IgG antibody titers against T. cruzi measured in terms of optical density (OD) were considered as real data populations. These data were obtained from the archive of samples processed between 1992 and 2014 by the Instituto de Biología Molecular de Parásitos (Institute of Molecular Biology of Parasites, BioMolP by its acronym in Spanish) and the Department of Parasitology of the Universidad de Carabobo, Valencia-Venezuela. Based on these records, mean (µ) and variance (σ²) were estimated for the results of both healthy individuals, µ_s and σ² ₅, and Chagasic patients, µ_e and σ² _E.

This sample was made up of the OD values obtained from the sera of individuals from non-endemic areas for Chagas disease with negative IFA, ELISA, and Western Blot tests.

This sample was made up of the OD values obtained from the sera of Chagasic patients from the endemic states of Carabobo and Cojedes, Venezuela, with positive results in at least 2 of the 3 tests mentioned above.^13-15

Both healthy individuals and Chagasic patients gave their informed consent to take part in epidemiological studies on T. cruzi. The ethical principles for medical research involving human subjects set out in the Declaration of Helsinki were respected. ¹⁶ This research was endorsed by the Bioethics Commission of the Faculty of Health Sciences, chaired by the Directorate of Research and Intellectual Production of the Faculty of Health Sciences of the Universidad de Carabobo, which guaranteed that the bioethics and biosafety principles were applied as stated in Minutes D1-058-11 of March 14, 2011.

The total proteins of T. cruzi epimastigotes of human origin were used as the antigen, which was identified using the discrete typing unit (DTU) named TcI based on the methodology outlined by De Lima et al.¹⁵ The TcI DTU was selected because it is the most common in Venezuela, representing about 95% of the isolates. ^17-19

The simulated data were obtained using the add-in for producing random numbers from a normal or Gaussian distribution of the Microsoft Excel program. ²⁰ On the other hand, the population parameters values used to generate the simulated samples were obtained from the characterization of the real data populations described above.

A population of healthy individuals (P_S) was defined using mean (µ_s) and variance (σ² ₅), as well as 3 sets of 5 populations of Chagasic patients (P_E): 1 set with the same variance of the population of healthy individuals (homoscedastic) and 2 sets with population variances different from that of the population of healthy individuals (heteroscedastic), for a total of 16 simulated populations.

As mentioned above, the variance in the homoscedastic set corresponded to that of the real data population of healthy individuals (σ² ₅). Regarding heteroscedastic sets, in the first case, the variance was obtained in the population from real data of Chagasic patients (σ² _E), while, for the second, the pooled or weighted variance (σ² _c) was calculated with the population variances of healthy individuals and Chagasic patients.

The means of the simulated populations of Chagasic patients were defined as a function of the mean of healthy individuals (µ_s) and the pooled standard deviation (σ_c). Thus, the mean values for Chagasic patients were defined by P_E1: µ_e1 = µ_s+0.5 σ_c; P_E2: µ_e2= µ_S + σ_c; P_E3: µ_E3 = µ_S + 2σ_c; P_E4:µ_E4=µ_S + 3σ_c y P_E5: µ_E5 = µ_S+4σ_c, to build up populations of Chagasic patients with means increasingly distant from those of the healthy 233were generated for the population P_S and for each population P_E. Each one consisted of n_S simulated observations coming from P_S, and of n_E simulated observations coming from P_E, i=1,2, 5. The size of n_S and n_E was set at n_S= n_E=30, because this is the most widely used sample size in practice (Table 1).

Table 1

Source: Own elaboration.

The set of results of healthy individuals and Chagasic patients was named scenario, and 5 scenarios were constructed for each of the variance assumptions: i=1,2,…5. Scenario-1: {P_S, P_E1}, Scenario-2: {P_S, P_E2}, Scenario-3: {P_S, P_E3}, Scenario-4: {P_S, P_E4} and Scenario-5: {P_S, P_E5}. In this way, 15 scenarios with simulated data were obtained.

These scenarios are fundamental to simulation since the sensitivity and specificity of the tests require information from healthy and Chagasic individuals on the ROC curve. The samples of healthy individuals were the same in each simulated scenario; only the simulated samples for the Chagasic patients varied, so this method allows maintaining the same point of comparison between Chagasic patients and healthy individuals.

The standard and ROC curve methodologies were applied to obtain the decision thresholds or critical values (V^C) for the real data and the simulated samples; the calculations were made using a routine written in Excel. For the standard methodology (Std), four V^C were established: StdM1 = µ+2σ, StdM2, µ+3σ, StdM3 = µ*+2σ* and StdM4=µ*+3σ*, where µ* and σ* are the trimmed arithmetic mean and the trimmed standard deviation, respectively.

The V^C of ROC curves were estimated using the minimum quadratic distance (MQD) and the Youden Index (I_Y). For MQD, V^C is min(MQD)=min{(1-sensitivity)²+(1-specifity)²}, and for I_Y, V^C is max (I_Y)=max{sensitivity+specifity-1}.^21,22

For the real data and the simulated scenarios, a k-th observation (y_K) was deemed healthy if y_K≤ V_p ^c and sick if , where V_p ^c is the decision threshold of the p test. For the simulated samples, the k-th observation y_ijk and the decision threshold V_i ^c _jp depended on the i scenario and the simulated sample j; i=1,2,..,5; j = 1,2,..,n*.

Sensitivity and specificity of a test were given by:

True positives are Chagasic patients declared positive through p test, while true negatives are healthy individuals declared negative through this same test.

The sensitivity and specificity of the methodologies applied in the simulated populations were compared based on the estimates given by

To compare discriminatory accuracy, it was established how many samples of the tests had sensitivity and specificity equal to 100%. Thus, for scenario i and test p, it was obtained:

The proportions of samples with specificity and sensitivity equal to 100% were given by:

Similarly, the number of samples in which the tests had sensitivity and specificity equal to 100% was determined. This result was named perfect-decision and was obtained with the equations

The proportions of samples with perfect-decision were given by

The population of healthy individuals was N_s=901 with the parameters µ_s=0.12226 and a2=0.0531. On the other hand, the Chagasic patients were N_e=342 with the parameters µ_e=0.4093 and σ_E=0.2234. The pooled standard deviation of both populations was σ_C=0.1255 The parameters were measured using the OD, and the absolute frequency distributions showed an overlapping response region for the OD of the 2 groups, with a total of 589 data (47.39%) (Figure 1).

Figure 1 Source: Own elaboration.

The thresholds of V^c for the ROC curve method were similar to each other and lower than all those of the standard method (I_y=0.186, MQD = 0.182, StdM1 = 0.229, StdM2=0.282, StdM3=0.194 and StdM4=0.230). Likewise, in the ROC curve, these values were located towards the center of the region of overlapping results, while they tended to be located towards the right in the standard method, favoring the specificity of the test. Sensitivity and specificity values were more balanced for the ROC curve methodology (sensitivity: 96%, specificity: 92%) than for the standard methodology (sensitivity: 67-87%, specificity: 99%). For the standard methodology, the most balanced equation was StdM3 with sensitivity of 87% and specificity of 98%.

The mean of the population of healthy individuals was set at µ_s=0.1226 and the variance for the condition of homoscedasticity was σ² ₅ =(0.0531)². Under heteroscedastic conditions, the variance for healthy observations was σ² ₅ =(0.0531)², while two values were considered for Chagasic populations: the variance of real data for Chagasic patients (heteroscedasticity-1), σ² _E1≡σ² _E=(0.2234)², and the pooled variance for groups of healthy and Chagasic individuals (heteroscedasticity-2), σ² _E2≡σ² _C=(0.1255)². The population means for Chagasic patients were established at P_E1: µ_E1=0.18535 ; P_E2: µ_E2=0.2481 ; P_E3: µ_E3= 0.3736 ; P_E4: µ _E4= 0.4991 ; P_E5: µ_E5=0.6246. As for real data, all these parameters correspond to OD readings.

The mean values obtained for V^c in the standard tests showed a fixed value for all three variance assumptions since they only depend on the population of healthy individuals. On the other hand, StdM2 and StdM4 showed the highest V^c, while those obtained with StdM3 were very close to those of the ROC curve in the second and third scenarios (Table 2).

Table 2

Equal variance assumption	Methodology	Scenario
		1	2	3	4	5
Homoscedasticity		0.1492	0.1804	0.2238	0.23	0.2301
	MQD	0.1526	0.1819	0.2238	0.23	0.2301
	StdM1	0.2273	0.2273	0.2273	0.2273	0.2273
	StdM2	0.2796	0.2796	0.2796	0.2796	0.2796
	StdM3	0.1953	0.1953	0.1953	0.1953	0.1953
	StdM4	0.2316	0.2316	0.2316	0.2316	0.2316
Heteroscedasticity-1		0.1994	0.2006	0.2057	0.2143	0.2223
	MQD	0.1719	0.1768	0.1925	0.2093	0.2215
	StdM1	0.2273	0.2273	0.2273	0.2273	0.2273
	StdM2	0.2796	0.2796	0.2796	0.2796	0.2796
	StdM3	0.1953	0.1953	0.1953	0.1953	0.1953
	StdM4	0.2316	0.2316	0.2316	0.2316	0.2316
Heteroscedasticity-2		0.1822	0.1865	0.2059	0.2226	0.2293
	MQD	0.1608	0.1739	0.2022	0.2227	0.2293
	StdM1	0.2273	0.2273	0.2273	0.2273	0.2273
	StdM2	0.2796	0.2796	0.2796	0.2796	0.2796
	StdM3	0.1953	0.1953	0.1953	0.1953	0.1953
StdM4	0.2316	0.2316	0.2316	0.2316	0.2316

I_Y: Youden index; MQD: minimum quadratic distance; StdM1 : standard methodology 1 ; StdM2: standard methodology 2; StdM3: standard methodology 3; StdM4: standard methodology 4.

The ROC curve methodologies showed V^c with little difference between them, which decreased when the mean of the Chagasic patients group moved away from the mean of healthy individuals and was higher under heteroscedasticity conditions. Likewise, V^c increased as a function of the mean of the Chagasic patients group (Table 2).

Homoscedasticity: For standard methodologies, specificity means were higher using StdM2, followed by StdM4 and Std1; StdM3 showed the lowest mean value. In addition, all estimators of this methodology revealed specificity values >90%. For ROC methodologies, I_Y and MQD showed similar specificity with a minimum of about 75% that increased as the average Chagasic patient population moved away from the mean of healthy individuals (Figure 2A).

Figure 2 Source: Own elaboration.

Sensitivity in all scenarios was higher in I_Y and MQD, ranging from 75% to 100%. Regarding standard methodologies, StdM3 showed the best behavior with 42% sensitivity in scenario-1, while StdM2 showed the lowest value with sensitivity of 5.21% in the same scenario (Figure 2B).

Heteroscedasticity-1: For specificity, both methodologies showed high values in all scenarios; the highest mean value was observed in StdM2 (approximately 100%), followed by StdM4 and StdM1 (values around 98%). For the ROC curve methodologies, heteroscedasticity affected MQD more than I_Y -the latter with 95% in scenario-1 and 99% in scenario-5. However, both showed a progressive increase according to the mean values of the Chagasic patient populations (Figure 2C).

For sensitivity, although the best behavior was obtained by MQD with a minimum value of 55% in scenario-1 and 97% in scenario-5, I_Y had a similar behavior. As for the standard methodology, StdM3 provided better sensitivity values and the mean values were similar to those of I_Y. The standard methodology that showed the lowest sensitivity values was StdM2, reaching values above 60% only from scenario-4 (Figure 2D). Heteroscedasticity-2: The mean specificity values were higher using the methodologies for StdM2 (99.95%), StdM1 and StdM4 (98% each). On the other hand, StdM3 caused a decrease in specificity by reaching an average of 92%. For the ROC methodologies, the I_Y had a better behavior than MQD (Figure 2E).

The best sensitivity values were observed with MQD, followed by I_Y and StdM3; the values were equal to MQD from scenario-3 onwards. The methodology that yielded the lowest mean sensitivity values was StdM2 (Figure 2F).

Homoscedasticity: StdM2 showed specificity=100% in almost all the simulated samples, followed by StdM4 and StdM1 with percentages around 50%. StdM3 showed specificity=100% in only 5% of cases. In the ROC curve methodologies, both showed a similar behavior, going from a low frequency of specificity=100% in the first two scenarios to a high percentage from scenario-3 onwards (87%) (Figure 3A).

Figure 3 Source: Own elaboration.

All methodologies showed lowfrequency sensitivity=100% in the first 2 scenarios; however, the percentages shown by I_Y and MQD were higher than the others. Similarly, all methodologies showed a notable increase in sensitivity 100% from scenario-3 onwards, except StdM2, with higher I_Y and StdM3 values (95%) (Figure 3B). Heteroscedasticity-1: The methodology that had the highest frequency of specificity= 100% was StdM2 with 98.6%; the others showed percentages ≤55% and the lowest value was observed in StdM3 with 5%. Both ROC curve methodologies showed a progressive increase but I_Y was less affected than MQD by heteroscedasticity (Figure 3C).

All methodologies obtained low percentages of sensitivity= 100% until scenario-4 and ≤50% in scenario-5. In the latter, the methodology that yielded the highest value was StdM3 (39.7%), followed by the ROC curve methodologies (35%); the one with the lowest value was StdM2 (Figure 3D).

Heteroscedasticity-2: The standard methodology with the highest accuracy for specificity=100% was StdM2 (98.6%); the others showed an accuracy <55%. Of the ROC curve methodologies, I_Y showed the best performance, although it presented low frequencies in the first 3 scenarios (42% maximum) and increased from scenario-4 onwards (Figure 3E).

With the exception of StdM2, the applied methodologies obtained values >10% sensitivity=100% from scenario-3 onwards, reaching a high percent-age in scenario-5. I_Y, MQD and StdM3 showed the best behavior; the latter had the highest values (Figure 3F).

Homoscedasticity: In the first 2 scenarios, no methodology yielded perfect-decision values, and the ROC curve methodologies showed the highest percentages from scenario-3 onwards; both I_Y and MQD showed the same values (between 83% and 99%). As for standard methodologies, only StdM2 showed similar values from scenario-4 onwards; the others reached a maximum of 55% decision-perfect (Figure 4A).

Figure 4 Source: Own elaboration.

Heteroscedasticity-1: From scenario-4 onwards, perfect decisions began to be observed. The highest percentages corresponded to I_Y and MQD (28%), which also showed twice the value of the methodologies StdMl and StdM4. In addition, few cases were observed with StdM3 (2%) (Figure 4B).

Heteroscedasticity-2: Perfect-decisions could be seen from scenario-3 onwards. The highest percentages corresponded to the ROC curve methodologies (figures between 60% and 96% without distinction between I_Y and MQD). Of the standard methodologies, only StdM2 reached figures >80%, while the lowest values were obtained by StdM3 (Figure 4C).