cnicholas · cnicholas · Sep 14, 2026 · Sep 14, 2026
diff --git a/CHANGELOG.md b/CHANGELOG.md
@@ -7,6 +7,21 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 
 ## [Unreleased]
 
+### Changed
+- **The R2 method for partial replication is reported as ``ma2``, not ``hybrid``.** The
+  docs described a per-cell hybrid for design state 3 (exact deviation where a cell has
+  n ≥ 2, moving average where n = 1) and said it was validated by Monte Carlo. The code
+  has not done that since cb037f8 (2026-03-05): with any singleton cell, R2 is Bishop's
+  ungrouped 2-point moving average over the full canonical sequence (Eq 13.7–13.9) for
+  every observation, and the 89 ADS 3 reference assertions hold against that. Only the word
+  survived, in six docs pages (three of which described it three different ways), the
+  detector's method token, the analysis-plan limitation text, two docstrings and two test
+  names. All now say what the code does. ``R2Method`` is ``Literal['exact', 'ma2']``;
+  ``SDSRegistry.get_r2_method`` returns ``'ma2'`` where it returned ``'hybrid'``;
+  ``get_sds_characteristics`` and ``SDSAnalysisPlan.residual_calculation_method`` use the
+  same two spellings in place of ``'within_cell'``, ``'moving_average'`` and ``'hybrid'``.
+  No arithmetic changed; none of these names is exported from the package top level.
+
 ### Fixed
 - **``evaluate`` no longer raises on a constant column.** Binning a column with no spread
   (every value identical) returned the documented no-spread result for ``equal_freq`` but

diff --git a/docs/appendix/wheeler-terminology.md b/docs/appendix/wheeler-terminology.md
@@ -94,7 +94,7 @@ Wheeler identifies six design states that determine valid analysis approaches.
 
 **ProcessBehavior Detection**: Some cells with n=1, others with n>=2
 
-**Capabilities**: Hybrid variance estimation
+**Capabilities**: R2 by moving average over the full sequence (mixed cell sizes)
 
 ---
 

diff --git a/docs/getting-started/key-concepts.md b/docs/getting-started/key-concepts.md
@@ -46,7 +46,7 @@ See [DS Definitions](../reference/sds_definitions.md) for the formal classificat
 
 The DS determines:
 - Which chart types are valid
-- How within-group variance is estimated (R2 method: exact, ma2, or hybrid)
+- How within-group variance is estimated (R2 method: exact or ma2)
 - Which VAS residuals can be computed
 - What conclusions you can draw
 

diff --git a/docs/reference/sds_definitions.md b/docs/reference/sds_definitions.md
@@ -52,7 +52,7 @@ For background on the lineage itself (Planned → Observed → Analytical), see
 - **Grid Status**: Semi-Complete
 - **Description**: Mix of cells - some have n=1, others have n≥2
 - **Implications**:
-  - Hybrid R2 calculation required (exact where n≥2, moving average where n=1)
+  - R2 by moving average over the full sequence (any singleton cell forces `ma2` for every observation)
   - **Most common in real-world data**
   - Partial interaction effect estimation
   - Requires careful handling of mixed replication
@@ -121,10 +121,10 @@ To determine the DS for a dataset:
 | 0 | N/A | No VAS decomposition |
 | 1 | Exact (within-cell) | Pooled within-cell variance |
 | 2 | Moving Average | Approximate via sequential differences |
-| 3 | Hybrid | Exact where n≥2, moving average where n=1 |
+| 3 | Moving Average | Any singleton cell: `ma2` over the full sequence, same as DS 2 |
 | 4 | Exact (present cells) | Within-cell for available data |
 | 5 | Moving Average | All present cells are unreplicated |
-| 6 | Hybrid | Complex handling of missing + mixed |
+| 6 | Moving Average | Collapses to ADS 3 after cleaning; `ma2` over the full sequence |
 
 ## References
 

diff --git a/docs/user-guide/chart-types.md b/docs/user-guide/chart-types.md
@@ -309,7 +309,7 @@ Do you have factors?
     ├── No → Use Xbar to compare factors
     └── Yes → Do you have replication (n>=2 per cell)?
         ├── All cells → DS 1: Full Xbar-S + VAS residuals
-        ├── Some cells → DS 3: Hybrid Xbar-S + limited VAS
+        ├── Some cells → DS 3: Xbar-S with MA2-based limits + limited VAS
         └── No cells → DS 2: Xbar-S with MR-based limits
 ```
 

diff --git a/docs/user-guide/residuals.md b/docs/user-guide/residuals.md
@@ -81,7 +81,7 @@ design states.
 **Formula by DS**:
 - **DS 1 (Full Replication)**: R2 = Y - Y̅<sub>kt</sub> (exact within-cell deviation)
 - **DS 2 (No Replication)**: R2 = (Y<sub>j</sub> - Y<sub>j-1</sub>) / 2 (backward 2-point moving average)
-- **DS 3 (Partial)**: Hybrid approach
+- **DS 3 (Partial)**: same moving average as DS 2, over the full sequence
 
 **Chart**: S chart with `value='R2'` (for replicated data) or X
 
@@ -255,17 +255,20 @@ Re-centering formulas:
 |-----|----|----|----|----|-----|-----|
 | 1 (Full Replication) | ✅ | ✅ Within-cell | ✅ | ✅ | ✅ | ✅ |
 | 2 (No Replication) | ✅ | ✅ MR-based | ✅ | ✅ | ✅ | ✅ |
-| 3 (Partial) | ✅ | ✅ Hybrid | ✅ | ✅ | ✅ | ✅ |
+| 3 (Partial) | ✅ | ✅ MR-based | ✅ | ✅ | ✅ | ✅ |
 | 4 (Incomplete, No Singletons) → ADS 1 | ✅ | ✅ Within-cell | ✅ | ✅ | ✅ | ✅ |
 | 5 (Incomplete, No Replication) → ADS 2 | ✅ | ✅ MR-based | ✅ | ✅ | ✅ | ✅ |
-| 6 (Incomplete, With Singletons) → ADS 3 | ✅ | ✅ Hybrid | ✅ | ✅ | ✅ | ✅ |
+| 6 (Incomplete, With Singletons) → ADS 3 | ✅ | ✅ MR-based | ✅ | ✅ | ✅ | ✅ |
 
 R6 requires factors (it is computed from R5 and R2 at `execute()` time).
 
-**Note on R2 calculation**: R2 adapts to your sampling structure:
-- **DS 1**: Within-cell deviation (`R2 = Y - Ȳ_kt`)
-- **DS 2, 6**: Moving average method (`R2 = Y - MA2`) for unreplicated/sparse designs
-- **DS 3, 4, 5**: Hybrid approach (within-cell for n>1 cells, zero for n=1 cells)
+**Note on R2 calculation**: R2 is the one residual whose formula depends on structure, and the
+choice is made on the analytical design state (ADS), after cleaning:
+- **ADS 1** (every cell n ≥ 2): within-cell deviation, `R2 = Y - Ȳ_kt` (Eq 59)
+- **ADS 2 and 3** (any singleton cell): 2-point moving average over the full canonical sequence,
+  `R2 = (Y_j - Y_{j-1}) / 2` (Eq 13.7–13.9), for every observation; no per-cell mixing
+
+DS 4, 5 and 6 are sampling states; they collapse to ADS 1, 2 and 3 once empty cells are dropped.
 
 ## Analysis Workflow with Residuals
 

diff --git a/docs/user-guide/sds-detection.md b/docs/user-guide/sds-detection.md
@@ -278,14 +278,14 @@ The design report shows:
 ```
 
 **Capabilities**:
-- ⚠️ Hybrid R2 estimation (exact where n >= 2, ma2 where n = 1)
+- ⚠️ R2 by the 2-point moving average over the full sequence (any singleton cell forces `ma2` for every observation)
 - ⚠️ VAS residuals available but interpretation requires care
-- ✅ Xbar-S analysis with hybrid limits
+- ✅ Xbar-S analysis with R2-based limits
 
 **Valid Charts**: Histogram, Xbar, S, X, mR
 
-!!! note "Why Mixed is treated conservatively"
-    For R2 calculation, DS 3 uses the hybrid method: exact within-cell deviation where cells have n >= 2, and the ma2 (moving average) method where cells have n = 1. This conservative approach was validated by Monte Carlo simulation — it produces more reliable variance estimates than attempting to use only the replicated cells. The recommended chart is X (not Xbar) because the mixed replication makes subgroup-mean interpretation less straightforward.
+!!! note "Why DS 3 does not mix methods"
+    For R2 in DS 3 the library does not combine an exact estimate from the replicated cells with a moving-average estimate from the singletons. Bishop's moving-average residual (Eq 13.7–13.9) is defined over the whole ordered sequence, j = 2 … J, with no grouping by cell. Once any cell is a singleton, every observation gets R2 = (Y_j − Y_{j−1}) / 2 on the canonical sort, and only the first observation in the sequence has no value. The replicated cells' within-cell deviations are not used for R2. This is the same calculation as DS 2, and the ADS 3 reference assertions in the [validation page](../reference/validation.md) hold against it. The recommended chart is X rather than Xbar because subgroup means over mixed cell sizes are uneven to interpret, not because the variance estimate differs.
 
 ## DS 4: Incomplete, No Singletons
 
@@ -379,7 +379,7 @@ The R2 method is determined by the tidy data structure (ADS), not the raw DS:
 |-----|-----------|-------------|
 | 1 | exact | Within-cell standard deviation (`R2 = Y - Ȳ_kt`) |
 | 2 | ma2 | 2-point moving average for unreplicated designs |
-| 3 | hybrid | Exact where n >= 2, ma2 where n = 1 |
+| 3 | ma2 | Any singleton cell: moving average over the full sequence (same as DS 2) |
 
 ### 2. Available Charts
 
@@ -389,7 +389,7 @@ The R2 method is determined by the tidy data structure (ADS), not the raw DS:
 |-----|--------|----------------|
 | 1 | ✅ | ✅ |
 | 2 | ✅ (MR-based limits) | ✅ |
-| 3 | ✅ (hybrid limits) | ✅ |
+| 3 | ✅ (R2-based limits) | ✅ |
 | 4 | ❌ | ✅ |
 | 5 | ❌ | ✅ |
 | 6 | ❌ | ✅ |
@@ -486,10 +486,10 @@ print(cell_counts['n'].value_counts())
 |----------------|-----|-----------------------|---------------|
 | Full replication (n>=2 per cell) | 1 | 1 | Xbar with full VAS |
 | One observation per cell | 2 | 2 | X with MA2-based R2 |
-| Mixed replication | 3 | 3 | X with hybrid R2 |
+| Mixed replication | 3 | 3 | X with MA2 R2 |
 | Incomplete grid, all observed replicated | 4 | 1 | Xbar with full VAS |
 | Incomplete grid, all observed n=1 | 5 | 2 | X with MA2-based R2 |
-| Incomplete grid, mixed observed | 6 | 3 | X with hybrid R2 |
+| Incomplete grid, mixed observed | 6 | 3 | X with MA2 R2 |
 
 ## Next Steps
 

diff --git a/processbehavior/analysis_dataset.py b/processbehavior/analysis_dataset.py
@@ -183,7 +183,7 @@ def tidy_structure_summary(self) -> dict | None:
             - n_cell_min: minimum observations per cell
             - n_cell_max: maximum observations per cell
             - K_obs: number of unique factor levels observed
-            - r2_method: 'exact', 'ma2', or 'hybrid'
+            - r2_method: 'exact' or 'ma2'
             Returns None if structure stats haven't been computed.
         """
         if self._structure_stats is None:

diff --git a/processbehavior/datasets/synthetic.py b/processbehavior/datasets/synthetic.py
@@ -416,13 +416,13 @@ def make_sds3(  # noqa: C901
     Characteristics:
     ---------------
     - Some cells have n=1, others have n≥2
-    - Requires hybrid R2 estimation
+    - Mixed cell sizes: R2 by MA2 across the full sequence
     - Reflects real-world data collection reality
     - Most challenging to analyze correctly
 
     Use Cases:
     ---------
-    - Testing hybrid variance estimation algorithms
+    - Testing R2 selection with mixed cell sizes
     - Demonstrating real-world data structures
     - Validating robustness to missing data
     - Development of SDS detection logic

diff --git a/processbehavior/residual_calculator.py b/processbehavior/residual_calculator.py
@@ -338,14 +338,14 @@ def calculate_r2(df: pd.DataFrame, y: str, r2_method: R2Method, n_per_cell: pd.S
     R2 is the ONLY structure-dependent residual. The method is determined by
     observed cell sizes (via SDSRegistry.get_r2_method()):
 
-    - exact (state 1): R2 = Y - Ȳ_kt (Eq 59), all cells have n >= 2
-    - ma2 (states 2 & 3): R2 = (Y_j - Y_{j-1}) / 2 (Eq 13.8-13.9),
+    - exact (ADS 1): R2 = Y - Ȳ_kt (Eq 59), every cell has n >= 2
+    - ma2 (ADS 2 and 3): R2 = (Y_j - Y_{j-1}) / 2 (Eq 13.7-13.9),
       any cell has n = 1
 
     When any cell has n=1, MA2 is applied to ALL observations across the
-    entire canonical-sorted stream — no grouping by rsg_key, no hybrid
-    per-cell selection. Bishop Eq 13.7-13.9 specify j=2,...,J with no
-    grouping; only j=1 gets R2=0.
+    entire canonical-sorted stream — no grouping by rsg_key, no per-cell
+    selection between exact and MA2. Bishop Eq 13.7-13.9 specify j=2,...,J
+    with no grouping; only j=1 has no value.
 
     Parameters
     ----------
@@ -354,8 +354,9 @@ def calculate_r2(df: pd.DataFrame, y: str, r2_method: R2Method, n_per_cell: pd.S
     y : str
         Name of response variable
     r2_method : R2Method
-        'exact', 'ma2', or 'hybrid' — retained for API compatibility.
-        Branching is driven by n_per_cell.
+        'exact' or 'ma2', as chosen by SDSRegistry.get_r2_method(). Kept for
+        callers and logging; the branch below re-derives it from n_per_cell so
+        the two can never disagree.
     n_per_cell : pd.Series, optional
         Pre-computed observations per cell. Pass from ADS to avoid recomputation.
 
@@ -427,7 +428,7 @@ def calculate_vas_residuals(
     spec : FormulationSpec
         Analysis specification
     r2_method : R2Method
-        'exact', 'ma2', or 'hybrid' - determines R2 calculation
+        'exact' or 'ma2' - determines R2 calculation
     n_per_cell : pd.Series, optional
         Pre-computed observations per cell. Pass from ADS to avoid recomputation.
     ybar_kt : pd.Series, optional

diff --git a/processbehavior/sds_detector.py b/processbehavior/sds_detector.py
@@ -39,7 +39,7 @@
 ]
 
 # R2 calculation method - structure-driven, not SDS-driven
-R2Method = Literal['exact', 'ma2', 'hybrid']
+R2Method = Literal['exact', 'ma2']
 
 
 @dataclass(frozen=True)
@@ -116,7 +116,7 @@ class SDSAnalysisPlan:
     residuals_available : list[str]
         Which residuals can be calculated
     residual_calculation_method : str
-        How R2 is calculated ('exact', 'moving_average', 'hybrid', or 'none')
+        How R2 is calculated ('exact', 'ma2', or 'none')
     main_effects_supported : bool
         Whether main effect calculations are supported
     interaction_effects_supported : bool
@@ -158,7 +158,7 @@ class SDSAnalysisPlan:
     # VAS residual capabilities
     vas_residuals_supported: bool
     residuals_available: list[str]
-    residual_calculation_method: str  # 'exact', 'moving_average', 'hybrid', 'none'
+    residual_calculation_method: str  # 'exact', 'ma2', 'none'
 
     # Effects and interactions
     main_effects_supported: bool
@@ -383,7 +383,7 @@ class SDSRegistry:
 
     **SDS 3**: Semi-Complete - Partial replication
         - Min N_kt = 1 and Max N_kt ≥ 2 (mixed)
-        - Hybrid variance estimation
+        - R2 by moving average across the full sequence (any singleton forces MA2)
 
     **Incomplete** (has empty cells, requires sampling plan to detect):
 
@@ -664,7 +664,7 @@ def get_sds_characteristics(self, sds: int) -> dict:
         >>> info['description']
         'Full replication (all cells n≥2)'
         >>> info['r2_method']
-        'within_cell'
+        'exact'
         >>> info['capabilities']
         ['full_vas', 'all_residuals', 'interactions', 'main_effects']
         """
@@ -680,23 +680,23 @@ def get_sds_characteristics(self, sds: int) -> dict:
             1: {
                 'description': 'Full replication (all cells n≥2)',
                 'replication_type': 'full',
-                'r2_method': 'within_cell',
+                'r2_method': 'exact',
                 'capabilities': ['full_vas', 'all_residuals', 'interactions', 'main_effects'],
                 'interaction_analysis': True,
                 'variance_decomposition': True,
             },
             2: {
                 'description': 'No replication (all cells n=1)',
                 'replication_type': 'none',
-                'r2_method': 'moving_average',
+                'r2_method': 'ma2',
                 'capabilities': ['all_residuals', 'limited_interactions', 'main_effects'],
                 'interaction_analysis': 'limited',
                 'variance_decomposition': True,
             },
             3: {
                 'description': 'Partial replication (mixed n=1 and n≥2)',
                 'replication_type': 'partial',
-                'r2_method': 'hybrid',
+                'r2_method': 'ma2',  # Any singleton: MA2 over the full sequence (Bishop Eq 13.7-13.9)
                 'capabilities': ['all_residuals', 'partial_interactions', 'main_effects'],
                 'interaction_analysis': 'partial',
                 'variance_decomposition': True,
@@ -720,7 +720,7 @@ def get_sds_characteristics(self, sds: int) -> dict:
             6: {
                 'description': 'Incomplete grid with singletons (has 0s, 1s, and ≥2s)',
                 'replication_type': 'partial',
-                'r2_method': 'hybrid',
+                'r2_method': 'ma2',
                 'capabilities': ['partial_vas', 'main_effects', 'stratification'],
                 'interaction_analysis': False,
                 'variance_decomposition': True,
@@ -751,23 +751,24 @@ def get_r2_method(self, stats: StructureStats) -> R2Method:
         Returns
         -------
         R2Method
-            'exact' if all cells have replication (n_cell_min >= 2)
-            'ma2' if all cells are singletons (n_cell_max == 1)
-            'hybrid' for mixed replication (exact where n >= 2, MA2 where n = 1)
+            'exact' if every cell has replication (n_cell_min >= 2)
+            'ma2' if any cell is a singleton (n_cell_min == 1), whether all
+            cells are singletons (ADS 2) or only some (ADS 3)
 
         Notes
         -----
         Deterministic rule based on Bishop methodology:
-        - Eq 59 (exact): R2 = Y - Ȳ_kt, requires replication
-        - Eq 66 (MA2): R2 = (Y_j - Y_{j-1}) / 2, for singletons
-        - Hybrid: exact where replicated, MA2 where singleton
+        - Eq 59 (exact): R2 = Y - Ȳ_kt, requires replication in every cell
+        - Eq 13.7-13.9 (MA2): R2 = (Y_j - Y_{j-1}) / 2 over the whole ordered
+          sequence, j = 2..J, with no grouping by cell. Once any cell is a
+          singleton this applies to every observation; the replicated cells'
+          within-cell deviations are not used for R2. There is no per-cell
+          mixing of the two (the earlier "hybrid" was removed in cb037f8, and
+          the ADS 3 reference assertions validate the ungrouped MA2).
         """
         if stats.n_cell_min >= 2:
             return 'exact'
-        elif stats.n_cell_max == 1:
-            return 'ma2'
-        else:
-            return 'hybrid'
+        return 'ma2'
 
     def validate_sds_for_analysis(self, sds: int, analysis_type: str) -> bool:
         """
@@ -1296,7 +1297,7 @@ def get_analysis_plan(sds: int, min_cell_size: int = 0) -> SDSAnalysisPlan:
                 invalid_charts=[],
                 vas_residuals_supported=True,
                 residuals_available=['R1', 'R2', 'R3', 'R4', 'R5', 'R6'],
-                residual_calculation_method='moving_average',
+                residual_calculation_method='ma2',
                 main_effects_supported=True,
                 interaction_effects_supported=True,
                 supports_stratification=True,
@@ -1324,7 +1325,7 @@ def get_analysis_plan(sds: int, min_cell_size: int = 0) -> SDSAnalysisPlan:
                 invalid_charts=[],
                 vas_residuals_supported=True,
                 residuals_available=['R1', 'R2', 'R3', 'R4', 'R5', 'R6'],
-                residual_calculation_method='hybrid',
+                residual_calculation_method='ma2',
                 main_effects_supported=True,
                 interaction_effects_supported=True,
                 supports_stratification=True,
@@ -1335,7 +1336,8 @@ def get_analysis_plan(sds: int, min_cell_size: int = 0) -> SDSAnalysisPlan:
                     'Pilot studies with targeted replication',
                 ],
                 limitations=[
-                    'R2 uses hybrid calculation (exact where possible, approximate elsewhere)',
+                    'R2 uses the 2-point moving average over the full sequence (Bishop Eq 13.7-13.9); '
+                    'the replicated cells contribute no separate within-cell estimate',
                     'Variance estimates less precise than SDS 1',
                     'May have unequal subgroup sizes',
                 ],
-Original file line number
+Diff line change
@@ Expand Up @@
     **ProcessBehavior Detection**: Some cells with n=1, others with n>=2
-    **Capabilities**: Hybrid variance estimation
+    **Capabilities**: R2 by moving average over the full sequence (mixed cell sizes)
     ---
@@ Expand Down @@