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.TH ANOVA 1 "March 5, 1985" "UNIX|STAT 5.0" "UNIX User's Manual"
.SH NAME
anova \- multi-factor analysis of variance
.SH SYNOPSIS
.B anova
[column names]
.SH DESCRIPTION
.I anova
does multi-factor analysis of variance on designs
with within groups factors, between groups factors, or both.
.I anova
reads from the standard input (via redirection with < or piped
with |) and writes to the standard output.
.I anova
allows variable numbers of replications
(averaged before analysis) on any factor.
All factors except the random factor must be crossed;
some nested designs are not allowed.
Unequal group sizes on between groups factors are allowed
and are solved with the weighted means solution,
however empty cells are not permitted.

.I "Input Format."
.I anova
uses an input format unlike conventional programs.
The input format was designed so that if a user specifies
the role individual data play in the overall design,
.I anova
would figure out program parameters that usually need to
be explicitly specified with a special control language.
This way, the error prone process of design specification
is taken out of the hands of the user.
The input to 
.I anova
consists of each datum on a separate line,
preceded by a list of indexes, one for each factor,
that specifies the level of each factor at which that datum was obtained.
By convention, data are always in the last column,
and indexes for the one allowable random factor must be in the first.
Data can be real numbers or integers.
Indexes can be any character string,
allowing mnemonic labels to simplify reading the output.
For example:
.ce
fred  3  hard  10
indicates that "fred" at level "3" of the factor indexed by column two
and at level "hard" of the factor indexed by column three, scored 10.
Indexes and data on a line can be separated by tabs or spaces for readability.
Data from an experiment consists of a series of lines like the one above.
The order of these lines does not matter, so additional data can
simply be appended to the end of a file.
Replications are coded by having more than one line
with the same list of leading indexes.

With this information, 
.I anova
determines the number of factors,
and the number and names of levels of each factor.
.I anova
also figures out whether a factor is between groups or within
groups so that it can correctly choose error terms for F-ratios.

Optionally, names of independent and dependent variables can
be specified to 
.I anova,
providing mnemonic labels for the output.
These names can be quite long but will be truncated in the output.
The names should have unique first characters because that is all
that is used in the F tables.
For example, in a three factor design, the call to 
.I anova:
.ce
anova subjects group difficulty errors
would give the name "subjects" to the random factor,
"group" and "difficulty" to the next two,
and "errors" to the dependent variable.
If names are not specified, the default name for the random factor
is RANDOM, for the dependent variable, DATA, and for the independent
variables, A, B, C, D, etc.

.I "Output Format."
The output from 
.I anova
consists of cell means and standard deviations
for each source not involving the random factor,
a summary of design information, and an F table.
Sums of squares, degrees of freedom, mean squares, F ratio and
significance level are all reported for each F test.
.ne 1i
.SH "LIMITATIONS AND DIAGNOSTICS
.PP
The maximum number of factors 
.I anova
allows is ten, and the maximum
number of levels of factors is 100 (on some versions, 500),
although both these
limitations can be changed by resetting just one parameter
and recompiling.
.PP
.I anova
checks its input to make sure that data are numerical,
and that each datum is preceded by a consistent number of indexes
(if not,
.I anova
will complain about "Ragged input").
.PP
.I anova
will not print its F tables if it cannot make sense out
of the the input specification
("Unbalanced factor or Empty cell").
This can happen if there is missing data
and is detected when the cell sizes of all the scores
for a source do not add up to the expected grand total,
the number of levels of the data column reported with the design information.
(This number is equal to the product of all the within subjects factors,
and the number of levels of the random factor.)
Unbalanced factors often are due to a typographical error,
but the empty cell size message can be due to an illegal nested design
(only the random factor can be nested).
.PP
.I anova
uses a temporary file to store its input,
will complain if it is unable to create it
("Can't open temporary file").
This is usually because you are in some other user's directory
that is "write protected."
.I anova
has to allocate space to compute means and for some designs
it will be unable due to lack of available memory
.SH EXAMPLE
.PP
Suppose you have done an experiment in which the two
factors of interest were difficulty of material to be learned,
and amount of knowledge a person brings with him or her
into the experiment.
Each person is given two learning tasks, one easy and one hard,
so task difficulty is a within groups factor.
Three people are experts in the task domain you have chosen,
while four are novices, so knowledge is a between groups factor
with unequal group sizes.
The dependent variable is the amount of time it takes a person
to correctly work through a problem.
Data is formatted as follows:
in column one is the name of the person;
in column two is the level of the difficulty factor;
in column three is the level of the knowledge factor;
and in column four is the time, in seconds, to solve the problem.
A set of fictitious data follow.
.nf
.in +.5i
.ta .75i +.75i +1iR
lucy	easy	novice	12
lucy	hard	novice	22
fred	easy	novice	15
fred	hard	novice	16
ethel	easy	novice	10
ethel	hard	novice	15
ricky	easy	novice	25
ricky	hard	novice	30
ernie	easy	expert	7
ernie	hard	expert	10
bert	easy	expert	12
bert	hard	expert	18
bigbird	easy	expert	15
bigbird	hard	expert	14
.in
.fi

The call to 
.I anova
to analyze the data would probably look like:

.ce
anova subjects difficulty knowledge time < data

"data" being the name of the file containing the above data.
Notice that subjects is the random factor so indexes for that
factor appear in the first column.
Data, here called "time", must appear in the last column.
You can see that "difficulty" is a within groups factor
because each person appears at every level of that factor.
In the third column are indexes for "knowledge", the between groups
factor, so identified because no person appears at more than one
level of that factor.
.SH FILES
/tmp/anova.????
.SH ALGORITHM
The program is based on a method of analysis discussed by
Keppel (1973) in
.I "Design and Analysis: A Researcher's Handbook."
.SH SEE\ ALSO
unixstat(1), abut(1)
.SH AUTHOR
Gary Perlman
.br
The input format was designed with Jay McClelland at UCSD.
.SH BUGS
Huge designs, such as those with hundreds of levels of the random factor,
cause some rounding errors to creep in.
This only seems to be true when the effects of a factor
are negligible, so it appears this is not a practical problem.

With UNequal cell sizes, the sums of squares can have large
rounding errors, sometimes causing strange results like
negative sums of squares.
No such problem has ever been encountered for equal cell designs.
.SH KEYWORDS
inferential statistics, data analysis
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