Authors’ reply to the letter to Editor (Annals of Clinical Biochemistry): ‘A simple approach to derive Z-score of reference change value involving more than two serial results’

Dear Editor

In their letter Liu et al. presented ‘A simple approach to derive Z-scores involving more than two serial results’. ¹ This stated approach was used to calculate Z_n-scores dependent on the number of serial results (n), and some calculated Z_n-scores were compared with the Z_n-scores calculated by the model based on 10,000 simulated data from Lund et al.^2,3 The total calculated Z_n-scores are listed in Table 1 for both uni- and bidirectional changes in serial results using both models. In general, the Z_n-scores are comparable, except for the Z_n-scores for 95% probability based on unidirectional changes, where the scores from Lund et al. ² are slightly lower than those calculated from the method of Liu et al. (see Table 1). Liu et al. ¹ explained the differences on the basis of the fact that Lund et al. ² used a ratio of two results being compared, instead of the percentage difference and assumed an underlying log-normal distribution of results. We think that this is unlikely, since all of the other Z_n-scores are comparable (Table 1). The minor differences observed may instead reflect a few extreme random numbers in the simulated calculations.

Table 1.

Z_n-score dependent on the number of uni- and bidirectional serial results (n), calculated using the model of Lund et al.^2,3 compared with Z_n-scores calculated by the method of Liu et al. ¹

	Unidirectional changes		Bidirectional changes
Number of results (n)	Lund et al.2,3	Liu et al. 1	Lund et al.2,3	Liu et al. 1
Zn-scores (95% probability)
2	1.64	1.65	1.97	1.96
3	1.91	1.95	2.23	2.24
4	2.07	2.12	2.33	2.38
5	2.18	2.23	2.43	2.49
6	2.23	2.31	2.53	2.58
7	2.28	2.38	2.58	2.63
8	2.33	2.43	2.63	2.68
9	2.38	2.49	2.63	2.73
10	2.43	2.53	2.68	2.76
15	2.58	2.68	2.83	2.90
20	2.68	2.78	2.97	2.99
Zn-scores (99% probability)
2	2.28	2.33	2.58	2.58
3	2.58	2.57	2.78	2.80
4	2.68	2.71	2.97	2.93
5	2.78	2.79	3.01	3.01
6	2.87	2.86	3.11	3.06
7	2.92	2.93	3.15	3.12
8	2.97	2.97	3.15	3.15
9	3.01	3.01	3.20	3.19
10	3.01	3.03	3.20	3.22
15	3.11	3.14	3.37	3.37
20	3.29	3.26	3.63	3.47

In conclusion, the two different calculation models of Z_n-scores for more than two serial results suggest that the Z_n-score values in Table 1 are correctly calculated. Both models confirm the need for using varying Z_n-scores as a function of number of results (n) (according to Table 1) in order to calculate the limits for constant percentage of false positives in several consecutive measurements.