center point runs

**Center point runs** are experimental runs performed at the **midpoint of all factor ranges** (the center of the design space) in a DOE. They serve multiple critical statistical purposes and are an essential component of well-designed factorial and response surface experiments. **What Center Points Are** In a $2^k$ factorial design where factors vary between low (−1) and high (+1) levels, center points are run at the **zero level (0)** for all factors simultaneously. - Factor A: (Low + High) / 2 - Factor B: (Low + High) / 2 - etc. Typically **3–5 center point replicates** are added to a factorial design. **Why Center Points Are Important** - **Curvature Detection**: The most important role. If the average response at the center point differs significantly from the average of the factorial points, this indicates **curvature** (nonlinear response) — meaning a linear model is inadequate and a response surface design (RSM) may be needed. - If center point average ≈ factorial average → linear model is adequate. - If center point average ≠ factorial average → curvature exists → consider RSM. - **Pure Error Estimation**: Because center points are **replicated** (run multiple times at the same conditions), they provide a direct estimate of experimental error (pure error) — independent of any model. - The variation among center point replicates reflects the inherent noise of the process. - This pure error estimate is used to test the significance of main effects and interactions. - **Process Insight**: The center point runs are at the nominal operating condition — they directly show process performance at the current baseline settings. **Center Points in Practice** - A typical $2^4$ factorial design (16 runs) might add **4 center points** = 20 total runs. - Center points should be **randomized** with the factorial points — interspersed throughout the run order, not grouped. - Good: runs 1,2,3(CP),4,5,6(CP),7,8(CP),... etc. - This also provides monitoring of **time-related drift** during the experiment. **Semiconductor Example** - Etch DOE with factors: Power (200–400W), Pressure (20–50 mTorr), Gas Flow (50–100 sccm). - Center point: Power=300W, Pressure=35 mTorr, Flow=75 sccm — run 3–4 times. - If the etch rate at center points differs significantly from the average of the 8 factorial corner points, the etch rate has a nonlinear dependence on one or more factors. Center points are the **cheapest and most informative** addition to any factorial DOE — they provide curvature detection, error estimation, and baseline verification for just a few extra runs.