Analyze a Box–Behnken design

Fit a curved response from edge and center runs, then check the model before using its predictions.

Guides · Design of experiments (DOE) · 5 min read

Example: ink mixing

15 independent ink batches: 12 edge-midpoint combinations and three center runs. An edge-midpoint run places two factors at low/high settings and the third at its midpoint.

Download the recorded measurements (CSV)

This is the same recorded example used in Help. The CSV contains factor settings and measured responses to import together. Use Open example in a practice copy in Help to prefill the settings.

1. Create the workspace

In the sidebar, open Design of Experiments and create an analysis. Name it, select New blank worksheet, then choose Create analysis.

2. Import the measurements

Download the example CSV above and import it into the worksheet. Choose Comma as the delimiter, Dot as the decimal separator and First row contains column names. Keep each row’s response, factor settings, point type and execution order together.

3. Choose the study and enter factors

On Inputs, choose Analyze existing worksheet for Task and Response surface designs for Design family. Choose Box–Behnken for Study type.

Under Factors, use Add factor or remove a row until there are 3 factors. Keep this order and set Type to Numeric for each factor.

Enter the Factor name, Unit, Low and High values below.

FactorFactor nameUnitLowHigh
Amix_temperature_c°C2040
Bimpeller_speed_rpmrpm400800
Cmixing_time_minmin816

4. Assign the input columns

Under Response mappings, open each column picker and select the matching worksheet column.

Worksheet columnInput field
mix_temperature_cFactor mix_temperature_c
impeller_speed_rpmFactor impeller_speed_rpm
mixing_time_minFactor mixing_time_min

Assign viscosity_mpa_s to Response 1 and enter 1 in Number of responses. Assign run_order to Execution order and point_type to Center point indicator. For Center point column convention, choose Point type (center / factorial / fractional / star / axial / edge / edge_midpoint).

Leave Block unassigned; this recorded study is unblocked.

5. Set the model and plots

Under Model, choose Default hierarchical model for Model terms and enter 95 in Confidence level (%). Under Predictions, check Calculate predictions. Enter 30 in mix_temperature_c (°C). Enter 600 in impeller_speed_rpm (rpm). Enter 12 in mixing_time_min (min).

Open Plot options. Check Effect estimates and Residual diagnostics and Response contours. Leave Pareto view, Effect probability plot, Interaction plot and 3D surface unchecked. Choose mix_temperature_c for Horizontal factor and impeller_speed_rpm for Vertical factor. Enter 12 in Hold mixing_time_min. Effect estimates is enabled, but this quadratic model produces no factorial-effects chart.

Design of Experiments: Inputs with the example’s assignments and settings
Inputs with the recorded-study mappings, factor bounds, model, confidence level and plot options. View full-size image (opens in a new tab)

6. Analyze worksheet

Choose Analyze worksheet, then open Results. This fits the recorded measurements; it does not generate or randomize a new run plan.

7. Interpret results

  1. Start with the recorded batches

    All 15 independently prepared ink batches are included: 12 edge-midpoint combinations and three center runs. The response is viscosity in mPa·s. The study target is 1,900 mPa·s. Combinations with all three factors at their extremes were not tested.

  2. Check the lack of fit first

    Lack of Fit has p = 0.04596. At the 5% level, the quadratic model misses a pattern beyond the variation seen in repeated batches. Even with a high R², its predictions need further investigation and confirmation.

  3. Read a residual

    A residual is measured minus fitted viscosity. Run 14 has about +11.54 mPa·s and run 15 about −11.54 mPa·s; their absolute sizes are approximately tied. Inspect both records and the whole residual cloud for curves or changing spread. Neither is a uniquely worst run, and the detected lack of fit still limits predictions.

    Residuals versus fits
    Residuals in mPa·s against the fitted response. View full-size chart (opens in a new tab)
  4. Check the run sequence

    Read the residuals in recorded execution order. Look for drift, cycles or abrupt changes, then check the preparation and measurement records. Investigate unusual runs before changing the model or excluding measurements.

    Residuals versus run order
    Residuals in mPa·s in recorded execution order. View full-size chart (opens in a new tab)
  5. Read the center prediction cautiously

    At 30 °C, 600 rpm and 12 minutes, the fitted mean is about 1,796 mPa·s, below the 1,900 target. The 95% mean interval is 1,779–1,813; the interval for one new batch is 1,762–1,830 mPa·s. Both depend on the model with detected lack of fit. They do not establish a validated recipe.

  6. Read the contour at a fixed time

    The horizontal axis is mixing temperature and the vertical axis is impeller speed. Each line joins settings with the same predicted viscosity, holding mixing time at 12 minutes. Check the marked extrapolation regions and remember the detected lack of fit. The displayed grid contains predictions, not additional measured batches.

    Response contour · surface-1
    Predicted response in mPa·s, holding mixing_time_min at 12 min. View full-size chart (opens in a new tab)
  7. Resolve the model issue before choosing a recipe

    Review the repeated-batch variation, residuals and experiment records. Plan additional feasible runs to investigate the missing pattern. Confirm any proposed setting with fresh batches and measured viscosity; a contour near the target is not enough.

More details: read each result section

Complete result tables use six significant digits, as in Help.

Model overview

Analyzed runs is 15; Missing runs is 0. DF counts the information available for the model and residual error. S is the residual standard deviation in mPa·s. R² describes variation fitted in these runs; adjusted R² accounts for model size. PRESS and predicted R² use leave-one-out prediction errors. A high R² does not establish model adequacy. Read the lack-of-fit test and residual plots.

Model overview · 12 rows
StatisticValue
Responseviscosity_mpa_s
Analyzed runs15
Missing responses0
Model rank10
Residual degrees of freedom5
S11.5187
R²0.996825
Adjusted R²0.99111
PRESS10332.8
Predicted R²0.950547
Confidence level0.95
Lack of fit availableYes

Adjusted term ANOVA

Source names the model term or error component. Type distinguishes adjusted term tests from totals and error decomposition. DF is degrees of freedom; SS is sum of squares; MS is SS divided by DF. F compares variation with its error estimate; p-value measures evidence against the tested null assumption. Read Lack of Fit before relying on individual effects. A dash means a statistic is not applicable.

Adjusted term ANOVA · 14 rows
SourceTypeDFSSMSFp-value
mix_temperature_cadjusted173095.873095.8550.9140.00000261327
impeller_speed_rpmadjusted128191.328191.3212.4740.00002744
mixing_time_minadjusted125764.525764.5194.1840.0000342059
mix_temperature_c × impeller_speed_rpmadjusted17849.967849.9659.16420.000592321
mix_temperature_c × mixing_time_minadjusted13393.063393.0625.57310.00391
impeller_speed_rpm × mixing_time_minadjusted13950.123950.1229.77160.00281165
mix_temperature_c²adjusted122806.122806.1171.8870.0000460787
impeller_speed_rpm²adjusted135999.735999.7271.3250.0000150515
mixing_time_min²adjusted116931.316931.3127.6090.0000950193
Modelsummary920828023142.2174.420.0000104963
Errorsummary5663.404132.681——
Totalsummary1420894314924.5——
Pure Errordecomposition220.486710.2433——
Lack of Fitdecomposition3642.917214.30620.92150.0459622

Coefficients

Term names a coded model term. Estimate is its fitted coefficient; SE measures uncertainty; t and p-value test a zero coefficient. CI low and CI high bound its 95% interval. Numeric low, midpoint and high settings are coded −1, 0 and +1. With interactions or squares, a main coefficient does not describe the change at every setting.

Coefficients · 10 rows
TermEstimateSEtp-valueCI lowCI high
Intercept1796.176.65033270.0871.32046e-111779.071813.26
mix_temperature_c95.58754.0724823.47160.0000026132785.1189106.056
impeller_speed_rpm59.36254.0724814.57650.0000274448.893969.8311
mixing_time_min-56.754.07248-13.9350.0000342059-67.2186-46.2814
mix_temperature_c × impeller_speed_rpm44.35.759367.691830.00059232129.495159.1049
mix_temperature_c × mixing_time_min-29.1255.75936-5.056990.00391-43.9299-14.3201
impeller_speed_rpm × mixing_time_min31.4255.759365.456340.0028116516.620146.2299
mix_temperature_c²78.59175.9945313.11060.000046078763.182294.0011
impeller_speed_rpm²98.74175.9945316.4720.000015051583.3322114.151
mixing_time_min²67.71675.9945311.29640.000095019352.307283.1261

Coefficient covariance

Read each row and column term together. Diagonal entries are the squared standard errors of the fitted coefficients. Off-diagonal entries describe shared estimation uncertainty. This matrix concerns model coefficients, not correlations between the measured factor columns.

Coefficient covariance · 10 rows
ColumnInterceptmix_temperature_cimpeller_speed_rpmmixing_time_minmix_temperature_c × impeller_speed_rpmmix_temperature_c × mixing_time_minimpeller_speed_rpm × mixing_time_minmix_temperature_c²impeller_speed_rpm²mixing_time_min²
Intercept44.2269-8.50468e-150-2.37713e-150-2.37713e-150-22.1135-22.1135-22.1135
mix_temperature_c-8.50468e-1516.5851-3.02425e-15-4.73612e-152.99945e-15-4.24231e-166.29613e-15-1.56822e-15-6.52517e-161.2779e-14
impeller_speed_rpm0-2.91183e-1516.5851-2.69128e-15-5.1223e-151.31912e-151.35458e-155.35377e-163.38424e-15-3.99719e-15
mixing_time_min-2.37713e-15-4.89798e-15-2.50689e-1516.58516.30649e-166.26751e-15-2.13614e-15-1.33722e-158.39623e-15-4.97718e-15
mix_temperature_c × impeller_speed_rpm02.99945e-15-5.1223e-156.30649e-1633.17021.72094e-151.03491e-155.28011e-15-5.74915e-157.20535e-16
mix_temperature_c × mixing_time_min-2.37713e-15-4.24231e-161.31912e-156.26751e-151.72094e-1533.17026.64018e-15-2.17563e-16-4.49906e-156.2007e-15
impeller_speed_rpm × mixing_time_min06.29613e-151.35458e-15-2.13614e-155.74578e-165.43475e-1533.1702-2.22912e-15-9.45378e-162.81433e-15
mix_temperature_c²-22.1135-1.56822e-155.35377e-16-1.33722e-155.28011e-159.87861e-16-2.22912e-1535.93442.764182.76418
impeller_speed_rpm²-22.1135-6.52517e-163.38424e-158.39623e-15-5.74915e-15-3.5784e-15-1.40571e-152.7641835.93442.76418
mixing_time_min²-22.11351.2779e-14-3.99719e-15-4.97718e-157.20535e-164.99527e-152.81433e-152.764182.7641835.9344

Predictions

Factor settings and Block identify the prediction requested. Fit is the estimated mean response. Mean SE and Mean CI describe uncertainty in that mean; Individual PI also includes variation in a new independent run, so it is wider. All intervals use 95% confidence and the fitted model. They are not acceptance limits. Detected lack of fit qualifies the viscosity prediction and both intervals.

Predictions · 1 row
PredictionFactor settingsBlockFitMean SEMean CI lowMean CI highIndividual PI lowIndividual PI high
prediction-1mix_temperature_c=30, impeller_speed_rpm=600, mixing_time_min=12—1796.176.650331779.071813.261761.981830.36

Surface values · surface-1

Each row gives the two plotted factor settings and their Fitted response in mPa·s, holding mixing_time_min at 12. Region marks extrapolation beyond the observed design hull. All grid rows are included; these are model predictions, not additional measured runs. Detected lack of fit qualifies this entire surface.

Surface values · surface-1 · 625 rows
mix_temperature_cimpeller_speed_rpmFitted responseRegion
204001862.85Observed slice region
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  • Held settings: mixing_time_min=12.
  • Region check: full numeric design hull within the selected categorical and block settings.
  • Sampled rows: help-doe-box-behnken-analysis-1 (different held numeric settings), help-doe-box-behnken-analysis-2 (different held numeric settings), help-doe-box-behnken-analysis-3, help-doe-box-behnken-analysis-4 (different held numeric settings), help-doe-box-behnken-analysis-5, help-doe-box-behnken-analysis-6, help-doe-box-behnken-analysis-7 (different held numeric settings), help-doe-box-behnken-analysis-8, help-doe-box-behnken-analysis-9 (different held numeric settings), help-doe-box-behnken-analysis-10 (different held numeric settings), help-doe-box-behnken-analysis-11, help-doe-box-behnken-analysis-12, help-doe-box-behnken-analysis-13, help-doe-box-behnken-analysis-14 (different held numeric settings), help-doe-box-behnken-analysis-15 (different held numeric settings).

Row diagnostics

Row ID identifies the worksheet row; Run order locates it in the experiment sequence. Status states whether it was analyzed. Observed is the measurement; Fit is the model value; Residual is Observed minus Fit. Standardized residual adjusts for residual variation and leverage. Leverage measures how unusual the model inputs are. All recorded rows are shown. Investigate an unusual row before changing or excluding it.

Row diagnostics · 15 rows
Row IDRun orderStatusObservedFitResidualStandardized residualLeverage
help-doe-box-behnken-analysis-11Analyzed1947.61952.19-4.5875-0.796530.75
help-doe-box-behnken-analysis-22Analyzed1809.81815.09-5.2875-0.9180710.75
help-doe-box-behnken-analysis-33Analyzed1975.31965.429.8751.71460.75
help-doe-box-behnken-analysis-44Analyzed2052.62047.315.28750.9180710.75
help-doe-box-behnken-analysis-55Analyzed21792172.756.251.085190.75
help-doe-box-behnken-analysis-66Analyzed1856.61862.85-6.25-1.085190.75
help-doe-box-behnken-analysis-77Analyzed1879.11874.514.58750.796530.75
help-doe-box-behnken-analysis-88Analyzed1799.41796.173.233330.3437890.333333
help-doe-box-behnken-analysis-99Analyzed19951996.66-1.6625-0.2886610.75
help-doe-box-behnken-analysis-1010Analyzed1993.11991.441.66250.2886610.75
help-doe-box-behnken-analysis-1111Analyzed1796.11796.17-0.0666667-0.007088430.333333
help-doe-box-behnken-analysis-1212Analyzed17931796.17-3.16667-0.3367010.333333
help-doe-box-behnken-analysis-1313Analyzed1883.11892.97-9.875-1.71460.75
help-doe-box-behnken-analysis-1414Analyzed1830.81819.2611.53752.003260.75
help-doe-box-behnken-analysis-1515Analyzed2112.42123.94-11.5375-2.003260.75

Factor coding

Factor names the input and Type its modeling type. Center and Scale translate numeric settings to coded values: (setting − center) / scale. Thus the midpoint is 0 and the entered low/high bounds are −1/+1. Levels / contrasts applies to categorical coding; a dash is not a missing numeric measurement. Keep the model-column, block and rank notes with this table.

Factor coding · 3 rows
FactorTypeCenterScaleLevels / contrasts
mix_temperature_cnumeric3010—
impeller_speed_rpmnumeric600200—
mixing_time_minnumeric124—
  • Model columns: Intercept, mix_temperature_c, impeller_speed_rpm, mixing_time_min, mix_temperature_c × impeller_speed_rpm, mix_temperature_c × mixing_time_min, impeller_speed_rpm × mixing_time_min, mix_temperature_c², impeller_speed_rpm², mixing_time_min²
  • Rank tolerance: 1e-10

8. Create a report

Keep Model overview, Adjusted term ANOVA, Coefficients, Coefficient covariance, Row diagnostics and Factor coding, plus Predictions, the complete Surface values grid, residual plots and contour. Include the response units, factor settings, recorded run order and all model notes. State the detected lack of fit when sharing conclusions or predictions.

Choose Add to report from the completed run. Open Choose individual items and advanced options to select tables and charts, then choose Add to draft. Hidden items stay in the saved analysis.

Name the report and add the purpose of the study and your interpretation. Choose Save, review Preview, then use Export → Download HTML. The example below contains the full selected content, including results beyond one screen.

Complete native output for this Help example, including applicable tables, charts and run notes. Report numbers use the app’s displayed precision. Exported charts are static. On narrow screens, scroll wide charts and tables sideways.
View example report: Analyze Box–Behnken: ink mixing

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