RSM Studio — SQalytics

1 Introduksi

1.1 Latar Belakang

Response Surface Methodology (RSM) adalah keluarga teknik untuk optimasi response berbasis fitted second-order polynomial (Box & Wilson, 1951; Myers, Montgomery, & Anderson-Cook, 2016). RSM diperkenalkan untuk industri kimia dan kini menjadi standar di food science, pharmaceutical, dan chemical engineering untuk optimasi multi-faktor multi-respons. Dua design RSM paling populer: Central Composite Design (CCD) (Box & Wilson, 1951) yang augment full factorial $2^k$ dengan axial points + center, dan Box-Behnken Design (BBD) (Box & Behnken, 1960) yang efisien untuk 3–5 faktor tanpa pure corner runs.

Setelah fit second-order polynomial $\hat{y} = \beta_0 + \sum \beta_i x_i + \sum \beta_{ii} x_i^2 + \sum_{i<j} \beta_{ij} x_i x_j$, canonical analysis mengidentifikasi stationary point (maximum/minimum/saddle), eigenvalues menentukan tipe surface, dan contour plots memvisualisasi region optimum.

1.2 Tujuan Modul

Modul RSM Studio di SQalytics ditujukan untuk:

Membangun CCD (rotatable atau face-centered) dengan $\alpha$ axial distance configurable.
Membangun Box-Behnken Design untuk 3, 4, 5 faktor.
Memfit second-order polynomial via OLS dengan alias-free estimation.
Menjalankan canonical analysis: stationary point coordinates + eigenvalues.
Menampilkan 3D surface plot dan 2D contour plot (slice).
Audiens: praktisi optimasi multi-faktor S2/S3, R&D yang menjalankan post-screening optimization.

1.3 Posisi di Antara Alternatif

Pilih RSM Studio untuk optimasi quadratic multi-faktor ($k = 2-5$). Untuk screening banyak faktor ($k \geq 6$), pakai Screening Design Builder dulu. Untuk mixture optimization, pakai Mixture Design Explorer. Untuk multi-response desirability, pakai Quick Design Optimizer setelah RSM fit.

2 Metode

2.1 Dasar Teoretis

Second-order polynomial model (Box & Wilson, 1951):

$$ \boxed{\, \hat{y} = \beta_0 + \sum_{i=1}^{k} \beta_i x_i + \sum_{i=1}^{k} \beta_{ii} x_i^2 + \sum_{i<j} \beta_{ij} x_i x_j + \varepsilon \,} $$

Jumlah parameter: $1 + k + k + k(k-1)/2 = (k+1)(k+2)/2$. Untuk $k=3$: 10 parameter; $k=4$: 15; $k=5$: 21.

Central Composite Design (CCD):

$$ N_{\text{CCD}} = 2^k + 2k + n_C $$

dengan $2^k$ factorial corners, $2k$ axial points di $\pm \alpha$, dan $n_C$ center points ($\geq 4$ rekomendasi).

$\alpha$ rotatable (variance constant pada radius dari center):

$$ \alpha = (2^k)^{1/4} $$

Untuk $k=2$: $\alpha = 1.414$; $k=3$: 1.682; $k=4$: 2.0; $k=5$: 2.378.

Face-centered CCD ($\alpha = 1$): semua titik di "face" cube — pakai bila axial points di luar cube tidak feasible.

Box-Behnken Design (BBD) (Box & Behnken, 1960):

$$ N_{\text{BBD},k=3} = 12 + n_C, \quad N_{\text{BBD},k=4} = 24 + n_C, \quad N_{\text{BBD},k=5} = 40 + n_C $$

BBD lebih efisien dari CCD untuk $k = 3, 4, 5$ dan menghindari corner points (relevan bila corner extreme tidak diinginkan, mis. T = 200 °C tidak feasible).

Canonical analysis: stationary point $x_s$ dari $\partial \hat{y} / \partial x = 0$:

$$ x_s = -\frac{1}{2} B^{-1} b $$

dengan $b$ vector first-order coefficients dan $B$ matrix second-order coefficients.

Eigenvalues $\lambda_i$ dari $B$ menentukan tipe surface:

Eigenvalues	Surface type
Semua $\lambda_i < 0$	Maximum
Semua $\lambda_i > 0$	Minimum
Mixed signs	Saddle point
Some $\lambda_i \approx 0$	Ridge / valley

Predicted response at stationary:

$$ \hat{y}_s = \beta_0 + \frac{1}{2} b^T x_s $$

2.2 Persamaan Inti

Second-order model: $\hat{y} = \beta_0 + \sum \beta_i x_i + \sum \beta_{ii} x_i^2 + \sum_{i<j} \beta_{ij} x_i x_j$

CCD size: $N = 2^k + 2k + n_C$

Rotatable $\alpha$: $\alpha = (2^k)^{1/4}$

Stationary point: $x_s = -B^{-1} b / 2$

Predicted at stationary: $\hat{y}_s = \beta_0 + b^T x_s / 2$

Eigenvalues of $B$ untuk surface classification.

2.3 Asumsi & Batas Validitas

Asumsi	Konsekuensi jika dilanggar	Cara cek di SQalytics
Second-order model adequate	Bias $x_s$ + $\hat{y}_s$	Lack-of-fit test (LOF F-test)
Center points $\geq 4$ untuk error + curvature	$df_{\text{error}}$ rendah	Modul default $n_C = 5$
Factor space includes optimum interior	Stationary at edge	Cek $\|x_s\| < \alpha$
Independent factors (no constraint $\sum = 1$)	Pakai Mixture Design instead	Modul cek
Residuals normal + homoskedastik	Inferensi koefisien bias	QQ-plot + residual vs fitted
Replicates di non-center points (opsional)	Pure error reduced	Trade-off vs more design points

3 Cara Kerja

3.1 Step-by-Step di SQalytics

Buka RSM Studio dari domain Desain Eksperimen.
Daftarkan faktor ($k = 2-5$): nama, satuan, low/high level.
Pilih design type: Central Composite Design (CCD) rotatable; Face-Centered CCD ($\alpha = 1$); Box-Behnken Design (BBD).
Atur center points $n_C$ (default 5).
Klik Generate RSM Design.
Eksekusi run di laboratorium, input response.
Klik Fit Second-Order Model.
Tinjau hasil: Tab Coefficients — $\beta_i$, $\beta_{ii}$, $\beta_{ij}$ dengan $p$-value; Tab ANOVA + LOF — model significance + lack-of-fit; Tab Canonical Analysis — stationary point + eigenvalues; Tab Surface Plot — 3D + 2D contour; Tab Optimization — link ke Quick Design Optimizer atau Response Overlay.

3.2 Template Tabel Input + Contoh Data Sintetis

CCD untuk optimasi ekstraksi flavonoid (3 faktor, rotatable $\alpha = 1.682$, 5 CP):

Factor	Low (−1)	Mid (0)	High (+1)	Low Axial (−$\alpha$)	High Axial (+$\alpha$)
T (°C)	60	70	80	53.18	86.82
Time (min)	20	30	40	13.18	46.82
Ratio (mL/g)	10	15	20	6.59	23.41

Total runs: $2^3 + 2 \cdot 3 + 5 = 19$ runs.

3.3 Contoh Luaran

Fitted coefficients (Yield = mg/g):

Term	Coefficient	SE	$t$	$p$
Intercept $\beta_0$	24.5	0.4	61.3	< 0.001
T $\beta_1$	+2.8	0.3	9.3	< 0.001
Time $\beta_2$	+1.5	0.3	5.0	0.001
Ratio $\beta_3$	+0.8	0.3	2.7	0.024
T² $\beta_{11}$	−1.2	0.4	−3.0	0.014
Time² $\beta_{22}$	−0.6	0.4	−1.5	0.16
Ratio² $\beta_{33}$	−0.4	0.4	−1.0	0.34
T·Time $\beta_{12}$	+0.5	0.4	1.3	0.23
T·Ratio $\beta_{13}$	+0.3	0.4	0.8	0.46
Time·Ratio $\beta_{23}$	+0.2	0.4	0.5	0.61
$R^2$	0.95	—	—	—
Lack-of-fit $p$	0.18	—	—	model adequate

Canonical analysis:

Parameter	Value (coded)	Decoded
Stationary point $x_s$	(+1.17, +1.25, +1.00)	T = 81.7 °C, Time = 42.5 min, Ratio = 20 mL/g
Predicted $\hat{y}_s$	28.5 mg/g	—
Eigenvalue $\lambda_1$	−1.4	—
Eigenvalue $\lambda_2$	−0.7	—
Eigenvalue $\lambda_3$	−0.5	—
Surface type	Maximum (semua $\lambda < 0$)	—

Kesimpulan ringkas: "RSM CCD 3-faktor untuk yield flavonoid memberikan fit excellent ($R^2 = 0.95$, LOF $p = 0.18$ — model adequate). Stationary point = maksimum (semua eigenvalues negatif): T = 81.7 °C, Time = 42.5 min, Ratio = 20 mL/g — predicted yield 28.5 mg/g. Stationary point berada di edge factor space ($x_s$ near +$\alpha$) — indikasi optimum bisa di luar design region, perlu augment design ke arah +T + +Time atau confirm via second-stage experiment. Recommendation: jalankan confirmatory triplicate pada stationary point; bila yield observasi within 95% PI dari model, optimum divalidasi. Lanjut ke Quick Design Optimizer untuk multi-response (Yield + Time + Cost desirability) atau Response Overlay untuk sweet spot visualisasi."

RSM CCD 3-faktor: surface plot 3D response surface dan contour 2D dengan stationary point ditandai star marker size 180 dibedakan dari marker observasi size 8, eigenvalue annotations, dan optimal callout navy bbox — Gambar 1. Response Surface Methodology — CCD $k = 3$ rotatable ($\alpha = 1.682$, $n_C = 5$, total 19 runs) untuk optimasi yield flavonoid (T × Time × Ratio), data sintetis ($R^2 = 0.95$). **(a)** 3D response surface plot: slice T–Time pada Ratio = 15 mL/g (optimal mid); sumbu-z = Yield (mg/g); surface warna viridis (biru rendah → kuning tinggi); wireframe menunjukkan quadratic curvature; titik observasi (CCD design) di-overlay pada bidang bawah. **(b)** 2D contour plot T–Time (Ratio = 15 mL/g held): iso-yield lines dari 20 hingga 28 mg/g; **stationary point (81.7 °C, 42.5 min)** ditandai *marker* "star" (size 180) membedakan dari *marker* observasi biasa (size 8); *label* "$\hat{y}_s = 28.5$" dan eigenvalue annotation "$\lambda_1 = -1.4, \lambda_2 = -0.7$" di kanan *marker* dengan clear margin; **optimal combo callout** "T = 81.7 °C, t = 42.5 min, Ratio = 20, Yield = 28.5 mg/g — Maximum" di **bottom dalam navy bbox**; *legend* faktor di lower-right. Peringatan tepi: stationary point di $x_s \approx +1.17\alpha$ (edge) — confirmatory run direkomendasikan sebelum scale-up (Box & Wilson, 1951; Myers et al., 2016; Bezerra et al., 2008).

4 Kesimpulan

4.1 Relevansi Real-World

Optimasi ekstraksi senyawa bioaktif (T × Time × Solvent).
Spray drying optimization (Inlet T × Outlet T × Feed rate).
Fermentation bioprocess (T × pH × Inoculum × Aeration).
Tablet formulation (Binder × Disintegrant × Compression force).
Encapsulation (Polymer ratio × Cross-linker × Temperature).
Heat treatment (T × Time × Pressure).

4.2 Where to Go from Here

Quick Design Optimizer — multi-response desirability dari fitted model.
Response Overlay / Sweet Spot Finder — visualisasi.
ANOVA Studio — drill down coefficient significance.
Regression Studio — alternative second-order fit dengan more diagnostics.
Bezerra et al. (2008) untuk RSM tutorial di chemistry.

⚙ Troubleshooting Cepat

Stationary point di edge factor space. Augment design ke arah optimum atau confirm dengan one-factor-at-a-time.

Eigenvalues mixed sign (saddle). Optimum tidak unique — pakai constrained optimization.

Lack-of-fit signifikan. Second-order model insufficient — pakai third-order atau split design ke region kecil.

Replicate $\sigma^2$ besar. Pure error tinggi — improve measurement precision atau tambah replicates.

i Riwayat Revisi

Tanggal	Revisi	Penulis
2026-05-12	Migrasi MD v2 → HTML final dengan figure publikasi + caption Elsevier-style (W2 batch 12 Mei)	Claude
2026-05-12	Draft v2 publikasi (KaTeX second-order + CCD/BBD + canonical analysis + APA Box-Wilson/Box-Behnken/Myers)	Claude

4 Referensi

Box, G. E. P., & Wilson, K. B. (1951). On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society: Series B, 13(1), 1–45. https://doi.org/10.1111/j.2517-6161.1951.tb00067.x
Box, G. E. P., & Behnken, D. W. (1960). Some new three level designs for the study of quantitative variables. Technometrics, 2(4), 455–475. https://doi.org/10.1080/00401706.1960.10489912
Myers, R. H., Montgomery, D. C., & Anderson-Cook, C. M. (2016). Response surface methodology: Process and product optimization using designed experiments (4th ed.). John Wiley & Sons.
Khuri, A. I., & Cornell, J. A. (1996). Response surfaces: Designs and analyses (2nd ed.). Marcel Dekker.
Bezerra, M. A., Santelli, R. E., Oliveira, E. P., Villar, L. S., & Escaleira, L. A. (2008). Response surface methodology (RSM) as a tool for optimization in analytical chemistry. Talanta, 76(5), 965–977. https://doi.org/10.1016/j.talanta.2008.05.019
Montgomery, D. C. (2017). Design and analysis of experiments (9th ed.). John Wiley & Sons.