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help for ^ordplot^
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Cumulative distribution plot of ordinal variable
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^ordplot^ ordvar [^if^ exp] [^in^ range] [fweight]
[ ^, by(^catvar^) miss^ing ^rev^erse ^sc^ale^(^scale^) pow^er^(^#^)^
^assc^ores^(^numlist^) le lt ge gt f^raction ^pla^bel^(^numlist^)^
^pli^ne^(^numlist^) pti^ck^(^numlist^)^ keyplot_options ]

Description
-----------

^ordplot^ produces a cumulative distribution plot for an ordinal numeric
variable ordvar. The cumulative probability is plotted on the y axis and
ordvar is plotted on the x axis.

^ordplot^ is designed primarily for data which are, or can be collapsed to,
a contingency table with frequencies for an ordinal response and an ordinal
or nominal covariate.

^ordplot^ may also be used with variables on interval or ratio scales.

^ordplot^ requires ^keyplot^ to be installed.

Remarks
-------

The cumulative probability P is here defined by default as

SUM counts in categories below + (1/2) count in this category
-------------------------------------------------------------.
SUM counts in all categories

With terminology from Tukey (1977, pp.496-497), this could be called a
`split fraction below'. It is also a `ridit' as defined by Bross (1958):
see also Fleiss (1981, pp.150-7) or Flora (1988). The numerator is a
`split count'. Using this numerator, rather than

SUM counts in categories below

or

SUM counts in categories below + count in this category,

means that more use is made of the information in the data. Either
alternative would always mean that some fractions are identically 0
or 1, which tells us nothing about the data. In addition, there are
fewer problems in showing the cumulative distribution on any
transformed scale (e.g. logit) for which the transform of 0 or 1 is
not plottable. If desired, these alternatives are available through
the ^lt^ and ^le^ options, respectively.

A plot of the complement of this cumulative probability, 1 - P, may be
obtained through the ^reverse^ option, in which case the pertinent
alternatives are available through the ^ge^ or ^gt^ options.

Further information on working with counted fractions and folded
transformations for probability scales is available in Tukey (1960,
1961, 1977), Atkinson (1985), Cox and Snell (1989) and Emerson (1991).
Some of the transformations used here appear as link functions in
the literature on generalized linear models (e.g. McCullagh and Nelder
1989; Aitkin et al. 1989).

Options
-------

^by( )^ specifies a categorical variable catvar, with 10 or fewer categories.
Cumulative distributions will be shown for each category of catvar.

^missing^ specifies that observations for which catvar is missing will
be included in the plot if ^by( )^ is specified. The default is to
exclude them.

^reverse^ specifies the use of reverse cumulative probabilities (1 - P
in notation above), a.k.a. the complementary distribution, reliability,
survival or survivor function.

^scale( )^ indicates a scale for plotting cumulative distributions.
^logit^ (synonym ^flog^ for `folded logarithm'), ^froot^, ^folded^,
^loglog^, ^cloglog^, ^normal^ or ^Gaussian^, ^percent^ and ^raw^ are
allowed.

^raw^ is the default.

Given cumulative probabilities P, defined as above, and using log
to denote natural logarithm (base e),

^logit^ or ^flog^ means log (P / (1 - P)) = log P - log (1 - P).

^froot^ for `folded root' means sqrt(P) - sqrt(1 - P).

^folded^ means `folded power' or P^^power - (1 - P)^^power.
The power to be used must be specified through the ^power( )^ option
and should be non-zero. For reference, note that, apart from scaling
constants, good emulations of the angular (arcsine square root)
transformation and of the probit transformation are obtained by
powers of 0.41 and 0.14 respectively. As power approaches 0, the
folded power tends to the logit.

^loglog^ means -log(-log P).

^cloglog^ means log(-log (1 - P)).

^normal^ or ^Gaussian^ means ^invnorm(^P^)^. See help on @functions@.

^percent^ means 100 P.

Under ^reverse^ P is replaced by 1 - P, and vice versa, in these
operations.

^power( )^ specifies a power for folded power transformation. See above.

^asscores( )^ specifies an ascending numlist to use as alternative scores in
plotting values on the x axis. The elements of the numlist must match
one-to-one with the distinct values of ordvar occurring in the
observations used and put into ascending order.

For example, if ordvar takes on values 1 2 3 4 5, ^asscores(1 4 5 6 9)^
will map 1 to 1, 2 to 4, etc.

^le^ (think "^l^ess than or ^e^qual to") specifies that probabilities are
to be calculated from counts for ordvar <= this category.

^lt^ (think "^l^ess ^t^han") specifies that probabilities are to be
calculated from counts for ordvar < this category.

^ge^ (think "^g^reater than or ^e^qual to") specifies that probabilities are
to be calculated from counts for ordvar >= this category. This
is allowed only with ^reverse^.

^gt^ (think "^g^reater ^t^han") specifies that probabilities are to be
calculated from counts for ordvar > this category. This is allowed
only with ^reverse^.

^fraction^ specifies use of the term "fraction" rather than "probability"
by the vertical axis of the plot. This option is cosmetic only
and not allowed with ^scale(percent)^.

^plabel(^numlist^)^, ^pline(^numlist^)^ and ^ptick(^numlist^)^ are for use if
the ^scale^ is ^logit^, ^flog^, ^froot^, ^folded^ with ^power^, ^loglog^,
^cloglog^, ^normal^ or ^Gaussian^. They specify labels, lines or ticks on t
> he
y axis on a probability or percent scale. Typically these will be more
intelligible and useful than labels, lines or ticks on the transformed
scales which are being plotted.

If the largest number in one or more of these numlists is >1, numbers
are treated as percents. Otherwise, numbers are treated as probabilities.
Numbers which are not plottable on the chosen scale, such as logit of
0 or 1, are ignored.

For ^scale^ of ^raw^ or ^percent^, use ^ylabel( )^, ^yline( )^ or ^ytick( )
> ^

^plabel( )^, ^pline( )^ or ^ptick( )^ may not be combined with the
corresponding ^ylabel( )^, ^yline( )^ or ^ytick( )^.

keyplot_options are options allowed with ^keyplot^.

Examples
--------

^. ordplot rep78^
^. ordplot rep78, by(foreign)^
^. ordplot rep78, by(foreign) yrev^
^. ordplot rep78, by(foreign) scale(logit)^
^pla(0.02 0.05 0.1(0.1)0.9 0.95 0.98)^
^. ordplot rep78, by(foreign) scale(logit) pla(2 5 10(10)90 95 98)^

Aitkin et al. (1989, p.242) reported data from a survey of student opinion
on the Vietnam War taken at the University of North Carolina in Chapel
Hill in May 1967. Students were classified by sex, year of study and
the policy they supported, given choices of

A The US should defeat the power of North Vietnam by widespread bombing
of its industries, ports and harbours and by land invasion.

B The US should follow the present policy in Vietnam.

C The US should de-escalate its military activity, stop bombing North
Vietnam, and intensify its efforts to begin negotiation.

D The US should withdraw its military forces from Vietnam immediately.

(They also report response rates (p.243), averaging 26% for males and 17%
for females.)

Suppose that, underneath the labels below, the value labels of ^sex^
are also called ^sex^ and ^policy^ runs 1/4.

^. l^

sex      year    policy          freq
1.      male         1         A           175
2.      male         1         B           116
3.      male         1         C           131
4.      male         1         D            17
5.      male         2         A           160
6.      male         2         B           126
7.      male         2         C           135
8.      male         2         D            21
9.      male         3         A           132
10.      male         3         B           120
11.      male         3         C           154
12.      male         3         D            29
13.      male         4         A           145
14.      male         4         B            95
15.      male         4         C           185
16.      male         4         D            44
21.    female         1         A            13
22.    female         1         B            19
23.    female         1         C            40
24.    female         1         D             5
25.    female         2         A             5
26.    female         2         B             9
27.    female         2         C            33
28.    female         2         D             3
29.    female         3         A            22
30.    female         3         B            29
31.    female         3         C           110
32.    female         3         D             6
33.    female         4         A            12
34.    female         4         B            21
35.    female         4         C            58
36.    female         4         D            10

^. ordplot policy [w=freq] if sex=="male":sex,^
^by(year) xla(1/4) yla(0(0.2)1) gap(3)^

^. ordplot policy [w=freq] if sex=="female":sex,^
^by(year) xla(1/4) yla(0(0.2)1) gap(3)^

Fienberg (1980, pp.54-55) reports data from Duncan, Schuman and Duncan
(1973) from 1959 and 1971 surveys of a large American city asking "Are
the radio and TV networks doing a good job, just a fair job, or a poor job?"

Suppose that, underneath the labels below, ^opinion^ runs 1/3.

^. l^

group    opinion      freq
1. 1959 Black       Good        81
2. 1959 Black       Fair        23
3. 1959 Black       Poor         4
4. 1959 White       Good       325
5. 1959 White       Fair       253
6. 1959 White       Poor        54
7. 1971 Black       Good       224
8. 1971 Black       Fair       144
9. 1971 Black       Poor        24
10. 1971 White       Good       600
11. 1971 White       Fair       636
12. 1971 White       Poor       158

. ^tab group opinion [w=freq], row^

^. ordplot opinion [w=freq], by(group) sc(logit) xla(1/3)^
^pla(20(10)90 95 98 99)^

This shows a clear shift of opinion towards Poor from 1959 to 1971, and
a narrowing gap between Black and White.

Clogg and Shihadeh (1994, p.156) give data from the 1988 General
Social Survey on answers to the question "When a marriage is troubled
and unhappy, do you think it is generally better for the children if
the couple stays together or gets divorced?". Responses "much better to
divorce", "better to divorce", "don't know", "worse to divorce" and
"much worse to divorce" were coded here as 1/5 with short value labels
"BETTER", "better", "?", "worse" and "WORSE", because ^graph^ in Stata 6.0
truncates value labels to the first 8 characters when shown as ^xlabel^s
or ^ylabel^s.

^. l^

sex      opinion          freq
1.     male       BETTER            84
2.     male       better           205
3.     male            ?           135
4.     male        worse           121
5.     male        WORSE            56
6.   female       BETTER           154
7.   female       better           330
8.   female            ?           178
9.   female        worse            72
10.   female        WORSE            49

It is not clear that the "don't know"s belong in the middle of the
scale. The point can be explored by graphs with and without those
values. The second uses scores 1 2 3 4 for 1 2 4 5. Either way, there
is a distinct separation between males and females, and logit gives
a more nearly linear pattern.

^. ordplot opinion [w=freq], by(sex) xla(1/5)^

^. ordplot opinion [w=freq], by(sex) xla(1/5) sc(logit)^
^pla(2 5 10(10)90 95 98)^

^. ordplot opinion [w=freq] if opinion != 3,^
^by(sex) xla(1/4) asscores(1/4)^

^. ordplot opinion [w=freq] if opinion != 3,^
^by(sex) xla(1/4) sc(logit) asscores(1/4)^
^pla(5 10(10)90 95)^

Knoke and Burke (1980, p.68) gave data from the 1972 General Social Survey
on church attendance. Suppose that, underneath the labels below, ^attend^
runs 1/3.

^. l^

group    attend          freq
1. young non-Catholic       low           322
2. young non-Catholic    medium           122
3. young non-Catholic      high           141
4.   old non-Catholic       low           250
5.   old non-Catholic    medium           152
6.   old non-Catholic      high           194
7.     young Catholic       low            88
8.     young Catholic    medium            45
9.     young Catholic      high           106
10.       old Catholic       low            28
11.       old Catholic    medium            24
12.       old Catholic      high           119

The ^reverse^ option ensures that higher attendance groups plot higher
on the graph. There are clear age and denomination effects and an indication
of an interaction between the two.

^. ordplot attend [w=freq], by(group) sc(logit) reverse^
^pla(0.1(0.1)0.9) xla(1/3)^

Box, Hunter and Hunter (1978, pp.145-9) gave data on five hospitals
on the degree of restoration (no improvement, partial functional
restoration, complete functional restoration) of certain joints impaired
by disease effected by a certain surgical procedure. (It is not clear
whether these data are real.) Hospital E is a referral hospital. Box
et al. carry out chi-square analyses, focusing on the difference between
Hospital E and the others.

Suppose that, underneath the labels below, ^restore^ runs 1/3.

^. l^

hospital    restore      freq
1.         A       none        13
2.         B       none         5
3.         C       none         8
4.         D       none        21
5.         E       none        43
6.         A    partial        18
7.         B    partial        10
8.         C    partial        36
9.         D    partial        56
10.         E    partial        29
11.         A   complete        16
12.         B   complete        16
13.         C   complete        35
14.         D   complete        51
15.         E   complete        10

^. ordplot restore [w=freq] , by(hospital)^
^pla(5 10(10)90 95) sc(logit) xla(1/3)^

References
----------

Aitkin, M., Anderson, D., Francis, B. and Hinde, J. 1989. Statistical
modelling in GLIM. Oxford: Oxford University Press.

Atkinson, A.C. 1985. Plots, transformations, and regression. Oxford:
Oxford University Press.

Box, G.E.P., Hunter, W.G. and Hunter, J.S. 1978. Statistics for
experimenters: an introduction to design, data analysis, and model
building. New York: John Wiley.

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

Clogg, C.C. and Shihadeh, E. 1994. Statistical models for ordinal
variables. Thousand Oaks, CA: Sage.

Cox, D.R. and Snell, E.J. 1989. Analysis of binary data. London:
Chapman and Hall.

Duncan, O.D., Schuman, H. and Duncan, B. 1973. Social change in a
metropolitan community. New York: Russell Sage Foundation.

Emerson, J.D. 1991. Introduction to transformation. In Hoaglin, D.C.,
Mosteller, F. and Tukey, J.W. (eds) Fundamentals of exploratory analysis
of variance. New York: John Wiley, 365-400.

Fienberg, S.E. 1980. The analysis of cross-classified categorical data.
Cambridge, MA: MIT Press.

Fleiss, J.L. 1981. Statistical methods for rates and proportions.
New York: John Wiley.

Flora, J.D. 1988. Ridit analysis. In Kotz, S. and Johnson, N.L. (eds)
Encyclopedia of statistical sciences. Wiley, New York, 8, 136-139.

Knoke, D. and Burke, P.J. 1980. Log-linear models. Beverly Hills, CA:
Sage.

McCullagh, P. and Nelder, J.A. 1989. Generalized linear models. London:
Chapman and Hall.

Tukey, J.W. 1960. The practical relationship between the common
transformations of percentages or fractions and of amounts. Reprinted in
Mallows, C.L. (ed.) 1990. The collected works of John W. Tukey. Volume VI:
More mathematical. Pacific Grove, CA: Wadsworth & Brooks-Cole, 211-219.

Tukey, J.W. 1961. Data analysis and behavioral science or learning to bear
the quantitative man's burden by shunning badmandments. Reprinted in
Jones, L.V. (ed.) 1986. The collected works of John W. Tukey. Volume III:
Philosophy and principles of data analysis: 1949-1964. Monterey, CA:

Author
------

Nicholas J. Cox, University of Durham, U.K.
n.j.cox@@durham.ac.uk

Acknowledgments
---------------