Anagram Experiments Critiqued

     This critique of the anagram experiments really resulted from the above quote in your text. Since the details of how to derive the "precise predictions" weren't worked out, I decided to try to work them out myself. My first step was to try to determine how to characterize the experimental task within the S-R framework.

An anagram problem is defined as one where the given sequence of letters must be rearranged in such a way that all of the letters are used; and, they form a word. (Typically the initial sequence of letters is not a word.) Abstractly then the problem is:

<letter sequence> ­­­> ?x

where x is a word

Now as good S - R psychologists, one of the first question we must ask is exactly what is the stimulus (S) and what it the response (R)?

     The most obvious answer would seem to be that the stimulus is the <letter sequence> and the response is the answer in the form of some word. But if we adopt this assumption then what can we assume about the strength of association between these two elements. For example, what is the strength of association between:

or between any of the other 116 other possibilities and beach? Presumably the strength of association should be the same for all 116 possibilities because we have probably not had the exciting experience of seeing these possibilities before taking this course! It was this observation that suggested the experiment that you did in the assignment.

     The initial experiments discussed in your text involved varying the frequency with which the answer, the word x, occurs in some sampling of texts. This frequency is taken as a "measure" of exactly what? We have already ruled out that it is a measure of the strength of association that a person would have between a particular sequence of letters and the word. And note, there was no mention of, nor explicit manipulation of the particular sequence of letters used in each of the problems. The experiment simply involved creating problems where the answers to some of the problems were high frequency and the answers to others were low frequency words. What we are left with is only the 'R side' of the S-R relation. The frequency of occurrence of a word is presumed to be a measure of the relative position of that word as a response in some response hierarchy. But what is the response hierarchy?

     If we have a particular stimulus, S, then our S-R theory assumes that there is a set of responses that are "elicited" by that stimulus. Each response in this set has some strength of association to the particular stimulus. In general, these responses can be partially or completely ordered using this measure of strength of association. This is what is meant by a response hierarchy (it is really not a hierarchy in the mathematical sense of the term). The first response(s) are those with the highest value of strength of association and those with the highest or high values are referred to as the dominant response(s). The probability that a response will be emitted is presumably a function of its position in this ordering along with other relevant variables.

     Now, we can give an S-R interpretation of the manipulation of the high and low frequency words as answers to an anagram problem. The interpretation would be, (I think), that regardless of the stimulus, the higher the word frequency of the answer then the higher the strength of association of that word to the stimulus. Note, for both high and low frequency words, they are probably the highest valued word in the response hierarchy that is an answer to the problem. The advantage of the high frequency word over the low frequency word rests not in this relative measure of strength but in its absolute value. That is, in order to predict the results we assume that:

(1)   a word need not ever be paired with a stimulus in order for that word to be in in the set of responses "elicited" by the stimulus;

and/or

(2)   the probability of any word in this set being emitted depends on its absolute value as determined by frequency of usage; not on its relative position in a particular response hierarchy.

     The first assumption would seem to predict that, in general, every word that we know is, in some sense, "elicited" by every sequence of letters (the stimulus) and that the "response hierarchy" is independent of the actual stimulus. This is a logical possibility and therefore we can't rule it our a priori. But, it would be quite surprising if true.

     The second assumption is not required if the first assumption is unconditionally made. In this case, every word occurs to every stimulus and thus every high frequency word has a greater probability of being elicited than every low frequency word. But what if the response hierarchy varied with the letter sequence? Then it could turn out to be the case that a low frequency word is the dominant response for some particular problem and a high frequency word was not among the dominant responses for some other particular problem. If the probability of response is dependent on the relative position in the response hierarchy, then the low frequency word would have the advantage in this situation. In order to avoid this we could assume that the probability of response is independent of the particular response hierarchy elicited.This seems rather strange but it would certainly make the theory more mathematically tractable.

     The next idea considered in your text is that the "response strength" of a word is measured by some function (e.g., the average) of the estimates of the frequency of each adjacent letter pair. For example, the pairs for 'beach' would be {be, ea, ac, ch}. This is essentially a way of estimating a words "strength" from its components!

     This is fairly ad hoc. Why not also use the triples? (e.g., {bea, eac, ach}); the quadruples? (e.g., {beac, each}). But more importantly, this has changed the rules of the game. Remember, in the above the R of the S-R was the word. Now the R has components! That is, strength(R) = f(ri,rj,...rn). Now intuitively this makes perfectly good sense. But it complicates our S-R theory.

     Recall the blank slate analogy that is often used to characterize the assumptions typically made in this S-R approach to the study of cognition. A slate has a smooth and homogeneous surface - it has no intrinsic structure or divisions. Whatever structure appears is the result of the way "the world" has written on this slate. If a word has divisions, then it must be the world that places these there. And, if the strength of an association is a function of the strength of the components of the response; then we must discover the components and the function that combines the values of the components to yield any prediction from this theory. (Remember these remarks when you read about Thorndike's view on transfer of learning as discussed in your text.)

     The final explanation offered in the text really has nothing at all to do with frequency. It is simply an assertion that "subjects tend to make dominant responses such as moving just one or two letters first; if that doesn't work, they try "weaker" responses such as moving all five letters."(p.27) Recall that a dominant response is the one that has the highest strength. Note that in this explanation we have shifted from talking about the word that is the answer to the anagram problem, e.g. "beach" to talking of the stimulus; i.e. the sequence of letters. Now, it may well be that we are more likely to move one letter than two or three or four; but the question for the S-R psychologist is to explain how this results from experience and why this set of responses is "elicited" by the stimulus. In the previous explanations the stimulus "elicited" words as responses; but now in this case the stimulus "elicits" letter-moving responses.

     A further problem arises with this explanation due to the fact that moving just one letter first is not really a well-defined description of a particular response in this context. With the five lettered stimuli used in these experiments, any one of the five different letters might be moved and there are many locations to which the letter might be moved. Thus, there are really many such "dominant" responses.

     A final comment...it is possible that there is no good general strategy for solving arbitrary anagrams problems. Systematically examining each of the possible permutations of the sequence of letters guarantees that you will find the solution if it exists. But this exhaustive strategy doesn't really require a great deal of thought...it simply requires an algorithm for generating all of the permutations and the patience to systematically look through the permutations. This leads one to ask whether this task is representative of the kinds of tasks that characterize human reasoning. Although some may find that doing anagrams is an interesting way to pass the time; I suspect they are not particularly characteristic of our reasoning activities. (Although one test of 'intelligence' involves only analogy problems!)

Associationism and Behaviorism