Proportion equations describe concurrent performance when alternatives differ only in rate of reinforcement. However, in complex environments, other factors also contribute to choice and preference. These factors arise from the biology and environmental history of the organism. For example, sources of error may include different amounts of effort for the responses, qualitative differences in reinforcement such as food versus water, a history of punishment, a tendency to respond to the right alternative rather than the left, and sensory capacities. To include these and other conditions within the matching law, it is useful to express the law in terms of ratios rather than proportions. When relative rate of response matches relative rate of reinforcement, the ratio equation is simply a restatement of the proportional form of the matching law. A generalized form of the ratio equation may, however, be used to handle the situation in which unknown factors influence the distribution of behavior. These factors produce systematic departures from ideal matching but may be represented as two constants (parameters) in the generalized matching equation. In this form, the matching equation is known as the generalized matching law. In the equation, the coefficient k and the exponent a are values that represent two sources of error for a given experiment. When these parameters are equal to 1, the equation is the simple ratio form of the matching law. See also matching law.