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30 35 40 45 50 55 60 65 70 75 Sensory intensity level

30 35 40 45 50 55 60 65 70 75 Sensory intensity level

Figure 3. Schematic relation between sensory attribute level and overall liking. The relation usually follows an inverted U shaped curve. Large areas under the curve (subtended by the curve) correspond to important attributes.

variables. Dependent variables can be sensory attributes, acceptance ratings (all from consumers), or physical measures (including cost of goods), etc.

Relations between variables are expressed by equations. Researchers can use one of many available equations, depending upon the expected relation between the independent variable and the dependent variable. Some of these equations are

The Linear Equation. Changes in the dependent variable correspond to linear changes in the independent variable. The linear equation often emerges when we vary one ingredient (eg, sucrose) in a simple food system (eg, sweetened carbonated beverage). Over the limited concentration range consumer ratings of perceived sweetness versus sweetener concentration follow a straight line. Although a straight line might not describe the "true, underlying" relation between the two variables (sucrose level versus per-

Quadratic Relation. As the independent variable increases the rating first increases, peaks and then drops down. The parabolic or quadratic equation is a good summary of how liking varies with physical level (33). The parameters of the quadratic equation are unique to each product. Parabolic equations allow for increases, peaks, and then decreases (the classical inverted U curve), or simply increase and flatten (or the corresponding decrease and flattening). The equation is expressed as:

Dependent Variable = A + B(Independent) + C(Independent)2

The Plane (for Two or More Independent Variables, Jointly Predicting the Dependent Variable). The plane generalizes the line. We can envision it easily as a flat sheet in 3 dimensions. The slope of the plane in any dimension (corresponding to a single independent variable) is given by the coefficient for that independent variable. The plane equation does not account for interactions among the variables. The plane assumes that each variable acts entirely independently to determine the dependent variable. We write the equation of the plane as a multiple linear equation:

Dependent

In the absence of theory about the relation between independent variables and dependent variable it is prudent to use the multiple linear equation. The equation makes no assumptions other than there exists a simple linear relation between the dependent variable and each of the independent variables.

Quadratic Surface. Like the plane the quadratic surface also relates several independent variables simultaneously to the dependent variable. We assume that there exists a parabolic or curvilinear relation between the independent variables and the dependent variable (eg, for overall liking). However, for a quadratic or parabolic surface not only do we have linear terms, but also square terms.

If the equation comprises only linear terms and square terms, then we assume that the variables act independently of each other to determine the value of the dependent variable. The square terms allow each independent variable to act in a nonlinear fashion. There may be a flattening of the surface, so that as an independent variable increases the dependent variable can first increase, but then flatten out, and then perhaps even decrease. If there are no interactions between the independent variables, then we write the equation as:

Dependent = A + B(Independent 1) -I- (¡¡(Independent If

+ D(Independent 2) + Etfndependent 2f

Like the plane equation, the parabolic or quadratic equation can be extended to accommodate many independent variables simultaneously. If there exists interaction between pairs of independent variables, we can write the equation as above, but also include a multiplicative term [(Independent 1) X (Independent 2)] to account for interactions (here between the first two independent variables). Many such pairwise interactions may exist. The equation can accommodate them.

There are many other equations that an investigator can use in order to model the relations between independent variables and dependent variables. For product testing, however, we generally do not really know the true form of the relation (viz, how variables interact with each other). It is prudent to use the most parsimonious equation. Most researchers opt for the linear equation, and add square terms and pairwise interactive (multiplicative) terms only when specifically required, usually for one of two reasons:

Table 17. Model (Equation) Relating Formula and Process Variables to Liking"

Variable6

Coefficient

Std error

T value0

P valued

Constant

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