Guthrie’s call was answered by William K. Estes. His stimulus sampling theory, which started as a form of stimulus-response associationism, attempted to formalize many of the ideas of Guthrie (Bower & Hilgard, 1981, p. 213). Estes believed that the interplay between theory and experiment was hindered “by the fact that none of the many current theories of learning commands general agreement among researchers,” that progress toward a common frame of reference would be slow “so long as most theories are built around verbally defined hypothetical constructs which are not susceptible to unequivocal verification,” and that while awaiting resolution of the disparities among competing theories it would “be advantageous to systematize well established empirical relationships at a peripheral, statistical level of analysis” (W. K. Estes, 1950, p. 94). He felt that the possibility of agreement on a theoretical framework would be maximized by defining concepts in terms of variables that could be experimentally manipulated and by developing consequences of assumptions through strict mathematical reasoning.
Estes, like Hull, developed a mathematical model of learning. His model treated learning and performance as a stochastic problem, and aimed to quantify the likelihood of a correct response—for example, the chance of a rat turning left in a T-maze. Initially, the probability of the rat turning to the left is assumed to be .5. However, unlike a coin toss, in which each toss has a 50/50 chance of resulting in heads or tails, as a result of the learning process, the rat becomes progressively more biased to turning left. This learning, he assumed, takes place over the course of successive trials.
An important assumption in the model is that the learning situation is made up of a large, but finite, number of stimulus elements. These elements include all things that the experimental subject experiences at the onset of a learning trial, including, “the experimenter, the room temperature, extraneous noises inside and outside the room, and conditions within the experimental subject, such as fatigue, or headache” (Hergenhahn, 1982, p. 222).
Estes’ model began with the basic premise of experimental behaviorism, namely, that a response, R, is a function of a stimulus, S:
R = f(S)
He then applied to this equation a mathematical expression of probability and restated this relationship to say that the probability of the occurrence of a response class, Rk, is a function of the ratio of x to S, where S represents the total number of stimulus elements effective in the stimulus situation, and x represents the number of those elements that are conditioned to the response class, Rk:
Probability of Rk = x/S
In other words, “the probability of any response is assumed to be equal to the proportion of sampled elements on that trial that are connected to that response” (Bower & Hilgard, 1981, p. 217).
It is important to understand how Estes’ concept of R and S were a departure from traditional definitions, and that his function was probabilistic rather than deterministic. Instead of considering R to represent a single response, he broke the concept of response into two parts: the R-class, and the R-occurrence. Each R-occurrence is a member of some R-class, and manifests the necessary characteristics required to satisfy some experimental definition of a ‘correct’ response. For example, in a bar-pressing experiment, the R-class might be defined to include any act that depresses the bar. “Any movement of the organism which results in sufficient depression of the bar to actuate the recording mechanism is counted as an instance of the class” (W. K. Estes, 1950, p. 96). Estes also noted that the response class might be further broken down into finer subclasses, e.g., one that includes bar presses made with the right forepaw, and one that includes bar presses made with the left forepaw. Reinforcement of right-paw bar pressing will increase the probability that instances of that subclass will occur, and will also increase the probability that responses of the more general class, bar-pressing, will occur.
Even more important than his definitions of response is Estes’ interpretation of the stimulus, for it is this interpretation that forms the defining core of his model, and from which it gets its name. Unlike the traditional behaviorist model in which the stimulus was regarded as a singular entity to which a response is conditioned, the stimulus condition in Estes’ model was regarded as “a finite population of relatively small, independent, environmental events, of which only a sample is effective at any given time” (W. K. Estes, 1950, p. 96). By “effective” Estes meant those stimulus elements that are experienced by the organism. Although all stimulus elements may be present “the experimental subject does not experience all of [them] on any given trial, but only a small proportion of them” (Hergenhahn, 1982, p .223). Bower and Hilgard (1981) attributed this fluctuation in momentary effective stimuli to both internal and external sources:
Two sources of random variation in stimulation may be indentified: the first arises from incidental changes in the environment during the experiment (extraneous noises, temperature fluctuations, stray odors, and so on); and the second arises from changes in the subject, either from changing orientation of her receptors (what she is looking at or listening to), from changes in her posture or response-produced stimuli, or from fluctuations in her sensory transmission system. When verbal stimuli are presented to human subjects, variability may occur due to different implicit associations or interpretations aroused by the material upon different occasions. (p. 215-216)
The phenomenon of momentary effective stimuli explains why multiple trials are necessary for each of the stimulus elements to become conditioned to the response.
The conditioning process can be illustrated by a hypothetical, simplified, example. Suppose that the total stimulus situation is made up of 100 possible stimulus elements. At the onset of the first trial, the experimental subject experiences 5 of the possible 100 stimulus elements. At the end of the trial—terminated by execution of the desired response—each of these five stimulus elements will become associated with whatever response is made. Any previous association of a given stimulus element to a different response class is thereby replaced by the new association. During the second trial, another five stimulus elements are experienced, each of which may or may not already be associated with the response to be made. After the trial-terminating response is made, any stimulus elements not previously associated with the exhibited response class now become so. This process continues from one trial to the next, until all stimulus elements in the situation become conditioned to the desired response.
To model learning across a series of trials, Estes set up an equation to predict the number of new elements that could be expected to become conditioned on any given trial. This equation was expressed in terms of (a) S0, a subpopulation of stimulus elements that may be manipulated independently of the remainder of the situation; (b) s0, the mean number of stimulus elements from the subpopulation that are effective on any one trial; and (c) x, the number of elements from the subpopulation that are conditioned to R at any given time:
∆x = s0[(S0 – x)/S0]
To provide intuitive access to the meaning behind this function, consider the following example. Suppose the subpopulation contains 100 stimulus elements, 20 of which are already conditioned to R. Then, if an average of 8 of those 100 elements are effective during any one trial, we can expect that the number of elements that will be conditioned during a trial is 8[(100-20)/100] = 8(80/100) = 640/100 = 6.4. Assuming that the change in x per trial is relatively small, the derivative of this function can be taken with respect to T, where T represents the number of trials. After taking this derivative, Estes substituted p for x/S0 to describe the probably of Rk as a function of the number of reinforced trials T:
p = 1 – (1 – p0)e-qT
He also derived a function to describe the probable duration of a trial in terms of the strength of Rk, and a function to predict the expected rate of occurrence of Rk as a function of time (for these derivations see W. K. Estes, 1950, pp. 100-103).
To summarize, in Estes’ model, learning is the successive association of stimulus elements to a response class, which occurs over multiple trials. When a response occurs, all stimulus elements effective at the time of the response, or just prior to it, become conditioned to the response. Learning takes place in an experimental situation by controlling the sampling of momentary effective stimulus elements to ensure that the desired response will take place on every single trial, while still introducing stimulus elements not yet conditioned to R in the stimulus element subpopulation so that they will also become conditioned to R.
This concept behind Estes’ model of learning is simple, and it has a defensibly intuitive practical utility. However, Estes’ model , which has come to be known as stimulus sampling theory, is given no attention whatsoever in present day books on educational psychology (see, for example, Bohlin et al., 2009; Driscoll, 2000; Eggen & Kauchak, 1999; Mowrer-Popiel & Woolfolk, 1998; O’Donnell et al., 2007; Ormrod, 2003; Sternberg & Williams, 2010; Woolfolk, 1998; Woolfolk, 2010). Perhaps, like Hull’s theory, this is because of the mathematical basis of the model. Or, perhaps it is because of the almost revolutionary turning of attention by practitioners to cognitive learning theory shortly after the publication of Toward a Statistical Sampling Theory (W. K. Estes, 1950).
 i.e., as though learning were a random process for which all possible response outcomes might be assigned a probability of occurrence.
 In this equation p0 represents the probability of a response other than a response of type Rk