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Friedman's test

`p = friedman(X,reps)p = friedman(X,reps,displayopt)[p,table] = friedman(...)[p,table,stats] = friedman(...)`

`p = friedman(X,reps)` performs
the nonparametric Friedman's test to compare column effects in a two-way
layout. Friedman's test is similar to classical balanced two-way ANOVA,
but it tests only for column effects after adjusting for possible
row effects. It does not test for row effects or interaction effects.
Friedman's test is appropriate when columns represent treatments that
are under study, and rows represent nuisance effects (blocks) that
need to be taken into account but are not of any interest.

The different columns of `X` represent changes
in a factor A. The different rows represent changes
in a blocking factor B. If there is more than one
observation for each combination of factors, input `reps` indicates
the number of replicates in each "cell," which must
be constant.

The matrix below illustrates the format for a set-up where column
factor A has three levels, row factor B has two levels, and there
are two replicates (`reps=2`). The subscripts indicate
row, column, and replicate, respectively.

Friedman's test assumes a model of the form

where μ is an overall location parameter,
represents the column effect,
represents the row effect, and
represents the error. This test
ranks the data within each level of B, and tests for a difference
across levels of A. The `p` that `friedman` returns
is the *p* value for the null hypothesis that
. If the *p* value
is near zero, this casts doubt on the null hypothesis. A sufficiently
small *p* value suggests that at least one column-sample
median is significantly different than the others; i.e., there is
a main effect due to factor A. The choice of a critical *p* value
to determine whether a result is "statistically significant"
is left to the researcher. It is common to declare a result significant
if the *p* value is less than 0.05 or 0.01.

`friedman` also displays a figure showing an
ANOVA table, which divides the variability of the ranks into two or
three parts:

The variability due to the differences among the column effects

The variability due to the interaction between rows and columns (if reps is greater than its default value of 1)

The remaining variability not explained by any systematic source

The ANOVA table has six columns:

The first shows the source of the variability.

The second shows the Sum of Squares (SS) due to each source.

The third shows the degrees of freedom (df) associated with each source.

The fourth shows the Mean Squares (MS), which is the ratio SS/df.

The fifth shows Friedman's chi-square statistic.

The sixth shows the

*p*value for the chi-square statistic.

`p = friedman(X,reps,displayopt)` enables
the ANOVA table display when

`[p,table] = friedman(...)` returns
the ANOVA table (including column and row labels) in cell array `table`.
(You can copy a text version of the ANOVA table to the clipboard by
selecting `Copy Text` from the **Edit** menu.

`[p,table,stats] = friedman(...)` returns
a `stats` structure that you can use to perform a
follow-up multiple comparison test. The `friedman` test
evaluates the hypothesis that the column effects are all the same
against the alternative that they are not all the same. Sometimes
it is preferable to perform a test to determine which pairs of column
effects are significantly different, and which are not. You can use
the `multcompare` function to perform
such tests by supplying the `stats` structure as
input.

Friedman's test makes the following assumptions about the data
in `X`:

All data come from populations having the same continuous distribution, apart from possibly different locations due to column and row effects.

All observations are mutually independent.

The classical two-way ANOVA replaces the first assumption with the stronger assumption that data come from normal distributions.

[1] Hogg, R. V., and J. Ledolter. *Engineering
Statistics*. New York: MacMillan, 1987.

[2] Hollander, M., and D. A. Wolfe. *Nonparametric
Statistical Methods*. Hoboken, NJ: John Wiley & Sons,
Inc., 1999.

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