# Posts filed under Denominator? (64)

March 17, 2015

## Bonus problems

If you hadn’t seen this graph yet, you probably would have soon.

The claim “Wall Street bonus were double the earnings of all full-time minimum wage workers in 2014″ was made by the Institute for Policy Studies (which is where I got the graph) and fact-checked by the Upshot blog at the New York Times, so you’d expect it to be true, or at least true-ish. It probably isn’t, because the claim being checked was missing an important word and is using an unfortunate definition of another word. One of the first hints of a problem is the number of minimum wage workers: about a million, or about 2/3 of one percent of the labour force.  Given the usual narrative about the US and minimum-wage jobs, you’d expect this fraction to be higher.

The missing word is “federal”. The Bureau of Labor Statistics reports data on people paid at or below the federal minimum wage of \$7.25/hour, but 29 states have higher minimum wages so their minimum-wage workers aren’t counted in this analysis. In most of these states the minimum is still under \$8/hr. As a result, the proportion of hourly workers earning no more than federal minimum wage ranges from 1.2% in Oregon to 7.2% in Tennessee (PDF).  The full report — and even the report infographic — say “federal minimum wage”, but the graph above doesn’t, and neither does the graph from Mother Jones magazine (it even omits the numbers of people)

On top of those getting state minimum wage we’re still short quite a lot of people, because “full-time” is defined by 35 or more hours per week at your principal job.  If you have multiple part-time jobs, even if you work 60 or 80 hours a week, you are counted as part-time and not included in the graph.

Matt Levine writes:

There are about 167,800 people getting the bonuses, and about 1.03 million getting full-time minimum wage, which means that ballpark Wall Street bonuses are 12 times minimum wage. If the average bonus is half of total comp, a ratio I just made up, then that means that “Wall Street” pays, on average, 24 times minimum wage, or like \$174 an hour, pre-tax. This is obviously not very scientific but that number seems plausible.

That’s slightly less scientific than the graph, but as he says, is plausible. In fact, it’s not as bad as I would have guessed.

What’s particularly upsetting is that you don’t need to exaggerate or use sloppy figures on this topic. It’s not even that controversial. Lots of people, even technocratic pro-growth economists, will tell you the US minimum wage is too low.  Lots of people will argue that Wall St extracts more money from the economy than it provides in actual value, with much better arguments than this.

By now you might think to check carefully that the original bar chart is at least drawn correctly.  It’s not. The blue bar is more than half the height of the red bar, not less than half.

February 16, 2015

## Pot and psychosis

The Herald has a headline “Quarter of psychosis cases linked to ‘skunk’ cannabis”, saying

People who smoke super-strength cannabis are three times more likely to develop psychosis than people who have never tried the drug – and five times more likely if they smoke it every day.

The relative risks are surprisingly large, but could be true; the “quarter” attributable fraction needs to be qualified substantially. As the abstract of the research paper (PDF) says, in the convenient ‘Interpretation’ section

Interpretation The ready availability of high potency cannabis in south London might have resulted in a greater proportion of first onset psychosis cases being attributed to cannabis use than in previous studies

Let’s unpack that a little.  The basic theory is that some modern cannabis is very high in THC and low in cannabidiol, and that this is more dangerous than more traditional pot. That is, the ‘skunk’ cannabis has a less extreme version of the same problem as the synthetic imitations now banned in NZ.

The study compared people admitted as inpatients in a particular area of London (analogous to our DHBs) to people recruited by internet and train advertisements, and leaflets (which, of course, didn’t mention that the study was about cannabis). The control people weren’t all that well matched to the psychosis cases, but it wasn’t too bad.  The psychosis cases were somewhat more likely to smoke cannabis, and much more likely to smoke the high-THC type. In fact, smoking of other cannabis wasn’t much different between cases and controls.

That’s where the relative risks of 3 and 5 come from.  It’s still possible that these are due at least in part to some other factor; you can’t tell from just this sort of data. The atttributable fraction (a quarter of cases) comes from combining the relative risk with the proportion of the population who are exposed.

Suppose ‘skunk-type’ cannabis triples your risk, and 20% of people in the population use it, as was seen for controls in the sample. General UK data (eg) suggest the rate in non-users might be 5 cases per 10,000 people per year. So, in 100,000 people, 80,000 would be non-users and you’d expect 40 cases per year. The other 20,000 would be users, and you’d expect a background rate of 10 cases plus 20 extra cases caused by the cannabis. So, in the 100,000 people, you’d get 70 cases per year, 50 of which would have happened anyway and 20 due to cannabis. That’s not exactly the calculation the researchers did — they used a trick where they don’t need the background rate as long as it’s low, and I rounded more — but it’s basically the same. I get 28%; they got 24%.

The figures illustrate two things. First, the absolute risk increase is roughly 20 cases per 100,000 20,000 people per year. Second, the ‘quarter’ estimate is very sensitive to the proportion exposed. If 5% of people used ‘skunk-type’ cannabis, you can run the numbers again and you get 5 cases due to cannabis out of 55 in 100,000 people: only 9% of cases due to exposure.

Now we’re at the ‘interpretation’ quote from the research paper.  In this South London area, 20% of people have used mostly the high-potency cannabis and 44% mostly have used other types, with 37% non-users. That’s a lot of pot.  Even if the relative risks are correct, the population attributable proportion will be much lower for the UK as a whole (or for NZ as a whole).

Still, the research does tend to support the idea of regulated legalisation, the sort of thing that Mark Kleiman advocates, where limits on THC and/or higher taxes for higher concentrations can be used to push cannabis supply to lower-risk varieties.

February 3, 2015

## Spotty coverage

Here’s a graph from the Economist showing the impact of the measles vaccine:

The number of measles cases fell from over half a million per year to about 100 per year when the vaccine was introduced. That’s a 99.98% reduction, in a disease that (in a healthy population) kills about two people in a thousand.

Here’s a graph from the Centers for Disease Control showing that little blip in 1990 on an expanded scale:

They say

The most important cause of the measles resurgence of 1989–1991 was low vaccination coverage. Measles vaccine coverage was low in many cities, including some that experienced large outbreaks among preschool-aged children throughout the early to mid-1980s. Surveys in areas experiencing outbreaks among preschool-aged children indicated that as few as 50% of children had been vaccinated against measles by their second birthday, and that black and Hispanic children were less likely to be age-appropriately vaccinated than were white children.

Vaccine coverage isn’t as bad as that now, but the profile of unvaccinated kids is different. Black and Hispanic children are just as likely as white children to have had at least one doses of the measles vaccine, and children in poverty have a rate only 1.5 percentage points lower. Now, a substantial chunk of the problem is parents who are anti-vaccine.

Kieran Healy has an interesting post on the ‘personal belief exemption’ data for kindergarten children in California.  They are only 3.36% of children, but they cluster.  That’s important because US is just on the edge of having high enough vaccine coverage to stop an epidemic from spreading, at least if the unvaccinated were evenly spread through the population. They aren’t:

the number of kindergarteners with PBEs, even in Berkeley, is not huge—about 67 kids out of 850 in the city. But 20 of those 67 are in the same school, and probably the same room.

Anti-vaccine hysteria is more prominent in the US than New Zealand: partly because our mainstream media don’t go in for it, and partly because everything is more prominent in the US. Similarly, reaction to the risks posed by unvaccinated children has been more prominent in the US. However,  New Zealand has a similar rate of measles vaccination. Our schools or early childhood services cannot refuse enrollment based on vaccination (no special paperwork is required as in California), and (like California) can only temporarily exclude unvaccinated children if they are known to have been exposed.

Last year, New Zealand had 283 cases of measles. Scaled for population, last year in NZ was about half as bad as the US in 1990, and about thirty times bigger than the current US outbreak (so far).

January 6, 2015

## Foreign drivers, again

The Herald has a poll saying 61% of New Zealanders want to make large subsets of foreign drivers sit written and practical tests before they can drive here (33.9%: people from right-hand drive countries; 27.4% everyone but Australians). It’s hard to tell how much of this is just the push effect of being asked the questions and how much is real opinion.

The rationale is that foreign drivers are dangerous:

Overseas drivers were found at fault in 75 per cent of 538 injury crashes in which they were involved. But although failure to adjust to local conditions was blamed for seven fatal crashes, that was the suspected cause of just 26 per cent of the injury crashes.

This could do with some comparisons.  75% of 538 is 403, which is about 4.5% of all injury crashes that year.  We get about 2.7 million visitors per year, with a mean stay of 20 days (PDF), so on average the population is about 3.3% short-term visitors.

Or, we can look at the ‘factors involved’ for all the injury crashes. I get 15367  drivers of motorised vehicles involved in injury crashes, and 9192 of them have a contributing factor that is driver fault (causes 1xx to 4xx in the Crash Analysis System). This doesn’t include things like brake failures.  So, drivers on average are at fault in about 60% of the injury crashes they are involved in.

Based on this, it looks as though foreign drivers are somewhat more dangerous, but that restricting them is very unlikely to prevent more than, say, 1-2% of crashes. If you consider all the ways we might reduce injury crashes by 1-2%, and think about the side-effects of each one, I don’t think this is going to be near the top of the list.

January 2, 2015

## Using the right denominator

We go on and on about denominators on StatsChat: the right way to report things that happen to people is usually a rate per capita rather than a total, otherwise you end up saying that Auckland has the highest number of whatever it is in New Zealand.  You do have to use the right denominator, though.

The Vatican City has the world’s highest crime rate.

That’s because the permanent population is less than 500, but the daily tourist population is about 100 times larger. The right denominator would be the tourist population.

In most countries this isn’t really an issue. For example,  in New Zealand,which has a lot of tourism, short-term visitors are only about 5% of the population. Even in the Cook Islands, residents outnumber tourists.

June 18, 2014

## Counts and proportions

Phil Price writes (at Andrew Gelman’s blog) on the impact of bike-share programs:

So the number of head injuries declined by 14 percent, and the Washington Post reporter — Lenny Bernstein, for those of you keeping score at home — says they went up 7.8%.  That’s a pretty big mistake! How did it happen?  Well, the number of head injuries went down, but the number of injuries that were not head injuries went down even more, so the proportion of injuries that were head injuries went up.

To be precise, the research paper found 638 hospitalised head injuries in 24 months before the bike share program, and 273 in the 12 months afterwards. In a set of control cities that didn’t start a bike-share program there were 712 head injuries in the 24 months before the matching date and 342 in the 12 months afterwards. That is, a 14.4% decrease in the cities that added bike-share programs and a 4% decrease in those that didn’t.

June 10, 2014

Stuff has a story about ‘Powerhouse Fruits and Vegetables”.  They get points for linking to the source, and for pointing out that although

“The rankings provide clarity on the nutrient quality of the different foods

the rankings are quite different from other rankings that are supposed to do the same thing.

They don’t point out that it’s silly to list scores to four digit accuracy when nutrient content varies enough to make the first digit somewhat unreliable.They also don’t point out that these are nutrient scores not per serving, but per 100 Calories of the food. Google thinks watercress has about 11 Calories/100g, so that’s 900g of watercress. The data are for raw watercress — the research paper doesn’t say how much the score goes down if you stir-fry it, Chinese-style.

If you much through a couple of pounds of watercress, it’s not surprising you’d pick up a few nutrients along the way. The applicability of this fact to NZ daily life must be a bit limited, though.

June 8, 2014

## Foreign drivers

From the ChCh Press

Foreign drivers cause more fatal and injury crashes in the South Island than the national average – and the West Coast is the worst spot.

They don’t actually mean “more,” they mean “a higher proportion of”.

New Zealand Transport Agency (NZTA) safety directions chief adviser Lisa Rossiter said its crash statistics for the past 10 years showed foreign drivers were involved in about 6 per cent of all fatal or injury crashes in New Zealand, and were at fault in about 2 per cent.

On average, short-term visitors make up roughly 2.5% of people in New Zealand (2.78 million visitors in the year to April 2014, median visit of 9 days, so I’m guessing mean visit about two weeks). About another 2% of people in New Zealand are international students, who are at least sometimes counted as foreign drivers.

So, the risk seems to be a bit higher for foreign drivers, but probably not twice as high. Some of the excess can probably be explained by age: international students, backpackers, and drunk Australians in Queenstown are younger than the population average.

It’s different in parts of the South Island

The tourist hot spots of Otago and the West Coast fared worst.

A foreign driver was identified as a factor in 13 per cent of fatal crashes on the coast, and 5 per cent of fatal crashes in Otago from 2004 to 2013.

A lot of this must be because tourists are over-represented in tourist hot spots: that’s what ‘tourist hot-spot’ means. The proportion of short-term visitors is about 2.5% nationwide, but it’s probably rather lower that than in Gisborne and rather higher on the West Coast.

It’s also worth noting that “identified as a factor” is fairly weak. If you go to the Ministry of Transport reports and add up the percentage of times different factors were involved in a crash, you get a lot more than 100% (for the 2010 report I get 225% for fatal crashes and 185% for injury crashes)

For crashes involving a tourist driver and more than one car, the foreign driver was fully or partly responsible two out of three times.

This at least gets rid of the denominator problem, but the “partly” responsible is still a problem. We aren’t told what proportion of the time the local driver was fully or partly responsible — based on the information given, that could also be two out of three times.

It’s quite likely that foreign drivers are at higher risk, especially those from countries that drive on the right, but the problem is not a big fraction of the NZ road toll. It’s worth considering things that can sensibly be done to reduce it — which doesn’t include withdrawing from the U.N. Convention on Road Traffic — but if you’re trying to stop road deaths it may be more effective to concentrate on interventions that don’t just affect foreign drivers.  Clearer signage, guard rails and median barriers, separated bike lanes, improved public transport… there are many things that might knock a percentage point off road deaths more easily than targetting foreign drivers.

May 23, 2014

## Is Roy Morgan weird?

There seems to be a view that the Roy Morgan political opinion poll is more variable than the others, even to the extent that newspapers are willing to say so, eg, Stuff on May 7

The National Party has taken a big hit in the latest Roy Morgan poll, shedding 6 points to 42.5 per cent in the volatile survey.

I was asked about this on Twitter this morning, so I went to get Peter Green’s data and aggregation model to see what it showed. In fact, there’s not much difference between the major polling companies in the variability of their estimates. Here, for example, are poll-to-poll changes in the support for National in successive polls for four companies

And here are their departures from the aggregated smooth trend

There really is not much to see here. So why do people feel that Roy Morgan comes out with strange results more often? Probably because Roy Morgan comes out with results more often.

For example, the proportion of poll-to-poll changes over 3 percentage points is 0.22 for One News/Colmar Brunton, 0.18 for Roy Morgan, and 0.23 for 3 News/Reid Research, all about the same, but the number of changes over 3 percentage points in this time frame is 5 for One News/Colmar Brunton, 14 for Roy Morgan, and 5 for 3 News/Reid Research.

There are more strange results from Roy Morgan than for the others, but it’s mostly for the same reason that there are more burglaries in Auckland than in the other New Zealand cities.

March 30, 2014

I saw this story in the Herald and didn’t read it in detail, just thought it was an interesting calculation to do

The Financial Times reported last week that the average global price of eight breakfast staples had risen almost 25 per cent this year.

The increases mainly affected coffee, orange juice, wheat, sugar, milk, butter, cocoa and pork.

We decided to create a Kiwi version of the Financial Times story and Statistics NZ food price figures reveal New Zealand families are not exempt from the trend.

David Farrar did read the story, and so was rather less impressed, as he also mentioned on Twitter.  The problem is that the calculation was done wrong.

If you served tomatoes, mushrooms, bacon, toast, eggs, tinned spaghetti and cereal, with coffee, tea and orange juice this weekend, it would have cost you 6.9 per cent more than the same meal in 2008, and almost 3 per cent more than in 2012. Breakfast food prices have risen more quickly than other prices.

Over the past five years, the compound average annual rate of inflation was 2.1 per cent.

If the average annual rate was 2.1%, which sounds about right, the total increase over five years would be 2.1% five times, which turns out to be 11%. Since 6.9% is less than 11%, breakfast food prices have risen less quickly than other prices. Quite a bit less. The story has it completely backward.

If you’re reading especially carefully, you might also notice that it’s more than five years from “this weekend” back to 2008 — for example, a comparison of end of March 2008 to this weekend would be a six year period.

This is the sort of thing that a subeditor should spot. It’s also the sort of thing the RBNZ inflation calculator is useful for — you put in a number and two years and it does the calculations.  If you use the calculator, you find that “this weekend” is apparently December 2013, and the 2008 comparison is December 2008, rather than March 2008 to March 2013. You’d also see that the sub-index for food had increased less than the total CPI, which would presumably make you more suspicious about the story.

There’s also some discussion of individual item prices. This doesn’t have the awful 5:1 error ratio of the main argument, but it still demonstrates where a bit of thinking could have helped

Mild Arabica coffee was trading on the commodity markets for US\$1.76 (\$2.03) a pound (453g) in February, up from US\$1.35 in January. Mild Arabica coffee was trading on the commodity markets for US\$1.76 (\$2.03) a pound (453g) in February, up from US\$1.35 in January.

If you go to the Countdown website you find that their Signature range coffee beans cost NZ\$6 for 200g, or roughly US\$12 per pound. Obviously most of the cost is not the wholesale commodity price. That’s presumably even more true for instant coffee (the authentic version of the beverage in a ‘traditional cooked breakfast’)

The components of the CPI that have increased fastest aren’t all that surprising if you read the Herald regularly. For example, the cost of home ownership was up 27% over that five-year period, insurance was up 26%, education, and cigarettes and tobacco were up 67%.

If some things go up faster than average, others must go up slower or even decrease. Household appliances and furniture are down a bit. Telecommunications equipment, computing equipment,  and telecommunications services have gotten much cheaper. You can hardly give away a 2008 phone or computer (though if you’re trying to, Te Whare Marama refuge will put it, and more recent kit, to good use)