Chart Types

ProcessBehavior supports multiple chart types, each suited for different data structures and analytical questions. This guide helps you choose the right chart.

Available Chart Types¶

Standard Charts¶

VAS Residual Charts¶

Use the value parameter to chart residuals instead of the response:

# Chart R5 residuals on Xbar
result = study.execute(chart='Xbar', value='R5')

# Chart R4 residuals on stratified X
result = study.execute(chart='X', by=['lane'], value='R4')

The by Parameter¶

The by parameter controls grouping and stratification:

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

# Xbar/S - aggregate by different levels
result = study.execute(chart='Xbar')                  # By all factors (default)
result = study.execute(chart='Xbar', by=['lane'])     # By single factor
result = study.execute(chart='Xbar', by=[])           # Collapse to grand mean

# X - stratify by factor(s)
result = study.execute(chart='X', by=['lane'])      # One chart per lane

Use study.valid_charts and study.residuals to see options.

Chart Selection Guide¶

Question: “Are my subgroups different from each other?”¶

Use: Xbar Chart

result = study.execute(chart='Xbar')
fig = result.plot(chart='Xbar', show_zones=True, highlight_signals=True)

Points beyond limits indicate subgroups that differ from the overall average.

Question: “Is within-group variation stable?”¶

Use: S Chart

result = study.execute(chart='S')
fig = result.plot(chart='S', show_zones=True)

Points beyond limits indicate subgroups with unusual variation.

Question: “How is my process behaving over time?”¶

Use: X Chart

result = study.execute(chart='X', by=['lane'])
fig = result.plot(show_zones=True, show_rules=True)

For stratified data, this creates one X chart per factor level.

Question: “Do factor effects change over time?”¶

Use: X with value='R3' (Interactions)

result = study.execute(chart='X', by=['lane'], value='R3')
fig = result.plot(title='Factor × Time Interactions')

Signals indicate that factor effects are not consistent over time.

Question: “Are there time trends after removing factor effects?”¶

Use: X with value='R4'

result = study.execute(chart='X', by=['lane'], value='R4')
fig = result.plot(title='Time Effects')

Signals indicate process drift or shifts over time.

Question: “Are there design condition differences after removing time effects?”¶

Use: X with value='R5'

result = study.execute(chart='X', by=['lane'], value='R5')
fig = result.plot(title='Design Condition Main Effects')

Signals indicate true differences between process design conditions.

Chart Validity by DS¶

Not all charts are valid for all Design States:

Residual availability depends on DS. Use study.residuals to check:

print(f"Valid charts: {study.valid_charts}")
print(f"Available residuals: {study.residuals}")

Note on R2: DS 2 and 5 use the moving average method; DS 1, 3, 4, 6 use within-cell deviation (R2 = Y - Ȳ_kt). See VAS Residuals for details.

Companion Charts¶

Wheeler recommends reading certain charts as pairs: Xbar with S, and X with mR. The companion parameter returns both charts in one result:

# Returns both Xbar and S charts
result = study.execute(chart='Xbar', companion=True)
result.plot(chart='Xbar')  # Xbar chart
result.plot(chart='S')     # S chart

# Returns both X and mR charts, stratified
result = study.execute(chart='X', by=['lane'], companion=True)

Either chart in the pair triggers the pair: chart='S', companion=True also returns Xbar+S.

Effects and Interaction Charts¶

When your study has factors, ProcessBehavior can visualize main effects and interactions. These charts help answer: Are the factor and time effects practically significant?

result = study.execute(chart='Xbar')

# All main effects combined
result.plot(chart='Effects')

# Factor effects only
result.plot(chart='MainEffects')

# Time effects only
result.plot(chart='TimeEffects')

# Factor x time interaction
result.plot(chart='TimeInteraction')

# Factor x factor interaction (requires 2+ factors)
result.plot(chart='FactorInteraction')

Effects charts require result.has_effects == True (i.e., the study must have factors). Interaction charts require the corresponding dimensions (factors + time for TimeInteraction, 2+ factors for FactorInteraction).

You can also access the raw effects data:

# Keys are named after your own factor columns, plus 'main_effect' and 'time'.
# For factors=['machine', 'shift']:
result.effects['machine']   # per-level main effects  (column: Main_Effect)
result.effects['time']      # time main effects       (column: PT_ME)
result.effects              # see AnalysisResult.effects for the full key list
result.interactions         # Dict of interaction terms

Understanding Xbar-S Charts¶

The Xbar Chart¶

Plots the mean of each subgroup (factor level at each time point).

• Centerline: Grand mean (Y̅)

• Control Limits: Based on within-subgroup variation

• Interpretation: Points beyond limits indicate subgroups with unusual means

!!! note “Limits for effect-carrying residuals (R4/R5)” When charting R4, R5, or their recentered variants (RCR4, RCR5) on Xbar, limits are based on R2’s within-group standard deviation (Sbar), not the plotted residual’s own standard deviation. This matters when by collapses factors — e.g., by=['factor 1'] in a two-factor study. At collapsed groupings, R5’s within-group std would include between-cell variance from the collapsed dimension, inflating limits. Using R2’s Sbar isolates within-cell noise as the limit basis. At the full RSG level (all factors in by), R5’s within-group std equals R2’s, so there is no difference. This follows Dr. Tom Bishop’s VAS methodology.

The S Chart¶

Plots the standard deviation of each subgroup.

• Centerline: Pooled within-subgroup standard deviation

• Control Limits: Based on chi-square distribution

• Interpretation: Points beyond limits indicate subgroups with unusual variation

!!! note “S chart with effect residuals (R3/R4/R5)” When charting an effect residual on S, the data points show R2’s within-group standard deviation, not the requested residual’s. The S chart always measures within-cell noise stability. See How Effect Residuals Are Charted.

Reading Order¶

1. First check S chart - Variation must be stable

2. Then interpret Xbar - Only meaningful if S is stable

3. Investigate signals - What makes that subgroup different?

Understanding X and mR Charts¶

The X (Individual) Chart¶

Plots each individual observation.

• Centerline: Average of all observations (X̅)

• Control Limits: X̅ ± 2.66 × R̅ (average moving range)

• Interpretation: Points beyond limits indicate special causes

The mR (Moving Range) Chart¶

Plots the absolute difference between consecutive observations.

• Centerline: Average moving range (R̅)

• UCL: 3.27 × R̅

• LCL: 0 (range cannot be negative)

• Interpretation: Large ranges indicate sudden changes

Stratified X Charts¶

When you have factors and time, use the by parameter to stratify X charts:

# Stratify by lane - creates one chart per lane
result = study.execute(chart='X', by=['lane'])

# Check the strata
print(result.charts['X']['strata'])  # ['A', 'B', 'C', 'D']

Each stratum has its own control limits based on its internal variation:

# View faceted plot with all lanes
fig = result.plot(chart='X', show_zones=True)

Lane Boundaries¶

When you collapse factors (use fewer factors in by than exist in the study), lane boundaries show where the collapsed factors change:

# Single X chart with lane boundaries
result = study.execute(chart='X', by=[])  # Collapse all factors
fig = result.plot(chart='X')  # Vertical lines show factor transitions

Re-centered Residual Charts¶

By default, residual charts are centered at zero. Use recentered=True to show residuals on the original measurement scale:

# Zero-centered (default)
result = study.execute(chart='X', by=['lane'], value='R4')

# Re-centered on original scale
result = study.execute(chart='X', by=['lane'], value='R4', recentered=True)

Re-centering uses:

• R4: RCR4 = R4 + Y̅_t (adds back time mean)

• R5: RCR5 = R5 + Y̅_k (adds back factor mean)

Decision Tree¶

Do you have factors?
├── No → Do you have time?
│   └── Yes → DS 6: Use X
└── Yes → Do you have time?
    ├── 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
        └── No cells → DS 2: Xbar-S with MR-based limits

Summary¶

Next Steps¶

• Plotting & Themes - Visualization options for all charts

• VAS Residuals - Deep dive into VAS residuals

• Coffee Shop - A complete analysis, start to finish