Logistics curve

Example 10 Logistics Curve We shall derive the logistics curve representing the ciimiilarive-freqiiency distrihiitions of the normal distrihiition curve defined by Eqs. (9-72) and (9-73). In this case, y varies between a cumulative probability of zero and unity as z varies from — to -l-oo. Since the upper bound is unity, c = 1. From Table 9-10 the area under the right-hand side of the curve between = 0 and z = z may be read. Since the frequency curve is symmetrical about the mean, this is also the area between = 0 and z=z. Hence, the area under the frequency curve, which represents the cumulative probability, is 0.50000 at = 0 and the 80 percentile, for which the area is 0.80000, corresponds to the value = 0.842. We substitute these values into Eqs. (9-92) through (9-94) to give... 

This example assumes that RIA was chosen. The principle behind RIA is the competition between the analyte A and a radioactively tagged control C (e.g., a /-marked ester of the species in question) for the binding site of an antibody specifically induced and harvested for this purpose. The calibration function takes on the shape of a logistic curve that extends over about three orders of magnitude. (Cf. Fig. 4.38a.) The limit of detection is near the B/Bo = 1 point (arrow ) in the upper left corner, where the antibody s binding sites are fully sequestered by C the nearly linear center portion is preferrably used for quantitation. 

The standard (four-parameter logistic) curve was prepared by the simplex method using absorbance values collected from each participating laboratory. 

Figure 6.8 The time-course of (a) N uptake, (b) soil solution NH4, and (c) and (d) root length density in pots of flooded soil planted with rice with (O) and without ( ) added N. In (a), lines are fitted logistic curves, slopes of which give values of dU/dt in Equation (6.10). In (b), solid horizontal lines are Cl broken lines Cls calculated with Equation (6.11). In (c) and (d), the lines indicate the minimum root length densities required to explain uptake calculated with the measured Cl values se (full lines) and Cl derived from exchangeable NH4+ values se (Kirk and Solivas, 1997). Reproduced by permission of Blackwell Publishing...

From the 1970s onwards, Cesare Marchetti and other system analysts have studied thousands of artifacts, and have discovered that their behaviour is described by the same equations that Lotka and Volterra found for the behaviour of predators and prey. The growth pattern of cars, for example, is a logistic curve. Cars spread in a market exactly as bacteria in a broth or rabbits in a prairie. Cultural novelties diffuse into a society as mutant genes in a population, and markets behave as their ecological niches. But why ... 

Figure 3. The logistic regression model used to estimate LD50 is represented on the left where 7C represents the proportion of dead plants. The logistic curve can be linearized by using the logit transformation shown on the right. LD50 values were estimated with the regression coefficients for logit 7C=0.0, as shown in the inset box.

Let us consider a product which is sold entirely on the basis of personal recommendation. The rate of sale will depend on the number of people who have already bought the product. Thus initially sales will increase exponentially. Eventually the market will be saturated, and only replacement purchases will be made. If the frequency curve may be assumed to be symmetrical about a single maximum value, the cumulative distribution curve is known as the logistics curve and is defined by Eq. (9-91) ... 

If the values of a obtained from Eq. (9-94) differ significantly, the logistics curve is not a suitable representation of the data. 

Example 10 Logistics Curve We shall derive the logistics curve representing the cumulative-frequency distrihutions of the normal distribution... 

When a cumulative-frequency curve can be satisfactorily represented by a logistics curve, the underlying frequency curve can be obtained by differentiation of Eq. (9-9l) as... 

Logistic regression modeling is used for predicting the probability of occurrence of an event by fitting data to a logistic curve.28 It describes the relationship between the categorical response variable and one or more continuous variables.29 Such a model can be described in Equation 3 ... 

This relationship is established by measurement of samples with known amounts of analyte (calibrators). One may distinguish between solutions of pure chemical standards and samples with known amounts of analyte present in the typical matrix that is to be measured (e.g., human serum). The first situation applies typically to a reference measurement procedure, which is not influenced by matrix effects, and the second case corresponds typically to a field method that often is influenced by matrix components and so preferably is calibrated using the relevant matrix. Calibration functions may be linear or curved, and in the case of immunoassays often of a special form (e.g., modeled by the four-parameter logistic curve) This model (logistic in log x) has been used for both radioimmunoassay and enzyme immunoassay techniques and can be written in several forms as shown (Table 14-1). Nonlinear regression analysis is applied to estimate the relationship, or a logit transforma-... 

Figure 1. Phytotoxicity of chlorotoluron and isoproturon incorporated in liquid medium of hydroponically grown resistant "Peldon" and susceptible "Rothamsted" black-grass. Standard error bars represent pooled errors of means, each determined from 10 measurements. Log ED50 values (plus standard errors) have been determined from a logistic curve fitted to the log-transformed data using a maximum likelihood program (67).

Measurement and Mathematical Fitting of Cell Growth. Richards (1960) used a generalized logistic curve for the mathematical fitting of the growth curve of plants. The equation is ... 

D Rodbard, DM Hutt. Statistical analysis of radioimmunoassays and immunoradio-metric (labelled antibody) assays A generalized weighted, iterative, least-squares method for logistic curve fitting. In Radioimmunoassay and Related Procedures in Medicine, Vol I. Vienna International Atomic Energy Agency, 1974, p 165. 

HE Grotjan Jr, E Steinberger. Radioimmunoassay and bioassay data processing using a logistic curve fitting routine adapted to a desk top computer. Comput Biol Med 7 159, 1977. 

A standard curve (a four-parameter logistic curve) is prepared using the absorbance value collected from each participant lab. 

Figure 13.5 Dimensionless representation of logistic curve for ceU growth in a batch bioreactor when X /Xq = 100.

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