Download A General Linear Models Approach for Comparing the Response of Several Species in Acute Toxicity Tests

A GENERAL LINEAR MODELS APPROACH FOR COMPARING THE RESPONSE
OF SEVERAL SPECIES IN ACUTE TO~JCITY TESTS.
Karen l. Daniels
Jonathan f. Goyert
Michael P. Farrell
Rodn~y H. Strand
Environmental Sciences Oivision
Lak Ridge National laboratory
Oak Ridge, Tn.
3783[
ABSTRACT
desfgned
Simply
to
estimate
the
concentration of a substan,e required to
produce
a
response in SOX of the
~opulation
(ttle LeSS value). Comparing
species on the basis of LC5C' vaLues is
inadequate because several species can
have
identical
values
while their
overall response to a substance may
differ
i~
either
the
threshold
concentration or the rate of response.
This pap~r presents a sequential general
linear mOdels approach for comparing the
responses of two or more species to a
particular compound using contrived data
as an example.
Acute
to~icity
tests
(bioassays)
estimate the concentration of a chemical
required to produce a response (usually
death)
in fifty percent of a population
<the lCSO). Simple conparisons of lC50
vaLues among several species are often
inadequate
because species can have
identical
LCSJ
values
while their
overall
response to a chemical may
differ
in
either
the
threshoLd
concentration (interce~t) or the rate of
response (slope). A sequentiaL approach
using
a
generaL
linear
model
is
presented for testing oifferences among
species in their overall response to a
chemical. This method tests for equality
of
slopes
followed by a test for
equality
of
regression lines. This
procedure
~mploys
the
Statistical
Analysis System's General Linear Models
procedUre
for
conductirg a weighted
least
squares
anal)sis
~ith
a
covariable.
STATISTICAL METHQOS
Data generated from bioassays consist of
the
test
concentrations
(t ci), the
control concentration (c ~i), the number
of orgdnisms in each te;t concentration
(t_ni), the number of organisms in the
controL concentration (c_nil, the number
of organisms responding in the test
concentration (t_ri), and the number of
organisms
responding in the control
concentration (c_ri).
*Research sponsored by the Office of
rlealth and EnvironmentaL Research, U.S.
Department of Energy, under contract
~-74r5-eng-26
with
tJnion
Carbide
Corporation.
Publication
No.
1888,
Environmental Sci~nces Division, ORNL.
8efore one can examine tre relationship
between concentration and response for
each
species, a number of variable
transformations are required. The first
transformation
corrects
for
the
oroportion of organisms responding to
the control conditions to isolate the
response to the test compound. Abbott's
(192S) formula
is frequently used in
toxicological
studies
to accomplish
this!
1NTROOUCT10N
Acute toxicity tests or bioassays are
widely used to asspss the potential
effect of to~ic substances released into
aquatic environments. Test species may
b~ selected for their sensitivity to the
substance,
for their importance to the
ecosystem or to the lccal community, or
for their success and widespr~~d use as
a
Laboratory
organism.
Questions
frequently
arise about which of these
criteria should be used for species
selection.
Data
from
comparative
toxicological studies must be analyzed
and evaluated to answer these Questions.
In the past, toxicological studies were
where pi is the proportion responding to
concentration
alonel p ti;t rilt ni,
the
proportion
respondi~g
- to
concentration i and to t~e control, and
p_ci;c_ri/c_ni,
the
proportion
responding to the control aLone. For the
733
nonconstant variance about the errors
can be fitted by the method of we;g~ted
Least squares, using the transformed
variables (Neter and Wasserman 1974).
With
this
method,
the
parameter
estimates are obtained by minimizing a
weighted sum of sQuares of residuals
where weights are inversely proportional
to the variance of the errors. For this
exaMple, the weighting variable (wi) may
be expressed as:
case where ti is less than c, the value
of
pi
is
set
to 2ero beca~se the
proportion
is
restricted to values
between 0 and 1.
The
relationship
bet~een
the
concentration
and
the
proportion
responding
(corrected for
control) is
examined
using
regression
analysis
tec~niques. The relationship between the
dependent variable
CV), the proportion
responding,
and a single independent
variable (X), the concentration, may be
formuLated as a linear model:
wi=niPi(l-Pi),
where ni=number of orqanisms in t~e ith
sample and pi;proportion responding in
the ith sample. This is in contrast with
ordinary Least squares where parameter
estimates are obtained by minimi2ing the
eQually weighted sums of SQuares of
residuals.
By minimizing the weighted
sum
of
squares
of
residuals, t~e
precision of the estimates is increased.
Yi=BO + R1Xi + ei, i=1,2, ••• ,n
where 80 and Bl are the modeL regression
parameters and ei
is
the residual or
error.
The relationship between the
proportion
responaing
and
the
concentration is nonlinear and generally
can be accurately
represented by the
logistic function
(Aerkson 1953). This
function,
often called the
logit, is
represented as follows:
The
Logistic transformations are not
defined for
~i=C or Pi=1, that is, for
those concentration levels where none or
all
of
the
subjects
responded.
Chatt·erjee and Frice
(1977) described
severaL ways to handle this situation.
for
example, these data points can be
omitted from the
regression analysis
because of
a considerable degree of
uncertainty
about
the
exact
concentration that caused all or none of
P;:exp(BC + B1Xi')/l + exp(BO + 81Xi'),
w~ere
Xi'=log(Xi).
An alternative model, the probit-model,
accurateLy
represents the reLationship
between
concentration and proportion
responding
(Finney 1978). T~e response
function in this model is represented by
the
inverse of
the cumulative nor~aL
distribution.
For
detaiLs refer to
Finney (1978)
and Hewlett and PLackett
(1979). The logistic f~nction is used in
this example based on the discussion by
Berkson (1951).
the subjects to respond.
An
alternative
proposed by Berkson
(19S3)
is
to
approximate Pi=C by
Pi=1/2xt_ni and Pi=1 by pi=1-1f2~t ni. A
finaL aLternative is to fit the-model
both with and without the extreme data
points.
If
the fitted modeLs differed
considerabLy,
then the ~xtreme data
points shouLd be reexamined. For this
example, Berkson·s rule was applied to
avoid deLetion of data.
To
linearize
the
logistic response
function,
it is necessary to transform
the response variable as folLows:
Our approach for testing differences in
concentration-response
relationships
among species is to fit a linear modeL
with
a
covariable
(Log10
of the
concentration)
using tte transformed
variables in a procedure often referred
to as an analysis of co~ariance (Neter
and Wasserman 1974). The key statistical
inferences of interest in this analysis
are
the
same as witt analysis of
variance modeLs, nameLy whether there
are any significant differences in the
concentration-response
relationship
among species.
Pi":ln(Pi/l-Pi).
The Linear modeL then becomes:
Pi"=BO +
B1~i'
+ ei,
where, Xi"=log1~(Xi) and the transformed
variable Pi'
has a ~ean approximateLy
equal to
Ln(Pi/l-Pi)
and a variance
V(Pi")=1/niP;(1-Pi)
(Chatterjee
and
Price 1977). Because the variance of the
transformed variabLe is a function of
its mean,
the error variance is not
constant (one of the modeL assumptions).
This can also bp observed by examining
the
residuaLs after fitting the Linear
model using the transformed variabLes:
the variance increases as one moves away
frop the mean or sex response point.
Linear
regression
models
with
An
important assumption in covariance
analysis
is that
all regression lines
have equal or parallel
sLopes. This
assumption
can be tested by examining
the
interaction between the covariable
and species. If there are no significant
differences,
then
the
slopes
are
734
The
equality
of
stopes
does not
necessarily
imply that the species have
the same regression lines because their
intercepts
may differ. To test for
equality
of
regression
lines among
species,
the full
and reduced models
described by ~eter and Wasserman (1974)
are compared.
considered the
same. The foLlowing SAS
statements
(SAS 1979) are used in this
example to
test
for differences among
three species in their rate of response
to a single compound~
DATA NEW; SET TOX_DATA;
in
Calculates
proportion
responding
treat ment s (PT)
*******************.*******************;
The full model (f) calculates a separate
concentration-response relationship for
eacn species in the model. The SAS
program statements used to generate the
output from the full model are~
******.************* •• **.***.** •••• **.**
~eneration of Full Model
responding
in
Calculates
proport ion
controls(PC)
*************.*****.** •• ********** •• **.;
GLM;
CLASS SPECIES;
PROC
PC=C RIIC NI;
~ODEL
/I.bbotts'
correction for the control
response
******.*******_.* •••• **.*._*.* •• *.** •• _;
LOGIT=SPECIES LOGCONC(SPECIES);
wEIGHT WI.
P=(PT-PC)I(l-PC);
The output
tram this analysis is shown
in Table 2. The separate slopes for each
species
are
specified as parameter
estimates.
****************************************
8erkson's rule 10r all or none responses
******-******.*.* •• **********.*.*.*.***;
The
reduced
model (r)
removes the
species effect
by combining them in a
simple linear reqression model. The SAS
program statements used to prOduce the
output for the reduced model are:
IF P=C THEN P=112*T_NI;
IF p=l THEN P=1-C1/2*T_NI);
Calculation
of
Loql0
of
the
concentration(LOGCONC)
**********--**.*****.*****.******** •• **;
***************************************.
Generation of Reduced ~odel
*************************************.*;
LOGCO~C~LOG10(CONC);
PROC GLM;
~OD[L LOGIT= LOGCONC;
.tIGHT WI;
****** •• **-****.**** •• _***.*.*.******.**
Calculation of the weighting vari~ble;
***************.***********************;
Table - 3
presents the results
from
executing
this
program.
The
test
statistic for
determining if there are
significant differences among regression
lines
requires the sums of squares for
the errOr
(SSE)
and the degrees of
freedom
tor
the error (DFf) from both
the fu l L (f) and reduced (r) model s. The
test statistic then is:
**********************************.*****
Transformatio~
of proportion responding
to logistic functionClOGIT)
***************************************;
LOGIT:LOG(P/(l-P));
***************************************.
Test for homogeneity of slopes
******************.********************;
F:
(SSEr-SS~f)/(DFEr-DFEf)
with
t..IROC GLM;
CLASS SPfCIES;
F(c-1,DFEf),
where
I(SSEf/OFEf),
c=number
of
speci~s.
MODEL
LOGIT:SPFCIES lOGCONC LOGCONC*SPECIES;
Far this exa~ple F(2,62)=4.0~ which is
greater than 3.15 (the 95th percentile
of the
F distribution
with 2 and 62
degrees of freedom).
We conclude that
there are significant differences among
the species regression
lines. Because
the
slopes are equal among the species,
we also conclude that the differences in
wFIGHT WI;
An example of
some of the output from
the executio~ of this program is shown
in Table 1. Because the interaction term
was nonsignificant, we conclude that the
regression
Lines for the three species
GIll have the same stope.
735
,,
the
regression
differences
in
investigate
lines
are
due
the
intercepts.
the
nature
of
to
To
Neter, John and william Wasserman. 1974.
Applied
linear
Statistical
models.
Richard
D.
Irwin,
Inc., Homewood,
Illinois. 842 pp.
these
differences,
pairwise comparisons or
more generaL contrasts among species may
be ~ade using a variety of multiple
1979 Edition.
SAS
SAS User's Guide,
Inc., Cary, North Carolina.
Institute,
494 pp.
comparison procedures (Chew 1977).
CONCLUSIONS
A methodology is described that examines
the concentration-res~onse relationship
among species in comparative toxicity
tests.
The species response data must
first
be
response
logistic
corrected
for
the
control
and then linearized using the
function and a logarithmic
transformation.
To statistically test
for
differences
in
responses among
species to a particular compound, a
covariance
model
was
principle of we;Qhteo
fitted
by
the
least squares in a
general linear modpls procedure.
ACKNO~LEDGEMENTS
authors wish to thank 8. ~alton and
Millelman for a critical re~iew of
the manuscript.
The
~.
LITERATURE CITED
Abbott,
w.
S.
1925.
A met nod of
comput ing
t ne
effectiveness
of an
insecticide.
J.
Econ.. Entomol. 18:
265-267.
Berkson, Joseph. 1951- :";hy I
prefer
logits
to
probits.
Piometrics. 7·
312-313.
Rerkson, Joseph. 1953. A statistically
precise and relatively SimpLe method of
estimating the bio-assay with Quantal
response,
based
on
the
logistic
function.
J.
Am.
Stat.
Assoc. 48:
565-599.
Chatterjee,
Samprit and B. Price. 1977.
Regression Analysis by fxample.
John
Wiley and Sons, New York. 228 pp.
Chew, Victor. 1977. Comparisons among
treatment
means
in an analysis of
~ariance.
U.S.
Department
of
Agriculture, APS/H/6. Washington, D. C.
Finney,
David
J.
1971:'.
Statistical
Method in Biological Assay. ~rd Edition.
MacM i l l an Pub. Co., Inc., New York.
~ewlett,
p. S., and R. L. Plackett.
1979.
The
Interpretation of Quantal
Responses in Biology. University Park
Press, Baltimore, Maryland. 82 Pp.
736
Table 1.
Output for testing equality (parallelism) of slopes
STATISTICAL ANALYSIS SYSTEM
GENERAL LINEAR MODELS PROCEDURE
Dependent Variable: LOGIT
Weight:
WI
OF
Sum of
Mean
Squares
Square
Model
5
24.368
4.874
Error
62
14.281
0.230
Corrected Tata 1
67
38.650
Source
Source
Species
Logconc*Species
PR
21.160
0.0001
OF
Type I
SS
F Value
2
1.868
4.06
0.0221
22.107
95.98
0.0001
0.393
0.85
0.4314
Logconc
2
PR
F Value
>
737
F
>
F
R-Square
C.V.
0.630
92.101
OF
Type IV
SS
F Value
2
0.256
0.55
0.5769
18.194
70.99
0.0001
0.393
0.85
0.4314
2
PR
>
F
Table 2.
Output for full model
STATISTICAL ANALYSIS SYSTEM
GENERAL LINEAR MODELS PROCEDURE
Dependent Varabile
LOGIT
~
Wei ght
WI
Sum of
Mean
OF
Squares
Square
Model
5
24.368
4.874
Error
62
14.281
0.230
Corrected Total
67
38.650
Source
Table 3.
F
Value
PR
>
F
0.000]
21.160
R-Square
C.V.
0.630
92.10]
Output for reduced model
STATISTICAL ANALYSIS SYSTEM
GENERAL LINEAR MODELS PROCEDURE
S
Dependent Varabile:
LOGIT
Weight:
WI
Source
OF
Sum of
Squares
Mean
Square
F Value
~~~.~
Model
23.608
23.608
0.273
Error
66
18.041
Corrected Total
67
38.650
738
86.74
PR
>
F
R-Square
C.V.
0.61
91.61
-------------~--
0.0001