Khatabook Growth Intelligence • Audit Dossier

Analytical & Methodological Audit Dossier

Mathematical proofs, variance identities, direct standardization formulas, statistical test specifications, and full code traceability for the Khatabook Growth Platform.

Author: Pranjal Sailwal

Scope: Mathematical & Statistical Lineage

Verification: 44 Automated Unit Tests Passing

Precision: Machine Epsilon ($10^{-6}$)

Section 1 • Mathematical Proofs

1. Mathematical Proofs & Variance Identities

Proof 1.1: Exact Three-Factor Efficiency Decomposition

Cost per Approved Conversion is expressed as the product of three distinct commercial levers:

$$\text{Cost per Approved} = \left(\frac{\text{Spend}}{\text{Impressions}}\right) \times \left(\frac{\text{Impressions}}{\text{Clicks}}\right) \times \left(\frac{\text{Clicks}}{\text{Approved}}\right) = \left(\frac{\text{CPM}}{1000}\right) \times \left(\frac{1}{\text{CTR}}\right) \times \left(\frac{1}{\text{ConvRate}}\right)$$

Let $C_0$ be baseline unit cost (Campaign 936, $15.811) and $C_1$ be scaled unit cost (Campaign 1178, $63.833). Under sequential substitution:

Step 1 = $\left(\frac{\text{CPM}_1}{1000}\right) \times \left(\frac{1}{\text{CTR}_0}\right) \times \left(\frac{1}{\text{ConvRate}_0}\right) = \$12.072$

→ Lever 1 (CPM Effect) = Step 1 - $C_0$ = $12.072 - $15.811 = -$3.739

Step 2 = $\left(\frac{\text{CPM}_1}{1000}\right) \times \left(\frac{1}{\text{CTR}_1}\right) \times \left(\frac{1}{\text{ConvRate}_0}\right) = \$16.732$

→ Lever 2 (CTR Effect) = Step 2 - Step 1 = $16.732 - $12.072 = +$4.660

Step 3 = $\left(\frac{\text{CPM}_1}{1000}\right) \times \left(\frac{1}{\text{CTR}_1}\right) \times \left(\frac{1}{\text{ConvRate}_1}\right) = C_1 = \$63.833$

→ Lever 3 (Conversion Effect) = Step 3 - Step 2 = $63.833 - $16.732 = +$47.101

$$\text{Total Explained Variance} = (-\$3.739) + (+\$4.660) + (+\$47.101) = +\$48.022$$ $$\text{Residual Error} = |\Delta C - (\text{Lever}_1 + \text{Lever}_2 + \text{Lever}_3)| = \$0.00000000$$

Validated in unit test tests/test_diagnostics.py::test_decomposition_bridge_mathematical_identity.

Proof 1.2: Direct Standardization & Confounding Adjustment

To adjust for demographic composition, we compute the pooled benchmark click distribution across all 4 age categories:

$$w_i = \frac{\sum_{c} \text{Clicks}_{c, i}}{\sum_{c} \sum_{i} \text{Clicks}_{c, i}}$$ $$\text{Cost}_{\text{adj}, c} = \sum_{i} w_i \times \text{Cost per Approved}_{c, i}$$

$$\text{Cost}_{\text{adj}, 936} = \$23.69 \quad \text{vs.} \quad \text{Cost}_{\text{adj}, 1178} = \$78.16 \quad (\text{Ratio: } 3.30\times)$$

Validated in unit test tests/test_diagnostics.py::test_mix_adjustment_direct_standardization.

Proof 1.3: Two-Proportion Hypothesis Testing & Normal Approximation

For test evaluation under null hypothesis $H_0: p_1 = p_2$:

$$\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}, \quad \text{SE}_{\text{pooled}} = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}$$ $$z = \frac{\hat{p}_1 - \hat{p}_2}{\text{SE}_{\text{pooled}}}, \quad p\text{-value} = 2 \times (1 - \Phi(|z|))$$ $$\text{CI}_{95\%} = (\hat{p}_1 - \hat{p}_2) \pm 1.96 \times \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$$

Validated in unit test tests/test_statistics.py::test_two_proportion_z_test_calculation.

Section 2 • Traceability Matrix

2. Code, Metric & File Traceability Matrix

Analytical Dimension	Python Source	SQL Transform	Export File	Verification Test
Ratio-of-Sums Scorecard	src/metrics.py	sql/03_metrics.sql	exports/campaign_scorecard.csv	tests/test_metrics.py
Efficiency Waterfall	src/diagnostics.py	sql/03_metrics.sql	reports/campaign_diagnostic.md	tests/test_diagnostics.py
4-Stage Funnel Drop-off	src/diagnostics.py	sql/06_funnel.sql	exports/funnel_analysis.csv	tests/test_funnel.py
Segment Stability (≥10, 25, 50)	src/diagnostics.py	sql/04_audience.sql	exports/segment_stability.csv	tests/test_diagnostics.py
Direct Standardization	src/diagnostics.py	sql/05_mix_analysis.sql	exports/mix_analysis.csv	tests/test_diagnostics.py
A/B Testing & Sample Sizing	src/statistics.py	—	Embedded in REST API	tests/test_statistics.py
11-Rule Quality Audit	src/validation.py	sql/07_quality.sql	exports/data_quality.csv	tests/test_quality.py
Consumer Session Intent	src/data_loader.py	sql/02_clean.sql	exports/consumer_behavior.csv	tests/test_metrics.py
FastAPI REST Endpoints	backend/main.py	—	OpenAPI Swagger /docs	tests/test_backend.py