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Posts tagged with magnitude

Research focuses on real wages—wages that are adjusted for inflation. Getting data on wages is tricky. But accounting for inflation is even harder. (For example, workers often paid rent informally, meaning that there are few records around).

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And so it is unsurprising that researchers differ in their estimations of real wages. Some, such as Peter Lindert and Jeffrey Williamson, suggest that full-time earnings for British common labourers, adjusted for inflation, more than doubled in the seventy years after 1780. But Charles Feinstein argued that over the same period, British real wages only increased by around 30%. It’s a bit of a … mess.

Most people agree that after about 1840, real wages did better. Nicholas Crafts and Terence Mills shows that from 1840 to 1910, real wages more than doubled. Their findings are mirrored by other researchers ….

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in almost all British cities, mortality conditions in the 1860s were no better—and were often worse—than in the 1850s. In Liverpool in the 1860s, the life expectancy fell to an astonishing 25 years. It was not until the two subsequent decades that rises in life expectancy were found

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Words that seem innocuous to me can set other people into anger mode. (I too, of course, have my own buttons that can be pushed—mostly economics-related.)

One woman who has raised more children than I can even fathom doing (9) will launch off if she hears the phrase “quality time”.

She says "Quality time" is short for "I don’t really know my kids—but I’ll make up for it by going to their sports games and dance recitals.”

To her, ∄ such thing as “quality time”—time spent with your kids that’s significantly better than, or can make up for, other, less important, time. For her you really get to know your kids during all the unimportant, mundane, pedestrian moments of life.




SETUP (CAN BE SKIPPED)

We start with data (how was it collected?) and the hope that we can compare them. We also start with a question which is of the form:

  • how much tax increase is associated with how much tax avoidance/tax evasion/country fleeing by the top 1%?
  • how much traffic does our website lose (gain) if we slow down (speed up) the load time?
  • how many of their soldiers do we kill for every soldier we lose?
  • how much do gun deaths [suicide | gang violence | rampaging multihomicide] decrease with 10,000 guns taken out of the population?
  • how much more fuel do you need to fly your commercial jet 1,000 metres higher in the sky?
  • how much famine [to whom] results when the price of low-protein wheat rises by $1?
  • how much vegetarian eating results when the price of beef rises by $5? (and again distributionally, does it change preferentially by people with a certain culture or personal history, such as they’ve learned vegetarian meals before or they grew up not affording meat?) How much does the price of beef rise when the price of feed-corn rises by $1?
  • how much extra effort at work will result in how much higher bonus?
  • how many more hours of training will result in how much faster marathon time (or in how much better heart health)?
  • how much does society lose when a scientist moves to the financial sector?
  • how much does having a modern financial system raise GDP growth? (here ∵ the X ~ branchy and multidimensional, we won’t be able to interpolate in Tufte’s preferred sense)
  • how many petatonnes of carbon per year does it take to raise the global temperature by how much?
  • how much does $1000 million spent funding basic science research yield us in 30 years?
  • how much will this MBA raise my annual income?
  • how much more money does a comparable White make than a comparable Black? (or a comparable Man than a comparable Woman?)
  • how much does a reduction in child mortality decrease fecundity? (if it actually does)

  • how much can I influence your behaviour by priming you prior to this psychological experiment?
  • how much higher/lower do Boys score than Girls on some assessment? (the answer is usually “low |β|, with low p" — in other words "not very different but due to the high volume of data whatever we find is with high statistical strength")

bearing in mind that this response-magnitude may differ under varying circumstances. (Raising morning-beauty-prep time from 1 minute to 10 minutes will do more than raising 110 minutes to 120 minutes of prep. Also there may be interaction terms like you need both a petroleum engineering degree and to live in one of {Naija, Indonesia, Alaska, Kazakhstan, Saudi Arabia, Oman, Qatar} in order to see the income bump. Also many of these questions have a time-factor, like the MBA and the climate ones.)

building up a nonlinear function from linear parts

As Trygve Haavelmo put it: using reason alone we can probably figure out which direction each of these responses will go. But knowing just that raising the tax rate will drive away some number of rich doesn’t push the debate very far—if all you lose is a handful of symbolic Eduardo Saverins who were already on the cusp of fleeing the country, then bringing up the Laffer curve is chaff. But if the number turns out to be large then it’s really worth discussing.

In less polite terms: until we quantify what we’re debating about, you can spit bollocks all day long. Once the debate is quantified then the discussion should become way more intelligent, less derailing to irrelevant theoretically-possible-issues-which-are-not-really-worth-wasting-time-on.

So we change one variable over which we have control and measure how the interesting thing responds. Once we measure both we come to the regression stage where we try to make a statement of the form “A 30% increase in effort will result in a 10% increase in wage” or “5 extra minutes getting ready in the morning will make me look 5% better”. (You should agree from those examples that the same number won’t necessarily hold throughout the whole range. Like if I spend three hours getting ready the returns will have diminished from the returns on the first five minutes.)

Correlation

Avoiding causal language, we say that a 10% increase in (your salary) is associated with a 30% increase in (your effort).

 
MAIN PART (SKIP TO HERE IF SKIMMING)

The two numbers that jump out of any regression table output (e.g., lm in R) are p and β.

  • β is the estimated size of the linear effect
  • p is how sure we are that the estimated size is exactly β. (As in golf, a low p is better: more confident, more sure. Low p can also be stated as a high t.)

Wary that regression tables spit out many, many numbers (like Durbin-Watson statistic, F statistic, Akaike Information, and more) specifically to measure potential problems with interpreting β and p naïvely, here are pictures of the textbook situations where p and β can be interpreted in the straightforward way:

First, the standard cases where the regression analysis works as it should and how to read it is fairly obvious:
(NB: These are continuous variables rather than on/off switches or ordered categories. So instead of “Followed the weight-loss regimen” or “Didn’t follow the weight-loss regimen” it’s someone quantified how much it was followed. Again, actual measurements (how they were coded) getting in the way of our gleeful playing with numbers.)

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Second, the case I want to draw attention to: a small statistical significance doesn’t necessarily mean nothing’s going on there.

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The code I used to generate these fake-data and plots.

If the regression measures a high β but low confidence (high p), that is still worth taking a look at. If regression picks up wide dispersion in male-versus-female wages—let’s say double—but we’re not so confident (high p) that it’s exactly double because it’s sometimes 95%, sometimes 180%, sometimes 310%, we’ve still picked up a significant effect.

The exact value of β would not be statistically significant or confidently precise due to a high p but actually this would be a very significant finding. (Try it the same with any of my other examples, or another quantitative-comparison scenario you think up. It’s either a serious opportunity, or a serious problem, that you’ve uncovered. Just needs further looking to see where the variation around double comes from.)

You can read elsewhere about how awful it is that p<.05 is the password for publishable science, for many reasons that require some statistical vocabulary. But I think the most intuitive problem is the one I just stated. If your geiger counter flips out to ten times the deadly level of radiation, it doesn’t matter if it sometimes reads 8, sometimes 0, and sometimes 15—the point is, you need to be worried and get the h*** out of there. (Unless the machine is wacked—but you’d still be spooked, wouldn’t you?)

 
FOLLOW-UP (CAN BE SKIPPED)

The scale of β is the all-important thing that we are after. Small differences in βs of variables that are important to your life can make a huge difference.

  • Think about getting a 3% raise (1.03) versus a 1% wage cut (.99).
  • Think about twelve in every 1000 births kill the mother versus four in every 1000.
  • Think about being 5 minutes late for the meeting versus 5 minutes early.

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linear maps as multiplication
linear mappings -- notice they're ALL straight lines through the origin!


Order-of-magnitude differences (like 20 versus 2) is the difference between fly and dog; between life in the USA and near-famine; between oil tanker and gas pump; between Tibet’s altitude and Illinois’; between driving and walking; even the Black Death was only a tenth of an order of magnitude of reduction in human population.




Keeping in mind that calculus tells us that nonlinear functions can be approximated in a local region by linear functions (unless the nonlinear function jumps), β is an acceptable measure of “Around the current levels of webspeed” or “Around the current levels of taxation” how does the interesting thing respond.



Linear response magnitudes can also be used to estimate global responses in a nonlinear function, but you will be quantifying something other than the local linear approximation.

Anscombes quartet  The four data sets are different, yet they have the same &#8220;line of best fit&#8221; as computed by ordinary least squares regression.




Readers of isomorphismes, you might enjoy powers of two tumblr.

powersoftwo:

2100 = 1,267,650,600,228,229,401,496,703,205,376 — one nonillion, two hundred sixty-seven octillion, six hundred fifty septillion, six hundred sextillion, two hundred twenty-eight quintillion, two hundred twenty-nine quadrillion, four hundred one trillion, four hundred ninety-six billion, seven hundred three million, two hundred five thousand, three hundred seventy-six (31 digits, 320 characters)

I think I’ve been subscribed since the 30’s. Never a letdown. And of course it’s only going to get more exciting.




A world where it takes 30 hours of minimum-wage work to pay the rent, is very different to a world where it takes 120 hours of minimum-wage work to pay the rent. It’s a factor of 4—just half an order of magnitude—and it already means

  • a leisurely life where you can eat out, drink beer, play music, chill with friends, basically a great mode of existence, versus
  • a life where you would need to work two jobs in order to pay for kids and a car; if you keep just the one job, you can fight for overtime so you also have enough money for food.

A factor of 30 is the difference in median incomes between U$A and the DRC. Basically one-and-a-half orders of magnitude from richest to poorest. Just one order of magnitude is the difference between man and mouse (length). And in finance, errors of a few basis points can, for certain structured or levered positions, mean the difference between a good bet and a bad bet.

1.3! Great!!!
0.7! Really Bad!!!

Parameters matter. Elasticities matter. Magnitudes matter. A lot. Think about that the next time you look at the coefficients in your linear regression.

See also: the nearly-invisible but all-important left-hand scale of the chart you’re reading.




Noticed:

  • It’s easier for me to grok statistical significance (p's and t's) from a scatterplot than magnitude (β's).
  • Even though magnitude can be the most important thing, it’s "hidden" off to the left.

    Note to self: look off to the left more, and for longer.
  • But I’m set up to understand the correlativeness in a sub_i, sub_j sense — which particular countries fit the pattern as well as how closely.

Questions:

  • Minute __:__ Do each of the dimensions of social problems correlate individually, or is this only a mass effect of the combination?

If it’s true that raising marginal tax rates on the rich lowers crime rates without paying for any anti-crime programmes, that’s almost a free lunch.

UPDATE: Oh, hey, six months after I watch this and 3 days after I put up the story, I see Harvard Business Review has a story corroborating the same effect, instead pointing out how economists don’t look at the p's and t's on a regression table. I feel like I “mentally cross out” any lines with a low t value and then wonder about the F value on a regression with the “worthless” line removed.




[I]n the late 1920’s and early 1930’s…. There were lots of deep thoughts [in economics], but a lack of quantitative results. … It is usually not of very great practical or even scientific interest to know whether the [causal] influence [of some factor] is positive or negative, if one does not know anything about the strength.


But much worse is the situation when an [outcome] is determined by many different factors at the same time, some factors working in one direction, others in the opposite directions. One could write long papers about so-called tendencies explaining how this … might work…. But what is the … total net effect of all the factors? This question cannot be answered without measures of … strength….

Trygve Haavelmo

Bank of Sweden pseudo-Dynamite Prize Laureate 1989, for work in econometrics

(Source: nobelprize.org)




The [foreign exchange] market is by far the largest and most liquid market in the world, with daily FX turnover estimated at around $2 trillion. If this seems like a lot to you, it is because it is. Compare FX volumes to the tiny $50 billion traded at the NYSE or the $800 billion traded in government debt and you get an idea of the size of the market.
Agustin Silvani, Beat the Forex Dealer