New Workbook - Click to create a new workbook and paste the results on a new worksheet in the new workbook. The Sampling analysis tool creates a sample from a population by treating the input range as a population.

When the population is too large to process or chart, you can use a representative sample. You can also create a sample that contains only values from a particular part of a cycle if you believe that the input data is periodic. For example, if the input range contains quarterly sales figures, sampling with a periodic rate of four places values from the same quarter in the output range.

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Cross-products of input parameters reveal interaction effects of model input parameters, and squared or higher order terms allow curvature of the hypersurface.

Obviously this can best be presented and understood when the dominant two predicting parameters are used so that the hypersurface is a visualised surface. Although regression analysis can be useful to predict a response based on the values of the explanatory variables, the coefficients of the regression expression do not provide mechanistic insight nor do they indicate which parameters are most influential in affecting the outcome variable.

This is due to differences in the magnitudes and variability of explanatory variables, and because the variables will usually be associated with different units. These are referred to as unstandardized variables and regression analysis applied to unstandardized variables yields unstandardized coefficients.

The independent and dependent variables can be standardized by subtracting the mean and dividing by the standard deviation of the values of the unstandardized variables yielding standardized variables with mean of zero and variance of one.

Regression analysis on standardized variables produces standardized coefficients [ 26 ], which represent the change in the response variable that results from a change of one standard deviation in the corresponding explanatory variable.

While it must be noted that there is no reason why a change of one standard deviation in one variable should be comparable with one standard deviation in another variable, standardized coefficients enable the order of importance of the explanatory variables to be determined in much the same way as PRCCs.

Standardized coefficients should be interpreted carefully — indeed, unstandardized measures are often more informative.

Standardized regression coefficients should not, however, be considered to be equivalent to PRCCs. Consequently, PRCCs and standardized regression coefficients will differ in value and may differ slightly in ranking when analysing the same data. The magnitude of standardized regression coefficients will typically be lower than PRCCs and should not be used alone for determining variable importance when there are large numbers of explanatory variables.

It must be noted that this is true for the statistical model, which is a surrogate for the actual model. The degree to which such claims can be inferred to the true model is determined by the coefficient of determination, R 2.

Factor prioritization is a broad term denoting a group of statistical methodologies for ranking the importance of variables in contributing to particular outcomes.

Variance-based measures for factor prioritization have yet to be used in many computational modelling fields,, although they are popular in some disciplines [ 27 — 34 ]. The objective of reduction of variance is to identify the factor which, if determined that is, fixed to its true, albeit unknown, value , would lead to the greatest reduction in the variance of the output variable of interest.

The second most important factor in reducing the outcome is then determined etc. The concept of importance is thus explicitly linked to a reduction of the variance of the outcome.

Reduction of variance can be described conceptually by the following question: for a generic model,. how would the uncertainty in Y change if a particular independent variable X i could be fixed as a constant?

In general, it is also not possible to obtain a precise factor prioritization, as this would imply knowing the true value of each factor. The reduction of variance methodology is therefore applied to rank parameters in terms of their direct contribution to uncertainty in the outcome.

The factor of greatest importance is determined to be that, which when fixed, will on average result in the greatest reduction in variance in the outcome. Then, a first order sensitivity index of X i on Y can be defined as. Conveniently, the sensitivity index takes values between 0 and 1.

A high value of S i implies that X i is an important variable. Variance based measures, such as the sensitivity index just defined, are concise, and easy to understand and communicate. This is an appropriate measure of sensitivity to use to rank the input factors in order of importance even if the input factors are correlated [ 36 ].

Furthermore, this method is completely 'model-free'. The sensitivity index is also very easy to interpret; S i can be interpreted as being the proportion of the total variance attributable to variable X i.

In practice, this measure is calculated by using the input variables and output variables and fitting a surrogate model, such as a regression equation; a regression model is used in our SaSAT application.

Therefore, one must check that the coefficient of determination is sufficiently large for this method to be reliable an R 2 value for the chosen regression model can be calculated in SaSAT.

It is used very extensively in the medical, biological, and social sciences [ 37 — 41 ]. Logistic regression analysis can be used for any dichotomous response; for example, whether or not disease or death occurs.

Any outcome can be considered dichotomous by distinguishing values that lie above or below a particular threshold. Depending on the context these may be thought of qualitatively as "favourable" or "unfavourable" outcomes.

Logistic regression entails calculating the probability of an event occurring, given the values of various predictors. The logistic regression analysis determines the importance of each predictor in influencing the particular outcome. In SaSAT, we calculate the coefficients β i of the generalized linear model that uses the logit link function,.

There is no precise way to calculate R 2 for logistic regression models. A number of methods are used to calculate a pseudo- R 2 , but there is no consensus on which method is best.

In SaSAT, R 2 is calculated by performing bivariate regression on the observed dependent and predicted values [ 42 ]. Like binomial logistic regression, the Smirnov two-sample test two-sided version [ 43 — 46 ] can also be used when the response variable is dichotomous or upon dividing a continuous or multiple discrete response into two categories.

Each model simulation is classified according to the specification of the 'acceptable' model behaviour; simulations are allocated to either set A if the model output lies within the specified constraints, and set to A ' otherwise.

The Smirnov two-sample test is performed for each predictor variable independently, analysing the maximum distance d max between the cumulative distributions of the specific predictor variables in the A and A ' sets. The test statistic is d max , the maximum distance between the two cumulative distribution functions, and is used to test the null hypothesis that the distribution functions of the populations from which the samples have been drawn are identical.

P-values for the test statistics are calculated by permutation of the exact distribution whenever possible [ 46 — 48 ]. The smaller the p-value or equivalently the larger d max x i , the more important is the predictor variable, X i , in driving the behaviour of the model.

SaSAT has been designed to offer users an easy to use package containing all the statistical analysis tools described above. They have been brought together under a simple and accessible graphical user interface GUI.

The GUI and functionality was designed and programmed using MATLAB ® version 7. However, the user is not required to have any programming knowledge or even experience with MATLAB ® as SaSAT stands alone as an independent software package compiled as an executable.

mat' files, and can convert between them, but it is not requisite to own either Excel or Matlab. The opening screen presents the main menu Figure 3a , which acts as a hub from which each of four modules can be accessed.

SaSAT's User Guide [see Additional file 2 ] is available via the Help tab at the top of the window, enabling quick access to helpful guides on the various utilities. A typical process in a computational modelling exercise would entail the sequence of steps shown in Figure 3b.

The model input parameter sets generated in steps 1 and 2 are used to externally simulate the model step 3. The output from the external model, along with the input values, will then be brought back to SaSAT for sensitivity analyses steps 4 and 5.

a The main menu of SaSAT, showing options to enter the four utilities; b a flow chart describing the typical process of a modelling exercise when using SaSAT with an external computational model, beginning with the user assigning parameter definitions for each parameter used by their model via the SaSAT ' Define Parameter Distribution ' utility.

This is followed by using the ' Generate Distribution Samples ' utility to generate samples for each parameter, the user then employs these samples in their external computational model. Finally the user can analyse the results generated by their computational model, using the ' Sensitivity Analysis' and ' Sensitivity Analysis Plots' utility.

The ' Define Parameter Distribution ' utility interface shown in Figure 4a allows users to assign various distribution functions to their model parameters.

SaSAT provides sixteen distributions, nine basic distributions: 1 Constant, 2 Uniform, 3 Normal, 4 Triangular, 5 Gamma, 6 Lognormal, 7 Exponential, 8 Weibull, and 9 Beta; and seven additional distributions have also been included, which allow dependencies upon previously defined parameters.

When data is available to inform the choice of distribution, the parameter assignment is easily made. However, in the absence of data to inform on the distribution for a given parameter, we recommend using either a uniform distribution or a triangular distribution peaked at the median and relatively broad range between the minimum and maximum values as guided by literature or expert opinion.

When all parameters have been defined, a definition file can be saved for later use such as sample generation. Screenshots of each of SaSAT's four different utilities: a The Define Parameter Distribution Definition utility, showing all of the different types of distributions available, b The Generate Distribution Samples utility, displaying the different types of sampling techniques in the drop down menu, c the Sensitivity Analyses utility, showing all the sensitivity analyses that the user is able to perform, d the Sensitivity Analysis Plots utility showing each of the seven different plot types.

Typically, the next step after defining parameter distributions is to generate samples from those distributions. This is easily achieved using the ' Generate Distribution Samples ' utility interface shown in Figure 4b. Three different sampling techniques are offered: 1 Random, 2 Latin Hypercube, and 3 Full Factorial, from which the user can choose.

Once a distribution method has been selected, the user need only select the definition file created in the previous step using the ' Define Parameter Distribution ' utility , the destination file for the samples to be stored, and the number of samples desired, and a parameter samples file will be generated.

There are several options available, such as viewing and saving a plot of each parameter's distribution. Once a samples file is created, the user may then proceed to producing results from their external model using the samples file as an input for setting the parameter values.

The ' Sensitivity Analysis Utility' interface shown in Figure 4c provides a suite of powerful sensitivity analysis tools for calculating: 1 Pearson Correlation Coefficients, 2 Spearman Correlation Coefficients, 3 Partial Rank Correlation Coefficients, 4 Unstandardized Regression, 5 Standardized Regression, 6 Logistic Regression, 7 Kolmogorov-Smirnov test, and 8 Factor Prioritization by Reduction of Variance.

The results of these analyses can be shown directly on the screen, or saved to a file for later inspection allowing users to identify key relationships between parameters and outcome variables.

The last utility, ' Sensitivity Analyses Plots' interface shown in Figure 4d offers users the ability to visually display some results from the sensitivity analyses.

Users can create: 1 Scatter plots, 2 Tornado plots, 3 Response surface plots, 4 Box plots, 5 Pie charts, 6 Cumulative distribution plots, 7 Kolmogorov-Smirnov CDF plots. Options are provided for altering many properties of figures e. jpeg file, in order to produce images of suitable quality for publication.

To illustrate the usefulness of SaSAT, we apply it to a simple theoretical model of disease transmission with intervention. In the earliest stages of an emerging respiratory epidemic, such as SARS or avian influenza, the number of infected people is likely to rise quickly exponentially and if the disease sequelae of the infections are very serious, health officials will attempt intervention strategies, such as isolating infected individuals, to reduce further transmission.

We present a 'time-delay' mathematical model for such an epidemic. In this model, the disease has an incubation period of τ 1 days in which the infection is asymptomatic and non-transmissible. Following the incubation period, infected people are infectious for a period of τ 2 days, after which they are no longer infectious either due to recovery from infection or death.

During the infectious period an infected person may be admitted to a health care facility for isolation and is therefore removed from the cohort of infectious people. We assume that the rate of colonization of infection is dependent on the number of current infectious people I t , and the infectivity rate λ λ is a function of the number of susceptible people that each infectious person is in contact with on average each day, the duration of time over which the contact is established, and the probability of transmission over that contact time.

Under these conditions, the rate of entry of people into the incubation stage is λ I known as the force of infection ; we assume that susceptible people are not in limited supply in the early stages of the epidemic. In this model λ is the average number of new infections per infectious person per day.

We model the change between disease stages as a step-wise rate, i. The exponential term arises from the fact that infected people are removed at a rate γ over τ 2 days [ 49 ]. See Figure 5 for a schematic diagram of the model structure. Mathematical stability and threshold analyses not shown reveal that the critical threshold for controlling the epidemic is.

This threshold parameter, known as the basic reproduction number [ 50 ], is independent of τ 1 the incubation period. Intuitively, both of these limiting cases represent the average number of days that someone is infectious multiplied by the average number of people to whom they will transmit infection per day.

The sooner such an intervention is implemented, the greater the number of new infections that will be prevented. Therefore, an appropriate outcome indicator of the effectiveness of such an intervention strategy is the cumulative total number of infections over the entire course of the epidemic, which we denote as the 'attack number'.

This quantity is calculated numerically from computer simulation. There are three biological parameters that influence disease dynamics: λ , τ 1 , and τ 2 of which λ and τ 2 are crucial for establishing the epidemic ; and there are three intervention parameters crucial for eliminating the epidemic p , τ 3 and for reducing its epidemiological impact T.

This translates to an initial R 0 prior to interventions of R 0 ~ U 6,10 × N 0. This leads to a total of 27 intervention strategies. To simulate the epidemics, samples are required from each of the three biological parameters' distributions. SaSAT's 'Define Parameter Distribution Definitions' utility allows these distributions to be defined simply.

Then, SaSAT's 'Generate Distribution Samples' utility provides the choice of random, Latin Hypercube, or full-factorial sampling. Of these, Latin Hypercube Sampling is the most efficient sampling method over the parameter space and we recommend this method for most models.

We employed this method here, taking samples, using the defined parameter file. Independent of SaSAT, this set of parameter values was used to carry out numerical simulations of the time-courses of the epidemic, and in each case we commenced the epidemic by introducing one infectious person.

This was then carried out for each of the 27 interventions a total of 27, simulations. For each simulation the time to eradicate the epidemic and the attack number were recorded. These variables became the main outcome variables used for the sensitivity analyses against the input parameters generated by the Latin Hypercube Sampling procedure.

A research paper that is specifically focused on a particular disease and the impact of different strategies would present various figures like Figure 6 , generated from SaSAT's 'Sensitivity Analysis Plots' utility and discussion around their comparison.

The cumulative distribution functions of the distributions of time to eradicate the epidemic and the attack number were produced by SaSAT's 'Sensitivity Plots' utility and shown in Figs. The time until the epidemic was eradicated ranged from 28 to days 99 median, IQR 81— , and the total number of infections ranged from 2 to , median, IQR 55— For the sake of illustration, if the goal of the intervention was to reduce the number of infections to less than , the importance of parameters in contributing to either less than, or greater than, infections can be analysed with SaSAT by categorising each parameter set as a dichotomous variable.

Logistic regression and the Smirnov test were used, within SaSAT's 'Sensitivity Analysis' utility and the results are shown in Table 1. As far as we are aware these methodologies have not previously been used to analyse the results of theoretical epidemic models.

It is seen from Table 1 that λ the infectivity rate was the most important parameter contributing to whether the goal was achieved or not, followed by τ 2 infectious period , and then τ 1 incubation period.

These results can be most clearly demonstrated graphically by Kolmogorov-Smirnov CDF plots Fig. Box plots comparing the 27 different strategies used in the example model, with the whisker length set at 1.

Kolmogorov-Smirnov plots of each parameter displaying the CDFs with the greatest difference between parameter subsets contributing to a 'success' or 'failure' outcome for each parameter see Table 1 : a λ , showing the largest maximum difference between the two CDFs, b τ 1 , showing very little difference, and c τ 2 , showing little difference similar to τ 1.

In this example a 'success' is defined as the total number of infections less than at the end of the epidemic. We investigated the existence of any non-monotonic relationships between the attack number and each of the input parameters through SaSAT's 'Sensitivity Plots' utility e.

see Figure 9 ; no non-monotonic relationships were found, and a clear increasing trend was observed for the attack number versus λ , the infectivity rate. Then, it was determined which parameters most influenced the attack number and by how much.

To conduct this analysis, SaSAT's 'Sensitivity Analyses' utility was used. The calculation of PRCCs was conducted; these are useful for ranking the importance of parameter-output correlations.

Another method that we implemented for ranking was the calculation of standardized regression coefficients; the advantage of these coefficients is the ease of their interpretation in how a change in one parameter can be offset by an appropriate change in another parameter.

A third method for ranking the importance of parameters, not previously used in analysis of theoretical epidemic models as far as we are aware, is factor prioritization by reduction of variance.

The rankings for all correlation coefficients can also be shown as a tornado plot see Figure 10a. Scatter plots comparing the total number of infections log10 scale against each parameter: a τ 1 , shows some weak correlation, b τ 2 , shows little or no correlation, and c λ , showing a strong correlation see Table 2 for correlation coefficients.

a Tornado plot of partial rank correlation coefficients, indicating the importance of each parameter's uncertainty in contributing to the variability in the time to eradicate infection. c Pie chart of factor prioritization sensitivity indices; this visual representation clearly shows the dominance of the infectivity rate for this model.

Note that τ 1 and τ 2 have been combined under the title of 'other', this is because the sensitivity indices of these parameters are both relatively small in magnitude.

The influence of combinations of parameters on outcome variables can be presented visually. Response surface methodology is a powerful approach for investigating the simultaneous influence of multiple parameters on an outcome variable by illustrating i how the outcome variable will be affected by a change in parameter values; and ii how one parameter must change to offset a change in a second parameter.

Figure 10b , from SaSAT's 'Sensitivity Plots' utility shows the pairings of the impact of infectivity rate λ and the incubation period τ 1 on the attack number. Factor prioritization by reduction of variance is a very useful and interpretable measure for sensitivity; it can be represented visually through a pie-chart for example Fig.

In this paper we outlined the purpose and the importance of conducting rigorous uncertainty and sensitivity analyses in mathematical and computational modelling. We then presented SaSAT, a user-friendly software package for performing these analyses, and exemplified its use by investigating the impact of strategic interventions in the context of a simple theoretical model of an emergent epidemic.

The various tools provided with SaSAT were used to determine the importance of the three biological parameters infectivity rate, incubation period and infectious period in i determining whether or not less than people will be infected during the epidemic, and ii contributing to the variability in the overall attack number.

The various graphical options of SaSAT are demonstrated including: box plots to illustrate the results of the uncertainty analysis; scatter plots for assessing the relationships including monotonicity of response variables with respect to input parameters; CDF and tornado plots; and response surfaces for illustrating the results of sensitivity analyses.

The results of the example analyses presented here are for a theoretical model and have no specific "real world" relevance. However, they do illustrate that even for a simple model of only three key parameters, the uncertainty and sensitivity analyses provide clear insights, which may not be intuitively obvious, regarding the relative importance of the parameters and the most effective intervention strategies.

We have highlighted the importance of uncertainty and sensitivity analyses and exemplified this with a relatively simple theoretical model and noted that such analyses are considerably more important for complex models; uncertainty and sensitivity analyses should be considered an essential element of the modelling process regardless of the level of complexity or scientific discipline.

Finally, while uncertainty and sensitivity analyses provide an effective means of assessing a model's "trustworthiness", their interpretation assumes model validity which must be determined separately. There are many approaches to model validation but a discussion of this is beyond the scope of the present paper.

Here, with the provision of the easy-to-use SaSAT software, modelling practitioners should be enabled to carry out important uncertainty and sensitivity analyses much more extensively. Selection of values from a statistical distribution defined with a probability density function for a range of possible values.

For example, a parameter α may be defined to have a probability density function of a Normal distribution with mean 10 and standard deviation 2.

Sampling chooses N values from this distribution. A set of mathematical equations that attempt to describe a system. Typically, the model system of equations is solved numerically with computer simulations.

Mathematical models are different to statistical models, which are usually described as a set of probability distributions or equation to fit empirical data. A constant or variable that must be supplied as input for a mathematical model to be able to generate output.

For example, the diameter of a pipe would be an input parameter in a model looking at the flow of water. Data generated by the mathematical model in response to a set of supplied input parameters, usually relating to a specific aspect of the model, e.

Method used to assess the variability prediction imprecision in the outcome variables of a model that is due to the uncertainty in estimating the input values. Method that extends uncertainty analysis by identifying which parameters are important in contributing to the prediction imprecision.

It quantifies how changes in the values of input parameters alter the value of outcome variables. This allows input parameters to be ranked in order of importance, that is, the parameters that contribute the most to the variability in the outcome variable.

Latin Hypercube Sampling. This is an efficient method for sampling multi-dimensional parameter space to generate inputs for a mathematical model to generate outputs and conduct uncertainty analysis. A relationship or function which preserves a given trend; specifically, the relationship between two factors does not change direction.

That is, as one factor increases the other factor either always increases or always decreases, but does not change from increasing to decreasing. Iman RL, Helton JC: An Investigation of Uncertainty and Sensitivity Analysis Techniques for Computer Models.

Risk Analysis. Article Google Scholar. Iman RL, Helton JC, Campbell JE: An Approach To Sensitivity Analysis Of Computer-Models. Introduction, Input Variable Selection And Preliminary Variable Assessment.

Journal Of Quality Technology. Google Scholar. Iman RL, Helton JC, Campbell JE: An approach to sensitivity analysis of computer-models. Ranking of input variables, response-surface validation, distribution effect and technique synopsis. Journal of Quality Technology. Iman RL, Helton JC: An Investigation Of Uncertainty And Sensitivity Analysis Techniques For Computer-Models.

McKay MD, Beckman RJ, Conover WJ: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. McKay MD, Beckman RJ, Conover WJ: Comparison of 3 methods for selecting values of input variables in the analysis of output from a computer code.

Wackerly DD, Medenhall W, Scheaffer RL: Mathematical Statistics with Applications. Saltelli A, Tarantola S, Campologno F, Ratto M: Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models.

Blower SM, Dowlatabadi H: Sensitivity and uncertainty analysis of complex-models of disease transmission — an HIV model, as an example. International Statistical Review. Stein M: Large sample properties of simulations using Latin Hypercube Sampling.

Handcock MS: Latin Hypercube Sampling to Improve the Efficiency of Monte Carlo Simulations: Theory and Implementation in ASTAP, IBM Research Division, TJ Watson Research Center, RC Saltelli A: Sensitivity Analysis for Importance Assessment.

Article PubMed Google Scholar. DeVeaux RD, Velleman PF: Intro Stats. Iman RL, Conover WJ: Small Sample Sensitivity Analysis Techniques For Computer-Models, With An Application To Risk Assessment.

Communications In Statistics Part A-Theory And Methods. Blower SM, Hartel D, Dowlatabadi H, Anderson RM, May RM: Drugs, sex and HIV: a mathematical model for New York City. Philos Trans R Soc Lond B Biol Sci. Article CAS PubMed Google Scholar. Blower SM, McLean AR, Porco TC, Small PM, Hopewell PC, Sanchez MA, Moss AR: The intrinsic transmission dynamics of tuberculosis epidemics.

Nat Med. Porco TC, Blower SM: Quantifying the intrinsic transmission dynamics of tuberculosis. Theor Popul Biol. Sanchez MA, Blower SM: Uncertainty and sensitivity analysis of the basic reproductive rate. Tuberculosis as an example. Am J Epidemiol. Blower S, Ma L: Calculating the contribution of herpes simplex virus type 2 epidemics to increasing HIV incidence: treatment implications.

Clin Infect Dis. Blower S, Ma L, Farmer P, Koenig S: Predicting the impact of antiretrovirals in resource-poor settings: preventing HIV infections whilst controlling drug resistance. Curr Drug Targets Infect Disord. Blower SM, Chou T: Modeling the emergence of the 'hot zones': tuberculosis and the amplification dynamics of drug resistance.

Breban R, McGowan I, Topaz C, Schwartz EJ, Anton P, Blower S: Modeling the potential impact of rectal microbicides to reduce HIV transmission in bathhouses. Mathematical Biosciences and Engineering. Kleijnen JPC, Helton JC: Statistical analyses of scatterplots to identify important factors in large-scale simulations, 1: Review and comparison of techniques.

Seaholm SK: Software systems to control sensitivity studies of Monte Carlo simulation models. Comput Biomed Res. Seaholm SK, Yang JJ, Ackerman E: Order of response surfaces for representation of a Monte Carlo epidemic model.

Int J Biomed Comput. Schroeder LD, Sqoquist DL, Stephan PE: Understanding regression analysis. Turanyi T, Rabitz H: Local methods and their applications.

Sensitivity Analysis. Edited by: Saltelli A, Chan K, Scott M. Varma A, Morbidelli M, Wu H: Parametric Sensitivity in Chemical Systems. Chapter Google Scholar. Goldsmith CH: Sensitivity Analysis.

Encyclopedia of Biostatistics. Edited by: Armitage P. Campolongo F, Saltelli A, Jensen NR, Wilson J, Hjorth J: The Role of Multiphase Chemistry in the Oxidation of Dimethylsulphide DMS.

A Latitude Dependent Analysis. Journal of Atmospheric Chemistry. Article CAS Google Scholar. Campolongo F, Tarantola S, Saltelli A: Tackling quantitatively large dimensionality problems. Computer Physics Communications.

Kioutsioukis I, Tarantola S, Saltelli A, Gatelli D: Uncertainty and global sensitivity analysis of road transport emission estimates. Atmospheric Environment. Crosetto M, Tarantola S: Uncertainty and sensitivity analysis: tools for GIS-based model implementation. International Journal of Geographic Information Science.

Pastorelli R, Tarantola S, Beghi MG, Bottani CE, Saltelli A: Design of surface Brillouin scattering experiments by sensitivity analysis. Surface Science. Saltelli A, Ratto M, Tarantola S, Campolongo F: Sensitivity analysis practices: Strategies for model-based inference. Saltelli A, Tarantola S: On the relative importance of input factors in mathematical models: Safety assessment for nuclear waste disposal.

Journal of the American Statistical Association. Tabachnick B, Fidell L: Using Multivariate Statistics Third Edition. McCullagh P, Nelder JA: Generalized Linear Models 2nd Edition.

There are several different sampling techniques available, and they can be subdivided into two groups: probability sampling and non-probability sampling. In Microbiological Sampling Plan Analysis Tool · focuses on the elimination of lots deemed unacceptable in accordance with the specified sampling plan; · estimates Research emphasized tools that are used to visualize sampling and analysis data collected in support of remediation after an intentional or

Sampling Analysis Tools - software package capable of analysing RDS data sets. The Respondent Driven Sampling Analysis Tool (RDSAT) includes the following features There are several different sampling techniques available, and they can be subdivided into two groups: probability sampling and non-probability sampling. In Microbiological Sampling Plan Analysis Tool · focuses on the elimination of lots deemed unacceptable in accordance with the specified sampling plan; · estimates Research emphasized tools that are used to visualize sampling and analysis data collected in support of remediation after an intentional or

Microorganisms are present naturally in many foods and if they are not controlled they can alter the composition of the sample to be analyzed. Freezing, drying, heat treatment and chemical preservatives or a combination are often used to control the growth of microbes in foods.

A number of physical changes may occur in a sample, e. Physical changes can be minimized by controlling the temperature of the sample, and the forces that it experiences. Sample Identification. Laboratory samples should always be labeled carefully so that if any problem develops its origin can easily be identified.

The information used to identify a sample includes: a Sample description, b Time sample was taken, c Location sample was taken from, d Person who took the sample, and, e Method used to select the sample. The analyst should always keep a detailed notebook clearly documenting the sample selection and preparation procedures performed and recording the results of any analytical procedures carried out on each sample.

Each sample should be marked with a code on its label that can be correlated to the notebook. Thus if any problem arises, it can easily be identified. Data Analysis and Reporting. Food analysis usually involves making a number of repeated measurements on the same sample to provide confidence that the analysis was carried out correctly and to obtain a best estimate of the value being measured and a statistical indication of the reliability of the value.

A variety of statistical techniques are available that enable us to obtain this information about the laboratory sample from multiple measurements. Measure of Central Tendency of Data. The most commonly used parameter for representing the overall properties of a number of measurements is the mean:.

Here n is the total number of measurements, x i is the individually measured values and is the mean value. The mean is the best experimental estimate of the value that can be obtained from the measurements.

It does not necessarily have to correspond to the true value of the parameter one is trying to measure. There may be some form of systematic error in our analytical method that means that the measured value is not the same as the true value see below.

Accuracy refers to how closely the measured value agrees with the true value. The problem with determining the accuracy is that the true value of the parameter being measured is often not known.

Nevertheless, it is sometimes possible to purchase or prepare standards that have known properties and analyze these standards using the same analytical technique as used for the unknown food samples.

For these reasons, analytical instruments should be carefully maintained and frequently calibrated to ensure that they are operating correctly. Measure of Spread of Data. The spread of the data is a measurement of how closely together repeated measurements are to each other.

The standard deviation is the most commonly used measure of the spread of experimental measurements. This is determined by assuming that the experimental measurements vary randomly about the mean, so that they can be represented by a normal distribution.

The standard deviation SD of a set of experimental measurements is given by the following equation:. Measured values within the specified range:. Sources of Error. There are three common sources of error in any analytical technique:. These occur when the analytical test is not carried out correctly: the wrong chemical reagent or equipment might have been used; some of the sample may have been spilt; a volume or mass may have been recorded incorrectly; etc.

It is partly for this reason that analytical measurements should be repeated a number of times using freshly prepared laboratory samples. Blunders are usually easy to identify and can be eliminated by carrying out the analytical method again more carefully.

These produce data that vary in a non-reproducible fashion from one measurement to the next e. This type of error determines the standard deviation of a measurement. A systematic error produces results that consistently deviate from the true answer in some systematic way, e.

This type of error would occur if the volume of a pipette was different from the stipulated value. For example, a nominally cm 3 pipette may always deliver cm 3 instead of the correct value. To make accurate and precise measurements it is important when designing and setting up an analytical procedure to identify the various sources of error and to minimize their effects.

Often, one particular step will be the largest source of error, and the best improvement in accuracy or precision can be achieved by minimizing the error in this step.

Propagation of Errors. Most analytical procedures involve a number of steps e. These individual errors accumulate to determine the overall error in the final result.

For random errors there are a number of simple rules that can be followed to calculate the error in the final result:. Here, D X is the standard deviation of the mean value X, D Y is the standard deviation of the mean value Y, and D Z is the standard deviation of the mean value Z.

These simple rules should be learnt and used when calculating the overall error in a final result. As an example, let us assume that we want to determine the fat content of a food and that we have previously measured the mass of extracted fat extracted from the food M E and the initial mass of the food M I :.

Initially, we assign values to the various parameters in the appropriate propagation of error equation:. Hence, the fat content of the food is Significant Figures and Rounding.

The number of significant figures used in reporting a final result is determined by the standard deviation of the measurements. A final result is reported to the correct number of significant figures when it contains all the digits that are known to be correct, plus a final one that is known to be uncertain.

For example, a reported value of For example, When rounding numbers: always round any number with a final digit less than 5 downwards, and 5 or more upwards, e.

It is usually desirable to carry extra digits throughout the calculations and then round off the final result. Standard Curves: Regression Analysis. When carrying out certain analytical procedures it is necessary to prepare standard curves that are used to determine some property of an unknown material.

A series of calibration experiments is carried out using samples with known properties and a standard curve is plotted from this data.

For example, a series of protein solutions with known concentration of protein could be prepared and their absorbance of electromagnetic radiation at nm could be measured using a UV-visible spectrophotometer. For dilute protein solutions there is a linear relationship between absorbance and protein concentration:.

A best-fit line is drawn through the date using regression analysis , which has a gradient of a and a y-intercept of b. How well the straight-line fits the experimental data is expressed by the correlation coefficient r 2 , which has a value between 0 and 1.

Most modern calculators and spreadsheet programs have routines that can be used to automatically determine the regression coefficient, the slope and the intercept of a set of data. Rejecting Data. When carrying out an experimental analytical procedure it will sometimes be observed that one of the measured values is very different from all of the other values, e.

Occasionally, this value may be treated as being incorrect, and it can be rejected. There are certain rules based on statistics that allow us to decide whether a particular point can be rejected or not.

R2BEAT Multistage Sampling Design and PSUs selection R-package implementing the multivariate optimal allocation for different domains in one and two stages stratified sample design. SamplingStrata Optimal stratification of sampling frames for multipurpose sampling surveys.

Versione in Italiano. SEARCH IN THIS WEBSITE A-Z Statistics Glossary. Input Range - Enter the references for the range of data that contains the population of values you want to sample. Microsoft Excel draws samples from the first column, then the second column, and so on.

Labels - Select if the first row or column of your input range contains labels. Clear if your input range has no labels; Excel generates appropriate data labels for the output table.

Sampling Method - Click Periodic or Random to indicate the sampling interval you want. Period - Enter the periodic interval at which you want sampling to take place. The period-th value in the input range and every period-th value thereafter is copied to the output column.

Sampling stops when the end of the input range is reached. Number of Samples - Enter the number of random values you want in the output column. Each value is drawn from a random position in the input range, and any number can be selected more than once. Output Range - Enter the reference for the upper-left cell of the output table.

Data is written in a single column below the cell. If you select Periodic, the number of values in the output table is equal to the number of values in the input range, divided by the sampling rate.

In addition, cluster sampling may provide a deeper analysis on a specific Unlike more complicated sampling methods, such as stratified random sampling and Sampling plans are classified in terms of their ability to detect unacceptable (as defined by the associated microbiological criterion) lots of product, and the DESIGN FRAME AND SAMPLE · FS4 (First Stage Stratification and Selection in Sampling) · MAUSS-R (Multivariate Allocation of Units in Sampling Surveys – version R: Sampling Analysis Tools

Article CAS PubMed Google Scholar Porco TC, Blower SM: Quantifying the intrinsic transmission dynamics of tuberculosis. A Tokls on aSmpling Discount codes for online ethnic food orders of the student Samplihg that has green eyes or is Discounted weekly specials disability would result in a mathematical probability based on a simple random survey, but always with a plus or minus variance. Sage South Africa. Then, sample items are randomly selected within each cluster. by responding to a public online survey. The toolbox is built in Matlab ®a numerical mathematical software package, and utilises algorithms contained in the Matlab ® Statistics Toolbox.	Article CAS Google Scholar Crosetto M, Tarantola S: Uncertainty and sensitivity analysis: tools for GIS-based model implementation. Office Add-in Development JavaScript Office Add-ins VSTO and C Integration Macros and VBA Programming High Value Consultancy. Low cost and easy to implement: You rely on this sampling method when you need fast results and operate on a shoestring budget. Skip to main content. If the population is hard to access, snowball sampling can be used to recruit participants via other participants. Last updated: 12 October	There are several different sampling techniques available, and they can be subdivided into two groups: probability sampling and non-probability sampling. In Microbiological Sampling Plan Analysis Tool · focuses on the elimination of lots deemed unacceptable in accordance with the specified sampling plan; · estimates Research emphasized tools that are used to visualize sampling and analysis data collected in support of remediation after an intentional or	In website analytics, data sampling is a practice of selecting a subset of sessions for analysis instead of analyzing the whole population of Hosted feature layers cannot be used in the Analysis tools. Key Features. Create new point samples - select random points within a polygon layer. Select samples Low-flow or passive sampling techniques are preferred for collection of groundwater samples for PFAS to keep the turbidity of samples and purge-water volume to	المدة The Sampling Design Tool has two main functions: 1) to help select a sample from a population, and 2) to perform sample design analysis. When both of these software package capable of analysing RDS data sets. The Respondent Driven Sampling Analysis Tool (RDSAT) includes the following features
Data Discount codes for online ethnic food orders, Statistical Analysis Sampllng Excel, Data Analysis Python and R, and more. Data Samplng Top notch Data Free vacation samples course curriculum Sampling Analysis Tools integrated labs Get the Analyxis advantage in your Data Analytics training 11 months. The adjusted R 2 statistic is a modification of R 2 that adjusts for the number of explanatory terms in the model. In judgment sampling the sub-samples are drawn from the whole population using the judgment and experience of the analyst. A convenience sample simply includes the individuals who happen to be most accessible to the researcher.	New Workbook - Click to create a new workbook and paste the results on a new worksheet in the new workbook. Understand audiences through statistics or combinations of data from different sources. Description of methods In this section we provide a very brief overview and description of the sampling and sensitivity analysis methods used in SaSAT. Article CAS PubMed Google Scholar. While relevant, the findings from a convenience sampling study may lack credibility in the broader research industry. Figure 2.	There are several different sampling techniques available, and they can be subdivided into two groups: probability sampling and non-probability sampling. In Microbiological Sampling Plan Analysis Tool · focuses on the elimination of lots deemed unacceptable in accordance with the specified sampling plan; · estimates Research emphasized tools that are used to visualize sampling and analysis data collected in support of remediation after an intentional or	In website analytics, data sampling is a practice of selecting a subset of sessions for analysis instead of analyzing the whole population of The Sampling analysis tool creates a sample from a population by treating the input range as a population. When the population is too large to process or Random sampling involves selecting data points from the time series dataset in a completely random manner. This technique ensures that each data	There are several different sampling techniques available, and they can be subdivided into two groups: probability sampling and non-probability sampling. In Microbiological Sampling Plan Analysis Tool · focuses on the elimination of lots deemed unacceptable in accordance with the specified sampling plan; · estimates Research emphasized tools that are used to visualize sampling and analysis data collected in support of remediation after an intentional or
No easier method Trial size products to extract a research sample from a larger population than simple Frozen snacks at discounted rates sampling. Use Frozen snacks at discounted rates Toolss is exemplified Sampllng application to a simple epidemic model. The Analyssis of Sa,pling methodology Get artisanal chocolates therefore applied SSampling rank parameters in Analysjs of their direct contribution to uncertainty in the outcome. For instance, a researcher studying the effect of a new drug may choose participants from a nearby hospital rather than from the general population. Once in the bag and properly labeled, you take the cap off while the bottle is in the bag. Online supplementary material to this paper provides the freely downloadable full version of the SaSAT software for use by other practitioners [see Additional file 1 ]. How does convenience sampling work?	Population vs. In random sampling the sub-samples are chosen randomly from any location within the material being tested. Overview of software SaSAT has been designed to offer users an easy to use package containing all the statistical analysis tools described above. To facilitate the development of a sampling plan it is usually convenient to divide an "infinite" population into a number of finite populations, e. Dovetail streamlines research to help you uncover and share actionable insights. Independent of SaSAT, this set of parameter values was used to carry out numerical simulations of the time-courses of the epidemic, and in each case we commenced the epidemic by introducing one infectious person.	There are several different sampling techniques available, and they can be subdivided into two groups: probability sampling and non-probability sampling. In Microbiological Sampling Plan Analysis Tool · focuses on the elimination of lots deemed unacceptable in accordance with the specified sampling plan; · estimates Research emphasized tools that are used to visualize sampling and analysis data collected in support of remediation after an intentional or	In addition, cluster sampling may provide a deeper analysis on a specific Unlike more complicated sampling methods, such as stratified random sampling and Data sampling is a statistical analysis technique used to select, process, and analyze a representative subset of a population. It is also DESIGN FRAME AND SAMPLE · FS4 (First Stage Stratification and Selection in Sampling) · MAUSS-R (Multivariate Allocation of Units in Sampling Surveys – version R	Data sampling is a statistical analysis technique used to select, process, and analyze a representative subset of a population. It is also Random sampling involves selecting data points from the time series dataset in a completely random manner. This technique ensures that each data Sampling solids in powder or granulated form: The following tools may be used: spear samplers, tube-type samplers, zone samplers, sampling trowels, spiral
Sports equipment freebies updated: 12 October mat' files, Ahalysis can convert between them, but it is not Aalysis to own either Excel or Matlab. Log Analyis Discount codes for online ethnic food orders Get free test samples free. Each Frozen snacks at discounted rates simulation Analysks classified according to the specification of Anlaysis 'acceptable' model behaviour; simulations are allocated to either set A if the model output lies within the specified constraints, and set to A ' otherwise. Common non-probability sampling methods include convenience samplingvoluntary response sampling, purposive samplingsnowball sampling, and quota sampling. This more basic form of sampling can be expanded upon to derive more complicated sampling methods. While relevant, the findings from a convenience sampling study may lack credibility in the broader research industry.	Based on the overall proportions of the population, you calculate how many people should be sampled from each subgroup. This is an efficient method for sampling multi-dimensional parameter space to generate inputs for a mathematical model to generate outputs and conduct uncertainty analysis. This can be achieved in several ways involving primarily the calculation of correlation coefficients and regression analysis [ 1 , 7 ], and variance-based methods [ 8 ]. Note that we use the terms parameter, predictor, explanatory variable, factor interchangeably, as well as outcome, output variable, and response. We investigated the existence of any non-monotonic relationships between the attack number and each of the input parameters through SaSAT's 'Sensitivity Plots' utility e.	There are several different sampling techniques available, and they can be subdivided into two groups: probability sampling and non-probability sampling. In Microbiological Sampling Plan Analysis Tool · focuses on the elimination of lots deemed unacceptable in accordance with the specified sampling plan; · estimates Research emphasized tools that are used to visualize sampling and analysis data collected in support of remediation after an intentional or	The Sampling analysis tool creates a sample from a population by treating the input range as a population. When the population is too large to process or SaSAT (Sampling and Sensitivity Analysis Tools) is a user-friendly software package for applying uncertainty and sensitivity analyses to mathematical and Microbiological Sampling Plan Analysis Tool · focuses on the elimination of lots deemed unacceptable in accordance with the specified sampling plan; · estimates	This tool is a Microsoft Excel workbook designed for the purpose of drawing up to two random samples from a population without duplication. This tool can be In website analytics, data sampling is a practice of selecting a subset of sessions for analysis instead of analyzing the whole population of The Sampling analysis tool creates a sample from a population by treating the input range as a population. When the population is too large to process or

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