Confidence ellipses -------------------
^ellip6^ ^yvar^ [^xvar^] [^if^ exp] [^in^ range] [^,^ ^l^evel^(^#^)^ {^means^ | ^coefs^ [^p^ool^(^#^)^]} ^c^onstant^(^string [#]^)^ ^g^enerate^(^newyvar newxvar^)^ ^a^dd^(^oldyvar oldxvar^)^ ^yf^orm[(%fm > t)] ^xf^orm[(%fmt)] ^np^oints^(^#^)^ ^evr(^#^)^ ^replace^ graph_options]
Description -----------
^ellip6^ is an update in Stata 6 of ^ellip^ (Alexandersson 1998), which was for Stata 5. The original program has been renamed ^ellip5^ to ensure backward compatibility and less name confusion. But ellip6 has more features and has been more tested than ellip5. ellip6 graphs confidence ellipses for approximately normally distributed data.
There are two options of centering the confidence ellipse. The default and the means option graph a confidence ellipse centered around the means of two variables, yvar and xvar. With this syntax, ellip6 is used as a stand-alone command. The coefs option graphs a confidence ellipse centered around the regression coefficients of yvar and xvar from an immediately preceding ^regress^. With this syntax, ellip6 remains a post-estimation command. If you restricted regress to a portion of the data using if or in, then you will generally want to use the same conditions with ellip..., coefs.
The size of the confidence ellipse is determined by the boundary constant c. In the overall default and in the the means option default, c is 4. In the coefs option default, c is 2 * F with (p, n-p) degrees of freedom.
Options -------
^l^evel^(^#^)^ specifies the confidence level, in percent, for calculation of the confidence ellipse; the default is 95. level() cannot be used with c(sd [#]).
^means^ | ^coefs^ specifies how to center the confidence ellipse. The default is the means option. If coefs is specified without xvar, then the _cons in regress is used for xvar.
^p^ool^(^#^)^ displays a "locus curve" of # weighted regressions for the two overlaid confidence ellipses due to gen() and add(). pool() also displays a separate table of the locus curve dots. The weights are fractional importance weights on the data not specified by if/in. The option must be used together with if/in, coefs, gen() and add(). See Alexandersson (1998) for the background of this option.
^g^enerate^(^newyvar newxvar^)^ generates two new variables, newyvar and newxvar, which define the confidence ellipse. If the current dataset contains fewer observations than the number of points in ^np^oints^(^#^)^ then the length of the dataset will be expanded accordingly with missing values, and a warning message is displayed.
^a^dd^(^oldyvar oldxvar^)^ adds an old confidence ellipse to the new confidence ellipse. The result is two overlaid confidence ellipses in the same graph. add() may be used with gen() but does not require gen().
^yf^orm and ^yf^orm[(%fmt)] specify the display format of the y-axis. The default is to use a %9.0g format. yf specifies that the y-axis uses the yvar display format. ^yf^[(%fmt)] specifies that the y-axis uses the %fmt display format. See [R] format.
^xf^orm and ^xf^orm[(%fmt)] specifies the display format of the x-axis; see the ^fy^ and ^fy^[(%fmt)] option above.
^np^oints^(^#^)^ specifies the number of points to be calculated for the confidence ellipse. The default is 400. You seldom have to use this option, but users with Small Stata may want to lower the number and if the output looks jagged try increasing the number.
^evr^(^#^)^ specifies the error variance ratio, where # is a floating point number between 0 and 10^36. The default is 1. evr(0) corresponds to regression of x on y, evr(1) to orthogonal regression, and a larger number, say evr(999) corresponds to regression of y on x. For a discussion of evr-regression, see McCartin (n.d.)
^replace^ will replace any existing variables in gen(newyvar newxvar).
graph_options are any options allowed with -graph, twoway- except by(). The by() bug has been fixed in ellip7. Defaults are: c(l) s(.) t1(" ") t2(" ") l1(yvar) b2(xvar) [or c(ll) s(..) if add() is specified, and l1(Estimated yvar) and b2(Estimated xvar) if coefs is specified].
^c^onstant^(^string [#]^)^ specifies the boundary constant c. The default is equivalent to c(sd 4) for means and to c(f 2) for coefs. The 8 allowed strings are:
^sd^ [#] displays a standard deviation (a.k.a covariance, concentration, inertia, or error) ellipse, where c = #^2. The confidence level, in percent, is (1 - exp^(-#/2)) * 100. The default is sd 4, which has the same mean and variance as the bivariate normal distribution. In this sense, it is the ellipse which is the most representative of the data points without any a priori statistical assumptions concerning their origin. The default corresponds to 95% INDIVIDUAL confidence intervals, or 86% JOINT confidence intervals. sd cannot be used with level(). Distances from the center to the ellipse boundary do NOT describe the standard deviation along directions other than along the principal axes; if you need to work with geographical units, see Gong (2002). ^tsq^ [#] or ^hotel^ [#] displays a Hotelling T-square confidence ellipse with # parameters, where c = # * (n-1) / (n-#) * F = T2. The default is tsq 2 or hotel 2. Use this option when mu is known but V is unknown.
^tsqn^ [#] or ^hoteln^ [#] displays a Hotelling T-square confidence ellipse with # parameters, where c = T2 / n. The default is tsq 2 or hotel 2. Use this option when mu and V are unknown. ^ptsqn^ [#] or ^photeln^ [#] displays a Hotelling T-square confidence ellipse for prediction with # parameters, where c = T2 * (n+1) / n. The default is ptsq 2 or photel 2. This confidence ellipse predicts the next observationin the population at a given confidence (aka prediction ellipse) and approximates a region containing a specified percentage of the population (aka tolerance ellipse). Use this option when mu and V are unknown. Other confidence ellipses for prediction are conceivable but little known and not implemented.
^chisq^ [#] displays a "true" Chi-square confidence ellipse, where c = chisq with # d.f. The default is chisq 2. Use this option when mu and V are known.
^chisqn^ [#] or ^pchisq^ [#] displays a Chi-square confidence ellipse, where c = chisq/n with # d.f. The default is chisq 2. Use this option when V is known but mu is unknown. It is more common to instead use chisq or f. ^f^ [#] displays a confidence ellipse based on F(#,n-#). The default is f 2, also known as Douglas confidence ellipse. ^fadj^ [#] displays an "F-adjusted" confidence ellipse, where c = F(2,n-#). The default is fadj 2, also known as Douglas confidence ellipse.
Remarks -------
This version 1.2 of ^ellip^ is the latest version for Stata 6 and is now named ^ellip6^. The new ellip6 improve the formatting options, and introduce the replace and evr(#) options and the ptsqn, chisqn and fadj arguments to constant(). ^ellip6^ saves in r().
ellip6 is a graphics command, but the option gen() may lengthen the dataset. There is no option for overlaid confidence ellipses. To create multiple overlaid confidence ellipses I recommend use a series of ellip ..., gen() followed by Nick Cox's ^muxyplot^ yvarlist xvarlist. The complementary command muxyplot with helpfile can be downloaded separately from SSC.
Stata 8 is now shipping, which has a new graphics programming language, and many new graphics features, including a new ||-separator and ()-binding notation for overlaid graphs.
^ellip7^ for Stata 7 is posted at the same time as this ellip6. Any new development by the author will be in Stata 8 or later.
Examples --------
. ellip y x graphs a standard ellipse around the means of y and x, where c = 4.
. ellip y x, g(sdy sdx) graphs a standard ellipse around the means of y and x, and generates the ellipse variables sdy and sdx with 400 obs. . ellip y x, c(hoteln) a(sdy sdx) graphs a 95% Hotelling confidence ellipse around the means of y and x, where c = T2 / n, and also graphs the previously generated standard ellipse. The result is an overlaid graph with two confidence ellipses.
. reg dv iv . ellip iv, coefs c(chisq) graphs a 95% Chi-square confidence ellipse around the regression coeffient for iv and around _cons in the preceding regression
Author ------
Anders Alexandersson Mississippi State University, MS, USA Email: aalex@@its.msstate.edu
References ----------
Alexandersson, A. 1998. gr32: Confidence ellipses. Stata Technical Bulletin 46: 10-13. Batschelet, E. 1981. Circular Statistics in Biology. London and New York: Academic Press. Gong, J. 2002. Clarifying the Standard Deviational Ellipse. Geographical Analysis 34(2): 155-167. Johnson, R., and D. Wichern. 2002. 5th ed. Applied Multivariate Statistical Analysis. Upper Saddle River, NJ: Prentice Hall. McCartin, B. n.d. A Geometric Characterization of Linear Regression. Statistics: A Journal of Theoretical and Applied Statistics.
Also see --------
Manual: [R] graph, [R] gph STB: STB-46 gr32, STB-34 gr20 Online: @graph@, @regress@, @gphdt@, @level@, @muxyplot@ (if installed)