APPENDIX

The Appendix consists of the following:

  1. Immunohistochemistry
  2. Bayesian methodology outline
  3. Generalities
  4. Specification of prior distributions
  5. Likelihood function
  6. Posterior distributions
  7. Appendix References
  8. Appendix Table 1
  9. Appendix Table 2
  10. Appendix Table 3

Immunohistochemistry

Sections were deparaffinized in xylene and rehydrated through graded concentrations of alcohol. Slides were then heated in a microwave oven for two circles of 15 minute each at 300W, in citrate buffer for antigen retrieval. Endogenous peroxidase’s activity was blocked with H2O2 solution in methanol (0,01M), for 30 minutes. After washing with PBS for 5 minutes, the primary antibodies (podoplanin, dilution 1:2000, D2-40,SIG-730, Signet Laboratories, dilution 1:40, CD34, AM353-5M, Biogenex USA, ready-to-use) were applied for 30 minutes in a humid chamber at room temperature. Then sections were washed for 10 minutes with PBS, and were visualized with the EnVision system using DAB (diaminobezidinetetrahydrochloride; chromogen). Finally, all sections were counterstained with hematoxylin.

Bayesian Methodology

Generalities

Let u, v, w and x be the observed number of vessels that are stained or unstained with either of the antibodies, as tabulated in a 2 by 2 contingency table. Let Y1, Y2,Y3 and Y4 be the corresponding true lymphatic vessels out of the observed counts. Y1, Y2,Y3 and Y4 represent the information that is missing when there is no gold standard and therefore have been termed “latent data” (1) (Appendix Table 1). Analyses using such data have been referred to as “latent class analyses” (2).

Let be the unknown true prevalence of the lymphatic vessels in the sections, S1, S2 be the unknown true sensitivities of podoplanin and D2-40 respectively, and C1, C2 the corresponding unknown true specificities of the two antibodies. It is possible to draw inferences on these unknown quantities using a Bayesian approach to this problem, as proposed by Joseph et al (3). The basic idea behind the Bayesian approach is to create a statistical model to link data to parameters and then formulate prior information about all unknown quantities. These two sources of information are combined through the likelihood function using Bayes’ theorem. This results to a posterior distribution allowing us to derive inferences on all parameters.

Specification of prior distributions

Following others (3-5), prior information in the form of a beta density was assumed for each of the five parameters. A random variable , has a beta distribution with parameters (, ) if it has a probability density given by

,, .

The parameters of the different priors and the results of the secondary analyses are summarized in Table 2 of the Appendix.

Likelihood function

The likelihood function of our data would then be

Posterior distributions

The posterior density is proportional to the likelihood function times the prior distribution. Let , , and represent the prior beta parameters for , Si, and Ci respectively (i=1 for podoplanin and i=2 for D2-40). Hence the posterior distribution of prevalence, sensitivities and specificities of each test will be in the Beta form.

Estimates of the prevalence of the lymphatics and the specificities and sensitivities of the two antibodies were obtained using Markov Chain Monte Carlo methods with Gibbs sampling, as previously proposed (3). Convergence of the chain was assessed using the coda package in R. We run one chain with a burn-in of 5000 iterations (to ensure convergence). The total number of iterations was 20 000.

The assumption that the two tests (antibodies) are statistically independent may not hold especially when both antibodies have the same biologic target (6). In a second set of analyses we used a model that allows for correlation between the two diagnostic tests. Technicalities are described in ref (6). In these calculations we used the same prior information for the prevalence, sensitivity and specificity, as described in the main analysis. The posterior distributions of the sensitivities and specificities of the two tests allowing for correlation between them are shown in Appendix Table 3.

Appendix References

  1. Tanner MA, Wong WH. The calculation of posterior densities by data augmentation. (with discussion) J Am Stat Assoc 1987;82:528-550
  2. Kaldor J, Clayton D. Latent class analysis in chronic disease epidemiology. Stat Med 1985;4:327-335
  3. Joseph L, Gyorkos TW, Coupal L. Bayesian estimation of disease prevalence and the parameters of diagnostic tests in the absence of a gold standard. Am J Epidemiol 1995;141:263-72.
  4. Gustafson P, LeND, Saskin R. Case control analysis with partial knowledge of exposure misclassifications probabilities. Biometrics 2001;57:598-609
  5. Johnson WO, Gastwirth JL and Pearson LM. Screening without a gold standard: The hui-walter paradigm revisited. Am J Epidemiol 2001;153:921-924
  6. Dendukuri N, Lawrence J. Bayesian approaches to modeling the conditional dependence between multiple diagnostic tests. Biometrics 2001;57:158-167.

AppendixTable 1: Tabulation of staining results for the two antibodies.

D2-40, stained / D2-40, unstained
Podoplanin, stained / u (Y1) / v (Y2)
Podoplanin, unstained / w (Y3) / x (Y4)

In the table, u, v, w, and x are the observed counts of the vessels stained with the two antibodies. Let Y1, Y2, Y3 and Y4 be the true lymphatic vessels out of the respective observed counts u, v, w, and x. Y1, Y2, Y3 and Y4 are unknown (latent) quantities.

Appendix Table 2: Sensitivity analyses.

Description of prior distribution
(indicative range [%]; density) / Posterior median (95% Credible Interval) (%)
Sensitivity / Specificity / Sensitivity / Specificity / Positive predictive value
Podoplanin / D2-40 / Podoplanin / D2-40 / Podoplanin / D2-40 / Podoplanin / D2-40 / Podoplanin / D2-40
60-100;
Beta(12,3) / 60-100;
Beta(12,3) / 90-100;
Beta(71.3,3.75) / 80-100;
Beta(31.5,3.5) / 92.6
(86.1-97.9) / 97.3
(94.9-99.2) / 99.7
(99.5-99.9) / 98.8
(98.3-99.5) / 96.3
(94.2-98.6) / 88.9
(83.9-95.7)
60-100;
Beta(12,3) / 60-100;
Beta(12,3) / 80-100;
Beta(31.5,3.5) / 80-100;
Beta(31.5,3.5) / 92.4
(85.8-97.8) / 97.2
(94.8-99.2) / 99.7
(99.5-99.9) / 98.9
(98.3-99.6) / 96.4
(94.2-98.7) / 89.2
(84.1-96.0)
0-100;
Beta(1,1) / 0-100;
Beta(1,1) / 90-100;
Beta(71.3,3.75) / 80-100;
Beta(31.5,3.5) / 96.2
(87.2-99.8) / 98.8
(95.8-99.9) / 99.5
(99.4-99.8) / 98.5
(98.1-99.3) / 94.9
(93.2-97.8) / 85.8
(82.0-94.6)
65-95;
Beta(21.9,5.5) / 65-95;
Beta(21.9,5.5) / 90-100;
Beta(71.3,3.75) / 80-100;
Beta(31.5,3.5) / 90.3
(85.1-95.9) / 96.3
(94.3-98.4) / 99.7
(99.5-99.9) / 99.1
(98.5-99.6) / 97.2
(95.0-99.0) / 91.2
(85.8-96.7)

The first row of the table shows the main analyses. The other rows are additional secondary explorations that document the robustness of the findings from the main analysis. Our ignorance about the prevalence was approximated by a uniform distribution on the range from 0 to 1 which is equivalent to a Beta density with parameters =1 and =1.

Appendix Table 3: Second set of analyses allowing for correlation between the two tests.

Description of prior distribution
(indicative range [%]; density) / Posterior median (95% Credible Interval) (%)
Covariances / Sensitivity / Specificity
Within stained vessels / Within unstained vessels / Podoplanin / D2-40 / Podoplanin / D2-40
0-80;
Beta(2, 3) / 0-80;
Beta(2, 3) / 82.6
(62.7-94.7) / 90.0
(74.0-97.6) / 95.6
(92.0-98.9) / 94.8
(90.9-98.3)
0-100;
Beta(1, 1) / 0-100;
Beta(1, 1) / 80.4
(60.1-94.3) / 85.7
(68.0-95.8) / 95.5
(92.1-98.3) / 94.6
(90.9-97.7)
0-50;
Beta(2.75, 8.25) / 0-50;
Beta(2.75, 8.25) / 90.8
(83.4-97.1) / 96.1
(91.0-98.9) / 99.4
(98.5-99.8) / 98.6
(97.7-99.4)