# Should one account for absolute precision producing variable number of significant digits in a dependent variable measurement? If so how?

Sometimes data for a variable are available in absolute precision, which can create different numbers of significant digits. For example, the CDC's estimates of state prevalences and incidence rates of HIV diagnoses report figures to a single decimal place of absolute precision, rather than, for example three significant digits (relative precision). For example, in 2017, the CDC reports that Alabama's prevalence* was 15.9 HIV diagnoses per 100,000 persons age 13+, but reports Alaska's prevalence was 4.8 HIV diagnoses per 100,000 persons age 13+. Alabama's measurement is thus using 3 significant digits, and Alaska's is using 2 significant digits** (contrast the value 4.8 with a value like 4.81, or 4.77).

Suppose I was modeling annual state prevalences of HIV diagnoses, say in a regression context:

1. Should I account for the resulting variability in significant digits? (What bias emerges in my estimates and inference if I do not?)

2. How? (I suppose I could truncate all figures to 2 significant digits, for example, rounding Alabama's 2017 prevalence to 16, but that necessarily discards information, decreasing precision. Are there alternatives I should consider?)

Bonus consideration: The distribution of values of the most significant digit of the CDC's figures skews strongly rightward, with most values being 0 or 1, some values being 2, and very few values being 3 or 5. Does this affect how I interpret any risk of bias in question (1) above?

NB: This is not a question about measurement error in a dependent variable in general—I am not asking about accounting for the reliability of the dependent variable figures generally.

* Actually, what the CDC terms "prevalence" the epidemiology textbooks I am familiar with term "count of prevalent cases," and what the CDC terms "prevalence rate" the epidemiology textbooks I am familiar with term "prevalence". I side strongly with the latter nomenclature, as "rate" means change (usually over time), but prevalences are static measures by definition (although one may certainly measure change in prevalence over time, that is not what the reported measures are), thus "prevalence rate" lands as an oxymoron to me.

** Assuming for the moment that Alaska's value in 3 significant digits is not actually 4.80.

References

CDC. (2018) Diagnoses of HIV Infection in the United States and Dependent Areas, 2017 (see Table 26).

• It's unclear that you need to do anything at all. I would characterize this precision as value independent. Significant figures reflect a precision relative to a value, whereas the CDC appears to be reporting results to a constant, absolute precision of 0.1 per 100,000. Unless some important values are close to 0 (relative to this precision), that can be safely absorbed into a homoscedastic error variance in any standard regression model.
– whuber
Commented Jan 15, 2020 at 18:43
• @whuber Thank you. That accords with my intuition that any bias will likely be pretty small. Some of the values are much closer to 0 than others. For example DC's prevalence was 56.3, but Maine's was 2.5. Will edit to reflect your point about absolute vs relative precision. Commented Jan 15, 2020 at 19:02
• I would view 2.5 as being far from zero, because it's 50 times the maximum possible absolute rounding error of 0.05.
– whuber
Commented Jan 15, 2020 at 19:06
• @whuber Solid! Thank you. Commented Jan 15, 2020 at 19:09

Alabama's total 2010 population was about 4.8 million. Let's say 4 million age 13+. 15.9 diagnoses per 100,000 then comes to 636 diagnoses. If these are rare, Poisson-distributed events then the standard deviation in that number is about 25 diagnoses, about 4% of the total.* Corresponding error in the value of 15.9 is about $$\pm 0.6$$

Alaska's total 2010 population was about 700,000. Let's say 600,000 age 13+. At 4.8 diagnoses per 100,000 that's about 29 diagnoses total. Under the above assumptions, the standard deviation in that number is about 5 diagnoses, or 17% of the mean. Corresponding error in the value of 4.8 is about $$\pm 0.8$$.

Those inherent errors arising from counts of rare events would seem to overwhelm any issue you might fear in terms of the numbers of significant digits of the reported values. For those two examples, at least, the truncation to a single decimal digit seems to represent the inherent errors pretty well.

*One could argue that almost all HIV cases are captured in the data, so it's not fair for this argument to assume that there is some broader population from which the cases are being sampled. This illustration would be for a hypothetical case where there is a large number of Alabamas with the same overall mean HIV prevalence; among such a set of Alabamas that would be the standard deviation among their prevalence values.

• "One could argue that almost all HIV cases are captured in the data, so it's not fair for this argument to assume that there is some broader population from which the cases are being sampled. " If the inferential target in the model was years-in-states actually observed, then sure. But if one argues that observations are a sample (not a complete enumeration) of HIV prevalences in years-in-states (some years-in-states are yet to be observed, and the models I am working with may make forecasts of future prevalences), then the argument you propose is irrelevant. Commented Jan 15, 2020 at 19:12
• @Alexis in the footnote I was trying to deal with a potential argument against my simple Poisson-based error estimates. I suspect that your modeling (if of HIV or some other low-prevalence disease) will necessarily have Poisson-type errors, and that those will often (in small states at least) be much more of an issue than the CDC's choice to report all prevalence data with a single decimal point.
– EdM
Commented Jan 15, 2020 at 19:17
• Ah right. Sorry, lost that point about the Poisson model. Yes. That makes more sense. :) Aside: actually my model is not a Poisson (or any) count model; I am using methodologies from other disciplines (I am not actually modeling rates and proportions, but modeling change in rates and proportions… still your example is a good one for my question as asked. Aside: negative binomial or GTFO! ;). Commented Jan 15, 2020 at 19:20