\chapter{Discrete variables}
\label{chap:discrete}
When a variable can take only a finite, typically small, number of
values, then the variable is said to be \emph{discrete}. Some
\app{gretl} commands act in a slightly different way when applied to
discrete variables; moreover, \app{gretl} provides a few commands that
only apply to discrete variables. Specifically, the \texttt{dummify}
and \texttt{xtab} commands (see below) are available only for discrete
variables, while the \texttt{freq} (frequency distribution) command
produces different output for discrete variables.
\section{Declaring variables as discrete}
\label{discr-declare}
\app{Gretl} uses a simple heuristic to judge whether a given variable
should be treated as discrete, but you also have the option of
explicitly marking a variable as discrete, in which case the heuristic
check is bypassed.
The heuristic is as follows: First, are all the values of the variable
``reasonably round'', where this is taken to mean that they are all
integer multiples of 0.25? If this criterion is met, we then ask
whether the variable takes on a ``fairly small'' set of distinct
values, where ``fairly small'' is defined as less than or equal to 8.
If both conditions are satisfied, the variable is automatically
considered discrete.
To mark a variable as discrete you have two options.
\begin{enumerate}
\item From the graphical interface, select ``Variable, Edit
Attributes'' from the menu. A dialog box will appear and, if the
variable seems suitable, you will see a tick box labeled ``Treat
this variable as discrete''. This dialog box can also be invoked
via the context menu (right-click on a variable) or by pressing the
F2 key.
\item From the command-line interface, via the \texttt{discrete}
command. The command takes one or more arguments, which can be
either variables or list of variables. For example:
\begin{code}
list xlist = x1 x2 x3
discrete z1 xlist z2
\end{code}
This syntax makes it possible to declare as discrete many
variables at once, which cannot presently be done via the graphical
interface. The switch \option{reverse} reverses the declaration of a
variable as discrete, or in other words marks it as continuous.
For example:
\begin{code}
discrete foo
# now foo is discrete
discrete foo --reverse
# now foo is continuous
\end{code}
\end{enumerate}
The command-line variant is more powerful, in that you can mark a
variable as discrete even if it does not seem to be suitable for
this treatment.
Note that marking a variable as discrete does not affect its content.
It is the user's responsibility to make sure that marking a variable
as discrete is a sensible thing to do. Note that if you want to
recode a continuous variable into classes, you can use the
\texttt{genr} command and its arithmetic functions, as in the
following example:
\begin{code}
nulldata 100
# generate a variable with mean 2 and variance 1
genr x = normal() + 2
# split into 4 classes
genr z = (x>0) + (x>2) + (x>4)
# now declare z as discrete
discrete z
\end{code}
Once a variable is marked as discrete, this setting is remembered when
you save the file.
\section{Commands for discrete variables}
\label{discr-commands}
\subsection{The \texttt{dummify} command}
\label{discr-dummify}
The \texttt{dummify} command takes as argument a series $x$ and creates
dummy variables for each distinct value present in $x$, which must
have already been declared as discrete. Example:
\begin{code}
open greene22_2
discrete Z5 # mark Z5 as discrete
dummify Z5
\end{code}
The effect of the above command is to generate 5 new dummy variables,
labeled \texttt{DZ5\_1} through \texttt{DZ5\_5}, which correspond to
the different values in \texttt{Z5}. Hence, the variable
\texttt{DZ5\_4} is 1 if \texttt{Z5} equals 4 and 0 otherwise. This
functionality is also available through the graphical interface by
selecting the menu item ``Add, Dummies for selected discrete variables''.
The \texttt{dummify} command can also be used with the following
syntax:
\begin{code}
list dlist = dummify(x)
\end{code}
This not only creates the dummy variables, but also a named list (see
section~\ref{named-lists}) that can be used afterwards. The
following example computes summary statistics for the variable \texttt{Y} for
each value of \texttt{Z5}:
\begin{code}
open greene22_2
discrete Z5 # mark Z5 as discrete
list foo = dummify(Z5)
loop foreach i foo
smpl $i --restrict --replace
summary Y
endloop
smpl --full
\end{code}
% $
Since \texttt{dummify} generates a list, it can be used directly
in commands that call for a list as input, such as \texttt{ols}. For
example:
\begin{code}
open greene22_2
discrete Z5 # mark Z5 as discrete
ols Y 0 dummify(Z5)
\end{code}
\subsection{The \texttt{freq} command}
\label{discr-freq}
The \texttt{freq} command displays absolute and relative frequencies
for a given variable. The way frequencies are counted depends on
whether the variable is continuous or discrete. This command is also
available via the graphical interface by selecting the ``Variable,
Frequency distribution'' menu entry.
For discrete variables, frequencies are counted for each distinct
value that the variable takes. For continuous variables, values are
grouped into ``bins'' and then the frequencies are counted for each
bin. The number of bins, by default, is computed as a function of the
number of valid observations in the currently selected sample via the
rule shown in Table~\ref{tab:bins}. However, when the command is
invoked through the menu item ``Variable, Frequency Plot'', this
default can be overridden by the user.
\begin{table}[htbp]
\centering
\begin{tabular}{cc}
\hline
Observations & Bins \\
\hline
$8 \le n < 16$ & 5 \\
$16 \le n < 50 $ & 7 \\
$50 \le n \le 850 $ & $\lceil \sqrt{n} \rceil$ \\
$n > 850 $ & 29 \\
\hline
\end{tabular}
\caption{Number of bins for various sample sizes}
\label{tab:bins}
\end{table}
For example, the following code
%
\begin{code}
open greene19_1
freq TUCE
discrete TUCE # mark TUCE as discrete
freq TUCE
\end{code}
%
yields
%
\begin{code}
Read datafile /usr/local/share/gretl/data/greene/greene19_1.gdt
periodicity: 1, maxobs: 32,
observations range: 1-32
Listing 5 variables:
0) const 1) GPA 2) TUCE 3) PSI 4) GRADE
? freq TUCE
Frequency distribution for TUCE, obs 1-32
number of bins = 7, mean = 21.9375, sd = 3.90151
interval midpt frequency rel. cum.
< 13.417 12.000 1 3.12% 3.12% *
13.417 - 16.250 14.833 1 3.12% 6.25% *
16.250 - 19.083 17.667 6 18.75% 25.00% ******
19.083 - 21.917 20.500 6 18.75% 43.75% ******
21.917 - 24.750 23.333 9 28.12% 71.88% **********
24.750 - 27.583 26.167 7 21.88% 93.75% *******
>= 27.583 29.000 2 6.25% 100.00% **
Test for null hypothesis of normal distribution:
Chi-square(2) = 1.872 with p-value 0.39211
? discrete TUCE # mark TUCE as discrete
? freq TUCE
Frequency distribution for TUCE, obs 1-32
frequency rel. cum.
12 1 3.12% 3.12% *
14 1 3.12% 6.25% *
17 3 9.38% 15.62% ***
19 3 9.38% 25.00% ***
20 2 6.25% 31.25% **
21 4 12.50% 43.75% ****
22 2 6.25% 50.00% **
23 4 12.50% 62.50% ****
24 3 9.38% 71.88% ***
25 4 12.50% 84.38% ****
26 2 6.25% 90.62% **
27 1 3.12% 93.75% *
28 1 3.12% 96.88% *
29 1 3.12% 100.00% *
Test for null hypothesis of normal distribution:
Chi-square(2) = 1.872 with p-value 0.39211
\end{code}
%
As can be seen from the sample output, a Doornik-Hansen test for
normality is computed automatically. This test is suppressed for
discrete variables where the number of distinct values is less than
10.
This command accepts two options: \option{quiet}, to avoid
generation of the histogram when invoked from the command line and
\option{gamma}, for replacing the normality test with Locke's
nonparametric test, whose null hypothesis is that the data follow a
Gamma distribution.
If the distinct values of a discrete variable need to be saved, the
\texttt{values()} matrix construct can be used (see chapter
\ref{chap:matrices}).
\subsection{The \texttt{xtab} command}
\label{discr-xtab}
The \texttt{xtab} command cab be invoked in either of the following
ways. First,
%
\begin{code}
xtab ylist ; xlist
\end{code}
%
where \texttt{ylist} and \texttt{xlist} are lists of discrete
variables. This produces cross-tabulations (two-way frequencies) of
each of the variables in \texttt{ylist} (by row) against each of the
variables in \texttt{xlist} (by column). Or second,
%
\begin{code}
xtab xlist
\end{code}
%
In the second case a full set of cross-tabulations is generated; that
is, each variable in \texttt{xlist} is tabulated against each other
variable in the list. In the graphical interface, this command is
represented by the ``Cross Tabulation'' item under the View menu,
which is active if at least two variables are selected.
Here is an example of use:
%
\begin{code}
open greene22_2
discrete Z* # mark Z1-Z8 as discrete
xtab Z1 Z4 ; Z5 Z6
\end{code}
which produces
\begin{code}
Cross-tabulation of Z1 (rows) against Z5 (columns)
[ 1][ 2][ 3][ 4][ 5] TOT.
[ 0] 20 91 75 93 36 315
[ 1] 28 73 54 97 34 286
TOTAL 48 164 129 190 70 601
Pearson chi-square test = 5.48233 (4 df, p-value = 0.241287)
Cross-tabulation of Z1 (rows) against Z6 (columns)
[ 9][ 12][ 14][ 16][ 17][ 18][ 20] TOT.
[ 0] 4 36 106 70 52 45 2 315
[ 1] 3 8 48 45 37 67 78 286
TOTAL 7 44 154 115 89 112 80 601
Pearson chi-square test = 123.177 (6 df, p-value = 3.50375e-24)
Cross-tabulation of Z4 (rows) against Z5 (columns)
[ 1][ 2][ 3][ 4][ 5] TOT.
[ 0] 17 60 35 45 14 171
[ 1] 31 104 94 145 56 430
TOTAL 48 164 129 190 70 601
Pearson chi-square test = 11.1615 (4 df, p-value = 0.0248074)
Cross-tabulation of Z4 (rows) against Z6 (columns)
[ 9][ 12][ 14][ 16][ 17][ 18][ 20] TOT.
[ 0] 1 8 39 47 30 32 14 171
[ 1] 6 36 115 68 59 80 66 430
TOTAL 7 44 154 115 89 112 80 601
Pearson chi-square test = 18.3426 (6 df, p-value = 0.0054306)
\end{code}
Pearson's $\chi^2$ test for independence is automatically displayed,
provided that all cells have expected frequencies under independence
greater than $10^{-7}$. However, a common rule of thumb states that
this statistic is valid only if the expected frequency is 5 or
greater for at least 80 percent of the cells. If this condition is not
met a warning is printed.
Additionally, the \option{row} or \option{column} options can be
given: in this case, the output displays row or column percentages,
respectively.
If you want to cut and paste the output of \texttt{xtab} to some other
program, e.g.\ a spreadsheet, you may want to use the \option{zeros}
option; this option causes cells with zero frequency to display the
number 0 instead of being empty.
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