More Quantitative Diet Analysis

Two Questions:

1. What is the correlation between calorie consumption and weight loss?
2. Is it happening the same day?

This Saturday, I skipped hiking up to the mountains to answer these naging questions my last diet analysis left over. To answer them, let me first present the data, carefully recorded over 35 days of dieting.

Weight-loss per day

Measured each morning before breakfast and after taking a crap, pardon my french.

How much did I lose on a regular day? 
Most of the time lost between  -0.07 and 0.5kg

The Percentiles are: 

Min.     1st Q. Median  Mean  3rd Q.  Max. 

-0.66kg  -0.07  0.16    0.2   0.5     1.5kg 

The 1.5 kg outlier draws the attention. Is it a measurement error?

Net caloric intake per day

This is the amount of calories I have eaten, minus the calories that I have burned doing sport.

When I say calories I mean the american metric which corresponds to the european kilo-calories (kcal).

Caloric Balance Sheet
How much calories did I take-in and consume in a day?
Most often 500 kcal to 960 kcal

  

The Percentiles are: 

Min.      1st Q  Median  Mean  3rd Q  Max. 

-1009kcal 490    660     630   960    1480 kcal 

My friend, the outlier, is here again. Seems reasonable: A day of heavy exercise and little eating... but 1.5 kg? That's a lot.

Caloric intake left to cover the daily base burn rate

Now its getting nasty. Our body consumes calories even if we lay in bed all day. This is the base rate. Whitout getting into details, it depends roughly on your body weight. When you take-in lesser calories than the base-rate (plus exercising) you lose fat. My formula for this spread goes like this:

kcal_left = base-rate - net_kcal

And the distribution of the not consumed calories is:

Distribution of calories-left
I normally underperform the base-rate by 250kcal - 730kcal.

   

   Min.   1st Q Median  Mean 3rd Q  Max. 

 -240kcal 250   520     570  730    2230kcal 

Median underperformance is 520kcal, roughly a spurned Pizza Magherita. Two times I've consumed slighty more than the base-rate. And the outlier from my sport weekend is here too!

Correlation between calories left and weight-loss

Back to question 1: What is the correlation between calorie consumption and weight loss?

Now I know that I have to maximize calories-left (to the sedentary base-rate) in order to lose weight.

What are the results of my effort?

Figuring a trend

The Linear regression line in blue 

shows there is a positive correlation
between calories-left and  weight loss.
The border of the darker region
is the mean deviation oft the data 

from the regression line,
a.k.a. the standard error.

If you are a genius you certainly have noticed that the outlier is away. It would have greatly influenced the regression and I felt that I want to answer what happened in normal days. Also the correlation is not high (R^2 = 0.3541857), there are other effects that influence the variables a lot and it's clear that grand part of them are measurement errors.

Coefficients of the regression line are:

(Intercept)   kCal_Left  

 0.1174047    0.0001555  

Basically this is the linear function:  

weightLoss = 0.1174047 +  0.0001555 * kCal_Left

For the median calorie underperformance of 520 kcal this means:

520 kcal/day * 0.0001555 kg/kcal + 0.1174 kg/day ~=  0.2 kg/day

Not bad at all, and its supported by the fact that really I lost 7kg in 34days. 7/34 = 0.20!

Do I lose weight the same day that I 'starve'?

Question 2 is remaining: Is it happening the same day?

Short answer: Yes. Long answer: Yes, but there might be a lag and prolonged effect for sport activities. The evidence in a picture:

Direct impact of 'fasting'
Size of points representing calories left to base-rate
The higher the line, the greater the weight-loss.
A calorie point sits on the weight-loss recorded the morning after.
That is: the loss-spike is caused by the calorie point it carries.

Often a bigger point sits on a higher spike, which leads to the satisfactory conclusion that indeed you lose the kilos the same day as you restrict your calorie intake. The big outlier we saw already before has a different behavior. Here, the big balls hang relatively low. Does sport have a weight loss lag? The mythical regeneration phase?

The days my sport activity burned more than 1000kcal: 

Day 12: 1587 kcal

Day 13: 1946 kcal

Day 29: 1099 kcal

This big outlier on day 12 and 13 could cause a after-burner effect, but the evidence is thin. 

More data needed!