Analysis of variance

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Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among group means in a sample. ANOVA was developed by statistician and evolutionary biologist Ronald Fisher. In the ANOVA setting, the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether the population means of several groups are equal, and therefore generalizes the t-test to more than two groups. ANOVA is useful for comparing (testing) three or more group means for statistical significance. It is conceptually similar to multiple two-sample t-tests, but is more conservative, resulting in fewer type I errors, and is therefore suited to a wide range of practical problems.

History

While the analysis of variance reached fruition in the 20th century, antecedents extend centuries into the past according to Stigler. These include hypothesis testing, the partitioning of sums of squares, experimental techniques and the additive model. Laplace was performing hypothesis testing in the 1770s. The development of least-squares methods by Laplace and Gauss circa 1800 provided an improved method of combining observations (over the existing practices then used in astronomy and geodesy). It also initiated much study of the contributions to sums of squares. Laplace knew how to estimate a variance from a residual (rather than a total) sum of squares. By 1827, Laplace was using least squares methods to address ANOVA problems regarding measurements of atmospheric tides. Before 1800, astronomers had isolated observational errors resulting from reaction times (the "personal equation") and had developed methods of reducing the errors. The experimental methods used in the study of the personal equation were later accepted by the emerging field of psychology  which developed strong (full factorial) experimental methods to which randomization and blinding were soon added. An eloquent non-mathematical explanation of the additive effects model was available in 1885.

Ronald Fisher introduced the term variance and proposed its formal analysis in a 1918 article The Correlation Between Relatives on the Supposition of Mendelian Inheritance. His first application of the analysis of variance was published in 1921. Analysis of variance became widely known after being included in Fisher's 1925 book Statistical Methods for Research Workers.

Randomization models were developed by several researchers. The first was published in Polish by Jerzy Neyman in 1923.

One of the attributes of ANOVA that ensured its early popularity was computational elegance. The structure of the additive model allows solution for the additive coefficients by simple algebra rather than by matrix calculations. In the era of mechanical calculators this simplicity was critical. The determination of statistical significance also required access to tables of the F function which were supplied by early statistics texts.

Motivating example

No fit.

 

Fair fit

 

Very good fit

 

The analysis of variance can be used as an exploratory tool to explain observations. A dog show provides an example. A dog show is not a random sampling of the breed: it is typically limited to dogs that are adult, pure-bred, and exemplary. A histogram of dog weights from a show might plausibly be rather complex, like the yellow-orange distribution shown in the illustrations. Suppose we wanted to predict the weight of a dog based on a certain set of characteristics of each dog. One way to do that is to explain the distribution of weights by dividing the dog population into groups based on those characteristics. A successful grouping will split dogs such that (a) each group has a low variance of dog weights (meaning the group is relatively homogeneous) and (b) the mean of each group is distinct (if two groups have the same mean, then it isn't reasonable to conclude that the groups are, in fact, separate in any meaningful way). 

In the illustrations to the right, groups are identified as X₁, X₂, etc. In the first illustration, the dogs are divided according to the product (interaction) of two binary groupings: young vs old, and short-haired vs long-haired (e.g., group 1 is young, short-haired dogs, group 2 is young, long-haired dogs, etc.). Since the distributions of dog weight within each of the groups (shown in blue) has a relatively large variance, and since the means are very similar across groups, grouping dogs by these characteristics does not produce an effective way to explain the variation in dog weights: knowing which group a dog is in doesn't allow us to predict its weight much better than simply knowing the dog is in a dog show. Thus, this grouping fails to explain the variation in the overall distribution (yellow-orange). 

An attempt to explain the weight distribution by grouping dogs as pet vs working breed and less athletic vs more athletic would probably be somewhat more successful (fair fit). The heaviest show dogs are likely to be big, strong, working breeds, while breeds kept as pets tend to be smaller and thus lighter. As shown by the second illustration, the distributions have variances that are considerably smaller than in the first case, and the means are more distinguishable. However, the significant overlap of distributions, for example, means that we cannot distinguish X₁ and X₂ reliably. Grouping dogs according to a coin flip might produce distributions that look similar. 

An attempt to explain weight by breed is likely to produce a very good fit. All Chihuahuas are light and all St Bernards are heavy. The difference in weights between Setters and Pointers does not justify separate breeds. The analysis of variance provides the formal tools to justify these intuitive judgments. A common use of the method is the analysis of experimental data or the development of models. The method has some advantages over correlation: not all of the data must be numeric and one result of the method is a judgment in the confidence in an explanatory relationship.

Background and terminology

ANOVA is a form of statistical hypothesis testing heavily used in the analysis of experimental data. A test result (calculated from the null hypothesis and the sample) is called statistically significant if it is deemed unlikely to have occurred by chance, assuming the truth of the null hypothesis. A statistically significant result, when a probability (p-value) is less than a pre-specified threshold (significance level), justifies the rejection of the null hypothesis, but only if the a priori probability of the null hypothesis is not high. 

In the typical application of ANOVA, the null hypothesis is that all groups are random samples from the same population. For example, when studying the effect of different treatments on similar samples of patients, the null hypothesis would be that all treatments have the same effect (perhaps none). Rejecting the null hypothesis is taken to mean that the differences in observed effects between treatment groups are unlikely to be due to random chance.

By construction, hypothesis testing limits the rate of Type I errors (false positives) to a significance level. Experimenters also wish to limit Type II errors (false negatives). The rate of Type II errors depends largely on sample size (the rate is larger for smaller samples), significance level (when the standard of proof is high, the chances of overlooking a discovery are also high) and effect size (a smaller effect size is more prone to Type II error). 

The terminology of ANOVA is largely from the statistical design of experiments. The experimenter adjusts factors and measures responses in an attempt to determine an effect. Factors are assigned to experimental units by a combination of randomization and blocking to ensure the validity of the results. Blinding keeps the weighing impartial. Responses show a variability that is partially the result of the effect and is partially random error. 

ANOVA is the synthesis of several ideas and it is used for multiple purposes. As a consequence, it is difficult to define concisely or precisely.

"Classical" ANOVA for balanced data does three things at once:

1. As exploratory data analysis, an ANOVA employs an additive data decomposition, and its sums of squares indicate the variance of each component of the decomposition (or, equivalently, each set of terms of a linear model).
2. Comparisons of mean squares, along with an F-test ... allow testing of a nested sequence of models.
3. Closely related to the ANOVA is a linear model fit with coefficient estimates and standard errors.

In short, ANOVA is a statistical tool used in several ways to develop and confirm an explanation for the observed data. 

Additionally:

4. It is computationally elegant and relatively robust against violations of its assumptions.
5. ANOVA provides strong (multiple sample comparison) statistical analysis.
6. It has been adapted to the analysis of a variety of experimental designs.

As a result: ANOVA "has long enjoyed the status of being the most used (some would say abused) statistical technique in psychological research." ANOVA "is probably the most useful technique in the field of statistical inference."

ANOVA is difficult to teach, particularly for complex experiments, with split-plot designs being notorious. In some cases the proper application of the method is best determined by problem pattern recognition followed by the consultation of a classic authoritative test.

Design-of-experiments terms

(Condensed from the "NIST Engineering Statistics Handbook": Section 5.7. A Glossary of DOE Terminology.)

Balanced design

An experimental design where all cells (i.e. treatment combinations) have the same number of observations.

Blocking

A schedule for conducting treatment combinations in an experimental study such that any effects on the experimental results due to a known change in raw materials, operators, machines, etc., become concentrated in the levels of the blocking variable. The reason for blocking is to isolate a systematic effect and prevent it from obscuring the main effects. Blocking is achieved by restricting randomization.

Design

A set of experimental runs which allows the fit of a particular model and the estimate of effects.

DOE

Design of experiments. An approach to problem solving involving collection of data that will support valid, defensible, and supportable conclusions.

Effect

How changing the settings of a factor changes the response. The effect of a single factor is also called a main effect.

Error

Unexplained variation in a collection of observations. DOE's typically require understanding of both random error and lack of fit error.

Experimental unit

The entity to which a specific treatment combination is applied.

Factors

Process inputs that an investigator manipulates to cause a change in the output.

Lack-of-fit error

Error that occurs when the analysis omits one or more important terms or factors from the process model. Including replication in a DOE allows separation of experimental error into its components: lack of fit and random (pure) error.

Model

Mathematical relationship which relates changes in a given response to changes in one or more factors.

Random error

Error that occurs due to natural variation in the process. Random error is typically assumed to be normally distributed with zero mean and a constant variance. Random error is also called experimental error.

Randomization

A schedule for allocating treatment material and for conducting treatment combinations in a DOE such that the conditions in one run neither depend on the conditions of the previous run nor predict the conditions in the subsequent runs.

Replication

Performing the same treatment combination more than once. Including replication allows an estimate of the random error independent of any lack of fit error.

Responses

The output(s) of a process. Sometimes called dependent variable(s).

Treatment

A treatment is a specific combination of factor levels whose effect is to be compared with other treatments.

Classes of models

There are three classes of models used in the analysis of variance, and these are outlined here.

Fixed-effects models

Main article: Fixed effects model

The fixed-effects model (class I) of analysis of variance applies to situations in which the experimenter applies one or more treatments to the subjects of the experiment to see whether the response variable values change. This allows the experimenter to estimate the ranges of response variable values that the treatment would generate in the population as a whole.

Random-effects models

Random-effects model (class II) is used when the treatments are not fixed. This occurs when the various factor levels are sampled from a larger population. Because the levels themselves are random variables, some assumptions and the method of contrasting the treatments (a multi-variable generalization of simple differences) differ from the fixed-effects model.

Mixed-effects models

A mixed-effects model (class III) contains experimental factors of both fixed and random-effects types, with appropriately different interpretations and analysis for the two types. 

Example: Teaching experiments could be performed by a college or university department to find a good introductory textbook, with each text considered a treatment. The fixed-effects model would compare a list of candidate texts. The random-effects model would determine whether important differences exist among a list of randomly selected texts. The mixed-effects model would compare the (fixed) incumbent texts to randomly selected alternatives. 

Defining fixed and random effects has proven elusive, with competing definitions arguably leading toward a linguistic quagmire.

Assumptions

The analysis of variance has been studied from several approaches, the most common of which uses a linear model that relates the response to the treatments and blocks. Note that the model is linear in parameters but may be nonlinear across factor levels. Interpretation is easy when data is balanced across factors but much deeper understanding is needed for unbalanced data.

Textbook analysis using a normal distribution

The analysis of variance can be presented in terms of a linear model, which makes the following assumptions about the probability distribution of the responses:

• Independence of observations – this is an assumption of the model that simplifies the statistical analysis.
• Normality – the distributions of the residuals are normal.
• Equality (or "homogeneity") of variances, called homoscedasticity — the variance of data in groups should be the same.

The separate assumptions of the textbook model imply that the errors are independently, identically, and normally distributed for fixed effects models, that is, that the errors (${\displaystyle \varepsilon }$) are independent and

${\displaystyle \varepsilon \thicksim N(0,\sigma ^{2}).\,}$

Randomization-based analysis

In a randomized controlled experiment, the treatments are randomly assigned to experimental units, following the experimental protocol. This randomization is objective and declared before the experiment is carried out. The objective random-assignment is used to test the significance of the null hypothesis, following the ideas of C. S. Peirce and Ronald Fisher. This design-based analysis was discussed and developed by Francis J. Anscombe at Rothamsted Experimental Station and by Oscar Kempthorne at Iowa State University. Kempthorne and his students make an assumption of unit treatment additivity, which is discussed in the books of Kempthorne and David R. Cox.

Unit-treatment additivity

In its simplest form, the assumption of unit-treatment additivity states that the observed response ${\displaystyle y_{i,j}}$ from experimental unit ${\displaystyle i}$ when receiving treatment ${\displaystyle j}$ can be written as the sum of the unit's response ${\displaystyle y_{i}}$ and the treatment-effect ${\displaystyle t_{j}}$, that is 

${\displaystyle y_{i,j}=y_{i}+t_{j}.}$

The assumption of unit-treatment additivity implies that, for every treatment ${\displaystyle j}$, the ${\displaystyle j}$th treatment has exactly the same effect ${\displaystyle t_{j}}$ on every experiment unit. 

The assumption of unit treatment additivity usually cannot be directly falsified, according to Cox and Kempthorne. However, many consequences of treatment-unit additivity can be falsified. For a randomized experiment, the assumption of unit-treatment additivity implies that the variance is constant for all treatments. Therefore, by contraposition, a necessary condition for unit-treatment additivity is that the variance is constant.

The use of unit treatment additivity and randomization is similar to the design-based inference that is standard in finite-population survey sampling.

Derived linear model

Kempthorne uses the randomization-distribution and the assumption of unit treatment additivity to produce a derived linear model, very similar to the textbook model discussed previously. The test statistics of this derived linear model are closely approximated by the test statistics of an appropriate normal linear model, according to approximation theorems and simulation studies. However, there are differences. For example, the randomization-based analysis results in a small but (strictly) negative correlation between the observations. In the randomization-based analysis, there is no assumption of a normal distribution and certainly no assumption of independence. On the contrary, the observations are dependent! 

The randomization-based analysis has the disadvantage that its exposition involves tedious algebra and extensive time. Since the randomization-based analysis is complicated and is closely approximated by the approach using a normal linear model, most teachers emphasize the normal linear model approach. Few statisticians object to model-based analysis of balanced randomized experiments.

Statistical models for observational data

However, when applied to data from non-randomized experiments or observational studies, model-based analysis lacks the warrant of randomization. For observational data, the derivation of confidence intervals must use subjective models, as emphasized by Ronald Fisher and his followers. In practice, the estimates of treatment-effects from observational studies generally are often inconsistent. In practice, "statistical models" and observational data are useful for suggesting hypotheses that should be treated very cautiously by the public.

Summary of assumptions

The normal-model based ANOVA analysis assumes the independence, normality and homogeneity of variances of the residuals. The randomization-based analysis assumes only the homogeneity of the variances of the residuals (as a consequence of unit-treatment additivity) and uses the randomization procedure of the experiment. Both these analyses require homoscedasticity, as an assumption for the normal-model analysis and as a consequence of randomization and additivity for the randomization-based analysis. 

However, studies of processes that change variances rather than means (called dispersion effects) have been successfully conducted using ANOVA. There are no necessary assumptions for ANOVA in its full generality, but the F-test used for ANOVA hypothesis testing has assumptions and practical limitations which are of continuing interest.

Problems which do not satisfy the assumptions of ANOVA can often be transformed to satisfy the assumptions. The property of unit-treatment additivity is not invariant under a "change of scale", so statisticians often use transformations to achieve unit-treatment additivity. If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify (in the protocol for the experiment or observational study) that the responses be transformed to stabilize the variance. Also, a statistician may specify that logarithmic transforms be applied to the responses, which are believed to follow a multiplicative model. According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.

Characteristics

ANOVA is used in the analysis of comparative experiments, those in which only the difference in outcomes is of interest. The statistical significance of the experiment is determined by a ratio of two variances. This ratio is independent of several possible alterations to the experimental observations: Adding a constant to all observations does not alter significance. Multiplying all observations by a constant does not alter significance. So ANOVA statistical significance result is independent of constant bias and scaling errors as well as the units used in expressing observations. In the era of mechanical calculation it was common to subtract a constant from all observations (when equivalent to dropping leading digits) to simplify data entry. This is an example of data coding.

Logic

The calculations of ANOVA can be characterized as computing a number of means and variances, dividing two variances and comparing the ratio to a handbook value to determine statistical significance. Calculating a treatment effect is then trivial, "the effect of any treatment is estimated by taking the difference between the mean of the observations which receive the treatment and the general mean".

Partitioning of the sum of squares

ANOVA uses traditional standardized terminology. The definitional equation of sample variance is ${\displaystyle s^{2}=\textstyle {\frac {1}{n-1}}\sum (y_{i}-{\bar {y}})^{2}}$, where the divisor is called the degrees of freedom (DF), the summation is called the sum of squares (SS), the result is called the mean square (MS) and the squared terms are deviations from the sample mean. ANOVA estimates 3 sample variances: a total variance based on all the observation deviations from the grand mean, an error variance based on all the observation deviations from their appropriate treatment means, and a treatment variance. The treatment variance is based on the deviations of treatment means from the grand mean, the result being multiplied by the number of observations in each treatment to account for the difference between the variance of observations and the variance of means. 

The fundamental technique is a partitioning of the total sum of squares SS into components related to the effects used in the model. For example, the model for a simplified ANOVA with one type of treatment at different levels.

${\displaystyle SS_{\text{Total}}=SS_{\text{Error}}+SS_{\text{Treatments}}}$

The number of degrees of freedom DF can be partitioned in a similar way: one of these components (that for error) specifies a chi-squared distribution which describes the associated sum of squares, while the same is true for "treatments" if there is no treatment effect.

${\displaystyle DF_{\text{Total}}=DF_{\text{Error}}+DF_{\text{Treatments}}}$

The F-test

The F-test is used for comparing the factors of the total deviation. For example, in one-way, or single-factor ANOVA, statistical significance is tested for by comparing the F test statistic

${\displaystyle F={\frac {\text{variance between treatments}}{\text{variance within treatments}}}}$

${\displaystyle F={\frac {MS_{\text{Treatments}}}{MS_{\text{Error}}}}={{SS_{\text{Treatments}}/(I-1)} \over {SS_{\text{Error}}/(n_{T}-I)}}}$

where MS is mean square, ${\displaystyle I}$ = number of treatments and ${\displaystyle n_{T}}$ = total number of cases to the F-distribution with ${\displaystyle I-1}$, ${\displaystyle n_{T}-I}$ degrees of freedom. Using the F-distribution is a natural candidate because the test statistic is the ratio of two scaled sums of squares each of which follows a scaled chi-squared distribution. 

The expected value of F is ${\displaystyle 1+{n\sigma _{\text{Treatment}}^{2}}/{\sigma _{\text{Error}}^{2}}}$ (where n is the treatment sample size) which is 1 for no treatment effect. As values of F increase above 1, the evidence is increasingly inconsistent with the null hypothesis. Two apparent experimental methods of increasing F are increasing the sample size and reducing the error variance by tight experimental controls. 

There are two methods of concluding the ANOVA hypothesis test, both of which produce the same result:

• The textbook method is to compare the observed value of F with the critical value of F determined from tables. The critical value of F is a function of the degrees of freedom of the numerator and the denominator and the significance level (α). If F ≥ F_Critical, the null hypothesis is rejected.
• The computer method calculates the probability (p-value) of a value of F greater than or equal to the observed value. The null hypothesis is rejected if this probability is less than or equal to the significance level (α).

The ANOVA F-test is known to be nearly optimal in the sense of minimizing false negative errors for a fixed rate of false positive errors (i.e. maximizing power for a fixed significance level). For example, to test the hypothesis that various medical treatments have exactly the same effect, the F-test's p-values closely approximate the permutation test's p-values: The approximation is particularly close when the design is balanced. Such permutation tests characterize tests with maximum power against all alternative hypotheses, as observed by Rosenbaum. The ANOVA F-test (of the null-hypothesis that all treatments have exactly the same effect) is recommended as a practical test, because of its robustness against many alternative distributions.

Extended logic

ANOVA consists of separable parts; partitioning sources of variance and hypothesis testing can be used individually. ANOVA is used to support other statistical tools. Regression is first used to fit more complex models to data, then ANOVA is used to compare models with the objective of selecting simple(r) models that adequately describe the data. "Such models could be fit without any reference to ANOVA, but ANOVA tools could then be used to make some sense of the fitted models, and to test hypotheses about batches of coefficients." "[W]e think of the analysis of variance as a way of understanding and structuring multilevel models—not as an alternative to regression but as a tool for summarizing complex high-dimensional inferences ..."

For a single factor

The simplest experiment suitable for ANOVA analysis is the completely randomized experiment with a single factor. More complex experiments with a single factor involve constraints on randomization and include completely randomized blocks and Latin squares (and variants: Graeco-Latin squares, etc.). The more complex experiments share many of the complexities of multiple factors. A relatively complete discussion of the analysis (models, data summaries, ANOVA table) of the completely randomized experiment is available.

For multiple factors

ANOVA generalizes to the study of the effects of multiple factors. When the experiment includes observations at all combinations of levels of each factor, it is termed factorial. Factorial experiments are more efficient than a series of single factor experiments and the efficiency grows as the number of factors increases. Consequently, factorial designs are heavily used. 

The use of ANOVA to study the effects of multiple factors has a complication. In a 3-way ANOVA with factors x, y and z, the ANOVA model includes terms for the main effects (x, y, z) and terms for interactions (xy, xz, yz, xyz). All terms require hypothesis tests. The proliferation of interaction terms increases the risk that some hypothesis test will produce a false positive by chance. Fortunately, experience says that high order interactions are rare. The ability to detect interactions is a major advantage of multiple factor ANOVA. Testing one factor at a time hides interactions, but produces apparently inconsistent experimental results.

Caution is advised when encountering interactions; Test interaction terms first and expand the analysis beyond ANOVA if interactions are found. Texts vary in their recommendations regarding the continuation of the ANOVA procedure after encountering an interaction. Interactions complicate the interpretation of experimental data. Neither the calculations of significance nor the estimated treatment effects can be taken at face value. "A significant interaction will often mask the significance of main effects." Graphical methods are recommended to enhance understanding. Regression is often useful. A lengthy discussion of interactions is available in Cox (1958). Some interactions can be removed (by transformations) while others cannot. 

A variety of techniques are used with multiple factor ANOVA to reduce expense. One technique used in factorial designs is to minimize replication (possibly no replication with support of analytical trickery) and to combine groups when effects are found to be statistically (or practically) insignificant. An experiment with many insignificant factors may collapse into one with a few factors supported by many replications.

Worked numeric examples

Numerous fully worked numerical examples are available in standard textbooks and online. A simple case uses one-way (a single factor) analysis.

Associated analysis

Some analysis is required in support of the design of the experiment while other analysis is performed after changes in the factors are formally found to produce statistically significant changes in the responses. Because experimentation is iterative, the results of one experiment alter plans for following experiments.

Preparatory analysis

The number of experimental units

In the design of an experiment, the number of experimental units is planned to satisfy the goals of the experiment. Experimentation is often sequential. 

Early experiments are often designed to provide mean-unbiased estimates of treatment effects and of experimental error. Later experiments are often designed to test a hypothesis that a treatment effect has an important magnitude; in this case, the number of experimental units is chosen so that the experiment is within budget and has adequate power, among other goals.

Reporting sample size analysis is generally required in psychology. "Provide information on sample size and the process that led to sample size decisions." The analysis, which is written in the experimental protocol before the experiment is conducted, is examined in grant applications and administrative review boards. 

Besides the power analysis, there are less formal methods for selecting the number of experimental units. These include graphical methods based on limiting the probability of false negative errors, graphical methods based on an expected variation increase (above the residuals) and methods based on achieving a desired confident interval.

Power analysis

Power analysis is often applied in the context of ANOVA in order to assess the probability of successfully rejecting the null hypothesis if we assume a certain ANOVA design, effect size in the population, sample size and significance level. Power analysis can assist in study design by determining what sample size would be required in order to have a reasonable chance of rejecting the null hypothesis when the alternative hypothesis is true.

Effect size

Several standardized measures of effect have been proposed for ANOVA to summarize the strength of the association between a predictor(s) and the dependent variable or the overall standardized difference of the complete model. Standardized effect-size estimates facilitate comparison of findings across studies and disciplines. However, while standardized effect sizes are commonly used in much of the professional literature, a non-standardized measure of effect size that has immediately "meaningful" units may be preferable for reporting purposes.

Follow-up analysis

It is always appropriate to carefully consider outliers. They have a disproportionate impact on statistical conclusions and are often the result of errors.

Model confirmation

It is prudent to verify that the assumptions of ANOVA have been met. Residuals are examined or analyzed to confirm homoscedasticity and gross normality. Residuals should have the appearance of (zero mean normal distribution) noise when plotted as a function of anything including time and modeled data values. Trends hint at interactions among factors or among observations. One rule of thumb: "If the largest standard deviation is less than twice the smallest standard deviation, we can use methods based on the assumption of equal standard deviations and our results will still be approximately correct."

Follow-up tests

A statistically significant effect in ANOVA is often followed up with one or more different follow-up tests. This can be done in order to assess which groups are different from which other groups or to test various other focused hypotheses. Follow-up tests are often distinguished in terms of whether they are planned (a priori) or post hoc. Planned tests are determined before looking at the data and post hoc tests are performed after looking at the data.

Often one of the "treatments" is none, so the treatment group can act as a control. Dunnett's test (a modification of the t-test) tests whether each of the other treatment groups has the same mean as the control.

Post hoc tests such as Tukey's range test most commonly compare every group mean with every other group mean and typically incorporate some method of controlling for Type I errors. Comparisons, which are most commonly planned, can be either simple or compound. Simple comparisons compare one group mean with one other group mean. Compound comparisons typically compare two sets of groups means where one set has two or more groups (e.g., compare average group means of group A, B and C with group D). Comparisons can also look at tests of trend, such as linear and quadratic relationships, when the independent variable involves ordered levels. 

Following ANOVA with pair-wise multiple-comparison tests has been criticized on several grounds. There are many such tests (10 in one table) and recommendations regarding their use are vague or conflicting.

Study designs

There are several types of ANOVA. Many statisticians base ANOVA on the design of the experiment, especially on the protocol that specifies the random assignment of treatments to subjects; the protocol's description of the assignment mechanism should include a specification of the structure of the treatments and of any blocking. It is also common to apply ANOVA to observational data using an appropriate statistical model.

Some popular designs use the following types of ANOVA:

• One-way ANOVA is used to test for differences among two or more independent groups (means),e.g. different levels of urea application in a crop, or different levels of antibiotic action on several different bacterial species, or different levels of effect of some medicine on groups of patients. However, should these groups not be independent, and there is an order in the groups (such as mild, moderate and severe disease), or in the dose of a drug (such as 5 mg/mL, 10 mg/mL, 20 mg/mL) given to the same group of patients, then a linear trend estimation should be used. Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test.^[65] When there are only two means to compare, the t-test and the ANOVA F-test are equivalent; the relation between ANOVA and t is given by F = t².
• Factorial ANOVA is used when the experimenter wants to study the interaction effects among the treatments.
• Repeated measures ANOVA is used when the same subjects are used for each treatment (e.g., in a longitudinal study).
• Multivariate analysis of variance (MANOVA) is used when there is more than one response variable.

Cautions

Balanced experiments (those with an equal sample size for each treatment) are relatively easy to interpret; Unbalanced experiments offer more complexity. For single-factor (one-way) ANOVA, the adjustment for unbalanced data is easy, but the unbalanced analysis lacks both robustness and power. For more complex designs the lack of balance leads to further complications. "The orthogonality property of main effects and interactions present in balanced data does not carry over to the unbalanced case. This means that the usual analysis of variance techniques do not apply. Consequently, the analysis of unbalanced factorials is much more difficult than that for balanced designs." In the general case, "The analysis of variance can also be applied to unbalanced data, but then the sums of squares, mean squares, and F-ratios will depend on the order in which the sources of variation are considered." The simplest techniques for handling unbalanced data restore balance by either throwing out data or by synthesizing missing data. More complex techniques use regression.

ANOVA is (in part) a test of statistical significance. The American Psychological Association holds the view that simply reporting statistical significance is insufficient and that reporting confidence bounds is preferred.

While ANOVA is conservative (in maintaining a significance level) against multiple comparisons in one dimension, it is not conservative against comparisons in multiple dimensions.

Generalizations

ANOVA is considered to be a special case of linear regression which in turn is a special case of the general linear model. All consider the observations to be the sum of a model (fit) and a residual (error) to be minimized. 

The Kruskal–Wallis test and the Friedman test are nonparametric tests, which do not rely on an assumption of normality.

Connection to linear regression

Below we make clear the connection between multi-way ANOVA and linear regression. 

Linearly re-order the data so that ${\displaystyle k^{\text{th}}}$ observation is associated with a response ${\displaystyle y_{k}}$ and factors ${\displaystyle Z_{k,b}}$ where ${\displaystyle b\in \{1,2,\ldots ,B\}}$ denotes the different factors and ${\displaystyle B}$ is the total number of factors. In one-way ANOVA ${\displaystyle B=1}$ and in two-way ANOVA ${\displaystyle B=2}$. Furthermore, we assume the ${\displaystyle b^{th}}$ factor has ${\displaystyle I_{b}}$ levels, namely ${\displaystyle \{1,2,\ldots ,I_{b}\}}$. Now, we can one-hot encode the factors into the ${\displaystyle \sum _{b=1}^{B}I_{b}}$ dimensional vector ${\displaystyle v_{k}}$. 

The one-hot encoding function ${\displaystyle g_{b}:\{1,2,\ldots ,I_{b}\}\mapsto \{0,1\}^{I_{b}}}$ is defined such that the ${\displaystyle i^{th}}$ entry of ${\displaystyle g_{b}(Z_{k,b})}$ is 

$${\displaystyle g_{b}(Z_{k,b})_{i}={\begin{cases}1&{\text{if }}i=Z_{k,b}\\0&{\text{otherwise}}\end{cases}}}$$

 

The vector ${\displaystyle v_{k}}$ is the concatenation of all of the above vectors for all ${\displaystyle b}$. Thus, ${\displaystyle v_{k}=[g_{1}(Z_{k,1}),g_{2}(Z_{k,2}),\ldots ,g_{B}(Z_{k,B})]}$. In order to obtain a fully general ${\displaystyle B}$-way interaction ANOVA we must also concatenate every additional interaction term in the vector ${\displaystyle v_{k}}$ and then add an intercept term. Let that vector be ${\displaystyle X_{k}}$. 

 

With this notation in place, we now have the exact connection with linear regression. We simply regress response ${\displaystyle y_{k}}$ against the vector ${\displaystyle X_{k}}$. However, there is a concern about identifiability. In order to overcome such issues we assume that the sum of the parameters within each set of interactions is equal to zero. From here, one can use F-statistics or other methods to determine the relevance of the individual factors.

Example

We can consider the 2-way interaction example where we assume that the first factor has 2 levels and the second factor has 3 levels. 

Define ${\displaystyle a_{i}=1}$ if ${\displaystyle Z_{k,1}=i}$ and ${\displaystyle b_{i}=1}$ if ${\displaystyle Z_{k,2}=i}$, i.e. ${\displaystyle a}$ is the one-hot encoding of the first factor and ${\displaystyle b}$ is the one-hot encoding of the second factor. 

With that,

$${\displaystyle X_{k}=[a_{1},a_{2},b_{1},b_{2},b_{3},a_{1}\times b_{1},a_{1}\times b_{2},a_{1}\times b_{3},a_{2}\times b_{1},a_{2}\times b_{2},a_{2}\times b_{3},1]}$$

 

where the last term is an intercept term. For a more concrete example suppose that

 

$${\displaystyle {\begin{aligned}Z_{k,1}&=2\\Z_{k,2}&=1\end{aligned}}}$$

 

Then, 

 

$${\displaystyle X_{k}=[0,1,1,0,0,0,0,0,1,0,0,1]}$$

Coalescent theory

From Wikipedia, the free encyclopedia

 

Coalescent theory is a model of how gene variants sampled from a population may have originated from a common ancestor. In the simplest case, coalescent theory assumes no recombination, no natural selection, and no gene flow or population structure, meaning that each variant is equally likely to have been passed from one generation to the next. The model looks backward in time, merging alleles into a single ancestral copy according to a random process in coalescence events. Under this model, the expected time between successive coalescence events increases almost exponentially back in time (with wide variance). Variance in the model comes from both the random passing of alleles from one generation to the next, and the random occurrence of mutations in these alleles.

 

The mathematical theory of the coalescent was developed independently by several groups in the early 1980s as a natural extension of classical population genetics theory and models, but can be primarily attributed to John Kingman. Advances in coalescent theory include recombination, selection, overlapping generations and virtually any arbitrarily complex evolutionary or demographic model in population genetic analysis.

The model can be used to produce many theoretical genealogies, and then compare observed data to these simulations to test assumptions about the demographic history of a population. Coalescent theory can be used to make inferences about population genetic parameters, such as migration, population size and recombination.

Theory

Time to coalescence

Consider a single gene locus sampled from two haploid individuals in a population. The ancestry of this sample is traced backwards in time to the point where these two lineages coalesce in their most recent common ancestor (MRCA). Coalescent theory seeks to estimate the expectation of this time period and its variance. 

The probability that two lineages coalesce in the immediately preceding generation is the probability that they share a parental DNA sequence. In a population with a constant effective population size with 2N_e copies of each locus, there are 2N_e "potential parents" in the previous generation. Under a random mating model, the probability that two alleles originate from the same parental copy is thus 1/(2N_e) and, correspondingly, the probability that they do not coalesce is 1 − 1/(2N_e).

At each successive preceding generation, the probability of coalescence is geometrically distributed—that is, it is the probability of noncoalescence at the t − 1 preceding generations multiplied by the probability of coalescence at the generation of interest:

${\displaystyle P_{c}(t)=\left(1-{\frac {1}{2N_{e}}}\right)^{t-1}\left({\frac {1}{2N_{e}}}\right).}$

For sufficiently large values of N_e, this distribution is well approximated by the continuously defined exponential distribution

${\displaystyle P_{c}(t)={\frac {1}{2N_{e}}}e^{-{\frac {t-1}{2N_{e}}}}.}$

This is mathematically convenient, as the standard exponential distribution has both the expected value and the standard deviation equal to 2N_e. Therefore, although the expected time to coalescence is 2N_e, actual coalescence times have a wide range of variation. Note that coalescent time is the number of preceding generations where the coalescence took place and not calendar time, though an estimation of the latter can be made multiplying 2N_e with the average time between generations. The above calculations apply equally to a diploid population of effective size N_e (in other words, for a non-recombining segment of DNA, each chromosome can be treated as equivalent to an independent haploid individual; in the absence of inbreeding, sister chromosomes in a single individual are no more closely related than two chromosomes randomly sampled from the population). Some effectively haploid DNA elements, such as mitochondrial DNA, however, are only carried by one sex, and therefore have one quarter the effective size of the equivalent diploid population (N_e/2)

Neutral variation

Coalescent theory can also be used to model the amount of variation in DNA sequences expected from genetic drift and mutation. This value is termed the mean heterozygosity, represented as ${\displaystyle {\bar {H}}}$. Mean heterozygosity is calculated as the probability of a mutation occurring at a given generation divided by the probability of any "event" at that generation (either a mutation or a coalescence). The probability that the event is a mutation is the probability of a mutation in either of the two lineages: ${\displaystyle 2\mu }$. Thus the mean heterozygosity is equal to

${\displaystyle {\begin{aligned}{\bar {H}}&={\frac {2\mu }{2\mu +{\frac {1}{2N_{e}}}}}\\[3pt]&={\frac {4N_{e}\mu }{1+4N_{e}\mu }}\\[3pt]&={\frac {\theta }{1+\theta }}\end{aligned}}}$

For ${\displaystyle 4N_{e}\mu \gg 1}$, the vast majority of allele pairs have at least one difference in nucleotide sequence.

Graphical representation

Coalescents can be visualised using dendrograms which show the relationship of branches of the population to each other. The point where two branches meet indicates a coalescent event.

Applications

Disease gene mapping

The utility of coalescent theory in the mapping of disease is slowly gaining more appreciation; although the application of the theory is still in its infancy, there are a number of researchers who are actively developing algorithms for the analysis of human genetic data that utilise coalescent theory.

A considerable number of human diseases can be attributed to genetics, from simple Mendelian diseases like sickle-cell anemia and cystic fibrosis, to more complicated maladies like cancers and mental illnesses. The latter are polygenic diseases, controlled by multiple genes that may occur on different chromosomes, but diseases that are precipitated by a single abnormality are relatively simple to pinpoint and trace – although not so simple that this has been achieved for all diseases. It is immensely useful in understanding these diseases and their processes to know where they are located on chromosomes, and how they have been inherited through generations of a family, as can be accomplished through coalescent analysis.

Genetic diseases are passed from one generation to another just like other genes. While any gene may be shuffled from one chromosome to another during homologous recombination, it is unlikely that one gene alone will be shifted. Thus, other genes that are close enough to the disease gene to be linked to it can be used to trace it.

Polygenic diseases have a genetic basis even though they don't follow Mendelian inheritance models, and these may have relatively high occurrence in populations, and have severe health effects. Such diseases may have incomplete penetrance, and tend to be polygenic, complicating their study. These traits may arise due to many small mutations, which together have a severe and deleterious effect on the health of the individual.

Linkage mapping methods, including Coalescent theory can be put to work on these diseases, since they use family pedigrees to figure out which markers accompany a disease, and how it is inherited. At the very least, this method helps narrow down the portion, or portions, of the genome on which the deleterious mutations may occur. Complications in these approaches include epistatic effects, the polygenic nature of the mutations, and environmental factors. That said, genes whose effects are additive carry a fixed risk of developing the disease, and when they exist in a disease genotype, they can be used to predict risk and map the gene. Both regular the coalescent and the shattered coalescent (which allows that multiple mutations may have occurred in the founding event, and that the disease may occasionally be triggered by environmental factors) have been put to work in understanding disease genes.

Studies have been carried out correlating disease occurrence in fraternal and identical twins, and the results of these studies can be used to inform coalescent modeling. Since identical twins share all of their genome, but fraternal twins only share half their genome, the difference in correlation between the identical and fraternal twins can be used to work out if a disease is heritable, and if so how strongly.

The genomic distribution of heterozygosity

The human single-nucleotide polymorphism (SNP) map has revealed large regional variations in heterozygosity, more so than can be explained on the basis of (Poisson-distributed) random chance.^[9] In part, these variations could be explained on the basis of assessment methods, the availability of genomic sequences, and possibly the standard coalescent population genetic model. Population genetic influences could have a major influence on this variation: some loci presumably would have comparatively recent common ancestors, others might have much older genealogies, and so the regional accumulation of SNPs over time could be quite different. The local density of SNPs along chromosomes appears to cluster in accordance with a variance to mean power law and to obey the Tweedie compound Poisson distribution. In this model the regional variations in the SNP map would be explained by the accumulation of multiple small genomic segments through recombination, where the mean number of SNPs per segment would be gamma distributed in proportion to a gamma distributed time to the most recent common ancestor for each segment.

History

Coalescent theory is a natural extension of the more classical population genetics concept of neutral evolution and is an approximation to the Fisher–Wright (or Wright–Fisher) model for large populations. It was discovered independently by several researchers in the 1980s.

Software

A large body of software exists for both simulating data sets under the coalescent process as well as inferring parameters such as population size and migration rates from genetic data.

• BEAST – Bayesian inference package via MCMC with a wide range of coalescent models including the use of temporally sampled sequences.
• BPP - software package for inferring phylogeny and divergence times among populations under a multispecies coalescent process.
• CoaSim – software for simulating genetic data under the coalescent model.
• DIYABC – A user-friendly approach to ABC for inference on population history using molecular markers.
• DendroPy – A Python library for phylogenetic computing, with classes and methods for simulating pure (unconstrained) coalescent trees as well as constrained coalescent trees under the multispecies coalescent model (i.e., "gene trees in species trees").
• GeneRecon – software for the fine-scale mapping of linkage disequilibrium mapping of disease genes using coalescent theory based on a Bayesian MCMC framework.
• genetree software for estimation of population genetics parameters using coalescent theory and simulation (the R package popgen). See also Oxford Mathematical Genetics and Bioinformatics Group
• GENOME – rapid coalescent-based whole-genome simulation
• IBDSim – A computer package for the simulation of genotypic data under general isolation by distance models.
• IMa – IMa implements the same Isolation with Migration model, but does so using a new method that provides estimates of the joint posterior probability density of the model parameters. IMa also allows log likelihood ratio tests of nested demographic models. IMa is based on a method described in Hey and Nielsen (2007 PNAS 104:2785–2790). IMa is faster and better than IM (i.e. by virtue of providing access to the joint posterior density function), and it can be used for most (but not all) of the situations and options that IM can be used for.
• Lamarc – software for estimation of rates of population growth, migration, and recombination.
• Migraine – A program which implements coalescent algorithms for a maximum likelihood analysis (using Importance Sampling algorithms) of genetic data with a focus on spatially structured populations.
• Migrate – Maximum likelihood and Bayesian inference of migration rates under the n-coalescent. The inference is implemented using MCMC
• MaCS – Markovian Coalescent Simulator – Simulates genealogies spatially across chromosomes as a Markovian process. Similar to the SMC algorithm of McVean and Cardin, and supports all demographic scenarios found in Hudson's ms.
• ms & msHOT – Richard Hudson's original program for generating samples under neutral models and an extension which allows recombination hotspots.
• msms – An extended version of ms that includes selective sweeps.
• msprime – A fast and scalable ms-compatible simulator, allowing demographic simulations, producing compact output files for thousands or millions of genomes.
• Recodon and NetRecodon – software to simulate coding sequences with inter/intracodon recombination, migration, growth rate and longitudinal sampling.
• CoalEvol and SGWE – software to simulate nucleotide, coding and amino acid sequences under the coalescent with demographics, recombination, population structure with migration and longitudinal sampling.
• SARG – Structure Ancestral Recombination Graph by Magnus Nordborg
• simcoal2 – software to simulate genetic data under the coalescent model with complex demography and recombination
• TreesimJ Forward simulation software allowing sampling of genealogies and data sets under diverse selective and demographic models.