Interpreting two TSH results from the same patient

Introduction

When the physician wants to confirm a normal analytical result, i.e. a concentration of an analyte within the reference interval of healthy individuals, the second analytical result may very well be different from the first. The second result may also be normal, but so different from the first that the physician may wonder if a change in the clinical condition has taken place. Reference change values (RCVs) are supposed to help the physician in this situation. RCVs are population reference limits for the change in analyte concentration from one sample to the next in the same subject [1, 2], but their estimation and clinical application are hampered by several problems [3]. However, if the patient’s mean (setpoint) concentration, i.e. the homeostatic set point, of an analyte is unknown and the physician tries to judge whether a certain pair of measured concentrations are commonly or rarely seen in clinically healthy and stable individuals, the physician really needs to know the distribution of paired values from individuals in a clinically healthy and stable population. This is a bivariate distribution. Just comparing the two measured values against univariate reference limits and their difference against RCVs may be misleading. In this work we simulated pairs of s-TSH values from clinically healthy and stable individuals and studied the performance of the combination of the 2.5 and 97.5 percentile univariate reference limits and the 2.5 and 97.5 percentile RCVs in delineating the central 95 % of the bivariate distribution.

Materials and methods

We simulated two s-TSH results for each of 100,000 individuals. The individuals were supposed to be euthyreot. First, we generated 100,000 lognormally distributed setpoint values by exponentiation of values from a standard normal distribution truncated at ±3 standard deviations. The 2.5, 50 and 97.5 percentiles of the lognormal distribution of setpoint values were 0.46, 1.40, and 4.21 mU/L, respectively. From each setpoint value (i.e. for each individual) we generated two randomly distributed values, x1 and x2. We assumed that s-TSH from each individual varied due to a combination of within-subject biological and analytical variation, and without autocorrelation between the two measurements. The combined biological and analytical variation was assumed to be Gaussian, and was truncated at ±3 standard deviations. The within-subject coefficient of variation (CV_I) was set to 17.9 % [4] and the analytical coefficient of variation (CV_A) to 3.4 %. In the same manner we generated a third series of measurements from the setpoint values, and defined the lower and upper normal reference limits as the 2.5 and 97.5 percentile in the distribution of values from this series.

We calculated the individual differences between x2 and x1, as well as the ratios x2/x1, and estimated the median of these values for data pairs with x1 below and above the median value of x1.

RCVs was calculated in two ways, parametrically as (exp(±z × 2^0.5 × (lnCV_I² + lnCV_A²)^0.5) − 1) [2], where z=1.96, and lnCV=(ln(1 + CV²))^0.5 [2]; and non-parametrically as the 2.5 and 97.5 percentile in the distribution of the ratio x2/x1.

We estimated the bivariate distribution of x1 and x2 with the Stata function robmv [5]. For each pair of x1 and x2, this function gives a robust distance estimate from the center of the bivariate distribution. We then estimated the 50, 60, 70, 80, 90, and 95 percentile in the distribution of the robust distances.

How well the 2.5 and 97.5 percentile univariate reference limits along with the 2.5 and 97.5 percentile RCVs were able to delineate the central 95 % of the pairs of x1 and x2 we evaluated both quantitatively and graphically. Quantitatively, we estimated the sensitivity and specificity of the combination of univariate reference limits and RCVs in classifying paired values according to the central 95 % of the bivariate distribution, taking those within the central 95 % to be normal and the others abnormal. Pairs of x1 and x2 where both values were within the univariate reference range and the difference between x1 and x2 was within RCVs were classified as normal according to those criteria, and otherwise as abnormal. We also calculated sensitivity and specificity for the univariate reference limits and RCVs without combining the two.

Graphically, we plotted x2 against x1, using different colors for the central 50, 60, 70, 80, 90, and 95 % of the bivariate distribution, along with the lines representing the 2.5 and 97.5 percentile univariate reference limits and the 2.5 and 97.5 percentile RCVs. In this plot the RCVs lines indicate a certain x2/x1 ratio, i.e. for any value of x1, 95 % of the x2-values are located in the space between the lines of the 2.5 and 97.5 percentile RCVs.

In order to graphically study the use of this principle on another analyte, with a completely different distribution of setpoint values, CV_I, and CV_A, we simulated 100,000 s-sodium setpoint values with a Gaussian distribution truncated at ±3 standard deviations. We then proceed as for s-TSH, using a CV_I of 0.5 % [6] and a CV_A of 0.7 %.

The Stata software, version 16 (StataCorp, College Station, TX 77845, USA) was used for simulations, calculations, and graphical work.

Results

The 2.5 and 97.5 percentile univariate reference limits of s-TSH were 0.428 and 4.38 mU/L, respectively. The RCVs calculated as 2.5 and 97.5 percentile in the distribution of x2/x1 ratio were 0.592 (95 % CI 0.589 to 0.595) and 1.69 (95 % CI 1.69 to 1.70), respectively. The corresponding parametrically estimated RCVs were 0.606 and 1.65. Figure 1 shows the distribution of the pairs of x2 and x1, the central 50, 60, 70, 80, 90, and 95 % of the bivariate distribution, along with the 2.5 and 97.5 percentile univariate reference limits and the lines of the 2.5 and 97.5 percentile RCVs. The agreement between the bivariate distribution and the combination of univariate reference limits and non-parametric RCVs is shown in Table 1. The sensitivity of the combination for identifying pairs of x1 and x2 lying outside the central 95 % of the bivariate distribution was 100 × (4,008/5,000) %=80.2 %. The specificity for identifying pairs of x1 and x2 lying inside the central 95 % of the bivariate distribution was 100 × (87,608/95,000) %=92.2 %. On their own, the univariate reference values had a sensitivity and specificity of 68.2 and 96.0 %, respectively. The corresponding figures for RCVs were 16.8 and 95.6 %.

Figure 1:

Two s-TSH values were simulated from each of 100,000 clinically healthy and stable individuals, and s-TSH in the second sample plotted against s-TSH in the first sample. The dark green points are the central 50 % of the bivariate distribution, and the lighter green/blue points are the central 60, 70, 80, 90, and 95 %, respectively. The red stippled line is the line of equality. The blue lines represent the 2.5 and 97.5 percentile RCVs of the nonparametric method. The black stippled lines are the corresponding RCVs of the parametric method; see text for an explanation of those methods. The red lines are the 2.5 and 97.5 percentile univariate reference limits (0.428 and 4.38 mU/L, respectively). For instance, a healthy and stable individual with a s-TSH equal to 1.4 mU/L in the first sample can be expected to have a value between approximately 0.43 (lower reference limit) and 2.9 mU/L in the second sample according to the bivariate distribution, and not between 0.83 and 2.4 mU/L as suggested by the RCVs (which is equal to the combined criteria of the RCVs and univariate reference limits in this case).

The median difference x2 – x1 was 0.0405 (95 % CI 0.0383 to 0.0427) for pairs with x1 less than the median value of x1, and −0.0846 (95 % CI −0.0899 to −0.0789) for those with x1 above the median. The median ratio x2/x1 was 1.05 (95 % CI 1.05 to 1.05) for pairs with x1 less than the median value of x1, and 0.960 (95 % CI 0.957 to 0.962) for those with x1 above the median.

The graphical study of the s-sodium data is presented in a supplementary figure. The sensitivity of the combination of univariate reference limits and RCVs for identifying pairs of x1 and x2 lying outside the central 95 % of the bivariate distribution of s-sodium was 100 %. The specificity for identifying pairs of x1 and x2 lying inside the central 95 % of the bivariate distribution was 92.5 %.

Discussion

If the physician does not know the patient’s setpoint value of s-TSH and wants to judge the clinical condition from s-TSH in two samples taken a time apart, we believe that the physician basically wants to assess whether the patient is healthy and stable. Then the two s-TSH values could be compared against the bivariate distribution in Figure 1, which represents a stable, euthyreot population. As clearly shown in Figure 1, the lines of the 2.5 and 97.5 percentile univariate reference limits in combination with the 2.5 and 97.5 percentile RCVs do not accurately delineate the central 95 % of the points of the bivariate distribution of x1 and x2. The space between the RCV lines contains 95 % of the points, as do the space between each set of reference limits, but the space between the RCVs and the reference limits is not congruent with the ellipse marking the central 95 % of the bivariate distribution. The RCV lines are approximately tangent to the ellipse marking the central 50 % of the distribution and cut through the other ellipses. Compared to the central 95 % of the bivariate distribution, the combination of univariate reference limits and RCVs had a fair specificity of 92.2 % but a lower sensitivity of 80.2 %. Without the assistance of univariate reference limits, the RCVs showed a particularly low sensitivity. Obviously, RCVs are not designed to detect healthiness, as the area between the limits of RCVs includes analyte concentrations from zero to infinity (Figure 1). These considerations are not limited to s-TSH; probability density contour plots for bivariate distributions are not straight lines for any analyte, as indicated in the Supplementary Figure. In the example of s-sodium, the diagnostic accuracy of the univariate reference limits and RCVs was considerably better than for s-TSH. Obviously, how well the combination of univariate reference limits and RCVs delineate the corresponding bivariate distribution must be studied for each analyte.

Looking at the ellipses marking the various central proportions of the bivariate distribution and the line of equality (Figure 1), it is obvious that regression towards the mean does occur in this scenario. If the measured value in the first specimen (x1) is relatively low, the measured value in the second specimen (x2) is most likely to be higher, and vice versa. The median values of the difference x2 − x1 and the ratio x2/x1 for pairs with x1 below and above the median value of x1 showed the same phenomenon, as expected, because a difference in percent is equivalent to a ratio. We prefer ratios in this setting.

Thus, the idea of RCVs as a constant fraction of the first measurement is flawed, a finding in accordance with a previous study [7]. We estimated the RCVs both parametrically and non-parametrically, to see whether the two methods gave different results. They did not; the two methods of estimation gave almost identical RCV lines. They were symmetrical about the equality line, and asymmetrical in the y direction. We simulated the s-TSH-values assuming a Gaussian distribution around the setpoints; still, the non-parametrically derived RCVs based on the simulated values were asymmetrical in the y direction.

We believe our data set was theoretically sound. Data were derived from a Gaussian distribution truncated at ±3 standard deviation, and transformed to lognormally distributed data as coming from an euthyreot (healthy) population. Each data pair was generated from the same, individual setpoint value, so the data represented a stable population. All data pairs were generated with the same CV_I of 17.9 % [4], thus the variance was homogeneous. The value of 3.4 % for CV_A is the total CV_A in our laboratory, estimated from quality control values over several months. Truncation at ±3 standard deviation when generating population setpoint and individual values were done because those distributions are not really Gaussian (they do not include values from minus to plus infinity) and often values outside ±3 standard deviation are regarded as outliers.

Note that this work deals with how the physician might interpret two s-TSH values from the same patient when the patient’s setpoint value is unknown. It has no relevance if the physician needs to judge a time series of three or more measured values of the same analyte. Neither does it go against the use of CV_I to estimate a personalized reference range using more than two values [8]. In fact, the construction of bivariate reference values for two results of the same analyte could be a new application of CV_I.

In conclusion, the combination of univariate reference limits and RCVs can not accurately describe the bivariate distribution of s-TSH values from many clinically healthy and stable individuals where each individual has contributed two values from samples taken a time apart.