Bishop’s design-state reference scale (1–6)

This document provides the formal definitions of the six structural design states defined by Thomas A. Bishop, Ph.D. in the Variation Analysis System (VAS) framework. These are the integer codes carried by each state in the PDS / ODS / ADS lineage via its .sds field.

For background on the lineage itself (Planned → Observed → Analytical), see Design-state detection.

Notation¶

• k: Factor level index (e.g., machine, lane, operator)

• t: Time period index (e.g., day, shift, batch)

• N_kt: Number of observations in cell (k, t) - the count of measurements for factor level k at time t

• Rational Subgroup: A (k, t) cell - observations sharing the same factor level and time period

DS Classification Table¶

Detailed Definitions¶

DS 1: Complete with Full Replication¶

• Condition: Min N_kt ≥ 2

• Grid Status: Complete

• Description: Every (factor × time) cell has at least 2 observations

• Implications:

  • True within-cell variance can be estimated directly

  • Full VAS residual decomposition (R1-R5) supported

  • Xbar-S charts are optimal

  • Interaction effects can be estimated precisely

DS 2: Semi-Complete with No Replication¶

• Condition: Min N_kt = 1 AND Max N_kt = 1

• Grid Status: Semi-Complete

• Description: Every (factor × time) cell has exactly 1 observation

• Implications:

  • No within-cell variance: R2 is estimated from the condition-and-period-adjusted series, differenced along the condition-then-time sequence and scaled by the R2 scale factor (Eqs 14-4..14-8)

  • Interaction effects confounded with pure error

  • VAS residuals available when the R2 scale factor is defined: at least 2 process design conditions and M ≥ K + 2 observations (see R2 calculation)

  • Common in unreplicated factorial designs

DS 3: Semi-Complete with Partial Replication¶

• Condition: Min N_kt = 1 AND Max N_kt ≥ 2

• Grid Status: Semi-Complete

• Description: Mix of cells - some have n=1, others have n≥2

• Implications:

  • R2 by the same scaled difference as DS 2, over every observation (Eqs 14-9..14-13); any one-observation cell selects ma2 for every observation

  • Xbar/S is the default chart

  • Most common in real-world data

  • Partial interaction effect estimation

  • Requires careful handling of mixed replication

DS 4: Incomplete with Replication Only¶

• Condition: Min N_kt = 0 AND N_kt ≠ 1 (no cells with exactly 1) AND Max N_kt ≥ 2

• Grid Status: Incomplete

• Description: Missing cells, but all present cells have n≥2

• Implications:

  • Incomplete grid structure

  • Where data exists, full replication is available

  • Can estimate within-cell variance for present cells

  • Common in nested/hierarchical designs with asynchronous coverage

DS 5: Incomplete with No Replication¶

• Condition: Min N_kt = 0 AND Max N_kt = 1

• Grid Status: Incomplete

• Description: Missing cells, and all present cells have exactly n=1

• Implications:

  • Most limited analytical case

  • Sparse, irregular data structure

  • No within-cell variance estimation possible

  • R2 by the scaled difference (ma2) throughout; collapses to ADS 2 after cleaning

DS 6: Incomplete with Mixed Replication¶

• Condition: Min N_kt = 0 AND some N_kt = 1 AND Max N_kt ≥ 2

• Grid Status: Incomplete

• Description: Missing cells, plus mix of n=1 and n≥2 in present cells

• Implications:

  • Incomplete (factor × time) grid

  • Some cells have no data at all

  • Present cells have mixed replication

  • Complex variance estimation required

Detection Algorithm¶

To determine the DS for a dataset:

1. Compute N_kt for all (factor, time) combinations
2. Determine:
   - min_n = minimum N_kt among PRESENT cells (excluding missing)
   - max_n = maximum N_kt
   - has_missing = whether any expected (k,t) cells have N_kt = 0
   - has_singles = whether any cells have N_kt = 1
   - has_multiples = whether any cells have N_kt ≥ 2

3. Classification:
   - If NOT has_missing (Complete/Semi-Complete grid):
     - If min_n ≥ 2 → DS 1
     - If min_n = 1 AND max_n = 1 → DS 2
     - If min_n = 1 AND max_n ≥ 2 → DS 3
   - If has_missing (Incomplete grid):
     - If NOT has_singles AND has_multiples → DS 4
     - If has_singles AND NOT has_multiples → DS 5
     - If has_singles AND has_multiples → DS 6

R2 Calculation Methods by DS¶

Equation numbers refer to Bishop’s VAS documentation manual dated 10-12-2026.

The scaled difference first removes the process mean and the condition and period effects, Z = Y − Ȳ_k − Ȳ_t + Ȳ (unweighted means of cell means), then differences Z along the condition-then-time sequence, across condition boundaries with no grouping, so only the first observation has no value. It divides by twice the R2 scale factor, computed from the layout:

c(K, M) = sqrt( (K - 1) / (2K) · (1 - K / (M - 1)) )

with K the number of process design conditions present and M the number of observations. This puts R2 on the noise (sigma) scale. The token ma2 names Bishop’s size-2 moving average of Z, of which R2 is the scaled deviation. When K < 2 or M < K + 2 the R2 scale factor is undefined and R2, with everything built on it, is reported unavailable.

References¶

• Wheeler, D. J. (1995). Advanced Topics in Statistical Process Control. SPC Press, Knoxville, TN.

• Wheeler, D. J. & Chambers, D. S. (1992). Understanding Statistical Process Control. SPC Press.

• Bishop, T. A. (2023). Personal communication — Variation Analysis System implementation.