Quadratic Response Surface Model Research Articles

Use of a Quadratic Response Model to Estimate Optimum Combinations of Nutritional Variables

Abstract Quadratic response surface modeling was used to estimate the optimum combination of feeding rate and percentage of dietary protein for growth of 3–6-week-old blue tilapia Tilapia aurea. Sixteen treatments consisting of all combinations of four dietary protein levels and four feeding rates were replicated in a recirculating-water aquarium system at 23°C for 3 weeks. Quadratic response surfaces fitted to the data were similar for the two replicates and identified treatment combinations supporting maximum growth. Dietary protein, feeding rate, and their interaction significantly affected food conversion by blue tilapia. The quadratic response surface model yielded results that generally agreed with published values for juvenile tilapias. This approach offers a promising research pathway to project optimum culture conditions from short-term trials.

A Paratransit Maintenance Cost-Allocation Model

The maintenance cost of public transit vehicles constitutes a significant portion of a transit system’s total operating cost. While a significant body of literature on the maintenance cost of bus transit systems is available in developed and developing nations, cost records concerning paratransit systems are incomplete at best. This paper reports on the development of a minibus paratransit system maintenance cost model. The physical, maintenance, and service costs of 92 paratransit minibuses, operating over six routes in Riyadh, Saudi Arabia are presented. A correlation matrix, indicating degrees of linear association between a set of explanatory variables (number of stops, route length, travel speed, kilometers of operation, number of service runs, hours of operation, vehicle age, and route pavement condition), land the response variable (maintenance cost), is given. The structure of quadratic response surface and multiple linear regression models are discussed. The model, which satisfies the needs of both accuracy and practical application, is recommended.

Thermal Inactivation of Aeromonas hydrophila As Affected by Sodium Chloride and Ascorbic Acid

The combined effects of sodium chloride (0, 1.0, 1.5, and 3.0%) and ascorbic acid (0, 1.0, and 2.0 mmol/liter) with mild heat (46 degrees C) on the survival of Aeromonas hydrophila were evaluated. Because of the nonlinear nature of the survivor curves obtained, several equations yielding an R (coefficient of multiple determination) of congruent with 1 were tested. The equation that most closely fit the curvature of the observed data set was a hyperbolic function. Equation coefficients were combined to obtain a so-called death value. This value (46.67% explained variance) was calculated by extracting the larger eigenvalue and the relative eigenvector from the correlation matrix of the coefficients. the effects of the experimental factors on the death value were described by a quadratic response surface model. Results revealed that the death value was not influenced by the presence of ascorbic acid. However, increased mortality resulted from the interaction between sodium chloride and ascorbic acid.

RESPONSE SURFACE MODELS FOR THE EFFECTS OF TEMPERATURE, pH, SODIUM CHLORIDE, AND SODIUM NITRITE ON THE AEROBIC AND ANAEROBIC GROWTH OF STAPHYLOCOCCUS AUREUS 196E1

ABSTRACT The effects and interactions of temperature (12–45C), initial pH (4.5–9.0), NaCl (0.5–16.5%), and sodium nitrite (1–200 μg/ml) on the aerobic and anaerobic growth of Staphylococcus aureus 196E were studied using 50 ml portions of Brain Heart Infusion Broth in 250ml unsealed and sealed trypsinizing flasks, respectively. The flasks were inoculated to a level of approximately 103 cfu/ml, incubated on a rotary shaker, sampled periodically, and enumerated on Tryptic Soy Agar. Growth curves were generated by fitting the data to the Gompertz function using nonlinear regression analysis. The general growth characteristics of S. aureus in response to the five environmental variables were similar to those observed by other investigators including (1) enhanced growth in the presence of oxygen, (3) ability to grow at high sodium chloride concentrations, and (3) dependence of the bacteriostatic activity of sodium nitrite on pH and oxygen availability. Supplemental studies indicated that growth kinetics were independent of inoculum size, which allowed the Gompertz A term to be treated as a constant. However, the maximum population density (MPD) achieved by the cultures was dependent on the independent variables, requiring that it be modeled in addition to the Gompertz B and M terms. The MPD was then used to calculate the Gompertz C term. Quadratic and cubic response surface models were generated using various data transformations. Quadratic models using and LN‐transformation provided reasonable predictions of the effects of the four variables on the growth kinetics of S. aureus, and should prove useful for providing initial estimates of the behavior of S. aureus in foods.

DISTANCE MEASURES IN POST HOC COMPARISONS OF TEMPERATURE GERMINATION QUADRATIC RESPONSE SURFACES

Generalized quadratic response surface models are used to describe seed germination at diurnally alternating cold and warm incubation temperatures for three Great Basin exotic plant species. The method of the F-statistic SSdrop is applied to determine whether the response surface models are equal. Two proposed distance measures are used as modified multiple comparison techniques for determining differences between surfaces. These measures prove useful in distinguishing between the species showing the highest germination response and the one showing the lowest response to the incubation temperature ranges studied.

Simulation Designs for Quadratic Response Surface Models in the Presence of Model Misspecification

This article considers the selection of experimental designs for the estimation of second-order response surface metamodels in a simulation environment. Rather than construct designs based on the premise that the postulated model exactly represents the simulated response, as is the case in optimal design theory, we assume that the estimation process may be biased by the presence of unfitted third-order terms. We therefore seek to specify experimental plans that address the bias due to possible model misspecification as well as traditional variance considerations. The performance measure used for this “fit-protect” scenario is Box and Draper's design criterion of average mean squared error of predicted response. Four important classes of response surface experimental plans are examined: (1) central composite designs, (2) Box-Behnken plans, (3) three-level factorial experiments, and (4) small central composite designs. Each design class is studied in conjunction with three pseudorandom number assignment strategies: (1) the use of a unique set of streams at each design point; (2) the assignment of a common stream set to all experimental points: and (3) the simultaneous use of common and antithetic streams through design blocking. Comparisons of both design classes and assignment strategies are presented to assist the user in the selection of an appropriate experimental strategy.

Comparison of a quadratic response surface model and a square root model for predicting the growth rate of Yersinia enterocolitic

Polynomial equations, relating the growth rate of Yersinia enterocolitica to temperature (0–25°C) and pH (4.5–6‐5) in a liquid medium were constructed for four different acidulants. The logarithm of the time for a 100‐fold increase in bacterial numbers could be represented by a quadratic response surface function of pH and temperature. The interactions between pH and temperature on growth rate were found to be additive. Values for a 2 log cycle increase in growth derived from the model were in good agreement with experimental values. Predictions from the quadratic model and from a square root model were compared with experimental values in laboratory media and UHT milk. The mean square error (MSE) for the quadratic response surface model was smaller than that for the square root model in 81% of cases. In UHT milk the square root model increasingly underestimated growth rate, as the temperature decreased and would ‘fail dangerous’ if used for predictive purposes. This indicated that the response surface model is more reliable for predicting the growth of Y. enterocolitica under conditions of sub‐optimal temperature and pH.

Nutrient uptake and partitioning by zucchini squash, head lettuce and potato in response to nitrogen

The effect of N application level on uptake and partitioning of nutrients by zucchini squash (Cucurbita pepo L.) cv. Blackjack, head lettuce (Lactuca sativa L.) cv. Monte110 and potato (Solanum tuberosum L.) cv. Sebago was investigated in sand culture experiments with a stable nutrient supply. Plants were grown during 14 (zucchini squash), 8 (lettuce) and 12 weeks (potato). Five nitrate N levels ranging from 2-36 mol m-3 for lettuce and 2-43 mol m-3 for zucchini squash and potato were applied in a complete nutrient solution. Gamma x quadratic response surface models fitted actual nutrient uptake data (R2> 0 . 9 5 ) . From these models, predicted nutrient uptake, partitioning of nutrients between plant parts and nutrient uptake rates were derived. Nitrogen had a pronounced effect on nutrient uptake of all species. The maximum predicted whole plant (excluding roots) uptake of N and K (parenthesis) was calculated (mol m-3 N) for zucchini squash (19.3, 17.7), lettuce (16.5, 12.1) and potato (12.7, 11.2) respectively. An N deficiency (2 mol m-3)did not result in remobilization of nutrients from vegetative to reproductive growth for zucchini squash and potato. Remobilization of N and K from outer leaves to head occurred for lettuce over the last week of the growth period at all N levels. An N deficiency increased partitioning of N and K by 36 and 54% respectively to reproductive growth for zucchini squash compared with adequate N for fresh yield (14 mol m-3), whereas an N deficiency had a negligible effect on partitioning to reproductive growth for potato. Partitioning, as measured by the ratio of nutrients in fruit, head or tuber to whole plant increased over the growth period, and at final harvest, partitioning was lowest for Ca (0.22-0.24 zucchini squash, 0.21-0.32 lettuce and 0.03-0.09 potato) and highest for K (0.39-0.60 zucchini squash, 0.59-0.68 lettuce and 0.63-0.86 potato). Potato tubers and lettuce head, in contrast to zucchini squash fruit, were dominant sinks for N, K and P.

The interaction of ion implantation with photoresist ashing: A statistical experimental design study

We applied statistical experimental design concepts to the study of the interaction of ion implantation with remote-plasma photoresist ashing. A 28–4IV fractional-factorial screening experiment revealed that implant species and resist hard bake conditions were not significant factors for resist ash rate and uniformity. We obtained quadratic polynomial response surface models using a G-optimal experimental design as a function of the remaining four ash variables (temperature, pressure, oxygen, and hydrogen mass flow rates) and two implant factors (implant dose and energy). Ion implant dose had a significant effect on ash rate while implant energy was an important factor for ash rate uniformity.

Determination of critical nitrogen concentrations of zucchini squash (Cucurbita pepo L.) cv. Blackjack grown in sand culture

A gamma x quadratic response surface model was used to predict the growth rate over the 14-week growth period of zucchini squash (Cucurbita pepo L.) cv. Blackjack in sand culture with nitrogen (N) levels of 2, 7, 14, 29 and 43 mmol/L. Growth rate relative to maximum was plotted against tissue N concentration every 2 weeks, to derive diagnostic petiole sap; leaf nitrate-N and leaf total-N in youngest fully opened leaf, youngest fully expanded leaf and oldest green leaf; and total N in bulked leaf samples. Critical concentrations corresponding to 90% maximum growth rate for deficiency and toxicity are presented. Petiole sap and leaf nitrate-N were much more responsive than leaf total N concentrations over the 2-14 mmol N/L range where positive growth responses were recorded. At 2 mmol N/L, plants were severely N-deficient and growth rate was low (1.6 g/plant.week at fruit set). Tissue nitrate concentrations were negligible, while leaf total N concentrations exceeded 2.6%. Salt toxicity occurred at 29 and 43 mmol N/L, and at the highest N level, tissue N concentrations were sometimes reduced so that concentration ranges for adequacy and toxicity overlapped. Critical tissue N concentrations always exceeded (P<0.05) levels recorded in plants receiving a marginally deficient N level (7 mmol/L). Critical petiole sap and leaf nitrate-N concentrations were much more variable between sampling periods than leaf total N concentrations. Adequate concentration ranges (values between critical concentrations for deficiency and toxicity) were determined for the pre-fruit harvest (weeks 2-6) and fruit harvest (weeks 8-14) growth stages where values were common for consecutive weeks within each sampling period. It was only possible to determine adequate concentrations over the entire growth period for bulked leaf total N (4.30440% prefruit harvest and 4.15-4.45% fruit harvest). Concentrations of potassium (K), phosphorus and sulfur were affected (P<0.05) by N application level, with the largest effect being recorded for K. This confirms the importance of optimising N supply when determining critical levels of these nutrients for zucchini squash. Determination of petiole sap nitrate-N concentrations in the field can be used to distinguish between a deficient and an adequate N supply, but the large variation in values between sampling periods renders this technique less reliable than leaf total N. Tissue N concentrations which exceed critical deficient levels can be interpreted as such because they were recorded when growth was depressed at high N levels. This will rarely occur under field conditions.

Effect of the Mode of Croscarmellose Sodium Incorporation on Tablet Dissolution and Friability

A computer-optimized experimental design was used to study the effect of incorporating a “super disintegrant”, croscarmellose sodium, intragranularly, extragranularly, or distributed equally between the two phases of a tablet in which a poorly soluble drug constituted at least 92.5% of the formulation. The results were analyzed by means of a general quadratic response surface model and suggest that tablets with the same total concentration of super disintegrant dissolve at a faster rate when the super disintegrant is included intragranularly. Tablet friability was not affected by the method of super disintegrant incorporation.

Selection for Vegetative Phase and Actual Filling Period Duration in Short Season Maize

Optimum use of the limited growing period for maize (Zea mays L.) in short season areas is essential to maximize grain yield. The objective of this study was to evaluate the effects of vegetative phase duration (VPD), i.e., time from planting to silk emergence, and actual filling period duration (AFPD) on grain yield and its components in short season maize germplasm. Eleven random mating populations were subjected to divergent selection to form four subpopulations in each that varied in VPD and AFPD, both measured on an Ontario Corn Heat Units (CHU) basis. The material was evaluated in three environments in Ontario, viz., Innerkip 1981, Woodstock 1982, and Ridgetown 1982. Soil type at each location was a Burford silt loam (Typic Hapludalf). The direct effects of selection, and the correlated responses of other agronomic traits, including grain yield, were significant for most traits. For example, at Ridgetown 1982 average differences between the two extreme subpopulation groups were 200 CHU for VPD, 57 CHU for AFPD, 46 g kg−1 for grain moisture at harvest, and 2.0 Mg ha−1 for grain yield. The joint effect of varying VPD and AFPD on grain yield and components, i.e., kernel number per plant and 1000 kernel mass, was assessed by fitting quadratic response surface models. Increased VPD had positive, significant effects on grain yield; increased AFPD had slight positive effects but these were generally not significant. Differences in kernel number per plant contributed most to grain yield variation within any environment, whereas 1000 kernel mass contributed most to grain yield differences across environments. These results indicate a vegetative size limitation for grain yield of short season maize in the material studied. This should not be interpreted strictly as a source (supply of photoassimilate) limitation during the grain filling period, because preanthesis source size may partly determine kernel number per plant and thus affect sink size.

Temperature and Photoperiod Influence Reproductive Development of Reduced-photoperiod-sensitive Mungbean Genotypes

Abstract Using the 9th and 10th International Mungbean Nursery (IMN) data, quadratic response surface models were developed to predict days to flowering (DF) of mungbean [Vigna radiata (L.) Wilczek] genotypes grown at 11-hr and 30 min and 13-hr and 30-min preflowering mean photoperiod (P) and 22 to 30C mean diurnal temperature (T) regimes. Both linear and quadratic effects were significant on DF and on the rate of progress towards flowering (1/DF); however, only the linear effect was significant in days to maturity (DM). The effect of T was more pronounced than that of P on DF of reduced-photoperiod-sensitive (RPS) genotypes. The earliest DF (flowering tendency) was estimated at 34 days after planting at the optimum mean diurnal temperature (T0) of 28C and the optimum mean photoperiod (P0) of 12 hr. At the suboptimal temperature (T < T0), the estimates of the base temperature Tb and the thermal time θr were 10C and 555 degree-days, respectively. Thus, flowering dates of these RPS mungbean lines can be predicted, which in turn will assist in the selection of proper planting dates.

Response surface analysis of the effects of seeding rates, N-rates and irrigation frequencies on durum wheat I. Grain yield and yield components

Interactive effects of nitrogen (N) rates, seeding (S) rates and irrigation frequencies on grain yield and yield components of durum wheat were studied for four years under field conditions at Tulelake, California. Each year the experiment was conducted using a split-plot design with 4 irrigation frequencies as main plots and combinations of 5 N-rates (0 to 360 kg/ha) and 5 S-rates (50 to 250 kg/ha) as subplot treatments replicated 4 times. A quadratic response surface model (RSM) was used to study the effects of these treatments on grain yield and yield components (tillers/area, kernel number/spike, kernel weight/spike and 100-seed weight). The RSM was very effective for analysis and data reduction for estimating the optimum combinations of N and S for maximizing the grain yield and yield components. The N utilization and uptake efficiency increased with each irrigation treatment and peaked at irrigation treatment C. Both N and uptake utilization efficiency decreased with each increment of N-rate. In most cases, the effect of irrigation was independent of N and S. One irrigation at tillering increased grain yield and yield components significantly over only a preplant irrigation. The response of additional irrigations were comparatively small and significant only in some cases. Both N and S had significant effects on grain yield and yield components, however, the response of N was larger than that of S. With increasing N-rate, grain yield and tiller number increased with the expected peak beyond 360kg N ha−1 but the increments beyond 180 kg N ha−1 were of progressively smaller magnitude. The kernel number and kernel weight per spike also increased with N-rate giving a peak between 270 and 360 kg N ha−1. With increasing S grain yield and tiller number/area increased while kernel number and kernel weight per spike decreased progressively. It was impossible to maximize yield and yield components at a given combination of N, S, and irrigation. According to the model, grain yield and tiller number were maximized at the highest level of N and S, while kernel number and kernel weight/spike were maximized at the lowest S (50 kg ha−1) and about 314 kg N ha−1 under adequate water supply. On the basis of the findings of this study and output of the model, 180–360 kg N ha−1, 150–250 kg S ha−1 and two post-sowing irrigations (at tillering and at boot stage) in addition to a preplant irrigation was recommended for optimum yield. An additional irrigation might be required depending on the weather conditions during the grain filling period.

Optimization of Tablet Friability, Maximum Attainable Crushing Strength, Weight Variation and In Vitro Dissolution by Establishing In-Process Variable Controls

AbstractThe level of intragranular microcrystalline cellulose, volume of granulating water, granulation moisture content and tablet crushing strength were used as in-process variables for optimizing tablet friability, maximum attainable crushing strength, weight variation and in vitro dissolution. A computer optimized experimental design (COED) allowed optimal characterization of the variables by designing 22 experiments. The results were analyzed by means of a general quadratic response surface model. Response surfaces were generated for tablet friability, maximum attainable crushing strength, weight variation and in vitro dissolution as a function of the in-process variables. The study provided a useful method in setting optimum ranges for the in-process variables in order to optimize the important tablet parameters.

Effect of Moisture and Crushing Strength on Tablet Friability and In Vitro Dissolution

The friability and dissolution of a formulation of compressed tablets were studied by varying the granulation moisture and tablet crushing strength. A general quadratic response surface model was used to analyze the data. The response surface contour plots of tablet friability consisted of a series of ellipsoidal curves. The optimum friability corresponding to a granulation moisture content and a tablet crushing strength was a simple minimum. The in vitro dissolution contour plots showed a stationary ridge system. Along the ridge, a large number of combinations of tablet crushing strength and granulation moisture represented 100% drug dissolution. The contour overlays of friability and dissolution contour plots showed a region where both the friability and dissolution requirement could be met. The analysis of the data by means of multiple linear regression was helpful in understanding the role of granulation moisture and tablet crushing strength on tablet friability and in vitro dissolution.

Branch-and-Bound Search for Experimental Designs Based onDOptimality and Other Criteria

This article presents a branch-and-bound algorithm that constructs a catalog of all D-optimal n-point designs for specified design region, linear model, and number of observations, n. While the primary design criterion is D optimality, the algorithm may also be. used to find designs performing well by other secondary criteria, if a small sacrifice in ellkiency as measured by D optimahty is accepted. Finally, some designs are supplied for a quadratic response surface model.

D-Optimal Fractions of Three-Level Factorial Designs

D-optimal fractions of three-level factorial designs for p factors are constructed for factorial effects models (2 ≤ p ≤ 4) and quadratic response surface models (2 ≤ p ≤ 5). These designs are generated using an exchange algorithm for maximizing |X′X| and an algorithm which produces D-optimal balanced array designs. The design properties for the DETMAX designs and the balanced array designs are tabulated. An example is given to illustrate the use of such designs.

D-Optimal Fractions of Three-Level Factorial Designs

D-optimal fractions of three-level factorial designs for p factors are constructed for factorial effects models (2 ≤ p ≤ 4) and quadratic response surface models (2 ≤ p ≤ 5). These designs are generated using an exchange algorithm for maximizing |X′X| and an algorithm which produces D-optimal balanced array designs. The design properties for the DETMAX designs and the balanced array designs are tabulated. An example is given to illustrate the use of such designs.

Discussion of: D-Optimal Fractions of Three-Level Factorial Designs

The Mitchell and Bayne paper gives some new designs for factorial models with four and fewer factors and some new designs for quadratic responsesurface models with five and fewer factors. The experimental region considered is a hypercube. It is interesting to examine the tables of properties for these designs (Table 3.2 for the factorial designs, and Table 4.3 for the response surface designs) to see how these designs compare with previously recommended designs. A valuable contribution of this paper clearly shows up in these tables. It is that practicing statisticians have been using good designs and that it is hard to develop better designs. The properties of previously recommended designs are close to the properties of the designs developed in this paper.