Design of experiments for Aspergillus awamori cellulase

Section 1 · Abstract

From a statistical design to a working enzyme

Background. Cellulases are expensive to make, and the cost of the enzyme is one of the main barriers to turning agricultural residue into sugars, fuels and textile finishes. Solid-state fermentation (SSF) of lignocellulosic waste is a low-cost route, but yield depends on many interacting process variables.

Methods. This study follows a sequential design-of-experiments strategy for SSF of Aspergillus awamori MK788209. A 12-run Plackett–Burman design (PBD) screened 11 variables. The two dominant variables, molokhia stem weight and incubation time, were carried into a 13-run rotatable central composite design (CCD) with a full quadratic response surface. A second three-factor CCD optimized saccharification of pea peels with the crude enzyme. Model quality was judged by ANOVA, R², adjusted R², predicted R² and residual diagnostics.

Results. Substrate weight (F = 99.5) and incubation time (F = 55.2) were the only variables beyond the Bonferroni limit in screening, lifting cellulase to 4.64 U/mL (2.51×). The CCD model (R² = 0.9878, adjusted 0.9792, predicted 0.9136) reached 9.00 U/mL, a 4.86-fold gain over the unoptimized condition, and showed a large substrate × time interaction that reverses the effect of incubation time depending on substrate loading. Pea peels were the best saccharification substrate (98.33 mg/mL), and CCD optimization raised the reported yield 3.44-fold to 338.18 mg/mL. The crude enzyme also biopolished, desized and improved the dyeability of cotton.

Conclusion. A small number of well-chosen experiments is enough to turn waste into a useful enzyme and then into sugar. This page also rebuilds the published models live from their design tables, so every coefficient, p-value and surface can be inspected and challenged.

Section 2 · Background and rationale

Why waste, why fungi, why designed experiments

Section 3 · Study design

A sequential strategy, one question per stage

Each stage answers a narrower question than the one before and hands its result to the next. This is the classical screen, then optimize logic of response-surface methodology [3,5].

Section 4 · Materials and methods

Strain, substrate, fermentation and assays

Section 5 · Statistical framework

The mathematics behind each design

This section defines every quantity that later tables and plots report, so each number can be traced to a formula.

5.1 Coded variables

Factors are centred and scaled so that the low and high levels sit at −1 and +1. Coding makes coefficients comparable across factors measured in different units and keeps the model matrix well conditioned.

5.2 Plackett–Burman screening

A Plackett–Burman design is a two-level orthogonal array with N = 12 runs that can estimate up to N − 1 = 11 main effects [2]. Each effect is the difference between the mean response at the high and low levels. Its standard error depends only on the error variance and N, so every factor is tested with the same precision.

The Pareto chart ranks |t| and draws two reference lines. The t-value limit is the two-sided critical value t_{1−α/2, ν}. The Bonferroni limit divides α by the number of simultaneous tests m = 11, giving t_{1−α/2m, ν}, which controls the family-wise error rate. Effects above the Bonferroni line are very likely real, those between the lines are possibly real, and those below are indistinguishable from noise.

5.3 Central composite design and the quadratic model

A CCD augments a two-level factorial with axial (star) points at ±α and replicated centre points [3,4]. For k = 2 factors there are 2² = 4 factorial points, 2k = 4 axial points and n₀ centre points. Choosing α = (2^k)^1/4 = √2 ≈ 1.414 makes the design rotatable, meaning prediction variance depends only on distance from the centre. With n₀ = 5 the study has N = 13 runs, and the replicated centre points provide an estimate of pure error.

5.4 ANOVA and fit statistics

Total variation splits into regression and residual parts, and the residual splits again into lack of fit and pure error when replicates exist. Three R² variants guard against over-fitting: plain R² never falls when terms are added, adjusted R² penalizes terms, and predicted R² uses leave-one-out residuals (PRESS) to measure how well the model predicts runs it has not seen. Adjusted and predicted R² should agree within about 0.2.

n = runs, p = model parameters, e_i = residual, h_ii = leverage. Adequate precision compares the range of fitted values with their average standard error, and values above 4 indicate a usable signal-to-noise ratio.

5.5 Canonical analysis of the surface

Writing the fitted surface as ŷ = b₀ + x^Tb + x^TBx, the stationary point is where the gradient vanishes. The signs of the eigenvalues of B classify it: all negative is a maximum, all positive a minimum, mixed signs a saddle, where no interior optimum exists [5].

Section 6 · Screening

Which substrate, and which variables matter?

6.1 Choosing the substrate

Cellulase production on each waste was compared before any optimization. Molokhia stem gave the highest titre, followed by AL, pea peel, rice straw, corn cob, lemon peel and onion peel. Molokhia stem was therefore used as the SSF substrate for all later optimization, while pea peel, the source of the fungus, became the substrate for the saccharification study.

Show the original chart from the source article

6.2 Plackett–Burman screening of 11 variables

Eleven variables were varied at two levels in 12 runs: substrate weight, incubation time, glucose, lactose, baker's yeast, peptone, ammonium sulfate, copper sulfate, iron sulfate, calcium chloride and potassium chloride. The panel below recomputes the analysis from the design table. Change the terms kept in the model to see how the error estimate, the limits and the p-values respond.

Design matrix and responses from Table 1 of [1]. Factor levels: A 2/4 g per flask, B 4/7 days, C and D 0/10 g/L, E, F and G 0/1 g/L, H, J, K and L 0/0.01 g/L.

Compare with the original Pareto chart from the source article

Section 7 · Response surface optimization

The cellulase response surface

The two factors identified by screening, molokhia stem weight (A, 4 to 8 g per flask at the ±1 levels) and incubation time (B, 7 to 12 days), were studied in a 13-run rotatable CCD. The panel below fits the quadratic model of equation (4) to the published responses. Drag the surface to rotate it, move the sliders to predict, and switch terms off to see what each one contributes.

7.1 The interaction changes the sign of the time effect

7.2 Model coefficients and ANOVA

The actual-unit equation is the coded model with x_A = (g − 6)/2 and x_B = (days − 9.5)/2.5 substituted and expanded, for use on the bench.

7.3 Canonical analysis: a saddle, and an optimum on the edge

7.4 Diagnostics

Three plots check the assumptions behind the inference. Points near the 45° line in the parity plot show the model reproduces the data. Residuals against fitted values should form a shapeless band, and studentized residuals should follow the normal reference line.

7.5 The design table

Design points and responses from Table 3 of [1]. The "published" column is the model prediction printed in that table. Run 2 prints B as 9.9 d in the source, but its coded level is 0 (9.5 d) and the published equation is reproduced only with the coded value, so the coded value is used here.

Compare with the original Design-Expert plots from the source article

7.6 In-silico comparison of experimental strategies

Method. The fitted surface f(x) of section 7.2 is taken as the true response. A strategy spends a budget of N flasks, each returning y = f(x) + ε with ε ~ N(0, (kσ̂)²), where σ̂ = U/mL is the residual standard error of the CCD fit and k is a noise multiplier. The strategy then recommends a setting x̂ inside the square 4 to 8 g by 7 to 12 d, and performance is the relative yield S = f(x̂) / max f. Following good practice for simulation studies [25], each contest is repeated R = 500 times from a fixed random seed and summarized by the mean with a 95% Monte Carlo interval, the spread, and the probability of reaching 95% of the optimum.

• One factor at a time (OFAT). Half the flasks vary substrate at a held time, the best level is fixed, and the other half vary time.
• Full grid. ⌊√N⌋ levels per factor at the centre of equal cells, as a practical grid would be laid out. The best observed flask is recommended.
• CCD plus model. Four factorial, four axial and N − 8 centre flasks. A quadratic model is fitted by least squares and its predicted maximum is recommended.

Figure 5. Top left, a single simulated experiment: dots are flasks, the orange ring is the recommendation, the white diamond is the best achievable setting. Top right and table, the distribution of S over 500 repeats. Bottom, the effect of budget. The CI column is the 95% Monte Carlo interval of the mean, ±1.96 SD/√R.

Reading the result. One factor at a time depends on where it starts. Because the substrate × time interaction reverses the effect of time, holding time at 7 days sends the search into the low-low corner, a good but lower region, whatever the budget. The grid spends flasks evenly and never tests the edge of the region where the best yield sits, so it improves only slowly with budget. The designed experiment places every flask where it helps estimate the whole surface and then reads the optimum from the model, which is why it leads at small budgets. Its advantage is robust to noise over the range offered: at four times the CCD residual error it still reaches about 99% of the optimum while the other two stay between 84% and 88%. Part of that robustness belongs to this surface, whose best setting is a corner fixed by a strong interaction term, so a flatter surface with an interior optimum would be a harder test. These are properties of the fitted surface and are shown as a demonstration of the principle, not as a new laboratory result.

7.7 Sequential search by hand, benchmarked

Task. Find the highest cellulase yield in 10 flasks. Each flask returns f(x) + ε with ε ~ N(0, σ̂²) from the fitted surface, and the surface stays hidden until you finish or reveal it. When the game ends, the page benchmarks your best observed yield against 400 simulated random-search players and 400 simulated designed experiments (a 10-flask CCD), and reports what a quadratic model fitted to your own flasks would have recommended.

Table 1 (generated at game end). The random-search benchmark places 10 flasks uniformly over the full slider range. The designed benchmark is a 10-flask CCD (four factorial, four axial, two centre flasks). Both report the best observed yield, the same quantity you are scored on. Settings beyond the explored square are model extrapolation, and a high score there would need confirmation in the laboratory.

7.8 Multi-objective trade-off analysis

Method. Three objectives are evaluated on the fitted surface over the square 4 to 8 g by 7 to 12 d: titre f, speed f/B (U/mL per day) and substrate efficiency f/A (U/mL per gram). Each is min-max normalized to [0, 1] across the square, and settings are ranked by the weighted sum S = Σ w_k z_k with Σ w_k = 1, the simplest scalarization of a multi-objective problem [24]. The Pareto set contains every setting that no other setting beats on all three objectives at once. Derringer–Suich desirability functions [20] are the usual alternative when targets or tolerances are known.

Figure 6. Left, weighted score over the square; ◇ peak corner (8 g, 12 d), □ fast alternative (4 g, 7 d), ring the optimum for the chosen weights. Right, every setting on a 0.05-coded-unit grid in objective space; blue points are Pareto-optimal over all three objectives (the plot shows two of them), grey points are dominated, and the ring marks the chosen optimum. Prediction intervals in the table are 95% intervals for a new flask from the CCD model.

Interpretation. The Pareto set is a thin ridge, not a cloud: improving speed or efficiency from the peak corner necessarily costs titre. The weighted-sum optimum is not a smooth function of the weights. It stays at the peak corner while titre carries enough weight and then jumps to the low-low corner at a critical weight, reported above for your current speed-to-efficiency ratio. Weighted sums can only reach Pareto points on the convex part of the front, so a compromise setting can be missed even when it is attractive. The weights encode priorities that the study did not measure, such as substrate price, reactor time and the value of a unit of enzyme, so the output is a decision aid and not an economic result.

7.9 Design your own experiment

Purpose. The designs above were fixed by the original study. This generator builds the run sheet for your own experiment: a central composite, Box–Behnken, two-level full factorial or 12-run Plackett–Burman design, with your factor names and ranges, a seeded random run order, and a CSV you can fill in at the bench and analyse with the models in this section.

Table 2. Run sheet for the chosen design. Coded units run from −1 (low) to +1 (high); a rotatable CCD adds axial runs at ±α beyond that range, so check the axial settings are physically possible before you start. The “Response” column is left empty for your measurements.

Reading the diagnostics. A quadratic model in k factors has (k+1)(k+2)/2 parameters; the error degrees of freedom are runs minus that number, and centre-point replicates supply the pure-error part used in the lack-of-fit test. Box–Behnken designs never place a run at a corner of the cube, which helps when the extreme combinations are unsafe or infeasible, at the price of not being able to extend a factorial sequentially.

7.10 How large an effect could 12 runs find? Power and detectable effect

Purpose. “Not significant” in a screening design is only informative if the design could have seen an effect of that size. For a two-level design with N runs the standard error of an effect is SE = 2σ/√N, and the test statistic t = E/SE is compared with the critical value t_1−α/2,ν. If the true effect is δ, t follows a non-central t distribution with non-centrality λ = δ/SE, so power = P(|T′| > c) with T′ = (Z + λ)/√(V/ν), Z ~ N(0,1) and V ~ χ²_ν. The panel evaluates this integral numerically in your browser and finds the smallest effect reached with 80% power.

Figure 7. Solid curve, Bonferroni-adjusted test over all screened factors (α/(N−1)); dashed, unadjusted test. Vertical marks are the estimated effects of the study (A, B, F, E); the 80% line crosses each curve at that test’s minimum detectable effect. The table gives the power the test would have had if the true effect equalled the estimate. That “observed power” is only a restatement of the p-value, so read it as a way to see the design’s resolution and not as evidence about the effect itself. σ defaults to the residual standard error of the 7-term screening model.

Interpretation. With the Bonferroni limit, 12 runs reliably detect only effects above roughly a fifth of the best titre, so the two large effects (A and B) are far inside the detectable range while peptone and baker’s yeast are not: if they were real at the size estimated, the test would have missed them most of the time. The honest reading of “not significant” for the nutrient additions is therefore “not shown to matter at this resolution”. Doubling the runs shrinks the detectable effect by roughly √2 through the standard error and a little more through the extra error degrees of freedom, which lower the critical value.

7.11 Analyse your own central composite design

Purpose. Everything above is computed live from the published data, and the same engine will analyse yours. Paste or upload a CSV with three columns (factor A, factor B, response) in natural units, one run per row. The panel fits the full quadratic model of equation (4), reports the ANOVA-level statistics, classifies the stationary point and finds the best setting inside your design region. The data never leave your browser.

Note. Coding is x = (value − centre)/step with centre = (min + max)/2 and step = (max − min)/(2α) for each factor, so the design needs its axial runs present and α set correctly. The map covers the circle through your outermost runs (a rotatable design’s region); outside it the model is extrapolating. With fewer than 7 runs there is no error estimate, and a model this size needs the usual care: replicate the centre point, and confirm any optimum with fresh runs.

7.12 How stable is the model? Leave-one-out and bootstrap

Method. Thirteen runs support six parameters, so a single odd run can move the fit. Two checks are computed live. Leave-one-out: each run is removed in turn and the model refitted, reporting R², predicted R², the prediction at the peak corner (8 g, 12 d) and the stationary point. Residual bootstrap: residuals are rescaled by 1/√(1−h_ii), resampled with replacement, added to the fitted values and the model refitted B times with a seeded generator. The spread of the refitted predictions gives a distribution-free interval to compare with the textbook t interval, and the refits show how often the saddle and the high-substrate, long-time optimum survive.

Figure 8. Left, bootstrap predictions at 8 g and 12 d; the shaded band is the 95% percentile interval and the dashed lines are the analytical 95% confidence interval for the mean from the t distribution. Right, leave-one-out refits: the row removed, the resulting R² and predicted R², the prediction at the peak corner and the stationary point. Removing one of the five identical centre runs changes little because the others carry the same information. The bootstrap assumes the residuals are exchangeable, which is least secure with only seven error degrees of freedom, so treat the interval as a robustness check on the t interval and not as a replacement for it.

Section 8 · Saccharification

Turning waste into fermentable sugar

The crude enzyme from the optimized fermentation was applied to the agricultural wastes themselves. Every substrate was hydrolyzed, and pea peels gave the highest reducing-sugar yield. Pea peels were then taken into a second, three-factor central composite design with pea peel weight, volume of culture filtrate and incubation time as factors.

Filtrate volume (B) dominates, and its coefficient is negative: within the range tested, adding more culture filtrate lowered the reported sugar concentration. Because the response is a concentration, dilution of the reaction mixture is one plausible contributor, alongside product inhibition and the enzyme-to-substrate ratio. The reported data cannot separate these explanations, and section 11 lists the experiments that would.

Section 9 · Textile applications

The same enzyme, a second market

Cellulases are used in textile finishing because they act on cotton's surface cellulose under mild conditions. The crude A. awamori enzyme was tested in three operations on cotton twill and satin: biopolishing, desizing and dyeing or printing.

Quantifying colour with K/S

Colour strength is measured from reflectance with the Kubelka–Munk function, which is proportional to the dye concentration in the fabric [18]. A higher K/S at the same dye bath means more dye was taken up, which is why K/S is the headline metric for dyeability.

R_∞ is the reflectance of an opaque sample at the wavelength of maximum absorption.

Section 10 · Poster and modelling toolkit

Five statistical models, one poster

The project poster surveys five modelling approaches used in design of experiments, each shown with outputs from statistical software. They go beyond the polynomial response surface used in section 7 and cover binary responses, batch effects, non-polynomial surfaces and time-to-event data. Select the poster to enlarge it.

Enlarge poster

The panels are software outputs from the project's worked examples. They illustrate each method and are separate from the cellulase dataset analysed in sections 6 and 7.

1 · Logistic regressionBinary responseMaximum likelihood

Modelling pass or fail

When the response is binary (growth or no growth, specification met or missed), a linear model can predict impossible probabilities. Logistic regression models the log-odds as linear in the factors and is fitted by maximum likelihood.

(12)ln [ p / (1 − p) ] = β₀ + β₁x₁ + … + β_kx_k

Factor significance comes from likelihood-ratio χ² tests. Because ordinary R² does not exist, fit is summarized by a pseudo-R², commonly McFadden's, 1 − ln L_model / ln L_null; values of 0.2 to 0.4 are conventionally read as a very good fit [15,16]. The poster pairs the significant p-values with decent pseudo-R² values, then checks residuals against predicted values, Cook's distance by run order against a threshold line [22], and actual against predicted.

In a bioprocess study. Contamination outcomes, spore germination, or whether a hydrolysis run exceeds a target yield.

2 · RSM with random blocksMixed modelREML

When runs come in batches

Experiments are often split into blocks such as days, operators or inoculum batches. Treating blocks as random effects assumes the batches in the study are a sample from a wider population of batches.

(13)y = Xβ + Zγ + ε,   γ ~ N(0, σ_b²I),   ε ~ N(0, σ²I)

The poster states the practical advantages: predictions can be made at block levels not included in the experiment, only one variance component has to be estimated instead of one coefficient per extra block, and estimates for the factors of interest become more precise [17]. Variance components are estimated by restricted maximum likelihood (REML), which also recovers information between blocks when the blocks are incomplete.

In a bioprocess study. Flask-to-flask and batch-to-batch variation in inoculum or substrate lot, separated from the factor effects.

3 · Gaussian process modelsNon-parametricSpace-filling designs

Letting the data draw the surface

A Gaussian process places a prior directly on the unknown response function instead of assuming a polynomial. A kernel states that nearby settings should give similar responses, and the posterior mean passes through, or close to, the observed points [12,13].

(14)k(x, x′) = σ_f² exp[ − Σ_j θ_j(x_j − x′_j)² ] + σ_n²δ

(15)ŷ(x) = μ + k(x)^TK⁻¹(y − μ1)   s²(x) = σ_f² − k(x)^TK⁻¹k(x)

Each θ_j acts as a sensitivity: a large value means the response changes quickly along that factor. The prediction variance gives uncertainty bands, which makes the method natural for sequential optimization. The poster's example shows several responses over two factors, with a desirability function [20] combining them, a surface plot for optimization and contour plots for analysis.

In a bioprocess study. A check on whether the saddle shape in section 7 is real or an artefact of forcing a quadratic form onto 13 points.

4 · Weibull regressionTime-to-eventCensoring

Modelling how long things take

For responses such as time to failure, time to germination or time to reach a threshold, a Weibull model describes the survival function and the hazard rate, and factors act on the scale parameter [14,23].

(16)S(t) = exp[ −(t/η)^β ]   h(t) = (β/η)(t/η)^β−1   ln η = x^Tγ

The shape β reads directly as a mechanism. With β < 1 the hazard falls with time (early failures), β = 1 gives a constant hazard (the exponential case), and β > 1 gives a rising hazard (wear-out). Runs that have not finished when observation stops are right-censored, and the likelihood uses them correctly instead of discarding them.

In a bioprocess study. Time for hydrolysis to reach a target conversion, or time to first visible sporulation, with unfinished flasks kept as censored data.

5 · Response surface designQuadratic modelCCD · Box–Behnken

The workhorse of process optimization

Response surface design fits a second-order polynomial to a purpose-built design, a central composite or a Box–Behnken design [3,4,21], and searches the fitted surface for an optimum. The poster shows reaction conversion against two continuous factors as a three-dimensional graph and a contour map with design points overlaid.

This is the method that produced the cellulase surface in section 7. Its limits are the ones listed there: the polynomial form is an approximation valid only near the explored region, and the stationary point can be a saddle rather than an optimum.

In a bioprocess study. Both CCD stages of the cellulase project.

Section 11 · Critical appraisal

What the data support, and what they do not

The study is a clean example of sequential experimental design, and its model statistics are strong. Reading the design tables closely also shows several points that limit how far the conclusions can be pushed.

Findings from re-analysing the design tables

1. The best condition sits on the edge of the design

The highest response (9.00 U/mL at 8 g and 12 days) is a corner of the factorial square, so the true optimum may lie beyond it. The model's predicted maximum inside the design region is also on its boundary (section 7.3). A short run of steepest-ascent experiments past that corner, followed by a smaller second CCD, is the standard next step [3].

2. The surface is a saddle, which opens a cheaper operating point

Because the stationary point is a saddle, there are two good regions rather than one. The table compares them using values from the design.

The low-low corner gives 87% of the peak titre in 58% of the time with half the substrate. Which is better depends on whether titre, throughput or raw-material cost is the objective, and a desirability function [20] is the tool for stating that trade-off formally.

3. Replicated centre points have zero variance

All five centre-point runs are tabulated at 6.85 U/mL, so the pure-error sum of squares is exactly zero and the lack-of-fit F-test cannot be formed. Either the replicates agreed to two decimals or the tabulated values are means. Reporting the individual replicate readings would restore a model-free estimate of experimental noise.

4. Selection inflates the screening p-values

In the Plackett–Burman analysis, terms were retained after inspecting the data and the residual then used for testing has only 4 degrees of freedom. The reduced-model p-values are therefore optimistic, which is why the Bonferroni limit, not the p-value, is the safer yardstick for the screen.

5. The unit basis of the saccharification response

A reducing-sugar value of 338.18 mg/mL is high for a hydrolysate, and many studies express yield per gram of substrate. When this result is reused, the basis of the calculation (per mL of reaction mixture, filtrate or substrate) should be stated, and it determines how the negative filtrate coefficient is interpreted.

Next experiments

1. Confirmation runs at the predicted optimum and at the low-low alternative, in triplicate, to give an external check on the model.
2. Steepest ascent from the high-high corner, then a second CCD or a Box–Behnken design [21] centred on the new region.
3. A multi-response desirability optimization over titre, time and substrate use.
4. Block terms for inoculum batch and substrate lot, analysed as random effects (section 10).
5. A Gaussian-process fit to the 13 CCD points to test whether the saddle survives a non-polynomial model.
6. Saccharification with filtrate dosed per gram of substrate, separating dilution from enzyme loading and from product inhibition.
7. Scale-up in tray or packed-bed bioreactors, where heat removal becomes the limiting variable, and a cost per unit of enzyme.

Section 12 · Reproducibility

Reported versus recomputed

Every number in the interactive panels is computed in your browser by ordinary least squares from the design tables printed in [1]. The table compares those results with the values published in the source article and in the project report. Small differences in the CCD come from the published responses being rounded to two decimals.

The CCD coefficients recomputed here reproduce the published equation ŷ = 6.85 + 0.81A − 0.40B + 1.34AB + 0.11A² + 0.21B² to two decimals.

Software and conventions

• Linear models are fitted with the normal equations solved by Gauss–Jordan elimination. p-values use the regularized incomplete beta function for the F and t distributions.
• Predicted R² uses leave-one-out PRESS residuals from the hat matrix. Studentized residuals are internally studentized.
• The Pareto limits use the t quantile at the error degrees of freedom of the selected model, with α = 0.05 and a Bonferroni divisor of 11.
• The 3-D surface is drawn on a 28 × 28 mesh with painter's-algorithm shading. The contour map uses marching squares on a 60 × 60 grid.

Section 13 · Glossary

Terms and abbreviations

ANOVA

Analysis of variance, partitioning total variation into model and residual parts.

CCD

Central composite design: factorial, axial and centre points for fitting quadratic models.

CMCase

Carboxymethyl-cellulase, the endoglucanase activity measured on CMC.

CV

Coefficient of variation, root mean square error as a percentage of the mean response.

DOE

Design of experiments, planning runs so effects can be estimated efficiently.

GP

Gaussian process, a probabilistic model over functions defined by a kernel.

K/S

Kubelka–Munk colour strength, proportional to dye concentration in fabric.

Leverage

Diagonal of the hat matrix, how far a run's settings lie from the centre of the design.

MS / PP

Molokhia stem and pea peel, the fermentation and saccharification substrates.

OFAT

One factor at a time, the approach that designed experiments replace.

PBD

Plackett–Burman design, a two-level screening design.

PRESS

Predicted residual error sum of squares, from leave-one-out residuals.

REML

Restricted maximum likelihood, the standard way to estimate variance components.

RSM

Response surface methodology, fitting and exploring a fitted polynomial surface.

Rotatable

A design whose prediction variance depends only on distance from the centre.

SSF

Solid-state fermentation, microbial growth on a moist solid with little free water.

Section 14 · References

References

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Section 15 · Code and contact

Reproduce everything

All code, data tables and figures are open. The static page you are reading, the Streamlit tool, and the executable notebook share the same published design tables.

Contact