### Introduction

^{3}bacilli/mL) in the clinical (sputum) sample from an infected person for the positivity of the test result [2]. However, a smear test can turn up a negative result, if there is only a small number of bacilli in the sputum sample. Concomitant to diagnosis by a smear test, sputum samples are routinely sent for culturing in Lowenstein–Jensen (L–J) medium. Unfortunately, in TB it takes 3–4 weeks for colonies to develop, during which time the disease becomes stable in the infected individual. Indeed, this test (culture test, cited herein) is regarded as the gold standard, because viable bacilli in the sputum sample grow into colonies in the L–Jmedium [3].

*false negative*(FN) cases (results where the smear test is negative and the culture test is positive) lead to cryptic invasions of bacilli that progress toward the establishment of the disease, as a negative smear test result prompts the decision for nontreatment (to control the infection), and when culture test results become available later (after 1 month or so), the person has already contracted the disease [4]. Therefore, to prevent this, clinicians usually recommend for patients to undergo empirical TB chemotherapy. When a patient is on chemotherapy, he/she may sometimes have a sufficient amount of dead bacilli to yield “smear test positivity along with culture test negativity”, giving rise to

*false positive*(FP) cases. The other two obvious possibilities are the positivity of both tests, i.e.,

*true positive*(TP) cases (smear test positivity and culture test positivity) and the negative results of both tests, i.e.,

*true negative*(TN) cases (smear test negativity and culture test negativity), which can be suitably taken care of by the clinician. The confusing ambivalence of FN and FP cases creates clinical ambiguity, i.e., persons with FN cases are not given chemotherapy unless multiple comorbidities are evident, leading to the establishment of the disease. By contrast, FP patients, particularly those with a small number of bacilli, are unnecessarily subjected to a rigorous regimen of chemotherapy. In addition, FP cases may arise from infection from mycobacteria other than tuberculosis (MOTT).

### 1.1. Why Bayesian analysis?

*post hoc*trial from 572 samples; in addition, the digital assessment of the credibility of the smear test could be performed. As the culture test also has its degree of fallibility, i.e., unviable bacilli as discussed above may give rise to positivity in the smear test and negativity in the culture test, its quantification also remains an obvious quest.

*a priori probability*or

*prevalence*or the prevalence of disease in the targeted population) is determined before using the data. Prevalence is computed as (TP + FN)/

*N*, where

*N*is the total number of samples. Additionally, several test statistics are associated in the analysis:

The

*sensitivity*(*TP rate*) is the proportion of people with the disease who will have positive smear test results, computed by [TP/(TP + FN)]. This value is the ability of the smear test to detect the infection status, when it is truly present, i.e., it is the probability of a positive test result, given that the samples were taken from sick individuals.The

*specificity*(*TN rate*) is the proportion of people without the disease who will have negative smear test results, obtained by [TN/(FP + TN)]. This value is the ability of the smear test to yield a negative result with samples from disease-free individuals, i.e., it is the probability of a negative test result.The

*FP rate*is the probability of errors in the culture test, computed as [FP/(FP + TN)].The

*FN rate*is the probability of errors in the smear test, computed as FN/(TP + FN).The

*positive predictivity*is the*posttest probability*of the disease that yielded a positive test result, or the probability of the portion of people with positive test results who actually had the disease, computed as [TP/(TP + FP)].The

*negative predictivity*is the*posttest probability*of the disease that gave a negative test result, or the probability of the proportion of people with negative test results who actually did not have the disease, computed as [TN/(FN + TN)].The

*diagnostic accuracy*(inherent validity or predictive validity) is the ability of the smear test to be correctly positive or negative, among the binary results of the culture test, computed as [(TP + TN)/*N*]. Additionally, this value estimates the accuracy of smear and culture tests together.The

*positive likelihood ratio*(LR) is the ratio between the TP rate and the FP rate, computed as [sensitivity/(1 − specificity)], when the smear test result was positive.The

*negative LR*is the ratio between the FN rate and the TN rate, computed as [(1 − sensitivity)/specificity], when the smear test result was negative. In fact, the larger is the positive LR value, the greater the likelihood of infection, and similarly, the lesser is the negative LR value, the lesser the likelihood of infection in a population.The

*a posteriori probability*is the value from posttest arithmetic computation of the data set for the diagnostic efficiency, and it clarifies the dependability of each test independently, with a numerical probability value in arriving at the truth, i.e., the sought-after conclusions from both tests.The area under the receiver operating characteristic (ROC) curve, drawn with values of

*sensitivity*and 1 −*specificity*, gives a graphical analysis for diagnostic efficiency. The graphical method additionally examines the predictive capability as another value of*a posteriori*probability, independent of the arithmetic computation.

### Materials and Methods

*g*for 20 minutes; the supernatant was discarded, and the residue was washed three times with sterile distilled water [12]. A smear was prepared using two droplets of the suspension on a glass slide, and this was air dried; drops of 1% carbol-fuchsin were poured onto the smear. Next, the slide was heated gently and was allowed to stand for 10 minutes for the coloration of the smear. The slide was gently washed with water and was decolorized with drops of 25% H

_{2}SO

_{4}. The smear was further counterstained with 0.1% methylene blue solution for 1 minute, and was gently washed before air-drying. At least 200–300 fields under an oil immersion objective were screened for red/pink AFB, and results were recorded as 0–1, 1–9, or 10–99 or more AFB per field (Figure 1). Results were reported, viewing under 100 fields, as follows: (1) negative with no red/pink bacteria, (2) scanty for 1–9 bacilli, (3) + for 10–99 bacilli, (4) ++ for more than 100 bacilli, or (5) +++ for bacilli more than 100 per field [13]. Furthermore, duplicate tubes of the L–J medium were inoculated from the prepared suspension and were incubated at 37° C for the growth of colonies that were checked later, in 6–8 weeks with weekly intervals.

### Results

*N*=1.0) obtained in a period of 19 months (March 2010 to September 2012) were performed with a smear test and culture test, in a parallel manner. It was found that from a total of 572 samples (

*N*= 1.0), 33 samples (0.05769) were TP cases; 22 samples (0.03846) were FP cases; 62 samples (0.10839) were FN cases; and 455 samples (0.79545) were TN cases. It was evident that there was mismatch of results in the two tests, so FN and FP cases arose (Table 1). Applying the Bayesian concept with the recorded data (Table 1), several other test statistics described earlier could be computed for additional probability values, with 95% confidence interval (CI) values (Table 2).

###
3.1. Computation of *a posteriori* probability mathematically and by ROC curve analysis

*a posteriori*probability or

*P*(

*E*

_{1}|

*E*), the probability value of a sample to be truly positive, can be calculated using the Bayesian formula, where

*E*is the event that the smear test result is positive;

*E*

_{1}is the event that the result of the culture test involving the same sample is positive;

*E*

_{1}′ is the partition of the sample space for all clinical samples from noninfected individuals, and it is a hypothetical value. This yields several probability values:

*a posteriori*probability, substituting the above values in its formula, we obtain

*a posteriori*probability were determined before drawing the graph for the ROC curve, and these values gave an idea that for all possible values of population and prevalence, the sensitivity patterns changed with a mean present value of 0.30 ± 0.13 (the original sensitivity value was 0.347), but the specificity values remained unchanged at 0.99 throughout. Values of

*a posteriori*probability also remained in the range at the mean value of 0.59 ± 0.05 (the original

*a posteriori*probability value was 0.6614) (Table 3).

*a posteriori*probability by the ROC curve (Figure 2), which was drawn by joining the cut-points represented by six values of each: sensitivity versus 1 − specificity; and the diagonal chance line, (45° line) through the coordinates (0, 0) and (1, 1), was drawn as the lower limit. The area of the upper triangle above the 45° diagonal line (called the chance line) was taken as the total value = 1.0, out of which the AUC (area under the ROC curve) was found to be 0.62 (95% CI, 0.473–0.767), determined by using the trapezoidal rule [14]. This means that the smear test has a 62% chance of correctly distinguishing a sample from an infected person and a sample from a noninfected individual. This is the second value of

*a posteriori*probability, the first one being 0.6614 or 66.14%.

### Discussion

*a priori*probability value of the test was 0.16608 or 16.6%, computed according to Zhou et al [15]. There were 95 positives (TP and FN cases) out of the total sputum samples, based on culture test results. Moreover, from both types of false cases (FN and FP), it was clear that each test was insufficient for the prognosis.

*Positive predictivity*is the conditional probability that a patient had the disease, given that the smear test result was positive. Similarly,

*negative predictivity*is the conditional probability, where the sample does not have the infection, given that the smear test result was negative. The

*positive predictivity*value, 0.6, and the

*negative predictivity*value, 0.88, computed herein are far from the absolute values of 0.4 and 0.12, respectively. However, both are dependable in terms of determining the

*prevalence*of the disease [16]. In other words, the

*negative predictivity*value is dependable for the smear test.

*Mycobacterium paratuberculosis*[17], including MOTT or prior Bacillus Calmette–Guérin vaccination, as noted [18]. A high value of FN cases (10.84%) should actually induce a progress in the infection that is present in the body, and it is matter of concern because TB chemotherapy has not been initiated in FN cases. Obviously, error in TB prognosis would cause an individual to become an outcast, because of drug-resistant infections, especially due to FN cases. Nevertheless, samples are concentrated before diagnostic steps are undertaken by default. Indeed, at least, 5–10 × 10

^{3}bacilli/mL must be present in a sample for a smear test result to be considered positive [2,19]. Thus, the insufficiency of the smear test could be attributed to the small number of bacilli in the sample. The pragmatic approach to TB prognosis would definitely be the nucleic acid amplification test with isolation of DNA from bacilli, meant for drug-resistant bacilli, which is not usually followed in resource-limited settings. Thus, a smear test would be inadequate in distinguishing a sick from a nonsick person with latent TB, as the latter would promote evasive FN or FP cases. The dependability of the culture test is challenged by the 22 FP cases; in other words, this test is dependable as the gold standard for 96.15% only. Virtually, the probability of the culture test result being positive would never be zero, but the probability of the smear test to be totally negative cannot be ruled out, when each sample contains an insufficient amount of bacilli. Moreover, the FP cases are 22 (3.85%), which suggests that the erroneous smear test results may be attributable to a patient undergoing chemotherapy, leading to unviable bacilli for the culture test, but the smear test would be positive because of the presence of dead bacilli. Thus, the FP rate is 0.046 or 4.6%.

*a posteriori*probability. The graphical representation value is 0.62 and the arithmetic value is 0.6648. Both values are in close proximity with a distance of 0.4% in derivation. Thus, statistically this signifies the dependability of the smear test with this binocular vision. Moreover, the values of associated test statistics generated in the Bayesian analysis clump around the data set facilitate a multiple evaluation of the ambivalence. Thus, this analysis would provide a methodological framework of quantitative assessment of two test results of diagnosis of pulmonary tuberculosis.

*post hoc*analysis of the data set generating two values of

*a posteriori*probability, falling within 62.0% and 66.48%, however, neither advocates strongly for, nor undermines both diagnostic methods. It should be noted, however, the recent outbreak of multidrug-resistant TB worldwide must be controlled with more rigorous measures, for which both these methods are insufficient.