adults for a year, counted every cold, and found two and a half colds per person-year. That's an incidence rate of 2.5 per person-year. **Sarah:** And that works because the denominator is person-time. One person can have several colds in a year. **Kiffer:** Right. A rate counts events per unit of time at risk, and each person can contribute more than one event, so nothing stops it from going above one. A risk is the proportion of people who have at least one event over a period, so it can never go above one. **Sarah:** So here's a question for everyone listening. If the rate of colds is 2.5 per person-year, what proportion of adults will catch at least one cold this year? Think about which formula applies and roughly what the answer should be. Take a few seconds. *(Pause)* **Kiffer:** The tempting answer is to multiply the rate by the time, the way the shortcut in the lesson does. Two point five times one year gives two point five, which would mean two hundred and fifty percent of adults. That can't be right. **Sarah:** Because the shortcut only works when the rate times the time is small. The lesson says less than about 0.1. **Kiffer:** Exactly, and here it's twenty-five times bigger than that. So we use the exact formula. Risk equals one minus e to the minus rate times time. E to the minus 2.5 is about 0.082, and one minus that is about 0.92. **Sarah:** So about ninety-two percent of adults would catch at least one cold. **Kiffer:** That's true if colds arrive at random, at the same steady rate for everyone. Real colds don't behave that way. A parent with two children in daycare might catch five in a year, while someone who lives alone and works from home might catch none. When the same average is made up of some people with many colds and some with very few, the share who catch at least one is lower than the formula says. **Sarah:** So for something like colds, the formula gives an upper limit. **Kiffer:** When the only difference is that people catch colds at different rates, yes. It's a good reminder that every formula in this lesson carries an assumption, and a steady rate that's the same for everyone is one of the big ones. **Sarah:** Question two. How can a disease become more common while fewer people are catching it? **Kiffer:** This is the difference between prevalence and incidence, and it's one of the most common sources of confusion in the course. Let me start with a real case. According to UNAIDS, new HIV infections worldwide have fallen by about two thirds since their peak in the mid-1990s. Over the same period, the number of people living with HIV has kept rising, to about forty-one million in 2025. **Sarah:** And the reason is treatment. **Kiffer:** Mostly, yes. Before effective antiretroviral therapy, the typical time from HIV infection to death was around ten years. Today, someone who is diagnosed early and treated can expect to live close to a normal lifespan. In the sink picture from the lesson, the tap has been turned down, but the death drain has narrowed a great deal, and there's still no recovery drain. So the pool of people living with HIV keeps growing. **Sarah:** So if I saw a headline that said more people than ever are living with HIV, I might read it as the epidemic getting worse, when part of it reflects a treatment success. **Kiffer:** Right. A rising prevalence can be good news or bad news, and the prevalence alone can't tell you which. You need the incidence and the duration as well. **Sarah:** Which brings me to a problem I want to try myself, because it's a calculation that trips a lot of people up. A health region reports that eight percent of its adults have diagnosed diabetes, and that there are six new diagnoses per one thousand person-years among adults without diabetes. What's the average duration of diabetes in this region? **Kiffer:** Before you start, let's give everyone a few seconds to try it. *(Pause)* **Sarah:** Okay. The formula from the lesson says prevalence equals I times D, divided by I times D plus one. I need to turn it around to get D. **Kiffer:** How would you do that? **Sarah:** If P equals I times D, divided by I times D plus one, then I times D equals P divided by the quantity one minus P. So I times D is 0.08 divided by 0.92, which is about 0.087. Then D is 0.087 divided by I, which is six. That gives D of about 0.0145 years. **Kiffer:** And how long is that? **Sarah:** About five days. Okay, that's clearly wrong. Nobody has diabetes for five days. **Kiffer:** That sanity check is worth doing every time. So where did it go wrong? **Sarah:** The incidence. It's six per one thousand person-years, so as a rate per person-year it's 0.006. I used six. **Kiffer:** Right. Try it again with 0.006. **Sarah:** 0.087 divided by 0.006 is about fourteen and a half. So the average duration is about fourteen and a half years. **Kiffer:** That's a believable number. Diabetes is a lifelong condition, so the duration here is mostly the time from diagnosis to death. There's one more question worth asking about any answer like this. What assumption does it depend on? **Sarah:** The steady state. The formula assumes the incidence and the duration have been stable for a long time, so the pool of cases is in balance. **Kiffer:** And for diabetes in Canada, that assumption is shaky. People with diabetes are living longer than they used to, so the share of adults living with diabetes has kept rising, even after allowing for the aging of the population. The pool hasn't settled into balance. Fourteen and a half years is a rough guide to the duration, and a good answer would say so. **Sarah:** There's one more consequence of all this that I think surprises people. If you study people who already have a disease to find its risk factors, you can be misled. **Kiffer:** Yes. Suppose you recruited people who currently have a disease and compared them with people who don't. Anything that helps people survive longer with the disease will be more common among the existing cases, so a study of existing cases can mistake a factor that prolongs survival for a cause of the disease. Epidemiologists call this prevalence-incidence bias, or Neyman bias. It's the main reason the lesson recommends incidence for risk factor research. **Sarah:** Question three. Why is the case fatality figure reported during an outbreak so unreliable? **Kiffer:** Let's build it with made-up numbers first. Imagine a new respiratory infection. Twenty days into the outbreak, there are one thousand confirmed cases. Thirty people have died, five hundred and seventy have recovered, and four hundred are still sick. **Sarah:** And the news reports thirty out of one thousand, which is a case fatality rate of three percent. **Kiffer:** So here's the question. Among these confirmed cases, is the final case fatality likely to be higher or lower than three percent? Take a few seconds. *(Pause)* **Kiffer:** It's likely to be higher. Four hundred people don't have an outcome yet. Some of them will die, and when they do, they'll be added to the numerator, but they're already in the denominator. People are counted as cases at the start of their illness, and deaths come at the end. Early in an outbreak, that lag pulls the estimate down. **Sarah:** So one fix would be to use only the people whose illness is over. **Kiffer:** Right. Thirty deaths among the six hundred people who have either died or recovered gives five percent. The final value can't be lower than three percent. If deaths and recoveries take about the same time, it will end up close to five percent, and if deaths tend to come sooner than recoveries, it will land somewhere between the two. **Sarah:** Is there a real example of this? **Kiffer:** SARS in 2003 is the classic one. Early in the outbreak, officials put the case fatality at under four percent. In May 2003, the World Health Organization revised its estimate to fourteen to fifteen percent, and to more than fifty percent among people aged sixty-five and over. By the end, about ten percent of the roughly eight thousand reported cases worldwide had died, and in both Canada and Hong Kong the figure was about seventeen percent. **Sarah:** That's a huge change. **Kiffer:** It is. And a second problem pushes in the other direction. The denominator is confirmed cases, and confirmed cases are mostly people who were sick enough to be tested. Many people with mild infections are never counted. **Sarah:** Which makes the case fatality figure too high, if what you want is the chance of dying for everyone who's infected. **Kiffer:** Exactly. That's the difference between the case fatality rate, which the lesson points out is really a proportion, and what's called the infection fatality ratio. In March 2020, the head of the World Health Organization said that about 3.4 percent of reported COVID-19 cases had died. Experts responding at the time pointed out both problems. Many of the reported cases were still ill, which would push the figure up over time. And many mild infections had never been tested, so the figure for everyone infected was probably much lower, with early estimates around one percent. **Sarah:** So the 3.4 percent was correct arithmetic for the question it answered, which was narrower than the question most people heard. **Kiffer:** That's the general point. Before you interpret any of these numbers, ask who is in the denominator, and whether everyone in it has had time to reach the outcome. **Sarah:** Question four is about standardization. When I first met a directly standardized rate, I didn't understand what the number meant, because it isn't the rate in any real place. **Kiffer:** That's the right question to ask. A directly standardized rate is a hypothetical number. It answers the question, what would this population's rate be if it had the age structure of the standard population? If you change the standard population, the number changes. **Sarah:** So a rate standardized to the 2011 Canadian population can't be compared with one standardized to the World Health Organization's standard population. **Kiffer:** Right. Standardized rates are comparable only when they use the same standard. A standardized rate is useful mainly for comparison with other rates standardized to the same population. **Sarah:** Here's a trap I've seen in published reports. Two towns each report a standardized mortality ratio, each calculated against provincial death rates. Town A has an SMR of 1.2, and Town B has an SMR of 1.1. Can we conclude that Town A has the higher mortality? Take a few seconds on that one. *(Pause)* **Kiffer:** Not safely. Each SMR compares one town with the provincial rates, using that town's own age structure as the weights. If Town A is mostly young people and Town B is mostly older people, the two ratios average over very different age groups. Each one is a fair comparison with the province. Comparing the two ratios with each other mixes the difference in age structure back in, unless each town's death rate is about the same multiple of the provincial rate in every age group. **Sarah:** So what would you do? **Kiffer:** If both towns have enough deaths in each age group, I'd directly standardize both to the same standard population and compare those rates. If they don't, I'd report each SMR as a comparison with the province, and I'd avoid ranking the towns against each other. I'd also check the confidence intervals, because 1.2 and 1.1 may not differ by more than chance. **Sarah:** This matters for real comparisons in Canada, too. Nova Scotia has an older population than Alberta, so it records more deaths per thousand residents each year. **Kiffer:** Which is why Statistics Canada compares provinces with age-standardized mortality rates. Most of the gap in crude death rates between those two provinces reflects their age structures. Once both are standardized, Nova Scotia's rate is still somewhat higher, but the gap is much smaller. **Sarah:** Okay. Now the part where we disagree. **Kiffer:** DALYs. **Sarah:** I have real reservations about disability weights. A disability weight is a number between zero and one that says how much of a year of healthy life is lost when that year is lived with a condition. When I first learned that, my reaction was, who decided how much a year lived with a disability is worth? **Kiffer:** That's a fair question, so let's be precise about where the weights come from. For the Global Burden of Disease 2010 study, they came from surveys of the general public. About fourteen thousand adults were surveyed in person or by phone in Bangladesh, Indonesia, Peru, Tanzania and the United States, and about sixteen thousand more answered an open web survey. They were shown pairs of short descriptions of health states and asked which person was healthier. **Sarah:** So most of the people judging the weights don't live with the conditions they're judging. **Kiffer:** That's right, and it was a deliberate choice. The researchers wanted the weights to describe how much health a condition takes away, as the general public judges it. They also found that people in very different countries ranked the health states in much the same way. **Sarah:** But people agreeing with each other doesn't tell us whether they're right. Many people living with serious disabilities rate their own quality of life much higher than outsiders expect. This is often called the disability paradox. If you asked people who live with a condition, some of the weights would probably be lower. **Kiffer:** I agree that's a real limitation, and the GBD researchers have acknowledged that the short descriptions people judge may not match the average experience of a condition. Where I'd push back is on the idea that the weight says anything about the value of a person. A disability weight is meant to describe health, and a DALY counts years of health lost. **Sarah:** In principle. But when DALYs are used to decide what to fund, a program that extends the lives of people with a disability averts fewer DALYs than one that extends the lives of people without one. That's a real consequence. **Kiffer:** That's true, and it's the strongest version of your argument. The method has also changed before. In the 2010 study, the GBD dropped two value choices. Earlier versions gave a year of life more weight at some ages than at others, which was called age weighting, and they discounted future years. Both were removed. **Sarah:** So where do you land? **Kiffer:** I think DALYs are the best tool we have for comparing conditions that mainly kill with conditions that mainly disable, and I'd keep using them for that. I wouldn't use them on their own to decide who gets care. I'd want them alongside equity considerations and the views of the people affected. **Sarah:** I'm more skeptical than Kiffer, but we agree on the practical rule. When you read a DALY figure, check where the weights came from and which version of the method produced it. **Kiffer:** Let's pull it together with three things to take away. **Sarah:** First, check the denominator. Whether it holds people, person-time, confirmed cases or everyone infected decides what the number means. **Kiffer:** Second, every formula in this lesson rests on an assumption. The risk formula assumes a constant rate, and for repeated events like colds, a rate that's the same for everyone. The prevalence formula assumes a steady state. State the assumption when you use the formula. **Sarah:** And third, read each measure alongside the others it depends on. A rising prevalence can reflect a treatment success, and a standardized rate is read by comparing it with another rate standardized to the same population. **Kiffer:** If you'd like more practice, the diabetes duration problem from this episode is a good model for the calculation questions in this course. Try it again with different numbers, and do the sanity check at the end. **Sarah:** Next time, it's Lesson five, Screening and Diagnostic Tests, where sensitivity, specificity and predictive values come in. **Kiffer:** Take care, everyone. **Sarah:** See you in Lesson five.