PACKAGE |STAT Data Manipulation and Analysis, by Gary Perlman
NAME anova - multi-factor analysis of variance
SYNOPSIS anova [-p] [-w width] [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 first part of the output from anova includes summary statistics and optional error bar plots for each source not involving the random factor. The summary statistics include: cell counts, means, standard deviations, and standard errors. The error bar plots place cell means between the grand minimum and grand maximum values and use the following characters to identify statistics, which are plotted in a specific order:

Example Plot: 

<    -----(--#--)----- >
Key: 
-   spanning one standard deviation around the mean,
    but within the minimum and maximum
(   one standard error below the mean
)   one standard error above the mean
<   minimum value
>   maximum value
#   mean value
With small plots and/or low variances the markers may overlap, some of the markers may be hidden by others. In particular, with no variability, only the mean will appear. A useful rule of thumb is that if the standard error envelopes of two means do not overlap, then they are significantly different, however, a post hoc test should be applied to verify this.

After the summary statistics and plots, a summary of design information and an F table testing main effects and interactions follow. Sums of squares, degrees of freedom, mean squares, F ratio and significance level are reported for each F test. The labels for interactions are based on the first character of the factor names involved, so it is wise to choose factor names with different first letters. For significance testing, one asterisk indicates a result significant at the .05 level, two *'s indicate .01, and three *'s indicate .001.

OPTIONS
-p
Show Error Bar Plots for N-Way Tests. Each use of the -p option will request plots for higher-order interactions. The first will request plots for main effects, the next will request them for two-way interactions, and so on.
-w
Width of Error Bar Plots. This option will let you increase the amount of detail in the plots.
The following standard help options are supported. The program exits after displaying the help.
-L
Display limits
-O
Display options and values
-V
Display version number and date
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.
SEE ALSO regress for multiple regression.
oneway for unpaired comparisons for a single factor.
pair for simple omparisons of paired data.
rankrel for analysis of related groups rank ordinal data.
rankind for paired analysis of independent groups rank ordinal data.
contab for multi-factor contingency table analysis.
UPDATED August 22, 1992