Non-linear Numerical Values “Binned” to Ordinal or Range

Author: Michael K. Martin, DVM, MPH, DACVPM
Date written: August 2022

Many laboratory and clinical observations begin as numerical values that are either non-linear or that have large enough random error that pure numerical (parametric) analyses, such as mean and standard deviation, are not useful. Such values are often converted to ordinal categories such as 1+, 2+, etc. or to ordered ranges. When more or less continuous values are grouped into smaller categories for reporting or analysis it is commonly called “binning” or sometimes “bucketing.” This conversion is useful for quick evaluation of a clinical case but it discards much information that may be useful in either more detailed evaluation or in aggregate with other instances.

A formal definition of linear here would be any set x (measurement value), and set y (precise amount of component) where we can define a function f and constants a and b such that y = a * f(x) + b. A simpler way of describing it is that we can meaningfully say that a value twice another has two times as many of the measured component or a log or some similar transformed scale such as a pH of 6 is ten times as acidic as a pH of 7.

Many, if not most, numerical data are true quantitative data and can be reported and evaluated using normal parametric analysis. These include counts, concentrations, most analytical measurements, etc. There is no controversy about calling these variables “quantitative.” Some data never exist as anything more quantitative than “yes/no” or “1+, 2+”, etc. There is no controversy about calling these “ordinal.” The challenge comes when dealing with data that start out, or could have started out, as continuous numerical values but are not appropriate for interpretation as true quantitative values. A 2+ reading on a dipstick should not be interpreted as twice as positive as a 1+. (Even a dipstick starts out as a potentially continuous value as anyone who has had one come out exactly between two sample colors can attest.) Other data, such as the Ct values returned by real-time PCR or optical densities in many automated analyzers, are true quantitative measures of some intermediate value but only non-linearly related to the component they measure. It would be inappropriate to treat these results as true quantitative data but there is still value in the continuous numerical values. Some data are only approximations of underlying true quantitative values. The most correct way to report these would be as a most likely value plus a measure of random error but that would be complicated and slow to use in most clinical settings.

Various tricks have been used to prevent over-interpretation of these “semi-quantitative” values. The most obvious is reporting only the ordinal interpretation as the “result” of the test. From the perspective of the clinical user, these are pure “ordinal” tests. Another technique is to report results as ranges. “Age somewhere between five and ten years” for example. This both prevents over interpretation as well as sometimes preserving deidentification. In both cases potentially valuable information is lost. True ranges are not based on binning of specific values but look very similar. Some examples of each technique may help.

An example of binning continuous data into clinically relevant ordinal categories is real-time PCR. In dealing with outbreaks of influenza A in poultry, labs across the country test thousands of samples using rtPCR. These are reported out as “Detected or Not Detected” as the interpretation. Clinicians and field epidemiologists use those ordinal results for case classification. But the Ct values are also reported. These can be useful for finding the most intensely infected barn on a farm, for example. The labs use the Ct value to triage the samples most likely to yield useful genetic sequencing as well. (Some real-time PCR tests have been carefully calibrated to give actual concentration of the target sequence. These would report true quantitative values.)

Ranges can be a form of binning or they can be true ranges. "I don't know the exact age but it is between 24 and 30 months" is binning of an unknown precise age. The example, “give the patient between 2 and 4 tablets,” is a true range. The prescriber says that the patient may take two, three or four tablets. The dates of the Tour de France, “July 1 - July 24" are also not binning. They are a true range of dates.

While titers are reported as a single dilution such as 1:8, they actually represent binning of a range. There is a precise level of antibody in the sample, but we only measure at specific dilutions. All we know is that the actual level is somewhere between the last dilution at which it reacted and the next one where it didn't. 1:8 in this example really means “greater than or equal to 1:8 and less than 1:16.” Newer methods, such as immunoassays, measuring antibody and antigen levels produce numerical values such as an optical density ratios but convert those to titer values via binning in the reporting algorithms. This conversion keeps the reported value in familiar terms but also helps prevent over reliance on the precise OD ratio values that may not be directly proportional to the exact level of the analyte.

The reporting of cells per low power field is interesting. If I counted 20 fields with the least being 2 and the most being 10 and reported the result as "2 to 10 per low power field" that wouldn't be binning. It is the range I measured. But if I count 80 cells in those 20 fields and choose "0 to 5 per low power field" from a set of ranges, that would be the binning I was talking about. Both are "semi-quantitative" the way I think of it in the real-world. Those 20 fields aren't precise enough to do any true quantitative analysis. They just give me a general ordered sense of scale.

Binning of continuous data into ranges to deal with uncertainty can fail to avoid overly precise interpretation. Following the U.S. Bovine Spongiform Encephalopathy “outbreak” (one case) in 2004, it was important to record the age of each cow/steer tested. Most of the time no precise date of birth was available and the age was estimated from teeth. International trade standards meant that a positive test in an animal less than 30 months of age would have billions of dollars worse impact on the country’s exports than one in an older animal. Because the age ranges reported broke at 30 months, the technicians making the call faced the same sharp cutoff that they would if they’d had to supply a single best estimate of “age in months.” If the form had then included best estimate and a “plus or minus” value, it would have done a much better job of dealing with the uncertainty.

Binning of ages for purposes of deidentification works well for any given study or clinical domain but when the same data are used in a different context, different bins might be needed. (I’m lacking specific examples here. I think I recall definitions of “child” for vaccination rules vs. psychological vs. OBGYN being different.)

We need mechanisms for safely avoiding over-interpretation of semi-quantitative results without losing valuable information. These are data elements that are in some way based on continuous numerical values. However, they cannot safely be treated as truly quantitative the way counts and direct measurements can. Binning on a set of fixed cutoff values is one potential solution but is often less than ideal. A designation of “semiquantitative” in that case, acknowledges that the underlying measurement was numeric. On-demand binning is another more flexible solution. Each setting would use data binned with cutoffs appropriate to its domain. A designation of “semiquantitative” there would provide a warning that the data should not be assumed to be true “linear” quantities.

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