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define analysis of variance

What Is Analysis of Variance (ANOVA)?

The formula for anova is:, what does the analysis of variance reveal, example of how to use anova, one-way anova versus two-way anova, analysis of variance (anova) explanation, formula, and applications.

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define analysis of variance

Xiaojie Liu / Investopedia

Analysis of variance (ANOVA) is an analysis tool used in statistics that splits an observed aggregate variability found inside a data set into two parts: systematic factors and random factors. The systematic factors have a statistical influence on the given data set, while the random factors do not. Analysts use the ANOVA test to determine the influence that independent variables have on the dependent variable in a regression study.

The t- and z-test methods developed in the 20th century were used for statistical analysis until 1918, when Ronald Fisher created the analysis of variance method. ANOVA is also called the Fisher analysis of variance, and it is the extension of the t- and z-tests. The term became well-known in 1925, after appearing in Fisher's book, "Statistical Methods for Research Workers." It was employed in experimental psychology and later expanded to subjects that were more complex.

Key Takeaways

What Is the Analysis of Variance (ANOVA)?

F = MST MSE where: F = ANOVA coefficient MST = Mean sum of squares due to treatment MSE = Mean sum of squares due to error \begin{aligned} &\text{F} = \frac{ \text{MST} }{ \text{MSE} } \\ &\textbf{where:} \\ &\text{F} = \text{ANOVA coefficient} \\ &\text{MST} = \text{Mean sum of squares due to treatment} \\ &\text{MSE} = \text{Mean sum of squares due to error} \\ \end{aligned} F = MSE MST where: F = ANOVA coefficient MST = Mean sum of squares due to treatment MSE = Mean sum of squares due to error

The ANOVA test is the initial step in analyzing factors that affect a given data set. Once the test is finished, an analyst performs additional testing on the methodical factors that measurably contribute to the data set's inconsistency. The analyst utilizes the ANOVA test results in an f-test to generate additional data that aligns with the proposed regression models.

The ANOVA test allows a comparison of more than two groups at the same time to determine whether a relationship exists between them. The result of the ANOVA formula, the F statistic (also called the F-ratio), allows for the analysis of multiple groups of data to determine the variability between samples and within samples.

If no real difference exists between the tested groups, which is called the null hypothesis , the result of the ANOVA's F-ratio statistic will be close to 1. The distribution of all possible values of the F statistic is the F-distribution. This is actually a group of distribution functions, with two characteristic numbers, called the numerator degrees of freedom and the denominator degrees of freedom.

A researcher might, for example, test students from multiple colleges to see if students from one of the colleges consistently outperform students from the other colleges. In a business application, an R&D researcher might test two different processes of creating a product to see if one process is better than the other in terms of cost efficiency.

The type of ANOVA test used depends on a number of factors. It is applied when data needs to be experimental. Analysis of variance is employed if there is no access to statistical software resulting in computing ANOVA by hand. It is simple to use and best suited for small samples. With many experimental designs, the sample sizes have to be the same for the various factor level combinations.

ANOVA is helpful for testing three or more variables. It is similar to multiple two-sample t-tests . However, it results in fewer type I errors and is appropriate for a range of issues. ANOVA groups differences by comparing the means of each group and includes spreading out the variance into diverse sources. It is employed with subjects, test groups, between groups and within groups.

There are two main types of ANOVA: one-way (or unidirectional) and two-way. There also variations of ANOVA. For example, MANOVA (multivariate ANOVA) differs from ANOVA as the former tests for multiple dependent variables simultaneously while the latter assesses only one dependent variable at a time. One-way or two-way refers to the number of independent variables in your analysis of variance test. A one-way ANOVA evaluates the impact of a sole factor on a sole response variable. It determines whether all the samples are the same. The one-way ANOVA is used to determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups.

A two-way ANOVA is an extension of the one-way ANOVA. With a one-way, you have one independent variable affecting a dependent variable. With a two-way ANOVA, there are two independents. For example, a two-way ANOVA allows a company to compare worker productivity based on two independent variables, such as salary and skill set. It is utilized to observe the interaction between the two factors and tests the effect of two factors at the same time.

Genetic Epidemiology, Translational Neurogenomics, Psychiatric Genetics and Statistical Genetics-QIMR Berghofer Medical Research Institute. " The Correlation Between Relatives on the Supposition of Mendelian Inheritance ."

Encyclopaedia Britannica. " Sir Ronald Aylmer Fisher ."

Ronald Fisher. " Statistical Methods for Research Workers ." Springer-Verlag New York, 1992.

Financial Analysis

Math and Statistics

Financial Ratios

Advanced Technical Analysis Concepts

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What is Analysis of Variance (ANOVA)?

Analysis of Variance ( ANOVA ) is a statistical formula used to compare variances across the means (or average) of different groups. A range of scenarios use it to determine if there is any difference between the means of different groups.

For example, to study the effectiveness of different diabetes medications, scientists design and experiment to explore the relationship between the type of medicine and the resulting blood sugar level. The sample population is a set of people. We divide the sample population into multiple groups, and each group receives a particular medicine for a trial period. At the end of the trial period, blood sugar levels are measured for each of the individual participants. Then for each group, the mean blood sugar level is calculated. ANOVA helps to compare these group means to find out if they are statistically different or if they are similar.

The outcome of ANOVA is the ‘F statistic’. This ratio shows the difference between the within group variance and the between group variance, which ultimately produces a figure which allows a conclusion that the null hypothesis is supported or rejected. If there is a significant difference between the groups, the null hypothesis is not supported, and the F-ratio will be larger.

ANOVA Ebook

ANOVA Terminology

Dependent variable : This is the item being measured that is theorized to be affected by the independent variables.

Independent variable/s : These are the items being measured that may have an effect on the dependent variable.

A null hypothesis (H0) : This is when there is no difference between the groups or means. Depending on the result of the ANOVA test, the null hypothesis will either be accepted or rejected.

An alternative hypothesis (H1) : When it is theorized that there is a difference between groups and means.

Factors and levels : In ANOVA terminology, an independent variable is called a factor which affects the dependent variable. Level denotes the different values of the independent variable that are used in an experiment.

Fixed-factor model : Some experiments use only a discrete set of levels for factors. For example, a fixed-factor test would be testing three different dosages of a drug and not looking at any other dosages.

Random-factor model : This model draws a random value of level from all the possible values of the independent variable.

What is the Difference Between One Factor and Two Factor ANOVA?

There are two types of ANOVA.

One-Way ANOVA

The one-way analysis of variance is also known as single-factor ANOVA or simple ANOVA. As the name suggests, the one-way ANOVA is suitable for experiments with only one independent variable (factor) with two or more levels. For instance a dependent variable may be what month of the year there are more flowers in the garden. There will be twelve levels. A one-way ANOVA assumes:

Full Factorial ANOVA (also called two-way ANOVA)

Full Factorial ANOVA is used when there are two or more independent variables. Each of these factors can have multiple levels. Full-factorial ANOVA can only be used in the case of a full factorial experiment, where there is use of every possible permutation of factors and their levels. This might be the month of the year when there are more flowers in the garden, and then the number of sunshine hours. This two-way ANOVA not only measures the independent vs the independent variable, but if the two factors affect each other. A two-way ANOVA assumes:

Why Does ANOVA work?

Some people question the need for ANOVA; after all, mean values can be assessed just by looking at them. But ANOVA does more than only comparing means.

Even though the mean values of various groups appear to be different, this could be due to a sampling error rather than the effect of the independent variable on the dependent variable. If it is due to sampling error, the difference between the group means is meaningless. ANOVA helps to find out if the difference in the mean values is statistically significant.

ANOVA also indirectly reveals if an independent variable is influencing the dependent variable. For example, in the above blood sugar level experiment, suppose ANOVA finds that group means are not statistically significant, and the difference between group means is only due to sampling error. This result infers that the type of medication (independent variable) is not a significant factor that influences the blood sugar level.

Limitations of ANOVA

ANOVA can only tell if there is a significant difference between the means of at least two groups, but it can’t explain which pair differs in their means. If there is a requirement for granular data, deploying further follow up statistical processes will assist in finding out which groups differ in mean value. Typically, ANOVA is used in combination with other statistical methods.

ANOVA also makes assumptions that the dataset is uniformly distributed, as it compares means only. If the data is not distributed across a normal curve and there are outliers, then ANOVA is not the right process to interpret the data.

Similarly, ANOVA assumes the standard deviations are the same or similar across groups. If there is a big difference in standard deviations, the conclusion of the test may be inaccurate.

How is ANOVA Used in Data Science?

One of the biggest challenges in machine learning is the selection of the most reliable and useful features that are used in order to train a model. ANOVA helps in selecting the best features to train a model. ANOVA minimizes the number of input variables to reduce the complexity of the model. ANOVA helps to determine if an independent variable is influencing a target variable.

An example of ANOVA use in data science is in email spam detection. Because of the massive number of emails and email features, it has become very difficult and resource-intensive to identify and reject all spam emails. ANOVA and f-tests are deployed to identify features that were important to correctly identify which emails were spam and which were not.

Questions That ANOVA Helps to Answer

Even though ANOVA involves complex statistical steps, it is a beneficial technique for businesses via use of AI. Organizations use ANOVA to make decisions about which alternative to choose among many possible options. For example, ANOVA can help to:

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What is ANOVA?

How does anova work, how can anova help, when might you use anova, examples of using anova, understanding anova assumptions, types of anova, how to conduct an anova test, what are the limitations of anova, additional considerations with anova, try qualtrics for free, what is anova (analysis of variance) and what can i use it for.

14 min read One-Way Analysis of Variance (ANOVA) tells you if there are any statistical differences between the means of three or more independent groups.

ANOVA stands for Analysis of Variance. It’s a statistical test that was developed by Ronald Fisher in 1918 and has been in use ever since. Put simply, ANOVA tells you if there are any statistical differences between the means of three or more independent groups.

One-way ANOVA is the most basic form. There are other variations that can be used in different situations, including:

Two-way ANOVA

Factorial ANOVA

Welch’s F-test ANOVA

Ranked ANOVA

Games-Howell pairwise test

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Like the t-test , ANOVA helps you find out whether the differences between groups of data are statistically significant. It works by analyzing the levels of variance within the groups through samples taken from each of them.

If there is a lot of variance (spread of data away from the mean) within the data groups, then there is more chance that the mean of a sample selected from the data will be different due to chance.

As well as looking at variance within the data groups, ANOVA takes into account sample size (the larger the sample, the less chance there will be of picking outliers for the sample by chance) and the differences between sample means (if the means of the samples are far apart, it’s more likely that the means of the whole group will be too).

All these elements are combined into a F value, which can then be analyzed to give a probability (p-vaue) of whether or not differences between your groups are statistically significant.

A one-way ANOVA compares the effects of an independent variable (a factor that influences other things) on multiple dependent variables. Two-way ANOVA does the same thing, but with more than one independent variable, while a factorial ANOVA extends the number of independent variables even further.

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The one-way ANOVA can help you know whether or not there are significant differences between the means of your independent variables.

Why is that useful?

Because when you understand how each independent variable’s mean is different from the others, you can begin to understand which of them has a connection to your dependent variable (such as landing page clicks) and begin to learn what is driving that behavior.

You could also flip things around and see whether or not a single independent variable (such as temperature) affects multiple dependent variables (such as purchase rates of suncream, attendance at outdoor venues, and likelihood to hold a cook-out) and if so, which ones.

You might use Analysis of Variance (ANOVA) as a marketer when you want to test a particular hypothesis. You would use ANOVA to help you understand how your different groups respond, with a null hypothesis for the test that the means of the different groups are equal. If there is a statistically significant result, then it means that the two populations are unequal (or different).

You may want to use ANOVA to help you answer questions like this:

Do age, sex, or income have an effect on how much someone spends in your store per month?

To answer this question, a factorial ANOVA can be used, since you have three independent variables and one dependent variable. You’ll need to collect data for different age groups (such as 0-20, 21-40, 41-70, 71+), different income brackets, and all relevant sexes. A two-way ANOVA can then simultaneously assess the effect on these variables on your dependent variable (spending) and determine whether they make a difference.

Does marital status (single, married, divorced, widowed) affect mood?

To answer this one, you can use a one-way ANOVA, since you have a single independent variable (marital status). You’ll have 4 groups of data, one for each of the marital status categories, and for each one you’ll be looking at mood scores to see whether there’s a difference between the averages.

When you understand how the groups within the independent variable differ (such as widowed or single, not married or divorced), you can begin to understand which of them has a connection to your dependent variable (mood).

However, you should note that ANOVA will only tell you that the average mood scores across all groups are the same or are not the same. It does not tell you which one has a significantly higher or lower average mood score.

Like other types of statistical tests , ANOVA compares the means of different groups and shows you if there are any statistical differences between the means. ANOVA is classified as an omnibus test statistic. This means that it can’t tell you which specific groups were statistically significantly different from each other, only that at least two of the groups were.

It’s important to remember that the main ANOVA research question is whether the sample means are from different populations. There are two assumptions upon which ANOVA rests:

From the basic one-way ANOVA to the variations for special cases, such as the ranked ANOVA for non-categorical variables, there are a variety of approaches to using ANOVA for your data analysis. Here’s an introduction to some of the most common ones.

What is the difference between one-way and two-way ANOVA tests?

This is defined by how many independent variables are included in the ANOVA test. One-way means the analysis of variance has one independent variable. Two-way means the test has two independent variables. An example of this may be the independent variable being a brand of drink (one-way), or independent variables of brand of drink and how many calories it has or whether it’s original or diet.

Factorial ANOVA is an umbrella term that covers ANOVA tests with two or more independent categorical variables. (A two-way ANOVA is actually a kind of factorial ANOVA.) Categorical means that the variables are expressed in terms of non-hierarchical categories (like Mountain Dew vs Dr Pepper) rather than using a ranked scale or numerical value.

Welch’s F Test ANOVA

Stats iQ recommends an unranked Welch’s F test if several assumptions about the data hold:

Unlike the slightly more common F test for equal variances, Welch’s F test does not assume that the variances of the groups being compared are equal. Assuming equal variances leads to less accurate results when variances are not, in fact, equal, and its results are very similar when variances are actually equal.

When assumptions are violated, the unranked ANOVA may no longer be valid. In that case, Stats iQ recommends the ranked ANOVA (also called “ANOVA on ranks”); Stats iQ rank-transforms the data (replaces values with their rank ordering) and then runs the same ANOVA on that transformed data.

The ranked ANOVA is robust to outliers and non-normally distributed data. Rank transformation is a well-established method for protecting against assumption violation (a “nonparametric” method) and is most commonly seen in the difference between the Pearson and Spearman correlation. Rank transformation followed by Welch’s F test is similar in effect to the Kruskal-Wallis Test.

Note that Stats iQ’s ranked and unranked ANOVA effect sizes (Cohen’s f) are calculated using the F value from the F test for equal variances.

Games-Howell Pairwise Test

Stats iQ runs Games-Howell tests regardless of the outcome of the ANOVA test (as per Zimmerman, 2010). Stats iQ shows unranked or ranked Games-Howell pairwise tests based on the same criteria as those used for ranked vs. unranked ANOVA, so if you see “Ranked ANOVA” in the advanced output, the pairwise tests will also be ranked.

The Games-Howell is essentially a t-test for unequal variances that accounts for the heightened likelihood of finding statistically significant results by chance when running many pairwise tests. Unlike the slightly more common Tukey’s b-test, the Games-Howell test does not assume that the variances of the groups being compared are equal. Assuming equal variances leads to less accurate results when variances are not in fact equal, and its results are very similar when variances are actually equal (Howell, 2012).

Note that while the unranked pairwise test tests for the equality of the means of the two groups, the ranked pairwise test does not explicitly test for differences between the groups’ means or medians. Rather, it tests for a general tendency of one group to have larger values than the other.

Additionally, while Stats iQ does not show results of pairwise tests for any group with less than four values, those groups are included in calculating the degrees of freedom for the other pairwise tests.

As with many of the older statistical tests, it’s possible to do ANOVA using a manual calculation based on formulae. You can also run ANOVA using any number of popular stats software packages and systems, such as R, SPSS or Minitab. A more recent development is to use automated tools such as Stats iQ from Qualtrics , which make statistical analysis more accessible and straightforward than ever before.

Stats iQ and ANOVA

Stats iQ from Qualtrics can help you run an ANOVA test. When you select one categorical variable with three or more groups and one continuous or discrete variable, Stats iQ runs a one-way ANOVA (Welch’s F test) and a series of pairwise “post hoc” tests (Games-Howell tests).

The one-way ANOVA tests for an overall relationship between the two variables, and the pairwise tests test each possible pair of groups to see if one group tends to have higher values than the other.

How to run an ANOVA test through Stats iQ

The Overall Stat Test of Averages in Stats iQ acts as an ANOVA, testing the relationship between a categorical and a numeric variable by testing the differences between two or more means. This test produces a p-value to determine whether the relationship is significant or not.

To run an ANOVA in StatsiQ , take the following steps:

Qualtrics Crosstabs and ANOVA

You can run an ANOVA test through the Qualtrics Crosstabs feature too. Here’s how:

Whilst ANOVA will help you to analyze the difference in means between two independent variables, it won’t tell you which statistical groups were different from each other. If your test returns a significant F-statistic (the value you get when you run an ANOVA test), you may need to run an ad hoc test (like the Least Significant Difference test) to tell you exactly which groups had a difference in means.

Read more about additional statistical analysis types:

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Analysis of Variance

The analysis of variance (ANOVA) is a method for performing simultaneous tests on data sets drawn from different populations.

From: Data Analysis Methods in Physical Oceanography (Third Edition) , 2014

Related terms:

Online Diagnosis of PEM Fuel Cell by Fuzzy C-Means Clustering

Damien Chanal , ... Marie-Cécile Péra , in Encyclopedia of Energy Storage , 2022

ANOVA F-Test

Analysis of variance (ANOVA) is a tool to compare the means of several populations, based on random, independent samples from each population. It provides a statistical test to determine if population means are equal or not (i.e. came from the same distribution). ANOVA is a parametric test that assumes a normal distribution of values (null hypothesis).

F-test is a class of statistical test that calculates the ratio between variances. F-Test is used with ANOVA to measure the ratio between explained and unexplained variances.

Three assumptions must be satisfied with ANOVA F-Test: samples are independents, from a normally distributed population and standard deviations of the groups are all equal (homoscedasticity). It permits the measure of the linear dependency between two variables. In Johnson and Synovec (2002) a feature selection based on ANOVA and PCA is done to classify jet fuel mixture. In Yakub et al. (2016) , a feature selection based on one way ANOVA is used to classify microarray data. The main advantage of ANOVA F-Test is its straightforward computation and interpretation. The limiting factor is that its applicability is only valid with the specific assumptions.

Process Optimization and Modeling of Hydraulic Fracturing Process Wastewater Treatment Using Aerobic Mixed Microbial Reactor via Response Surface Methodology

Thirugnanasambandham Karchiyappan , Rama Rao Karri , in Soft Computing Techniques in Solid Waste and Wastewater Management , 2021

3.3 Statistical analysis of response surface model

Analysis of variance (ANOVA) was used to examine the statistical significance of the developed nonlinear quadratic model. The ANOVA results shown in Table 21.6 for the suggested quadratic model indicated that model factors are highly significant. The high F value (46) with a low probability value implies that the regression model is statistically significant. The predicted correlation coefficient (pred. R 2 = 0.9527) is also in good agreement with the adjusted correlation coefficient (adj. R 2 = 0.9611). Higher R 2 indicates that the model predictions are very close to the experimental values, which is further confirmed in the scatter plot shown in Fig. 21.5 . Overall, the ANOVA analysis confirms the superiority of the developed second-order quadratic model to forecast the COD degradation ( Oller, Malato, & Sánchez-Pérez, 2011 ).

Table 21.6 . ANOVA results for response.

define analysis of variance

Figure 21.5 . Scatter plot showing the model predictions versus experimental values.

Structure as Groups of Objects/Variables

Valérie David , in Data Treatment in Environmental Sciences , 2017

3.2.2 Analysis of variance

One-factor analysis of variance (ANOVA) allows us to explain a quantitative variable by means of a qualitative one. In this sense, it allows us to analyze descriptors that present significant differences between groups , e.g. the groups of stations that present different phytoplankton communities. ANOVA presents a parametric approach (classical ANOVA), non-parametric approaches (Kruskal–Wallis test and ANAlysis of SIMilarities (ANOSIM)), and an intermediate approach with less constraints in terms of applicability conditions (permutational ANOVA) than classical ANOVA. Section 5.2.1.7 describes in detail the application of these four general types of ANOVA as well as the verification of the relative applicability conditions.

Normal One-Way ANOVA

Marc Kéry , in Introduction to WinBUGS for Ecologists , 2010

Publisher Summary

Analysis of variance (ANOVA) is the generalization of a t-test to more than two groups. There are different kinds of ANOVA: one-way, with just a single factor, and two- or multiway, with two or more factors, and main- and interaction-effects models. This chapter presents a one-way ANOVA and introduces the concept of random effects along the way. In random-effects models, a set of effects is constrained to come from some distribution, which is most often a normal, although it may be a Bernoulli, a Poisson, or yet another distribution. There are three reasons for making distributional assumptions about a set of effects in a model: extrapolation of inference to a wider population, improved accounting for system uncertainty, and efficiency of estimation. First, viewing the studied effects as a random sample from some population enables one to extrapolate to that population. This generalization can only be achieved by modeling the process that generates the realized values of the random effects. Second, declaring factor effects as random acknowledges that when repeating our study, we obtain a different set of effects, so the resulting parameter estimates will differ from those in our current study. Random-effects modeling properly accounts for this added uncertainty in our inference about the analyzed system. Third, when making a random-effects assumption about a factor, these effects are no longer estimated independently; instead, estimates are influenced by each other and therefore are dependent. Random-effects modeling can also be viewed as a compromise between assuming no effects and fully independent effects of the levels of a factor.

Toxicology Testing and Evaluation

S.C. Gad , in Comprehensive Toxicology , 2010

3.13.3.2.8 Analysis of variance (ANOVA)

ANOVA is used for the comparison of three or more groups of continuous data when the variances are homogeneous and the data are independent and normally distributed. This is the most commonly used statistical test in the biomedical sciences. In comparing multiple (more than two at a time) groups, it will generally tell if there are ( or are not) significant differences between any of the groups, though on its own it will not identify which groups are different. This latter task is performed by one of the post hoc tests which will be considered next.

Unsupervised learning

Horst Langer , ... Conny Hammer , in Advantages and Pitfalls of Pattern Recognition , 2020

Appendix 3.1. Analysis of variance (ANOVA)

Analysis of variance (ANOVA) is used to analyze the differences among group means and their associated procedures (such as “variation” among and between groups). In its simplest form, ANOVA provides a statistical test of whether or not the means of several groups are equal, and therefore generalizes the t -test to more than two groups.

The term “variance” is a bit sloppy, as ANOVA is based on the dispersion rather than the variance.

The fundamental technique is a partitioning of the total sum of squares into components related to the effects used in the model. A fundamental concept is the split of the total dispersion in terms regarding the variation between groups and the one encountered within ( Table A3.1 ).

Table A3.1 . ANOVA table (see Davis, 1986).

where x ¯ is the global mean, x ¯ j is the mean of the j -th group, and m j is the number of samples in that group. The two relations are nothing else than the Eqs. (3.7) and (3.8) for the univariate case. The term

which is the univariate version of Eq. (3.9) . The total dispersion corresponds to the sum

we can apply an F-test on MS B / MS W where MS B = S B / (M − 1) , MS W = S W /( N − M), checking whether at least one of the groups differs from the total population with respect to its mean. ANOVA requests that group members are randomly sampled, all groups have the same variance and follow a normal distribution. More details can be found in the referenced textbooks.

Blue carbon storage comparing mangroves with saltmarsh and seagrass habitats at a warm temperate continental limit

Sinegugu P. Banda , ... Jacqueline L. Raw , in Dynamic Sedimentary Environments of Mangrove Coasts , 2021

2.6 Data analysis

Analysis of variance (ANOVA) was used to test for significant differences in soil parameters (organic matter, bulk density, organic and inorganic carbon, and carbon density) with changes in depth (depth section as a fixed factor) using the MASS package in RStudio, R version 3.5.1 (2018, The R Foundation for Statistical Computing).

Equations were created to determine the relationship between soil organic matter (LOI%) and organic carbon concentration (C org %) by plotting the values against each other. Natural logarithmic transformations were performed for Z. capensis soil values to improve the fit of the linear relationship between organic matter and organic carbon. The values were then further transformed from negative to positive values using the modulus. A similar linear equation was done for saltmarsh to determine the relationship between stem height and stem biomass (as dry weight).

Contemporary Methods for Statistical Design and Analysis

D.R. Fox , in Marine Ecotoxicology , 2016

2.3 Design Considerations

Much has been written about the design and analysis of environmental data in general ( USEPA, 2002, 2006 ) and ecotoxicological data more specifically ( Environment Canada, 2005; CCME, 2007; European Commission, 2011; Newman, 2012; ANZECC/ARMCANZ, 2000a,b; OECD, 2012, 2014 ) although most of this is based on and repeats standard frequentist principles that are taught in all introductory statistics courses ( Sparks, 2000 ). The OECD (2012) and Environment Canada (2005) documents exemplify this point. Topics covered in both documents include hypothesis testing, Type I/II errors, statistical power, randomization, replication, outliers, and data transformations. While these statistical concepts are important, it could be argued that the emphasis placed on classical/frequentist statistics has stifled the development of strategies and procedures that are better equipped to handle the many and varied perturbations of assumed conditions encountered in ecotoxicology. The assumption of randomness is a case in point. It is safe to say that the majority of statistical methods are predicated on the joint notions of randomness and independence. Indeed statistical theory demands that toxicity data used in SSD modeling be obtained from a randomly selected sample of species. The well-accepted reality is that this is never the case ( Fox, 2015 and references therein). The irony is that while guideline documents stress the importance of randomness (eg, “randomization should prevail in all aspects of the design and procedures for a toxicity test,” Environment Canada, 2005 ), they simultaneously recommend procedures that ensure samples are biased. For example, the revised Australian and New Zealand Water Quality Guidelines recommend using toxicity data from at least eight species from at least four taxonomic groups ( Batley et al., 2014 ). Such purposive sampling is the antithesis of randomness and while the distinction between probability sampling and judgmental sampling for ecotoxicological studies is not always made clear, good advice does exist albeit in the broader context of environmental monitoring ( USEPA, 2002, 2006 ).

So while not diminishing the importance of sound statistical design to guide all aspects of the data collection and analysis process, the reality is that ecotoxicological studies tend to be severely constrained by (1) high cost of data acquisition; (2) inability (or compromised ability) to invoke the core statistical principles of randomness , replication , and blocking ; and (3) nonconformity. The high cost of data acquisition is a function of the logistics of field sampling coupled with the expensive laboratory analyses required to generate toxicity data. The strict definition of randomness means that every “unit” in the target “population” under investigation has an equal chance of being included in the sample and we have already seen that protocols exist (eg, Australian Water Quality Guidelines, ANZECC/ARMCANZ, 2000a,b ) which ensure this cannot happen. In addition, standardized protocols for laboratory-based toxicity tests tend to be available for only a relatively small number of animals or organisms thus ensuring another layer of nonrandom selection. Replication improves the quality of estimation and inference but is a casualty of (1), while control through the use of “blocking” where experimental units are organized according to some other exogenous variable(s) is only an option if the major sources of extraneous variation are known in advance. “Nonconformity” refers to the propensity of ecotoxicological data to violate many of the prerequisites or assumptions required by most statistical tests and procedures described in the various guideline documents. These include, but are not limited to, violations of assumptions concerning: independence; distributional form; variance structures; sample size; outliers; censoring; and response-generating mechanism.

Rather than summarizing standard statistical design theory, which is readily available in textbooks (eg, Hinkelmann and Kempthorne, 2008; Gad, 2006 ) and the aforementioned guideline documents, the remainder of this section is devoted to the exploration of some more “contemporary” aspects of experimental design in ecotoxicology.

ANOVA techniques have (and continue) to play a significant role in the analysis of ecotoxicological data. Even though the use of this technique is expected to diminish as scientists move away from generating NOEC data, ANOVA methods still have an important role to play in testing hypotheses concerning the toxic effects of chemicals in the environment. The identification of an appropriate experimental design is a critical first step in the use of ANOVA and related tools of statistical inference. At the very least the experimental design should be such that it:

allows for the unbiased and efficient estimation of all effects of interest;

controls (to the extent possible) sources of extraneous variation likely to affect the measured response (for example, a temperature or salinity effect in a C-R experiment); and

makes minimal use of limited resources.

Reducing bias, improving precision, and controlling extraneous variation tend to result in a greater number of treatment combinations and/or increased replication—both of which increase the cost of the experiment. It is therefore surprising that orthogonal fractional factorial designs have not been used more widely in ecotoxicology ( Dey, 1985 ). While it is not possible to provide a comprehensive treatment of this important topic in this book, we illustrate the potential benefits with the use of a simple example.

In a recent paper, Webb et al. (2014) described a toxicology experiment that “did not succumb to standard experimental design.” The challenge was to satisfy the requirements of the three dot-points above in a way that accommodated unique physical and logistical constraints. Their solution relied upon advanced mathematical and computational skills—the detail of which is beyond the scope of this book. The situation described in Webb et al. (2014) motivates the following hypothetical example illustrating the use of a fractional factorial design.

2.3.1 Example

A study into the potential impacts associated with the discharge of waste water from a proposed desalination plant relied on toxicity testing. For one of these tests, researchers were interested in whether or not the hypersaline effluent was toxic to marine organisms. Other factors thought to be important were the time of day (TOD) when exposure to the toxicant commenced as well as the temperature (temp) and salinity (salin) of the waste stream. As in the Webb et al. (2014) study, the manner in which test samples were stored was potentially another source of variation that needed to be controlled for. In this case, beakers could be placed on shelves that were arranged in three racks each having four shelves. The positioning of beakers (representing different combinations of dose, TOD, temp, and salin) on shelves was important due to the potential influences of light levels, proximity to the door, and thermal stratification. The basic experiment involved the direct manipulation of the four factors: dose (present/absent); TOD (am/pm); temp (15°C/25°C); and salin (ambient/elevated) coupled with the two factors determining beaker position (racks and shelves). A “full factorial” experiment would require all 384 combinations of these factors to be tested at least once. Not only is such an experiment time-consuming and expensive, it may be unnecessary. If information is only sought on the “main effects” (ie, the effect of each factor separately) and higher-order interaction effects are (or can be assumed to be) negligible, then significant savings in experimental effort can be realized with the use of a fraction of the treatments in the full design—hence the name fractional factorial design . In addition, if the treatments comprising this fraction are carefully selected, it is possible to estimate the main effects independently of each other—a property that is clearly desirable but not guaranteed by either a random or subjective selection. In statistics, the independence of two random variables has the geometrical interpretation of orthogonality (ie, being at right angles to each other). Hence, fractional factorial designs which permit the independent estimation of effects are referred to as orthogonal fractional factorial designs . These are not new and date back to the work of Adelman (1961), Bose and Bush (1952), Rao (1950), Kempthorne (1947) , and others. As one might expect, the method of identifying how many and which treatments to include in the fractional design such that the orthogonality requirement is met is far from simple and requires a good understanding of advanced mathematical concepts such as linear algebra, Hadamard matrices, and Galois Field theory. Thankfully, statistical software tools make this task easier although more complex designs tend not to be included. R ( R Development Core Team, 2004 ) is the only free package of such tools; it has a large and rapidly increasing library of user-contributed functions including packages for creating and analyzing fractional factorial designs. Readers interested in learning more should consult the R website ( CRAN, 2015 ) and the texts by Lawson (2015) and Gad (2006) .

Returning to the present example, an illustration of the savings in experimental effort is indicated by the experimental design represented by the allocations in Table 2.1 which has reduced the number of treatments from 384 to a mere 25. This design is what is referred to as Resolution III , meaning that main effects are estimated independently of each other but not independently of interactions (hence the need to know or assume that interaction effects are negligible). Other resolution designs may be available which are less restrictive but require greater experimental resources (typically in the form of more treatment combinations).

Table 2.1 . Orthogonal Fractional Factorial Design for Effluent Toxicity Study

More will be said about “optimal” designs in the context of planning a C-R experiment in Section 2.6 .

Entropy and MTOPSIS assisted central composite design for preparing activated carbon toward adsorptive defluoridation of wastewater

Kumar Anupam , ... Rama Rao Karri , in Green Technologies for the Defluoridation of Water , 2021

5.3.4.2 Model validation and diagnostics

ANOVA for model equations is presented in Table 5.10 . The model F -value of 63.10 with P -value <0.0001 demonstrates that the model is significant. There is only a 0.01% possibility that an F -value of this high could arise because of noise. P -values <0.05 implies that model terms are significant. In the present case, h , t , T , tT , h 2 , t 2 , and T 2 are significant model terms. P -values >0.1000 indicate that the model terms are not significant. If there are many insignificant model terms (not counting those required to support hierarchy), model reduction may improve the model. The Lack of Fit F -value 0.5431 denotes that the lack of fit is not significant concerning the pure error. There is a 74.05% likelihood that a Lack of Fit F -value this large could arise due to noise. Nonsignificant lack of fit is excellent because it signifies a fit model. The R 2 , adjusted R 2 , and predicted R ² of this equation are 98.27%, 96.71%, and 93.71%, respectively. The predicted R ² is in sufficient accord with the adjusted R² since the difference between them is <20%. The signal-to-noise ratio as indicated by Adeq precision is 23.9825 in the present case. This ratio indicates an adequate signal because it is > 4. The standard deviation, mean, and coefficient of variation of the equation are 0.0496%, 0.5242%, and 9.46%, respectively. Fig. 5.2 A–F, respectively, illustrate the normal plot of residuals, residuals versus the experimental run, predicted, H 3 PO 4 concentration, carbonization time, and carbonization temperature. Fig. 5.2 A shows that the residuals are normally distributed and follow a straight line. Further, Fig. 5.2 B–F illustrates that the residuals are randomly scattered across the x -axis and lies between ±3.00 ( Yi et al., 2011 ). All the aforesaid diagnostics validate that the MTOPSIS score model formulated for the performance evaluation of the coconut shell based activated carbon preparation process in terms of H 3 PO 4 concentration, carbonization time, and carbonization temperature is adequately fit to navigate the design space.

Table 5.10 . ANOVA for quadratic model of activated carbon preparation.

Fig. 5.2 . Model diagnostics (A) normal plot of residuals (B) residuals versus run numbers (C) residuals versus predicted (D) residuals versus H 3 PO 4 (E) residuals versus time (F) residuals versus temperature.

Analysis of Population Indices

John R. Skalski , ... Joshua J. Millspaugh , in Wildlife Demography , 2005

8.5.1 Example: Forest Birds, New South Wales, Australia

Shields (1990) investigated the effects of varying rates of logging (i.e., removal of tree cover) on the abundance of forest bird populations through a replicated and randomized manipulative experiment in New South Wales, Australia. Twelve forested sites designated for logging were randomly assigned to serve as control (0% logging), 50% forest removal, or 90% forest removal sites in a balanced design. In 1987, before logging, bird abundance was surveyed by using circular-transect counts replicated over 4 days during January ( Table 8-8 ). Treatment plots were logged in January 1989, and bird abundance was resurveyed in January 1990 ( Table 8-8 ).

Table 8-8 . Mean count of old-growth species of birds pre- and postlogging, and the fraction of trees logged at old-growth plots, New South Wales, Australia, 1987–1990.

(Data from Shields 1990 .)

The purpose of the study was to assess whether forest birds were lost proportional to the fraction of trees removed in old-growth forests. The premise was that birds would be retained at a rate greater than a 1 : 1 proportion of birds to trees. The response variable was the postlogging bird abundance in 1990. A response model was constructed that assumed the number of birds seen in 1990 at the control plots would be proportional to the baseline counts seen in 1987, where

The “phase effect” is the fractional change in bird abundance by the naturally occurring effects of time. At the logged sites, bird abundance was thought to be related not to the fraction of trees removed but rather to the fraction of trees remaining (i.e., more habitat remaining, more birds remaining), such that

where β modifies the effect of removal. Specifically, if there is a 1 : 1 correspondence between trees removed and birds lost, then β = 1. If this relationship is not as severe and more birds are retained than the fraction of trees left standing, then β < 1. The response model suggests the hypotheses,

The null hypothesis states that birds are lost at a rate equal to or greater than the loss of trees; the alternative hypothesis states the rate of bird loss is less.

Log-transforming both sides of Eq. (8.96) produces the log-linear model

The intercept will characterize the fn(Phase effect), and the independent variable is ln(1 – Fraction logged). In addition, model (8.97) has the covariate ln N Pre , but it is not an ordinary covariate. Based on the model, there is no regression coefficient to estimate. Instead the regression coefficient, by definition, has the value 1. A variable that is entered into the model with a prespecified regression coefficient (or no regression coefficient) is called an “offset” in GLM. No degree of freedom is lost by including these covariates because no regression coefficient is estimated. The GLM analyses were based on a normal error structure, because the bird counts ( Table 8-8 ) were an average of four replicate surveys. The Central Limit theorem suggests the averages may be approximately normally distributed.

The ANOVA table for the New South Wales bird analysis is presented below:

The mean bird count is significantly related ( P = 0.0001) to the fraction of trees retained post-logging. The fitted response model is

or back-transforming

The estimated phase effect of 0.9801 ( SE ˆ = 0.05189 ) is not significantly different from 1, suggesting bird counts at the control sites (i.e., nonlogged) have not varied between 1987 and 1990. Modeling the response without an intercept results in the simplified model

with SE ˆ ( β ˆ ) = 0.02544 .

A comparison (below) of the postlogging bird counts with the predicted values from the final regression model (8.98) shows good agreement:

The test of the null hypotheses (i.e., H o : β ≥ 1) can be performed by using a t -statistic of the form:

with a significance level of P ( t 11 ≤ – 29.9096) ≈ 0. Thus, we conclude that forest birds were retained at a higher fraction than the fraction of old-growth trees left standing.

The fitted response model can also be used to estimate the fraction of birds retained under the different treatment levels, specifically:

The model suggests that with 50% of the trees removed 85% of the bird count is retained; for 90% tree removal, 58% of the bird count is retained. Shields (1990) performed a similar analysis examining the count of invading bird species into the old-growth forest.

In this example and others like it, the anthropogenic effect under investigation may affect animal abundance and behavior. There is also the possibility the logging treatment, by virtue of changing the habitat structure, may also have directly affected observation rates. Without auxiliary information on observation distances, there is the possibility that the estimates of relative abundance may be confounding with increases in observational rates owing to opening the forest canopy. Ambiguous and possibly faulty interpretations linger over all index studies regardless of how well they are designed and implemented.

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