Investigation: Hypobaric Hypoxia Causes Body Weight Reduction by Lippl et al. (2010)

I. 

One of the mysterious aspects of obesity is that it is correlated with altitude. People tend to be leaner at high altitudes and fatter near sea level. Colorado is the highest-altitude US state and also the leanest, with an obesity rate of only 22%. In contrast, low-altitude Louisiana has an obesity rate of about 36%. This is pretty well documented in the literature, and isn’t just limited to the United States. We see the same thing in countries around the world, from Spain to Tibet

A popular explanation for this phenomenon is the idea that hypoxia, or lack of oxygen, leads to weight loss. The story goes that because the atmosphere is thinner at higher altitudes, the body gets less oxygen, and this ends up making people leaner.

One paper claims to offer final evidence in favor of this theory: Hypobaric Hypoxia Causes Body Weight Reduction in Obese Subjects by Lippl, Neubauer, Schipfer, Lichter, Tufman, Otto, & Fischer in 2010. Actually, the webpage says 2012, but the PDF and all other sources say 2010, so whatever.

This paper isn’t terribly famous, but as of this writing it’s been cited 171 times, and it was covered by WIRED magazine in 2010, so let’s take a look.

This study focused on twenty middle-aged obese German men (mean age 55.7, mean BMI 33.7), all of whom normally lived at a low altitude — 571 ± 29 meters above sea level. Participants were first given a medical exam in Munich, Germany (530 meters above sea level) to establish baseline values for all measures. A week later, all twenty of the obese German men, as well as (presumably) the researchers, traveled to “the air‐conditioned Environmental Research Station Schneefernerhaus (UFS, Zugspitze, Germany)”, a former hotel in the Bavarian Alps (2,650 meters above sea level). The hotel/research station “was effortlessly reached by cogwheel train and cable car during the afternoon of day 6.”

Patients stayed in the Schneefernerhaus research station for a week, where they “ate and drank without restriction, as they would have at home.” Exercise was “restricted to slow walks throughout the station: more vigorous activity was not permitted.” They note that there was slightly less activity at the research station than there was at low altitudes, “probably due to the limited walking space in the high‐altitude research station.” Sounds cozy.

During this week-long period at high altitude, the researchers continued collecting measurements of the participants’ health. After the week was through, everyone returned to Munich (530 meters above sea level). At this point the researchers waited four weeks (it’s not clear why) before conducting the final health examinations, at which point the study concluded. We’re not sure what to say about this study design, except that it’s clear the film adaptation should be directed by Wes Anderson.

Schneefernerhaus Research Station. Yes, really.

II.

While this design is amusing, the results are uninspiring. 

To begin with, the weight loss was minimal. During the week they spent at 2,650 meters, patients lost an average of 3 pounds (1.5 kg). They were an average of 232 lbs (105.1 kg) to begin with, so this is only about 1% of their body weight. Going from 232 lbs (105.1 kg) to 229 lbs (103.6 kg) doesn’t seem clinically relevant, or even all that noticeable. The authors, surprisingly, agree: “the absolute amount of weight loss was so small.”

More importantly, we’re not convinced that this tiny weight loss result is real, because the paper suffers from serious multiple comparison problems. Also known as p-hacking or “questionable research practices”, multiple comparisons are a problem because they can make it very likely to get a false positive. If you run one statistical test, there’s a small chance you will get a false positive, but as you run more tests, false positives get more and more likely. If you run enough tests, you are virtually guaranteed to get a false positive, or many false positives. If you try running many different tests, or try running the same test many different ways, and only report the best one, it’s possible to make pure noise look like a strong finding.

We see evidence of multiple comparisons in the paper. They collect a lot of measures and run a lot of tests. The authors report eight measures of obesity alone, as well many other measures of health. 

The week the patients spent at 2,650 meters — Day 7 to Day 14 — is clearly the interval of interest here, but they mostly report comparisons of Day 1 to the other days, and they tend to report all three pairs (D1 to D7, D1 to D14, and D1 to D42), which makes for three times the number of comparisons. It’s also confusing that there are no measures for D21, D28, and D35. Did they not collect data those days, or just not report it? We think they just didn’t collect data, but it’s not clear.

The authors also use a very unusual form of statistical analysis — for each test, first they conducted a nonparametric Friedmann procedure. Then, if that showed a significant rank difference, they did a Wilcoxon signed‐rank method test. It’s pretty strange to run one test conditional on another like this, especially for such a simple comparison. It’s also not clear what role the Friedmann procedure is playing in this analysis. Presumably they are referring to the Friedman test (we assume they don’t mean this procedure for biodiesel analysis) and this is a simple typo, but it’s not clear why they want to rank the means. In addition, the Wilcoxon signed‐rank test seems like a slightly strange choice. The more standard analysis here would be the humble paired t-test. 

Even if this really were best practice, there’s no way to know that they didn’t start by running paired t-tests, throwing those results out when they found that they were only trending in the right direction. And in fact, we noticed that if we compare body weight at D7 to D14 using a paired t-test, we find a p-value of .0506, instead of the p < .001 they report when comparing D1 to D14 with a Wilcoxon test. We think that this is the more appropriate analysis, and as you can see, it’s not statistically significant.

Regardless, the whole analysis is called into question by the number of tests they ran. By our count they conducted at least 74 tests in this paper, which is a form of p-hacking and makes the results very hard to interpret. It’s also possible that they conducted even more tests that weren’t reported in the paper. This isn’t really their fault — p-hacking wasn’t described until 2011 (and the term itself wasn’t invented until a few years later), so like most people they were almost certainly unfamiliar with issues of multiple comparisons when they did their analysis. While we don’t accuse the authors of acting in bad faith, we do think this seriously undermines our ability to interpret their results. When we ran the single test that we think was most appropriate, we found that it was not significant. 

And of course, the sample size was only 20 people, though perhaps there wasn’t room for many more people in the research station. On one hand this is pretty standard for intensive studies like this, but it reduces the statistical power. 

There appear to be about 68 statistical tests in this table alone. Every little star (*) indicates a significant test against the number in D1. It’s hard to tell for sure how many tests they performed here (due to their very weird procedure) but it’s as high as 68.

III.

The authors claim to show that hypoxia causes weight loss, but this is overstating their case. They report that people brought to 2,650 meters lost a small amount of weight and had lower blood oxygen saturation [1], but we think the former result is noise and the latter result is unsurprising. Obviously if you bring people to 2,650 meters they will have lower blood oxygen, and there’s no evidence linking that to the reported weight loss. 

Even more concerning is the fact that there’s no control group, which means that this study isn’t even an experiment. Without a control group, there can be no random assignment, and with no random assignment, a study isn’t an experiment. As a result, the strong causal claim the authors draw from their results is pretty unsubstantiated. 

There isn’t an obvious fix for this problem. A control group that stayed in Munich wouldn’t be appropriate, because oxygen is confounded with everything else about altitude. If there were a difference between the Munich group and the Schneefernerhaus group, there would be no way to tell if that was due to the amount of oxygen or any of the other thousand differences between the two locations. A better approach would be to bring a control group to the same altitude, and give that control group extra oxygen, though that might introduce its own confounds — for example, the supplemental-oxygen group would all be wearing masks and carrying canisters. I guess the best way to do this would be to bring both groups to the Alps, give both of them canisters and masks, but put real oxygen in the canisters for one group and placebo oxygen (nitrogen?) in the canisters for the other groups.

We’re sympathetic to inferring causal relationships from correlational data, but the authors don’t report a correlation between blood oxygen saturation and weight loss, even though that would be the relevant test given the data that they have. Probably they don’t report it because it’s not significant. They do report, “We could not find a significant correlation between oxygen saturation or oxygen partial pressure, and either ghrelin or leptin.” These are tests that we might expect to be significant if hypoxia caused weight loss — which suggests that it does not. 

Unfortunately, the authors report no evidence for their mechanism and probably don’t have an effect to explain in the first place. This is too bad — the study asks an interesting question, and the design looks good at first. It’s only on reflection that you see that there are serious problems.


Thanks to Nick Brown for reading a draft of this post. 

[1] One thing that Nick Brown noticed when he read the first draft of this post is that the oxygen saturation percentages reported for D7 and D14 seem to be dangerously low. We’ve all become more familiar with oxygen saturation measures because of COVID, so you may already know that a normal range is 95-100%. Guidelines generally suggest that levels below 90% are dangerous, and should be cause to seek medical attention, so it’s a little surprising that the average for these 20 men was in the mid-80’s during their week at high altitude. We found this confusing so we looked into it, and it turns out that this is probably not a issue. Not only are lower oxygen saturation levels normal at higher altitudes, the levels can apparently be very low by sea-level standards without becoming dangerous. For example, in this study of residents of El Alto in Bolivia (an elevation of 4018 m), the mean oxygen saturation percentages were in the range of 85-88%. So while this is definitely striking, it’s probably not anything to worry about.

Investigation: Ultra-Processed Diets by Hall et al. (2019)

[This is Part One of a two-part analysis in collaboration with Nick Brown. Part Two is on Nick’s blog.]

I. 

Recently we came across a 2019 paper called Ultra-Processed Diets Cause Excess Calorie Intake and Weight Gain: An Inpatient Randomized Controlled Trial of Ad Libitum Food Intake, by Kevin D. Hall and colleagues. 

Briefly, Hall et al. (2019) is a metabolic ward study on the effects of “ultra-processed” foods on energy intake and weight gain. The participants were 20 adults, an average of 31.2 years old. They had a mean BMI of 27, so on average participants were slightly overweight, but not obese.

Participants were admitted to the metabolic ward and randomly assigned to one of two conditions. They either ate an ultra-processed diet for two weeks, immediately followed by an unprocessed diet for two weeks — or they ate an unprocessed diet for two weeks, immediately followed by an ultra-processed diet for two weeks. The study was ad libitum, so whether they were eating an unprocessed or an ultra-processed diet, participants were always allowed to eat as much as they wanted — in the words of the authors, “subjects were instructed to consume as much or as little as desired.”

The authors found that people ate more on the ultra-processed diet and gained a small amount of weight, compared to the unprocessed diet, where they ate less and lost a small amount of weight.

We’re not in the habit of re-analyzing published papers, but we decided to take a closer look at this study because a couple of things in the abstract struck us as surprising. Weight change is one main outcome of interest for this study, and several unusual things about this measure stand out immediately. First, the two groups report the same amount of change in body weight, the only difference being that one group gained weight and the other group lost it. In the ultra-processed diet group, people gained 0.9 ± 0.3 kg (p = 0.009), and in the unprocessed diet group, people lost 0.9 ± 0.3 kg (p = 0.007). (Those ± values are standard errors of the mean.) It’s pretty unlikely for the means of both groups to be identical, and it’s very unlikely that both the means and the standard errors would be identical.

It’s not impossible for these numbers to be the same (and in fact, they are not precisely equal in the raw data, though they are still pretty close), especially given that they’re rounded to one decimal place. But it is weird. We ran some simple simulations which suggest that this should only happen about 5% of the time — but this is assuming that the means and SDs of the two groups are both identical in the population, which itself is very unlikely.

Another test of interest reported in the abstract also seemed odd. They report that weight changes were highly correlated with energy intake (r = 0.8, p < 0.0001). This correlation coefficient struck us as surprising, because it’s pretty huge. There are very few measures that are correlated with one another at 0.8 — these are the types of correlations we tend to see between identical twins, or repeated measurements of the same person. As an example, in identical twins, BMI is correlated at about r = 0.8, and height at about r = 0.9.

We know that these points are pretty ticky-tacky stuff. By themselves, they’re not much, but they bothered us. Something already seemed weird, and we hadn’t even gotten past the abstract.

Even the authors found these results surprising, and have said so on a couple of occasions. As a result, we decided to take a closer look. Fortunately for us, the authors have followed best practices and all their data is available on the OSF.

To conduct this analysis, we teamed up with Nick Brown, with additional help from James Heathers. We focused on one particular dependent variable of this study, weight change, while Nick took a broader look at several elements of the paper.

II. 

Because we were most interested in weight change, we decided to begin by taking a close look at the file “deltabw”. In mathematics, delta usually means “change” or “the change in”, and “bw” here stands for “body weight”, so this title indicates that the file contains data for the change in participants’ body weights. On the OSF this is in the form of a SAS .sas7bdat file, but we converted it to a .csv file, which is a little easier to work with.

Here’s a screenshot of what the deltabw file looks like:

In this spreadsheet, each row tells us about the weight for one participant on one day of the 4-week-long study. These daily body weight measurements were performed at 6am each morning, so we have one row for every day. 

Let’s also orient you to the columns. “StudyID” is the ID for each participant. Here we can see that in this screenshot we are looking just at participant ADL001, or participant 01 for short. The “Period” variable tells us whether the participant was eating an ultra-processed (PROC) or an unprocessed (UNPROC) diet on that day. Here we can see that participant 01 was part of the group who had an unprocessed diet for the first two weeks, before switching to the ultra-processed diet for the last two weeks. “Day” tells us which day in the 28-day study the measurement is from. Here we show only the first 20 days for participant 01. 

“BW” is the main variable of interest, as it is the participant’s measured weight, in kilograms, for that day of the study. “DayInPeriod” tells us which day they are on for that particular diet. Each participant goes 14 days on one diet then begins day 1 on the other diet. “BaseBW” is just their weight for day 1 on that period. Participant 01 was 94.87 kg on day one of the unprocessed diet, so this column holds that value as long as they’re on that diet. “DeltaBW” is the difference between their weight on that day and the weight they were at the beginning of that period. For example, participant 01 weighed 94.87 kg on day one and 94.07 kg on day nine, so the DeltaBW value for day nine is -0.80.

Finally, “DeltaDaily” is a variable that we added, which is just a simple calculation of how much the participant’s weight changed each day. If someone weighed 82.85 kg yesterday and they weigh 82.95 kg today, the DeltaDaily would be 0.10, because they gained 0.10 kg in the last 24 hours.

To begin with, we were able to replicate the authors’ main findings. When we don’t round to one decimal place, we see that participants on the ultra-processed diet gained an average of 0.9380 (± 0.3219) kg, and participants on the unprocessed diet lost an average of 0.9085 (± 0.3006) kg. That’s only a difference of 0.0295 kg in absolute values in the means, and 0.0213 kg for the standard errors, which we still find quite surprising. Note that this is different from the concern about standard errors raised by Drs. Mackerras and Blizzard. Many of the standard errors in this paper come from GLM analysis, which assumes homogeneity of variances and often leads to identical standard errors. But these are independently calculated standard errors of the mean for each condition, so it is still somewhat surprising that they are so similar (though not identical).  

On average these participants gained and lost impressive, but not shocking amounts of weight. A few of the participants, however, saw weight loss that was very concerning. One woman lost 4.3 kg in 14 days which, to quote Nick Brown, “is what I would expect if she had dysentery” (evocative though perhaps a little excessive). In fact, according to the data, she lost 2.39 kg in the first five days alone. We also notice that this patient was only 67.12 kg (about 148 lbs) to begin with, so such a huge loss is proportionally even more concerning. This is the most extreme case, of course, but not the only case of such intense weight change over such a short period.

The article tells us that participants were weighed on a Welch Allyn Scale-Tronix 5702 scale, which has a resolution of 0.1 lb or 100 grams (0.1 kg). This means it should only display data to one decimal place. Here’s the manufacturer’s specification sheet for that model. But participant weights in the file deltabw are all reported to two decimal places; that is, with a precision of 0.01 kg, as you can clearly see from the screenshot above. Of the 560 weight readings in the data file, only 55 end in zero. It is not clear how this is possible, since the scale apparently doesn’t display this much precision. 

To confirm this, we wrote to Welch Allyn’s customer support department, who confirmed that the model 5702 has 0.1 kg resolution.

We also considered the possibility that the researchers measured people’s weight in pounds and then converted to kilograms, in order to use the scale’s better precision of 0.1 pounds (45.4 grams) rather than 100 grams. However, in this case, one would expect to see that all of the changes in weight were multiples of (approximately) 0.045 kg, which is not what we observe.

III.

As we look closer at the numbers, things get even more confusing. 

As we noted, Hall et al. report participant weight to two decimal places in kilograms for every participant on every day. Kilograms to two decimal places should be pretty sensitive, but there are many cases where the exact same weight appears two or even three times in a row. For example, participant 21 is listed as having a weight of exactly 59.32 kg on days 12, 13, and 14, participant 13 is listed as having a weight of exactly 96.43 kg on days 10, 11, and 12, and participant 06 is listed as having a weight of exactly 49.54 kg on days 23, 24, and 25. 

Having the same weight for two or even three days in a row may not seem that strange, but it is very remarkable when the measurement is in kilograms precise to two decimal places. After all, 0.01 kg (10 grams) is not very much weight at all. A standard egg weighs about 0.05 kg (50 grams). A shot of liquor is a little less, usually a bit more than 0.03 kg (30 grams). A tablespoon of water is about 0.015 kg (15 grams). This suggests that people’s weights are varying by less than the weight of a tablespoon of water over the course of entire days, and sometimes over multiple days. This uncanny precision seems even more unusual when we note that body weight measurements were taken at 6 am every morning “after the first void”, which suggests that participants’ bodily functions were precise to 0.01 kg on certain days as well. 

The case of participant 06 is particularly confusing, as 49.54 kg is exactly one kilogram less, to two decimal places, than the baseline for this participant’s weight when they started, 50.54 kg. Furthermore, in the “unprocessed” period, participant 06 only ever seems to lose or gain weight in full increments of 0.10 kilograms. 

We see similar patterns in the data from other participants. Let’s take a look at the DeltaDaily variable. As a reminder, this variable is just the difference between a person’s weight on one day and the day before. These are nothing more than daily changes in weight. 

Because these numbers are calculated from the difference between two weight measurements, both of which are reported to two decimal places of accuracy, these numbers should have two places of accuracy as well. But surprisingly, we see that many of these weight changes are in full increments of 0.10.

Take a look at the histograms below. The top histogram is the distribution of weight changes by day. For example, a person might gain 0.10 kg between days 15 and 16, and that would be one of the observations in this histogram. 

You’ll see that these data have an extremely unnatural hair-comb pattern of spikes, with only a few observations in between. This is because the vast majority (~71%) of the weight changes are in exact multiples of 0.10, despite the fact that weights and weight changes are reported to two decimal places. That is to say, participants’ weights usually changed in increments like 0.20 kg, -0.10 kg, or 0.40 kg, and almost never in increments like -0.03 kg, 0.12 kg, or 0.28 kg. 

For comparison, on the bottom is a sample from a simulated normal distribution with identical n, mean, and standard deviation. You’ll see that there is no hair-comb pattern for these data.

As we mentioned earlier, there are several cases where a participant stays at the exact same weight for two or three days in a row. The distribution we see here is the cause. As you can see, the most common daily change is exactly zero. Now, it’s certainly possible to imagine why some values might end up being zero in a study like this. There might be a technical incident with the scale, a clerical error, or a mistake when recording handwritten data on the computer. A lazy lab assistant might lose their notes, resulting in the previous day’s value being used as the reasonable best estimate. But since a change of exactly zero is the modal response, a full 9% of all measurements, it’s hard to imagine that these are all omissions or technical errors.

In addition, there’s something very strange going on with the trailing digits:

On the top here we have the distribution of digits in the 0.1 place. For example, a measurement of 0.29 kg would appear as a 2 here. This follows about the distribution we would expect, though there are a few more 1’s and fewer 0’s than usual. 

The bottom histogram is where things get weird. Here we have the distribution of digits in the 0.01 place. For example, a measurement of 0.29 kg would appear as a 9 here. As you can see, 382/540 of these observations have a 0 in their 0.01’s place — this is the same as that figure of 71% of measured changes being in full increments of 0.10 kg that we mentioned earlier. 

The rest of the distribution is also very strange. When the trailing digit is not a zero, it is almost certainly a 1 or a 9, possibly a 2 or an 8, and almost never anything else. Of 540 observed weight changes, only 3 have a trailing digit of 5.

We can see that this is not what we would expect from (simulated) normally distributed data:

It’s also not what we would expect to see if they were measuring to one decimal place most of the time (~70%), but to two decimal places on occasion (~30%). As we’ve already mentioned, this doesn’t make sense from a methodological standpoint, because all daily weights are to two decimal places. But even it somehow were a measurement accuracy issue, we would expect an equal distribution across all the other digits besides zero, like this:

This is certainly not what we see in the reported data. The fact that 1 and 9 are the most likely trailing digit after 0, and that 2 and 8 are most likely after that, is especially strange.

IV. 

When we first started looking into this paper, we approached Retraction Watch, who said they considered it a potential story. After completing the analyses above, we shared an early version of this post with Retraction Watch, and with our permission they approached the authors for comment. The authors were kind enough to offer feedback on what we had found, and when we examined their explanation, we found that it clarified a number of our points of confusion. 

The first thing they shared with us was this erratum from October 2020, which we hadn’t seen before. The erratum reports that they noticed an error in the documented diet order of one participant. This is an important note but doesn’t affect the analyses we present here, which have very little to do with diet conditions.

Kevin Hall, the first author on this paper, also shared a clarification on how body weights were calculated:

I think I just discovered the likely explanation about the distribution of high-precision digits in the body weight measurements that are the main subject of one of the blogs. It’s kind of illustrative of how difficult it is to fully report experimental methods! It turns out that the body weight measurements were recorded to the 0.1 kg according to the scale precision. However, we subtracted the weight of the subject’s pajamas that were measured using a more precise balance at a single time point. We repeated subtracting the mass of the pajamas on all occasions when the subject wore those pajamas. See the example excerpted below from the original form from one subject who wore the same pajamas (PJs) for three days and then switched to a new set. Obviously, the repeating high precision digits are due to the constant PJs! 😉

This matches what is reported in the paper, where they state, “Subjects wore hospital-issued top and bottom pajamas which were pre-weighed and deducted from scale weight.” 

Kevin also included the following image, which shows part of how the data was recorded for one participant: 

If we understand this correctly, the first time a participant wore a set of pajamas, the pajamas were weighed to three decimals of precision. Then, that measurement was subtracted from the participant’s weight on the scale (“Patient Weight”) on every consecutive morning, to calculate the participant’s body weight. For an unclear reason, this was recorded to two decimals of precision, rather than the one decimal of precision given by the scale, or the three decimals of precision given by the PJ weights. When the participant switched to a new set of pajamas, the new set was weighed to three decimals of precision, and that number was used to calculate participant body weight until they switched to yet another new set of pajamas, etc.

We assume that the measurement for the pajamas is given in kilograms, even though they write “g” and “gm” (“qm”?) in the column. I wish my undergraduate lab TAs were as forgiving as the editors at Cell Metabolism.

This method does account for the fact that participant body weights were reported to two decimal places of precision, despite the fact that the scale only measures weight to one decimal place of precision. Even so, there were a couple of things that we still found confusing.

The variable that interests us the most is the DeltaDaily variable. We can easily calculate that variable for the provided example, like so:

We can see that whenever a participant doesn’t change their pajamas on consecutive days, there’s a trailing zero. In this way, the pajamas can account for the fact that 71% of the time, the trailing digits in the DeltaDaily variable were zeros. 

We also see that whenever the trailing digit is not zero, that lets us identify when a participant has changed their pajamas. Note of course that about ten percent of the time, a change in pajamas will also lead to a trailing digit of zero. So every trailing digit that isn’t zero is a pajama change, though a small number of the zeros will also be “hidden” pajama changes.

In any case, we can use this to make inferences about how often participants change their pajamas, which we find rather confusing. Participants often change their pajamas every day for multiple days in a row, or go long stretches without apparently changing their pajamas at all, and sometimes these are the same participants. It’s possible that these long stretches without any apparent change of pajamas are the result of the “hidden” changes we mentioned, because about 10% of the time changes would happen without the trailing digit changing, but it’s still surprising.

For example, participant 05 changes their pajamas on day 2, day 5, and day 10, and then apparently doesn’t change their pajamas again until day 28, going more than two weeks without a change in PJs. Participant 20, in contrast, changes pajamas at least 16 times over 28 days, including every day for the last four days of the study. The record for this, however, has to go to participant 03, who at one point appears to have switched pajamas every day for at least seven days in a row. Participant 03 then goes eight days in a row without changing pajamas before switching pajamas every day for three days in a row. 

Participant 08 (the participant from the image above) seems to change their pajamas only twice during the entire 28-day study, once on day 4 and again on day 28. Certainly this is possible, but it doesn’t look like the pajama-wearing habits we would expect. It’s true that some people probably want to change their pajamas more than others, but this doesn’t seem like it can be entirely attributed to personality, as some people don’t change pajamas at all for a long time, and then start to change them nearly every day, or vice-versa.

We were also unclear on whether the pajamas adjustment could account for the most confusing pattern we saw in the data for this article, the distribution of digits in the .01 place for the DeltaDaily variable:

The pajamas method can explain why there are so many zeros — any day a participant didn’t change their pajamas, there would be a zero, and it’s conceivable that participants only changed their pajamas on 30% of the days they were in the study. 

We weren’t sure if the pajamas method could explain the distribution of the other digits. For the trailing digits that aren’t zero, 42% of them are 1’s, 27% of them are 9’s, 9% of them are 2’s, 8% of them are 8’s, and the remaining digits account for only about 3% each. This seems very strange.

You’ll recall that the DeltaDaily values record the changes in participant weights between consecutive days. Because the weight of the scale is only precise to 0.1 kg, the data in the 0.01 place records information about the difference between two different pairs of pajamas. For illustration, in the example Kevin Hall provided, the participant switched between a pair of pajamas weighing 0.418 kg and a pair weighing 0.376 kg. These are different by 0.042 kg, so when they rounded it to two digits, the difference we see in the DeltaDaily has a trailing digit of 4. 

We wanted to know if the pajama adjustment could explain why the difference (for the digit in the 0.01’s place) between the weights of two pairs of pajamas are 14x more likely to be a 1 than a 6, or 9x more likely to be a 9 than a 3. 

Verbal arguments quickly got very confusing, so we decided to run some simulations. We simulated 20 participants, for 28 days each, just like the actual study. On day one, simulated participants were assigned a starting weight, which was a random integer between 40 and 100. Every day, their weight changed by an amount between -1.5 and 1.5 by increments of 0.1 (-1.5, -1.4, -1.3 … 1.4, 1.5), with each increment having an equal chance of occuring. 

The important part of the simulation were the pajamas, of course. Participants were assigned a pajama weight on day 1, and each day they had a 35% chance of changing pajamas, and being assigned a new pajama weight. The real question was how to generate a reasonable distribution of pajama weights. We didn’t have much to go off of, just the two values in the image that Kevin Hall shared with us. But we decided to give it a shot with just that information. Weights of 418 g and 376 g have a mean of just under 400 g and a standard deviation of 30 g, so we decided to sample our pajama weights from a normal distribution with those parameters.

When we ran this simulation, we found a distribution of digits in the 0.01 place that didn’t show the same saddle-shaped distribution as in the data from the paper:

We decided to run some additional simulations, just to be sure. To our surprise, when the SD of the pajamas is smaller, in the range of 10-20 g, you can sometimes get saddle-shaped distributions just like the ones we saw in data from the paper. Here’s an example of what the digits can look like when the SD of the pajamas is 15 g:

It’s hard for us to say whether a standard deviation of 15 g or of 30 g is more realistic for hospital pajamas, but it’s clear that under certain circumstances, pajama adjustments can create this kind of distribution (we propose calling it the “pajama distribution”).

While we find this distribution surprising, we conclude that it is possible given what we know about these data and how the weights were calculated.

V. 

When we took a close look at these data, we originally found a number of patterns that we were unable to explain. Having communicated with the authors, we now think that while there are some strange choices in their analysis, most of these patterns can be explained when we take into account the fact that pajama weights were deducted from scale weights, and the two weights had different levels of precision.

While these patterns can be explained by the pajama adjustment described by Kevin Hall, there are some important lessons here. The first, as Kevin notes in his comment, is that it can be very difficult to fully record one’s methods. It would have been better to include the full history of this variable in the data files, including the pajama weights, instead of recording the weights and performing the relevant comparisons by hand. 

The second is a lesson about combining data of different levels of precision. The hair-comb pattern that we observed in the distribution of DeltaDaily scores was truly bizarre, and was reason for serious concern. It turns out that this kind of distribution can occur when a measure with one decimal of precision is combined with another measure with three decimals of precision, with the result being rounded to two decimals of precision. In the future researchers should try to avoid combining data in this way to avoid creating such artifacts. While it may not affect their conclusions, it is strange for the authors to claim that someone’s weight changed by (for example) 1.27 kg, when they have no way to measure the change to that level of precision.

There are some more minor points that this explanation does not address, however. We still find it surprising how consistent the weight change was in this study, and how extreme some of the weight changes were. We also remain somewhat confused by how often participants changed (or didn’t change) their pajamas. 

This post continues in Part Two over at Nick Brown’s blog, where he covers several other aspects of the study design and data.

Thanks again to Nick Brown for comparing notes with us on this analysis, to James Heathers for helpful comments, and to a couple of early readers who asked to remain anonymous. Special thanks to Kevin Hall and the other authors of the original paper, who have been extremely forthcoming and polite in their correspondence. We look forward to ongoing public discussion of these analyses, as we believe the open exchange of ideas can benefit the scientific community.