Analyze a central composite design

Fit a curved response to experimental runs and read predictions at stated factor settings.

Guides · Design of experiments (DOE) · 6 min read

Example: coating durability

20 independently prepared panels: eight factorial corners, six face-centered axial points, and six center runs. Corners combine low/high settings for all factors. Axial points move one factor to its bound while the others stay at their midpoints. Center runs use all midpoints.

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 Central composite 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
Adrying_temperature_c°C5575
Bair_speed_m_per_sm/s12
Cdrying_time_minmin1220

4. Assign the input columns

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

Worksheet columnInput field
drying_temperature_cFactor drying_temperature_c
air_speed_m_per_sFactor air_speed_m_per_s
drying_time_minFactor drying_time_min

Assign abrasion_mass_loss_mg 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 65 in drying_temperature_c (°C). Enter 1.5 in air_speed_m_per_s (m/s). Enter 16 in drying_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 drying_temperature_c for Horizontal factor and air_speed_m_per_s for Vertical factor. Enter 16 in Hold drying_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 panels

    All 20 panels are included: eight factorial corners, six face-centered axial points and six center runs. The response is abrasion mass loss in mg. Lower loss means less material removed in this test. The quadratic model allows bends and interactions as settings change.

  2. Check whether the model misses a pattern

    Lack of Fit has p = 0.8313. This test does not detect a missing pattern at the 5% level. It does not prove the model is correct; read the residual plots before using its predictions.

  3. Read the prediction at the center

    At 65 °C, 1.5 m/s and 16 minutes, the fitted mean loss is 42.03 mg. The 95% interval for the mean is 41.31–42.75 mg. The interval for one new panel is wider, 39.83–44.24 mg, because it also includes panel-to-panel variation. These are model intervals, not acceptance limits.

  4. Read a residual

    A residual is the measured response minus the fitted value. Panel run 18 measured 40.65 mg; the model fitted 42.03 mg, leaving about −1.38 mg unexplained. This is the largest absolute raw residual. Look for curves, changing spread or unusual points.

    Residuals versus fits
    Residuals in mg against the fitted response. View full-size chart (opens in a new tab)
  5. Check the run sequence

    Read the residuals in recorded execution order. Look for drift, cycles or abrupt changes during preparation and testing. Check the experiment records for any pattern; the chart alone does not identify its cause.

    Residuals versus run order
    Residuals in mg in recorded execution order. View full-size chart (opens in a new tab)
  6. Read the contour at a fixed time

    The horizontal axis is drying temperature and the vertical axis is air speed. Each line joins settings with the same predicted loss in mg, holding drying time at 16 minutes. Check the marked extrapolation regions. Changing the held time changes this view; the grid is not a set of additional tested panels.

    Response contour · surface-1
    Predicted response in mg, holding drying_time_min at 16 min. View full-size chart (opens in a new tab)
  7. Confirm a useful setting

    Choose a feasible setting within the studied region and test fresh independent panels. Compare their measured loss with the prediction and the study requirement. The contour does not automatically choose or validate an operating recipe.

More details: read each result section

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

Model overview

Analyzed runs is 20; Missing runs is 0. DF counts the information available for the model and residual error. S is the residual standard deviation in mg. R² describes variation fitted in these runs; adjusted R² accounts for model size. See Prediction and coefficient detail for PRESS and predicted R². A high R² does not establish model adequacy. Read the lack-of-fit test and residual plots.

Model overview · 12 rows
StatisticValue
Responseabrasion_mass_loss_mg
Analyzed runs20
Missing responses0
Model rank10
Residual degrees of freedom10
S0.935936
R²0.988928
Adjusted R²0.978964
PRESS26.9097
Predicted R²0.965988
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
drying_temperature_cadjusted191.324891.3248104.2550.0000013128
air_speed_m_per_sadjusted121.815321.815324.9040.000545004
drying_time_minadjusted171.770471.770481.93190.00000393051
drying_temperature_c × air_speed_m_per_sadjusted134.944834.944839.89240.0000872558
drying_temperature_c × drying_time_minadjusted143.524443.524449.68680.0000350335
air_speed_m_per_s × drying_time_minadjusted12.69122.69123.072230.110185
drying_temperature_c²adjusted163.696463.696472.71480.0000067035
air_speed_m_per_s²adjusted117.710217.710220.21760.00114948
drying_time_min²adjusted154.401354.401362.10360.0000134296
Modelsummary9782.43286.936899.24561.37644e-8
Errorsummary108.759760.875976——
Totalsummary19791.19141.6417——
Pure Errordecomposition56.25681.25136——
Lack of Fitdecomposition52.502960.5005930.4000390.831291

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
Intercept42.03090.321752130.6311.69614e-1741.31442.7478
drying_temperature_c-3.0220.295969-10.21050.0000013128-3.68146-2.36254
air_speed_m_per_s-1.4770.295969-4.990390.000545004-2.13646-0.81754
drying_time_min-2.6790.295969-9.051620.00000393051-3.33846-2.01954
drying_temperature_c × air_speed_m_per_s2.090.3309036.316040.00008725581.35272.8273
drying_temperature_c × drying_time_min-2.33250.330903-7.048890.0000350335-3.0698-1.5952
air_speed_m_per_s × drying_time_min0.580.3309031.752780.110185-0.1572991.3173
drying_temperature_c²4.812730.5643918.52730.00000670353.555196.07027
air_speed_m_per_s²2.537730.5643914.49640.001149481.280193.79527
drying_time_min²4.447730.5643917.880580.00001342963.190195.70527

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.

Predictions · 1 row
PredictionFactor settingsBlockFitMean SEMean CI lowMean CI highIndividual PI lowIndividual PI high
prediction-1drying_temperature_c=65, air_speed_m_per_s=1.5, drying_time_min=16—42.03090.32175241.31442.747839.825744.2361

Surface values · surface-1

Each row gives the two plotted factor settings and their Fitted response in mg, holding drying_time_min at 16. Region marks extrapolation beyond the observed design hull. All grid rows are included; these are model predictions, not additional measured runs.

Surface values · surface-1 · 625 rows
drying_temperature_cair_speed_m_per_sFitted responseRegion
55155.9704Observed slice region
55.8333154.7757Observed slice region
56.6667153.6478Observed slice region
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  • Held settings: drying_time_min=16.
  • Region check: full numeric design hull within the selected categorical and block settings.
  • Sampled rows: help-doe-face-centered-ccd-analysis-1 (different held numeric settings), help-doe-face-centered-ccd-analysis-2, help-doe-face-centered-ccd-analysis-3 (different held numeric settings), help-doe-face-centered-ccd-analysis-4, help-doe-face-centered-ccd-analysis-5, help-doe-face-centered-ccd-analysis-6 (different held numeric settings), help-doe-face-centered-ccd-analysis-7, help-doe-face-centered-ccd-analysis-8, help-doe-face-centered-ccd-analysis-9 (different held numeric settings), help-doe-face-centered-ccd-analysis-10, help-doe-face-centered-ccd-analysis-11, help-doe-face-centered-ccd-analysis-12, help-doe-face-centered-ccd-analysis-13 (different held numeric settings), help-doe-face-centered-ccd-analysis-14 (different held numeric settings), help-doe-face-centered-ccd-analysis-15 (different held numeric settings), help-doe-face-centered-ccd-analysis-16 (different held numeric settings), help-doe-face-centered-ccd-analysis-17, help-doe-face-centered-ccd-analysis-18, help-doe-face-centered-ccd-analysis-19 (different held numeric settings), help-doe-face-centered-ccd-analysis-20 (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 · 20 rows
Row IDRun orderStatusObservedFitResidualStandardized residualLeverage
help-doe-face-centered-ccd-analysis-11Analyzed59.5559.49160.05840910.1372270.793182
help-doe-face-centered-ccd-analysis-22Analyzed44.7843.82160.9583641.435110.490909
help-doe-face-centered-ccd-analysis-33Analyzed55.755.7856-0.0855909-0.2010880.793182
help-doe-face-centered-ccd-analysis-44Analyzed43.1643.09160.06836360.1023720.490909
help-doe-face-centered-ccd-analysis-55Analyzed41.5642.0309-0.470909-0.5357980.118182
help-doe-face-centered-ccd-analysis-66Analyzed55.7155.8516-0.141591-0.3326560.793182
help-doe-face-centered-ccd-analysis-77Analyzed46.3146.04560.2643640.3958750.490909
help-doe-face-centered-ccd-analysis-88Analyzed49.2449.8656-0.625636-0.9368670.490909
help-doe-face-centered-ccd-analysis-99Analyzed48.8249.1576-0.337636-0.5055980.490909
help-doe-face-centered-ccd-analysis-1010Analyzed42.6342.03090.5990910.6816430.118182
help-doe-face-centered-ccd-analysis-1111Analyzed43.342.03091.269091.443970.118182
help-doe-face-centered-ccd-analysis-1212Analyzed42.6942.03090.6590910.7499110.118182
help-doe-face-centered-ccd-analysis-1313Analyzed61.5561.34460.2054090.4825910.793182
help-doe-face-centered-ccd-analysis-1414Analyzed44.4743.79960.6703641.003840.490909
help-doe-face-centered-ccd-analysis-1515Analyzed53.5253.51760.002409090.005659950.793182
help-doe-face-centered-ccd-analysis-1616Analyzed53.4153.05060.3594090.8444010.793182
help-doe-face-centered-ccd-analysis-1717Analyzed40.6942.0309-1.34091-1.525680.118182
help-doe-face-centered-ccd-analysis-1818Analyzed40.6542.0309-1.38091-1.571190.118182
help-doe-face-centered-ccd-analysis-1919Analyzed46.746.9886-0.288591-0.6780190.793182
help-doe-face-centered-ccd-analysis-2020Analyzed44.1644.6026-0.442591-1.039830.793182

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
drying_temperature_cnumeric6510—
air_speed_m_per_snumeric1.50.5—
drying_time_minnumeric164—
  • Model columns: Intercept, drying_temperature_c, air_speed_m_per_s, drying_time_min, drying_temperature_c × air_speed_m_per_s, drying_temperature_c × drying_time_min, air_speed_m_per_s × drying_time_min, drying_temperature_c², air_speed_m_per_s², drying_time_min²
  • Rank tolerance: 1e-10

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
ColumnInterceptdrying_temperature_cair_speed_m_per_sdrying_time_mindrying_temperature_c × air_speed_m_per_sdrying_temperature_c × drying_time_minair_speed_m_per_s × drying_time_mindrying_temperature_c²air_speed_m_per_s²drying_time_min²
Intercept0.103524000000-0.0398171-0.0398171-0.0398171
drying_temperature_c00.0875976-8.47919e-20-1.96937e-174.14187e-17-1.0291e-172.77171e-172.09269e-174.50318e-17-5.50199e-17
air_speed_m_per_s09.9423e-190.08759761.45876e-175.5875e-17-1.47892e-173.74576e-174.4901e-17-3.91899e-17-4.69049e-18
drying_time_min0-1.76527e-171.43707e-170.08759765.2068e-18-6.78675e-18-6.43757e-18-1.44606e-173.41623e-17-2.52274e-17
drying_temperature_c × air_speed_m_per_s04.14187e-175.5875e-175.2068e-180.109497-3.43178e-18-8.49181e-18-2.58837e-174.72145e-17-2.13307e-17
drying_temperature_c × drying_time_min0-1.0291e-17-1.47892e-17-6.78675e-18-2.81173e-180.109497-2.60668e-17-3.4447e-171.58095e-171.86375e-17
air_speed_m_per_s × drying_time_min02.77171e-173.74576e-17-6.43757e-18-1.08227e-17-2.86498e-170.109497-8.21318e-176.1826e-172.03059e-17
drying_temperature_c²-0.03981712.09269e-174.4901e-17-1.44606e-17-2.58837e-17-3.4447e-17-8.21318e-170.318537-0.119451-0.119451
air_speed_m_per_s²-0.03981714.50318e-17-3.91899e-173.41623e-174.72145e-171.58095e-176.1826e-17-0.1194510.318537-0.119451
drying_time_min²-0.0398171-5.50199e-17-4.69049e-18-2.52274e-17-2.13307e-171.86375e-172.03059e-17-0.119451-0.1194510.318537

Prediction and coefficient detail

PRESS sums leave-one-out prediction errors: each run is predicted by a model fitted without that run. Predicted R² uses those errors. Coefficient covariance records estimation uncertainty; its diagonal entries are squared coefficient standard errors. These diagnostics supplement, rather than replace, lack-of-fit and residual checks.

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. Keep the lack-of-fit result with the residual checks; it does not validate the model on its own.

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 Central composite: coating durability

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