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Curve Fitting
Fitting a curve to the points on a graph implies that there is some sort of relationship between the variables x and y, such as dose of drug versus pharmacologic effect (eg, lowering of blood pressure). Moreover, when using curve fitting, the relationship is not confined to isolated points but is a continuous function of x and y. In many cases, a hypothesis is made concerning the relationship between the variables x and y. Then, an empirical equation is formed that best describes the hypothesis. This empirical equation must satisfactorily fit the experimental or observed data. If the relationship between x and y is linearly related, then the relationship between the two can be expressed as a straight line.
Physiologic variables are not always linearly related. However, the data may be arranged or transformed to express the relationship between the variables as a straight line. Straight lines are very useful for accurately predicting values for which there are no experimental observations. The general equation of a straight line is y = mx + b (2.5) where m = slope and b = y intercept. Equation 2.5 could yield any one of the graphs shown in Fig. 2-5, depending on the value of m. The absolute magnitude of m gives some idea of the steepness of the curve. For example, as the value of m approaches 0, the line becomes more horizontal. As the absolute value of m becomes larger, the line slopes farther upward or downward, depending on whether m is positive or negative, respectively.
Linear Regression/Least Squares Method This method is often encountered and used in clinical pharmacy studies to construct a linear relationship between an independent variable (also known as the input factor or the x factor) and a dependent variable (commonly known as an output variable, an outcome, or the y factor). In pharmacokinetics, the relationship between the plasma drug concentrations versus time can be expressed as a linear function. Because of the availability of computing devices (computer programs, scientific calculators, etc), the development of a linear equation has indeed become a simple task.
A general format for a linear relationship is often expressed as:
y = mx + b (2.6)
where y is the dependent variable, x is the independent variable, m is the slope, and b is the y intercept. The value of the slope and the y intercept may be positive, negative, or zero. A positive linear relationship has a positive slope, and a negative slope belongs to a negative linear relationship (Gaddis and Gaddis, 1990; Munro, 2005).
The strength of the linear relationship is assessed by the correlation coefficient (r). The value of r is positive when the slope is positive and it is negative when the slope is negative. When r takes the value of either +1 or −1, a perfect relationship exists between the variables. A zero value for the slope (or for r) indicates that there is no linear relationship existing between y and x. In addition to r, the coefficient of determination (r2) is often computed to express how much variability in the outcome is explained by the input factor. For example, if r is 0.90, then r2 equals to 0.81. This means that the input variable explains 81% of the variability observed in y. It should be noted, however, that a high correlation between the two variables does not necessarily mean causation. For example, the passage of time is not really the cause for the drug concentration in the plasma to decrease. Rather it is the distribution and the elimination functions that cause the level of the drug to decrease over time (Gaddis and Gaddis, 1990; Munro, 2005).
The linear regression/least squares method assumes, for simplicity, that there is a linear relationship between the variables. If a linear line deviates substantially from the data, it may suggest the need for a nonlinear regression model, although several variables (multiple linear regression) may be involved.
Nonlinear regression models are complex mathematical procedures that are best performed with a computer program.
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