Precision of bone density measurement

Question

I just did a precision study scanning 30 patients twice. The calculator presented the least significant change score, but how do I calculate my precision? For instance, the ISCD states that the minimum acceptable precision for a technologist at the lumbar spine site is 1.9%. How do I calculate my precision? Do I just average the 30 score changes between each patient's scores?

Answer 1

There is no such thing as a least significant change, it is a self-contradictory concept. The concept of least significant difference exists. That is, if a numerical difference is significant, it may represent a change, or not, depending on the circumstances. In the case of your data, it does not. Your measurements are a collection of bone density machine and patient slightly variable results obtained on the same day. Neither of those change on the same day. That is, the drift in machine calibration takes, typically, months to occur. Changes in patient bone mineral density typically occur over months to years.

If differences, even >5 IQR beyond quartile outliers were seen on the same day, it would require an acute event occurrence, like an interval lumbar spine compression fracture between measurements, mistaken comparison between two different patients, a machine failure like slippage of an x-ray mask, or mistakenly using a different machine for the duplicate scan, to be called a change. It that case, we might ask if we can detect a change (although then correct, it would still be pointless). The concept of least significance change is an invented quantity (by BMD machine manufacturers: Lunar, Hologic) that appears in the BMD literature, and nowhere else. "Least significant change" would likely not pass muster during statistical review. For example, it hasn't lasted for six inches into text on this post, which is heavily peer reviewed.

The same day measurements you have can indeed be used to establish machine precision (read as machine+patient precision). You likely have 30 or so same-day (or nearly same-day) repeat BMD measurements from which differences can be constructed. ANSWER: To find the precision from that data, you cannot average the standard deviations of those differences without performing small number correction of standard deviation.

A simpler alternative is to perform root mean square of the standard deviations of those differences. That is, 1) sum the squares of the standard deviations of the pairs of repeat measurements. 2) divide that by $n$, the number of repeated measurements. 3) Take the square root to get a relatively unbiased standard deviation. That standard deviation is then a single standard deviation of same day machine error, i.e., the precision you asked for.

Caution: 1) The precision number you get from only 30 patients is itself not very precise, it has a standard deviation of (very) approximately 25%. 2) BMD measurements are not of proportional error type. The errors are linear and should be expressed in projected density units (gm/cm$^2$), i.e., not in percentages.

The better way to approach this problem is as in the abstract below, where thousands of patients have been used to obtain results. In that abstract, from Lunar BMD data, the difference numbers listed (as TDD) correspond to approximately $p=0.05$ significance, but only after an elapsed time of approx. 1 or 2 years.

The statistically correct method for examining BMD differences has been presented.