Evaluation of data transformations used with the square root and schoolfield models for predicting bacterial growth rate.

A comparison was made between mathematical variations of the square root and Schoolfield models for predicting growth rate as a function of temperature. The statistical consequences of square root and natural logarithm transformations of growth rate use in several variations of the Schoolfield and square root models were examined. Growth rate variances of Yersinia enterocolitica in brain heart infusion broth increased as a function of temperature. The ability of the two data transformations to correct for the heterogeneity of variance was evaluated. A natural logarithm transformation of growth rate was more effective than a square root transformation at correcting for the heterogeneity of variance. The square root model was more accurate than the Schoolfield model when both models used natural logarithm transformation.     

New modified Square Root and Schoolfield models for predicting bacterial growth rate as a function of temperature

Summary A new modified Square Root model and two new modified Schoolfield models were evaluated for their ability to predict the growth rate ofYersinia enterocolitica as a function of temperature. The new Square Root model fits the data better than both the original Square Root model and the Zwietering Square Root model. Both new Schoolfield models, a six-and a four-parameter equation, fit the data better than the original Schoolfield model. The new four-parameter Schoolfield model was developed by removing the term describing low temperature inactivation from the new six-parameter Schoolfield model. Inclusion of the two extra parameters in the new six-parameter Schoolfield model (F=318) did not significantly improve the fit compared to the new fourparameter Schoolfield model (F=488).     

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.     

Evaluation of Data Transformations and Validation of a Model for the Effect of Temperature on Bacterial Growth

The temperature of chilled foods is an important variable for controlling microbial growth in a production and distribution chain. Therefore, it is essential to model growth as a function of temperature in order to predict the number of organisms as a function of temperature and time. This article deals with the correct variance-stabilizing transformation of the growth parameters A (asymptotic level), μ (specific growth rate), and λ (lag time). This is of importance for the regression analysis of the data. A previously gathered data set and model for the effect of temperature on the growth of Lactobacillus plantarum (M. H. Zwietering, J. T. de Koos, B. E. Hasenack, J. C. de Wit, and K. van 't Riet, Appl. Environ. Microbiol. 57:1094-1101, 1991) is extended with new data. With the total data set (original and new data), a variance-stabilizing transformation is selected in order to determine which transformation should precede fitting. No transformation for the asymptote data, a square root for the growth rate, and a logarithmic transformation for the lag time were found to be appropriate. After these transformations, no significant correlation was found between the variance and the magnitude of the variable. Model corrections were made and model parameters were estimated by using the original data. With the new data, the models were validated by comparing the lack of fit of the models with the measurement error, using an F test. The predictions of the models for μ and λ were adequate. The model for A showed a systematic deviation, and therefore a new model for A is proposed.     

Approximate variance-stabilizing transformations for gene-expression microarray data

MOTIVATION: A variance stabilizing transformation for microarray data was recently introduced independently by several research groups. This transformation has sometimes been called the generalized logarithm or glog transformation. In this paper, we derive several alternative approximate variance stabilizing transformations that may be easier to use in some applications. RESULTS: We demonstrate that the started-log and the log-linear-hybrid transformation families can produce approximate variance stabilizing transformations for microarray data that are nearly as good as the generalized logarithm (glog) transformation. These transformations may be more convenient in some applications.     

Transformations of count data for tests of interaction in factorial and split-plot experiments

In applied entomological experiments, when the response is a count-type variable, certain transformation remedies such as the square root, logarithm (log), or rank transformation are often used to normalize data before analysis of variance. In this study, we examine the usefulness of these transformations by reanalyzing field-collected data from a split-plot experiment and by performing a more comprehensive simulation study of factorial and split-plot experiments. For field-collected data, significant interactions were dependent upon the type of transformation. For the simulation study, Poisson distributed errors were used for a 2 by 2 factorial arrangement, in both randomized complete block and split-plot settings. Various sizes of main effects were induced, and type I error rates and powers of the tests for interaction were examined for the raw response values, log-, square root-, and rank-transformed responses. The aligned rank transformation also was investigated because it has been shown to perform well in testing interactions in factorial arrangements. We found that for testing interactions, the untransformed response and the aligned rank response performed best (preserved nominal type I error rates), whereas the other transformations had inflated error rates when main effects were present. No evaluations of the tests for main effects or simple effects have been conducted. Potentially these transformations will still be necessary when performing these tests.     

Dynamic modeling of Listeria monocytogenes growth in pasteurized milk

AIMS: The development and validation of a dynamic model for predicting Listeria monocytogenes growth in pasteurized milk stored at both static and dynamic temperature conditions. METHODS AND RESULTS: Growth of inoculated L. monocytogenes in a commercial pasteurized whole milk product was monitored at various isothermal conditions from 1.5 to 16 degrees C. The kinetic parameters of the pathogen were modelled as a function of temperature using a square root type model, which was further validated using data from 92 published growth curves from eight different milk products. Compared to four published models for L. monocytogenes growth, the model developed in this study performed better, with a per cent discrepancy and bias of 49.1 and -1.01%, respectively. The performance of the model in predicting growth at dynamic temperature conditions was evaluated at four different fluctuating temperature scenarios with periodic temperature changes from -2 to 16 degrees C. The prediction of growth at dynamic storage temperature was based on the square root model in conjunction with the differential equations of the Baranyi and Roberts model, which were numerically integrated with respect to time. The per cent relative errors between the observed and the predicted growth of L. monocytogenes were less than 10% for all temperature scenarios tested. CONCLUSIONS: Available models from experiments conducted in laboratory media may result in significant overestimation of L. monocytogenes growth in pasteurized milk because they do not take into account factors such as milk composition (e.g. natural antimicrobial compounds present in milk) and the interactions of the pathogen with the natural microflora. The product-targeted model developed in the present study showed a high performance in predicting growth of L. monocytogenes in pasteurized milk under both static and dynamic temperature conditions. SIGNIFICANCE AND IMPACT OF THE STUDY: Temperature fluctuations often occur during the transportation and storage of pasteurized milk. A high performance, dynamic model for the growth of L. monocytogenes can be a useful tool for effective management and optimization of product safety and can lead to more realistic estimations of pasteurized-milk related safety risks.     

Estimating biophysical parameters of rice with remote sensing data using support vector machines

Hyperspectral reflectance (350–2500 nm) measurements were made over two experimental rice fields containing two cultivars treated with three levels of nitrogen application. Four different transformations of the reflectance data were analyzed for their capability to predict rice biophysical parameters, comprising leaf area index (LAI; m² green leaf area m⁻² soil) and green leaf chlorophyll density (GLCD; mg chlorophyll m⁻² soil), using stepwise multiple regression (SMR) models and support vector machines (SVMs). Four transformations of the rice canopy data were made, comprising reflectances (R), first-order derivative reflectances (D1), second-order derivative reflectances (D2), and logarithm transformation of reflectances (LOG). The polynomial kernel (POLY) of the SVM using R was the best model to predict rice LAI, with a root mean square error (RMSE) of 1.0496 LAI units. The analysis of variance kernel of SVM using LOG was the best model to predict rice GLCD, with an RMSE of 523.0741 mg m⁻². The SVM approach was not only superior to SMR models for predicting the rice biophysical parameters, but also provided a useful exploratory and predictive tool for analyzing different transformations of reflectance data.     

Relationship between temperature and growth rate of bacterial cultures.

The Arrhenius Law, which was originally proposed to describe the temperature dependence of the specific reaction rate constant in chemical reactions, does not adequately describe the effect of temperature on bacterial growth. Microbiologists have attempted to apply a modified version of this law to bacterial growth by replacing the reaction rate constant by the growth rate constant, but the modified law relationship fits data poorly, as graphs of the logarithm of the growth rate constant against reciprocal absolute temperature result in curves rather than straight lines. Instead, a linear relationship between in square root of growth rate constant (r) and temperature (T), namely, square root = b (T - T0), where b is the regression coefficient and T0 is a hypothetical temperature which is an intrinsic property of the organism, is proposed and found to apply to the growth of a wide range of bacteria. The relationship is also applicable to nucleotide breakdown and to the growth of yeast and molds.     

Development of predictive mathematical model for the growth kinetics of Staphylococcus aureus by response surface model

A response surface model was developed for predicting the growth rates of Staphylococcus aureus in tryptic soy broth (TSB) medium as a function of combined effects of temperature, pH, and NaCl. The TSB containing six different concentrations of NaCl (0, 2, 4, 6, 8, and 10%) was adjusted to an initial of six different pH levels (pH 4, 5, 6, 7, 8, 9, and 10) and incubated at 10, 20, 30, and 40 degrees C. In all experimental variables, the primary growth curves were well (r2=0.9000 to 0.9975) fitted to a Gompertz equation to obtain growth rates. The secondary response surface model for natural logarithm transformations of growth rates as a function of combined effects of temperature, pH, and NaCl was obtained by SAS's general linear analysis. The predicted growth rates of the S. aureus were generally decreased by basic (pH 9-10) or acidic (pH 5-6) conditions and higher NaCl concentrations. The response surface model was identified as an appropriate secondary model for growth rates on the basis of correlation coefficient (r=0.9703), determination coefficient (r2=0.9415), mean square error (MSE=0.0185), bias factor (B(f)=1.0216), and accuracy factor (A(f)=1.2583). Therefore, the developed secondary model proved reliable for predictions of the combined effect of temperature, NaCl, and pH on growth rates for S. aureus in TSB medium.     

Bivariate random‐effects meta‐analysis models for diagnostic test accuracy studies using arcsine‐based transformations

Diagnostic or screening tests are widely used in medical fields to classify patients according to their disease status. Several statistical models for meta‐analysis of diagnostic test accuracy studies have been developed to synthesize test sensitivity and specificity of a diagnostic test of interest. Because of the correlation between test sensitivity and specificity, modeling the two measures using a bivariate model is recommended. In this paper, we extend the current standard bivariate linear mixed model (LMM) by proposing two variance‐stabilizing transformations: the arcsine square root and the Freeman–Tukey double arcsine transformation. We compared the performance of the proposed methods with the standard method through simulations using several performance measures. The simulation results showed that our proposed methods performed better than the standard LMM in terms of bias, root mean square error, and coverage probability in most of the scenarios, even when data were generated assuming the standard LMM. We also illustrated the methods using two real data sets.     

A response surface model based on absorbance data for the growth rates of Salmonella enterica serovar typhimurium as a function of temperature, NaCl, and pH

Response surface model was developed for predicting the growth rates of Salmonella enterica sv. Typhimurium in tryptic soy broth (TSB) medium as a function of combined effects of temperature, pH, and NaCl. The TSB containing six different concentrations of NaCl (0, 2, 4, 6, 8, and 10%) was adjusted to an initial of six different pH levels (pH 4, 5, 6, 7, 8, 9, and 10) and incubated at 10 or 20 degrees C. In all experimental variables, the primary growth curves were well (r2 = 0.900 to 0.996) fitted to a Gompertz equation to obtain growth rates. The secondary response surface model for natural logarithm transformations of growth rates as a function of combined effects of temperature, pH, and NaCl was obtained by SAS's general linear analysis. The predicted growth rates of the S. Typhimurium were generally decreased by basic (9, 10) or acidic (5, 6) pH levels or increase of NaCl concentrations (0-8%). Response surface model was identified as an appropriate secondary model for growth rates on the basis of coefficient determination (r2 = 0.960), mean square error (MSE = 0.022), bias factor (B(f) = 1.023), and accuracy factor (A(f) = 1.164). Therefore, the developed secondary model proved reliable predictions of the combined effect of temperature, NaCl, and pH on growth rates for S. Typhimurium in TSB medium.     

Nonlinear Mixed-Effects (NLME) Diameter Growth Models for Individual China-Fir (Cunninghamia lanceolata) Trees in Southeast China

An individual-tree diameter growth model was developed for Cunninghamia lanceolata in Fujian province, southeast China. Data were obtained from 72 plantation-grown China-fir trees in 24 single-species plots. Ordinary non-linear least squares regression was used to choose the best base model from among 5 theoretical growth equations; selection criteria were the smallest absolute mean residual and root mean square error and the largest adjusted coefficient of determination. To account for autocorrelation in the repeated-measures data, we developed one-level and nested two-level nonlinear mixed-effects (NLME) models, constructed on the selected base model; the NLME models incorporated random effects of the tree and plot. The best random-effects combinations for the NLME models were identified by Akaike''s information criterion, Bayesian information criterion and −2 logarithm likelihood. Heteroscedasticity was reduced with two residual variance functions, a power function and an exponential function. The autocorrelation was addressed with three residual autocorrelation structures: a first-order autoregressive structure [AR(1)], a combination of first-order autoregressive and moving average structures [ARMA(1,1)] and a compound symmetry structure (CS). The one-level (tree) NLME model performed best. Independent validation data were used to test the performance of the models and to demonstrate the advantage of calibrating the NLME models.     

In vitro effects of water activity, temperature and solutes on the growth rate of P. italicum Wehmer and P. digitatum Sacc

AIMS: To evaluate the effect of water activity (a(w) 0.98-0.89, adjusted with glycerol, sorbitol, glucose, or NaCl) and temperature (5-25 degrees C) on the lag phase and radial growth rate (mm day(-1)) of the important citrus spoilage fungi, such as Penicillium italicum and Penicillium digitatum grown in potato dextrose agar (PDA) medium. To select, among models based on the use of different solutes, a model fitting accurately the growth of these species in relation to a(w) and temperature. METHODS AND RESULTS: Extensive data analyses showed for both Penicillium species a highly significant effect of a(w), temperature, solutes and their interactions on radial growth rate (P < 0.0001). Radial growth rate was inhibited and the lag phase (i.e. the time required for growth) lengthened as the a(w) of the medium decreased. NaCl appeared to causes the greatest stress on growth when compared with other nonionic solutes. Penicillium italicum stopped growing at 0.96 a(w) and P. digitatum at 0.93 a(w). Under the dry conditions where growth was observed, P. italicum grew faster than P. digitatum at low temperature and P. digitatum remained more active at ambient temperature. Multiple regression analysis applied to the square roots of the growth rates observed in the presence of each solute showed that both the 'glycerol model' and the 'sorbitol model' yielded a good prediction of P. italicum growth and the 'sorbitol model' gave an accurate fit for P. digitatum growth, offering high-quality prediction within the experimental limits described. CONCLUSIONS: Mathematical models describing and predicting, as a function of a(w) and temperature, the square root of the radial growth rate of the agents responsible for blue and green decays are important tools for understanding the behaviour of these fungi under natural conditions and for predicting citrus fruit spoilage. SIGNIFICANCE AND IMPACT OF THE STUDY: Implementation of these results should contribute towards a more rational control strategy against citrus spoilage fungi.     

Gastric evacuation in horse mackerel. I. The effects of meal size, temperature and predator weight

Gastric evacuation experiments were performed on horse mackerel Trachurus trachurus. A nearly full matrix experimental design with respect to the variables predator weight (<10–400 g) meal size (up to 7·8% body weight) and temperature (10–20°) was covered with 0-group smelt Osmerus eperlanus as prey. A general evacuation model without meal size as a variable was fitted to the data on wet weights as well as on dry weights by means of non-linear regression technique. Two methods of data transformation, relative data and square root transformation, were applied to improve variance homogeneity. The most reliable model fit was achieved on dry weight data applying the square root transformation technique: where S_t=stomach content (g wet weight) at time t after ingestion, S₀=the initial meal size, W =predator (g wet weight), and T =temperature. The estimated coefficient of the exponential temperature function, δ=00·032, corresponds to a Q ₁₀ value of 1·4 which is outstandingly low in comparison with results on other species. However additional experiments to determine maximum daily food rations indicated that appetite in contrast to gastric evacuation is strongly temperature dependent.     

Utility of phenomenological models for describing temperature dependence of bacterial growth.

We compared three unstructured mathematical models, the master reaction, the square root, and the damage/repair models, for describing the relationship between temperature and the specific growth rates of bacteria. The models were evaluated on the basis of several criteria: applicability, ease of use, simple interpretation of model parameters, problem-free determination of model parameters, statistical evaluation of goodness of fit (chi 2 test), and biological relevance. Best-fit parameters for the master reaction model could be obtained by using two consecutive nonlinear least-square fits. The damage/repair model proved to be unsuited for the data sets considered and was judged markedly overparameterized. The square root model allowed nonproblematical parameter estimation by a nonlinear least-square procedure and, together with the master reaction model, was able to describe the temperature dependence of the specific growth rates of Klebsiella pneumoniae NCIB 418, Escherichia coli NC3, Bacillus sp. strain NCIB 12522, and the thermotolerant coccobacillus strain NA17. The square root and master reaction models were judged to be equally valid and superior to the damage/repair model, even though the square root model is devoid of a conceptual basis.     

Comparison of Arrhenius-type and Bêlehrádek-type models for prediction of bacterial growth in foods

D.A. RATKOWSKY, T. ROSS, T.A. WCMEEKIN AND J. OLLEY. 1991. The development of Arrhenius-type ('Schoolfield') and Bêlehrádek-type (square root) models that describe microbial growth rates is briefly described. Both types of model have been advocated for use in predictive microbiology. On the basis of published data sets for the growth of bacteria, the consequences of mathematical transformation of data and the use of invalid stochastic assumptions upon model predictions are demonstrated. Mean square error is shown to be an inappropriate criterion by which to compare the performance of predictive models. The data show that bacterial growth responses such as generation time and lag time become more variable as their mean magnitude increases. The practical consequences of such variability for predictive microbiology are discussed.     

Utility of phenomenological models for describing temperature dependence of bacterial growth.

We compared three unstructured mathematical models, the master reaction, the square root, and the damage/repair models, for describing the relationship between temperature and the specific growth rates of bacteria. The models were evaluated on the basis of several criteria: applicability, ease of use, simple interpretation of model parameters, problem-free determination of model parameters, statistical evaluation of goodness of fit (chi 2 test), and biological relevance. Best-fit parameters for the master reaction model could be obtained by using two consecutive nonlinear least-square fits. The damage/repair model proved to be unsuited for the data sets considered and was judged markedly overparameterized. The square root model allowed nonproblematical parameter estimation by a nonlinear least-square procedure and, together with the master reaction model, was able to describe the temperature dependence of the specific growth rates of Klebsiella pneumoniae NCIB 418, Escherichia coli NC3, Bacillus sp. strain NCIB 12522, and the thermotolerant coccobacillus strain NA17. The square root and master reaction models were judged to be equally valid and superior to the damage/repair model, even though the square root model is devoid of a conceptual basis.     

Regression models for overdispersed jejunal surviving crypts data

The dose-response model concerns to establish a relationship between a dose and the magnitude of the response produced by the dose. A common complication in the dose-response model for jejunal crypts cell surviving data is overdispersion, where the observed variation exceeds that predicted from the binomial distribution. In this study, two different methods for analyzing jejunal crypts cell survival after regimens of several fractions are contrasted and compared. One method is the logistic regression approach, where the numbers of surviving crypts are predicted by the logistic function of a single dose of radiation. The other one is the transform-both-sides approach, where the arcsine transformation family is applied based on the first-order variance-stabilizing transformation. This family includes the square root, arcsine, and hyperbolic arcsine transformations, which have been used for Poisson, binomial, and negative binomial count data, as special cases. These approaches are applied to a data set from radiobiology. Simulation study indicates that the arcsine transformation family is more efficient than the logistic regression when there exists moderate overdispersion.