## Calories in, calories out revisited

“All models are wrong, but some are useful.” George E. P. Box

A couple of months ago, I wrote about my experience “counting calories,” particularly about the accuracy of a very simple model of daily weight changes as a recurrence relation, converting each day’s net calorie deficit (or excess) into a 3500 calorie-per-pound weight loss (or gain).  The resulting predictions agreed very closely with my actual weight loss… however, I raised some additional questions at the end of the post, and the post itself generated some interesting comments as well.  Since it’s New Year’s resolution time, I thought this would be an appropriate time to follow up on these.

The following figure shows my predicted weight loss (in blue) over a period of 136 days, compared with each day’s actual measured weight (in red).  The details of the predictive model are provided in the earlier post; briefly, the idea is simple: start with a “zero day” measured weight, then predict weight changes for all subsequent days using only daily calorie intake (from eating) and expenditure (based on weight and, in my case, running).

Predicted and actual measured weight over 136 days.

For side-by-side comparison, the following figure shows the corresponding estimated daily calorie intake over the same time period.

Daily calorie intake over the same 136 days.

The first 75 days (the focus of the earlier post) show reasonably consistent behavior: relatively aggressive calorie deficits that yield almost 2.5 pounds lost each week.  But what if I were not so consistent?  How well does this simple model handle more radical changes in diet?

As you can see from both figures, days 76 through 79 get messy.  During that long weekend I had to travel to give a talk, and thus I didn’t have ready access to a scale, hence the missing weight measurements; and I also had less convenient control over the food that I ate.  There is a banquet buffet lunch in there, some restaurant food, etc., where my best-effort calorie estimates are obviously much less accurate.

But although I certainly expected to have gained weight after returning from my trip, I was surprised at how much I had gained, much more than the predictive model could possibly account for.  However, over the next several weeks, when I returned to a more well-behaved diet (more on this later), my weight seemed to “calm down,” returning to reasonably close agreement with the simple “calories in, calories out” model.  It’s not clear to me what causes wild swings like this.  For example, there are also a couple of 4 am wake-up calls for early flights, late nights, and the general stress of travel during those four days.  Perhaps those departures from my normally routine lifestyle might also contribute to the fluctuation in some way?

Also, note the planned incremental increases in calorie intake over the last couple of months, and the resulting slowdown in the rate of weight loss.  I didn’t stop losing weight, I just started losing less weight each week as I approached my goal.  This ability to eat more while still losing weight may be counter-intuitive, but the math makes sense: it’s really hard to lose weight… but it’s much easier to maintain weight once you’re where you want to be.  (On the other hand, it’s an exercise for the reader to verify using the model that it’s also dangerously easy to gain weight, and to do so much more quickly than you lost it.)

Finally, this subject generated quite a bit of discussion about the initial question of whether “it’s really as simple as calories in, calories out.”  In particular, several commenters insisted that no, it is not that simple, that the human body “is not a bomb calorimeter,” but a much more complex machine where weight is influenced by many other factors, including genetic variation, gut flora, etc.

I don’t disagree with this.  In fact, while we’re at it, let’s point out several other limitations as well: this model treats calorie expenditure as a linear function of total body weight, instead of lean body mass, which is arguably a better fit (but is not nearly as convenient to actually measure).  It also treats calorie expenditure from running as a function of weight and distance, but not speed.

Which brings me to the quotation at the top of this post.  No, this simple recurrence relation does not reflect the full complexity of the biological processes occurring in the human body that contribute to weight loss or gain… but so what?  Don’t use a complicated model when a simple one will do.  In this case, most of that additional complexity is “rolled up” into the single coefficient $\alpha$ reflecting the individual’s “burn rate” based on gender, genetic variation, flora in the gut, etc.  Granted, that coefficient may be unknown ahead of time, but at worst it can be estimated using a procedure similar to that described in the original post.

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### 6 Responses to Calories in, calories out revisited

1. matthew mcdonald says:

Re the weight gain during your trip – i would have guess that this might be mostly changes in water retention triggered by changes in glycogen levels. A quick google gives this link, which suggests that might be more or less right: http://www.phlaunt.com/diabetes/43067073.php

2. Kim says:

Thanks for your original post and follow-up!! Are you still tracking calories and, if so, has your model continued to accurately predict your weight? I’ve found it very useful. Occasionally it won’t accurately predict my weight if I stop working out for a while or if I drastically increase the intensity of my exercise but those situations are rare and, like you said in your previous post, just a result of relying on body weight instead of something like lean body mass.

• Thanks! I did continue to track calories closely for a while after this, but now that I’m holding relatively steady at my desired weight, I have found that I don’t need to count and record calories as closely… but I still weigh myself every morning.

You’re right about the model’s accuracy suffering through radical changes in behavior, either diet or exercise. I have also found that, as in the “blip” mentioned in this post, the model doesn’t accurately reflect what seems to be water retention (see the other comment here) due to eating more than usual even for just a short period. Because of this, I’ve found that it’s better when starting out, to begin “behaving” better eating-wise… but to wait a week or two after that to mark the “zero day” (w_0 in the notation of the previous post) of the recurrence model. In that first week or two it’s not uncommon to lose a *lot* of weight, largely water, that messes up the subsequent fit.

• Kim says:

That makes sense. If, after you’ve been behaving well for a while, you change your behavior drastically enough so that the predicted and actual data don’t fit as well anymore (for example, you gained 2 pounds of muscle and unlike the “blip” your weight didn’t return to the overall trend), would you remove or replace that data or would you leave it in? If I leave the data in then there isn’t a nice fit anymore but I can still see the correlation between relative weight gain or loss on most days. There probably isn’t a “right” way of handing occurrences like these but I’m curious what you would do with them.

3. @Kim, Good question. I think this model is simple enough that it probably doesn’t accurately reflect weight behavior through any radical changes, in which case my recommendation– and my past practice– has been to simply “start over,” so to speak. That is, definitely not removing any subset of data, but instead remove it all :), and start with a new zero-day weight (and possibly a new alpha parameter (*)) and go from there.

(*) This is one of the interesting unanswered questions about this whole approach. I personally have not observed any measurable change in my alpha value, despite aging several years since I first did this. But I also haven’t significantly changed my eating habits or exercise behavior (although I run longer distances now). But you or others may find the need to change this value, by changing diet, gaining a lot of additional muscle mass by lifting weights, etc.