help iquantile

Interpolated quantiles

Syntax

iquantile
varlist
[if] [in]
[weight]
[
,
allobs
by(byvarlist)
format(format)
p(numlist)
list_options
]

fweights and aweights are allowed.

Description

iquantile calculates and displays quantiles estimated by linear
interpolation in the mid-distribution function. The user may specify one
or more numeric variables, one or more grouping variables and one or
more quantiles.

iquantile also displays the number of observations with non-missing values used in each calculation.

Remarks

By quantiles here are meant those summaries defined by the fact that
some percent of a batch of values is lower. Thus the median (50%) and
the quartiles (25% and 75%) are examples. Most commands in Stata that
calculate such summaries select particular sample values or at most
average two sample values. That is often sufficient for the purpose
intended. iquantile offers an alternative, which is perhaps most
useful when the number of distinct values is small. For example,
although the variable in question may be measured coarsely, say on an
integer scale, and many ties may be observed, it may be hoped or
imagined that a property on a continuous scale lies beneath. Note that
iquantile performs no white magic, just elementary linear
interpolation.

The cumulative probability is here defined as

SUM counts of values below + (1/2) count of this value

.
SUM counts of all values

With terminology from Tukey (1977, 496-497), this could be called a
`split fraction below'. It is also a `ridit' as defined by Bross (1958):
see also Fleiss et al. (2003, 198-205), Beder and Heim (1990) or Flora (1988). Yet again, it
is also the mid-distribution function of Parzen (1993, 3295) and the
grade function of Haberman (1996, 240-241). Parzen's term appears best for
the purposes of this command. The numerator is a `split count'. Using
this numerator, rather than

SUM counts of values below

or

SUM counts of values below + count of this value,

treats distributions symmetrically. For applications to plotting ordinal
categorical data, see Cox (2004). For yet more discussion and references,
see the help for distplot as updated in Cox (2019) (or any later update).

The technique used in iquantile is illustrated by a worked example using Mata
calculator-style. We first enter the data as values and frequencies:

: y = 2, 3, 4, 5

: f = 2, 9, 8, 8

Then we can work out the cumulative frequencies:

: runningsum(f)
1 2 3 4
+---------------------+
1 | 2 11 19 27 |
+---------------------+

Subtract half the frequencies and get the cumulative proportions,
symmetrically considered, i.e. the mid-distribution function:

: runningsum(f) :- f/2
1 2 3 4
+-------------------------+
1 | 1 6.5 15 23 |
+-------------------------+

: (runningsum(f) :- f/2) / 27
1 2 3 4
+---------------------------------------------------------+
1 | .037037037 .2407407407 .5555555556 .8518518519 |
+---------------------------------------------------------+

: cup = (runningsum(f) :- f/2) / 27

To get the median, we need to interpolate between the 2nd and 3rd
values of y.

: y[2] + (y[3] - y[2]) * (0.5 - cup[2]) / (cup[3] - cup[2])
3.823529412

iquantile uses list to show results.

iquantile issues a warning if any quantile was calculated by
extrapolation, i.e. it lies in one or other tail of the distribution
beyond the observed mid-distribution function. Such results should be
treated with extreme caution.

If the data consist of a single distinct value, then exactly that value
is always returned as a quantile.

iquantile uses Mata for its innermost calculations.
Thus Stata 9 up is required.

Options

allobs specifies use of all possible observations when missing
values are present. Suppose you ask for results for two variables
x y and there are missing values in either variable or in both. Then by
default iquantile works only with observations with non-missing
values on both variables. The purpose of allobs is to override
this default and to insist on using all observations with non-missing
values, so ignoring missing values separately for each variable
specified.

by() specifies that calculations are to be carried out separately
for the distinct groups defined by byvarlist. The variable(s) in
byvarlist may be numeric or string.

format() specifies a numeric format to be used to display the
quantiles. This option has no lasting effect.

p() specifies a numlist of integers betweem 1 and 99 to
indicate the p% quantiles. If p() is not specified, it
defaults to 50, i.e. the 50% point or median is calculated.
p(25(25)75) specifies the median and quartiles.

list_options are options of list other than noobs and
subvarname. They may be specified to tune the display of quantiles.

Examples

. sysuse auto, clear

. iquantile mpg

. iquantile mpg, p(25 50 70)

. iquantile mpg, p(25 50 70) format(%2.1f)

. iquantile mpg, p(25 50 70) format(%2.1f) by(rep78)

. iquantile mpg weight price

. iquantile mpg rep78, by(foreign)

. iquantile mpg rep78, allobs by(foreign)

Saved results

Saved results are best explained by example. After iquantile mpg,
saved results include r(mpg_50_1) and r(mpg_50_1_epolate).
The elements of the name for both are first, the variable name (if
necessary, abbreviated to 16 characters); second, the percent defining
the quantile; third, the number of the group in question in the
observations processed (here, the first of one). The extra flag
epolate indicates whether extrapolation was needed (1 for true, 0
for false).

In addition, the number of non-missing values is saved as r(n_1)
upwards by default, depending on whether there is just one group or there are
more groups; and as something like r(n_mpg_1) upwards if the
allobs option is specified (that is, the number of non-missing
values is saved for each variable as well as each group specified).

Author

Nicholas J. Cox, Durham University, UK

n.j.cox@durham.ac.uk

Acknowledgments

This command grew out of a thread on Statalist started by
Taggert J. Brooks. See
http://www.stata.com/statalist/archive/2009-01/msg00652.html

Cees van der Eijk asked how the command handles observations with missing values and thereby prompted the addition of the allobs option and of
display of the number of observations used. Thanks to Cees for
wide-ranging discussion.

References

Beder, J. H. and R. C. Heim.
1990. On the use of ridit analysis. Psychometrika 55: 603-616.

Bross, I. D. J. 1958. How to use ridit analysis. Biometrics
14: 38-58.

Cox, N. J. 2004. Speaking Stata: Graphing categorical and
compositional data. Stata Journal 4(2): 190-215.
See Section 5.
http://www.stata-journal.com/sjpdf.html?articlenum=gr0004

Cox, N. J. 2019. Software update: Distribution function plots.
Stata Journal 19(1): 260.

Fleiss, J. L., B. Levin, and M. C. Paik. 2003.
Statistical Methods for Rates and Proportions.
Hoboken, NJ: Wiley.

Flora, J. D. 1988. Ridit analysis. In
Encyclopedia of Statistical Sciences,
ed. S. Kotz and N. L. Johnson, (8) 136-139.
New York: Wiley.

Haberman, S. J. 1996.
Advanced Statistics Volume I: Description of Populations.
New York: Springer.

Parzen, E. 1993. Change PP plot and continuous sample quantile
function. Communications in Statistics -Theory and Methods
22: 3287-3304.

Tukey, J. W. 1977. Exploratory Data Analysis.
Reading, MA: Addison-Wesley.

Also see

help for summarize, centile, pctile,
tabstat, distplot (if installed), hdquantile (if installed)