.- help for ^ordplot^ .-Cumulative distribution plot of ordinal variable ------------------------------------------------

^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( ) > ^ instead.

^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 17. male Graduate A 118 18. male Graduate B 176 19. male Graduate C 345 20. male Graduate D 141 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 37. female Graduate A 19 38. female Graduate B 27 39. female Graduate C 128 40. female Graduate D 13

^. 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: Wadsworth & Brooks-Cole, 187-389.

Tukey, J.W. 1977. Exploratory data analysis. Reading, MA: Addison-Wesley.

Author ------

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

Elizabeth Allred and Ronan Conroy made very helpful comments. The implementation of ^plabel^, ^pline^ and ^ptick^ is based on an idea of Patrick Royston.

Also see --------

On-line: help for @graph@, @functions@, @keyplot@, @distplot@ (if installed)