Calories in, calories out

This article was discussed on Hacker News.


How do we lose (or gain) weight?  Is it really as simple as “calories in, calories out” (i.e., eat less than you burn), or is what you eat more important than how much?  Is “3500 calories equals one pound” a useful rule of thumb, or just a myth?  I don’t think I would normally find these to be terribly interesting questions, except for the fact that there seems to be a lot of conflicting, confusing, and at times downright misleading information out there.  That can be frustrating, but I suppose it’s not surprising, since there is money to be made in weight loss programs– whether they are effective or not– particularly here in the United States.

Following is a description of my attempt to answer some of these questions, using a relatively simple mathematical model, in an experiment involving daily measurement of weight, caloric intake, and exercise over 75 days.  The results suggest that you can not only measure, but predict future weight loss– or gain– with surprising accuracy.  But they also raise some interesting open questions about how all this relates to the effectiveness of some currently popular diet programs.

(Edit 2017-08-11: When I originally posted this article 3 years ago, I quoted and linked to some sources whose calculations and/or arguments I disagreed with.  One of those sources was professor of exercise science Gregory Hand, whose linked blog post has since been locked as private.  Another was nutritionist Zoë Harcombe, whose linked blog post is still accessible, but has since been modified to no longer contain the argument as quoted here… but with a new “footnote” argument that indicates the same misunderstanding of the model described here.  I’m leaving all of the original text and quotes here, but see below for an additional edit to comment on Harcombe’s footnote.)

The model and the experiment

Here was my basic idea: given just my measured starting weight w_0, and a sequence (c_n) of measurements of subsequent daily caloric intake, how accurately could I estimate my resulting final weight, weeks or even months later?

More precisely, consider the sequence (\hat{w}_n) of predicted daily weights given by the following recurrence relation:

\hat{w}_0 = w_0

\hat{w}_{n+1} = \hat{w}_n + \frac{c_n - \alpha \hat{w}_n - 0.63 \hat{w}_n d_n}{3500}

Intuitively, my weight tomorrow morning \hat{w}_{n+1} should be my weight this morning \hat{w}_n, plus the effect of my net intake of calories that day, assuming 3500 calories per pound.  Net calorie intake is modeled with three components:

  • c_n is the number of calories consumed.
  • -\alpha\hat{w}_n is the number of calories burned due to normal daily activity.  Note that this is a function of current weight, with typical values for \alpha of 12 to 13 calories per pound for men, or 10 to 11 for women; I used 12.5 (more on this later).
  • -0.63 \hat{w}_n d_n is the number of additional calories burned while running (my favorite form of exercise), where d_n is the number of miles run that day.  Note that we don’t really have to account for exercise separately like this; especially if duration and intensity don’t change much over time, we could skip this term altogether and just roll up all daily activity into the (larger) value for \alpha.

(Aside: I am intentionally sticking with U. S. customary units of pounds, miles, etc., to be consistent with much of the related literature.)

So, at the start of my experiment, my initial weight was w_0=251.8 pounds (for reference, I am a little over 6’4″ tall, 40-ish years old).  Over each of the next 75 days, I recorded:

  • My actual weight w_n, first thing in the morning after rolling out of bed, using a digital scale with a display resolution of 0.1 pound.
  • My total calories consumed for the day c_n.
  • My running mileage for the day d_n.

Plugging in c_n and d_n to the recurrence relation above, I computed the sequence of predicted weights (\hat{w}_n), and compared with the sequence of my actual weights (w_n).


The following figure shows the resulting comparison of predicted weight \hat{w}_n (in blue) with measured actual weight w_n (in red).  See the appendix at the end of this post for all of the raw data.

Predicted and actual weight over 75 days.
Predicted and actual weight over 75 days.

I was surprised at just how well this worked.  Two and a half months and nearly 30 pounds later, the final predicted weight differed from the actual weight by less than a pound!

There are a couple of useful observations at this point.  First, the “3500 calories per pound” rule of thumb is perfectly valid… as long as it is applied correctly.  Zoë Harcombe, a “qualified nutritionist,” does a great job of demonstrating how to apply it incorrectly:

“Every person who didn’t have that [55-calorie] biscuit every day should have lost 141 pounds over the past 25 years.”

This seems to be a common argument– professor of exercise science Gregory Hand makes a similar but slightly more vivid claim using the same reasoning about a hypothetical dieter, that “if she will lose 1 lb for every 3,500 calorie deficit [my emphasis], our individual will completely disappear from the face of the earth in 300 days.”

The problem in both cases is the incorrect assumption that an initial calorie deficit, due to skipping a biscuit, for example, persists as the same deficit over time, causing a linear reduction in weight.  But that’s not how it works: as weight decreases, calorie expenditure also decreases, so that an initial reduced diet, continued over time, causes an asymptotic reduction in weight.  (In the context of the recurrence relation above, Harcombe and Hand’s calculation effectively replaces the varying \alpha \hat{w}_n in the numerator with the constant \alpha w_0.)

(Edit 2017-08-11: Harcombe’s blog post no longer contains the above quote, but has since been edited to include the following footnote:

“If we think one pound equals 3,500 calories and in fact one pound equals 2,843 calories, over a year, 657 ‘extra’ calories a day, simply from the formula ‘being wrong’, would add up to 239,805 extra calories and this, divided by 2,843 gives 84 pounds, or six stone. Adjust the calculations for women more typically maintaining at 2,000 calories a day and men more typically at 2,600 calories a day and the inaccuracy of the formula still creates wide disparity.”

This argument still exhibits the same problem.  It is unclear how we should interpret the calculation of 84 pounds; the word “extra” seems to suggest the idea that a person whose diet was structured assuming an incorrect 3500 calories per pound (when the “real” value is 2843) would… what?  Gain 84 pounds over the course of a year?  If anything, I would expect to lose more weight if the “cost” of losing each pound of fat was burning fewer calories.

But even ignoring the sign issue, the real problem as mentioned above is not the accuracy of choice of denominator in the division, but the fixed numerator.  To see why, consider the example of an overweight man who weighs 250 pounds, burns 12.5 calories per pound per day, and plans to eat 2600 calories per day for the next year.  Using the recurrence relation described here, assuming “3500 calories per pound,” he expects to lose a little over 30 pounds in that year.  If the correct value were really 2843 calories per pound, then his predicted final weight changes by… less than 3 pounds, over the course of an entire year.)

Estimating \alpha

The second– and, I think, most important– observation is that I arguably “got lucky” with my initial choice of \alpha=12.5 calories burned per pound of body weight.  If I had instead chosen 12, or 13, the resulting predictions would not agree nearly as well.  And your value of \alpha is likely not 12.5, but something different.  This seems to be one of the stronger common arguments against calorie-counting: even if you go to the trouble of religiously measuring calories in, you can never know calories out exactly, so why bother?

The Harris-Benedict equation is often used in an attempt to remedy this, by incorporating not only weight, but also height, age, gender, and activity level into a more complex calculation to estimate total daily calorie expenditure.  But I think the problem with this approach is that the more complex formula is merely a regression fit of a population of varying individuals, none of whom are you.  That is, even two different people of exactly the same weight, height, age, gender, and activity level do not necessarily burn calories at the same rate.

But even if you don’t know your personal value of \alpha ahead of time, you can estimate it, by measuring calories in (c_n) and actual weight (w_n) for a few weeks, and then finding the corresponding \alpha that yields a sequence of predicted weights (\hat{w}_n) that best fits the actual weights over that same time period, in a least-squares sense.

The following figure shows how this works: as time progresses along the x-axis, and we collect more and more data points, the y-axis indicates the corresponding best estimate of \alpha so far.

Estimating burn rate (alpha) over time. Early estimates are overwhelmed by the noisy weight measurements.
Estimating burn rate (alpha) over time. Early estimates are overwhelmed by the noisy weight measurements.

Here we can see the effect of the noisiness of the measured actual weights; it can take several weeks just to get a reasonably settled estimate.  But keep in mind that we don’t necessarily need to be trying to lose weight during this time.  This estimation approach should still work just as well whether we are losing, maintaining, or even gaining weight.  But once we have a reasonably accurate “personal” value for \alpha, then we can predict future weight changes assuming any particular planned diet and exercise schedule.

(One final note: recall the constant 0.63 multiplier in the calculation of calories burned per mile run.  I had hoped that I could estimate this value as well using the same approach… but the measured weights turned out to be simply too noisy.  That is, the variability in the weights outweighs the relatively small contribution of running to the weight loss on any given day.)

Edit: In response to several requests for a more detailed description of a procedure for estimating \alpha, I put together a simple Excel spreadsheet demonstrating how it works.  It is already populated with the time series of my recorded weight, calories, and miles from this experiment (see the Appendix below) as an example data set.

Given a particular calories/pound value for \alpha, you can see the resulting sequence of predicted weights, as well as the sum of squared differences (SSE) between these predictions and the corresponding actual measured weights.

Or you can estimate \alpha by minimizing SSE.  This can either be done “manually” by simply experimenting with different values of \alpha (12.0 is a good starting point) and observing the resulting SSE, trying to make it as small as possible; or automatically using the Excel Solver Add-In.  The following figure shows the Solver dialog in Excel 2010 with the appropriate settings.

Excel Solver dialog showing the desired settings to estimate alpha minimizing SSE.
Excel Solver dialog showing the desired settings to estimate alpha minimizing SSE.

Conclusions and open questions

I learned several interesting things from this experiment.  I learned that it is really hard to accurately measure calories consumed, even if you are trying.  (Look at the box and think about this the next time you pour a bowl of cereal, for example.)  I learned that a chicken thigh loses over 40% of its weight from grilling.  And I learned that, somewhat sadly, mathematical curiosity can be an even greater motivation than self-interest in personal health.

A couple of questions occur to me.  First, how robust is this sort of prediction to abrupt changes in diet and/or exercise?  That is, if you suddenly start eating 2500 calories a day when you usually eat 2000, what happens?  What about larger, more radical changes?  I am continuing to collect data in an attempt to answer this, so far with positive results.

Also, how much does the burn rate \alpha vary over the population… and even more interesting, how much control does an individual have over changing his or her own value of \alpha?  For example, I intentionally paid zero attention to the composition of fat, carbohydrates, and protein in the calories that I consumed during this experiment.  I ate cereal, eggs, sausage, toast, tuna, steak (tenderloins and ribeyes), cheeseburgers, peanut butter, bananas, pizza, ice cream, chicken, turkey, crab cakes, etc.  There is even one Chipotle burrito in there.

But what if I ate a strict low-carbohydrate, high-fat “keto” diet, for example?  Would this have the effect of increasing \alpha, so that even for the same amount of calories consumed, I would lose more weight than if my diet were more balanced?  Or is it simply hard to choke down that much meat and butter, so that I would tend to decrease c_n, without any effect on \alpha, but with the same end result?  These are interesting questions, and it would be useful to see experiments similar to this one to answer them.  (Edit: See this later post for a follow-up with an additional two months of recorded data.)

Appendix: Data collection

The following table shows my measured actual weight in pounds over the course of the experiment:

Mon     Tue     Wed     Thu     Fri     Sat     Sun
251.8   251.6   250.6   249.8   248.4   249.8   249.0
250.4   249.0   247.8   246.6   246.6   247.8   246.2
246.6   244.0   244.6   243.6   243.6   244.0   244.8
242.0   240.6   240.4   240.2   240.2   239.4   238.6
238.0   238.0   237.6   238.0   238.0   238.6   238.6
237.4   239.0   237.6   235.8   236.0   235.0   236.0
233.8   232.4   232.6   233.4   233.4   232.0   233.2
232.6   231.6   232.2   232.2   231.2   231.2   229.6
229.6   229.6   230.6   230.4   229.8   228.0   227.4
227.6   226.2   226.4   225.6   225.8   225.8   226.0
228.0   225.8   225.4   224.6   223.8

The following table shows my daily calorie intake:

Mon     Tue     Wed     Thu     Fri     Sat     Sun
1630    1730    1670    1640    2110    2240    1980
1630    1560    1690    1700    2010    1990    2030
1620    1710    1590    1710    2180    2620    2100
1580    1610    1610    1620    1690    2080    1930
1620    1680    1610    1610    1810    2550    2430
1710    1660    1630    1710    1930    2470    1970
1660    1750    1710    1740    2020    2680    2100
1740    1750    1750    1610    1990    2290    1940
1950    1700    1730    1640    1820    2230    2280
1740    1760    1780    1650    1900    2470    1910
1570    1740    1740    1750

And finally, the following table shows the number of miles run on each day:

Mon     Tue     Wed     Thu     Fri     Sat     Sun
2.5     0.0     2.5     0.0     0.0     2.5     0.0
2.5     0.0     2.5     0.0     0.0     2.5     0.0
2.5     0.0     2.5     0.0     0.0     3.0     0.0
2.5     0.0     2.5     0.0     0.0     3.0     0.0
2.5     0.0     3.0     0.0     0.0     3.0     0.0
2.5     0.0     3.0     0.0     0.0     3.0     0.0
3.0     0.0     3.0     0.0     0.0     3.0     0.0
3.0     0.0     3.0     0.0     0.0     3.0     0.0
3.0     0.0     3.0     0.0     0.0     3.5     0.0
3.0     0.0     3.0     0.0     0.0     3.5     0.0
3.0     0.0     3.5     0.0
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64 Responses to Calories in, calories out

  1. This is a crazy coincidence. You mentioned you wanted to know if keto affected α. I’m already sitting on this exact data and haven’t done anything with it yet. I can send it to you if you like. I’m interested to see how it compares.

    I started keto at the beginning of 2014, a New Years resolution of sorts. The goal was to keep under 20g of net carbs per day. To make it a proper experiment I recorded daily weight, blood pressure, and a fat/protein/netcarb triple of everything I ate, as accurately as I could manage with common kitchen tools. That involved rewashing the measuring cups/spoons regularly. I maintain a very steady exercise schedule, so while I didn’t track exercise directly in the log, it would be easy to compute from a calendar. I kept up the log for just over 5 months, at which point I got sick of writing it down daily. I lost 37 pounds over that period. To date it’s now over 50 pounds.

    Sorry I didn’t notice any of your weight loss! I think I see you regularly enough around the halls that I gradually got used to it.

    • This sounds great, I would be very interested to see your data. I think it would be particularly useful to find some single person willing to do *both*– that is, someone to self-monitor for, say, four months, two with a 25/15/60 fat/protein/carb diet, and two with a low-carb diet, holding exercise schedule constant.

      • dug says:

        or maybe 30 people in each cell, so it could be all sciency and stuff

      • Mandi says:

        I track calories through myfitnesspal and they give you your daily c/f/p, which you can export as a .csv file. Looking at the relationship between at alpha and macro ratios might lead to some interesting conclusions? I’m female and my alpha is quite high (average = 15.5 over the past 3 weeks), I’m assuming because I heat a high protein/low carb diet.

  2. Jen says:

    Could you please explain how to calculate an estimated alpha for those of us who are neither particularly strong in math nor fresh out of school? I have a fair grasp of excel, but no Matlab or anything like that… I have no problem tracking calories and weight, but am not sure what to do with the predicted weight term to figure out alpha. Thank you 😡

    • @Jen, you’re right, I glossed over how to do that estimation (which I actually did in Mathematica)… partly because it isn’t straightforward. The problem is that, as a function of alpha, the predicted weight on the n-th day ends up being a polynomial of approximately degree n. So to minimize the sum of squared differences from the measured weights, it’s simpler just to do a bounded search– basically search for values of alpha until you find one that yields the smallest sum of squared errors.

      Let me poke around in Excel and try to put together a spreadsheet that will demonstrate how it works…

    • @Jen and @Chuck, I have updated the post to include a link to a spreadsheet and a description of how to use it to estimate alpha.

      • Jen says:

        You’re the coolest!

        But could you explain a bit, maybe I’m missing something–I don’t see any formula in the “alpha” cell in the spreadsheet, it looks like “12.6735…” was just typed in. So, I’m still not sure how you got that number :} Thank you again!

    • @Jen– if you want to use the Solver to compute the value for alpha, then follow these steps:

      1. Install the Solver Add-In. To do this (I’m in Excel 2010, but a web search should yield instructions for older/newer versions of Excel), select File/Options/Add-Ins/Go, then check the Solver box and click Ok.

      2. In the spreadsheet, from the Data menu, in the Analysis group, click Solver; this will bring up the dialog box shown in the post edit above. Edit the settings as shown (the objective function is $F$2 (the sum of squared errors), which we want to *minimize*, by changing variable $E$2 (alpha). Click Solve, and it should compute an optimal value for alpha.

      Let me know if any of this doesn’t work.

    • Wes_W says:

      The much less mathematically rigorous version of calculating alpha is:
      1) Pick a calorie intake that you can stick to. An estimated maintenance level from an online calculator is fine.
      2) For some period of time (at least two weeks for a good estimate), stick to that intake goal. The more precisely you can track your intake here, the better.
      3) See how much your weight changed.

      If your weight was unchanged, then congratulations: that’s your TDEE. If you gained or lost, your TDEE is lower or higher respectively, by 3500 calories per pound gained or lost, divided by the number of days your experiment ran. For example, losing 2 pounds in 4 weeks would be -2*3500/28 = -250; you on average at 250 calories less than you expended.

      Once you know your TDEE, you can divide by your weight to get an alpha value.

      Your total weight shouldn’t change very much over the course of this experiment, so the distinction between measuring TDEE and TDEE-per-pound should be negligible.

      Either way, you may need to re-measure your alpha after significant changes in body composition; it’s not fixed for life.

  3. Chuck says:

    Or better yet, share your spreadsheet! Please.

  4. cutit says:

    > This seems to be one of the stronger common arguments against calorie-counting: even if you go to the trouble of religiously measuring calories in, you can never know calories out exactly, so why bother?

    You can not know a priori, but you can find out in not too much time experimentally. Just find out how many calories will have you keep your weight. When I started weight training, I believed my TDEE was 2200 and tried to bulk at 2500 to build muscle. Instead, I lost weight and figured out that my TDEE is 2700-2800 and to build muscle I need 3000+. Later to cut excess fat I did a -1000 (which in hindsight was too extreme, I lost a lot of strength) at 1800 calories and lost pretty much exactly the predicted 1kg/week.

  5. As a physician and a nutritionist, i love this 🙂

  6. Dave says:

    The data on the burn rate over time is interesting. What’s also interesting is that, as exercising over time increases muscle mass (both density and likely overall body weight), the burn rate actually tends to decline over time. This would make sense from a biological/survival standpoint, I suppose.

    • Tim says:

      Running doesn’t increase muscle mass. It probably decreases it

      • Dave says:

        Exercise of any kind increases muscle mass, at least for awhile until it plateaus. And, notice how the burn curve trends down at the end. It’s true that obese people will shed weight quickly as the burn rate increases early on in exercise, but over time, the burn rate slightly declines, while weight stabilizes = decline in burn rate over time.
        If you look at long-term studies of weight loss, there is always a point where weight loss is resisted and slows down, no matter what you do. This must be some sort of survival mechanism.

  7. James says:

    Very cool, and thanks for sharing the data!

    I was curious about trying to estimate the calories burned per lb per mile (I called it beta), and you’re right about how noisy the weights are. To figure out what the relationship between alpha and beta might be, I did some Bayesian MCMC so I could pull the MCMC trace plot of the parameters. The recurrence relationship was the mean for a 1 lb std. deviation normal, and alpha and beta were defined with uniform priors .

    You can see that plot here:

    It makes sense at how linear the relationship is, since you ran so consistently. The MCMC picked alpha as 13 and beta as .32. I suppose those values aren’t as motivational for getting out to run, though, so we can ignore them in favor of yours.

    Thanks again for sharing the data, it was fun to play with.

    • Thanks! This is a great visualization of the problem– that “curve” in the (alpha,beta) plane is also essentially a contour of those parameter values that yield a locally minimal sum of squared errors in predicted weight… which unfortunately is minimized when beta=0 (and alpha=13.426). But that just means that, if running distances didn’t increase significantly in the future (they are, though), then it’s just as accurate to simply “roll up” all daily activity into the single parameter alpha.

  8. Evo Terra says:

    Fantastic writeup! I’m excited to learn of others who are taking a scientific approach and challenging the BS that surrounds “diet” these days. If my data help, I’d happily share them with you. I kept careful (as possible, as you mentioned) details of my dietary intake, made all the more fun as it was based on beer and sausage. And as a bonus, I have info on my blood serum levels, as I tested before, during, and after my 31-day regiment.

    This year (the fourth year of my Beer Diet), I’m making it more mainstream, showing that beer is a perfectly acceptable part of a sensible diet. One beer with breakfast, one with lunch, and one with dinner. Everything tracked on a spreadsheet, with photographic evidence to boot. It’s working quite well… again!

  9. Jason L says:

    I love seeing this type of analysis! You question the need to even bother counting calories, but there is a psychological reason: the very act of counting calories makes you much more conscious of what you are eating, and of serving sizes. The act of measuring itself helps restrict consumption.

    When I’m on a weight loss kick, I use a Withings scale with I force myself to get on the scale every single day, for much the same reason I measure calories — the knowledge that I’ll be weighing myself the next day makes me more aware of what I am eating. Using trendweight, I can focus only on the weighted trend value (effectively making day-to-day variations less of a problem). Do a google search on the hacker’s diet for more — but I’m sure your familiar.

  10. Paul Sutter says:

    Great work, but I’d propose that percent bodyfat is a better “north star” metric for health than weight itself. I’d like to have more muscle and less fat, and I think that’s true of most men. Focus on weight loss is one of the strangest widespread beliefs. Easy to test, ask a panel of women what male bodies are most attractive, and measure the results against bodyfat and weight.

    My own experience is that calories in/out controls weight, yes.

    But I’ve only been able to lose fat and gain muscle by eating a diet that keeps my blood sugar stable. I think this is the real benefit of avoiding high-glycemic carbs, not some magical exemption from thermodynamics.

    I’d be interested to help test this.

  11. Pingback: Re: diet & exercise, “mathematical curiosity c… | shawncampbell

  12. Bene says:

    John Walker made the same discovery and wrote a book with additional online and excel tools to track your weight. You can set targets and can see your progress toward the aspired weight loss.

  13. I started reading then it went over my head. Good job. This doesn’t happen often lol.

    Is there a TL;DR version?

  14. Brent Noorda says:

    After keeping similar data for most of a decade, I think your formula is overly simplistic and just happens to work well for two reasons: (1) You’re BMI is in the high range (but getting better, congrats) where simpler rules apply, and (2) you have only tested it in one direction (a fairly-constant downward slope).

    For (1, your BMI is still high) I expect you’ll start to see plateau periods within the next 20 lbs, where your body’s clever mechanisms will start to fight you and the formula will have to radically change (mostly your α value).

    But, before you lose any more weight, if you really want to test the formula you will need to try different directions. As you said “…keep in mind that we don’t necessarily need to be trying to lose weight during this time. This estimation approach should still work just as well whether we are losing, maintaining, or even gaining weight”. It would be VERY informative if you were to stop at 220 lbs and see if by using your formula you could simply maintain that 220 lbs for a month or so, and verify whether the same α applies to maintaining weight as to losing it. And then, purely in the interest of science, try using your formula for a weight gain (at the same rate you lost weight) and again see if the same α applies. My belief, based on my own tests over the years, is that α is not a simple value but instead changes a lot when you’re gaining versus losing (i.e. your body loves to gain weight, and hates to lose it, so will adjust α to get what it wants).

    I’m very interested in whatever results you continue to get, whether they match my own or not.

    More of my own weight-loss thoughts are here:

    • @Brent, these are all great points. You’re right that the formula in the OP makes a lot of simplifying assumptions– lean body mass would be a better predictor of calories burned than total weight, etc. What I found interesting was how well it worked *despite* those simplifications.

      I am not interested in *increasing* my weight again :), but as I approach a target of around 215 lbs I plan to monitor looking for exactly the behavior you describe: can I increase my calorie intake to *maintain* a constant weight using that same estimated alpha, or does that “overshoot” resulting in weight gain? I have already started increasing (about 400 calories/day on average) and predictions still look good, but we’ll see what happens when I actually try to level off.

  15. quickthought says:

    Just one kick thought, human body is not a car, there are a lot of chemical reactions affecting each other. Just begin eating less calories can make you loose weight, but it can affect your health. If you only eat 1000 kcal. of sugar you may loose weight, but you will be probably dead in a a few years. Also it’s not the same loosing weight from muscles that loosing weight from fat.

  16. Juan de los Palotes says:

    One of the factors that most impact day by day weight is the amount of water in/water out, as it weights a kilo per liter and barely has any calories. Have you considered to include it in your calculations? I realize that measuring water loss is complex, specially during/after sports, but a quite accurate method would be to weight yourself before and after the activity.

  17. Eric says:

    There is a part of the calories in calories out equation that no one seems to pay any attention to. Sorry to bring it up, but I can’t find any more pleasant way to say it… faecal matter is always assumed to have zero calories, is that really true? I find it doubtful, given that undigested food can occasionally be found in it. Food choices must have an effect here, I would have thought that extracting calories from soda is much easier to do for your digestive system than it is from a steak for example.

  18. x4mmmx4m says:

    kilocalories != calories ?

  19. crisfole says:

    I’m willing to net there’s another confounding factor on the first term, call it gamma if you will, an individual’s fat storage mechanism. For some people it’s weak, for other people (like women with PCOS) it might be very strong. I’m not 100% site that alpha can fully take whatever that mechanism is into account. I’m also willing to bet that as you lose weight that term will start to dominate the equation leading to the plateau that James mentioned. Keep us posted! 🙂

    Very cool article and research project.

  20. Mike says:

    Great post. I thought you might be interested in this Reddit submission, this fellow took a similar approach to yours and had similar results, and in the comments section describes his experiment in similar detail. That’s a lot of “similars”…more coffee.

  21. Derek C. says:

    What about beverages? If I have 1 pint of beer, that’s 500ml = 500g = 1lb. Surely there are not 3500 calories in a beer!

    I like the 3500cal/lb rule, and might implement it, but I’m concerned about errant milk and/or beer consumption, both of which could account for ~20% of my caloric intake.

    • Mike says:

      Hey Derek, it’s not that every food item of 1 lb. is 3500 calories, it’s that ~3500 calories is equal to a pound of body weight gained or lost via surplus or deficit, respectively. So if you cut your weekly caloric intake by 3500 calories (500 calories a day of something you didn’t eat below your baseline Total Daily Energy Expenditure or TDEE) you would lose 1 lb. of body weight that week. If you overate by 500 calories per day, you’d gain 1 lb. of body weight per week. Check out for a calculator based on some of the equations the author mentions above.

  22. Excellent post. By the way, are you on Twitter? I tweeted a link to this post and did not know your handle.

  23. Ryan says:

    So based on what you said you were eating it seems like your macros: protein, carbohydrates, and fat were somewhat balanced. You were not doing a keto diet or anything similar. As you mentioned, I wonder how your alpha would change if you did a ketogenic diet.

    I also see a lot of studies that factor in glycemic index or glycemic load of the foods consumed. If you spike your blood sugar, then your body will convert the sugar to stored fat. Your experiment sets out to show that none of this is a really factor, which is definitely intriguing. The laws of thermodynamics are only what really applies: calories in versus calories out.

    I would like to see if your trend continues, or you plateau. Like you, I am only a sample size of one. My experience is that if I have a calorie deficit that is too high, my body gets fatigued rather than burning the stored fat. I’d like to see the results if you did a keto diet. I have trouble accepting that all calories are created equal. If you eat 2000 calories of table sugar, I don’t think it would be equivalent to you eating 2000 calories of chicken breast. Also, what about the indigestible calories such as fiber.

    Love this post, and I hope to see more.

    • @Ryan, your “all-sugar” example is a good one. As another commenter also put it, I am not prepared to predict what would happen if consuming 2000 calories worth of, say, gasoline.

      These are both what I would call “abrupt” or “radical” changes in behavior :). I think *anything* extreme– whether consuming much *more* than you usually do, consuming much *less*, or consuming different *composition* (fat/protein/carb) than you usually do, could all result in potential changes to alpha.

      But even less *abrupt* transitions, such as to my soon-to-be-leveling-off weight maintenance, might be difficult to “predict through.” For example, can I eventually start consuming over 2800 calories *every day* (which seems like a lot) as my estimated alpha and current running schedule would suggest, and not gain, but just maintain weight? We’ll see…

  24. gtownescapee says:

    Wow, this post has generated a lot of discussion. Congrats! I’m only disappointed that in all that time, you only consumed ONE Chipotle burrito! I also assume, based on geographic location, there was no Bojangles in your diet which makes all this analysis inapplicable to me 😉 .

    • Great to hear from you! Yeah, just one burrito– not so much because of the *number* of calories, but because of the *variance*. I don’t know if the Chipotle near us is unique in this respect, but it seems like you can end up with anything from a mini-burrito to a brick depending on who happens to be serving you. So in the interest of accuracy, I have tried to stay away from anything I can’t reasonably accurately measure.

  25. Kim says:

    Thanks for the post! I wonder if there’s any variance in the intensity of your exercise (and if there is none, how you do it!). I’ve found that monitoring my heart rate during exercise is the only way I can make sure I’m burning a certain amount of calories.

    • @Kim, you seem to be the first to bring this up– namely, that I did not provide any data on the *speed* at which I ran. In fact, I am gradually increasing my pace (along with my distance). Fortunately, one nice thing about running is that calorie expenditure depends more on *distance* than intensity. That is, “calories per mile” remains relatively constant over a reasonable range of running speeds (this is *not* true for walking, however). See here for a relevant recent study, for example.

      Having said that, as mentioned in the post, my experiment does *not* back this up– because of the variability in my weight measurements, this data does not provide a clear relationship between running distance and calorie expenditure.

  26. I love energy metabolism says:

    Hey, everyone. Research biochemist here. Forgive my english. This is all very interesting, and believe me, I love data, but there are a few things that are already well understood about metabolic systems. It’s unknown to me why the foundations of biochemistry, when applied to humans through clinical science, is forgotten or overlooked, or simply misunderstood. It must be an aversion to thinking of humans as animals. Continuing…

    Laws of thermodynamics keep us all in line when counting energy. Given. Every metabolic process in every cell satisfies them. Granted. The cellular level, and specifically the cell membrane is really the only place where it’s appropriate to apply this accountancy, and not your mouth. That is not to say that between your mouth and your cells, between whole food and molecules this process breaks thermodynamic laws, only to satisfy it at the last second. No. It is to say that the process of getting food to molecular size, absorbing it, and ultimately utilising or storing it is an enormous process, each with its own energy balance sheet that cannot be abstracted away with a “calories in – calories out” handwaving. Your body is not a bomb calorimeter, oxidizing all carbon and hydrogen to water and CO2, measuring liberated heat. Indeed, the very process of metabolism releases energy gradually, at mild levels, so that we may continue to live and not burn to a crisp. Energy metabolism is more or less enzymatically and hormonally controlled inside the cell, and basically hormonally controlled outside the cell. There are “sensors” of sorts at many points along the cellular pathways. Secondary messengers and intermediate metabolic products give the cell a notion of how much energy it has on hand, and influence processes inside the cell. Macro control is exerted by numerous hormones, insulin being a well known one.

    The metabolic fate of specific molecules in your food cannot be overlooked when counting calories. That’s the basis for “all calories are not created equal,” because they are not. Protein calories measured in a calorimeter look no different to carbohydrate calories or fat calories in the same instrument, but the metabolic fate they have in metabolism is worlds apart. If it isn’t used in an energy transaction, the calories are effectively lost to your accounting. How many calories does your lean body mass actually hold in molecules? How much of that protein that you ate went to the structural components of muscle? These are some of the non-trivial complexities of trying to account for the laws of thermodynamics at the mouth.

    I see some mentions of the keto diet in these posts. Those focused on a keto diet have achieved a practical understanding of what controls fat loss and fat gain in a metabolic way, and they are manipulating their metabolism to achieve a desired goal, namely: body fat loss. There is one thing that is blatantly obvious to any lover of biochemistry or energy metabolism (you need not be a scientist!) once you gain a working knowledge of fatty acid synthesis (FAS), beta-oxidation of fatty acids, transport of fatty acids across the cell membrane, and the hormonal control of each. That is literally all you need to study, but you must know it molecularly. Enzymatically. Enjoying the cyclic nature of FAS and beta-ox. You, like every other student of energy metabolism, will reach the same conclusion: Insulin is the switch that controls the movement of fats.

    It is not your energy accounting at the end of the day. Your body does not keep a cash till while you are awake, subsequently adding to or removing from your fat stores at the close of business. It is not your [(calories in) – (basal calories used) + (activity calories)] = deltaE equation. That does not explain the movement of fatty acids.

    It is this:

    The length of time your cells are exposed to a high insulin level determines for how long and how much of the in/out cycle of fatty acids at the cell membrane gets shunted towards IN, and being stored, retained as body fat. Additionally, fatty acid synthesis is increased and beta-oxidation is decreased.

    In contrast, the length of time your cells are exposed to a low insulin level determines for how long and how much of the in/out cycle of fatty acids at the cell membrane gets shunted towards OUT, being shuttled toward the bloodstream, the liver, and utilized through direct energy, or converted to glucose, topping off blood sugar, or refilling glycogen storage. Additionally, fatty acid synthesis is decreased and beta-oxidation is increased.

    That is it. These are the first principles. All things must begin with this, otherwise you are lost.

    By way of example, you could feed a pet mouse a large calorie deficit, but in tiny doses, and every 20 minutes, thus keeping the mouse’s insulin levels constantly high over the course of the day. Traffic of molecules will submit to the mighty insulin hormone, and this mouse’s metabolism will forsake his lean body mass to maintain or even add to his body fat. This is old news in biochemistry. It is either unheard of or heresy in clinical science circles, but it is something that cannot be denied or worked around.

    • Dave says:

      Good information, but I don’t anything you said contradicts the data in the article.

      The running events, indicated above by the author would, would have clearly put him into a state of ketosis (fat burning) to support the energy demands of his muscles. So, reduced insulin, and increased glucagon, growth hormone, etc. – as you are saying. Thus, weight loss (and fat loss) resulted, apparently more so than increased in muscle mass.

      • I love energy metabolism says:

        I applaud experimentation. It is the labor of a curious mind. As discovered, often experiments leave more or open questions.

        A question was asked and left unanswered in the conclusion: “Is it really as simple as “calories in, calories out” (i.e., eat less than you burn), or is what you eat more important than how much?”

        My post was intended to say that these questions are answered in biochemistry, but there is a strange amnesia when applying it to humans through clinical science. I would submit that to the question “calories in vs calories out,” the answer is “no, it is not as simple as calories in vs calories out.” To the question “is what you eat more important than how much,” I would submit that it is indeed the case.

      • Dave says:

        @I love energy metabolism, it’s both. If you eat a diet high in refined carbs, and low in protein and fat, you likely eat more food/kcal’s each day. This is due to the satiety factor of carbs vs. protein/fat. So, what you eat affects how much you eat over time…

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  28. Jack says:

    Aren’t you recalibrating your weight every single day? The next day’s weight is today’s weight + some stuff. In that case, doesn’t most of the accuracy come from your actual measured weight? Am I missing something here? Couldn’t I say I’m 99% accurate if I said tomorrow’s predicted weight = today’s measured wight * 0.99. Doesn’t matter if 0.99 is too far off because tomorrow I’m going to be using my next measured weight again.

    • No– note the “hat,” or caret, on top of the w’s in the equation, where “w hat” means “predicted/estimated value of w”, and just “w” (without the caret) means “measured/actual weight.”

      That is, each next day’s *estimated* weight is today’s *estimated* weight + some stuff. We only “bootstrap” the whole process on day zero, where the first day’s estimated weight *equals* the first day’s measured *actual* weight. From then on, we don’t use any of the actual weight measurements to compute the sequence of predicted daily weights.

  29. Lawrence Kesteloot says:

    My personal experience actually matches better with John Walker’s graph here (from The Hacker’s Diet):

    where your body’s metabolism adapts daily, and you have to get outside its range in order to lose or gain weight. I’ve not seen that graph (or concept) anywhere else, though, so I’ve always been a bit skeptical of it. Do you think your data disproves it?

    • @Lawrence, I don’t think so. That is, *throughout* the duration of my experiment, I was *always* well into the “eat too little” region of Walker’s graph. If I understand you correctly, you’re asking whether the “flat” part of his graph makes sense– i.e., if you only vary your “maintenance” calories by small amounts, does your “alpha” value (from my post) fluctuate accordingly?

      I am only now transitioning to a point where I can collect data to answer this. That is, as I continue to lose weight, I am gradually *increasing* my calorie intake to “level off” at a target of around 210-215 pounds. Once I am consistently eating a fixed amount, and neither gaining nor losing weight, *then* small deviations in my diet could address whether there is such a “flat spot” in Walker’s graph, and if so, how wide that flat spot is.

      • Lawrence Kesteloot says:

        Exactly, I’m asking whether the plateau exists. My experience (and I think that of many others) is that reducing intake by 500 calories per day has no effect on weight, when in fact it should cause one pound per week loss. My guess (if the plateau is right) is that we’re always at the right knee in the graph, and dropping 500 calories (or whatever) brings us to the left knee. Relatedly, again if plateau is true, I wonder if being at the right knee and always having “full metabolism” is unhealthy. It may be healthier to be at the left knee even if this has no effect on weight.

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  31. brightmorningdew says:

    Just wanted to say that this was an excellent read!

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  34. GregoryPE says:

    Did you track water intake, or just assume that you had roughly the same level of hydration at your weigh in time every day?

  35. Ketu says:

    Well done, you lost weight by losing weight (calorie deficit) google photos on concentration camps during WW2 to see it on a larger and more dramatic scale. You could also create your pretty graph gaining weight, unfortunately it will still say nothing about causation other than starvation or force feeding; just like the calorie equation. Is pregnancy caused by having a calorie surplus? Is cancer tumor growth caused by a calorie surplus? Is infant growth caused by a calorie surplus?. No, they are all complex biological processes just like fat accumulation, unless you can prove that fat accumulation is different to every other biological process of mass (weight) gain and only requires an overly simplistic physics equation to explain causation.

  36. Abe says:

    That’s an interesting experiment! Are you familiar with the research of Hall and of Forbes on the matter (like the sources cited here) and which formula would you say is more accurate? You can also link any newer articles you have on the topic!

    • Thanks for the link; the model described there seems to incorporate an interesting combination of very specific information– like neck and waist measurements– with very vague information, like “number of times exercising per week” (what counts as “one” exercise unit? A one-mile walk around the neighborhood, an hour of yoga, interval sprints, etc.?). The larger number of parameters would presumably yield a better fit model of weight-loss behavior than the simpler, essentially single-parameter model described here; except that that single parameter is estimated/fitted for a *single individual*. I don’t have neck/waist measurements from before starting this experiment; it would be interesting to compare to see how well this works.

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