Functions logistic

Most corrected characteristic separation curves (27) fit the following logistic function ... 

FIGURE 10.5 Full agonist potency ratios, (a) Data fit to individual three-parameter logistic functions. Potency ratios are not independent of level of response. At 20%, PR = 2.4 at 50%, PR = 4.1 and at 80%, PR = 6.9. (b) Curves refit to logistic with common maximum asymptote and slope. PR = 4.1. The fit to common slope and maximum is not statistically significant from individual fit. 

Ideally, there should be >4 data points for each estimated parameter. Under this guideline, a 3 parameter logistic function should have 12 data points. At the least, the number of data points minus the number of parameters should be > 3. 

The responses are values of y, and n is the number of responses. A calculated SSq value will have associated with it a value for the degrees of freedom. If there are no fitting parameters involved in applying the model, the number of degrees of freedom will be n. For the data in Figure 11.13, dfs = 10. A more complex model for these data is a four-parameter logistic function of the form... 

This value is identified in F tables for the corresponding dfc and dfs. For example, for the data in Figure 11.13, F = 7.26 for df=6, 10. To be significant at the 95% level of confidence (5% chance that this F actually is not significant), the value of F for df = 6, 10 needs to be > 4.06. In this case, since F is greater than this value there is statistical validation for usage of the most complex model. The data should then be fit to a four-parameter logistic function to yield a dose-response curve. 

FIGURE 11.13 A collection of 10 responses (ordinates) to a compound resulting from exposure of a biological preparation to 10 concentrations of the compound (abscissae, log scale). The dotted line indicates the mean total response of all of the concentrations. The sigmoidal curve indicates the best fit of a four-parameter logistic function to the data points. The data were fit to Emax = 5.2, n = 1, EC5o = 0.4 pM, and basal = 0.3. The value for F is 9.1, df=6, 10. This shows that the fit to the complex model is statistically preferred (the fit to the sigmoidal curve is indicated). 

FIGURE 11.14 Data set consisting of a control dose-response curve and curves obtained in the presence of three concentrations of antagonist. Panel a curves fit to individual logistic functions (Equation 11.29) each to its own maximum, K value, and slope. Panel b curves fit to the average maximum of the individual curves (common maximum) and average slope of the curves (common n) with only K fit individually. The F value for the comparison of the two models is 2.4, df = 12,18. This value is not significant at the 95% level. Therefore, there is no statistical support for the hypothesis that the more complex model of individual maxima and slopes is required to fit the data. In this case, a set of curves with common maximum and slope can be used to fit these data. 

The mean maximal response for the five curves is 96.1 and the mean slope is 1.27. The five curves are then refit to three-parameter logistic functions utilizing the mean maximal response and mean slope. The ECso for the curves fit in this manner are shown in Table 12.6b (ECS0 values in column labeled Mean). 

FIGURE 12.13 Calculation of a pA2 value for an insurmountable antagonist, (a) Conner ation-response curve for control (filled circles) and in the presence of 2 jiM antagonist (open circles), (b) Data points fit to logistic functions. Dose ratio measured at response value 0.3 (dotted line). In this case, the DR = (200nM/50nM = 4). 

Sigmoid, the characteristic S-shaped curves defined by functions such as the Langmuir isotherm and logistic function (when plotted on a logarithmic abscissal scale). 

Remember that proponents of airline deregulation argued for price benefits of competition, but they did not foresee how it would revolutionize the logistics functions of corporate America through the rise of companies such as Federal Express. Similarly, those who advocated telecommunications competition did not anticipate the new value-added services provided at the switch and whole new categories of customer-owned equipment connected to the network. 

A commonly used sigmoidal function is the logistic function (Figure 2.22). 

When the logistic function is used we can make use of the identity that ... 

These tests are based on the generalized logistic function (Cox, 1972). Specifically one can use the Cocrhan-Armitage test (or its parallel, Mantel-Haenszel verson) for monotonic trend as heterogeneity test. 

Tumor prevalence is modeled as logistic function of dose and polynomial in age. 

FIGURE 5.9 Visualization of Equations 5.17 and 5.18 for modeling the posterior probabilities with LR. The left-hand plot pictures the logistic function P z) = ez/( 1 + ez) the right-hand plot shows P2(z) = 1 - Pi(z). 

As the last section implies, a pharmaceutical excipient distributor can greatly complement the procurement and logistical functions of an excipient user by taking on... 

Outsourcing by chemical producers as chemical producers faced up to increasing competition over the past decades, they attempted not only to expand their market positions geographically, but also to improve operational efficiency by focusing on their core activities. As a result, suppliers have continuously outsourced sales and logistics functions as well as related value-added services to third-party distributors. 

Physico-chemical measurements using chromatographic methods produce responses that are linear to the concentrations. As IA measures the resulting signals of a reaction, however, the response is a nonlinear function of the analyte concentration. Often, the regression model used to describe this relationship is a four- or five-parameter logistic function, as shown in the sigmoid shape standard curve in Fig. 6.4. 

This is a combination of a proportional control (first term of the parentheses) and a differential (second term). The beta cell is thus able to respond with a fast, but transient, response dependent on the rate of glucose change. This is demonstrated in Fig. 6.5b, where the glucose concentration increases from 5 mM to 10 mM as a logistic function with different slopes. The figure shows that there is an increasing overshoot for increasing slope. This type of differential control explains part of the so-called first phase of insulin release [33-35]. 

A simple example is found in the Logistic function discussed above ... 

In their dichotomous fit, Walker and Duncan transform the Logistic function to an equal variance space by dividing each data point by its variance. The variance of a probability value, P, is P (1-P). For the first iteration, P (1-P) is equal to w. This suggests minimizing the function V ap 9 Y... 

The idealized step function (Heaviside function) described above is only one of many functions used in basic neural network units. One problem with this function is that it is discontinuous, and does not a continuous derivative derivatives are at the heart of many training methods, a topic to be discussed later. An approximation to a step function that is easier to handle mathematically is the logistic function, shown in Figure 2.6 below. 

The output of a sigmoid function, such as the logistic function, is not 0 or 1 but somewhere between. Therefore a decision must be made as to what value will be called on this value is called, by convention in this book, a critical value. Typically, if the output of the function is > 0.5, then the unit is said to be on (equivalent to the Heaviside function output of 1) otherwise it s off. Depending upon the application this can be changed to other critical values like 0.8 or 0.9. Higher critical values can be said to be less sensitive (it takes more input to turn them on) and lower critical values more sensitive (they turn on at lower input levels). 

---