[pspp] / doc / statistics.texi

@node Statistics
@chapter Statistics

This chapter documents the statistical procedures that @pspp{} supports so
far.

@menu
* DESCRIPTIVES::                Descriptive statistics.
* FREQUENCIES::                 Frequency tables.
* EXAMINE::                     Testing data for normality.
* CORRELATIONS::                Correlation tables.
* CROSSTABS::                   Crosstabulation tables.
* FACTOR::                      Factor analysis and Principal Components analysis
* MEANS::                       Average values and other statistics.
* NPAR TESTS::                  Nonparametric tests.
* T-TEST::                      Test hypotheses about means.
* ONEWAY::                      One way analysis of variance.
* QUICK CLUSTER::               K-Means clustering.
* RANK::                        Compute rank scores.
* REGRESSION::                  Linear regression.
* RELIABILITY::                 Reliability analysis.
* ROC::                         Receiver Operating Characteristic.
@end menu

@node DESCRIPTIVES
@section DESCRIPTIVES

@vindex DESCRIPTIVES
@display
DESCRIPTIVES
        /VARIABLES=var_list
        /MISSING=@{VARIABLE,LISTWISE@} @{INCLUDE,NOINCLUDE@}
        /FORMAT=@{LABELS,NOLABELS@} @{NOINDEX,INDEX@} @{LINE,SERIAL@}
        /SAVE
        /STATISTICS=@{ALL,MEAN,SEMEAN,STDDEV,VARIANCE,KURTOSIS,
                     SKEWNESS,RANGE,MINIMUM,MAXIMUM,SUM,DEFAULT,
                     SESKEWNESS,SEKURTOSIS@}
        /SORT=@{NONE,MEAN,SEMEAN,STDDEV,VARIANCE,KURTOSIS,SKEWNESS,
               RANGE,MINIMUM,MAXIMUM,SUM,SESKEWNESS,SEKURTOSIS,NAME@}
              @{A,D@}
@end display

The @cmd{DESCRIPTIVES} procedure reads the active dataset and outputs
descriptive
statistics requested by the user.  In addition, it can optionally
compute Z-scores.

The VARIABLES subcommand, which is required, specifies the list of
variables to be analyzed.  Keyword VARIABLES is optional.

All other subcommands are optional:

The MISSING subcommand determines the handling of missing variables.  If
INCLUDE is set, then user-missing values are included in the
calculations.  If NOINCLUDE is set, which is the default, user-missing
values are excluded.  If VARIABLE is set, then missing values are
excluded on a variable by variable basis; if LISTWISE is set, then
the entire case is excluded whenever any value in that case has a
system-missing or, if INCLUDE is set, user-missing value.

The FORMAT subcommand affects the output format.  Currently the
LABELS/NOLABELS and NOINDEX/INDEX settings are not used.  When SERIAL is
set, both valid and missing number of cases are listed in the output;
when NOSERIAL is set, only valid cases are listed.

The SAVE subcommand causes @cmd{DESCRIPTIVES} to calculate Z scores for all
the specified variables.  The Z scores are saved to new variables.
Variable names are generated by trying first the original variable name
with Z prepended and truncated to a maximum of 8 characters, then the
names ZSC000 through ZSC999, STDZ00 through STDZ09, ZZZZ00 through
ZZZZ09, ZQZQ00 through ZQZQ09, in that sequence.  In addition, Z score
variable names can be specified explicitly on VARIABLES in the variable
list by enclosing them in parentheses after each variable.

The STATISTICS subcommand specifies the statistics to be displayed:

@table @code
@item ALL
All of the statistics below.
@item MEAN
Arithmetic mean.
@item SEMEAN
Standard error of the mean.
@item STDDEV
Standard deviation.
@item VARIANCE
Variance.
@item KURTOSIS
Kurtosis and standard error of the kurtosis.
@item SKEWNESS
Skewness and standard error of the skewness.
@item RANGE
Range.
@item MINIMUM
Minimum value.
@item MAXIMUM
Maximum value.
@item SUM
Sum.
@item DEFAULT
Mean, standard deviation of the mean, minimum, maximum.
@item SEKURTOSIS
Standard error of the kurtosis.
@item SESKEWNESS
Standard error of the skewness.
@end table

The SORT subcommand specifies how the statistics should be sorted.  Most
of the possible values should be self-explanatory.  NAME causes the
statistics to be sorted by name.  By default, the statistics are listed
in the order that they are specified on the VARIABLES subcommand.  The A
and D settings request an ascending or descending sort order,
respectively.

@node FREQUENCIES
@section FREQUENCIES

@vindex FREQUENCIES
@display
FREQUENCIES
        /VARIABLES=var_list
        /FORMAT=@{TABLE,NOTABLE,LIMIT(limit)@}
                @{AVALUE,DVALUE,AFREQ,DFREQ@}
        /MISSING=@{EXCLUDE,INCLUDE@}
        /STATISTICS=@{DEFAULT,MEAN,SEMEAN,MEDIAN,MODE,STDDEV,VARIANCE,
                     KURTOSIS,SKEWNESS,RANGE,MINIMUM,MAXIMUM,SUM,
                     SESKEWNESS,SEKURTOSIS,ALL,NONE@}
        /NTILES=ntiles
        /PERCENTILES=percent@dots{}
        /HISTOGRAM=[MINIMUM(x_min)] [MAXIMUM(x_max)] 
                   [@{FREQ[(y_max)],PERCENT[(y_max)]@}] [@{NONORMAL,NORMAL@}]
        /PIECHART=[MINIMUM(x_min)] [MAXIMUM(x_max)]
                  [@{FREQ,PERCENT@}] [@{NOMISSING,MISSING@}]

(These options are not currently implemented.)
        /BARCHART=@dots{}
        /HBAR=@dots{}
        /GROUPED=@dots{}
@end display

The @cmd{FREQUENCIES} procedure outputs frequency tables for specified
variables.
@cmd{FREQUENCIES} can also calculate and display descriptive statistics
(including median and mode) and percentiles,
@cmd{FREQUENCIES} can also output
histograms and pie charts.  

The VARIABLES subcommand is the only required subcommand.  Specify the
variables to be analyzed.

The FORMAT subcommand controls the output format.  It has several
possible settings:  

@itemize @bullet
@item
TABLE, the default, causes a frequency table to be output for every
variable specified.  NOTABLE prevents them from being output.  LIMIT
with a numeric argument causes them to be output except when there are
more than the specified number of values in the table.

@item
Normally frequency tables are sorted in ascending order by value.  This
is AVALUE.  DVALUE tables are sorted in descending order by value.
AFREQ and DFREQ tables are sorted in ascending and descending order,
respectively, by frequency count.
@end itemize

The MISSING subcommand controls the handling of user-missing values.
When EXCLUDE, the default, is set, user-missing values are not included
in frequency tables or statistics.  When INCLUDE is set, user-missing
are included.  System-missing values are never included in statistics,
but are listed in frequency tables.

The available STATISTICS are the same as available in @cmd{DESCRIPTIVES}
(@pxref{DESCRIPTIVES}), with the addition of MEDIAN, the data's median
value, and MODE, the mode.  (If there are multiple modes, the smallest
value is reported.)  By default, the mean, standard deviation of the
mean, minimum, and maximum are reported for each variable.

@cindex percentiles
PERCENTILES causes the specified percentiles to be reported.
The percentiles should  be presented at a list of numbers between 0
and 100 inclusive.  
The NTILES subcommand causes the percentiles to be reported at the
boundaries of the data set divided into the specified number of ranges.
For instance, @code{/NTILES=4} would cause quartiles to be reported.

@cindex histogram
The HISTOGRAM subcommand causes the output to include a histogram for
each specified numeric variable.  The X axis by default ranges from
the minimum to the maximum value observed in the data, but the MINIMUM
and MAXIMUM keywords can set an explicit range.  Specify NORMAL to
superimpose a normal curve on the histogram.  Histograms are not
created for string variables.

@cindex piechart
The PIECHART adds a pie chart for each variable to the data.  Each
slice represents one value, with the size of the slice proportional to
the value's frequency.  By default, all non-missing values are given
slices.  The MINIMUM and MAXIMUM keywords can be used to limit the
displayed slices to a given range of values.  The MISSING keyword adds
slices for missing values.

The FREQ and PERCENT options on HISTOGRAM and PIECHART are accepted
but not currently honored.

@node EXAMINE
@section EXAMINE

@vindex EXAMINE
@cindex Exploratory data analysis
@cindex Normality, testing for

@display
EXAMINE
        VARIABLES= @var{var1} [@var{var2}] @dots{} [@var{varN}]
           [BY @var{factor1} [BY @var{subfactor1}]
             [ @var{factor2} [BY @var{subfactor2}]]
             @dots{}
             [ @var{factor3} [BY @var{subfactor3}]]
            ]
        /STATISTICS=@{DESCRIPTIVES, EXTREME[(@var{n})], ALL, NONE@}
        /PLOT=@{BOXPLOT, NPPLOT, HISTOGRAM, ALL, NONE@}
        /CINTERVAL @var{p}
        /COMPARE=@{GROUPS,VARIABLES@}
        /ID=@var{identity_variable}
        /@{TOTAL,NOTOTAL@}
        /PERCENTILE=[@var{percentiles}]=@{HAVERAGE, WAVERAGE, ROUND, AEMPIRICAL, EMPIRICAL @}
        /MISSING=@{LISTWISE, PAIRWISE@} [@{EXCLUDE, INCLUDE@}] 
                [@{NOREPORT,REPORT@}]

@end display

The @cmd{EXAMINE} command is used to perform exploratory data analysis.
In particular, it is useful for testing how closely a distribution follows a
normal distribution, and for finding outliers and extreme values.

The VARIABLES subcommand is mandatory.  
It specifies the dependent variables and optionally variables to use as
factors for the analysis.
Variables listed before the first BY keyword (if any) are the 
dependent variables.
The dependent variables may optionally be followed by a list of
factors which tell @pspp{} how to break down the analysis for each
dependent variable. 

Following the dependent variables, factors may be specified.
The factors (if desired) should be preceeded by a single BY keyword.
The format for each factor is 
@display
@var{factorvar} [BY @var{subfactorvar}].
@end display
Each unique combination of the values of  @var{factorvar} and
@var{subfactorvar} divide the dataset into @dfn{cells}.
Statistics will be calculated for each cell
and for the entire dataset (unless NOTOTAL is given).

The STATISTICS subcommand specifies which statistics to show.
DESCRIPTIVES will produce a table showing some parametric and
non-parametrics statistics.
EXTREME produces a table showing the extremities of each cell.
A number in parentheses, @var{n} determines
how many upper and lower extremities to show.
The default number is 5.

The subcommands TOTAL and NOTOTAL are mutually exclusive.
If TOTAL appears, then statistics will be produced for the entire dataset
as well as for each cell.
If NOTOTAL appears, then statistics will be produced only for the cells
(unless no factor variables have been given).
These subcommands have no effect if there have  been no factor variables
specified.

@cindex boxplot
@cindex histogram
@cindex npplot
The PLOT subcommand specifies which plots are to be produced if any.
Available plots are HISTOGRAM, NPPLOT and BOXPLOT.
They can all be used to visualise how closely each cell conforms to a 
normal distribution.
Boxplots will also show you the outliers and extreme values.

The COMPARE subcommand is only relevant if producing boxplots, and it is only 
useful there is more than one dependent variable and at least one factor.
If 
/COMPARE=GROUPS is specified, then one plot per dependent variable is produced,
each of which contain boxplots for all the cells.
If /COMPARE=VARIABLES is specified, then one plot per cell is produced,
each containing one boxplot per dependent variable.
If the /COMPARE subcommand is omitted, then @pspp{} behaves as if
/COMPARE=GROUPS were given.
 
The ID subcommand is relevant only if /PLOT=BOXPLOT or 
/STATISTICS=EXTREME has been given.
If given, it shoule provide the name of a variable which is to be used
to labels extreme values and outliers.
Numeric or string variables are permissible.  
If the ID subcommand is not given, then the casenumber will be used for
labelling.

The CINTERVAL subcommand specifies the confidence interval to use in
calculation of the descriptives command.  The default is 95%.

@cindex percentiles
The PERCENTILES subcommand specifies which percentiles are to be calculated, 
and which algorithm to use for calculating them.  The default is to
calculate the 5, 10, 25, 50, 75, 90, 95 percentiles using the
HAVERAGE algorithm.

The TOTAL and NOTOTAL subcommands are mutually exclusive.  If NOTOTAL
is given and factors have been specified in the VARIABLES subcommand,
then then statistics for the unfactored dependent variables are
produced in addition to the factored variables.  If there are no
factors specified then TOTAL and NOTOTAL have no effect.


The following example will generate descriptive statistics and histograms for
two variables @var{score1} and @var{score2}.
Two factors are given, @i{viz}: @var{gender} and @var{gender} BY @var{culture}.
Therefore, the descriptives and histograms will be generated for each
distinct  value
of @var{gender} @emph{and} for each distinct combination of the values
of @var{gender} and @var{race}.
Since the NOTOTAL keyword is given, statistics and histograms for 
@var{score1} and @var{score2} covering the  whole dataset are not produced.
@example
EXAMINE @var{score1} @var{score2} BY 
        @var{gender}
        @var{gender} BY @var{culture}
        /STATISTICS = DESCRIPTIVES
        /PLOT = HISTOGRAM
        /NOTOTAL.
@end example

Here is a second example showing how the @cmd{examine} command can be used to find extremities.
@example
EXAMINE @var{height} @var{weight} BY 
        @var{gender}
        /STATISTICS = EXTREME (3)
        /PLOT = BOXPLOT
        /COMPARE = GROUPS
        /ID = @var{name}.
@end example
In this example, we look at the height and weight of a sample of individuals and
how they differ between male and female.
A table showing the 3 largest and the 3 smallest values of @var{height} and 
@var{weight} for each gender, and for the whole dataset will be shown.
Boxplots will also be produced.
Because /COMPARE = GROUPS was given, boxplots for male and female will be
shown in the same graphic, allowing us to easily see the difference between
the genders.
Since the variable @var{name} was specified on the ID subcommand, this will be
used to label the extreme values.

@strong{Warning!}
If many dependent variables are specified, or if factor variables are
specified for which
there are many distinct values, then @cmd{EXAMINE} will produce a very
large quantity of output.

@node CORRELATIONS
@section CORRELATIONS

@vindex CORRELATIONS
@display
CORRELATIONS
     /VARIABLES = varlist [ WITH varlist ]
     [
      .
      .
      .
      /VARIABLES = varlist [ WITH varlist ]
      /VARIABLES = varlist [ WITH varlist ]
     ]

     [ /PRINT=@{TWOTAIL, ONETAIL@} @{SIG, NOSIG@} ]
     [ /STATISTICS=DESCRIPTIVES XPROD ALL]
     [ /MISSING=@{PAIRWISE, LISTWISE@} @{INCLUDE, EXCLUDE@} ]
@end display    

@cindex correlation
The @cmd{CORRELATIONS} procedure produces tables of the Pearson correlation coefficient
for a set of variables.  The significance of the coefficients are also given.

At least one VARIABLES subcommand is required. If the WITH keyword is used, then a non-square
correlation table will be produced.
The variables preceding WITH, will be used as the rows of the table, and the variables following
will be the columns of the table.
If no WITH subcommand is given, then a square, symmetrical table using all variables is produced.


The @cmd{MISSING} subcommand determines the handling of missing variables.  
If INCLUDE is set, then user-missing values are included in the
calculations, but system-missing values are not.
If EXCLUDE is set, which is the default, user-missing
values are excluded as well as system-missing values. 
This is the default.

If LISTWISE is set, then the entire case is excluded from analysis
whenever any variable  specified in any @cmd{/VARIABLES} subcommand
contains a missing value.   
If PAIRWISE is set, then a case is considered missing only if either of the
values  for the particular coefficient are missing.
The default is PAIRWISE.

The PRINT subcommand is used to control how the reported significance values are printed.
If the TWOTAIL option is used, then a two-tailed test of significance is 
printed.  If the ONETAIL option is given, then a one-tailed test is used.
The default is TWOTAIL.

If the NOSIG option is specified, then correlation coefficients with significance less than
0.05 are highlighted.
If SIG is specified, then no highlighting is performed.  This is the default.

@cindex covariance
The STATISTICS subcommand requests additional statistics to be displayed.  The keyword 
DESCRIPTIVES requests that the mean, number of non-missing cases, and the non-biased
estimator of the standard deviation are displayed.
These statistics will be displayed in a separated table, for all the variables listed
in any /VARIABLES subcommand.
The XPROD keyword requests cross-product deviations and covariance estimators to 
be displayed for each pair of variables.
The keyword ALL is the union of DESCRIPTIVES and XPROD.

@node CROSSTABS
@section CROSSTABS

@vindex CROSSTABS
@display
CROSSTABS
        /TABLES=var_list BY var_list [BY var_list]@dots{}
        /MISSING=@{TABLE,INCLUDE,REPORT@}
        /WRITE=@{NONE,CELLS,ALL@}
        /FORMAT=@{TABLES,NOTABLES@}
                @{PIVOT,NOPIVOT@}
                @{AVALUE,DVALUE@}
                @{NOINDEX,INDEX@}
                @{BOX,NOBOX@}
        /CELLS=@{COUNT,ROW,COLUMN,TOTAL,EXPECTED,RESIDUAL,SRESIDUAL,
                ASRESIDUAL,ALL,NONE@}
        /STATISTICS=@{CHISQ,PHI,CC,LAMBDA,UC,BTAU,CTAU,RISK,GAMMA,D,
                     KAPPA,ETA,CORR,ALL,NONE@}
        
(Integer mode.)
        /VARIABLES=var_list (low,high)@dots{}
@end display

The @cmd{CROSSTABS} procedure displays crosstabulation
tables requested by the user.  It can calculate several statistics for
each cell in the crosstabulation tables.  In addition, a number of
statistics can be calculated for each table itself.

The TABLES subcommand is used to specify the tables to be reported.  Any
number of dimensions is permitted, and any number of variables per
dimension is allowed.  The TABLES subcommand may be repeated as many
times as needed.  This is the only required subcommand in @dfn{general
mode}.  

Occasionally, one may want to invoke a special mode called @dfn{integer
mode}.  Normally, in general mode, @pspp{} automatically determines
what values occur in the data.  In integer mode, the user specifies the
range of values that the data assumes.  To invoke this mode, specify the
VARIABLES subcommand, giving a range of data values in parentheses for
each variable to be used on the TABLES subcommand.  Data values inside
the range are truncated to the nearest integer, then assigned to that
value.  If values occur outside this range, they are discarded.  When it
is present, the VARIABLES subcommand must precede the TABLES
subcommand.

In general mode, numeric and string variables may be specified on
TABLES.  In integer mode, only numeric variables are allowed.

The MISSING subcommand determines the handling of user-missing values.
When set to TABLE, the default, missing values are dropped on a table by
table basis.  When set to INCLUDE, user-missing values are included in
tables and statistics.  When set to REPORT, which is allowed only in
integer mode, user-missing values are included in tables but marked with
an @samp{M} (for ``missing'') and excluded from statistical
calculations.

Currently the WRITE subcommand is ignored.

The FORMAT subcommand controls the characteristics of the
crosstabulation tables to be displayed.  It has a number of possible
settings:

@itemize @bullet
@item
TABLES, the default, causes crosstabulation tables to be output.
NOTABLES suppresses them.

@item
PIVOT, the default, causes each TABLES subcommand to be displayed in a
pivot table format.  NOPIVOT causes the old-style crosstabulation format
to be used.

@item
AVALUE, the default, causes values to be sorted in ascending order.
DVALUE asserts a descending sort order.

@item
INDEX/NOINDEX is currently ignored.

@item
BOX/NOBOX is currently ignored.
@end itemize

The CELLS subcommand controls the contents of each cell in the displayed
crosstabulation table.  The possible settings are:

@table @asis
@item COUNT
Frequency count.
@item ROW
Row percent.
@item COLUMN
Column percent.
@item TOTAL
Table percent.
@item EXPECTED
Expected value.
@item RESIDUAL 
Residual.
@item SRESIDUAL
Standardized residual.
@item ASRESIDUAL
Adjusted standardized residual.
@item ALL
All of the above.
@item NONE
Suppress cells entirely.
@end table

@samp{/CELLS} without any settings specified requests COUNT, ROW,
COLUMN, and TOTAL.  If CELLS is not specified at all then only COUNT
will be selected.

The STATISTICS subcommand selects statistics for computation:

@table @asis
@item CHISQ
@cindex chisquare
@cindex chi-square

Pearson chi-square, likelihood ratio, Fisher's exact test, continuity
correction, linear-by-linear association.
@item PHI
Phi.
@item CC
Contingency coefficient.
@item LAMBDA
Lambda.
@item UC
Uncertainty coefficient.
@item BTAU
Tau-b.
@item CTAU
Tau-c.
@item RISK
Risk estimate.
@item GAMMA
Gamma.
@item D
Somers' D.
@item KAPPA
Cohen's Kappa.
@item ETA
Eta.
@item CORR
Spearman correlation, Pearson's r.
@item ALL
All of the above.
@item NONE
No statistics.
@end table

Selected statistics are only calculated when appropriate for the
statistic.  Certain statistics require tables of a particular size, and
some statistics are calculated only in integer mode.

@samp{/STATISTICS} without any settings selects CHISQ.  If the
STATISTICS subcommand is not given, no statistics are calculated.

@strong{Please note:} Currently the implementation of CROSSTABS has the
followings bugs:

@itemize @bullet
@item
Pearson's R (but not Spearman) is off a little.
@item
T values for Spearman's R and Pearson's R are wrong.
@item
Significance of symmetric and directional measures is not calculated.
@item
Asymmetric ASEs and T values for lambda are wrong.
@item
ASE of Goodman and Kruskal's tau is not calculated.
@item
ASE of symmetric somers' d is wrong.
@item
Approximate T of uncertainty coefficient is wrong.
@end itemize

Fixes for any of these deficiencies would be welcomed.

@node FACTOR
@section FACTOR

@vindex FACTOR
@cindex factor analysis
@cindex principal components analysis
@cindex principal axis factoring
@cindex data reduction

@display
FACTOR  VARIABLES=var_list

        [ /METHOD = @{CORRELATION, COVARIANCE@} ]

        [ /EXTRACTION=@{PC, PAF@}] 

        [ /ROTATION=@{VARIMAX, EQUAMAX, QUARTIMAX, NOROTATE@}]

        [ /PRINT=[INITIAL] [EXTRACTION] [ROTATION] [UNIVARIATE] [CORRELATION] [COVARIANCE] [DET] [KMO] [SIG] [ALL] [DEFAULT] ]

        [ /PLOT=[EIGEN] ]

        [ /FORMAT=[SORT] [BLANK(@var{n})] [DEFAULT] ]

        [ /CRITERIA=[FACTORS(@var{n})] [MINEIGEN(@var{l})] [ITERATE(@var{m})] [ECONVERGE (@var{delta})] [DEFAULT] ]

        [ /MISSING=[@{LISTWISE, PAIRWISE@}] [@{INCLUDE, EXCLUDE@}] ]
@end display

The FACTOR command performs Factor Analysis or Principal Axis Factoring on a dataset.  It may be used to find
common factors in the data or for data reduction purposes.

The VARIABLES subcommand is required.  It lists the variables which are to partake in the analysis.

The /EXTRACTION subcommand is used to specify the way in which factors (components) are extracted from the data.
If PC is specified, then Principal Components Analysis is used.  If PAF is specified, then Principal Axis Factoring is
used. By default Principal Components Analysis will be used.

The /ROTATION subcommand is used to specify the method by which the extracted solution will be rotated.
Three methods are available: VARIMAX (which is the default), EQUAMAX, and QUARTIMAX.
If don't want any rotation to be performed, the word NOROTATE will prevent the command from performing any
rotation on the data. Oblique rotations are not supported.

The /METHOD subcommand should be used to determine whether the covariance matrix or the correlation matrix of the data is
to be analysed.  By default, the correlation matrix is analysed.

The /PRINT subcommand may be used to select which features of the analysis are reported:

@itemize
@item UNIVARIATE
      A table of mean values, standard deviations and total weights are printed.
@item INITIAL
      Initial communalities and eigenvalues are printed.
@item EXTRACTION
      Extracted communalities and eigenvalues are printed.
@item ROTATION
      Rotated communalities and eigenvalues are printed.
@item CORRELATION
      The correlation matrix is printed.
@item COVARIANCE
      The covariance matrix is printed.
@item DET
      The determinant of the correlation or covariance matrix is printed.
@item KMO
      The Kaiser-Meyer-Olkin measure of sampling adequacy and the Bartlett test of sphericity is printed.
@item SIG
      The significance of the elements of correlation matrix is printed.
@item ALL
      All of the above are printed.
@item DEFAULT
      Identical to INITIAL and EXTRACTION.
@end itemize

If /PLOT=EIGEN is given, then a ``Scree'' plot of the eigenvalues will be printed.  This can be useful for visualizing
which factors (components) should be retained.

The /FORMAT subcommand determined how data are to be displayed in loading matrices.  If SORT is specified, then the variables
are sorted in descending order of significance.  If BLANK(@var{n}) is specified, then coefficients whose absolute value is less
than @var{n} will not be printed.  If the keyword DEFAULT is given, or if no /FORMAT subcommand is given, then no sorting is 
performed, and all coefficients will be printed.

The /CRITERIA subcommand is used to specify how the number of extracted factors (components) are chosen.  If FACTORS(@var{n}) is
specified, where @var{n} is an integer, then @var{n} factors will be extracted.  Otherwise, the MINEIGEN setting will
be used.  MINEIGEN(@var{l}) requests that all factors whose eigenvalues are greater than or equal to @var{l} are extracted.
The default value of @var{l} is 1.    The ECONVERGE and ITERATE settings have effect only when iterative algorithms for factor
extraction (such as Principal Axis Factoring) are used.   ECONVERGE(@var{delta}) specifies that iteration should cease when
the maximum absolute value of the communality estimate between one iteration and the previous is less than @var{delta}. The
default value of @var{delta} is 0.001.
The ITERATE(@var{m}) setting sets the maximum number of iterations to @var{m}.  The default value of @var{m} is 25.

The @cmd{MISSING} subcommand determines the handling of missing variables.  
If INCLUDE is set, then user-missing values are included in the
calculations, but system-missing values are not.
If EXCLUDE is set, which is the default, user-missing
values are excluded as well as system-missing values. 
This is the default.
If LISTWISE is set, then the entire case is excluded from analysis
whenever any variable  specified in the @cmd{VARIABLES} subcommand
contains a missing value.   
If PAIRWISE is set, then a case is considered missing only if either of the
values  for the particular coefficient are missing.
The default is LISTWISE.

@node MEANS
@section MEANS

@vindex MEANS
@cindex means

@display 
MEANS [TABLES =] 
      @{varlist@} 
        [ BY @{varlist@} [BY @{varlist@} [BY @{varlist@} @dots{} ]]]

      [ /@{varlist@} 
         [ BY @{varlist@} [BY @{varlist@} [BY @{varlist@} @dots{} ]]] ]

      [/CELLS = [MEAN] [COUNT] [STDDEV] [SEMEAN] [SUM] [MIN] [MAX] [RANGE]
        [VARIANCE] [KURT] [SEKURT] 
        [SKEW] [SESKEW] [FIRST] [LAST] 
        [HARMONIC] [GEOMETRIC] 
        [DEFAULT]
        [ALL]
        [NONE] ]

      [/MISSING = [TABLE] [INCLUDE] [DEPENDENT]]
@end display 

You can use the @cmd{MEANS} command to calculate the arithmetic mean and similar
statistics, either for the dataset as a whole or for categories of data.

The simplest form of the command is
@example
MEANS @var{v}.
@end example
@noindent which calculates the mean, count and standard deviation for @var{v}.
If you specify a grouping variable, for example
@example
MEANS @var{v} BY @var{g}.
@end example
@noindent then the means, counts and standard deviations for @var{v} after having
been grouped by @var{g} will be calculated.
Instead of the mean, count and standard deviation, you could specify the statistics
in which you are interested:
@example
MEANS @var{x} @var{y} BY @var{g}
      /CELLS = HARMONIC SUM MIN.
@end example
This example calculates the harmonic mean, the sum and the minimum values of @var{x} and @var{y}
grouped by @var{g}.

The CELLS subcommand specifies which statistics to calculate.  The available statistics
are:
@itemize
@item MEAN
@cindex arithmetic mean
      The arithmetic mean.
@item COUNT
      The count of the values.
@item STDDEV
      The standard deviation.
@item SEMEAN
      The standard error of the mean.
@item SUM
      The sum of the values.
@item MIN
      The minimum value.
@item MAX
      The maximum value.
@item RANGE
      The difference between the maximum and minimum values.
@item VARIANCE
      The variance.
@item FIRST
      The first value in the category.
@item LAST
      The last value in the category.
@item SKEW
      The skewness.
@item SESKEW
      The standard error of the skewness.
@item KURT
      The kurtosis
@item SEKURT
      The standard error of the kurtosis.
@item HARMONIC
@cindex harmonic mean
      The harmonic mean.
@item GEOMETRIC
@cindex geometric mean
      The geometric mean.
@end itemize

In addition, three special keywords are recognized:
@itemize
@item DEFAULT
      This is the same as MEAN COUNT STDDEV
@item ALL
      All of the above statistics will be calculated.
@item NONE
      No statistics will be calculated (only a summary will be shown).
@end itemize


More than one @dfn{table} can be specified in a single command. 
Each table is separated by a @samp{/}. For
example
@example
MEANS TABLES =
      @var{c} @var{d} @var{e} BY @var{x}
      /@var{a} @var{b} BY @var{x} @var{y}
      /@var{f} BY @var{y} BY @var{z}.
@end example
has three tables (the @samp{TABLE =} is optional).
The first table has three dependent variables @var{c}, @var{d} and @var{e}
and a single categorical variable @var{x}.
The second table has two dependent variables @var{a} and @var{b}, 
and two categorical variables @var{x} and @var{y}.
The third table has a single dependent variables @var{f}
and a categorical variable formed by the combination of @var{y} and @var{z}.


By default values are omitted from the analysis only if missing values
(either system missing or user missing)
for any of the variables directly involved in their calculation are 
encountered.
This behaviour can be modified with the  /MISSING subcommand.
Three options are possible: TABLE, INCLUDE and DEPENDENT.

/MISSING = TABLE causes cases to be dropped if any variable is missing 
in the table specification currently being processed, regardless of 
whether it is needed to calculate the statistic.

/MISSING = INCLUDE says that user missing values, either in the dependent
variables or in the categorical variables should be taken at their face
value, and not excluded.

/MISSING = DEPENDENT says that user missing values, in the dependent
variables should be taken at their face value, however cases which 
have user missing values for the categorical variables should be omitted 
from the calculation.

@node NPAR TESTS
@section NPAR TESTS

@vindex NPAR TESTS
@cindex nonparametric tests

@display 
NPAR TESTS
     
     nonparametric test subcommands
     .
     .
     .
     
     [ /STATISTICS=@{DESCRIPTIVES@} ]

     [ /MISSING=@{ANALYSIS, LISTWISE@} @{INCLUDE, EXCLUDE@} ]

     [ /METHOD=EXACT [ TIMER [(n)] ] ]
@end display

NPAR TESTS performs nonparametric tests. 
Non parametric tests make very few assumptions about the distribution of the 
data.
One or more tests may be specified by using the corresponding subcommand.
If the /STATISTICS subcommand is also specified, then summary statistics are 
produces for each variable that is the subject of any test.

Certain tests may take a long time to execute, if an exact figure is required.
Therefore, by default asymptotic approximations are used unless the
subcommand /METHOD=EXACT is specified.  
Exact tests give more accurate results, but may take an unacceptably long 
time to perform.  If the TIMER keyword is used, it sets a maximum time,
after which the test will be abandoned, and a warning message printed.
The time, in minutes, should be specified in parentheses after the TIMER keyword.
If the TIMER keyword is given without this figure, then a default value of 5 minutes 
is used.


@menu
* BINOMIAL::                Binomial Test
* CHISQUARE::               Chisquare Test
* COCHRAN::                 Cochran Q Test
* FRIEDMAN::                Friedman Test
* KENDALL::                 Kendall's W Test
* KOLMOGOROV-SMIRNOV::      Kolmogorov Smirnov Test
* KRUSKAL-WALLIS::          Kruskal-Wallis Test
* MANN-WHITNEY::            Mann Whitney U Test
* MCNEMAR::                 McNemar Test
* MEDIAN::                  Median Test
* RUNS::                    Runs Test
* SIGN::                    The Sign Test
* WILCOXON::                Wilcoxon Signed Ranks Test
@end menu


@node    BINOMIAL
@subsection Binomial test
@vindex BINOMIAL
@cindex binomial test

@display 
     [ /BINOMIAL[(p)]=var_list[(value1[, value2)] ] ]
@end display 

The /BINOMIAL subcommand compares the observed distribution of a dichotomous 
variable with that of a binomial distribution.
The variable @var{p} specifies the test proportion of the binomial 
distribution.  
The default value of 0.5 is assumed if @var{p} is omitted.

If a single value appears after the variable list, then that value is
used as the threshold to partition the observed values. Values less
than or equal to the threshold value form the first category.  Values
greater than the threshold form the second category. 

If two values appear after the variable list, then they will be used
as the values which a variable must take to be in the respective
category. 
Cases for which a variable takes a value equal to neither of the specified  
values, take no part in the test for that variable.

If no values appear, then the variable must assume dichotomous
values.
If more than two distinct, non-missing values for a variable
under test are encountered then an error occurs.

If the test proportion is equal to 0.5, then a two tailed test is
reported.   For any other test proportion, a one tailed test is
reported.   
For one tailed tests, if the test proportion is less than
or equal to the observed proportion, then the significance of
observing the observed proportion or more is reported.
If the test proportion is more than the observed proportion, then the
significance of observing the observed proportion or less is reported.
That is to say, the test is always performed in the observed
direction. 

@pspp{} uses a very precise approximation to the gamma function to
compute the binomial significance.  Thus, exact results are reported
even for very large sample sizes.



@node    CHISQUARE
@subsection Chisquare Test
@vindex CHISQUARE
@cindex chisquare test


@display
     [ /CHISQUARE=var_list[(lo,hi)] [/EXPECTED=@{EQUAL|f1, f2 @dots{} fn@}] ]
@end display 


The /CHISQUARE subcommand produces a chi-square statistic for the differences 
between the expected and observed frequencies of the categories of a variable. 
Optionally, a range of values may appear after the variable list.  
If a range is given, then non integer values are truncated, and values
outside the  specified range are excluded from the analysis.

The /EXPECTED subcommand specifies the expected values of each
category.  
There must be exactly one non-zero expected value, for each observed
category, or the EQUAL keywork must be specified.
You may use the notation @var{n}*@var{f} to specify @var{n}
consecutive expected categories all taking a frequency of @var{f}.
The frequencies given are proportions, not absolute frequencies.  The
sum of the frequencies need not be 1.
If no /EXPECTED subcommand is given, then then equal frequencies 
are expected.


@node COCHRAN
@subsection Cochran Q Test
@vindex Cochran
@cindex Cochran Q test
@cindex Q, Cochran Q

@display
     [ /COCHRAN = varlist ]
@end display

The Cochran Q test is used to test for differences between three or more groups.
The data for @var{varlist} in all cases must assume exactly two distinct values (other than missing values). 

The value of Q will be displayed and its Asymptotic significance based on a chi-square distribution.

@node FRIEDMAN
@subsection Friedman Test
@vindex FRIEDMAN
@cindex Friedman test

@display
     [ /FRIEDMAN = varlist ]
@end display

The Friedman test is used to test for differences between repeated measures when there is no indication that the distributions are normally distributed.

A list of variables which contain the measured data must be given.  The procedure prints the sum of ranks for each variable, the test statistic and its significance.

@node KENDALL
@subsection Kendall's W Test
@vindex KENDALL
@cindex Kendall's W test
@cindex coefficient of concordance

@display
     [ /KENDALL = varlist ]
@end display

The Kendall test investigates whether an arbitrary number of related samples come from the 
same population.
It is identical to the Friedman test except that the additional statistic W, Kendall's Coefficient of Concordance is printed.
It has the range [0,1] --- a value of zero indicates no agreement between the samples whereas a value of
unity indicates complete agreement.


@node KOLMOGOROV-SMIRNOV
@subsection Kolmogorov-Smirnov Test
@vindex KOLMOGOROV-SMIRNOV
@vindex K-S
@cindex Kolmogorov-Smirnov test

@display
     [ /KOLMOGOROV-SMIRNOV (@{NORMAL [@var{mu}, @var{sigma}], UNIFORM [@var{min}, @var{max}], POISSON [@var{lambda}], EXPONENTIAL [@var{scale}] @}) = varlist ]
@end display

The one sample Kolmogorov-Smirnov subcommand is used to test whether or not a dataset is
drawn from a particular distribution.  Four distributions are supported, @i{viz:}
Normal, Uniform, Poisson and Exponential.

Ideally you should provide the parameters of the distribution against which you wish to test
the data. For example, with the normal distribution  the mean (@var{mu})and standard deviation (@var{sigma})
should be given; with the uniform distribution, the minimum (@var{min})and maximum (@var{max}) value should
be provided.
However, if the parameters are omitted they will be imputed from the data. Imputing the
parameters reduces the power of the test so should be avoided if possible.

In the following example, two variables @var{score} and @var{age} are tested to see if
they follow a normal distribution with a mean of 3.5 and a standard deviation of 2.0.
@example
  NPAR TESTS
        /KOLMOGOROV-SMIRNOV (normal 3.5 2.0) = @var{score} @var{age}.
@end example
If the variables need to be tested against different distributions, then a separate
subcommand must be used.  For example the following syntax tests @var{score} against
a normal distribution with mean of 3.5 and standard deviation of 2.0 whilst @var{age}
is tested against a normal distribution of mean 40 and standard deviation 1.5.
@example
  NPAR TESTS
        /KOLMOGOROV-SMIRNOV (normal 3.5 2.0) = @var{score}
        /KOLMOGOROV-SMIRNOV (normal 40 1.5) =  @var{age}.
@end example

The abbreviated subcommand  K-S may be used in place of KOLMOGOROV-SMIRNOV.

@node KRUSKAL-WALLIS
@subsection Kruskal-Wallis Test
@vindex KRUSKAL-WALLIS
@vindex K-W
@cindex Kruskal-Wallis test

@display
     [ /KRUSKAL-WALLIS = varlist BY var (lower, upper) ]
@end display

The Kruskal-Wallis test is used to compare data from an 
arbitrary number of populations.  It does not assume normality.
The data to be compared are specified by @var{varlist}.
The categorical variable determining the groups to which the
data belongs is given by @var{var}. The limits @var{lower} and
@var{upper} specify the valid range of @var{var}. Any cases for
which @var{var} falls outside [@var{lower}, @var{upper}] will be
ignored.

The mean rank of each group as well as the chi-squared value and significance
of the test will be printed.
The abbreviated subcommand  K-W may be used in place of KRUSKAL-WALLIS.


@node MANN-WHITNEY
@subsection Mann-Whitney U Test
@vindex MANN-WHITNEY
@vindex M-W
@cindex Mann-Whitney U test
@cindex U, Mann-Whitney U

@display
     [ /MANN-WHITNEY = varlist BY var (group1, group2) ]
@end display

The Mann-Whitney subcommand is used to test whether two groups of data come from different populations.
The variables to be tested should be specified in @var{varlist} and the grouping variable, that determines to which group the test variables belong, in @var{var}.
@var{Var} may be either a string or an alpha variable.
@var{Group1} and @var{group2} specify the
two values of @var{var} which determine the groups of the test data.
Cases for which the @var{var} value is neither @var{group1} or @var{group2} will be ignored.

The value of the Mann-Whitney U statistic, the Wilcoxon W, and the significance will be printed.
The abbreviated subcommand  M-W may be used in place of MANN-WHITNEY.

@node MCNEMAR
@subsection McNemar Test
@vindex MCNEMAR
@cindex McNemar test

@display
     [ /MCNEMAR varlist [ WITH varlist [ (PAIRED) ]]]
@end display

Use McNemar's test to analyse the significance of the difference between
pairs of correlated proportions.

If the @code{WITH} keyword is omitted, then tests for all
combinations of the listed variables are performed.
If the @code{WITH} keyword is given, and the @code{(PAIRED)} keyword
is also given, then the number of variables preceding @code{WITH}
must be the same as the number following it.
In this case, tests for each respective pair of variables are
performed.
If the @code{WITH} keyword is given, but the
@code{(PAIRED)} keyword is omitted, then tests for each combination
of variable preceding @code{WITH} against variable following
@code{WITH} are performed.

The data in each variable must be dichotomous.  If there are more
than two distinct variables an error will occur and the test will
not be run.

@node MEDIAN
@subsection Median Test
@vindex MEDIAN
@cindex Median test

@display
     [ /MEDIAN [(value)] = varlist BY variable (value1, value2) ]
@end display

The median test is used to test whether independent samples come from 
populations with a common median.
The median of the populations against which the samples are to be tested
may be given in parentheses immediately after the 
/MEDIAN subcommand.  If it is not given, the median will be imputed from the 
union of all the samples.

The variables of the samples to be tested should immediately follow the @samp{=} sign. The
keyword @code{BY} must come next, and then the grouping variable.  Two values
in parentheses should follow.  If the first value is greater than the second,
then a 2 sample test is performed using these two values to determine the groups.
If however, the first variable is less than the second, then a @i{k} sample test is
conducted and the group values used are all values encountered which lie in the
range [@var{value1},@var{value2}].


@node RUNS
@subsection Runs Test
@vindex RUNS
@cindex runs test

@display 
     [ /RUNS (@{MEAN, MEDIAN, MODE, value@})  = varlist ]
@end display

The /RUNS subcommand tests whether a data sequence is randomly ordered.

It works by examining the number of times a variable's value crosses a given threshold. 
The desired threshold must be specified within parentheses.
It may either be specified as a number or as one of MEAN, MEDIAN or MODE.
Following the threshold specification comes the list of variables whose values are to be
tested.

The subcommand shows the number of runs, the asymptotic significance based on the
length of the data.

@node SIGN
@subsection Sign Test
@vindex SIGN
@cindex sign test

@display
     [ /SIGN varlist [ WITH varlist [ (PAIRED) ]]]
@end display

The /SIGN subcommand tests for differences between medians of the 
variables listed.
The test does not make any assumptions about the
distribution of the data.

If the @code{WITH} keyword is omitted, then tests for all
combinations of the listed variables are performed.
If the @code{WITH} keyword is given, and the @code{(PAIRED)} keyword
is also given, then the number of variables preceding @code{WITH}
must be the same as the number following it.
In this case, tests for each respective pair of variables are
performed.
If the @code{WITH} keyword is given, but the
@code{(PAIRED)} keyword is omitted, then tests for each combination
of variable preceding @code{WITH} against variable following
@code{WITH} are performed.

@node WILCOXON
@subsection Wilcoxon Matched Pairs Signed Ranks Test
@comment  node-name,  next,  previous,  up
@vindex WILCOXON
@cindex wilcoxon matched pairs signed ranks test

@display
     [ /WILCOXON varlist [ WITH varlist [ (PAIRED) ]]]
@end display

The /WILCOXON subcommand tests for differences between medians of the 
variables listed.
The test does not make any assumptions about the variances of the samples.
It does however assume that the distribution is symetrical.

If the @code{WITH} keyword is omitted, then tests for all
combinations of the listed variables are performed.
If the @code{WITH} keyword is given, and the @code{(PAIRED)} keyword
is also given, then the number of variables preceding @code{WITH}
must be the same as the number following it.
In this case, tests for each respective pair of variables are
performed.
If the @code{WITH} keyword is given, but the
@code{(PAIRED)} keyword is omitted, then tests for each combination
of variable preceding @code{WITH} against variable following
@code{WITH} are performed.

@node T-TEST
@comment  node-name,  next,  previous,  up
@section T-TEST

@vindex T-TEST

@display
T-TEST
        /MISSING=@{ANALYSIS,LISTWISE@} @{EXCLUDE,INCLUDE@}
        /CRITERIA=CIN(confidence)


(One Sample mode.)
        TESTVAL=test_value
        /VARIABLES=var_list


(Independent Samples mode.)
        GROUPS=var(value1 [, value2])
        /VARIABLES=var_list


(Paired Samples mode.)
        PAIRS=var_list [WITH var_list [(PAIRED)] ]

@end display


The @cmd{T-TEST} procedure outputs tables used in testing hypotheses about 
means.  
It operates in one of three modes:
@itemize
@item One Sample mode.
@item Independent Groups mode.
@item Paired mode.
@end itemize

@noindent
Each of these modes are described in more detail below.
There are two optional subcommands which are common to all modes.

The @cmd{/CRITERIA} subcommand tells @pspp{} the confidence interval used
in the tests.  The default value is 0.95.


The @cmd{MISSING} subcommand determines the handling of missing
variables.  
If INCLUDE is set, then user-missing values are included in the
calculations, but system-missing values are not.
If EXCLUDE is set, which is the default, user-missing
values are excluded as well as system-missing values. 
This is the default.

If LISTWISE is set, then the entire case is excluded from analysis
whenever any variable  specified in the @cmd{/VARIABLES}, @cmd{/PAIRS} or 
@cmd{/GROUPS} subcommands contains a missing value.   
If ANALYSIS is set, then missing values are excluded only in the analysis for
which they would be needed. This is the default.


@menu
* One Sample Mode::             Testing against a hypothesized mean
* Independent Samples Mode::    Testing two independent groups for equal mean
* Paired Samples Mode::         Testing two interdependent groups for equal mean
@end menu

@node One Sample Mode
@subsection One Sample Mode

The @cmd{TESTVAL} subcommand invokes the One Sample mode.
This mode is used to test a population mean against a hypothesized
mean. 
The value given to the @cmd{TESTVAL} subcommand is the value against
which you wish to test.
In this mode, you must also use the @cmd{/VARIABLES} subcommand to
tell @pspp{} which variables you wish to test.

@node Independent Samples Mode
@comment  node-name,  next,  previous,  up
@subsection Independent Samples Mode

The @cmd{GROUPS} subcommand invokes Independent Samples mode or
`Groups' mode. 
This mode is used to test whether two groups of values have the
same population mean.
In this mode, you must also use the @cmd{/VARIABLES} subcommand to
tell @pspp{} the dependent variables you wish to test.

The variable given in the @cmd{GROUPS} subcommand is the independent
variable which determines to which group the samples belong.
The values in parentheses are the specific values of the independent
variable for each group.
If the parentheses are omitted and no values are given, the default values 
of 1.0 and 2.0 are assumed.

If the independent variable is numeric, 
it is acceptable to specify only one value inside the parentheses.
If you do this, cases where the independent variable is
greater than or equal to this value belong to the first group, and cases
less than this value belong to the second group.
When using this form of the @cmd{GROUPS} subcommand, missing values in
the independent variable are excluded on a listwise basis, regardless
of whether @cmd{/MISSING=LISTWISE} was specified.


@node Paired Samples Mode
@comment  node-name,  next,  previous,  up
@subsection Paired Samples Mode

The @cmd{PAIRS} subcommand introduces Paired Samples mode.
Use this mode when repeated measures have been taken from the same
samples.
If the @code{WITH} keyword is omitted, then tables for all
combinations of variables given in the @cmd{PAIRS} subcommand are
generated. 
If the @code{WITH} keyword is given, and the @code{(PAIRED)} keyword
is also given, then the number of variables preceding @code{WITH}
must be the same as the number following it.
In this case, tables for each respective pair of variables are
generated.
In the event that the @code{WITH} keyword is given, but the
@code{(PAIRED)} keyword is omitted, then tables for each combination
of variable preceding @code{WITH} against variable following
@code{WITH} are generated.


@node ONEWAY
@comment  node-name,  next,  previous,  up
@section ONEWAY

@vindex ONEWAY
@cindex analysis of variance
@cindex ANOVA

@display
ONEWAY
        [/VARIABLES = ] var_list BY var
        /MISSING=@{ANALYSIS,LISTWISE@} @{EXCLUDE,INCLUDE@}
        /CONTRAST= value1 [, value2] ... [,valueN]
        /STATISTICS=@{DESCRIPTIVES,HOMOGENEITY@}
        /POSTHOC=@{BONFERRONI, GH, LSD, SCHEFFE, SIDAK, TUKEY, ALPHA ([value])@}
@end display

The @cmd{ONEWAY} procedure performs a one-way analysis of variance of
variables factored by a single independent variable.
It is used to compare the means of a population
divided into more than two groups. 

The dependent variables to be analysed should be given in the @code{VARIABLES}
subcommand.  
The list of variables must be followed by the @code{BY} keyword and
the name of the independent (or factor) variable.

You can use the @code{STATISTICS} subcommand to tell @pspp{} to display
ancilliary information.  The options accepted are:
@itemize
@item DESCRIPTIVES
Displays descriptive statistics about the groups factored by the independent
variable.
@item HOMOGENEITY
Displays the Levene test of Homogeneity of Variance for the
variables and their groups.
@end itemize

The @code{CONTRAST} subcommand is used when you anticipate certain
differences between the groups.
The subcommand must be followed by a list of numerals which are the
coefficients of the groups to be tested.
The number of coefficients must correspond to the number of distinct
groups (or values of the independent variable).
If the total sum of the coefficients are not zero, then @pspp{} will
display a warning, but will proceed with the analysis.
The @code{CONTRAST} subcommand may be given up to 10 times in order
to specify different contrast tests.
The @code{MISSING} subcommand defines how missing values are handled.
If LISTWISE is specified then cases which have missing values for 
the independent variable or any dependent variable will be ignored.
If ANALYSIS is specified, then cases will be ignored if the independent
variable is missing or if the dependent variable currently being 
analysed is missing.  The default is ANALYSIS.
A setting of EXCLUDE means that variables whose values are
user-missing are to be excluded from the analysis. A setting of
INCLUDE means they are to be included.  The default is EXCLUDE.

Using the @code{POSTHOC} subcommand you can perform multiple
pairwise comparisons on the data. The following comparison methods
are available:
@itemize
@item LSD
Least Significant Difference.
@item TUKEY
Tukey Honestly Significant Difference.
@item BONFERRONI
Bonferroni test.
@item SCHEFFE
Scheff@'e's test.
@item SIDAK
Sidak test.
@item GH
The Games-Howell test.
@end itemize

@noindent
The optional syntax @code{ALPHA(@var{value})} is used to indicate
that @var{value} should be used as the
confidence level for which the posthoc tests will be performed.
The default is 0.05.

@node QUICK CLUSTER
@comment  node-name,  next,  previous,  up
@section QUICK CLUSTER
@vindex QUICK CLUSTER

@cindex K-means clustering
@cindex clustering

@display
QUICK CLUSTER var_list
      [/CRITERIA=CLUSTERS(@var{k}) [MXITER(@var{max_iter})]]
      [/MISSING=@{EXCLUDE,INCLUDE@} @{LISTWISE, PAIRWISE@}]
@end display

The @cmd{QUICK CLUSTER} command performs k-means clustering on the
dataset.  This is useful when you wish to allocate cases into clusters
of similar values and you already know the number of clusters.

The minimum specification is @samp{QUICK CLUSTER} followed by the names
of the variables which contain the cluster data.  Normally you will also
want to specify @samp{/CRITERIA=CLUSTERS(@var{k})} where @var{k} is the
number of clusters.  If this is not given, then @var{k} defaults to 2.

The command uses an iterative algorithm to determine the clusters for
each case.  It will continue iterating until convergence, or until @var{max_iter}
iterations have been done.  The default value of @var{max_iter} is 2.

The @cmd{MISSING} subcommand determines the handling of missing variables.  
If INCLUDE is set, then user-missing values are considered at their face
value and not as missing values.
If EXCLUDE is set, which is the default, user-missing
values are excluded as well as system-missing values. 

If LISTWISE is set, then the entire case is excluded from the analysis
whenever any of the clustering variables contains a missing value.   
If PAIRWISE is set, then a case is considered missing only if all the
clustering variables contain missing values.  Otherwise it is clustered
on the basis of the non-missing values.
The default is LISTWISE.


@node RANK
@comment  node-name,  next,  previous,  up
@section RANK


@vindex RANK
@display
RANK
        [VARIABLES=] var_list [@{A,D@}] [BY var_list]
        /TIES=@{MEAN,LOW,HIGH,CONDENSE@}
        /FRACTION=@{BLOM,TUKEY,VW,RANKIT@}
        /PRINT[=@{YES,NO@}
        /MISSING=@{EXCLUDE,INCLUDE@}

        /RANK [INTO var_list]
        /NTILES(k) [INTO var_list]
        /NORMAL [INTO var_list]
        /PERCENT [INTO var_list]
        /RFRACTION [INTO var_list]
        /PROPORTION [INTO var_list]
        /N [INTO var_list]
        /SAVAGE [INTO var_list]
@end display

The @cmd{RANK} command ranks variables and stores the results into new
variables. 

The VARIABLES subcommand, which is mandatory, specifies one or
more variables whose values are to be ranked.  
After each variable, @samp{A} or @samp{D} may appear, indicating that
the variable is to be ranked in ascending or descending order.
Ascending is the default.
If a BY keyword appears, it should be followed by a list of variables
which are to serve as group variables.  
In this case, the cases are gathered into groups, and ranks calculated
for each group.

The TIES subcommand specifies how tied values are to be treated.  The
default is to take the mean value of all the tied cases.

The FRACTION subcommand specifies how proportional ranks are to be
calculated.  This only has any effect if NORMAL or PROPORTIONAL rank
functions are requested.

The PRINT subcommand may be used to specify that a summary of the rank
variables created should appear in the output.

The function subcommands are RANK, NTILES, NORMAL, PERCENT, RFRACTION,
PROPORTION and SAVAGE.  Any number of function subcommands may appear.
If none are given, then the default is RANK.
The NTILES subcommand must take an integer specifying the number of
partitions into which values should be ranked.
Each subcommand may be followed by the INTO keyword and a list of
variables which are the variables to be created and receive the rank
scores.  There may be as many variables specified as there are
variables named on the VARIABLES subcommand.  If fewer are specified,
then the variable names are automatically created.

The MISSING subcommand determines how user missing values are to be
treated. A setting of EXCLUDE means that variables whose values are
user-missing are to be excluded from the rank scores. A setting of
INCLUDE means they are to be included.  The default is EXCLUDE.

@include regression.texi


@node RELIABILITY
@section RELIABILITY

@vindex RELIABILITY
@display
RELIABILITY
        /VARIABLES=var_list
        /SCALE (@var{name}) = @{var_list, ALL@}
        /MODEL=@{ALPHA, SPLIT[(N)]@}
        /SUMMARY=@{TOTAL,ALL@}
        /MISSING=@{EXCLUDE,INCLUDE@}
@end display

@cindex Cronbach's Alpha
The @cmd{RELIABILTY} command performs reliability analysis on the data.

The VARIABLES subcommand is required. It determines the set of variables 
upon which analysis is to be performed.

The SCALE subcommand determines which variables reliability is to be 
calculated for.  If it is omitted, then analysis for all variables named
in the VARIABLES subcommand will be used.
Optionally, the @var{name} parameter may be specified to set a string name 
for the scale.

The MODEL subcommand determines the type of analysis. If ALPHA is specified, 
then Cronbach's Alpha is calculated for the scale.  If the model is SPLIT, 
then the variables  are divided into 2 subsets.  An optional parameter 
@var{N} may be given, to specify how many variables to be in the first subset.
If @var{N} is omitted, then it defaults to one half of the variables in the 
scale, or one half minus one if there are an odd number of variables.
The default model is ALPHA.

By default, any cases with user missing, or system missing values for 
any variables given 
in the VARIABLES subcommand will be omitted from analysis.
The MISSING subcommand determines whether user missing values are to 
be included or excluded in the analysis.

The SUMMARY subcommand determines the type of summary analysis to be performed.
Currently there is only one type: SUMMARY=TOTAL, which displays per-item
analysis tested against the totals.



@node ROC
@section ROC

@vindex ROC
@cindex Receiver Operating Characteristic
@cindex Area under curve

@display
ROC     @var{var_list} BY @var{state_var} (@var{state_value})
        /PLOT = @{ CURVE [(REFERENCE)], NONE @}
        /PRINT = [ SE ] [ COORDINATES ]
        /CRITERIA = [ CUTOFF(@{INCLUDE,EXCLUDE@}) ]
          [ TESTPOS (@{LARGE,SMALL@}) ]
          [ CI (@var{confidence}) ]
          [ DISTRIBUTION (@{FREE, NEGEXPO @}) ]
        /MISSING=@{EXCLUDE,INCLUDE@}
@end display


The @cmd{ROC} command is used to plot the receiver operating characteristic curve 
of a dataset, and to estimate the area under the curve.
This is useful for analysing the efficacy of a variable as a predictor of a state of nature.

The mandatory @var{var_list} is the list of predictor variables.
The variable @var{state_var} is the variable whose values represent the actual states, 
and @var{state_value} is the value of this variable which represents the positive state.

The optional subcommand PLOT is used to determine if and how the ROC curve is drawn.
The keyword CURVE means that the ROC curve should be drawn, and the optional keyword REFERENCE,
which should be enclosed in parentheses, says that the diagonal reference line should be drawn.
If the keyword NONE is given, then no ROC curve is drawn.
By default, the curve is drawn with no reference line.

The optional subcommand PRINT determines which additional tables should be printed.
Two additional tables are available. 
The SE keyword says that standard error of the area under the curve should be printed as well as
the area itself.
In addition, a p-value under the null hypothesis that the area under the curve equals 0.5 will be
printed.
The COORDINATES keyword says that a table of coordinates of the ROC curve should be printed.

The CRITERIA subcommand has four optional parameters:
@itemize @bullet
@item The TESTPOS parameter may be LARGE or SMALL.
LARGE is the default, and says that larger values in the predictor variables are to be 
considered positive.  SMALL indicates that smaller values should be considered positive.

@item The CI parameter specifies the confidence interval that should be printed.
It has no effect if the SE keyword in the PRINT subcommand has not been given.

@item The DISTRIBUTION parameter determines the method to be used when estimating the area
under the curve.  
There are two possibilities, @i{viz}: FREE and NEGEXPO.
The FREE method uses a non-parametric estimate, and the NEGEXPO method a bi-negative 
exponential distribution estimate.
The NEGEXPO method should only be used when the number of positive actual states is
equal to the number of negative actual states.
The default is FREE.

@item The CUTOFF parameter is for compatibility and is ignored.
@end itemize

The MISSING subcommand determines whether user missing values are to 
be included or excluded in the analysis.  The default behaviour is to
exclude them.
Cases are excluded on a listwise basis; if any of the variables in @var{var_list} 
or if the variable @var{state_var} is missing, then the entire case will be 
excluded.