Nicholas J. Cox

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Research Papers in Economics

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The Stata Journal, electronic version (ISSN 1536-8734) is a publication of Stata Press, and Stata is a registered
trademark of StataCorp LP.The Stata Journal (2004)
4, Number 4, pp. 488–489
Stata tip 15: Function graphs on the ﬂy
Nicholas J. Cox
University of Durham, UK
n.j.cox@durham.ac.uk
[G] graph twoway function gives several examples of how function graphs may
be drawn on the ﬂy. The manual entry does not quite explain the full ﬂexibility and
versatility of the command. Here is a further advertisement on its behalf. To underline
a key feature: you do not need to create variables following particular functions. The
command handles all of that for you. We will look at two more examples.
A common simple need is to draw a circle. A trick with twoway function is to draw
two half-circles, upper and lower, and combine them. If you are working in some scheme
using color, you will usually also want to ensure that the two halves are shown in the
same color. The aspect() option was explained in Cox (2004).
. twoway function sqrt(1 - x * x), ra(-1 1) ||
> function -sqrt(1 - x * x), ra(-1 1) aspect(1)
> legend(off) yla(, ang(h)) ytitle(, orient(horiz)) clp(solid)
￿1
￿.5
0
.5
1
y
￿1
￿.5 0 .5 1
x
MacKay (2003, 316) asserts that, if we transform beta distributions of variables P
between 0 and 1 to the corresponding densities over logit P =l n [ P/(1 − P)], then
we ﬁnd always pleasant bell-shaped densities. In contrast, densities over P may have
singularities at P =0a n dP = 1. This is the kind of textbook statement that should
provoke some play with friendly statistical graphics software.
To explore MacKay’s assertion, we need a standard result on changing variables (see,
for example, Evans and Rosenthal 2004, theorems 2.6.2 and 2.6.3). Suppose that P is
an absolutely continuous random variable with density function fP, h is a function that
is diﬀerentiable and monotone, and X = h(P). The density function of X is then
c ￿ 2004 StataCorp LP gr0010N. J. Cox 489
fX(x)=
fP{h−1(x)}
|h′{h−1(x)}|
In our case, h(P) = logit P, so that h−1(X)=e x p ( X)/{1+exp(X)} and h′{h−1(X)} =
{1 + exp(X)}2/exp(X). In Stata terms, beta densities transformed to the logit scale
are the product of betaden(p) or betaden(invlogit(x)) and exp(x)/(1+exp(x))^2.
The latter term may be recognized as a logistic density function, which always has a
bell shape.
An example pair of original and transformed distributions is given by the commands
below. To explore further in parameter space, you need only vary the parameters from
0.5 and 0.5 (and, if desired, to vary the range).
. twoway function betaden(0.5,0.5,x), ytitle(density) xtitle(p)
. twoway function betaden(0.5,0.5,invlogit(x)) * (exp(x) / (1 + exp(x))^2),
> ra(-10 10) ytitle(density) xtitle(logit p)
0
2
4
6
d
e
n
s
i
t
y
0 .2 .4 .6 .8 1
p
0
.
0
5
.
1
.
1
5
d
e
n
s
i
t
y
￿10
￿5 0 5 10
logit p
References
Cox, N. J. 2004. Stata tip 12: Tuning the plot region aspect ratio. Stata Journal 4(3):
357–358.
Evans, M. J. and J. S. Rosenthal. 2004. Probability and Statistics: The Science of
Uncertainty. New York: W. H. Freeman.
MacKay, D. J. C. 2003. Information Theory, Inference, and Learning Algorithms. Cam-
bridge: Cambridge University Press. Also available at
http://www.inference.phy.cam.ac.uk/mackay/itprnn/book.html.

Stata tip 15: Function graphs on the fly

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Research Papers in Economics