Designation: E1847 − 96 (Reapproved 2013)Standard Practice forStatistical Analysis of Toxicity Tests Conducted UnderASTM Guidelines1This standard is issued under the fixed designation E1847; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice covers guidance for the statistical analysisof laboratory data on the toxicity of chemicals or mixtures ofchemicals to aquatic or terrestrial plants and animals. Thispractice applies only to the analysis of the data, after the testhas been completed.All design concerns, such as the statementof the null hypothesis and its alternative, the choice of alphaand beta risks, the identification of experimental units, possiblepseudo replication, randomization techniques, and the execu-tion of the test are beyond the scope of this practice. Thispractice is not a textbook, nor does it replace consultation witha statistician. It assumes that the investigator recognizes thestructure of his experimental design, has identified the experi-mental units that were used, and understands how the test wasconducted. Given this information, the proper statistical analy-ses can be determined for the data.1.1.1 Recognizing that statistics is a profession in whichresearch continues in order to improve methods for performingthe analysis of scientific data, the use of statistical methodsother than those described in this practice is acceptable as longas they are properly documented and scientifically defensible.Additional annexes may be developed in the future to reflectcomments and needs identified by users, such as more detaileddiscussion of probit and logistic regression models, or statisti-cal methods for dose response and risk assessment.1.2 The sections of this guide appear as follows:Title SectionReferenced Documents 2Terminology 3Significance and Use 4Statistical Methods 5Flow Chart 6Flow Chart Comments 7Keywords 8References1.3 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2E178 Practice for Dealing With Outlying ObservationsE456 Terminology Relating to Quality and StatisticsE1241 Guide for Conducting Early Life-Stage Toxicity Testswith FishesE1325 Terminology Relating to Design of ExperimentsIEEE/ASTM SI 10 American National Standard for Use ofthe International System of Units (SI):The Modern MetricSystem3. Terminology3.1 Definitions of Terms Specific to This Standard:3.1.1 The following terms are defined according to thereferences noted:3.1.2 analysis of variance (ANOVA)—a technique that sub-divides the total variation of a set of data into meaningfulcomponent parts associated with specific sources of variationfor the purpose of testing some hypothesis on the parameters ofthe model or estimating variance components (1).33.1.3 categorical data—variates that take on a limitednumber of distinct values (2).3.1.4 censored data—some subjects have not experiencedthe event of interest at the end of the study or time of analysis.The exact survival times of these subjects are unknown (3).3.1.5 central limit theorem—whatever the shape of thefrequency distribution of the original populations of X’s, thefrequency distribution of the mean, in repeated randomsamples of size n tends to become normal as n increases (2).1This practice is under the jurisdiction of ASTM Committee E50 on Environ-mental Assessment, Risk Management and Corrective Action and is the directresponsibility of Subcommittee E50.47 on Biological Effects and EnvironmentalFate.Current edition approved March 1, 2013. Published March 2013. Originallyapproved in 1996. Last previous edition approved in 2008 as E1847–96(2008). DOI:10.1520/E1847-96R13.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

[email protected] For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.3The boldface numbers given in parentheses refer to a list of references at theend of the text.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States13.1.6 central tendency measure—a statistic that measuresthe central location of the sample observations (4).3.1.7 concentration-response testing—the quantitative rela-tion between the amount of factor X and the magnitude of theeffect it causes is determined by performing parallel sets ofoperations with various known amounts, or doses, of the factorand measuring the result, that is called the response (5).3.1.8 continuous data—a variable that can assume a con-tinuum of possible outcomes (4).3.1.9 control—an experiment in which the subjects aretreated as in a parallel experiment except for omission of theprocedure or agent under test and that is used as a standard ofcomparison in judging experimental effects (6).3.1.10 dichotomous data—variates that have only 2 mutu-ally exclusive outcomes, binary data, success or failure data(3).3.1.11 dispersion measure—a statistic that measures thecloseness of the independent observations within groups, orrelative to a sample’s central value (4).3.1.12 distribution—a set of all the various values thatindividual observations may have and the frequency of theiroccurrence in the sample or population (1).3.1.13 duplication—the execution of a treatment at leasttwice under similar conditions (1).3.1.14 experimental unit—a portion of the experimentalspace to which a treatment is applied or assigned in theexperiment (1).3.1.15 homogeneity—lack of significant differences amongmean squares of an analysis (2).3.1.16 hypothesis test—a decision rule (strategy, recipe)which, on the basis of the sample observations, either acceptsor rejects the null hypothesis (4).3.1.17 independence—having the property that the jointprobability (as of all events or samples) or the joint probabilitydensity function (as of random variables) equals the product ofthe probabilities or probability density functions of separateoccurrence (6).3.1.18 mean—a measure of central tendency or location thatis the sum of the observations divided by the number ofobservations (1).3.1.19 model—an equation that is intended to provide afunctional description of the sources of information which maybe obtained from an experiment (1).3.1.20 nonparametric statistic—a statistic which has certaindesirable properties that hold under relatively mild assump-tions regarding the underlying populations (4).3.1.21 normality—having the characteristics of a normaldistribution (2).3.1.22 outlier—an outlying observation is one that appearsto deviate markedly from other members of the sample inwhich it occurs (see Practice E178).3.1.23 parametric statistic—a statistic that estimates anunknown constant associated with a population (4).3.1.24 probit logit—when the response Y in binary, theprobit/logit equation is as follows:p 5 Pr~Y 5 0! 5 C1~1 2 C! F~x b! (1)where:b = vector of parameter estimates,F = cumulative distribution function (normal, logistic),x = vector of independent variables,p = probability of a response, andC = natural (threshold) response rate.The choice of the distribution function, F, (normal for theprobit model, logistic for the logit model) determines the typeof analysis (7).3.1.25 regression analysis—the process of estimating theparameters of a model by optimizing the value of an objectivefunction (for example, by the method of least squares) and thentesting the resulting predictions for statistical significanceagainst an appropriate null hypothesis model (1).3.1.26 replication—the repetition of the set of all the treat-ment combinations to be compared in an experiment. Each ofthe repetitions is called a replicate (1).3.1.27 residual—Yobsminus Ypred− the difference betweenthe observed response variable value and the response variablevalue that is predicted by the model that is fit to the data (8).3.1.28 scedasticity—variance (5).3.1.29 significance level—the probability at which the nullhypothesis is falsely rejected, that is, rejecting the null hypoth-esis when in fact it is true (4).3.1.30 transformation—the transformation of the observa-tions Xij into another scale for purposes of allowing thestandard analysis to be used as an adequate approximation (2).3.1.31 treatment—a combination of the levels of each of thefactors assigned to an experimental unit (see TerminologyE456).3.1.32 variance—a measure of the squared dispersion ofobserved values or measurements expressed as a function ofthe sum of the squared deviations from the population mean orsample average (see Terminology E456).4. Significance and Use4.1 The use of statistical analysis will enable the investiga-tor to make better, more informed decisions when using theinformation derived from the analyses.4.1.1 The goals when performing statistical analyses, are tosummarize, display, quantify, and provide objective measuresfor assessing the relationships and anomalies in data. Statisticalanalyses also involve fitting a model to the data and makinginferences from the model. The type of data dictates the type ofmodel to be used. Statistical analysis provides the means to testdifferences between control and treatment groups (one form ofhypothesis testing), as well as the means to describe therelationship between the level of treatment and the measuredresponses (concentration effect curves), or to quantify thedegree of uncertainty in the end-point estimates derived fromthe data.E1847 − 96 (2013)24.1.2 The goals of this practice are to identify and describecommonly used statistical procedures for toxicity tests. Fig. 1,Section 6, following statistical methods (Section 5), presents aflow chart and some recommended analysis paths, with refer-ences. From this guideline, it is recommended that eachinvestigator develop a statistical analysis protocol specific tohis test results. The flow chart, along with the rest of thisguideline, may provide both useful direction, and service as aquality assurance tool, to help ensure that important steps in theanalysis are not overlooked.5. Statistical Methods5.1 Exploratory Data Analysis—The first step in any dataanalysis is to look at the data and become familiar with theircontent, structure, and any anomalies that might be present.5.1.1 Plots:5.1.1.1 Histograms are unidimensional plots that show thedistributional shapes in the data and the frequencies of indi-vidual values. These diagrams allow the investigator to checkfor unusual observations and also visually check the validity ofsome assumptions that are necessary for several statisticalanalyses that may be used (9).5.1.1.2 Scatter plots of two or more variables demonstratethe relationships among the variables, so that correlations canbe observed and interactions can be studied. These plots arevery useful when looking for concentration effect relationships(9).5.1.1.3 Normality and box plots are additional plots thatgive distributional information, quantiles and pictures of thedata, either as a whole or by treatment group (9).5.1.2 Outliers—On occasion, some data points in thehistogram, scatter plot, or box plot, appear to be quite differentfrom the majority of points. These data, known as outliers, canbe tested to determine if they are truly different from thedistribution of the experimental data (10). The Z or t scores areusually used for testing, with a confidence level chosen by theinvestigator. If they are different and can be attributed to anerror in the execution of the study (violation of protocol, dataentry error, and so forth), then they can be removed from theanalyses. However, if there is no legitimate reason to removethem, then they must be kept in the analyses. It is recom-mended that the analyses can be conducted on two data sets,the complete one and one with the outliers removed. In thisway, the outliers’ influence on the analyses can be studied.FIG. 1 Flow Chart for Practice for Statistical AnalysisE1847 − 96 (2013)3FIG. 1 Flow Chart for Practice for Statistical Analysis (continued)FIG. 1 Flow Chart for Practice for Statistical Analysis (continued)E1847 − 96 (2013)45.1.3 Non-Detected Data:5.1.3.1 Data that fall below a chemical analysis thresholdlevel of detection, in an analytical technique used to measure avalue, are called non-detected. Values that occur above thedetection limit but are below the limit of quantitation, arecalled non-estimable. Occasionally, the two terms are usedinterchangeably. Essentially, these data are results for which noreliable number can be determined.5.1.3.2 In analyzing a data set containing one or morenon-detects, several methods can be used. If the amount ofnon-detects is below approximately 25 % of the entire data set,then the non-detects can be replaced by one half the detectionlimit (or quantitation limit, whichever is appropriate) andanalysis proceeds (11). One half the detection or quantitationlimit is often used to prevent undue bias from entering theanalysis. In some cases, the full detection limit may be moreappropriate for the analyses, or substituting values derivedfrom a distribution function fit to the non-detected range, thatis appropriate given the distribution of the detected values.Zero is not usually used as a substitute because of the bias itintroduces to the analyses, and potential underestimation of thestatistics involved. However, zero may be the most appropriatevalue in certain situations, as determined by best professionaljudgment. One example is the analysis of control samples, thatare known with a very high degree of confidence to be free ofthe chemical being analyzed, that is, zero concentration. Ifthere are more than approximately 25 % non-detects in the dataset, then the proportions of non-detects to the total sample sizefor each group are analyzed on a present/absent basis, and theanalysis is done on the proportions. If there are more thanapproximately 50 % non-detects in the data set, the proportionscan be analyzed as above, or the data can be partitioned intodetects and non-detects. The detects group is then analyzed byitself, to reveal the information it holds.5.1.4 Descriptive Statistics—The next step is to summarizethe information contained in the data, by means of descriptivestatistics. First and foremost is the sample size or number ofobservations in the test, broken out by treatment groups,experimental units, or blocks, whatever is appropriate for thetest being analyzed. Other most common ones are measures ofcentral tendency and of dispersion within the data. Centraltendency measures are the mean, median (also