A DISCOUNTED LEAST SQUARES QUADRATIC ... Either a discounted linear or a discounted quadratic growth pre- .... Percent time administered dose within ... 3'18. 0·18. ]0 mg/kg d'. 100. 96,7. 70,5. 36·1. 9,97. 0,59. 10 mg/kg ~. 100. 95,].
Laboratory Animals (]975) 9, 289-296.
A DISCOUNTED
289
LEAST SQUARES QUADRATIC
GROWTH PREDICTION
MODEL FOR USE
IN 2 YEAR TOXICITY STUDIES by RONALD
L. IMAN
and MARSHALL
N. BRUNDEN
The Upjohn Company, Kalamazoo, Michigan 49001, United States of America SUMMARY
Weekly weighings of the laboratory rats are required to determine the correct dosage for mixing in the food. This creates problems in that the food mixing must be done immediately after the weighings and staff are often heavily taxed to perform the task. A discounted least squares growth prediction model allows for prediction of weights a week ahead of time, obviating the necessity for instantaneously processing the weight data. When dosages were prepared based on these predictions, for 10 treatment combinations 100% of the doses proved to be within 8'0% of the required dosage; 98'4% were within 5% of the required dosage; 78'7% were within 2% of the required dosage; and 51'6% were within 1% of the required dosage. The quadratic weight prediction model can also be incorporated into a model for predicting food consumption. Current 2-year toxicity studies using Sprague-Dawley rats require dosages to be based on bodyweight (mg/kg). The drug is then mixed into the food. The procedure for mixing uses 2 steps: I amount of drug needed for 1 week (mg) dose (mg/kg/day) X 7 days; II amount
=
total bodyweight
X
required
of drug to be weighed (mg) and put in food to be mixed
weight of food to be mixed (g) I
X ------------
last week's food consumption
(g)
In this procedure the weight of the food to be mixed is set high enough to insure an adequate food supply for all animals. In order to determine the correct mixture it is necessary to first weigh the rats and then mix the food immediately to provide the correct dosage. This procedure taxes the personnel involved to the utmost as the weight data, once obtained, must be processed with the minimum of delay so that the food can be properly mixed.
290
R. L. IMAN
AND
M. N. BRUNDEN
This paper describes a technique for predicting weights a week ahead of time, which eliminates the need for hurried calculations between the time the rats are weighed and the food mixed. The nature of the growth curve is likely to be influenced by the drug-dosage combination administered to the rats so that its true form may be unique to the particular treatment. The model used for prediction is well suited to deal with each group on its own and does not rely on data other than those for that particular group. Discounted least square techniques are discussed in detail in Brown (1963) and a linear growth prediction model was given by Brunden, Clapp & Heun (1974) for use in the rat rearing complex with regard to inventory control and production planning. The model used by Brunden et al. regarded the growth curve as being composed of straight-line segments whose slope within different time periods would be continually changing. Sample data were gathered weekly and the updated coefficients for the linear equations were generated and stored in the computer for use in rat weight prediction for any given period. This approach will not work for 2-year toxicity studies because the growth curves must be considered as unique to the experiment. Hence, a prediction equation that will continually update itself based on weekly weights is needed. Either a discounted linear or a discounted quadratic growth prediction model could be used here, and both were tried. The quadratic prediction model was found to do a much better job as it allows for the 'curves' that occur naturally in the growth data, while the linear prediction model makes predictions based on tangents to the curve which are not generally as good as the quadratic. Also, when the data are essentially linear, the coefficient of the 2nd degree term of the quadratic is essentially zero, resulting in a linear prediction. THE QUADRATIC
The general
GROWTH
form of the quadratic
Y = a
PREDICTION
equation
MODEL
1S
+ bX + CX2
where X represents age or time, Y weight and a, band c parameters to be estimated. The usual least squares approach chooses a, band e such that the following quantity is minimized: n :E (Yi -
a-
bXi - eXi 2) 2
i=1 The partial derivatives with respect to the usual normal equations:
a,
band
c when
set equal to zero yield
QUADRATIC (1)
a=
MODEL
291
Y - bX - cI.X 2/n
(2) I.XY
=
(3) I.X2Y
nYX-nbX2-cXI.X2 YI.X2 - bXI.X 2 -
=
For computer (4)
GROWTH PREDICTION
+ bI.X2 + CI.X3 c(I.X2)2jn + bI.X3 + cI.X4
usage these equations
are more conveniently
rewritten
as
c=
[I.X 2Y-I.X 2I.Y/n][I.X 2_(I.X 2_(I.X) 2jn]-(I.X
3_I.XI.X 2/n][I.XY-I.XI.Yjn]
[I.X 4Y-(I.X 2) 2jn][I.X 2_(I.X) 2/n]-[I.X 3_I.XI.X 2/n] 2 (5)
b=
[I.XY -I.XI.Y /n]-e[I.X 3_I.XI.X 2/n] I.X 2_(:tX) 2/n (6)
a=
Y - bX - cI.X 2/n
The usual least squares technique would then make use of all of the data to calculate the estimates of a, band c. The discounted least squares approach modifies this by discounting the older information and thereby making the estimates reflect the latest trend in the data. We choose some discount factor y such that 0< y< 1 and make the following calculations for the estimates of a, band c up through k weeks of data: I.X
k nj I. yk-j[I. Xij] j=1 i=1
=
k nj I.X 3 = I. yk-j[I. Xij 3]
I.Y
j=1
i=1
k
nj
= I. yk-j[I. Yij] j=1
k nj I.X 4 = I. yk-j[I. Xij 4] j=1 i=l k I.XY
j= I
j==1 i=l
nj
= 1:yk-i[I. Xij 2Yij] i=l
nj
= I. yk-i[I. XijYij]
i= I k
!X 2y
k nj I.X 2 = I. yk-j [I. Xij 2] j=l i=l
n
=
k I. yk-jnj
j=l
Substitution of these quantities into the equations (4), (5) and (6) yields the estimates of a, band c needed for the weight prediction for week k I. When the animals are next weighed the above sums would be updated and
+
R. L. IMAN
292
AND
M. N. BRUNDEN
new estimates of a, band c obtained. Hence very little computer storage is needed for the week-to-week storage of these sums. H is obvious that for a choice of y = I the estimates of a, band c are the same as those obtained from not using the discounted least squares while a choice of y = 0 uses only the data from the very latest week. Hence, a choice of 0 < Y < I is used. For example, with y = 0,3 all of the latest data are used while data 1 week old are discounted to 0.31 = 30%; data 2 weeks old are discounted to O'32 = 9 %; data 3 weeks old are discounted to O'33 = 2.7%, etc. Hence, the effect of older data is quickly diminished. RESULTS
To test the discounted least squares approach, weekly consumption data from a 2-year toxicity study in 360 treated for a period of 104 weeks at dosage rates of the O'32, I, 3,2 and IO mg/kg/day were used. Control rats
bodyweight and food Sprague-Dawley rats drug under test of 0, were given the same
75
65
55 .9 c:
o
'';::;
()
~Q) 45
a. •... Ol 'w
..c:
~ 35
25
15
,
10
20
30
I
40
I
50
I
60
I
70
Age (weeks) Fig. 1. Predicted and observed weights of male rats (dose 1 mgjkg).
I
80
QUADRATIC
GROWTH
PREDICTION
MODEL
293
volume of solution on a bodyweight basis as the treated groups. The age range of rats actually used to fit the quadratic model was] 0-75 weeks as some pretesting prior to 10 weeks precluded the use of the data before that time and weighing was done on a biweekly rather than weekly basis after 75 weeks. However, the time span from ]0 to 75 weeks of age is more than adequate to demonstrate the technique. The choice of y was made by examining the residual sum of squares for predicted and observed weights for y = 0,05, 0·10, 0,] 5, ... 0'95. The residual sum of squares was found to be reasonably invariant to choices of y over a wide range from 0·05 to 0·75. A choice of y = 0·3 was used in the study. Figs] and 2 show a plot of age versus weight for observed (0) weights and predicted (X) weights for male and female groups treated at ] mg/kg. The average error in the difference between predicted and observed average weight from 10 to 75 weeks of age is given in Table 1 for each of the 10 treatment combinations considered. Table 1 shows the magnitude of the error to be 75
65
55
25
15 t------r----,---.,.------r------,.---.-----, 10 20 30 40 5060 70 Age (weeks) Fig. 2. Predicted and observed weights of female rats (dose 1 mg/kg).
80
R. L. IMAN
294
M. N. BRUNDEN
AND
very small in all cases and that the prediction over- or always under-predict the weight.
is not biased so as to always
Table 1. Summary of weight and food consumption predictions for Sprague-Dawley rats from age 13 weeks to 75 weeks. Group
No. rats
Control b' Control ~ 0·32 mg/kg b' 0·32 mg/kg ~ 1 mg/kg b' 1 mg/kg ~ 3·2 mg/kg b' 3·2 mg/kg ~ 10 mg/kg b' 10 mg/kg ~
60 60 30 30 30 30 30 30 30 30
Av. wt. prediction error (g)
0·220 0,072 0·136 0'192 0'008 -0,226 0·285 -0,208 0·152 -0,205
Root mean Av.food square error consumption error (g)
Root mean square error
7·281 5,724 6·207 5,033 8,730 4,096 7·780 5,001 8·358 7'135
8,729 8,138 9·655 6·241 8·671 6'144 9,383 6'444 8·948 6'752
1·696 0,704 2·000 0,953 1·831 0·962 1·511 0·779 0·794 0,454
Table 2. Summary of dose administered and dose received based on predicted values. Percent time administered dose within ± of required dose
Av. dose (mg/kg) received based on weight and food predictions
Group
±8%
±5%
±2%
actual dose
±1%
0·32 mg/kg d' 0·32 mg/kg ~
100 100
100,0 98,4
93,4 77-1
75-4 50,8
1 mg/kg b' 1 mg/kg ~
100 100
98,4 100'0
82'0 90·2
67·2 59,0
3·2 mg/kg b' 3·2 mg/kg ~
100 100
100,0 98,4
77-1 77-1
44,3 37,7
]0 mg/kg d' 10 mg/kg ~
100 100
96,7 95,]
70,5 62·3
36·1 42·6
100
98-4
78·7
51·6
Totals
root mean square error
0·32 0·32 0,99 0,99
0'02 0·02 0,05 0,05
3'17 3'18 9,97 9,98
0·18 0·18 0,59 0,55
If the dosage be prepared based on the predicted weight, then the actual dosage administered can be calculated when the true weight is recorded: actual dosage administered
predicted weight X
true weight
required dosage
QUADRATIC
GROWTH
PREDICTION
MODEL
295
If the weight prediction is high the dosage administered will be high, and a low prediction results in a low dosage. Table 2 is a performance summary of actual dosage administered based on the weight predictions. FOOD
CONSUMPTION
PREDICTION
A prediction for food consumption can be made on the basis of the previous ratio of food consumption and weight and the predicted weight: predicted food consumption for next week A modified for week k
this week's food consumption
predicted weight for next week
-----------x this week's weight
ad hoc discount model for food consumption 1 can be made:
(Fe)
prediction
for 0 < 0 < 1. Table 1 also gives the re"sults in food consumption from the sample data for a choice of 0 = 0,3.
prediction
+
--------
X
. ~ (
ok-j
'" Wt.k+
I
Wt.j )
J=1
Unlike the weight prediction, the food consumption prediction tends to be slightly biased towards over estimation. However, the magnitude of the bias is reasonably small. Once the predicted and actual weight and food consumption values are known, the following computation will yield the average dosages actually received by the animals if the dosages are prepared based on the weight and food consumption predictions: drug actually received
=
(wt
'"
--
(kg))
wt (kg)
(FC
---
(gms) )
X
required dosage (mg/kg)
F"'C(gms)
These results are given in Table 2. Further
computation
'" of Dk+
1
=
'" 7 . Wtk+
I
(mg/kg
drug)
and
C'" k+ I
'" I/FC'" k+ I provide for the predicted total drug usage and concentration Dk+ drug in food, respectively, for week k 1.
+
= of
296
R. L. IMAN
AND
M. N. BRUNDEN
CONCLUSION
The discounted least squares quadratic growth prediction model is accurate and easy to use with long-term toxicity studies. The amount of computer storage needed for the sums required to provide updated estimates of a, b and c is minimal. The implementation of food consumption prediction can be worked into the model very easily. Most importantly, the quadratic prediction model would eliminate the need for instantaneous processing of weight data in order that food dosage can be properly mixed. For implementation of the model 3 weeks of base data must be available before a prediction can be made (for the current study data was available starting at 10 weeks of age; the first prediction of weight was made starting at 13 weeks of age), and on some computers it may be necessary for the success of the model that all calculations be carried out in double precis.on.
REFERENCES
Brown, R. G. (1963). Smoothing, forecasting and prediction of discrete time series.
New Jersey:
Prentice-Hall. Brunden, M. N., Clapp, H. W. & Heun, L. L. (1974). A growth model for laboratory rats. Laboratory Animals 8, 329-335.
