zip-reader.c: Fix memory leak

[pspp] / doc / tutorial.texi

@alias prompt = sansserif

@include tut.texi

@node Using PSPP
@chapter Using @pspp{}

@pspp{} is a tool for the statistical analysis of sampled data.
You can use it to discover patterns in the data,
to explain differences in one subset of data in terms of another subset
and to find out
whether certain beliefs about the data are justified.
This chapter does not attempt to introduce the theory behind the 
statistical analysis,
but it shows how such analysis can be performed using @pspp{}.

For the purposes of this tutorial, it is assumed that you are using @pspp{} in its 
interactive mode from the command line.
However, the example commands can also be typed into a file and executed in 
a post-hoc mode by typing @samp{pspp @var{filename}} at a shell prompt,
where @var{filename} is the name of the file containing the commands.
Alternatively, from the graphical interface, you can select
@clicksequence{File @click{} New @click{} Syntax} to open a new syntax window
and use the @clicksequence{Run} menu when a syntax fragment is ready to be 
executed.
Whichever method you choose, the syntax is identical.

When using the interactive method, @pspp{} tells you that it's waiting for your
data with a string like @prompt{PSPP>} or @prompt{data>}.
In the examples of this chapter, whenever you see text like this, it
indicates the prompt displayed by @pspp{}, @emph{not} something that you 
should type.

Throughout this chapter reference is made to a number of sample data files.
So that you can try the examples for yourself,
you should have received these files along with your copy of @pspp{}.@c
@footnote{These files contain purely fictitious data.  They should not be used
for research purposes.}
@note{Normally these files are installed in the directory
@file{@value{example-dir}}.
If however your system administrator or operating system vendor has
chosen to install them in a different location, you will have to adjust
the examples accordingly.}


@menu
* Preparation of Data Files::   
* Data Screening and Transformation::  
* Hypothesis Testing::          
@end menu

@node Preparation of Data Files
@section Preparation of Data Files


Before analysis can commence,  the data must be loaded into @pspp{} and
arranged such that both @pspp{} and humans can understand what
the data represents.
There are two aspects of data:

@itemize @bullet
@item The variables --- these are the parameters of a quantity
 which has been measured or estimated in some way.
 For example height, weight and geographic location are all variables.
@item The observations (also called `cases') of the variables ---
 each observation represents an instance when the variables were measured
 or observed. 
@end itemize

@noindent
For example, a data set which has the variables @var{height}, @var{weight}, and
@var{name}, might have the observations:
@example
1881 89.2 Ahmed
1192 107.01 Frank
1230 67 Julie
@end example
@noindent
The following sections explain how to define a dataset.

@menu
* Defining Variables::          
* Listing the data::            
* Reading data from a text file::  
* Reading data from a pre-prepared PSPP file::  
* Saving data to a PSPP file.::  
* Reading data from other sources::  
@end menu

@node Defining Variables
@subsection Defining Variables
@cindex variables

Variables come in two basic types, @i{viz}: @dfn{numeric} and @dfn{string}.
Variables such as age, height and satisfaction are numeric,
whereas name is a string variable.
String variables are best reserved for commentary data to assist the 
human observer.
However they can also be used for nominal or categorical data.


@ref{data-list} defines two variables @var{forename} and @var{height},
and reads data into them by manual input. 

@float Example, data-list
@cartouche
@example
@prompt{PSPP>} data list list /forename (A12) height.
@prompt{PSPP>} begin data.
@prompt{data>} Ahmed 188
@prompt{data>} Bertram 167
@prompt{data>} Catherine 134.231
@prompt{data>} David 109.1
@prompt{data>} end data
@prompt{PSPP>}
@end example
@end cartouche
@caption{Manual entry of data using the @cmd{DATA LIST} command.
Two variables
@var{forename} and @var{height} are defined and subsequently filled
with  manually entered data.}
@end float

There are several things to note about this example.

@itemize @bullet
@item
The words @samp{data list list} are an example of the @cmd{DATA LIST}
command. @xref{DATA LIST}.
It tells @pspp{} to prepare for reading data.
The word @samp{list} intentionally appears twice.
The first occurrence is part of the @cmd{DATA LIST} call,
whilst the second
tells @pspp{} that the data is to be read as free format data with
one record per line.

@item
The @samp{/} character is important. It marks the start of the list of
variables which you wish to define.

@item
The text @samp{forename} is the name of the first variable,
and @samp{(A12)} says that the variable @var{forename} is a string
variable and that its maximum length is 12 bytes.
The second variable's name is specified by the text @samp{height}.
Since no format is given, this variable has the default format.
For more information on data formats, @pxref{Input and Output Formats}.


@item
Normally, @pspp{} displays the  prompt @prompt{PSPP>} whenever it's
expecting a command.
However, when it's expecting data, the prompt changes to @prompt{data>}
so that you know to enter data and not a command.

@item
At the end of every command there is a terminating @samp{.} which tells
@pspp{} that the end of a command has been encountered.
You should not enter @samp{.} when data is expected (@i{ie.} when 
the @prompt{data>} prompt is current) since it is appropriate only for
terminating commands.
@end itemize

@node Listing the data
@subsection Listing the data
@vindex LIST

Once the data has been entered,
you could type
@example
@prompt{PSPP>} list /format=numbered.
@end example
@noindent
to list the data.
The optional text @samp{/format=numbered} requests the case numbers to be 
shown along with the data.
It should show the following output:
@example
@group
Case#     forename   height
----- ------------ --------
    1 Ahmed          188.00 
    2 Bertram        167.00 
    3 Catherine      134.23 
    4 David          109.10 
@end group
@end example
@noindent
Note that the numeric variable @var{height} is displayed to 2 decimal 
places, because the format for that variable is @samp{F8.2}.
For a complete description of the @cmd{LIST} command, @pxref{LIST}.

@node Reading data from a text file
@subsection Reading data from a text file
@cindex reading data

The previous example showed how to define a set of variables and to 
manually enter the data for those variables.
Manual entering of data is tedious work, and often
a file containing the data will be have been previously 
prepared.
Let us assume that you have a file called @file{mydata.dat} containing the
ascii encoded data:
@example
Ahmed          188.00 
Bertram        167.00 
Catherine      134.23 
David          109.10 
@              .
@              .
@              .
Zachariah      113.02
@end example
@noindent
You can can tell the @cmd{DATA LIST} command to read the data directly from
this file instead of by manual entry, with a command like:
@example
@prompt{PSPP>} data list file='mydata.dat' list /forename (A12) height.
@end example
@noindent
Notice however, that it is still necessary to specify the names of the
variables and their formats, since this information is not contained
in the file.
It is also possible to specify the file's character encoding and other 
parameters.
For full details refer to @pxref{DATA LIST}.

@node Reading data from a pre-prepared PSPP file
@subsection Reading data from a pre-prepared @pspp{} file
@cindex system files
@vindex GET

When working with other @pspp{} users, or users of other software which
uses the @pspp{} data format, you may be given the data in
a pre-prepared @pspp{} file.
Such files contain not only the data, but the variable definitions,
along with their formats, labels and other meta-data.
Conventionally, these files (sometimes called ``system'' files) 
have the suffix @file{.sav}, but that is
not mandatory.
The following syntax loads a file called @file{my-file.sav}.
@example
@prompt{PSPP>} get file='my-file.sav'.
@end example
@noindent
You will encounter several instances of this in future examples.


@node Saving data to a PSPP file.
@subsection Saving data to a @pspp{} file.
@cindex saving
@vindex SAVE

If you want to save your data, along with the variable definitions so
that you or other @pspp{} users can use it later, you can do this with
the @cmd{SAVE} command.

The following syntax will save the existing data and variables to a
file called @file{my-new-file.sav}.
@example
@prompt{PSPP>} save outfile='my-new-file.sav'.
@end example
@noindent
If @file{my-new-file.sav} already exists, then it will be overwritten.
Otherwise it will be created.


@node Reading data from other sources
@subsection Reading data from other sources
@cindex comma separated values
@cindex spreadsheets
@cindex databases

Sometimes it's useful to be able to read data from comma 
separated text, from spreadsheets, databases or other sources.
In these instances you should
use the @cmd{GET DATA} command (@pxref{GET DATA}).

  
@node Data Screening and Transformation
@section Data Screening and Transformation

@cindex screening
@cindex transformation

Once data has been entered, it is often desirable, or even necessary,
to transform it in some way before performing analysis upon it.
At the very least, it's good practice to check for errors.

@menu
* Identifying incorrect data::  
* Dealing with suspicious data::  
* Inverting negatively coded variables::  
* Testing data consistency::    
* Testing for normality ::      
@end menu

@node Identifying incorrect data
@subsection Identifying incorrect data
@cindex erroneous data
@cindex errors, in data

Data from real sources is rarely error free.
@pspp{} has a number of procedures which can be used to help 
identify data which might be incorrect.

The @cmd{DESCRIPTIVES} command (@pxref{DESCRIPTIVES}) is used to generate
simple linear statistics for a dataset.  It is also useful for
identifying potential problems in the data.
The example file @file{physiology.sav} contains a number of physiological
measurements of a sample of healthy adults selected at random.
However, the data entry clerk made a number of mistakes when entering
the data.
@ref{descriptives} illustrates the use of @cmd{DESCRIPTIVES} to screen this
data and identify the erroneous values.

@float Example, descriptives
@cartouche
@example
@prompt{PSPP>} get file='@value{example-dir}/physiology.sav'.
@prompt{PSPP>} descriptives sex, weight, height.
@end example

Output:
@example
DESCRIPTIVES.  Valid cases = 40; cases with missing value(s) = 0.
+--------#--+-------+-------+-------+-------+
|Variable# N|  Mean |Std Dev|Minimum|Maximum|
#========#==#=======#=======#=======#=======#
|sex     #40|    .45|    .50|    .00|   1.00|
|height  #40|1677.12| 262.87| 179.00|1903.00|
|weight  #40|  72.12|  26.70| -55.60|  92.07|
+--------#--+-------+-------+-------+-------+
@end example
@end cartouche
@caption{Using the @cmd{DESCRIPTIVES} command to display simple 
summary information about the data.
In this case, the results show unexpectedly low values in the Minimum
column, suggesting incorrect data entry.}
@end float

In the output of @ref{descriptives},
the most interesting column is the minimum value.
The @var{weight} variable has a minimum value of less than zero,
which is clearly erroneous.
Similarly, the @var{height} variable's minimum value seems to be very low.
In fact, it is more than 5 standard deviations from the mean, and is a 
seemingly bizarre height for an adult person.
We can examine the data in more detail with the @cmd{EXAMINE}
command (@pxref{EXAMINE}):

In @ref{examine} you can see that the lowest value of @var{height} is
179 (which we suspect to be erroneous), but the second lowest is 1598
which
we know from the @cmd{DESCRIPTIVES} command 
is within 1 standard deviation from the mean.
Similarly the @var{weight} variable has a lowest value which is
negative but a plausible value for the second lowest value.
This suggests that the two extreme values are outliers and probably 
represent data entry errors. 

@float Example, examine
@cartouche
[@dots{} continue from @ref{descriptives}]
@example
@prompt{PSPP>} examine height, weight /statistics=extreme(3).    
@end example

Output:
@example
#===============================#===========#=======#
#                               #Case Number| Value #
#===============================#===========#=======#
#Height in millimetres Highest 1#         14|1903.00#
#                              2#         15|1884.00#
#                              3#         12|1801.65#
#                     ----------#-----------+-------#
#                       Lowest 1#         30| 179.00#
#                              2#         31|1598.00#
#                              3#         28|1601.00#
#                     ----------#-----------+-------#
#Weight in kilograms   Highest 1#         13|  92.07#
#                              2#          5|  92.07#
#                              3#         17|  91.74#
#                     ----------#-----------+-------#
#                       Lowest 1#         38| -55.60#
#                              2#         39|  54.48#
#                              3#         33|  55.45#
#===============================#===========#=======#
@end example
@end cartouche
@caption{Using the @cmd{EXAMINE} command to see the extremities of the data
for different variables.  Cases 30 and 38 seem to contain values
very much lower than the rest of the data.
They are possibly erroneous.}
@end float

@node Dealing with suspicious data
@subsection Dealing with suspicious data

@cindex SYSMIS
@cindex recoding data
If possible, suspect data should be checked and re-measured.
However, this may not always be feasible, in which case the researcher may
decide to disregard these values.
@pspp{} has a feature whereby data can assume the special value `SYSMIS', and
will be disregarded in future analysis. @xref{Missing Observations}.
You can set the two suspect values to the `SYSMIS' value using the @cmd{RECODE}
command.
@example
@pspp{}> recode height (179 = SYSMIS).
@pspp{}> recode weight (LOWEST THRU 0 = SYSMIS).
@end example
@noindent
The first command says that for any observation which has a
@var{height} value of 179, that value should be changed to the SYSMIS
value.
The second command says that any @var{weight} values of zero or less
should be changed to SYSMIS.
From now on, they will be ignored in analysis.
For detailed information about the @cmd{RECODE} command @pxref{RECODE}.

If you now re-run the @cmd{DESCRIPTIVES} or @cmd{EXAMINE} commands in
@ref{descriptives} and @ref{examine} you
will see a data summary with more plausible parameters.
You will also notice that the data summaries indicate the two missing values.

@node Inverting negatively coded variables
@subsection Inverting negatively coded variables

@cindex Likert scale
@cindex Inverting data
Data entry errors are not the only reason for wanting to recode data.
The sample file @file{hotel.sav} comprises data gathered from a 
customer satisfaction survey of clients at a particular hotel.
In @ref{reliability}, this file is loaded for analysis.
The line @code{display dictionary.} tells @pspp{} to display the
variables and associated data.
The output from this command has been omitted from the example for the sake of clarity, but
you will notice that each of the variables
@var{v1}, @var{v2} @dots{} @var{v5}  are measured on a 5 point Likert scale,
with 1 meaning ``Strongly disagree'' and 5 meaning ``Strongly agree''.
Whilst variables @var{v1}, @var{v2} and @var{v4} record responses
to a positively posed question, variables @var{v3} and @var{v5} are 
responses to negatively worded questions.
In order to perform meaningful analysis, we need to recode the variables so
that they all measure in the same direction.
We could use the @cmd{RECODE} command, with syntax such as:
@example
recode v3 (1 = 5) (2 = 4) (4 = 2) (5 = 1).
@end example
@noindent
However an easier and more elegant way uses the @cmd{COMPUTE}
command (@pxref{COMPUTE}).
Since the variables are Likert variables in the range (1 @dots{} 5),
subtracting their value  from 6 has the effect of inverting them:
@example
compute @var{var} = 6 - @var{var}.
@end example
@noindent
@ref{reliability} uses this technique to recode the variables 
@var{v3} and @var{v5}.
After applying  @cmd{COMPUTE} for both variables,
all subsequent commands will use the inverted values.


@node Testing data consistency
@subsection Testing data consistency

@cindex reliability
@cindex consistency

A sensible check to perform on survey data is the calculation of
reliability.
This gives the statistician some confidence that the questionnaires have been 
completed thoughtfully.
If you examine the labels of variables @var{v1},  @var{v3} and @var{v5},
you will notice that they ask very similar questions.
One would therefore expect the values of these variables (after recoding) 
to closely follow one another, and we can test that with the @cmd{RELIABILITY} 
command (@pxref{RELIABILITY}).
@ref{reliability} shows a @pspp{} session where the user (after recoding
negatively scaled variables) requests reliability statistics for
@var{v1}, @var{v3} and @var{v5}.

@float Example, reliability
@cartouche
@example
@prompt{PSPP>} get file='@value{example-dir}/hotel.sav'.
@prompt{PSPP>} display dictionary.
@prompt{PSPP>} * recode negatively worded questions.
@prompt{PSPP>} compute v3 = 6 - v3.
@prompt{PSPP>} compute v5 = 6 - v5.
@prompt{PSPP>} reliability v1, v3, v5.
@end example

Output (dictionary information omitted for clarity):
@example
1.1 RELIABILITY.  Case Processing Summary
#==============#==#======#
#              # N|   %  #
#==============#==#======#
#Cases Valid   #17|100.00#
#      Excluded# 0|   .00#
#      Total   #17|100.00#
#==============#==#======#

1.2 RELIABILITY.  Reliability Statistics
#================#==========#
#Cronbach's Alpha#N of Items#
#================#==========#
#             .86#         3#
#================#==========#
@end example
@end cartouche
@caption{Recoding negatively scaled variables, and testing for
reliability with the @cmd{RELIABILITY} command. The Cronbach Alpha
coefficient suggests a high degree of reliability among variables
@var{v1}, @var{v2} and @var{v5}.}
@end float

As a rule of thumb, many statisticians consider a value of Cronbach's Alpha of 
0.7 or higher to indicate reliable data.
Here, the value is 0.86 so the data and the recoding that we performed 
are vindicated.


@node Testing for normality 
@subsection Testing for normality
@cindex normality, testing 

Many statistical tests rely upon certain properties of the data.
One common property, upon which many linear tests depend, is that of
normality --- the data must have been drawn from a normal distribution.
It is necessary then to ensure normality before deciding upon the
test procedure to use.  One way to do this uses the @cmd{EXAMINE} command.

In @ref{normality}, a researcher was examining the failure rates
of equipment produced by an engineering company.
The file @file{repairs.sav} contains the mean time between
failures (@var{mtbf}) of some items of equipment subject to the study.
Before performing linear analysis on the data, 
the researcher wanted to ascertain that the data is normally distributed.

A normal distribution has a skewness and kurtosis of zero.
Looking at the skewness of @var{mtbf} in @ref{normality} it is clear
that the mtbf figures have a lot of positive skew and are therefore
not drawn from a normally distributed variable.
Positive skew can often be compensated for by applying a logarithmic 
transformation.
This is done with the @cmd{COMPUTE} command in the line
@example
compute mtbf_ln = ln (mtbf).
@end example
@noindent
Rather than redefining the existing variable, this use of @cmd{COMPUTE}
defines a new variable @var{mtbf_ln} which is
the natural logarithm of @var{mtbf}.
The final command in this example calls @cmd{EXAMINE} on this new variable,
and it can be seen from the results that both the skewness and
kurtosis for @var{mtbf_ln} are very close to zero.
This provides some confidence that the @var{mtbf_ln} variable is 
normally distributed and thus safe for linear analysis.
In the event that no suitable transformation can be found,
then it would be worth considering 
an appropriate non-parametric test instead of a linear one.
@xref{NPAR TESTS}, for information about non-parametric tests.

@float Example, normality
@cartouche
@example
@prompt{PSPP>} get file='@value{example-dir}/repairs.sav'.
@prompt{PSPP>} examine mtbf 
                /statistics=descriptives.
@prompt{PSPP>} compute mtbf_ln = ln (mtbf).
@prompt{PSPP>} examine mtbf_ln
                /statistics=descriptives.
@end example

Output:
@example
1.2 EXAMINE.  Descriptives
#====================================================#=========#==========#
#                                                    #Statistic|Std. Error#
#====================================================#=========#==========#
#mtbf    Mean                                        #   8.32  |   1.62   #
#        95% Confidence Interval for Mean Lower Bound#   4.85  |          #
#                                         Upper Bound#  11.79  |          #
#        5% Trimmed Mean                             #   7.69  |          #
#        Median                                      #   8.12  |          #
#        Variance                                    #  39.21  |          #
#        Std. Deviation                              #   6.26  |          #
#        Minimum                                     #   1.63  |          #
#        Maximum                                     #  26.47  |          #
#        Range                                       #  24.84  |          #
#        Interquartile Range                         #   5.83  |          #
#        Skewness                                    #   1.85  |    .58   #
#        Kurtosis                                    #   4.49  |   1.12   #
#====================================================#=========#==========#

2.2 EXAMINE.  Descriptives
#====================================================#=========#==========#
#                                                    #Statistic|Std. Error#
#====================================================#=========#==========#
#mtbf_ln Mean                                        #   1.88  |    .19   #
#        95% Confidence Interval for Mean Lower Bound#   1.47  |          #
#                                         Upper Bound#   2.29  |          #
#        5% Trimmed Mean                             #   1.88  |          #
#        Median                                      #   2.09  |          #
#        Variance                                    #   .54   |          #
#        Std. Deviation                              #   .74   |          #
#        Minimum                                     #   .49   |          #
#        Maximum                                     #   3.28  |          #
#        Range                                       #   2.79  |          #
#        Interquartile Range                         #   .92   |          #
#        Skewness                                    #   -.16  |    .58   #
#        Kurtosis                                    #   -.09  |   1.12   #
#====================================================#=========#==========#
@end example
@end cartouche
@caption{Testing for normality using the @cmd{EXAMINE} command and applying
a logarithmic transformation.
The @var{mtbf} variable has a large positive skew and is therefore
unsuitable for linear statistical analysis.
However the transformed variable (@var{mtbf_ln}) is close to normal and
would appear to be more suitable.}
@end float


@node Hypothesis Testing
@section Hypothesis Testing

@cindex Hypothesis testing
@cindex p-value
@cindex null hypothesis

One of the most fundamental purposes of statistical analysis
is hypothesis testing.
Researchers commonly need to test hypotheses about a set of data.
For example, she might want to test whether one set of data comes from
the same distribution as another,
or
whether the mean of a dataset significantly differs from a particular
value.
This section presents just some of the possible tests that @pspp{} offers.

The researcher starts by making a @dfn{null hypothesis}.
Often this is a hypothesis which he suspects to be false.
For example, if he suspects that @var{A} is greater than @var{B} he will 
state the null hypothesis as @math{ @var{A} = @var{B}}.@c
@footnote{This example assumes that it is already proven that @var{B} is
not greater than @var{A}.}

The @dfn{p-value} is a recurring concept in hypothesis testing.
It is the highest acceptable probability that the evidence implying a
null hypothesis is false, could have been obtained when the null
hypothesis is in fact true.
Note that this is not the same as ``the probability of making an
error'' nor is it the same as ``the probability of rejecting a
hypothesis when it is true''.



@menu
* Testing for differences of means::  
* Linear Regression::           
@end menu

@node Testing for differences of means
@subsection Testing for differences of means

@cindex T-test
@vindex T-TEST

A common statistical test involves hypotheses about means.
The @cmd{T-TEST} command is used to find out whether or not two separate 
subsets have the same mean.

@ref{t-test} uses the file @file{physiology.sav} previously
encountered.
A researcher suspected that the heights and core body 
temperature of persons might be different depending upon their sex.
To investigate this, he posed two null hypotheses: 
@itemize @bullet
@item The mean heights of males and females in the population are equal.
@item The mean body temperature of males and
      females in the population are equal.
@end itemize
@noindent
For the purposes of the investigation the researcher
decided to use a  p-value of 0.05.

In addition to the T-test, the @cmd{T-TEST} command also performs the 
Levene test for equal variances.
If the variances are equal, then a more powerful form of the T-test can be used.
However if it is unsafe to assume equal variances,
then an alternative calculation is necessary.
@pspp{} performs both calculations.

For the @var{height} variable, the output shows the significance of the 
Levene test to be 0.33 which means there is a 
33% probability that the  
Levene test produces this outcome when the variances are unequal.
Such a probability is too high
to assume that the variances are equal so the row
for unequal variances should be used.
Examining this row, the two tailed significance for the @var{height} t-test 
is less than 0.05, so it is safe to reject the null hypothesis and conclude
that the mean heights of males and females are unequal.

For the @var{temperature} variable, the significance of the Levene test 
is 0.58 so again, it is unsafe to use the row for equal variances.
The unequal variances row indicates that the two tailed significance for
@var{temperature} is 0.19.  Since this is greater than 0.05 we must reject
the null hypothesis and conclude that there is insufficient evidence to 
suggest that the body temperature of male and female persons are different.

@float Example, t-test
@cartouche
@example
@prompt{PSPP>} get file='@value{example-dir}/physiology.sav'.
@prompt{PSPP>} recode height (179 = SYSMIS).
@prompt{PSPP>} t-test group=sex(0,1) /variables = height temperature.
@end example
Output:
@example
1.1 T-TEST.  Group Statistics
#==================#==#=======#==============#========#
#              sex | N|  Mean |Std. Deviation|SE. Mean#
#==================#==#=======#==============#========#
#height      Male  |22|1796.49|         49.71|   10.60#
#            Female|17|1610.77|         25.43|    6.17#
#temperature Male  |22|  36.68|          1.95|     .42#
#            Female|18|  37.43|          1.61|     .38#
#==================#==#=======#==============#========#
1.2 T-TEST.  Independent Samples Test
#===========================#=========#===============================   =#
#                           # Levene's| t-test for Equality of Means      #
#                           #----+----+------+-----+------+---------+-   -#
#                           #    |    |      |     |      |         |     #
#                           #    |    |      |     |Sig. 2|         |     #
#                           #  F |Sig.|   t  |  df |tailed|Mean Diff|     #
#===========================#====#====#======#=====#======#=========#=   =#
#height      Equal variances# .97| .33| 14.02|37.00|   .00|   185.72| ... #
#          Unequal variances#    |    | 15.15|32.71|   .00|   185.72| ... #
#temperature Equal variances# .31| .58| -1.31|38.00|   .20|     -.75| ... #
#          Unequal variances#    |    | -1.33|37.99|   .19|     -.75| ... #
#===========================#====#====#======#=====#======#=========#=   =#
@end example                                                          
@end cartouche
@caption{The @cmd{T-TEST} command tests for differences of means. 
Here, the @var{height} variable's two tailed significance is less than
0.05, so the null hypothesis can be rejected.
Thus, the evidence suggests there is a difference between the heights of
male and female persons. 
However the significance of the test for the @var{temperature}
variable is greater than 0.05 so the null hypothesis cannot be
rejected, and there is insufficient evidence to suggest a difference
in body temperature.}
@end float

@node Linear Regression
@subsection Linear Regression
@cindex linear regression
@vindex REGRESSION

Linear regression is a technique used to investigate if and how a variable 
is linearly related to others.
If a variable is found to be linearly related, then this can be used to 
predict future values of that variable.

In example @ref{regression}, the service department of the company wanted to
be able to predict the time to repair equipment, in order to improve
the accuracy of their quotations.
It was suggested that the time to repair might be related to the time
between failures and the duty cycle of the equipment.
The p-value of 0.1 was chosen for this investigation.
In order to investigate this hypothesis, the @cmd{REGRESSION} command
was used.
This command not only tests if the variables are related, but also
identifies the potential linear relationship. @xref{REGRESSION}.


@float Example, regression 
@cartouche
@example
@prompt{PSPP>} get file='@value{example-dir}/repairs.sav'.
@prompt{PSPP>} regression /variables = mtbf duty_cycle /dependent = mttr.
@prompt{PSPP>} regression /variables = mtbf /dependent = mttr.
@end example
Output:
@example
1.3(1) REGRESSION.  Coefficients
#=============================================#====#==========#====#=====#
#                                             #  B |Std. Error|Beta|  t  #
#========#====================================#====#==========#====#=====#
#        |(Constant)                          #9.81|      1.50| .00| 6.54#
#        |Mean time between failures (months) #3.10|       .10| .99|32.43#
#        |Ratio of working to non-working time#1.09|      1.78| .02|  .61#
#        |                                    #    |          |    |     #
#========#====================================#====#==========#====#=====#

1.3(2) REGRESSION.  Coefficients
#=============================================#============#
#                                             #Significance#
#========#====================================#============#
#        |(Constant)                          #         .10#
#        |Mean time between failures (months) #         .00#
#        |Ratio of working to non-working time#         .55#
#        |                                    #            #
#========#====================================#============#
2.3(1) REGRESSION.  Coefficients
#============================================#=====#==========#====#=====#
#                                            #  B  |Std. Error|Beta|  t  #
#========#===================================#=====#==========#====#=====#
#        |(Constant)                         #10.50|       .96| .00|10.96#
#        |Mean time between failures (months)# 3.11|       .09| .99|33.39#
#        |                                   #     |          |    |     #
#========#===================================#=====#==========#====#=====#

2.3(2) REGRESSION.  Coefficients
#============================================#============#
#                                            #Significance#
#========#===================================#============#
#        |(Constant)                         #         .06#
#        |Mean time between failures (months)#         .00#
#        |                                   #            #
#========#===================================#============#
@end example
@end cartouche
@caption{Linear regression analysis to find a predictor for
@var{mttr}.
The first attempt, including @var{duty_cycle}, produces some
unacceptable high significance values.
However the second attempt, which excludes @var{duty_cycle}, produces
significance values no higher than 0.06.
This suggests that @var{mtbf} alone may be a suitable predictor
for @var{mttr}.}
@end float

The coefficients in the first table suggest that the formula
@math{@var{mttr} = 9.81 + 3.1 \times @var{mtbf} + 1.09 \times @var{duty_cycle}}
can be used to predict the time to repair.
However, the significance value for the @var{duty_cycle} coefficient
is very high, which would make this an unsafe predictor.
For this reason, the test was repeated, but omitting the
@var{duty_cycle} variable.
This time, the significance of all coefficients no higher than 0.06,
suggesting that at the 0.06 level, the formula
@math{@var{mttr} = 10.5 + 3.11 \times @var{mtbf}} is a reliable
predictor of the time to repair.


@c  LocalWords:  PSPP dir itemize noindent var cindex dfn cartouche samp xref
@c  LocalWords:  pxref ie sav Std Dev kilograms SYSMIS sansserif pre pspp emph
@c  LocalWords:  Likert Cronbach's Cronbach mtbf npplot ln myfile cmd NPAR Sig
@c  LocalWords:  vindex Levene Levene's df Diff clicksequence mydata dat ascii
@c  LocalWords:  mttr outfile