Results

Pathology Sampling Adequacy Analysis

JamoviTest::pathsampling(    data = data,    analysisContext = "omentum",    totalSamples = cassette_number,    firstDetection = first_cassette_tumor_identified,    showBinomialModel = TRUE,    showBootstrap = TRUE,    showDetectionCurve = TRUE,    showSensitivityCI = TRUE,    showClinicalSummary = TRUE,    showProbabilityExplanation = TRUE,    showKeyResults = TRUE,    showRecommendText = TRUE,    showInterpretText = TRUE,    showReferencesText = TRUE)

⚠️ SELECTION BIAS WARNING: Bootstrap resamples only DETECTED cases. Sensitivity estimates assume 100% of true positives were detected and will be OVEROPTIMISTIC if any cases were missed. These are CONDITIONAL estimates (probability of detection given lesion was eventually found). For unbiased population-level sensitivity, you must provide gold-standard total positive count or use external validation. Interpret sample size recommendations conservatively.

Data Summary
Measure	Value
Total cases supplied	1096
Cases analyzed	1096
Total samples (input)	4395 (recorded)
Total samples analyzed	4162 (up to first detection)
Mean samples per analyzed case	4.01
Median first detection	1
Cases without detected lesion	1035
Binomial estimator	Geometric MLE (first detection only)

📊 Two Ways to Measure Detection Performance

This analysis reports probabilities in two ways, depending on the clinical question:

1️⃣ Conditional Detection (Sensitivity) - "If metastasis is present"

Clinical question: If metastasis is truly present, how many blocks do I need to detect it?

Best used for: Setting minimum blocks per tumour-positive case, validating adequacy targets, or counselling surgeons on block counts required to avoid false reassurance.

Formula: P(detect | metastasis present) = 1 - (1-q)ⁿ

In Your Data: Among 61 cases with detected metastasis (q = 0.560):

With 3 samples: Detects 91.5% of positive cases
With 5 samples: Detects 98.3% of positive cases
With 10 samples: Detects 100.0% of positive cases

Clinical Use: "How many samples do I need to confidently rule out metastasis?" This is the probability shown in the Diagnostic Yield Curve.

2️⃣ Population-Level Detection - "Overall detection rate"

Clinical question: Across all specimens submitted (positive + negative), how often do we detect tumour with n blocks?

Best used for: Monitoring service-level performance, comparing surgeons/protocols, or highlighting when low prevalence—not sampling—limits detection.

Formula: P(detect overall) = Prevalence × Sensitivity = π × [1 - (1-q)ⁿ]

In Your Data: Observed prevalence = 5.6% (61/1096 cases had metastasis):

With 3 samples: Detects metastasis in 5.1% of all specimens
With 5 samples: Detects metastasis in 5.5% of all specimens
With 10 samples: Detects metastasis in 5.6% of all specimens

Clinical Use: "What percentage of incoming specimens will test positive?" This is useful for workload planning and quality metrics.

⚠️ Important: These are fundamentally different quantities!

• Conditional (sensitivity) assumes metastasis is present

• Population-level includes cases without metastasis

The ratio between them equals the prevalence (5.6% in your data). This module focuses on conditional probability (sensitivity) because that's what determines sampling adequacy.

Target confidence: 95%%

Recommended minimum samples: 4 (Bootstrap) achieving 96.7%% sensitivity (10000 iterations; 95% CI 62.2%-100.0%).

Model comparison:

Bootstrap: Meets target at 4 samples (96.7%) (10000 iterations; 95% CI 62.2%-100.0%)
Binomial: Meets target at 4 samples (96.2%) (p = 0.5596)
Empirical: Meets target at 4 samples (96.7%) (Conditional probability among 61 positive cases)

Clinical Summary

Analysis Overview:

Pathology sampling adequacy analysis of 1096 cases to determine the minimum number of samples required to reliably detect lesions.

Key Findings:

Detection probability per sample: 56.0%
Recommended samples for 95%% sensitivity: 4 samples
Bootstrap validation (10k iterations): 96.7% sensitivity (95%% CI: 91.8%-100.0%)
First 3 samples detected: 88.5% of all cases

Copy-ready text for reports:

"Sampling adequacy analysis of 1096 cases showed a per-sample detection probability of 56.0%. To achieve 95%% sensitivity, a minimum of 4 samples is recommended based on binomial probability modeling and bootstrap validation (10000 iterations, 95%% CI: 91.8%%-100.0%%). Observed data showed 88.5% of lesions detected within first 3 samples."

Clinical Recommendation: Submit a minimum of 4 samples to ensure adequate diagnostic sensitivity in routine practice.

Binomial Probability Model (Conditional Sensitivity)

Estimated per-sample detection probability: q = 0.5596

Estimation Method: Geometric MLE (first detection only)

Based on 61 positive cases with mean first detection at sample 1.79.

Formula: P(detect ≥ 1 in n samples | lesion present) = 1 - (1-q)ⁿ

Note: This estimates sensitivity (detection given lesion is present), not population-level detection rate.

Binomial Model Predictions
Number of Samples	Sensitivity (P(detect \| present))	Marginal Gain
1	56.0%	56.0%
2	80.6%	24.6%
3	91.5%	10.9%
4	96.2%	4.8%
5	98.3%	2.1%
6	99.3%	0.9%
7	99.7%	0.4%
8	99.9%	0.2%
9	99.9%	0.1%
10	100.0%	0.0%

Minimum Samples for Target Confidence
Target Confidence	Minimum Samples Required
80.0%	2
90.0%	3
95.0%	4
99.0%	6

Bootstrap Resampling Analysis

Empirical sensitivity estimates based on 10000 bootstrap iterations.

This method resamples cases with replacement to estimate sensitivity and confidence intervals without parametric assumptions.

Reference: Skala SL, Hagemann IS. Int J Gynecol Pathol. 2015;34(4):374-378.

Bootstrap Sensitivity Estimates (Conditional)
Number of Samples	Mean Sensitivity (given present)	95% CI Lower	95% CI Upper
1	57.4%	44.3%	68.9%
2	78.6%	67.2%	88.5%
3	88.5%	80.3%	95.1%
4	96.7%	91.8%	100.0%
5	100.0%	100.0%	100.0%
6	100.0%	100.0%	100.0%
7	100.0%	100.0%	100.0%
8	100.0%	100.0%	100.0%
9	100.0%	100.0%	100.0%
10	100.0%	100.0%	100.0%

Diagnostic Yield Curve

Sensitivity with Confidence Intervals

Omentum Sampling Recommendations

Recommended minimum samples for 95% sensitivity: 4 (based on Bootstrap model) (10000 iterations; 95% CI 62.2%-100.0%).

This plan achieves an estimated sensitivity of 96.7% using the Empirical resampling of cases.

Observed cumulative detection:

1 sample: 57.4%%
2 samples: 78.7%%
3 samples: 88.5%%
4 samples: 96.7%%

Statistical Interpretation

This analysis addresses the question: How many tissue samples are necessary to reliably detect a lesion?

Key Concepts:

Sensitivity: Probability of detecting the lesion if it is present
Cumulative detection probability: Increases with each additional sample
Diminishing returns: At some point, additional samples provide minimal gain
Target confidence: User-selected detection probability used to determine minimum sampling

Assumptions and Limitations:

Complete cohorts: Record every case in the dataset, including those with no detected lesion. Use NA for the first-detection column when a lesion is never observed so the model can treat the case as a confirmed negative.
Binomial model: Assumes the per-sample detection probability is constant across all cases and samples. Heterogeneous disease burden can violate this assumption.
Hypergeometric model: Requires per-case totals and positives. It still simplifies within-case variability; the beta-binomial model relaxes this by allowing overdispersion.
Residual false negatives: Even when all submitted samples are examined, lesions may still be missed. The reported sensitivities therefore represent an upper bound conditional on the submitted material.
Bootstrap CIs: Assume the analyzed sample is representative of the broader population of interest.

Statistical Methods & References

Methods:

Binomial probability distribution for detection modeling
Hypergeometric probability distribution for finite population sampling
Beta-Binomial probability distribution for sampling with overdispersion
Bootstrap resampling with replacement (10,000 iterations default)
95% confidence intervals using percentile method
Lymph node ratio (LNR) classification with X-tile optimized cutpoints
Non-parametric effect size measures (Cliff's delta, Hodges-Lehmann)

Key References:

Lymph Node Adequacy & LNR Classification:
- Tomlinson JS, et al. Accuracy of Staging Node-Negative Pancreas Cancer. Arch Surg. 2007;142(8):767-774. minELN=15, 8-month survival (SEER n=1,150, HR=0.63)
- Pu N, et al. An Artificial Neural Network Improves Prediction of Observed Survival in Patients with Pancreatic Cancer. J Natl Compr Canc Netw. 2021;19(9):1029-1036. minELN=12, binomial model, LNR thresholds 0.1 & 0.3 (n=1,837)
- Yoon SJ, et al. Optimal Number of Lymph Nodes Retrieved to Lower False N0 Risk. Ann Surg Oncol. 2025. doi:10.1245/s10434-025-18029-7 minELN=12-16, false N0 modeling, Korean dual-cohort (n=1,252)
Sampling Adequacy Methods:
- Skala SL, Hagemann IS. Pathologic Sampling of the Omentum. Int J Gynecol Pathol. 2015;34(4):374-378.
- Gönen M, et al. Nodal Staging Score for Colon Cancer. J Clin Oncol. 2009;27(36):6166-6171.
- Zhou J, et al. Beta-binomial model for lymph node yield. Front Oncol. 2022;12:872527.
- Buderer NM. Statistical methodology for diagnostic test sample size. Stat Med. 1996;15(6):649-652.
Distribution Pattern Analysis:
- Ates D, et al. Lymphovascular Space Invasion in Endometrial Cancer. Mod Pathol. 2025;38:100885. Single vs summed cassette distribution patterns

Pathology Sampling Adequacy Analysis

JamoviTest::pathsampling(    data = data,    analysisContext = "omentum",    totalSamples = cassette_number,    firstDetection = first_cassette_tumor_identified,    sampleType = Location,    showBinomialModel = TRUE,    showBootstrap = TRUE,    showDetectionCurve = TRUE,    showSensitivityCI = TRUE,    showClinicalSummary = TRUE,    showProbabilityExplanation = TRUE,    showKeyResults = TRUE,    showRecommendText = TRUE,    showInterpretText = TRUE,    showReferencesText = TRUE,    duplicate = 2)

Data Summary
Measure	Value
Total cases supplied	1096
Cases analyzed	1096
Total samples (input)	4395 (recorded)
Total samples analyzed	4162 (up to first detection)
Mean samples per analyzed case	4.01
Median first detection	1
Cases without detected lesion	1035
Binomial estimator	Geometric MLE (first detection only)

📊 Two Ways to Measure Detection Performance

This analysis reports probabilities in two ways, depending on the clinical question:

1️⃣ Conditional Detection (Sensitivity) - "If metastasis is present"

Clinical question: If metastasis is truly present, how many blocks do I need to detect it?

Best used for: Setting minimum blocks per tumour-positive case, validating adequacy targets, or counselling surgeons on block counts required to avoid false reassurance.

Formula: P(detect | metastasis present) = 1 - (1-q)ⁿ

In Your Data: Among 61 cases with detected metastasis (q = 0.560):

With 3 samples: Detects 91.5% of positive cases
With 5 samples: Detects 98.3% of positive cases
With 10 samples: Detects 100.0% of positive cases

Clinical Use: "How many samples do I need to confidently rule out metastasis?" This is the probability shown in the Diagnostic Yield Curve.

2️⃣ Population-Level Detection - "Overall detection rate"

Clinical question: Across all specimens submitted (positive + negative), how often do we detect tumour with n blocks?

Best used for: Monitoring service-level performance, comparing surgeons/protocols, or highlighting when low prevalence—not sampling—limits detection.

Formula: P(detect overall) = Prevalence × Sensitivity = π × [1 - (1-q)ⁿ]

In Your Data: Observed prevalence = 5.6% (61/1096 cases had metastasis):

With 3 samples: Detects metastasis in 5.1% of all specimens
With 5 samples: Detects metastasis in 5.5% of all specimens
With 10 samples: Detects metastasis in 5.6% of all specimens

Clinical Use: "What percentage of incoming specimens will test positive?" This is useful for workload planning and quality metrics.

⚠️ Important: These are fundamentally different quantities!

• Conditional (sensitivity) assumes metastasis is present

• Population-level includes cases without metastasis

The ratio between them equals the prevalence (5.6% in your data). This module focuses on conditional probability (sensitivity) because that's what determines sampling adequacy.

Target confidence: 95%%

Recommended minimum samples: 4 (Bootstrap) achieving 96.7%% sensitivity (10000 iterations; 95% CI 62.2%-100.0%).

Model comparison:

Bootstrap: Meets target at 4 samples (96.7%) (10000 iterations; 95% CI 62.2%-100.0%)
Binomial: Meets target at 4 samples (96.2%) (p = 0.5596)
Empirical: Meets target at 4 samples (96.7%) (Conditional probability among 61 positive cases)

Clinical Summary

Analysis Overview:

Pathology sampling adequacy analysis of 1096 cases to determine the minimum number of samples required to reliably detect lesions.

Key Findings:

Detection probability per sample: 56.0%
Recommended samples for 95%% sensitivity: 4 samples
Bootstrap validation (10k iterations): 96.7% sensitivity (95%% CI: 91.8%-100.0%)
First 3 samples detected: 88.5% of all cases

Copy-ready text for reports:

Clinical Recommendation: Submit a minimum of 4 samples to ensure adequate diagnostic sensitivity in routine practice.

Binomial Probability Model (Conditional Sensitivity)

Estimated per-sample detection probability: q = 0.5596

Estimation Method: Geometric MLE (first detection only)

Based on 61 positive cases with mean first detection at sample 1.79.

Formula: P(detect ≥ 1 in n samples | lesion present) = 1 - (1-q)ⁿ

Note: This estimates sensitivity (detection given lesion is present), not population-level detection rate.

Binomial Model Predictions
Number of Samples	Sensitivity (P(detect \| present))	Marginal Gain
1	56.0%	56.0%
2	80.6%	24.6%
3	91.5%	10.9%
4	96.2%	4.8%
5	98.3%	2.1%
6	99.3%	0.9%
7	99.7%	0.4%
8	99.9%	0.2%
9	99.9%	0.1%
10	100.0%	0.0%

Minimum Samples for Target Confidence
Target Confidence	Minimum Samples Required
80.0%	2
90.0%	3
95.0%	4
99.0%	6

Bootstrap Resampling Analysis

Empirical sensitivity estimates based on 10000 bootstrap iterations.

This method resamples cases with replacement to estimate sensitivity and confidence intervals without parametric assumptions.

Reference: Skala SL, Hagemann IS. Int J Gynecol Pathol. 2015;34(4):374-378.

Bootstrap Sensitivity Estimates (Conditional)
Number of Samples	Mean Sensitivity (given present)	95% CI Lower	95% CI Upper
1	57.4%	44.3%	68.9%
2	78.6%	67.2%	88.5%
3	88.5%	80.3%	95.1%
4	96.7%	91.8%	100.0%
5	100.0%	100.0%	100.0%
6	100.0%	100.0%	100.0%
7	100.0%	100.0%	100.0%
8	100.0%	100.0%	100.0%
9	100.0%	100.0%	100.0%
10	100.0%	100.0%	100.0%

Diagnostic Yield Curve

Sensitivity with Confidence Intervals

Omentum Sampling Recommendations

Recommended minimum samples for 95% sensitivity: 4 (based on Bootstrap model) (10000 iterations; 95% CI 62.2%-100.0%).

This plan achieves an estimated sensitivity of 96.7% using the Empirical resampling of cases.

Observed cumulative detection:

1 sample: 57.4%%
2 samples: 78.7%%
3 samples: 88.5%%
4 samples: 96.7%%

Statistical Interpretation

This analysis addresses the question: How many tissue samples are necessary to reliably detect a lesion?

Key Concepts:

Sensitivity: Probability of detecting the lesion if it is present
Cumulative detection probability: Increases with each additional sample
Diminishing returns: At some point, additional samples provide minimal gain
Target confidence: User-selected detection probability used to determine minimum sampling

Assumptions and Limitations:

Complete cohorts: Record every case in the dataset, including those with no detected lesion. Use NA for the first-detection column when a lesion is never observed so the model can treat the case as a confirmed negative.
Binomial model: Assumes the per-sample detection probability is constant across all cases and samples. Heterogeneous disease burden can violate this assumption.
Hypergeometric model: Requires per-case totals and positives. It still simplifies within-case variability; the beta-binomial model relaxes this by allowing overdispersion.
Residual false negatives: Even when all submitted samples are examined, lesions may still be missed. The reported sensitivities therefore represent an upper bound conditional on the submitted material.
Bootstrap CIs: Assume the analyzed sample is representative of the broader population of interest.

Statistical Methods & References

Methods:

Binomial probability distribution for detection modeling
Hypergeometric probability distribution for finite population sampling
Beta-Binomial probability distribution for sampling with overdispersion
Bootstrap resampling with replacement (10,000 iterations default)
95% confidence intervals using percentile method
Lymph node ratio (LNR) classification with X-tile optimized cutpoints
Non-parametric effect size measures (Cliff's delta, Hodges-Lehmann)

Key References:

Lymph Node Adequacy & LNR Classification:
- Tomlinson JS, et al. Accuracy of Staging Node-Negative Pancreas Cancer. Arch Surg. 2007;142(8):767-774. minELN=15, 8-month survival (SEER n=1,150, HR=0.63)
- Pu N, et al. An Artificial Neural Network Improves Prediction of Observed Survival in Patients with Pancreatic Cancer. J Natl Compr Canc Netw. 2021;19(9):1029-1036. minELN=12, binomial model, LNR thresholds 0.1 & 0.3 (n=1,837)
- Yoon SJ, et al. Optimal Number of Lymph Nodes Retrieved to Lower False N0 Risk. Ann Surg Oncol. 2025. doi:10.1245/s10434-025-18029-7 minELN=12-16, false N0 modeling, Korean dual-cohort (n=1,252)
Sampling Adequacy Methods:
- Skala SL, Hagemann IS. Pathologic Sampling of the Omentum. Int J Gynecol Pathol. 2015;34(4):374-378.
- Gönen M, et al. Nodal Staging Score for Colon Cancer. J Clin Oncol. 2009;27(36):6166-6171.
- Zhou J, et al. Beta-binomial model for lymph node yield. Front Oncol. 2022;12:872527.
- Buderer NM. Statistical methodology for diagnostic test sample size. Stat Med. 1996;15(6):649-652.
Distribution Pattern Analysis:
- Ates D, et al. Lymphovascular Space Invasion in Endometrial Cancer. Mod Pathol. 2025;38:100885. Single vs summed cassette distribution patterns

Pathology Sampling Adequacy Analysis

JamoviTest::pathsampling(    data = data,    analysisContext = "omentum",    totalSamples = cassette_number,    firstDetection = first_cassette_tumor_identified,    sampleType = TumorType,    showBinomialModel = TRUE,    showBootstrap = TRUE,    showDetectionCurve = TRUE,    showSensitivityCI = TRUE,    showClinicalSummary = TRUE,    showProbabilityExplanation = TRUE,    showKeyResults = TRUE,    showRecommendText = TRUE,    showInterpretText = TRUE,    showReferencesText = TRUE,    duplicate = 2)

Data Summary
Measure	Value
Total cases supplied	1096
Cases analyzed	1096
Total samples (input)	4395 (recorded)
Total samples analyzed	4162 (up to first detection)
Mean samples per analyzed case	4.01
Median first detection	1
Cases without detected lesion	1035
Binomial estimator	Geometric MLE (first detection only)

📊 Two Ways to Measure Detection Performance

This analysis reports probabilities in two ways, depending on the clinical question:

1️⃣ Conditional Detection (Sensitivity) - "If metastasis is present"

Clinical question: If metastasis is truly present, how many blocks do I need to detect it?

Best used for: Setting minimum blocks per tumour-positive case, validating adequacy targets, or counselling surgeons on block counts required to avoid false reassurance.

Formula: P(detect | metastasis present) = 1 - (1-q)ⁿ

In Your Data: Among 61 cases with detected metastasis (q = 0.560):

With 3 samples: Detects 91.5% of positive cases
With 5 samples: Detects 98.3% of positive cases
With 10 samples: Detects 100.0% of positive cases

Clinical Use: "How many samples do I need to confidently rule out metastasis?" This is the probability shown in the Diagnostic Yield Curve.

2️⃣ Population-Level Detection - "Overall detection rate"

Clinical question: Across all specimens submitted (positive + negative), how often do we detect tumour with n blocks?

Best used for: Monitoring service-level performance, comparing surgeons/protocols, or highlighting when low prevalence—not sampling—limits detection.

Formula: P(detect overall) = Prevalence × Sensitivity = π × [1 - (1-q)ⁿ]

In Your Data: Observed prevalence = 5.6% (61/1096 cases had metastasis):

With 3 samples: Detects metastasis in 5.1% of all specimens
With 5 samples: Detects metastasis in 5.5% of all specimens
With 10 samples: Detects metastasis in 5.6% of all specimens

Clinical Use: "What percentage of incoming specimens will test positive?" This is useful for workload planning and quality metrics.

⚠️ Important: These are fundamentally different quantities!

• Conditional (sensitivity) assumes metastasis is present

• Population-level includes cases without metastasis

The ratio between them equals the prevalence (5.6% in your data). This module focuses on conditional probability (sensitivity) because that's what determines sampling adequacy.

Target confidence: 95%%

Recommended minimum samples: 4 (Bootstrap) achieving 96.7%% sensitivity (10000 iterations; 95% CI 62.2%-100.0%).

Model comparison:

Bootstrap: Meets target at 4 samples (96.7%) (10000 iterations; 95% CI 62.2%-100.0%)
Binomial: Meets target at 4 samples (96.2%) (p = 0.5596)
Empirical: Meets target at 4 samples (96.7%) (Conditional probability among 61 positive cases)

Clinical Summary

Analysis Overview:

Pathology sampling adequacy analysis of 1096 cases to determine the minimum number of samples required to reliably detect lesions.

Key Findings:

Detection probability per sample: 56.0%
Recommended samples for 95%% sensitivity: 4 samples
Bootstrap validation (10k iterations): 96.7% sensitivity (95%% CI: 91.8%-100.0%)
First 3 samples detected: 88.5% of all cases

Copy-ready text for reports:

Clinical Recommendation: Submit a minimum of 4 samples to ensure adequate diagnostic sensitivity in routine practice.

Binomial Probability Model (Conditional Sensitivity)

Estimated per-sample detection probability: q = 0.5596

Estimation Method: Geometric MLE (first detection only)

Based on 61 positive cases with mean first detection at sample 1.79.

Formula: P(detect ≥ 1 in n samples | lesion present) = 1 - (1-q)ⁿ

Note: This estimates sensitivity (detection given lesion is present), not population-level detection rate.

Binomial Model Predictions
Number of Samples	Sensitivity (P(detect \| present))	Marginal Gain
1	56.0%	56.0%
2	80.6%	24.6%
3	91.5%	10.9%
4	96.2%	4.8%
5	98.3%	2.1%
6	99.3%	0.9%
7	99.7%	0.4%
8	99.9%	0.2%
9	99.9%	0.1%
10	100.0%	0.0%

Minimum Samples for Target Confidence
Target Confidence	Minimum Samples Required
80.0%	2
90.0%	3
95.0%	4
99.0%	6

Bootstrap Resampling Analysis

Empirical sensitivity estimates based on 10000 bootstrap iterations.

This method resamples cases with replacement to estimate sensitivity and confidence intervals without parametric assumptions.

Reference: Skala SL, Hagemann IS. Int J Gynecol Pathol. 2015;34(4):374-378.

Bootstrap Sensitivity Estimates (Conditional)
Number of Samples	Mean Sensitivity (given present)	95% CI Lower	95% CI Upper
1	57.4%	44.3%	70.5%
2	78.7%	68.9%	88.5%
3	88.5%	80.3%	95.1%
4	96.7%	91.8%	100.0%
5	100.0%	100.0%	100.0%
6	100.0%	100.0%	100.0%
7	100.0%	100.0%	100.0%
8	100.0%	100.0%	100.0%
9	100.0%	100.0%	100.0%
10	100.0%	100.0%	100.0%

Diagnostic Yield Curve

Sensitivity with Confidence Intervals

Omentum Sampling Recommendations

Recommended minimum samples for 95% sensitivity: 4 (based on Bootstrap model) (10000 iterations; 95% CI 62.2%-100.0%).

This plan achieves an estimated sensitivity of 96.7% using the Empirical resampling of cases.

Observed cumulative detection:

1 sample: 57.4%%
2 samples: 78.7%%
3 samples: 88.5%%
4 samples: 96.7%%

Statistical Interpretation

This analysis addresses the question: How many tissue samples are necessary to reliably detect a lesion?

Key Concepts:

Sensitivity: Probability of detecting the lesion if it is present
Cumulative detection probability: Increases with each additional sample
Diminishing returns: At some point, additional samples provide minimal gain
Target confidence: User-selected detection probability used to determine minimum sampling

Assumptions and Limitations:

Complete cohorts: Record every case in the dataset, including those with no detected lesion. Use NA for the first-detection column when a lesion is never observed so the model can treat the case as a confirmed negative.
Binomial model: Assumes the per-sample detection probability is constant across all cases and samples. Heterogeneous disease burden can violate this assumption.
Hypergeometric model: Requires per-case totals and positives. It still simplifies within-case variability; the beta-binomial model relaxes this by allowing overdispersion.
Residual false negatives: Even when all submitted samples are examined, lesions may still be missed. The reported sensitivities therefore represent an upper bound conditional on the submitted material.
Bootstrap CIs: Assume the analyzed sample is representative of the broader population of interest.

Statistical Methods & References

Methods:

Binomial probability distribution for detection modeling
Hypergeometric probability distribution for finite population sampling
Beta-Binomial probability distribution for sampling with overdispersion
Bootstrap resampling with replacement (10,000 iterations default)
95% confidence intervals using percentile method
Lymph node ratio (LNR) classification with X-tile optimized cutpoints
Non-parametric effect size measures (Cliff's delta, Hodges-Lehmann)

Key References:

Lymph Node Adequacy & LNR Classification:
- Tomlinson JS, et al. Accuracy of Staging Node-Negative Pancreas Cancer. Arch Surg. 2007;142(8):767-774. minELN=15, 8-month survival (SEER n=1,150, HR=0.63)
- Pu N, et al. An Artificial Neural Network Improves Prediction of Observed Survival in Patients with Pancreatic Cancer. J Natl Compr Canc Netw. 2021;19(9):1029-1036. minELN=12, binomial model, LNR thresholds 0.1 & 0.3 (n=1,837)
- Yoon SJ, et al. Optimal Number of Lymph Nodes Retrieved to Lower False N0 Risk. Ann Surg Oncol. 2025. doi:10.1245/s10434-025-18029-7 minELN=12-16, false N0 modeling, Korean dual-cohort (n=1,252)
Sampling Adequacy Methods:
- Skala SL, Hagemann IS. Pathologic Sampling of the Omentum. Int J Gynecol Pathol. 2015;34(4):374-378.
- Gönen M, et al. Nodal Staging Score for Colon Cancer. J Clin Oncol. 2009;27(36):6166-6171.
- Zhou J, et al. Beta-binomial model for lymph node yield. Front Oncol. 2022;12:872527.
- Buderer NM. Statistical methodology for diagnostic test sample size. Stat Med. 1996;15(6):649-652.
Distribution Pattern Analysis:
- Ates D, et al. Lymphovascular Space Invasion in Endometrial Cancer. Mod Pathol. 2025;38:100885. Single vs summed cassette distribution patterns

Pathology Sampling Adequacy Analysis

JamoviTest::pathsampling(    data = data,    analysisContext = "omentum",    totalSamples = cassette_number,    firstDetection = first_cassette_tumor_identified,    positiveCount = total_cassettes_with_metastasis,    showBinomialModel = TRUE,    showBootstrap = TRUE,    showDetectionCurve = TRUE,    showSensitivityCI = TRUE,    showClinicalSummary = TRUE,    showEmpiricalCumulative = TRUE,    showPopulationDetection = TRUE,    showIncrementalYield = TRUE,    duplicate = 2)

Data Summary
Measure	Value
Total cases supplied	1096
Cases analyzed	1096
Total samples (input)	4395 (recorded)
Total samples analyzed	4162 (up to first detection)
Mean samples per analyzed case	4.01
Median first detection	1
Cases without detected lesion	1035
Binomial estimator	Empirical Proportion (uses all positive samples)

Clinical Summary

Analysis Overview:

Pathology sampling adequacy analysis of 1096 cases to determine the minimum number of samples required to reliably detect lesions.

Key Findings:

Detection probability per sample: 60.2%
Recommended samples for 95%% sensitivity: 4 samples
Bootstrap validation (10k iterations): 96.7% sensitivity (95%% CI: 91.8%-100.0%)
First 3 samples detected: 88.5% of all cases

Copy-ready text for reports:

"Sampling adequacy analysis of 1096 cases showed a per-sample detection probability of 60.2%. To achieve 95%% sensitivity, a minimum of 4 samples is recommended based on binomial probability modeling and bootstrap validation (10000 iterations, 95%% CI: 91.8%%-100.0%%). Observed data showed 88.5% of lesions detected within first 3 samples."

Clinical Recommendation: Submit a minimum of 4 samples to ensure adequate diagnostic sensitivity in routine practice.

⚠️ DATA QUALITY WARNINGS

⚠️ MODERATE HETEROGENEITY: Detection probability shows moderate variation across cases (CV=0.49). Interpret pooled estimate with caution.

Binomial Probability Model (Conditional Sensitivity)

Estimated per-sample detection probability: q = 0.6023

Estimation Method: Empirical Proportion (uses all positive samples)

Based on 61 positive cases with 206 positive samples out of 342 total samples examined.

Formula: P(detect ≥ 1 in n samples | lesion present) = 1 - (1-q)ⁿ

Note: This estimates sensitivity (detection given lesion is present), not population-level detection rate.

Binomial Model Predictions
Number of Samples	Sensitivity (P(detect \| present))	Marginal Gain
1	60.2%	60.2%
2	84.2%	24.0%
3	93.7%	9.5%
4	97.5%	3.8%
5	99.0%	1.5%
6	99.6%	0.6%
7	99.8%	0.2%
8	99.9%	0.1%
9	100.0%	0.0%
10	100.0%	0.0%

Minimum Samples for Target Confidence
Target Confidence	Minimum Samples Required
80.0%	2
90.0%	3
95.0%	4
99.0%	5

Bootstrap Resampling Analysis

Empirical sensitivity estimates based on 10000 bootstrap iterations.

This method resamples cases with replacement to estimate sensitivity and confidence intervals without parametric assumptions.

Reference: Skala SL, Hagemann IS. Int J Gynecol Pathol. 2015;34(4):374-378.

Bootstrap Sensitivity Estimates (Conditional)
Number of Samples	Mean Sensitivity (given present)	95% CI Lower	95% CI Upper
1	57.4%	44.3%	68.9%
2	78.7%	67.2%	88.5%
3	88.5%	80.3%	95.1%
4	96.7%	91.8%	100.0%
5	100.0%	100.0%	100.0%
6	100.0%	100.0%	100.0%
7	100.0%	100.0%	100.0%
8	100.0%	100.0%	100.0%
9	100.0%	100.0%	100.0%
10	100.0%	100.0%	100.0%

Diagnostic Yield Curve

Sensitivity with Confidence Intervals

Empirical Cumulative Detection Analysis

Non-parametric estimation of detection probability based on actual observed data. Does not assume geometric distribution - uses bootstrap resampling for confidence intervals.

Based on: 61 positive cases with first detection positions ranging from 1 to 5.

Empirical Detection Rates by Sample Threshold
Samples Examined	Cumulative Detection	95% CI Lower	95% CI Upper	Incremental Yield
1	57.3%	44.3%	70.5%	57.3%
2	78.6%	68.9%	88.5%	21.3%
3	88.5%	80.3%	95.1%	9.9%
4	96.7%	91.8%	100.0%	8.2%
5	100.0%	100.0%	100.0%	3.3%
6	100.0%	100.0%	100.0%	0.0%
7	100.0%	100.0%	100.0%	0.0%
8	100.0%	100.0%	100.0%	0.0%
9	100.0%	100.0%	100.0%	0.0%
10	100.0%	100.0%	100.0%	0.0%

Empirical Cumulative Detection Curve

Incremental Diagnostic Yield Analysis

Marginal benefit of examining each additional sample. Helps identify the optimal stopping point where yield diminishes.

Interpretation: High value (≥10%), Moderate (5-10%), Diminishing (<5%), Low (<2%).

Marginal Benefit per Additional Sample
From N Samples	To N+1 Samples	Additional Detection Rate	Additional Cases per 100	Cost-Benefit Rating
1	2	21.3%	21.31	High value
2	3	9.8%	9.84	Moderate value
3	4	8.2%	8.20	Moderate value
4	5	3.3%	3.28	Diminishing returns
5	6	0.0%	0.00	Low yield
6	7	0.0%	0.00	Low yield
7	8	0.0%	0.00	Low yield
8	9	0.0%	0.00	Low yield
9	10	0.0%	0.00	Low yield

Population-Level vs Conditional Detection

Distinguishes between:

Conditional (Sensitivity): P(detect | lesion present) - assumes disease exists
Population (Overall): P(detect in general) = prevalence × sensitivity

Observed prevalence: 5.6% (61/1096 cases positive)

Note: Prevalence reflects this specific dataset and may not generalize.

Conditional vs Population Detection
Samples	Prevalence	Sensitivity (given present)	Detection Rate (overall)
1	5.6%	60.2%	3.4%
2	5.6%	84.2%	4.7%
3	5.6%	93.7%	5.2%
4	5.6%	97.5%	5.4%
5	5.6%	99.0%	5.5%
6	5.6%	99.6%	5.5%
7	5.6%	99.8%	5.6%
8	5.6%	99.9%	5.6%
9	5.6%	100.0%	5.6%
10	5.6%	100.0%	5.6%

Pathology Sampling Adequacy Analysis

JamoviTest::pathsampling(    data = data,    analysisContext = "omentum",    totalSamples = cassette_number,    firstDetection = first_cassette_tumor_identified,    positiveCount = total_cassettes_with_metastasis,    positiveSamplesList = cassettes_with_metastasis,    showBinomialModel = TRUE,    showBootstrap = TRUE,    showDetectionCurve = TRUE,    showSensitivityCI = TRUE,    showClinicalSummary = TRUE,    showEmpiricalCumulative = TRUE,    showSpatialClustering = TRUE,    showPopulationDetection = TRUE,    showIncrementalYield = TRUE,    showMultifocalAnalysis = TRUE,    duplicate = 2)

Data Summary
Measure	Value
Total cases supplied	1096
Cases analyzed	1096
Total samples (input)	4395 (recorded)
Total samples analyzed	4162 (up to first detection)
Mean samples per analyzed case	4.01
Median first detection	1
Cases without detected lesion	1035
Binomial estimator	Empirical Proportion (uses all positive samples)

Clinical Summary

Analysis Overview:

Pathology sampling adequacy analysis of 1096 cases to determine the minimum number of samples required to reliably detect lesions.

Key Findings:

Detection probability per sample: 60.2%
Recommended samples for 95%% sensitivity: 4 samples
Bootstrap validation (10k iterations): 96.7% sensitivity (95%% CI: 91.8%-100.0%)
First 3 samples detected: 88.5% of all cases

Copy-ready text for reports:

Clinical Recommendation: Submit a minimum of 4 samples to ensure adequate diagnostic sensitivity in routine practice.

⚠️ DATA QUALITY WARNINGS

⚠️ MODERATE HETEROGENEITY: Detection probability shows moderate variation across cases (CV=0.49). Interpret pooled estimate with caution.

Binomial Probability Model (Conditional Sensitivity)

Estimated per-sample detection probability: q = 0.6023

Estimation Method: Empirical Proportion (uses all positive samples)

Based on 61 positive cases with 206 positive samples out of 342 total samples examined.

Formula: P(detect ≥ 1 in n samples | lesion present) = 1 - (1-q)ⁿ

Note: This estimates sensitivity (detection given lesion is present), not population-level detection rate.

Binomial Model Predictions
Number of Samples	Sensitivity (P(detect \| present))	Marginal Gain
1	60.2%	60.2%
2	84.2%	24.0%
3	93.7%	9.5%
4	97.5%	3.8%
5	99.0%	1.5%
6	99.6%	0.6%
7	99.8%	0.2%
8	99.9%	0.1%
9	100.0%	0.0%
10	100.0%	0.0%

Minimum Samples for Target Confidence
Target Confidence	Minimum Samples Required
80.0%	2
90.0%	3
95.0%	4
99.0%	5

Bootstrap Resampling Analysis

Empirical sensitivity estimates based on 10000 bootstrap iterations.

This method resamples cases with replacement to estimate sensitivity and confidence intervals without parametric assumptions.

Reference: Skala SL, Hagemann IS. Int J Gynecol Pathol. 2015;34(4):374-378.

Bootstrap Sensitivity Estimates (Conditional)
Number of Samples	Mean Sensitivity (given present)	95% CI Lower	95% CI Upper
1	57.3%	44.3%	68.9%
2	78.6%	67.2%	88.5%
3	88.5%	80.3%	95.1%
4	96.7%	91.8%	100.0%
5	100.0%	100.0%	100.0%
6	100.0%	100.0%	100.0%
7	100.0%	100.0%	100.0%
8	100.0%	100.0%	100.0%
9	100.0%	100.0%	100.0%
10	100.0%	100.0%	100.0%

Diagnostic Yield Curve

Sensitivity with Confidence Intervals

Empirical Cumulative Detection Analysis

Non-parametric estimation of detection probability based on actual observed data. Does not assume geometric distribution - uses bootstrap resampling for confidence intervals.

Based on: 61 positive cases with first detection positions ranging from 1 to 5.

Empirical Detection Rates by Sample Threshold
Samples Examined	Cumulative Detection	95% CI Lower	95% CI Upper	Incremental Yield
1	57.5%	44.3%	70.5%	57.5%
2	78.7%	67.2%	88.5%	21.3%
3	88.6%	80.3%	95.1%	9.8%
4	96.7%	91.8%	100.0%	8.2%
5	100.0%	100.0%	100.0%	3.3%
6	100.0%	100.0%	100.0%	0.0%
7	100.0%	100.0%	100.0%	0.0%
8	100.0%	100.0%	100.0%	0.0%
9	100.0%	100.0%	100.0%	0.0%
10	100.0%	100.0%	100.0%	0.0%

Empirical Cumulative Detection Curve

Incremental Diagnostic Yield Analysis

Marginal benefit of examining each additional sample. Helps identify the optimal stopping point where yield diminishes.

Interpretation: High value (≥10%), Moderate (5-10%), Diminishing (<5%), Low (<2%).

Marginal Benefit per Additional Sample
From N Samples	To N+1 Samples	Additional Detection Rate	Additional Cases per 100	Cost-Benefit Rating
1	2	21.3%	21.31	High value
2	3	9.8%	9.84	Moderate value
3	4	8.2%	8.20	Moderate value
4	5	3.3%	3.28	Diminishing returns
5	6	0.0%	0.00	Low yield
6	7	0.0%	0.00	Low yield
7	8	0.0%	0.00	Low yield
8	9	0.0%	0.00	Low yield
9	10	0.0%	0.00	Low yield

Population-Level vs Conditional Detection

Distinguishes between:

Conditional (Sensitivity): P(detect | lesion present) - assumes disease exists
Population (Overall): P(detect in general) = prevalence × sensitivity

Observed prevalence: 5.6% (61/1096 cases positive)

Note: Prevalence reflects this specific dataset and may not generalize.

Conditional vs Population Detection
Samples	Prevalence	Sensitivity (given present)	Detection Rate (overall)
1	5.6%	60.2%	3.4%
2	5.6%	84.2%	4.7%
3	5.6%	93.7%	5.2%
4	5.6%	97.5%	5.4%
5	5.6%	99.0%	5.5%
6	5.6%	99.6%	5.5%
7	5.6%	99.8%	5.6%
8	5.6%	99.9%	5.6%
9	5.6%	100.0%	5.6%
10	5.6%	100.0%	5.6%

Spatial Clustering Analysis

Analyzes how positive samples are distributed spatially:

Clustered (index < 0.7): Positive samples grouped together (focal/contiguous)
Random (0.7-1.3): Positive samples evenly distributed
Dispersed (> 1.3): Positive samples widely separated (multifocal)

Clinical significance: Clustered patterns may allow more targeted sampling; dispersed patterns require broader sampling strategy.

Spatial Distribution Patterns
Pattern	Cases	Percentage	Mean Clustering Index
Clustered (focal)	8	17.4%	0.487
Random	38	82.6%	0.938
Dispersed (multifocal)	0	0.0%	.

Multifocal Detection Analysis

Estimates number of separate foci based on spatial distribution of positive samples. Gaps > 2 samples suggest separate foci.

Clinical note: Multifocal involvement may indicate more advanced disease and can affect staging/treatment decisions.

Number of Foci Distribution
Number of Foci	Cases	Percentage	Mean First Detection
Unifocal (1 focus)	55	90.2%	1.82
Bifocal (2 foci)	3	4.9%	1.67
Multifocal (3+ foci)	3	4.9%	1.33

Multifocal Detection: Probability of detecting multiple lesions in 'n' samples, assuming per-sample detection probability q = 0.602.

Useful for planning sampling when the goal is to find multiple foci (e.g., multifocal tumor).

Probability of Detecting Multiple Lesions
Samples	P(≥1 lesion)	P(≥2 lesions)	P(≥3 lesions)
1	60.2%	0.0%	0.0%
2	84.2%	36.3%	0.0%
3	93.7%	65.1%	21.9%
4	97.5%	82.3%	47.9%
5	99.0%	91.5%	68.7%
6	99.6%	96.0%	82.4%
7	99.8%	98.2%	90.6%
8	99.9%	99.2%	95.2%
9	100.0%	99.6%	97.6%
10	100.0%	99.8%	98.8%
1	60.2%	0.0%	0.0%
2	84.2%	36.3%	0.0%
3	93.7%	65.1%	21.9%
4	97.5%	82.3%	47.9%
5	99.0%	91.5%	68.7%
6	99.6%	96.0%	82.4%
7	99.8%	98.2%	90.6%
8	99.9%	99.2%	95.2%
9	100.0%	99.6%	97.6%
10	100.0%	99.8%	98.8%

References

[1] The jamovi project (2025). jamovi. (Version 2.7) [Computer Software]. Retrieved from https://www.jamovi.org.

[2] R Core Team (2025). R: A Language and environment for statistical computing. (Version 4.5) [Computer software]. Retrieved from https://cran.r-project.org. (R packages retrieved from CRAN snapshot 2025-05-25).