Homoscedastic and heteroscedastic—if nothing else, these are two good adjectives for word games. They’re also good terms for those who make measurements, because the words can be used to describe the noise structure of the data. This can be important for identifying the possible need for “weighting” the data during mathematical analysis, and for deciding whether to report the standard deviation or the relative standard deviation when describing the fundamental behavior of a measurement method.
The Greek roots homo and hetero mean “the same” and “different,” respectively. The Greek root skedastikos has to do with “scattering” or “dispersion.” So homoscedastic means “the same scatter” (or constant noise) and heteroscedastic means “different scatter” (or variable noise). But “the same” or “different” with respect to what?
In our context, we’re looking at the magnitude of noise with respect to the amount of analyte (also called the “substance being determined” or the “determinand”), so we need to look at the noise structure on a calibration relationship to decide if we’re working with a homoscedastic measurement method or a heteroscedastic measurement method (or perhaps a measurement method that exhibits both types of variability).
Figure 1 illustrates a purely homoscedastic measurement method with constant noise (the green squiggles above and below the black fitted calibration relationship). The analyte amount x varies from 0 to 100 units from left to right across the graphic; the measured response y ranges from 0 to 100 units from bottom to top. The extra label at the right of the graphic shows the magnitudes of the standard deviation σ (red, measured in response units) and the percent relative standard deviation %RSD (blue, measured in percent), also called the %CV (percent coefficient of variation). At the right edge of the graphic, σ = 3 units and %RSD = 100% × (σ / x) = 100% × (3/100) = 3%.
Figure 1 – A calibration relationship with constant (homoscedastic) noise (green). Light grey lines are drawn ±1σ
from the calibration line. Note that the standard deviation σ
is a constant (3 signal units), but the %RSD varies inversely with analyte amount.Some persons seem to believe it’s necessary to report the %RSD for every measurement method they develop, as if it is the single fundamental descriptor of variability for any measurement method. From Figure 1 it should be clear that homoscedastic measurement methods do not have a single value of %RSD—the %RSD depends on where you’re working along the calibration relationship. For a homoscedastic measurement method, the standard deviation σ is the single fundamental descriptor of the measurement method—it is constant across the calibration domain.
Figure 2 illustrates a purely heteroscedastic measurement method with variable noise (the purple squiggles above and below the black fitted calibration relationship). There is only one kind of homoscedastic noise structure, but the number of heteroscedastic noise structures is infinite. The most common of these is proportional noise—as the amount of analyte increases, the standard deviation σ increases proportionally, as shown in Figure 2. In this example, when the amount of analyte is 100 units, σ = 9 response units.
Figure 2 – A calibration relationship with variable (heteroscedastic, in this case proportional) noise. Light grey lines are drawn ±1σ from the calibration line. Note that the %RSD is a constant (9%), but the standard deviation σ
varies directly with analyte amount.Is the standard deviation σ the single fundamental descriptor of the variability of this heteroscedastic measurement method? Clearly not. Almost by definition, the %RSD is the fundamental descriptor for the variability of proportionally heteroscedastic measurement methods. The standard deviation depends on where you’re working along the calibration relationship.
So for homoscedastic measurement methods, report the fundamental standard deviation σ; for proportionally heteroscedastic measurement methods, report the fundamental percent relative standard deviation %RSD. In this example, the %RSD = 9%.
Unfortunately, real life isn’t this simple. While investigating the between laboratory variability of measurements,1 William Horwitz discovered that both components of variation seem to be present in most measurement methods. Statistically, the two sources of variation combine as:
.
where σcombined is the resulting standard deviation, σhomoscedastic is the constant noise component, and σheteroscedastic is the variable noise component (variances—standard deviations squared—are additive in statistics, not the standard deviations themselves). For our example, the result is shown in Figure 3. The blue curve showing %RSD as a function of analyte is known as the “Horwitz curve,” sometimes called the “Horwitz horn” (I think because if you were to rotate the curve about the x-axis, it would generate a three-dimensional image that would look like the bell of a trumpet).
Figure 3 – A realistic calibration relationship with both constant (homoscedastic) and variable (heteroscedastic) noise. Light grey lines are drawn ±1σ from the calibration line. Note that both the %RSD and the standard deviation σ vary with analyte amount. At the left, the noise is purely homoscedastic; at the right, the homoscedastic noise contributes less than 5% of the total standard deviation.At low amounts of analyte (e.g., trace analysis), the variability is fairly constant—the heteroscedastic component isn’t large enough yet to make its presence felt. (In previous modules on limits of detection and related concepts, this was one of the assumptions we made, and it’s generally a good assumption.) As the amount of analyte increases, the heteroscedastic component grows and adds to the homoscedastic component until at large amounts of analyte, the homoscedastic component becomes almost insignificant in comparison, and the noise behavior of the measurement method becomes almost purely heteroscedastic.
Each measurement method has its own noise structure, of course, but the existence of both types of noise, homoscedastic and heteroscedastic, seems to be common in most measurement systems. If they can be estimated, perhaps reporting both the constant homoscedastic component (σ) and the variable heteroscedastic component (%RSD) would be appropriate.
As a final note, when statisticians fit mathematical models to data, they prefer to emphasize those data points that have little uncertainty and de-emphasize those data points that have large uncertainty, a process called weighting. There are many different approaches, but most use a “reciprocal variance weighting”—that is, each data point is weighted by 1/σ2, where σ is the standard deviation associated with a data point. In Figure 3, for example, the data points at low amounts of analyte would receive more weight than those at higher amounts of analyte.
Next up is “pooling,” a technique for combining several uncertain measurements of uncertainty into a more certain estimate of uncertainty. (Yes, I said it correctly.) If it’s appropriate for your measurement method, you can save a lot of money by using it.
Reference
1. Horwitz, W. Evaluation of analytical methods used for regulation of foods and drugs. Anal. Chem. 1982, 54(1), 67A–76A.
Stanley N. Deming, Ph.D., is an analytical chemist masquerading as a statistician at Statistical Designs, El Paso, Texas, U.S.A.; e-mail: [email protected]; www.statisticaldesigns.com