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Correlations

Correlation tells us how well two measurements of the same thing compare to each other. Correlation has two nice properties:

(1) It is a simple number without dimension (ex., feet or seconds or calories).  This means we can use it to compare different types of measurements.
(2) Its magnitude falls between 0 and 1.  This means there is a clear best value and a clear worst value.

What's not so clear is what the in-between values mean. How good is good enough? That depends on the application. To get some intuition for correlation we can look at several examples. First, let me explain where these examples come from.

Cheaper Measurements

One way to evaluate the precision of a measurement is to compute the correlation of the device's measurement with a known, "true" value. Since genuinely "true" values are unknowable, usually a measurement from a more precise -- but usually also more expensive -- reference measuring device stands in for the true value.

The interest in evaluating measurements this way usually stems from the desire to take a certain kind of measurement more cheaply. (I'm using "cheaply" in a generic sense: less complex, less difficult, or just costing less money.)

For example, the motivation for studying the Body Mass Index (BMI) -- a proxy for body fat -- was that BMI is very easily computed from height and weight. Alternative measures of body fat can be labor intensive when applied to many people (ex., skin calipers) or even when applied to one person (ex., weighing a person under water).

To evaluate the quality of BMI, one could compare BMI for many individuals to one of the more expensive methods of measuring body fat. If BMI were to correlate well with the more expensive measurement, you could then rely on BMI alone for future measurements. This was the approach taken by Indices of relative weight and obesity in 1972. BMI is still widely used today.

In the blog posts Precision and Dynamics we took the same approach with Pertinacity's fist-sized portion proxy for calories and Pertinacity's method of setting a limit for proxy calories. In these measurements we used as the "true", or reference, values the calorie values found in a calorie database.

Examples

Below are some examples of published correlation values for familiar measurement devices and indicators along with Pertinacity's correlations.

Measurement, Reference MeasurementCorrelation
Breathalyzer (Cooperative Patients), Serum Blood Alcohol Level
Gibb KA, Yee AS, Johnston CC, Martin SD, Nowak RM. Accuracy and usefulness of a breath alcohol analyzer. Ann Emerg Med. 1984;13(7):516-20.
.96
Ear Thermometer, Pulmonary Artery Temperature
Erickson RS, Meyer LT. Accuracy of infrared ear thermometry and other temperature methods in adul ts. Am J Crit Care. 1994;3(1):40-54.
~.89 (.87-.91)
Jawbone UP Steps, Handcounted Steps
Stackpool, Caitlin M. Accuracy of various activity trackers in estimating steps taken and energy expenditure. 2013
~.83 (.34, .98, .99, .99)
BMI, Water-Weighing for Density
Keys A, Fidanza F, Karvonen MJ, Kimura N, Taylor HL. Indices of relative weight and obesity. J Chronic Dis. 1972;25(6):329-43.
~.76 (.66, .85)
BMI, Skin Calipers
Keys A, Fidanza F, Karvonen MJ, Kimura N, Taylor HL. Indices of relative weight and obesity. J Chronic Dis. 1972;25(6):329-43.
~.73 (.61 - .85)
Breathalyzer (Uncooperative Patients), Serum Blood Alcohol Level
Gibb KA, Yee AS, Johnston CC, Martin SD, Nowak RM. Accuracy and usefulness of a breath alcohol analyzer. Ann Emerg Med. 1984;13(7):516-20.
.73
Fitbit Ultra Steps, Handcounted Steps
Stackpool, Caitlin M. Accuracy of various activity trackers in estimating steps taken and energy expenditure. 2013
~.73 (.44, .49, .99, .99)
Nike Fuelband Steps, Handcounted Steps
Stackpool, Caitlin M. Accuracy of various activity trackers in estimating steps taken and energy expenditure. 2013
~.67 (.17, .55, .97, .98)
Pertinacity's Fist-Sized Portions, Calories from DB
Precision
.65
Jawbone UP Calories, Portable Metabolic Gas Analyzer
Stackpool, Caitlin M. Accuracy of various activity trackers in estimating steps taken and energy expenditure. 2013
~.63 (.40, .57, .69, .87)
Calories Reported, Calories Fed
Stackpool, Caitlin M. Accuracy of various activity trackers in estimating steps taken and energy expenditure. 2013
.61
Changes in Pertinacity's Fist-Sized Portions, Changes in Calories from DB
Dynamics
.59
SAT, First-Year College GPA
Validi ty of the SAT® for Predicting First-Year College Grade Point Average
.53
Fitbit Ultra Calories, Portable Metabolic Gas Analyzer
Stackpool, Caitlin M. Accuracy of various activity trackers in estimating steps taken and energy expenditure. 2013
~.49 (.24, .41, .63, .67)
Nike Fuelband Calories, Portable Metabolic Gas Analyzer
Stackpool, Caitlin M. Accuracy of various activity trackers in estimating steps taken and energy expenditure. 2013
~.44 (.08, .47, .49, .72)
GRE Subject, First-Year Graduate GPA
A Comprehensive Meta-Analysis of the Predictive Validity of ...
.34
GRE Verbal, First-Year Graduate GPA
A Comprehensive Meta-Analysis of the Predictive Validity of ...
.24
Great Stock Picker, Future Stock Returns
Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk
.10

(The numbers prefixed by ~ are mean values of the numbers that follow in parentheses. The means were calculated by me, whereas the numbers in parentheses are given in the reference publication.)