Weibull / Survival Shelf-Life Modeling

1 Introduksi

1.1 Latar Belakang

Survival analysis dengan Weibull distribution adalah pendekatan elegant untuk shelf life dari sensory perspective: bukan model rate degradation, tetapi model waktu hingga produk dinyatakan unacceptable oleh konsumen (failure-time data) (Gámbaro, Fiszman, Giménez, Varela, & Salvador, 2004; Hough, 2010). Diadopsi dari engineering reliability (Weibull, 1951), survival analysis memberi probability survival $S(t) = P(T > t)$ + hazard rate $h(t)$.

Aplikasi sains pangan: shelf life ditentukan oleh % konsumen menolak produk pada usia tertentu (rejection threshold 50%). Lebih realistis dari kinetic chemistry endpoint (mis. PV = 10) karena directly consumer-perception-based.

1.2 Tujuan Modul

Menerima survival data: panelis × storage time × accept/reject.
Memfit Weibull distribution ke survival curve.
Menghitung median shelf life ($t_{50}$) dan 75% confidence shelf life ($t_{25}$).
Mendukung right-censored data (panelis tidak reject sampai end of study).
Memberikan Kaplan-Meier non-parametric sebagai komparasi.
Audiens: peneliti senior shelf life consumer test, R&D yang reporting best-before.

1.3 Posisi di Antara Alternatif

Pilih Weibull / Survival Shelf-Life Modeling untuk consumer rejection-based shelf life. Untuk kinetic chemistry shelf life, pakai ASLT / Shelf-Life Prediction. Untuk microbial shelf life, pakai Predictive Microbiology / Shelf-Life Modeling. Untuk time-to-event general, modul ini cocok.

2 Metode

2.1 Dasar Teoretis

Weibull distribution untuk failure time $T$ (Weibull, 1951):

$$\boxed{\, f(t) = \frac{\beta}{\alpha}\left(\frac{t}{\alpha}\right)^{\beta-1} \exp\!\left[-\left(\frac{t}{\alpha}\right)^\beta\right] \,}$$

dengan:

$\alpha$ = scale parameter (~ median lifetime).
$\beta$ = shape parameter — bentuk hazard:
- $\beta < 1$: decreasing hazard (infant mortality).
- $\beta = 1$: constant hazard (exponential).
- $\beta > 1$: increasing hazard (aging — typical untuk pangan).

Survival function:

$$S(t) = \exp\!\left[-\left(\frac{t}{\alpha}\right)^\beta\right]$$

Hazard rate:

$$h(t) = \frac{f(t)}{S(t)} = \frac{\beta}{\alpha}\left(\frac{t}{\alpha}\right)^{\beta-1}$$

Percentile shelf life $t_p$ saat $S(t_p) = 1 - p$:

$$t_p = \alpha [-\ln(1 - p)]^{1/\beta}$$

Mis. $t_{50}$ (median, 50% reject): $t_{50} = \alpha (\ln 2)^{1/\beta}$.

Maximum Likelihood Estimation (MLE) untuk $\alpha$ dan $\beta$ dari survival + censored data (Lawless, 2003).

Kaplan-Meier (KM) non-parametric estimator (Kaplan & Meier, 1958):

$$\hat{S}(t) = \prod_{t_i \leq t} \left(1 - \frac{d_i}{n_i}\right)$$

dengan $d_i$ deaths/rejections pada $t_i$, $n_i$ at-risk count.

Right censoring — panelis tidak mengevaluasi sampai produk dinyatakan reject (sampai end of test). KM dan Weibull handle censored properly via likelihood.

Log-rank test untuk compare 2 groups survival (Mantel, 1966).

2.2 Persamaan Inti

Weibull PDF: $f(t) = (\beta/\alpha)(t/\alpha)^{\beta-1} \exp[-(t/\alpha)^\beta]$
Survival: $S(t) = \exp[-(t/\alpha)^\beta]$
Hazard: $h(t) = (\beta/\alpha)(t/\alpha)^{\beta-1}$
$t_p$: $\alpha [-\ln(1-p)]^{1/\beta}$
Kaplan-Meier: $\hat{S}(t) = \prod (1 - d_i/n_i)$

2.3 Asumsi & Batas Validitas

Asumsi	Konsekuensi jika dilanggar	Cara cek di SQalytics
Right-censored data correctly recorded	Bias likelihood	Verify input
Survival times independent	Bias model	Independent panelists
Censoring uninformative (random)	Bias estimate	Cek causes of censoring
Weibull fit adequate	Bias prediction	Cek with KM non-parametric
$n \geq 60$ panelis untuk Weibull	CI lebar	Modul flag
Storage T konstan	Drift kinetic	Time-T logger

3 Cara Kerja

3.1 Step-by-Step di SQalytics

Buka Weibull / Survival Shelf-Life Modeling dari domain Mutu dan Analisis Lanjutan.
Muat tabel: Panelist, Time_day (when evaluated), Reject (0 = still acceptable / censored, 1 = reject event).
(Multi-group) Tambah kolom Group (mis. formulation, packaging) untuk Kaplan-Meier comparison.
Pilih distribution: Weibull (default), Exponential, Log-normal.
Pilih target percentile: $t_{50}$ (median), $t_{25}$ (75% still acceptable).
Klik Run Survival Analysis.
Tinjau: Tab Weibull Parameters — $\alpha$, $\beta$, CI; Tab Survival Curve — Weibull + KM overlay; Tab Hazard Plot — $h(t)$ over time; Tab Group Comparison — log-rank test bila multi-group.

3.2 Template Tabel Input + Contoh Data Sintetis

Sensory rejection survival data (cookie 50 panelis, 12-week test):

Panelist	Time_day	Reject (0=censored, 1=event)
P01	21	1 (reject pada hari 21)
P02	35	1
P03	84	0 (still acceptable at end of test, censored)
P04	28	1
P05	42	1
… (50 panelis total, sebagian censored)

SYNTHETIC Survival data cookie — 50 panelis, 12-week test, sebagian right-censored. CSV setara: docs/assets/example-data/id/quality-advanced/template_quality_weibull_survival.csv.

3.3 Contoh Luaran

Weibull fit (MLE):

Parameter	Estimate	95% CI
$\alpha$ (scale)	52.3 day	[45.1, 60.5]
$\beta$ (shape)	2.4	[1.9, 3.0]
Median lifetime $t_{50}$	47.5 day	[40.2, 55.8]
$t_{25}$ (75% still acceptable)	30.8 day	[25.6, 36.5]
$t_{75}$ (only 25% still acceptable)	64.2 day	[54.8, 75.1]

Interpretation: $\beta > 1$ → increasing hazard (rejection rate naik dengan time, klasik aging).

Kaplan-Meier comparison (validasi Weibull fit): KM curve overlaps Weibull within CI — Weibull appropriate.

Multi-group comparison (regular vs improved packaging):

Group	$t_{50}$ (day)	$\beta$	Median
Regular packaging	47.5	2.4	47.5 d
Improved O₂ barrier	78.5	2.1	78.5 d
Log-rank test	—	—	$\chi^2 = 18.2$, $p < 0.001$

Weibull / Survival Shelf-Life Modeling — figure 01 — Gambar 1. *Panel* (a) Survival curves — Weibull fit + Kaplan-Meier overlay untuk 2 packaging types (regular vs improved O₂ barrier); *panel* (b) hazard rate $h(t)$ vs time (increasing hazard, $\beta = 2.4$).

Kesimpulan ringkas: "Survival analysis cookie regular: $\alpha = 52.3$ day, $\beta = 2.4$ (increasing hazard). Median shelf life $t_{50}$ = 47.5 day (50% panelis akan reject by day 47), $t_{25}$ = 31 day (75% masih acceptable). Untuk label best-before, $t_{25}$ = 31 day rekomendasi konservatif. Improved O₂ barrier packaging extends median shelf life dari 47.5 → 78.5 day (+65% improvement), log-rank test signifikan ($p < 0.001$). Recommendation: pakai improved packaging untuk label 8 minggu best-before vs current 4 minggu. Lanjut ke ASLT / Shelf-Life Prediction untuk chemistry-based parallel validation (PV + browning kinetics)."

4 Kesimpulan

4.1 Relevansi Real-World

Sensory shelf life consumer-based — directly correlate dengan consumer rejection.
Reliability engineering untuk packaging.
Pharmaceutical drug stability — failure-time analysis.
Comparison antar formulasi — log-rank rigorous statistical test.
Quality risk assessment — hazard rate over time.

4.2 Where to Go from Here

ASLT / Shelf-Life Prediction — kinetic chemistry parallel.
Predictive Microbiology / Shelf-Life Modeling — microbial complement.
Compare Many Groups — bila multi-group rejection rate.
Hough (2010) untuk sensory survival shelf life dedicated.

⚙ Troubleshooting Cepat

Weibull poor fit (KM diverges). Coba log-normal atau exponential; cek bimodal failure pattern.

Banyak censored di end of test. Test perlu lebih lama atau accelerate.

$\beta < 1$ unexpected. Indikasi infant mortality (formulation problem early failure).

i Riwayat Revisi

Tanggal	Revisi	Penulis
2026-05-12	Draft v2 publikasi (KaTeX Weibull + survival + Kaplan-Meier + log-rank + APA Weibull/Gámbaro/Hough/Kaplan-Meier)	Claude
2026-05-12	Konversi MD → HTML (W5 quality-advanced batch)	Claude

4 Referensi

Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18(3), 293–297. https://doi.org/10.1115/1.4010337
Gámbaro, A., Fiszman, S., Giménez, A., Varela, P., & Salvador, A. (2004). Survival analysis of sensory shelf life of brown pan bread. Journal of Texture Studies, 35(2), 167–175. https://doi.org/10.1111/j.1745-4603.2004.tb00832.x
Hough, G. (2010). Sensory shelf life estimation of food products. CRC Press. https://doi.org/10.1201/9781420092945
Kaplan, E. L., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53(282), 457–481. https://doi.org/10.1080/01621459.1958.10501452
Lawless, J. F. (2003). Statistical models and methods for lifetime data (2nd ed.). John Wiley & Sons.
Mantel, N. (1966). Evaluation of survival data and two new rank order statistics arising in its consideration. Cancer Chemotherapy Reports, 50(3), 163–170.