Nonlinear regression.
Common types of curve.
Defi nition of an exponential curve.
The negative decay curve.
Logarithmic transformation.
Goodness of fi t to an exponential curve.
21.1 INTRODUCTION
In Statnotes 15 , 17 , and 18 , the use of correlation and regression methods to analyze a linear relationship between two variables was described. Hence, Pearson ’ s correlation coeffi cient r is used to establish whether there is a signifi cant linear correlation between two variables. Once a linear correlation has been established, a regression line can be fi tted using the method of least squares to describe the relationship in more detail. Linear regres-sion may be adequate for many purposes, but some variables in microbiology may not be connected by such a simple relationship. The discovery of the precise relation between two or more variables is a problem of curve fi tting known as nonlinear or curvilinear regression , and the fi tting of a straight line to data is the simplest case of this general principle. Curve fi tting may be appropriate in a variety of experimental circumstances. For example, an investigator may be interested in the pattern of decline in the number of fungal
Statistical Analysis in Microbiology: Statnotes, Edited by Richard A. Armstrong and Anthony C. Hilton Copyright © 2010 John Wiley & Sons, Inc.
110 NONLINEAR REGRESSION: FITTING AN EXPONENTIAL CURVE
colonies with depth in the soil or the pattern of penetration of an antiseptic compound into the skin. This statnote discusses the common types of curve that can arise in microbiologi-cal research and specifi microbiologi-cally describes the fi tting of a curve in which the relationship between Y and X can be described by an exponential decay function . Subsequent statnotes will describe the fi tting of a general polynomial - type curve (see Statnote 22 ) and curves that require more complex fi tting methods such as nonlinear estimation (see Statnote 23 ).
21.2 COMMON TYPES OF CURVE
There are four types of curve that commonly occur in microbiological research, and these are illustrated in Figure 21.1 . The fi rst curve is the compound interest law or exponential curve and is given by the following relation:
Y =a
(
expbx)
(21.1)where a and b are constants to be estimated. This is the type of curve exhibited by the growth of a bacterial population in a culture when it is increasing in numbers most rapidly. The second curve is the exponential decay curve where Y declines to zero from an initial value a and represents the way in which quantities often decay or decline with time:
Y =a
(
exp−bx)
(21.2) The third type of curve is the asymptotic curve which increases from a value a − b and then steadily approaches a maximum value a known as the asymptote . The asymptotic curve is given by the following relation:Y= −a b
(
exp−cx)
(21.3)
Figure 21.1. Common types of curve that occur in microbiological research.
Exponential growth Y
X
Exponential decay Y
X Asymptotic
Y
X
Logistic growth law Y
X
DATA 111
TA B L E 21.1 Number of Fungal Colonies Derived by Dilution Plate Method from a Gram of Soil Collected from Different Depths in Consolidated Sand a
Finally, the fourth type of curve is the logistic growth law , the most common sigmoid type curve and a relationship that has played a prominent part in the study of the growth of microorganisms in batch culture. Hence, the initial lag stage is followed by approximately exponential growth, but then as saturation approaches, growth slows down and reaches a maximum stationary value. The logistic growth law is given by the following equation:
Y a
b cx
=1+ exp− (21.4)
21.3 SCENARIO
Soil has various horizons and forms a complex series of microhabitats. If soil has a uniform structure, soil organisms decline markedly within a few centimeters of the surface and continue to decrease with depth. This pattern of decline results in a typical curve exhibited by populations of bacteria in soil of uniform composition (Burges, 1967 ). It has been postulated that the decline in microbial numbers could be attributable to the reduction in available carbon compounds with depth, but a similar decline is also seen in peaty soils, which vary less in available carbon with depth. To determine whether soil fungi exhibit a similar curve to bacteria, the number of fungal colonies at different depths was measured in sandy soil at a site in the West Midlands, United Kingdom. The number of fungal colonies was estimated by the dilution plate method from a profi le dug into the consolidated sand at the site. Samples of soil of 1 g were taken at varying depths down to 100 cm.
21.4 DATA
The data are presented in Table 21.1 and comprise measurements of the number of fungal colonies ( Y ) in relation to soil depth ( X ).
112 NONLINEAR REGRESSION: FITTING AN EXPONENTIAL CURVE
Figure 21.2. Fitting an exponential decay model to the data in Table 21.1 . Circular data points represent the untransformed data and exhibit the typical features of a negative exponential curve. Square data points represent the Y values transformed to logarithms.
260000 13.0 is plotted against X . If the relationship is exponential (positive or negative), the graph will be linear, and a straight line can be fi tted using the methods of linear regression described in Statnote 18 . The method is similar to that which was used to fi t a power law model in Statnote 15 . Fitting a regression by transformation of the Y variable makes similar assumptions to those described for linear regression (see Statnote 18 ). Most sta-tistical software will carry out this analysis as it requires only that the Y variable is transformed to logarithms and then the fi tting of a conventional straight line.
21.5.2 Interpretation
The data are plotted on the original scale and on a log scale in Figure 21.2 . A linear regres-sion appears to be a good fi t to the data ( r = − 0.90, P < 0.001, r 2 = 0.81), suggesting that the number of fungal colonies does decline exponentially with depth of soil, a curve similar to that shown by the soil bacteria.
21.6 CONCLUSION
Nonlinear relationships are common in microbiological research and necessitate the use of the statistical methods of nonlinear regression or curve fi tting. In some circumstances, the investigator may wish to fi t an exponential model to the data, that is, to test the hypothesis that a quantity Y either increases or decays exponentially with increasing X . This type of model is straightforward to fi t, as taking logarithms of the Y variable makes the relationship linear, which can then be treated by the method of linear regression.