Analysis of variance

In statistics, analysis of variance (ANOVA) is a collection of statistical
models and their associated procedures which compare means by splitting the
overall observed variance into different parts. The initial techniques of
the analysis of variance were pioneered by the statistician and geneticist
Ronald Fisher in the 1920s and 1930s. There are three conceptual classes of
such models:

  1. The fixed effects model assumes that the data come from normal
     populations which differ in their means.
  2. Random effects models assume that the data describe a hierarchy of
     different populations whose differences are constrained by the
     hierarchy.
  3. Mixed models describe situations where both fixed and random effects
     are present.

The fundamental technique is a partitioning of the total sum of squares into
components related to the effects in the model used. For example, we show
the model for a simplified ANOVA with one type of treatment at different
levels. (If the treatment levels are quantitative and the effects are
linear, a linear regression analysis may be appropriate.)

     SSTotal = SSError + SSTreatments

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.

     dfTotal = dfError + dfTreatments

Fixed effects model

The fixed effects model of analysis of variance applies to situations in
which the experimenter has subjected his experimental material to several
treatments, each of which affects only the mean of the underlying normal
distribution of the response variable.

Random effects model

Random effects models are used to describe situations in which incomparable
differences in experimental material occur. The simplest example is that of
estimating the unknown mean of a population whose individuals differ from
each other. In this case, the variation between individuals is confounded
with that of the observing instrument.

Degrees of freedom

Degrees of freedom indicates the effective number of observations which
contribute to the sum of squares in an ANOVA, the total number of
observations minus the number of linear constraints in the data.

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