ANOVA(1)                           |STAT                   January 29, 1987

NAME
     anova - multi-factor analysis of variance

SYNOPSIS
     anova [factor names]

DESCRIPTION
     _a_n_o_v_a does multi-factor analysis of variance on designs with within
     groups factors, between groups factors, or both.  _a_n_o_v_a 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_n_p_u_t _F_o_r_m_a_t. The input format was designed so that when the user
     specifies the role individual data play in the overall design, _a_n_o_v_a
     figures out the experimental design.  This helps reduce design
     specification errors.  The input to _a_n_o_v_a 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, _a_n_o_v_a 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 _a_n_o_v_a,
     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 _a_n_o_v_a:
                   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.

     _O_u_t_p_u_t _F_o_r_m_a_t. The output from _a_n_o_v_a 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
     _a_n_o_v_a will complain about "Ragged input" if the number of variables in
     its input varies.  _a_n_o_v_a 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).

     _a_n_o_v_a 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 _a_n_o_v_a 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) _D_e_s_i_g_n _a_n_d _A_n_a_l_y_s_i_s: _A _R_e_s_e_a_r_c_h_e_r'_s _H_a_n_d_b_o_o_k.

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.
