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Growth is a fundamental property of biological systems, occuring at
the level of populations, individual animals and plants, and within
organisms.
<P>
Growth is a fundamental property of biological systems, occuring
at the level of populations, individual animals and plants, and
within organisms. Much research has been devoted to modelling growth
processes, and there are many ways of doing this: mechanistic
models, time series, stochastic differential equations etc.
<p>
Sometimes we simply wish to summarise growth observations in terms
of a few parameters, perhaps in order to compare individuals or groups.
Many growth phenomena in nature show an "S" shaped pattern, with initially
slow growth speeding up before slowing down to approach a limit. <p>
These patterns can be modelled using several mathematical functions.
Here we explore the generalised logistic and Gompertz
curves.
<P>
The lower asymptote is the starting level. <br>
The upper asymptote is the mature level. <br>
The point of inflexion is the point of maximum growth. <p>
The curve can be modelled in a number of ways.
<P>
The generalised logistic (or Richard's) curve is a widely used and flexible
function for growth modelling. <p>
The A parameter controls the lower asymptote. The C parameter controls the upper asymptote.
The M parameter controls the time of maximum growth.
The B parameter controls the growth rate. It is at the point of
inflexion.
The T parameter controls where maximum growth occurs - nearer the lower or
upper asumptote.
We will fit the generalised logistic curve to body weight data for cows
<p>
<a href="cowplot.htm"> <img alt="cowplot"
src="../images/cowplot2.gif" WIDTH=100 HEIGHT=77> </a>
Cow live weights have been recorded every two weeks for 232 weeks, i.e. 116
observations in total. <p>
The curve is fitted separately for each cow, so that parameters can be
compared. <p>
<pre>
***** Nonlinear regression analysis *****
Response variate: weight Explanatory: age
Fitted Curve: A + C/(1 + T*EXP(-B*(X - M)))**1/T
*** Estimates of parameters ***
estimate s.e.
B 0.01955 0.00184
M 20.1 18.9
T 0.15 1.26
C 968. 519.
A -243. 522.
*** Summary of analysis ***
d.f. s.s. m.s. v.r.
Regression 4 3819540. 954885.00 9788.66
Residual 111 10828. 97.55
Total 115 3830368. 33307.55
Percentage variance accounted for 99.7
Standard error of observations is estimated to be 9.88
</pre>
<H3>
THE FITTED GENERALISED LOGISTIC CURVE
</H3>
<BR>
<CENTER>
<img alt="graph" src="../images/cowfit.gif" WIDTH=683 HEIGHT=542>
</CENTER>
<p>
<BR>
It is useful to look at a plot of residuals versus fitted values.
<pre>
-+---------+---------+---------+---------+-----
I * I
2.5 I * I
I * * I
I 3 I
I *** 3 I
I ** * *2* 2 * 24 I
I * * ***** * 2* * 26* I
0.0 I * ** * ** * * *** *2 ** I
I *** * 2 *** 2* * *2*24 I
I * 2 * 2 *22 * I
I *3 2 I
I * *** I
I 3 * I
-2.5 I I
I * I
I I
I I
I I
I I
-5.0 I I
-+---------+---------+---------+---------+-----
0.0 200.0 400.0 600.0 800.0
res v. fit using symbol *
</pre>
Clearly the residuals are not randomly distributed about zero. This is because
observations over time are correlated, and the cows weight fluctuates about
the fitted curve. Unless these fluctuations are of interest, we will not
attempt to model them.
<p>
The <a href="gomp.htm"> Gompertz curve </a> is a limiting case of the
generalised logistic as T becomes very small or very large. <p>
<img src="../images/gomp.gif" WIDTH=225 HEIGHT=40> <p>
where Y=weight, height, size etc; X = time.
<p>
<BR>
We use <a href="slide12a.htm"> data </a> from a different cow to illustrate
fitting this curve <p>
<pre>
***** Nonlinear regression analysis *****
Response variate: weight
Explanatory: age
Fitted Curve: A + C*EXP(-EXP(-B*(X-M)))
Constraints: C > 0
*** Estimates of parameters ***
estimate s.e.
B 0.031022 0.000848
M 35.63 1.57
C 664.7 19.3
A 8.2 17.5
*** Summary of analysis ***
d.f. s.s. m.s. v.r.
Regression 3 2545361. 848453.69 12336.39
Residual 78 5365. 68.78
Total 81 2550726. 31490.44
Percentage variance accounted for 99.8
Standard error of observations is estimated to be 8.29
</pre>
<H3>
FITTED COW DATA
</H3>
<BR>
<CENTER>
<img src="../images/cowfit2.gif" alt="Fitted cow data" WIDTH=692 HEIGHT=534>
</CENTER>
<p>
<CENTER>
<pre>
-+---------+---------+---------+---------+-----
I * I
2.5 I * I
I * * I
I 3 I
I *** 3 I
I ** * *2* 2 * 24 I
I * ***** * 2* * 26* I
0.0 I ** 2* * ** * * *** *3 2* I
I ** * 2 *** ** * ****4 I
I * 2 * 2* *22 * I
I *3 2 I
I * *** I
I 3 * I
-2.5 I I
I * I
I I
I I
I I
I I
-5.0 I I
-+---------+---------+---------+---------+-----
0.0 200.0 400.0 600.0 800.0
res v. fit using symbol *
</pre>
</CENTER>
<p>
The generalized logistic model is a flexible model that can accommodate most
growth data. <p>
The Gompertz curve can be thought of as a limiting version of the generalized
logistic (i.e. as T approaches zero or infinity). <p>
The Gompertz curve has one parameter fewer than the generalized logistic and
should be used if :
<blockquote>
* it is recommended by the statistics package, or <p>
* it is difficult to estimate T in the generalized logistic or it does
not differ significantly from zero, or <p>
* it is the conventional model for a particular type of growth data
</blockquote>
<p>
<blockquote>
* The generalised logistic and Gompertz functions provide a flexible family
of curves for summarising and comparing growth. <p>
* The estimated parameters can be used to study differences in growth
patterns in different population treatment groups. <p>
</blockquote>
<p>
Several programs can be used to fit growth curve models: <p>
<blockquote>
<a TARGET="_top" href="http://www.sas.com/"> SAS </a> <br>
<a TARGET="_top" href="http://www.mathsoft.com/"> S-PLUS </a> <br>
<a TARGET="_top" href="http://www.nag.co.uk:80/stats/TT.html"> GENSTAT 5 </a>
</blockquote>
The easiest of these packages to use for fitting non-linear models is:
<blockquote>
<a href="slide17a.htm"> GENSTAT 5 </a>
</blockquote>
for which instructions are included here.
<p>
<H3>
FITTING GENERALISED LOGISTIC IN GENSTAT
</H3>
<BR>
Models are fitted with these two instructions
<pre>
MODEL weight; RESIDUALS=res; FITTED=fit
FITCURVE [CURVE=glog] age
</pre>
<p>
The MODEL statement specifies the response variable. <p>
The FITCURVE statement specifies the curve to fit and the independent (time)
variable. <p>
The option <code> CURVE=glog </code> specifies that we want to fit the
generalised logistic curve. Other options could be
<ul>
<li> <code> CURVE=log </code> for the logistic curve. (See
<a href="slide07.htm"> T </a> parameter.)
<li> <code> CURVE=gomp </code> for the <a href="gomp.htm"> Gompertz </a> curve.
</ul>
If you would like to explore various models you can obtain the
<a href="../example/cow.htm"> data </a> and the Genstat
<a href="../example/cowgen.htm"> code </a> used above.
<p>
A further example is <a href="slide17c.htm"> available </a> which illustrates
the fitting of separate growth curves for each experimental unit.
The coefficients are averaged for the various combinations of
treatment factors in order to
identify possible effects of treatments on the shape of the curves.
<p>
<h3> FITTING GOMPERTZ CURVES IN GENSTAT<br>
<img alt="line" src="../../icons/coloured.gif">
</h3>
<p>
Curves are fitted with these instructions : <p>
MODEL weight; RESIDUALS=res; FITTED=fit <br>
FITCURVE [CURVE=gomp] age <p>
The MODEL statement specifies the response variable. <p>
The FITCURVE statement specifies the curve to fit and the independent (time)
variable. <p>
This experiment examined the effects of various factors on the rate of
germination of chickweed seeds.
<p>
The experiment consisted of a three-replicate complete factorial
arrangement with the factors :
<ul>
<li> Seed populations (3) ; <br>
<li> Collections within populations (3) ; <br>
<li> Sowing temperatures (2). <br>
</ul>
<p>
In total there were 3 x 3 x 3 x 2 = 54 experimental units for which the
number of seeds germinating was recorded daily over a period of 24 days.
<p>
The <a href="../example/chick.htm"> data </a> and the Genstat
<a href="../example/chickgen.htm"> code </a> , which fits growth curves
for each experimental unit and forms a table of the medians for
each of the five parameters of the Gompertz curve , is provided here.
<p>
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