Design-state lineage (PDS / ODS / ADS)

ProcessBehavior reports the design state at three points in your analysis lifecycle, and routes chart selection, residual availability, and variance decomposition by the third one (ADS).

The three states¶

The integer 1–6 codes are Bishop’s reference scale (“Bishop Table 1”). Each of PDS, ODS, ADS carries a value on that scale via its .sds field.

Access the full lineage on a formulated study:

study.plan_design_state        # PDS (or None when no plan was supplied)
study.observed_design_state    # ODS — Bishop's classification of the raw N_kt distribution
study.analytical_design_state  # ADS — what the analysis is fit for
study.design()                 # DesignReport — all three in one report object

Expected vs Observed: the underlying model¶

The lineage above falls out of ProcessBehavior’s expected-vs-observed model:

• Observed structure comes from your data — the actual factor levels, time points, and cell sizes present.

• Expected structure comes from your sampling plan — what the experimental design intended.

The system operates effectively with or without an explicit plan=:

ODS detection runs on raw data before NA rows are dropped, so cells where all response values are NA are counted as empty (N_kt=0). This enables ODS 4–6 detection even without a plan. With a plan, ProcessBehavior can additionally compare what was collected against what should have been collected — catching entirely absent factor levels and providing detailed reports showing exactly what’s missing.

Bishop’s six design states (the 1–6 scale)¶

Each of PDS, ODS, ADS reports a value on this scale.

Complete/Semi-Complete (no empty cells):

Incomplete (has empty cells — applies to ODS only; collapses during tidying):

How DS is Detected¶

When you call formulate(), ProcessBehavior examines:

1. Presence of factors - Do you have grouping variables?

2. Presence of time - Do you have a time/sequence variable?

3. Replication - How many observations per (factor, time) cell?

4. Structure - Is the design balanced or irregular?

5. Grid coverage - What proportion of expected cells are observed?

study = pb.formulate(
    response='weight',
    factors=['lane'],
    time='batch'
)

print(f"ODS: {study.observed_design_state.sds}  ADS: {study.analytical_design_state.sds}")
print(f"Reason: {study.ads_reason}")
print(f"Description: {study.ads_description}")

The VAS Problem Formulation¶

Following Bishop’s Variance Analysis System (VAS) methodology, a rigorous problem formulation includes:

• Unit of Analysis: The fundamental entity being measured (e.g., “filled cup”, “loan contract”)

• Response: The measurement variable

• Factors: Grouping variables that define rational subgroups

• Time: The sequencing variable

from processbehavior import ProcessBehavior

pb = ProcessBehavior(df)

# Basic formulation - factors inferred from observed data
study = pb.formulate(
    response='weight',
    factors=['lane', 'shift'],
    time='batch'
)

print(f"DS: {study.observed_design_state.sds}")
print(f"Charts: {study.valid_charts}")

This works for all DS types (1-6). DS detection is determined by properties of the observed data—replication levels, cell sizes, and factor structure—including cells where all response values are NA (counted as empty).

Extending Formulation with a Sampling Plan¶

A sampling plan extends the formulation by explicitly defining what was expected in the experimental design:

plan = {
    'factors': {
        'Machine': ['Machine1', 'Machine2', 'Machine3'],
        'Shift': ['Day', 'Night']
    },
    'T': 80,   # Expected time points
    'N': 2     # Expected observations per cell
}

study = pb.formulate(response='weight', time='time', plan=plan)

The plan provides three key benefits:

1. Catches entirely absent factor levels - Factor levels or time points not present in data at all are only visible with a plan

2. Design reports - See exactly what’s missing from your data (planned vs observed K, T, N, R)

3. Documentation - Captures the intended experimental design for reproducibility

DS 4-6 Detection: With and Without a Plan¶

ProcessBehavior can detect DS 4-6 with or without a plan. DS detection runs on raw data before NA rows are dropped, so cells where all response values are NA (e.g., from garbage values like * or ND) are counted as empty cells (N_kt=0).

A plan adds value by catching gaps that factors alone cannot see—for example, if a factor level was planned but never collected at all, no rows exist for that level, so factors won’t see it. But when the data contains all planned factor levels (even if some cells have all-NA responses), both approaches detect the same DS.

# With factors: ProcessBehavior sees all-NA cells as empty → DS 4-6 detected
# With plan: additionally catches factor levels entirely absent from data

Expected vs Observed: How It Works¶

When you provide a plan, ProcessBehavior compares:

design = study.design()
print(f"K: planned={design.K}, observed={design.K_observed}, missing={design.K_missing}")
print(f"T: planned={design.T}, observed={design.T_observed}, missing={design.T_missing}")
print(f"R: planned={design.R}, observed={design.R_observed}, missing={design.R_missing}")

Plan Parameter Structure¶

plan = {
    'factors': {
        'factor_name': [list of expected levels],
        ...
    },
    'T': expected_time_points,      # optional
    'N': expected_obs_per_cell      # optional
}

Notes:

• Factor levels should include ALL expected levels, even if some are missing from data

• T enables detection of missing time points

• N documents expected replication (useful for design reports)

Viewing the Design Report¶

After formulation, inspect how your data compares to the plan:

design = study.design()
print(design)

# DesignReport(2 factors, with plan)
#   SDS reason: incomplete_grid
#   K: planned=6, observed=6, missing=0
#   T: planned=80, observed=8, missing=72
#   R: planned=480, observed=34, missing=446
#
#   Factors:
#     Machine: planned=['Machine1','Machine2','Machine3'], observed=[...]
#     Shift: planned=['Day','Night'], observed=[...]
#
#   Structure: Incomplete: 72 time points missing

The design report shows:

• K, T, R: Planned vs observed counts with missing tallies

• N: Planned vs observed (min, median, max) cell sizes

• missing_combos: Which factor combinations are missing

• sds_reason: Why this DS was detected

• structure_summary: Overall assessment

Summary: When Plans Add Value¶

Best practice: Always use a plan for rigorous VAS analysis. While not required for DS detection, plans document your experimental design, catch entirely absent factor levels, and enable rich design reports.

DS 1: Full Replication¶

Structure: Every (factor, time) combination has 2+ observations.

Example: 4 lanes × 10 batches × 3 replicates = 120 observations

# Every lane-batch combination has exactly 3 measurements
# Lane A, Batch 1: [99.2, 100.1, 99.8]
# Lane A, Batch 2: [100.3, 99.9, 100.5]
# ...

Capabilities:

• ✅ Exact within-cell variance estimation

• ✅ All VAS residuals (R1-R5)

• ✅ Full interaction analysis

• ✅ Most powerful statistical tests

Valid Charts: Xbar, S, X + residual charts (R2 via S, R3-R5 via Xbar/S)

DS 2: No Replication¶

Structure: Every (factor, time) combination has exactly 1 observation.

Example: 4 lanes × 10 batches × 1 measurement = 40 observations

# Each lane-batch combination has exactly 1 measurement
# Lane A, Batch 1: [99.5]
# Lane A, Batch 2: [100.2]
# ...

Capabilities:

• ⚠️ Variance estimated via 2-point moving average

• ⚠️ R2 residuals approximate (using backward moving range)

• ✅ Main effects analysis

• ✅ Xbar-S analysis (with MR-based limits)

Valid Charts: Xbar, S, X

DS 3: Partial Replication¶

Structure: Mix of n=1 and n>=2 cells (most common in practice).

Example: Some batches have 3 replicates, others have 1

# Lane A, Batch 1: [99.2, 100.1, 99.8]  # 3 reps
# Lane A, Batch 2: [100.3]               # 1 rep
# Lane A, Batch 3: [99.7, 100.0]         # 2 reps
# ...

Capabilities:

• ⚠️ Hybrid R2 estimation (exact where n >= 2, ma2 where n = 1)

• ⚠️ VAS residuals available but interpretation requires care

• ✅ Xbar-S analysis with hybrid 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.

DS 4: Incomplete, No Singletons¶

Structure: Incomplete grid — some cells have no data, but all present cells have N_kt >= 2.

Example: A planned 4-lane x 10-batch study where 3 batches were skipped entirely, but all collected batches have 3+ replicates.

After cleansing: Collapses to ADS 1 (Full Replication) — the empty cells are removed and all remaining cells have full replication.

Valid Charts: Histogram, X, mR (plus Xbar, S after collapse to ADS 1)

DS 5: Incomplete, No Replication¶

Structure: Incomplete grid — some cells have no data, and all present cells have N_kt = 1.

Example: A sparse unreplicated factorial where some factor-time combinations were never measured.

After cleansing: Collapses to ADS 2 (No Replication) — the empty cells are removed, leaving an unreplicated structure.

Valid Charts: Histogram, X, mR (plus Xbar, S after collapse to ADS 2)

DS 6: Incomplete, With Singletons¶

Structure: Incomplete grid — some cells have no data, and present cells have a mix of N_kt = 1 and N_kt >= 2.

Example: A manufacturing study where some shifts had one measurement, others had three, and some were skipped entirely.

After cleansing: Collapses to ADS 3 (Partial Replication) — the empty cells are removed, leaving a mixed-replication structure.

Valid Charts: Histogram, X, mR (plus Xbar, S after collapse to ADS 3)

From Observation to Analysis: ODS → ADS¶

ProcessBehavior tracks three Design States as data flows through the system. Understanding this pipeline is key to interpreting your results correctly.

The Pipeline¶

1. Plan Design State (PDS) — Computed from your sampling plan parameters (K × T × N). Always DS 1 or DS 2. Only available when you provide a plan to formulate().

2. Observed Design State (ODS) — Detected on raw data before any NA filtering. Response rows with garbage values (*, ND, etc.) are preserved during detection, so cells where all responses are NA count as empty cells (N_kt = 0). This enables detection of DS 4-6 (Incomplete designs).

3. Analytical Design State (ADS) — Computed on tidy data after data cleansing removes invalid response rows. The ADS reflects the structure that is actually fit for analysis and drives all analysis decisions: valid charts, R2 calculation method, residual availability, and interaction analysis.

Why ODS and ADS Can Differ¶

After data cleansing, empty cells disappear. DS 4-6 (Incomplete designs) collapse to their Complete/Semi-Complete equivalents:

This separation means the system correctly identifies incomplete data collection (via ODS) while performing the most powerful analysis the clean data supports (via ADS).

Checking Your Design States¶

study = pb.formulate(
    response=pb.cols.weight,
    factors=[pb.cols.lane],
    time=pb.cols.batch
)

# The ADS drives analysis — use these for chart selection
print(f"ADS: {study.analytical_design_state.sds}")  # e.g., 1
print(f"Reason: {study.ads_reason}")                # e.g., "full_replication"
print(f"Description: {study.ads_description}")       # Human-readable

# The ODS captures the raw data structure (diagnostic)
print(f"ODS: {study.observed_design_state.sds}")    # e.g., 6

# Chart validity is determined by the ADS
print(f"Valid charts: {study.valid_charts}")
print(f"Recommended: {study.recommended_chart}")
print(f"Available residuals: {study.residuals}")

# Full design lineage via study.design()
print(study.design())

Impact on Analysis¶

The Analytical Design State determines three key aspects of the analysis:

1. Variance Estimation (R2 Method)¶

The R2 method is determined by the tidy data structure (ADS), not the raw DS:

2. Available Charts¶

Standard Charts¶

VAS Residual Charts¶

The available chart types for each residual depend on the rational subgrouping structure:

*When cells have n≥2. DS 2 and 5 use X only.

Key insight: R4 and R5 use different rational subgrouping than R2/R3:

• R4: Aggregates observations across factor levels for each time point (N_.t = Σ_k N_kt)

• R5: Aggregates observations across time for each factor level (N_k. = Σ_t N_kt)

This enables Xbar/S analysis for R4 and R5 even when individual cells have n=1, because the aggregated subgroups have larger sample sizes.

Note on R2 calculation: R2 adapts to your sampling structure:

• DS 1: Within-cell deviation (R2 = Y - Ȳ_kt)

• DS 2, 5: Moving average method (R2 = Y - MA2) for unreplicated/sparse designs

• DS 3, 4, 6: Within-cell deviation (R2=0 for cells with n=1)

3. Signal Detection Rules¶

Improving Your DS¶

To get a higher DS (more analytical power):

Move from DS 2 → DS 1¶

Add replicates to each cell:

# Instead of 1 measurement per lane-batch
# Take 3 measurements per lane-batch

Move from DS 3 → DS 1¶

Ensure all cells have the same number of replicates:

# Standardize data collection
# Every lane-batch combination gets exactly n measurements

Move from DS 6 → DS 1-3¶

Add factor levels:

# Instead of one sensor
# Monitor multiple sensors simultaneously

Example: Diagnosing DS¶

import pandas as pd
from processbehavior import ProcessBehavior

# Check cell sizes to understand your DS
study = pb.formulate(
    response=pb.cols.weight,
    factors=[pb.cols.lane],
    time=pb.cols.batch
)

# Get cell counts
cell_counts = (study.dataset
    .groupby(['lane', 'batch'])
    .size()
    .reset_index(name='n'))

print("Observations per cell:")
print(cell_counts['n'].value_counts())

# If all n >= 2: DS 1
# If all n == 1: DS 2
# If mixed: DS 3

Summary¶

Next Steps¶

• Chart Types - Choosing the right chart for your DS

• VAS Residuals - VAS residual interpretation by DS

• API Reference - Complete DS detection API