Analysis of variance

# Overview
In statistics, analysis of variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance is partitioned into components due to different explanatory variables.  The initial techniques of the analysis of variance were developed by the statistician and geneticist R. A. Fisher in the 1920s and 1930s, and is sometimes known as Fisher's ANOVA or Fisher's analysis of variance, due to the use of Fisher's F-distribution as part of the test of statistical significance.
There are three conceptual classes of such models:
- Fixed-effects model assumes that the data come from normal populations which may differ only in their means. (Model 1)
- Random-effects models assume that the data describe a hierarchy of different populations whose differences are constrained by the hierarchy. (Model 2)
- Mixed effects models describe situations where both fixed and random effects are present. (Model 3)
In practice, there are several types of ANOVA depending on the number of treatments and the way they are applied to the subjects in the experiment:
- One-way ANOVA is used to test for differences among two or more independent groups. Typically, however, the One-way ANOVA is used to test for differences among three or more groups, with the two-group case relegated to the t-test (Gossett, 1908), which is a special case of the ANOVA. The relation between ANOVA and t is given as
F = t squared.
- One-way ANOVA for repeated measures is used when the subjects are subjected to repeated measures; this means that the same subjects are used for each treatment.  Note that this method can be subject to carryover effects.
- Factorial ANOVA is used when the experimenter wants to study the effects of two or more treatment variables. The most commonly used type of factorial ANOVA is the 22 (read: two by two) design, where there are two independent variables and each variable has two levels or distinct values. Factorial ANOVA can also be multi-level such as 33, etc. or higher order such as 222, etc. but analyses with higher numbers of factors are rarely done because the calculations are lengthy and the results are hard to interpret.
- When one wishes to test two or more independent groups subjecting the subjects to repeated measures, one may perform a factorial mixed-design ANOVA, in which one factor is independent and the other is repeated measures.  This is a type of mixed effect model.
- Multivariate analysis of variance (MANOVA) is used when there is more than one dependent variable.

# Models

## Fixed-effects model
The fixed-effects model of analysis of variance applies to situations in which the experimenter applies several treatments to the subjects of the experiment to see if the  response variable values change. This allows the experimenter to estimate the range of response variable values that the treatment would generate in the population as a whole.

## Random-effects model
Random effects models are used when the treatments are not fixed. This occurs when the various treatments (also known as factor levels) are sampled from a larger population. Because the treatments themselves are random variables, some assumptions and the method of contrasting the treatments differ from Anova model 1.
Most random-effects or mixed-effects models are not concerned with making inferences concerning the particular sampled factors. For example, consider a large manufacturing plant in which many machines  produce the same product. The statistician studying this plant would have very little interest in comparing the three particular machines to each other. Rather, inferences that can be made for all machines are of interest, such as their variability and the overall mean.

# Assumptions
- Independence of cases - this is a requirement of the design.
- Normality - the distributions in each of the groups are normal (use the Kolmogorov-Smirnov and Shapiro-Wilk normality tests to test it). Some say that the F-test is extremely non-robust to deviations from normality (Lindman, 1974) while others claim otherwise (Ferguson & Takane 2005: 261-2).  The Kruskal-Wallis test is a nonparametric alternative which does not rely on an assumption of normality.
- Homogeneity of variances - the variance of data in groups should be the same (use Levene's test for homogeneity of variances).
These together form the common assumption that the error residuals are independently, identically, and normally distributed for fixed effects models, or:
Anova 2 and 3 have more complex assumptions about the expected value and variance of the residuals since the factors themselves may be drawn from a population.

# Logic of ANOVA

## Partitioning of the sum of squares
The fundamental technique is a partitioning of the total sum of squares into components related to the effects used in the model. For example, we show the model for a simplified ANOVA with one type of treatment at different levels.
The number of degrees of freedom (abbreviated df) can be partitioned in a similar way and specifies the chi-square distribution which describes the associated sums of squares.

## The F-test
The F-test is used for comparisons of the components of the total deviation. For example, in one-way, or single-factor Anova, statistical significance is tested for by comparing the F test statistic
to the F-distribution with I-1,nT  degrees of freedom. Using the F-distribution is a natural candidate because the test statistic is the quotient of two mean sums of squares which have a chi-square distribution.

## ANOVA on ranks
As first suggested by Conover and Iman in 1981, in many cases when the data do not meet the assumptions of ANOVA, one can replace each original data value by its rank from 1 for the smallest to N for the largest, then run a standard ANOVA calculation on the rank-transformed data.  "Where no equivalent nonparametric methods have yet been developed such as for the two-way design, rank transformation results in tests which are more robust to non-normality, and resistant to outliers and non-constant variance, than is ANOVA without the transformation." (Helsel & Hirsch, 2002, Page 177).
However Seaman et al. (1994) noticed that the rank transformation of Conover and Iman (1981) is not appropriate for testing interactions among effects in a factorial design as it can cause an increase in Type I error (alpha error). Furthermore, if both main factors are significant there is little power to detect interactions
- Conover, W. J., Iman, R. L. (1981). Rank transformations as a bridge between parametric and nonparametric statistics. American Statistician, 35, 124-129.  
- Helsel, D.R. and R. M. Hirsch, 2002. Statistical Methods in Water Resources: Techniques of Water Resourses Investigations, Book 4, chapter A3. U.S. Geological Survey. 522 pages.
- Seaman, J. W., Walls, S. C., Wide, S.E. and Jaeger, R.G.(1994)  Caveat emptor:  rank transform methods and interactions. Trends Ecol. Evol. 9, 261-263.

# Examples
Group A is given vodka, Group B is given gin, and Group C is given a placebo. All groups are then tested with a memory task. A one-way ANOVA can be used to assess the effect of the various treatments (that is, the vodka, gin, and placebo).
Group A is given vodka and tested on a memory task. The same group is allowed a rest period of five days and then the experiment is repeated with gin. The procedure is repeated using a placebo. A one-way ANOVA with repeated measures can be used to assess the effect of the vodka versus the impact of the placebo.
In an experiment testing the effects of expectations, subjects are randomly assigned to four groups:
- expect vodka-receive vodka
- expect vodka-receive placebo
- expect placebo-receive vodka
- expect placebo-receive placebo (the last group is used as the control group)
Each group is then tested on a memory task. The advantage of this design is that multiple variables can be tested at the same time instead of running two different experiments. Also, the experiment can determine whether one variable affects the other variable (known as interaction effects). A factorial ANOVA (22) can be used to assess the effect of expecting vodka or the placebo and the actual reception of either.