ANOVA(1)

NAME
anova - multi-factor analysis of variance

SYNOPSIS
anova [factor names]

DESCRIPTION
anova does multi-factor analysis of variance on designs with within groups factors, between groups factors, or both. 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.

Input Format. The input format was designed so that when the user specifies the role individual data play in the overall design, anova figures out the experimental design. This helps reduce design specification errors. The input to anova consists of each datum on a separate line, preceded by a list of index labels, 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, so mnemonic labels can simplify reading the output. For example:
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 be appended to the end of files. Replications are coded by having more than one line with the same list of leading indexes. With this information, anova determines the number of factors, the number and names of levels of each factor, and whether a factor is between groups or within groups so that error terms for F-ratios can be chosen.

Names of independent and dependent variables can be supplied to anova, providing mnemonic labels for the output. These names may be truncated in the output. The names should have unique first characters because that is all that is used in parts of F tables. For example, in a three factor design, the call to anova:
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.

Output Format. The output from anova includes cell counts, means, standard deviations, and standard errors for each source not involving the random factor, a summary of design information, and an F table testing main effects and interactions. Sums of squares, degrees of freedom, mean squares, F ratio and significance level are reported for each F test.

DIAGNOSTICS

anova will complain about "Ragged input" if the number of variables in its input varies. 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 are missing data (detected when the cell sizes of all the scores for a source do not add up to the expected grand total). 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).

anova uses a temporary file to store its input and will complain if it is unable to create it. This may be because you are in some other user's directory that is "write protected."

EXAMPLE

An experiment has two experimental factors: difficulty of material to be learned, and amount of knowledge a person brings with him or her. (This design is due to Naomi Miyake.) Each person is given two learning tasks, one easy and one hard, so task difficulty is a within groups factor. Two people are experts in the task domain, while three 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 (the random factor); 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. Fictitious data follow.

	lucy	easy	novice	12
	lucy	hard	novice	22
	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

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

anova subjects difficulty knowledge time < data

"data" is the name of the file containing the above data. "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. "difficulty" is a within groups factor because each person appears at every level of that factor. In the third column are indexes for "knowledge", a between groups factor, because no person appears at more than one level of that factor.

FILES
UNIX	/tmp/anova.????
MSDOS	anova.tmp

ALGORITHM
Keppel (1973) Design and Analysis: A Researcher's Handbook.

WARNING
When unequal sized cell designs are used, the cell sizes must be in the same proportion across all rows and columns of interactions, or there may be marked distortions and the analysis may be invalid. This applies only to designs with more than one between groups factor. See Keppel's discussion of unequal cell designs.

LIMITS
Use the -L option to determine the program limits.

MISSING VALUES
Missing data values (NA) are counted but not included in the analysis.