DATA TRANSFORMATIOOS IN REX;RESSIOO ANALYSIS WITH APPLICATIOOS
REIATIOOSHIPS
ro
STOCK - REX::RUI'lMENl'
David Ruppert
•
and
Rayrrond J. carroll
Department of Statistics
University of North carolina
Chapel Hill, NC 27514/USA
Abstract
we prcpose a nethodo1ogy for fitting theoretical m:xle1s to data. The dependent
variable (or response) and the m:xle1 are transformed in the
sam:!
way.
TWo types of
transformations, power transformation and weighting, are used together to rerove
skewness and to induce oonstant variance.
recruitment data of foor fish stocks.
conditional
•
1.
•
Irean
01r nethod is applied to the stock-
Also discussed are estinates of the
and the conditional quantiles of the original response.
Introduction
In regression analysis one seeks to establish a relationship between a response
y and independent variables
~
=
(X1' ••• '~)'.
Often the physical or biological
system generating the data suggest that in the absense of randan error y =
where f is a knCMl'l function and
~
is an unknCMl'l paraneter.
f(~,~)
In reality randan
variability, m:xleling errors, and IreaSurement errors will cause y to deviate fran
f(~,~).
Usually
~
is estinated by the (possibly nonlinear) least-squares estinator
~
which minimizes
N
"\
L
~
t - f(~,~»
2
(Y
t=l
where yt and
~
= (X
1t ,.·· ,Xkt ) , are the t-th OOservations of the response and the
independent variables, t=l, ••• ,N. The nethod of least squares is not uniquely
determined in the follO'Ning sense.
inplies that hey)
= h(f(~,~».
h(f(~,~»
= f(x,9)
In the absense of randan error, there is an infinity
of possible m:xle1s, one for each h.
and
If hey) is a IlOnotonic function then y
Taking hey) to be the new dependent
~riab1e
to be the new regression m:xle1, the 1east-squares estinate 9(h),
depending on h, minimizes
This paper addresses the question "heM should h be chosen?" In the ~t h was
often chosen so that the model was linear in the parameters, rot with the wide
•
availability of nallinear least squares software :inearization is no longer
necessary.
Instead h sho.lld be selected so that 9(h) has high statistical
efficiency in sane sense, say low asynptotic mean square error.
•
For nonlinear
models finite-saJ'll)le mean square errors are seldan known and large-saJ'll)le, or
asynptotic, mean square errors nust be used.
Least-squares estimation is asyrrptotically efficient. when the errors are
additive, normally distriroted, and haroscedastic, that is, if
(1)
where ~, ••• ,~ are independent N(O,~2) randan variables for sane ~2
> O.
If
(1)
fails to hold for the original response and model, it uay hold after these have been
transformed.
For exarrple, if the errors are nultiplicative and lognonnal, then
•
so that hey) = log(y) is the awropriate transfonnation.
In general, it is inpossible to know a priori heM the randan errors affect y.
It is, however, often reasonable to assune as an appraxination that for an unknown h
and then to estinate h fran the data.
Carroll and Ruppert (1984) introduced a methodology where h is assuned to
belong to a parametric class such as the class of power functions.
hey)
= h(y, \)
parameter.
Specifically,
where hey, \) is a known parametric family and \ is an unknown
Then~,~,
and \ are estinated simlltaneously, for exaIiple by naxiITn.Im
likelihood.
Box and Cox (1964) used a modified power transformation family
y(\)
=
(y\-l)/\
= log(y)
va
\=0,
which includes the log transforuation in a natural, Le., continuoos, fashion.
It
should be mentioned that Box and Cox were concerned with a quite different
transformation methodology.
Carroll and R1.JI:.t;lert (1984).
'!bey transform only the response, not the model: see
•
Another transformation family which, as we will see, is useful in stockrecruitment analysis is division by (ut)cx where u
is a known constant and cx is an
t
Since u t is a constant which does not depend on yt' a
transformation of this kind has no effect on skewness rot it induces
unknown parameter.
•
f
halDscedasticity if (Ut)cx is the standard deviation of Y • '!be constant u can be
t
t
one of the independent variables, a variable not depending on !t. or Yt' or a
function of such variables.
transformation".
of u
t
Division by (Ut)CX will be called a "power weighting
Note that U~ denotes ordinary, not Bax-eac, power transformation
so that U~ = 1 when cx = o.
This paper is restricted to a IOOdel canbining a Bax-eac power transformation
with a power weighting transformation:
(2)
However, the nethodology that is developed here can be applied to other
transformation families.
According by lOOdel (2),
Y~)..) is syIlI'Ietric:ally distributed aboot f()..)(!.t,~).
Therefore, f()..) (!.t'~) gives both the conditional (given ~i) lEan and Iredian of y~)..)
This inplies further that the untransforned IOOdel
f(!t.,~)
gives the conditional
Iredian of the untransforIred response Yt • However, the conditional
f(!t.,~) except if )., = 1.
The conditional nean is discussed below.
that
f(!t.,~)
lEan
of Yt is not
we see then
has two interpretations; it gives the value of Yt if there is no error,
and otherwise it gives the conditional Iredian of Yt.
In our exanples, !.t = x
Yt is the total return R •
t
(1954) IOOdel
is the size of a spawning stock St and the response
lt
camonly used IOOdel functions f(~,~) include the Ricker
(3)
and the Beverton-Holt (1957) IOOdel
(4)
We will let u
t
= X •
t
Then (3) can be transforned to
(5)
by letting )..=1 and a:=l.
By
using ).,=0 and a:=O on (5), one obtains
(6)
-,!
E>quation (6) is often favored since it is linear in the paraneters.
Equation
(4)
can be linearized to
(7)
or
(8)
oorresponding to )..=-1 and at=O and )..-1 and cx=-l, respectively.
By estinating ).. and
ex we can tell when these linearizing transformations are
appropriate for a given set of data, and we can find IIDre suitable transformations
when the linearization is not appropriate.
Although we advocate transforming the response to achieve an error structure
where least-squares is' efficient, often one 1l'l1St estinate characteristics (such as
the conditional
conditional
2.
~
Dean)
Dean
of the original response.
In section 5 estinates of the
and the conditional quantiles of Yt given
~
are presented.
transformations, skewness, and heteroscedasticity
In this section, we discuss how lXJW& transformations affect, and in particular
raoove, skewness and heteroscedasticity.
~ and variance o-~
= g(~);
SUppose that the randan variable Yt has
the sane function g awlies for each t.
If Yt is
transformed to h(Yt)' then by a Taylor approximation as in Bartlett (1947), the
Dean
variance of h(Yt) is
E{h(Yt) - h(~)}2
•
:!: (h(~»
where h(y) = d/dt h(y).
proportional to g-i (y).
2
E{Y
t
- mt}
2
The variance of h(Yt) is appraxinately constant if h(y) is
For exanple, gIro) = m for Poissondistributed data and g(m) is proportion to m2 if the coefficient of variation (CV)
is cxnstant. Also, g(m) = m2 for exponentially distributed data. When
g(m) ex m2 (1-)..) then hIm) ex g4 (m) if h(y) = y()..). In the case ).. = 0 (cxnstant
In many cases g is a lXJW& function.
CV), the log transformation stabilizes the variance, and for Poisson data) )..
the square-root transformation is indicated.
The effect of h on skewness is also revealed by h.
= 112)
SUppose that y is
positively sk~. The extended right tail in the distribution of y is reduced
through transformation to h(y) if large values of y are "OCIIpI'essed together" IIDre
r,
than snall values: I'IOre precisely, if for all 6,
> 0,
1~(Yl+6,)
- h(Yl)
I<
Ih(y +6,)
2
- h(Y2) I whenever Yl > Y2' SUCh canpression occurs if hey) is a decreasing
function, in which Clse h is ClUed oonClve.
When h(y) is a
BCIIc--cae
pcMer
transformation then hey) = (dldy) )..-l(Y~l) when
}.j(l or hey) = (dldy) log y when ).. = O.
oonsequently h is ooncave if )..
•
when )..
<1
~
1.
In either case, hey) = y)..-l, and
Bac--cae transformations reduce right-skewness
and the aroount of reduction increases as ).. decreases •
Left skewness if reduced by transformations that are oonvex, that is, which
have an increasing derivative. If).. ~ 1, then y()..) is oonvex.
3.
Simultaneous
~
transformation and weighting !?y maximum likelihood.
Although power transformations can reduce both skewness and heteroscedasticity,
the value of ).. inducing rormality, or at least a reasonably symnetric distriwtion,
need rot be the
saIre )..
which transforms to oonstant variance.
A I'IOre flexible
approach to l'IOdeling cx:mbines a BCIIc--cae power transformation to rormality with a
power weighting transformation to txm:>scedasticity as in
~tion
(2).
In this
section we discuss the estimation of ).., ex, 9, and ()- by maximun likelihood and the
oonstruction of a oonfidence region for ).. and ex by likelihood-ratio testing.
To £loo the likelihood, we first rote that the density of €t is
•
Therefore, the ooooitional density of
;TT
2 -i
)..-1 ex
()..)
()..)
2
2 2ex
(211 ()-)
(Yt /u ) exp(-[Yt
- f
(!.t,!)] /(2 ()- u t ».
t
If x t and u t are fixed oonstants, not depending 00 Yl'''''Yt-l' then the loglikelihood of Yl""'YN is
L(}..,ex,9,()-2) = -N/2 log(21T ()-2)
N
,.
+ ()..-l)
.L
t=l
.:
N
log(y ) - ex
t
.L
log (u )
t
t=l
N
- (1/2 ()-2)
.L
{y~}..) - f(}..) (!.t,!)}2/u~ex •
t=l
If
x t and u t depend 00 previous values of y, then L(}..,ex,9,()-2) is still the
conditional log-likelihood of Y1'''.'Yu given x 1 ,x ,x_ , ... and u ,u ,u_ , ... , and
1 O l
O l
maximization of L()..,CX,e,~2) should still produce good estimates of )..,cx,e,~2. The
distinction between conditional and unconditional estimates fran tine series data is
discussed in Box and Jenkins <1976, Chapter 7).
Y IIBY, however, produce biases.
and u on past
t
t
walters (1985) foond sizeable biases in a Monte
carlo study with small semple sizes, N
and equal to O.
= 10.
The dependence of x
His study asSUllEd that ).. was known
His formulas sOOw that the bias should decrease as N increases.
Following Box and Cox (1964), we maximize L()..,CX,e,~2) in two stages.
First,
for fixed ).. and cx the MLE of e is the nonlinear, weighted 1east-squares estimator
e()..,cx) which minimizes
•
•
sse )..,cx,e)
~2
and the MLE of "
is
~
2 ()..,cx) = N-1 SS()..,cx, el)..,cx».
-
Define
to be the log-likelihood maximized with respect to e and ~ for fixed ).. and cx. An
approximate MLE is found by eatp.lting L
()..,cx) on a grid; in section 4 we use
max
cx = -1(.25)1 and ).. = -1<'25)1. If the exact MLE is needed then Lmax()..,CX) can be
maximized by a numerical optimization technique using the approximate MLE as an
initial value.
HGilever, an exact MLE is probably unnecessary for IOOSt applications.
The function
~l)..,cx)
can be used to test hypotheses and to construct
confidence regions for ()..,cx).
()..,cx)
= 1>..0,
•
For a given null value ()..O' CX ), one can test H :
O
O
# ()..O' CX O) by the likelihood ratio test. The
cx ) against HI: ().., cx)
O
log-likelihood ratio is
Here ().., cx) is the MLE or approximate MLE.
H is rejected at level € i f
O
2 LRl)..O' CX ) exceeds ~(l, l-€), the 100<1-€) percentile of the chi-square
O
distriootion with one degree of freedan (Rae, 1973, section 6e). This test can be
applied to each value
(>..0'
CX ) on the grid, and a 100 (l-€) percent confidence
O
region for ().., cx) consists of all null values which are not rejected.
4.
4.1
.,
Examples
Popllation ~
These data and the poJ;Ulation B data discussed later were obtained through
Professor carl Walters (pers. conn.).
refused by the original source.
Permission to identify the stocks has been
There are twenty-eight years of data.
The
·0
variables, R
t
= total
= spawner
return and St
~c:apement,
•
are plotted in figure 1•
,..
,..
,....
200000
,..
.c
III
....
....
,..
....o
'"
Gl
.0
!
,..
....~ 100000
e
Mean-Ricker
.. k-Median-Ricker
_---....s::=-~--
,..
u
Gl
c.:
,.
.1
-,"
o
--------------------
~~~~~------~~~~~~~~*
~-::--
,..
,..
""
o
~
,..
,..
,..
,..
10000
Median-BH
.--
,..
.
,..
20000
Spawners
Fig. 1:
,..
...
30000
(n~~bers
'10000
of fish)
Population A. Actual recruitment from 1940 to 1967,
estimated median recruitment based on the Ricker and
Beverton-Holt (BH) models, and estimated mean recruitment based on the Ricker model.
we first fit the (transformed) Ricker rocx:1e1:
(9)
The appraxinate MlE (on the grid)..
= -1<.25)1
and cx = -1<.25)1) is ().., cx)
(.25,0) and the maximized log-likelihood is -338.3106.
With
()..,CX)
equal to the
MLE, rocx:1el (9 ) becanes
Confidence regions for ().., cx) are given in Table 1.
The 95% univariate confidence
regions for).. and cx are {0,.25,.5} and {-.25,0,.25,.5,.75}, respectiVely.
Alpha
-.25
0
0
*
*
*
+
.5
**
*
+
++
+
+
*
*
*
*
+
*
.75
+
*
**
+
.5
*
+
+
.25
.25
+
,
+
+
= 95\ oonfidence region assuming the Ricker IOOdel.
= MLE assuming the Ricker IOOdel.
= 95\ oonfidence region assuming the Beverton-Holt
++ = MLE assuming the Beverton-Holt IOOdel.
+
Table 1: Pcp.1lation A.
laml:rla- aOO. alpha.
IOOdel.
Coofidence regions aOO. maximum likelihood estimates of
Next the transfonred Beverton-Holt IOOdel
was fit.
The MLE is 0'1
-337.0919.
(l[)
= (.25,.25)
aOO. the maximum log-likelihood is
The difference between the maximlll\ log-likelihood for the Beverton-Holt
and Ricker IOOdels is ooly 1. 2187 which indicates that both IOOdels fit alJoost as
well, though by this criterion the Beverton-Holt m:::rlel does provide a slightly
better fit.
The estimated median return, calculated as described in section 5, is plotted
in figure 1 for both the Ricker and Beverton-Holt IOOdels, as is the "snearing
estimate (section 5) of the nean assuming the Ricker lTOdel.
A researcher with ooly linear regression software might be tempted to linearize
both the Ricker aOO. Beverton-Holt lTOdels aOO. then to canpare than
linearizing scales.
00
their
When the linearized Ricker lTOdel
log (Rt.!'St)
= 91
+ 9 2 St
is, fit R2 = 0.207 and the F-value for testing overall significance of the m:::rlel is
6.79 (p
= 0.015).
If the linearized Beverton-Holt lTOdel
l/Rt
is fit then R2
IOOdel
= 0.000549,
F
= 0.01,
p
= 91 + 9/St
= .9058,
and if the alternative lineariZed
2
is fit then R
= 0.110,
llIJdels ooly
their linearizing scales might easily be tempted into oonc100i.ng that
00
F = 0.29, and p
= .5950.
the Ricker llIJdel p:ovides a far better fit.
•
A researcher cxmpa.ring these
On the oontrary, we believe that the
Beverton-Halt JD:ldel is very slightly better fitting, rot the log transformation is
vastly superior to the inverse transformation.
It is interesting to see how well the MLE transformation achieves both
haooscedasticity and near normality.
In table 2 the skewness and kurtosis are given
for the three transformed Ricker lOOdels:
(I)·
R/S = eKp(6 + 6 S) ().,=1, «=1)
1
2
(II)
log(R) = log(S) + 6
(III)
~= ~eKp{(61 + 6 2 S)/4}
that is, no
rower
1
+ 6 S
2
(}.,=O, «=0)
(}.,=.25, «=0),
transformation, log transformation and the MEE, respectively.
Alpha
Skewness
Kurtosis
Spearman
Cccrelation
e
1
1
.99
2.69
.39
0
.25
0
-1.04
1.22
.12
0
-.27
-.23
.14
Table
1:
Pcp.1lation A.
Skewness and kurtosis of residuals and Spearman rank
correlation between the absolute residuals and the p:edicted values.
The Ricker
model was used.
For model (III) both skewness and. kurtosis are closest to 0, their theoretical value
under normality.
Also in table 2 are the Spearman rank correlations between the
absolute residuals and the fitted values.
Since the fitted values are an increasing
function of S, the rank correlation is unchanged if the fitted values are replaced
by S.
Clear1y,}.,=O and }.,=.25 both transform to near l'xlooscedasticity where there
is little or no relationship between the mean and variance of R. HOIIleVer, the MLE
}.,=.25 is p:eferable to }.,=O since }.,=O Rovertransforms" to negative skewness.
.
Normality and hanoscedasticity of the residuals can be checked graphically by a
•
lX>nDa1 plot of the residuals and a plot of the residuals against the fitted values •
The graphical analysis of residuals should be oone routinely for transformation
llIJdels as for ordinary regression models.
Draper and Smith (1981) provide an
excellent ac<xlUnt of graphical residual analysis. We plotted the residuals fran the
MLE and found no evidence of non-normality or heteroscedasticity. A normal
probability plot of the residuals is roughly linear, thoogh the very slight negative
skewness (table 2) is evident.
Far these data, a plot of the residuals versus the
predicted values is oot easy to
interpret~
the Ricker and Beverton-Holt curves are
nearly oonstant over the observed rar1ge of 5 so the predicted values are quite
similar except for those oorrespooo.ing to the smallest values of 5.
4.2 5keena River sockeye salIron
Far this stock, effective escapement (5) and total return (R) are given for
years 1940 to 1967 in Ricker and smith (1975).
These data are plotted in figure 2
along with the estinBted medians (section 5) for the Ricker and Beverton-Holt
trode1s.
Conpared with the Pc.p.llation A stock, the Skeena River data shCM less
positive skewness bIt
~re
heteroscedasticity.
Figure 2 shCMS several particularly
lCM values of R associated with high values of S bIt mly ooe particularly high
value. This Day ioo.icate overcxxapensation, ar it Day simply be due to the high
variability in R when S is large.
*
*
*.*
lfOO
*
Beverton-Holt
800
1200
Spawners (thousands of fish)
Fig. 2:
Skeena River sockeye. Actual recruitment from
1940 to 1967 and estimated median recruitment
based on Beverton-Holt and Ricker models.
The 95\ oonfidenoe re;ions and the MLE of 0"
ex)
are the same for the Ricker
and Beverton-Holt mdels and they are given in table 3.
The maximum log-likelihoods
for the Ricker and for the Beverton-Holt trodels differ by cnly 0.164, so there is
little to su9geBt ooe DDdel over the other.
strong evidence of overcxxapensation.
In particular, the data provide
00
LanI:x3a
•
Al~
0
.25
-.25
0
*
*
*
*
*
*
.5
.75
1
* = 95% oonfidence
** = MLE
Table
1.:
.25
.5
.75
1
*
*
*
*
*
*
**
*
*
*
*
*
*
region.
Skeena River sockeye sa1Ioon stock.
95% oonfidence region arxi maximum
likelihood estimate assuming either the Ricker or Beverton-Holt IOOdel.
Ccmpared wit;.h the Pcp.llation A Adata, the MLE here results in less
transformation 0,
instead of at
=0
= .75
instead of }..
or .25).
= .25
as before) rot rore weighting (at
= .5
This is oonsistent with the observation that the
untransformed Skeena data exhibit little skewness rot considerable
heteroscedasticity.
4.3 Pacific Cod, Necate Strait (Walters, et al., 1982)
The twenty-one observations, 1959 to 1979, on this stock are plotted in figure
3 along with predicted neiians
for bIo IOOdels.
Recruitment is virtually independent of spawning stock except that the bIo
unusually large recruitments occur when S is rather snall.
Beverton-Holt rode1 fits poorly.
For this reason the
For this stock we will consider the Ricker IOOdel
arxi the power rodel
With
Bat~
and weighting transformations the power IOOdel becmtes
,.
Confidence regions for (}..,
(1;)
are given in table 4.
The maxi.Im:Jn log-likelihood is
-34.9690 arxi -37.2541 for the power and Ricker IOOdels, respectively, and the large
difference (2.2851) indicates lack-of-fit for the Ricker IOOdel.
...
•
...
..... 6
.c
III
.....
....
....o
...
III
~
.....o
...
----
....
....
.....
...
...
Power
- ...
e
'-'
...
------ .. -
...
...
...
...
j
R'lC ker
t",-, -',
...
...
'
.
...
a
o
2
6
8
Spawners (millions of fish)
Fig. 3:
Necate Strait Pacific Cod. Actual recruitment
from 1959 to 1979 and estimated median recruitment based on the Power and Ricker models.
AlIfla
-1
-.75
-1
-.75
-.5
-.25
0
.25
.5
.75
1
* = 95% Calfidence
** = MLE or within
*
*
-.5
*
*
*
*
*
*
-.25
*
*
*
*
*
*
*
0
*
*
*
*
**
*
*
*
.25
.5
.75
1
*
*
*
*
*
**
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
**
*
*
*
region.
.1 of naximizing the log-likelihood.
Table !. He::ate Strait pacific cod stock. 95% oonfidence region and naximl.lll
likelihood estillate assuming the power nodel.
We can test the Ricker lIOdel as follows.
The general m:xiel
6
R ... St3 eKp(6 + 6 St)
t
1
2
inclmes the Ricker lIOdel (6 ... 1) and the power lIOdel (6 = 0) as special cases.
3
2
... 1 against H : 6 rio 1. The naxim\lll
O 3
l
3
log-likelihood is -34.9022 and the log-likelihood maximized subject to the
One tests the Ricker m:xiel by testing H : 6
•
oonstraint 6 = 1 is -37.2541 (see above). Twice the difference is 4.704 which
3
exceeds X2 (1, .95) = 3.84. The Ricker lIOdel is reject at level .05. By the saIle
reasoning the power lIOdel is accepted since 34.9690 - 34.9022 = .0668 is very small.
The 95\ oonfidence region for ().., ex) assuming the power lIOdel is quite large
because (a) there are ally 21 data points and, IlDre importantly, (b) the variability
in both spawning stock and return is small canpared to the previously examined
stocks.
The oonfidence region for the Ricker lIOdel is even larger, rot this is to
be expected oonsidering the lack of fit.
For the power lIOdel, the MLE is ().., ex)
).. =
!:.!
.25 and ex
= .25
=
(0, .5), rot the log-likelihood at
or .5 is within 0.1 of the maximum.
Population!! (fran Professor carl Walter,
~
cx:mn.)
For this stock, Rt and St vary over a much wider range than for the three
preceding stocks, and it seems preferable to use the return to spa.wner ratio, R~St'
•
as the response. A plot of log (R~St) against St is rather linear, rot IlDSt of the
data are b.mched together in the range 0
scattered CNer 300, 000
log (5).
~
St
~
~
St
~
300,000 while the remainder are
3,300,000, so in figure 4 log (R/5) is plotted against
The ooly lIOdel studied here is the transforned Ricker lIOdel
Also the grid ex
= -1(1/4)1
is replaced by ex = -1/4 (1/16 )1/4 ~becaus: values of ex
far fran 0 fit poorly and lead to oonvergence problE!llS when 6
and 6 are cxrnp.1ted.
1
2
The 95\ oonfidence region for ().., ex) and the MLE are found in table 5. Clearly,
the oonfidence region is quite small.
,
Because Rt and St vary over extremely wide
ranges, ).. and ex are well determined by the data.
~
Estimating the Conditional Distribltion of ":l.
We have seen how to utilize the lIOdel
to efficiently estimate 6.
The lIOdel expresses a transforned response as a function
of
!,
~i' and
the nonnally (ar approximately nonnally) distril:uted €i.
interest centers
00
the untransformed response y t.
to estimate the oonditional (gi ven
~i)
Typically
In this section it is shown how
man of y i as well as oonditional quantiles
such as the median.
•
...
Lf
...
,......
...
...
1Il
""Cll
i
...
~
... ..*
til
......
1Il
.......
*;
* Ricker
;:l
""
u
Cll
e:,
-
'"
t>Il
0
.....l
...
...
...
...
...
...
...
...
,
...
'" ... '"
'"
'"
...
-2
-l.,.-~...----.--------.---;-~---.-----.---.---,.......-..........
15
10
5
Fig. 4:
Log (Spawners)
population B. Estimated median production
ratio based on the Ricker model. Recruits
and spawners are in numbers of fish. The
production ratio and spawners are expressed
in logarithms.
."
.
Alpha
-~16
-3/16
*
*
-.25
f
-2116
-1/16
0
1/16
*
**
*
*
o
.25
*
*
2116
3/16
*
*
~16
*
= in 95% oonfidence regions
** = MLE
Table~.
~lation
B.
95% oonfidence region assuming the Ricker IOOdel R/S =
exp(9 + 9 S).
2
1
Now
ei
cannot" be exactly normally distribIted in IIOSt cases, for example wen
VO. I t is better to suppose that ei is nearly
Fortunately, for pt"esent prrposes we need not nake
Yi and f(x, 9) are non-negative and
nomel bIt with a bounded range.
any assumption about the
.
{e
i
} except that they are independent and identically
distribIted.
If y (}.,)
y = exp(x)
= x, then the inverse relationship
if }., = O. In what follows we assume
simply replaces
(1
+ ).,x)1/)" by exp(x).
ei
response Yi as a function of ~i' ui'
Using
is y
=
(1
+ ).,x)1/)" if }., .,. 0 and
that }., .,. O.
(2)
E(Y.lx.)
1. -1.
,
Let
~
= J{l
... ,
~-.
~
-1.
then ooe
Let F be the
The oorditional mean of y.1. given -1.
x. is
+ }., f(}.,) (x., 9) + }., u~ e}1/}., dF(e) •
-
be the pth quantile of F, Le.,
of Yi given ~i is
I
e1 ,
= 0,
and the );BI"ameters }."a:,9:
With this representation we can study the untransforned yi •
distribItion function of
If}.,
we can ecpress the original
1.
F(~)
= p.
(10)
Then the conditional pth quantile
N
-1 ' \
A
E(Yil.!i) = N
-where
et
/
t=l
A
{l
+ ~ fo..> (.!i' !) + ~ u~ €t}l/~
(11)
is the t-th residual,
A
A
€t = {y(~)
t
The relationship between (10) and
A
f(~) (x &)}
==to' -
-
cl.ear~
is
(11)
/
u«
t .
all parameters are replaced by
their estinates and averaging with res~ to th: theoretical distril:11tion of ei is
replacing by averaging 0ITer the sample e , ••• ,~. Even if F were known, (9) oou1d
1
only be evaluated by numerical integration, I:11t (10) does not require integration.
•
This is a distinct advantage of the srwaaring estinate.
Let q be the p-th sample quantile of {e.}. Then we estinate q_(y.lx.) by
A
A
p.-
-p
1.
_
A
q (Y.lx.)
P
1. -1.
=
{l
+
A
~
f
(~)
(x., 9) +
-1.
-
A
~
«A
u.
1.
1.-1.
A
<L}
-p
1/~
•
In the case of the median (p = .5), the residuals should have a median close to 0
and we can replace q.5 by O.
The estinated oonditional median is then
A
A
m(y.,
x.)
1. -1.
= q • 5(y.lx.)
1. -1.
,
A
A
As mentioned before, in figure 1 E(yl.!i) and m(y/.!) are plotted for Pcp1lation
A assuming the Ricker node1, and lft(Ylx) is plotted for the Beverton-Holt nodel as
well.
The mean return is always oonsiderably larger than the median return.
This
reflects the oonsiderab1e positive ske.mess seen in the actual recruitments and
evident in the MLE of ~, ~ = •25 •
Because recruitment is so highly variable arty realistic IIBIlagement nodel will
be stochastic, and when a stochastic nodel is oonstructed it is vital that the
entire oonditional distril:11tion of R given St be estinated. This can be done by
t
estinating oonditional quantiles as above. Ruppert, Reish, Deriso, and Carroll
(1985) use a closely related method for estinating oonditiona1 quantiles when
oonstructing a stochastic nodel of the Atlantic menhaden p:)pUlation.
Transformation
of the menhaden stock-recruitment data is discussed further in Carroll and Ruppert
(1984) •
'"
Smmary and Conclusions
\
A theoretical nodel relating a response y to independent variables .! and
parameters! Day not be suitable in· its original form for
1east-~es
estination.
This is the case if the response exhibits ske.mess or nonoonstant variance.
HCM!VeI', if the response and the m:xiel function are transformed in the same way,
then the transformed response nay be awralCi.nately normally distrimted with a
nearly oonstant variance and then 1east-s:}UlU"es estimation will be efficient.
In this piper we pl:'opose cxmbining the BaIe-<ac power transformation with
weighting by u~, where u t is a variable not depending 00 yt and ex is a parameter to
be estimated. Another pos~ibility, ooe that is not explored here, is to have ~t be
the pl:'edicted value f(.!i' !). Weighting the untransformed response by I£<.!i' Q) lex
is stooied in Bale and Hill (1974), Pritchard, Domie, and Bacon (1977) and Carroll
and Ruppert (1982).
The BaIe-<ac parameter }.. and the power weighting parameter ex can
simultaneously with
!
and 0- by naxi.mum likelihood.
~
estimated
A oonfidence region for (}.., ex)
can be oonstructed by likelihood-ratio testing.
FalI" stock-recruitment data sets were analyzed by alI" transfoIlllation
methodology.
The original response, return, is in each case skewed or
heteroscedastic.
Only for the Pacific ood stock is }..=1 and ex=O, oorresponding to
no transfoIlllation, in the 95% oonfidence region.
On the other hand }..=O and ex=O,
which is the linearizing transfoIlllation for the Ricker m:xiel, is in the 95%
oonfidence region for all four stocks.
Also, the inverse transformation ().. = -1)
which linearizes the Bel7erton-Holt is not in the 95% region for any of the four
stocks.
If both the Ricker and Bel7erton-Holt m:xiels are linearized, then the Ricker
m:xiel will appear better fitting, not because it is necessarily superior mt because
the log transformation is uore suitable than the inverse transfoIlllation.
There are
examples where }..=-1 is quite suitable, for example the Atlantic menhaden stock
(Carroll and Ruppert, 1984).
After }.. and ex have been estimated by naximun likelihood, the fitted uodel
should be checked by residual analysis as described in, for example, Draper and
Smith (1981).
Maximun likelihood estimatiion is highly sensitive to aJt1ying
observations.
Such influential points nay be evident fran the residuals.
Rd:ust
estimators for alI" transformation uodel is an important area for future research.
At pl:'esent, robJst estimation has been stOOied ooly for the rather different
methodology where ally the response, not the uodel, is transformed; see Carroll and
Ruppert (1985), Carroll (1980), and Bickel and Doksum <1981>.
The uodel function gives the oonditiona1 median of the response, and the
oonditiona1 mean and the other oonditional quantiles of the response can be easily
estimated.
By
estimating ooooitional quanti1es ooe in effect milds a m:xiel of the
skewness and heteroscedasticity in the untransformed response.
crucial
plrt
SUCh a m:xiel is a
of a realistic stochastic nanagement uodel of the stock.
We have used four fish stocks as examples of alI" pl:'qlOSed statistical
methodology, mt alI" analyses shalld not be oonsidered definitive.
For example, we
did not consider the effects 00 the Skenna River stock of the 1951-2 rock slide,
changes in exploitation rates, an artifica1 spawning d1annel cpened in 1965, or
\
interactions between year classes: see Ricker and Smith (1975).
We believe,
however, that power and weighting transfonnations will be EqUally as useful for IIOre
elaborate m:xlels as for the basic ales that we have enployed for illustrative
p,trpOSes.
Acknowledcznents:
analysis.
Rod Reish and Rick Deriso introduced us stock-recruitment
carl Walters supplied the data, and we benefitted fran his helpful
discussions.
Bartlett, M.S., 1947.
The use of transfanations.
Bianetrics, 3, 39-52.
Belrerton, R.J.H., and Holt, S.J., 1957. On the dynamics of
Her Majesty's Stationery Office, Lcndon, 533 p.
~
fish popiLations.
Bickel, P.J., and Dcltsum, K.A., 1981. An analysis of transfonnations revisted.
Journal of the American Statistical Association, 76 296-311.
Bale, George E.P. and Coe, David R., 1964.
discussion).
Journal of the
~
An analysis of transfonnations (with
Statistical Society, Series!!r.. 26, 211-246.
Bale, George E.P. and Hill, William J., 1974.
Correcting inhaoogeneity of variance
with power transfonnation weighting. Technanetrics, 16, 385-389.
Bale, George E.P. and Jenkins, G.M., 1976.
Control, Revised Edition.
Time Series Analysis:
Holden-Day: San Francisco.
Forecasting and
carroll, R.J., 1980. A rob.1st method for testing transfonnations to achieve
approximate normality. Journal of the ~ Statistical Society, Series!! 42,
71-78.
carroll, R.J. and Ruppert, D., 1982. Rcb.1st estimation in heteroscedastic linear
m:xlels. Annals of Statistics 10, 429-441.
carroll, R.J. and Ruppert, D., 1984. Pcwer transfonnations when fitting theoretical
m:xlels to data. Journal of the American Statistical Association, 79, 321-328.
carroll, R.J. and Ruppert, D., 1985. Transfonnations in regression:
analysis. Technametrics, 27, 1-12.
Draper, Norman and Smith, Harry, 1981.
wiley: NEW York.
a rob.1st
Applied Regression Analysis, 2nd Edition.
Duan, Naihua, 1983.
Slrearing estimate: a nonpararnetric retransfonnation method.
Journal of the American Statistical Association, 78, 605-610.
Pritchard, D.J., Dcwni.e, J., and Bacon, D.W., 1977. Further consideration of
heteroscedasticity in fitting kinetic m:xlels. Technanetrics, 19, 227-236.
Ricker, W.E., 1954.
623.
Stock and recruitment.
~
of Fish. Res. Board Can., 11, 559-
Ricker, W.E. and Smith, H.D., 1975. A revised interpretation of the history of the
Speena River sockeye "salIOOn (Oncorhynchus nerka). J. the Fish. Res. Board of
Can., 32, 1369-1381.
- - - -- --- -
;
Ruppert, D., Reish, R.L., Deriso, R.B., and carroll, R.J., 1985. A Stochastic
~lation lIDdel for uanaging the atlantic IIBnhaden fishery and assessing
IIBlIagerial risks.
Can. ~ Fish. and ~ Sc. - to appear.
Walters, C.F., 1985. Bias in the estimation of functional relationships fran time
series data. Can. ~ Fish. and ~ Sc., 42, 147-149.
Walters, C.F., Hiltx>rn, R., Staley, M.J., Wcng, F., 1982 ~ to the Pearse
carmi.ssion 2!! Pacific Fisheries Policy, Vancouver, B.C.