Position Analysis 1: Discrete Content-Independent Schemes
In evaluating the discrete content-independent positional schemes we began by asking, for each scheme, whether CM's and LSS's perseveration errors maintained source position more often than expected by chance.
Analysis 1a: Observed vs. Chance Position Matches
Analysis 1a was carried out over all potential perseveration-source pairs. A potential perseveration was defined as a letter intrusion error in which the intruded letter appeared in one or more of the preceding responses within the perseveration window (4 prior responses for CM and 2 for LSS). The preceding responses that contained the intruded letter were considered potential source responses. For example, shows CM's error “blunt” → BLANT, as well as the preceding responses in the four-trial perseveration window (FARM, NOSE, ORANGE, and PIANO). BLANT is a potential perseveration response, because the intruded letter A appears in at least one (and in fact three) of the four responses within the perseveration window (FARM, ORANGE, PIANO). For CM, therefore, the set of potential perseveration-source pairs included BLANT-FARM, BLANT-ORANGE, and BLANT-PIANO. The total number of these pairs was 3265 for CM and 1993 for LSS.
CM's Error BLUNT ? BLANT, and the Four Preceding Responses within the Perseveration Window
The set of potential perseveration-source pairs for each participant unavoidably includes not only true perseveration-source pairs (i.e., pairs associating a true perseveration with its true source), but also what may be called pseudo perseveration-source pairs. In a pseudo pair the intruded letter is either not a perseveration from a prior response, or is not paired with its true source. Suppose, for example, that CM intruded an A into the response BLANT for reasons having nothing to do with the presence of As in prior responses. In this case BLANT-FARM, BLANT-ORANGE, and BLANT-PIANO would all be pseudo perseveration-source pairs. Suppose, alternatively, that the A in BLANT was a true perseveration, and that the only source for the perseveration was the immediately preceding response FARM. The pair BLANT-FARM would then be a true perseveration-source pair, but the pairs BLANT-ORANGE and BLANT-PIANO would be pseudo pairs.
Because we had no way of knowing which particular pairs were true perseveration-source pairs and which were pseudo pairs, we included all potential perseveration-source pairs in the analysis. Fortunately, however, we can determine from this mixture of true and pseudo pairs whether true perseverations maintain the position of the perseverated letter in the true source more often than expected by chance. For pseudo pairs, the letter intrusion in the potential perseveration response has nothing to do with the presence of the letter in the potential source, and therefore the intruded letter should maintain source position no more often than expected by chance. Consequently, if we find that position is maintained at an above-chance rate in the full set of potential perseveration-source pairs, we can conclude that true perseverations maintain true source position more often than expected by chance.
For each participant and for each discrete content-independent scheme a computer program calculated the proportion of potential perseveration-source pairs in which the intruded letter was in the same position in the perseveration and source responses. For example, shows how the left- and right-edge representational schemes assign position to the A in the perseveration response BLANT and the source response FARM. When position is defined by the right-edge scheme, the A is in the same position in the perseveration and source responses. According to the left-edge scheme, however, the position of the A does not match in the perseveration and source responses. Hence, for BLANT-FARM the analysis program tallied a position match for the right-edge scheme but not for the left-edge scheme.
Position of A in Source and Perseveration Responses According to Left-Edge and Right-Edge Positional Schemes
The program also estimated for each positional scheme the proportion of position matches expected by chance. The chance likelihood of an intruded letter having the same position in the perseveration response as in the source response can be estimated by examining source control responses. A source control response was defined as a response that had the same number of letters as the source response and contained the intruded letter, but was not produced in close proximity to the perseveration response. (As in the perseveration analysis, close proximity was defined to encompass the five preceding through five following responses.) For each perseveration-source pair, the analysis program generated a pool of source control responses. A Monte Carlo analysis used the source control responses to estimate for each representational scheme the proportion of position matches expected by chance. For each potential perseveration-source pair, the program randomly sampled a response from the source control pool, and tabulated for each positional scheme whether the perseverated letter was in the same position in that response as in the perseveration response. For example, on one run of the analysis CAPERT was randomly chosen as the control for the source response ORANGE in the BLANT-ORANGE pair. According to both left-edge and right-edge schemes the A in the perseveration response (BLANT) and the A in the source control response (CAPERT) are in different positions (L+3 vs. L+2, respectively, for the left-edge scheme, and R-3 vs. R-5 for the right-edge scheme). The result of each run of the Monte Carlo analysis is a single estimate of proportion of perseverated letters expected by chance to match position in the perseveration and source responses. The entire process was carried out 10,000 times, yielding a distribution of chance proportions for each of the schemes.
The results are presented in . The black bars show the proportion of perseveration-source pairs in which the position of the intruded letter in the perseveration response matched the position of the letter in the actual source response. For example, the .51 proportion for the left-edge representational scheme observed for LSS reflects the finding that for 1013 of the 1993 perseveration-source pairs (51%), the intruded letter appeared in the same left-edge position in the perseveration and source responses. The white bars in the figure show the proportions expected by chance (more specifically, the mean chance proportion across the 10,000 runs of the Monte Carlo analysis).
Figure 7 Proportion of perseveration-source response pairs in which an intruded letter in the perseveration response appear in the same position in the source response, as defined by various content-independent positional schemes (black bars) compared to the proportion (more ...)
For both CM and LSS, the observed proportion of positional matches between perseveration and source responses was greater than each of the 10,000 chance estimates for all five content-independent representation schemes (p < .0001 for both participants for all schemes). These results demonstrate, for both participants, that letters were more likely than expected by chance to maintain the same content-independent position when perseverating from a source response to a perseveration response.
The results of Analysis 1a are robust over variation in criteria for inclusion of potential perseveration-source pairs. In the reported analyses we excluded potential perseveration responses containing more than one instance of an intruded letter, because of potential ambiguities concerning which instance(s) should be treated as intrusions. For example, in CM's error “umbrella” → UBLERRER the response includes two extra Rs, but it is not clear which of the Rs should be considered intruded letters, leading in turn to uncertainty about the positions in which intruded letters appeared. For similar reasons, source and control responses were limited to responses containing only one instance of the intruded letter. However, analyses that included responses with multiple instances of the intruded letter yielded the same pattern of results. We also conducted analyses in which each potential perseveration response was paired with only a single potential source (the most recent response in the perseveration window that contained the intruded letter). Once again the pattern of results was the same.
The finding that position was maintained at above-chance rates for all five content-independent schemes does not necessarily mean that all of these schemes contribute to the representation of letter position. The content-independent schemes agree with one another regarding which position in a source response matches the position of an intruded letter in a perseveration response for any pair in which the potential perseveration response has the same number of letters as the potential source response. Consider the pair RINT-REST, where RINT has an intruded R. The position of the R in RINT differs by scheme (e.g., L+1, C-2, R-4), but all of the content-independent schemes agree that the R occupies the same position in RINT (potential perseveration response) and REST (potential source response). Hence, if letter position were encoded according to any of the content-independent schemes one would expect all of the schemes to show above-chance proportions of positional matches. Additional analyses were therefore conducted to identify which of the content-independent positional schemes best accounted for the pattern of responses. Analysis 1b defined a measure of success for candidate schemes, and Analysis 1c applied this measure in evaluating the various content-independent schemes.
Analysis 1b: Proportion of Position Matches for True Perseveration-Source Pairs
The observed proportions of position matches reported in Analysis 1a were calculated over all potential perseveration-source pairs. As previously noted, however, the pool of potential perseveration-source pairs includes not only true perseveration-source pairs (i.e., pairs consisting of a true perseveration and its true source), but also pseudo perseveration-source pairs (i.e., pairs in which the intruded letter is present in the potential source by chance).
In Analysis 1b we estimated the proportion of position matches for the true perseveration-source pairs alone. For each positional scheme the estimated proportion of position matches for true perseveration-source pairs (henceforth pPMforTrPS) provides a measure of how well the scheme performs in using the position of a perseverated letter in a perseveration response to predict the position of the corresponding source letter in the true source response. For example, a pPMforTrPS of .60 would indicate that in 60% of the true perseveration-source pairs the scheme in question correctly predicted the position of the source letter from the position of the perseverated letter. As we shall see, a scheme's pPMforTrPS value provides a basis for assessing that scheme's contribution to letter position encoding.
To estimate pPMforTrPS values, we first estimate the numbers of true and pseudo perseveration-source pairs in the pool of potential perseveration-source pairs. Next, we estimate for each scheme, the number of position matches for true perseveration-source pairs, and then convert these values to proportions.
The perseveration analysis provided three relevant values to estimate the number of true perseveration-source pairs: (1) IP
, the total number of intrusion-prior response pairs, (2) PotentialPS
, the number of potential perseveration-source pairs (i.e., the number of intrusion-prior response pairs in which the prior response contained the intruded letter), and (3) pChanceILinP
, the estimated probability of the intruded letter appearing in the prior response by chance.4
PotentialPS is the sum of TruePS and PseudoPS, the numbers of true and pseudo perseveration-source pairs, respectively:
TruePS and PseudoPS are both unknowns. However, we can substitute another expression for PseudoPS and thereby eliminate this unknown. A pseudo perseveration-source pair is any intrusion-prior response pair that is not a true-perseveration-source pair, but includes the intruded letter in the prior response by chance. Accordingly, the expected number of pseudo pairs is the number of intrusion-prior response pairs that are not true perseveration-source pairs (IP -TruePS) times the probability of the intruded letter being present in a prior response by chance:
Substituting this expression for PseudoPS into equation (1)
Inserting LSS's values for PotentialPS, IP, and ChanceILinP yields
Solving for TruePS gives us 1595 as the estimated number of true perseveration-source pairs. From this value we straightforwardly obtain 398 as the estimate for PseudoPS, the number of pseudo perseveration-source pairs. Performing the same calculations for CM yields estimates of 1392 for TruePS and 1873 for PseudoPS.
Using these values we can estimate the number of position matches for true perseverations (PsnMtchforTruePS) for any given positional scheme. We first express the number of observed position matches for the full set of potential perseveration-source pairs in Analysis 1a (PsnMtchforPotentialPS) as a sum of position matches for true perseveration-source pairs and position matches for pseudo perseveration-source pairs:
For pseudo perseveration-source pairs position matches will occur only by chance. Hence, the number of position matches for the pseudo pairs should equal the number of pseudo pairs (PsuedoPS) times the probability of a position match by chance (pChancePsnMtch):
Substituting this expression into equation (4)
and solving for PsnMtchforTruePS yields
We have values for all of the variables on the right side of the equation. The PsnMtchforPotentialPS and pChancePsnMtch values for each discrete content-independent position scheme are available from Analysis 1a; For any given scheme we have calculated the observed number of position matches (PsnMtchforPotentialPS) for that scheme, and the chance proportion of a position match (pChancePsnMtch). The value for PseudoPS does not depend upon the positional scheme.
Consider, for example, LSS's results for the left-edge scheme. The observed number of position matches for this scheme across the full set of potential perseveration-source pairs was 1013, and the chance proportion of a position match was .252. Given the estimate of 398 for the number of pseudo perseveration-source pairs, we can estimate the number of position matches for true perseveration-source pairs as follows:
Finally, we can convert the number of position matches for true perseverations to a proportion by dividing by the estimated number of true perseverations:
For LSS the estimated number of true perseverations (TruePS) was 1595, and so his estimated proportion of position matches for true perseverations was 913/1595, or .57. In other words we estimate that in 57% of LSS's true perseverations the perseverated letter occupied the same left-edge position in the perseveration response as in the true source response.
presents for each participant the estimated pPMforTrPS value for each of the discrete content-independent schemes. For both CM and LSS pPMforTrPS was highest for the both-edges scheme. In more than three-fourths of the true perseveration-source pairs (78% for CM and 87% for LSS) this scheme was successful in using the position of the perseverated letter in the perseveration response to predict the position of the source letter in the source response.
Estimated Proportions of Position Matches for True Perseveration-Source Pairs (pPMforTrPS) for Discrete Content-Independent Position Schemes
Results from the Monte Carlo procedure in Analysis 1a provide a basis for assessing the statistical reliability of differences among schemes in pPMforTrPS. For each positional scheme each Monte Carlo run provided a different estimate of the chance probability of a position match (pChancePsnMtch). Using these pChancePsnMtch estimates, we estimated pPMforTrPS for each scheme on each of the 10,000 Monte Carlo runs. For both CM and LSS, the estimated pPMforTrPS was highest for the both-edges scheme on all 10,000 runs; hence, for each participant the difference between the both-edges scheme and each of the other schemes was significant at p < .0001.
These results would seem to indicate that within the class of discrete content-independent schemes, the both-edges scheme most closely approximates the actual scheme for representing letter position in graphemic spelling representations. However, before drawing this conclusion we must address an issue arising from the nature of the both-edges scheme.
Analysis 1c: The Both-Edges Scheme and its Components
The left-edge, right-edge, center, and closer-edge schemes are simple schemes; each assigns only position representation to each letter in a word. However, the both-edges scheme is a combination of left-edge and right-edge schemes, and so assigns both left- and right-edge position representations to each letter (e.g., L+1 and R-4 for the F in FACE). We refer to such schemes as compound schemes. In evaluating compound schemes one must ask whether each component scheme (e.g., the left-edge and right-edge components for the both-edges scheme) plays a role in the success of the overall scheme.
For each potential perseveration-source pair, a simple scheme predicts at most one position in the source response where the intruded letter should be found.5
Consider our example pair ERGE-FRENCE. According to the left-edge scheme, the intruded R occupies position L+2 in ERGE, and therefore the scheme predicts that the R in the source response will be found at position L+2. For the source response FRENCE the R does in fact occupy position L+2, and so a position match would be tallied for the left-edge scheme.
Compound schemes can make multiple predictions about what positions in a source response count as position matches. For the ERGE-FRENCE pair, the both-edges scheme assigns position representations L+2 and R-3 to the intruded R in ERGE. As a consequence, the scheme predicts that the R will be found in either position L+2 or position R-3 of the source response FRENCE, and a position match would be tallied if the R occupied either of these source positions.
On average the number of predicted source positions is greater for the both-edges scheme than for either the left- or right-edge scheme. As a consequence, the both-edges scheme has more opportunities for tallying position matches—that is, more opportunities for finding the intruded letter at a predicted source position. This point has implications for interpreting the finding of higher pPMforTrPS values for the both-edges scheme than for the component left- and right-edge schemes. Consider the comparison between both-edges and right-edge schemes. For all potential perseveration-source pairs the both-edges scheme predicts the same source position as the right-edge scheme (e.g., the R-3 position for ERGE-FRENCE). However, for many pairs the both-edges scheme also predicts an additional source position on the basis of left-edge coding (e.g., the L+2 position in FRENCE). Now suppose that the left-edge scheme is not implicated in letter position encoding, and that as a consequence the additional predicted left-edge positions are completely uncorrelated with the source positions predicted by the scheme that actually encodes letter position in graphemic spelling representations. In other words, suppose that the additional source positions predicted by the left-edge scheme are random relative to the positions predicted by the actual scheme for letter position representation. Even a random guess will occasionally be correct, and so we expect the additional left-edge predictions to yield occasional position matches for true perseveration-source pairs (assuming that the right-edge scheme does not always predict source position correctly for true perseveration-source pairs). As a consequence we expect the proportion of position matches for true perseveration-source pairs to be higher for the both-edges scheme than for the right-edge scheme, even if the left-edge scheme makes no systematic (i.e., non-random) contribution to predicting the source positions of perseverated letters. Hence, the finding that pPMforTrPS was higher for the both-edges scheme than for the right-edge scheme does not necessarily mean that the left-edge component of the both-edges scheme makes any systematic contribution over and above that of the right-edge scheme in predicting the source position of perseverated letters.
To determine whether the left-edge component of the both-edges scheme makes a systematic contribution beyond that of the right-edge component we compare the additional predictions made by the left-edge scheme with an equal number of additional predictions generated randomly. That is, we can compare a right-edge+left-edge (i.e., both-edges) scheme to a right-edge+random scheme that makes the same number of source position predictions, and so has the same number of opportunities for position matches. If the proportion of position matches for true perseverations (pPMforTrPS) is significantly higher for the right-edge+left-edge scheme than for the right-edge+random scheme, we can conclude that the left-edge component of the both-edges scheme makes a systematic contribution over and above that of the right-edge component in predicting the source position of perseverated letters. Similarly, we can assess the contribution of the right-edge component by comparing the both-edges scheme to a left-edge+random scheme.
The methods for these analyses may be explained by considering the both-edges vs. right-edge+random comparison. For potential perseveration-source pairs in which the perseveration and source responses are of the same length, the right-edge and both-edges schemes do not differ in number of predicted source positions. For these pairs, no random source predictions were generated for the right-edge+random scheme. However, when the perseveration and source responses have different numbers of letters, the both-edges scheme usually predicts two source positions, whereas the right-edge scheme usually predicts only one. For every pair in which the both-edges scheme predicted an additional source position, the source predictions of the right-edge+random scheme consisted of the right-edge prediction plus an additional source position selected at random by the analysis program. For example, in the case of the ERGE-FRENCE pair, the right-edge scheme designates one predicted position in the source FRENCE (i.e., the R-3 position, occupied by the letter N). However, the both-edges scheme designates not only this position but also the L+2 position (occupied by the R in FRENCE). Accordingly, an additional position was selected randomly for the right-edge plus random scheme; the chosen position could be the same as, or different from, the left-edge position predicted by the both-edges scheme. Through this method the number of predicted source positions, and hence the number of opportunities for position matches, were equated on a pair-by-pair basis across the both-edges and right-edge+random schemes. On each of 10,000 Monte Carlo runs, a new set of random source predictions was generated.
Using the procedures described for Analyses 1a and 1b, proportions of position matches for true perseverations were estimated on each Monte Carlo run for the both-edges and right-edge+random schemes. The results were straightforward: For both CM and LSS the pPMforTrPS estimate was higher for the both-edges scheme than for the right-edge+random scheme on all 10,000 runs of the Monte Carlo analysis. For LSS the mean pPMforTrPS was .87 for both-edges but only .74 for right-edge plus random; for CM the values were .78 and .66 (p <.0001 for both participants). These results demonstrate that the left-edge component of the both-edges scheme made a systematic contribution to the scheme's success, over and above the contribution of the right-edge component.
A similar analysis demonstrated that the converse was also true: The right-edge component made a systematic contribution beyond that of the left-edge component to the success of the both-edges scheme. This analysis found that that pPMforTrPS values for both CM and LSS were significantly higher for the both-edges scheme than for a left-edge plus random scheme (see ), ps <.0001.
Estimated Proportions of Position Matches for True Perseveration-Source Pairs (pPMforTrPS) for the Discrete Both-Edges Scheme and Component Schemes Plus Random Source Predictions
The both-edges scheme can also be decomposed into closer-edge and farther-edge schemes. As shown in the pPMforTrPS values were significantly higher for the both-edges scheme than for a closer-edge+random scheme, ps <.0001, indicating that farther-edge coding (i.e., left-edge coding for letters in the right half of a word and vice versa) contributed systematically to the success of the both-edges scheme. Not surprisingly, a both-edges vs. farther-edge+random comparison revealed that the closer-edge component also played a significant role over and above that of the farther-edge component, ps <.0001.
These analyses demonstrate that the success of the compound both-edges scheme relative to its component left-, right-, closer- and farther-edge schemes cannot be explained by assuming that one of the scheme's components (e.g., the right-edge component) does all of the systematic work in predicting source positions of perseverated letters, with the complementary component (e.g., the left-edge component) contributing only in the sense of making essentially random source position predictions that occasionally happen to yield position matches for true perseverations. Rather, each component (left- and right-edge components given one decomposition of the scheme, and closer- and farther-edge components given an alternative decomposition) plays a systematic and significant role in the success of the both-edges scheme.
Position Analysis 2: Graded vs. Discrete Position Representations
Analysis 1 revealed that for most of the true perseverations made by each participant (78% CM and 87% for LSS) the perseverated letter appeared at a source location that exactly matched the left-edge or right-edge position of the letter in the perseveration response. However, for the remaining true perseverations (22% for CM and 13% for LSS) the discrete both-edges scheme did not correctly predict the source location of the perseverated letter. Consider the pair ZEROBA-FLOWER, and assume that this pair is a true perseveration-source pair in which the intruded O in ZEROBA (a misspelling of ZEBRA) is a perseveration from the source response FLOWER. In ZEROBA the O occupies positions L+4 and R-3, and so the discrete both-edges scheme predicts that the O will appear in position L+4 or R-3 in the source FLOWER. In fact, the O in FLOWER occupies positions L+3 and R-4. Why did the perseveration and source positions not match?
Analysis 2a: Graded vs. Discrete Both-Edges Schemes
One possibility is that the both-edges representations are graded rather than discrete. Considering first the left-edge component of the both-edges scheme, the O in FLOWER might be represented by activating the O unit strongly in the L+4 pool, moderately in the L+3 and L+5 pools, and perhaps more weakly in more distant positions. As a consequence, the O would be most likely to perseverate into position L+4 of a subsequent response, but might also perseverate into an adjacent position (L+3 or L+5) or, rarely, into a more distant position. The same point applies to the right-edge component of the both-edges scheme: The O in FLOWER would be represented by activating the O unit not only in the R-3 pool, but also (less strongly) in the R-2 and R-4 pools, and so forth. Accordingly, the O would be most likely to perseverate into position R-3 of a subsequent response, but might also perseverate into positions R-2 or R-4 (or perhaps even a more distant position).
By the same logic, when we use the position of a perseverated letter in a perseveration response to predict the position of the source letter in the source response (as in our analyses), we expect sometimes to find the letter not at an exactly matching source position, but instead at a nearby position. Consider once again the pair ZEROBA-FLOWER. Given a graded both-edges scheme, we expect most often to find the O at one of the exactly matching source positions (i.e., L+4 or R-3); however, we also expect sometimes to find the O instead at one of the nearby source positions (e.g., L+3, L+5, R-2, or R-4).
In the present analysis we evaluated a version of graded both-edges scheme that defined a predicted source position to be either an exactly matching position, or an immediately adjacent position. For the ZEROBA-FLOWER pair, the graded scheme predicted that the O would be found at position L+3, L+4, L+5, R-2, R-3, or R-4. In this example, the left-edge and right-edge positions coincide, and therefore the graded both-edges scheme identifies three predicted source positions (those occupied by the O, W, and E). Because the O appears at one of these positions, the graded scheme scores a position match for the ZEROBA-FLOWER pair.
The graded both-edges scheme may be viewed as a combination of the discrete both-edges scheme (which predicts an exact match between perseveration and source positions) and what we will call an adjacent both-edges scheme (which predicts that the source position will be adjacent to an exact-match position). Therefore, the question we need to ask is whether the addition of the adjacent both-edges component makes a systematic contribution over and above that of the discrete component in predicting the source positions of perseverated letters.
To answer this question we compared the graded both-edges scheme to a discrete both-edges+random scheme that was matched to the graded scheme in number of opportunities for position matches. The results reveal that for both CM and LSS the proportion of position matches for true perseverations (pPMforTrPS) was significantly higher for the graded scheme than for the discrete+random scheme (p < .001 for both participants). This result demonstrates that the adjacent component of the graded both-edges scheme was significantly better than a random position selection process at predicting the source positions of perseverated letters, and hence that the adjacent component contributed systematically to the success of the graded scheme. Additional analyses confirmed that even within the context of a graded positional scheme, the various subcomponents of the both-edges scheme contributed systematically to its success: pPMforTrPS was significantly higher for the graded both-edges scheme than for graded left-edge plus random, graded right-edge plus random, graded closer-edge plus random, and graded farther-edge plus random schemes (p < .0001 for all comparisons for each participant).
For both participants the pPMforTrPS for the graded scheme was remarkably high: .94 for CM and .97 for LSS. In other words, in almost all of each participant's true perseverations the perseverated letter appeared at a predicted position in the source response. The graded both-edges scheme did treat a large number of positions as predicted positions: a mean of 3.2 positions per word for CM and 3.1 for LSS, constituting about 60% (CM) and 55% (LSS) of the positions in a word. Nevertheless, it is impressive that the actual source position for nearly 100% of the true perseverations fell within the 55-60% of source positions picked out by the graded both-edges scheme.
Analysis 2b: The Center Scheme
The final issue to be addressed in our analyses of content-independent positional schemes concerns the center scheme. Analysis 1b showed much lower pPMforTrPS values for the discrete center scheme than for the discrete both-edges scheme. A similar analysis contrasting graded both-edges and graded center schemes yielded the same result: pPMforTrPS was significantly higher for the both-edges than for the center scheme (p < .0001 for both participants). In conjunction with the evidence that each component of the both-edges scheme contributes to its success, these results indicate that the center scheme does not merit consideration as an alternative to the both-edges scheme. However, one can nevertheless ask whether the center scheme might conceivably play a role in addition to that of the both-edges scheme in coding letter position in graphemic spelling representations.
We addressed this question by comparing a graded both-edges+center scheme to a graded both-edges+random scheme. For both participants the pPMforTrPS value for the both-edges+center scheme was not significantly different from that of the both-edges+random scheme (p > .3 for both participants), indicating that the graded center scheme made no systematic contribution beyond that of the graded both-edges scheme in predicting the source positions of perseverated letters. Given that the graded both-edges scheme alone predicted nearly all of the source positions, this result is entirely unsurprising.
We conclude that within the class of content-independent positional schemes, the graded both-edges scheme most closely approximates the scheme for encoding letter position in graphemic spelling representations. In Analyses 3 and 4 we examine letter-context and syllabic schemes, asking whether these schemes mediate letter-position coding instead of, or in addition to, the both-edges scheme.
Position Analysis 3: Letter-Context Schemes
Letter-context schemes represent the position of a letter relative to other letters in the word. Analysis 3 considered three letter-context schemes. In the preceding-letter scheme the position of each letter is defined by the immediately-preceding letter. According to this scheme, the A in FACE occupies the F_ position (i.e., the immediately-preceded-by-an-F position), and therefore could be represented by activating the A unit in an F_ pool (see ). The following-letter scheme similarly represents each letter relative to the immediately following letter. In this scheme the A in FACE occupies the _C position. Finally, in the trigram scheme the position of a letter is defined jointly by the immediately-preceding and immediately-following letters. In this scheme the A in FACE occupies the F_C position.
Applying methods from previous analyses, Analysis 3 evaluated the letter-context schemes and compared these schemes with the graded both-edges scheme. The set of potential perseveration-source pairs was limited in these analyses in the following way: For each letter-context scheme we excluded an intruded letter from the position analysis if a context letter in the perseveration response was also an intruded letter. For example, in CM's error STUB → SLAB, we excluded the intruded A from the preceding-letter analysis because the context letter L was also an intruded letter. For the following-letter scheme we excluded intruded letters when the following letter was also an intruded letter, and for the trigram analyses we excluded an intruded letter when either the preceding or following letter was also an intrusion. Including these intrusions would lead to overestimation of the pPMforTrPS value. When two letters intrude into adjacent positions, these letters may often be perseveration from adjacent positions in the source. For example, CM wrote PLAY correctly on the trial immediately preceding the STUB → SLAB error, and hence the intruded L and A in SLAB may both have been perseverated from PLAY. If two adjacent intrusions are both perseverations from adjacent positions in the source, the second intrusion will always
maintain preceding letter position (the A is in the position L_ in both SLAB and PLAY), even if the preceding letter scheme is completely unrelated to the brain's scheme for representing letter position in graphemic spelling representations. To obtain unbiased estimates of pPMforTrPS for the preceding letter scheme we must exclude all but the first intruded letter in adjacent-letter intrusions6
. By the same logic, in analyses of the following-letter scheme, we must exclude all but the last intruded letter in adjacent-letter intrusions, and in trigram-scheme analyses we must exclude all of the letters in adjacent-letter intrusions. Given this restriction, for CM and LSS, respectively, the number of potential perseveration-source pairs was: 1679 and 699 for preceding letter, 925 and 455 for following letter, and 600 and 207 for the trigram scheme.
Analysis 3a: pPMforTrPS for Letter-Context and Both-Edges Schemes
In this analysis we estimated pPMforTrPS values for each of the letter-context schemes, and compared these values to values obtained by applying the graded both-edges scheme to the same data sets. For example, results for the preceding-letter scheme were compared with both-edges results obtained from the set of potential perseveration-source pairs used in the preceding-letter analysis.
The results are presented in . The pPMforTrPS values for the letter-context schemes were generally quite low, and in all cases were far lower than the values for the graded both-edges scheme (p < .0001 for all comparisons). These results demonstrate clearly that none of the letter-context schemes is a viable alternative to the graded both-edges scheme. Nevertheless, letter-context schemes might conceivably make some contribution, in addition to that of the both-edges scheme, to coding of letter position in graphemic spelling representations. Analysis 3b explored this possibility.
Estimated Proportions of Position Matches for True Perseveration-Source Pairs (pPMforTrPS) for the Letter-Context Schemes and the Graded Both-Edges Scheme
Analysis 3b: Both-Edges+Letter-Context Schemes
This analysis applied the method used to evaluate potential contributions of the center scheme in Analysis 2b. We defined three graded both-edges+letter-context schemes, one for each of the three letter-context schemes (e.g., graded both-edges+preceding-letter), and compared each combined scheme to a graded both-edges+random scheme that made the same number of source predictions. The results, presented in , are extremely clear: The pPMforTrPS values for the graded both-edges plus letter-context schemes were in all cases indistinguishable from the values for the matched graded both-edges plus random schemes (p > .25 for all comparisons). This finding demonstrates that none of the letter-context schemes made a systematic contribution beyond that of the graded both-edges scheme in predicting source locations of perseverated letters. In fact the pPMforTrPS values in were little if at all higher than the values shown in for the both-edges scheme alone. This result reflects the fact that the context schemes made very few source predictions that were not also made by the both-edges scheme, and therefore were unlikely to achieve additional position matches even by chance.
Estimated pPMforTrPS for Graded Both-Edges Plus Letter-Context Schemes and Matched Graded Both-Edges Plus Random Schemes
The results from Analysis 3 make a strong case that simple bigram and trigram schemes (e.g., Brown & Loosemore, 1994
; Seidenberg & McClelland, 1989
) do not play a role in coding letter position in graphemic spelling representations. Furthermore, the results argue against open bigram schemes (e.g., Grainger & van Heuven, 2003
; Whitney, 2001
) for graphemic spelling representations. Any plausible open bigram scheme includes at least one of the simple bigram schemes as a key subcomponent. For example, an open-bigram preceding-letter scheme (in which, for example, the C in FACE has position representations F_ and A_) includes the simple preceding-letter scheme. Hence, if an open-bigram scheme played a role representing letter position in spelling representations, we should have observed systematic contributions of one or both of the simple bigram schemes. No hint of such contributions was observed.
Position Analysis 4: Syllabic Representations
In this analysis we evaluated syllabic positional schemes, and compared these schemes with the graded both-edges scheme. A computer program syllabified each participant's responses according to the principle of orthographic onset maximization, and assigned each letter a position defined jointly by the position of the syllable in which the letter appeared, and the role of the letter within that syllable. The position of a syllable within a word was defined by a both-edges scheme in which the syllable's position was encoded in terms of its distance (in syllables) from both the left and the right edges of the word. In the word HYDRANT, for example, DRANT is the second syllable from the left (Syllable L+2) and the first syllable from the right (Syllable R-1). Other methods for specifying the position of the syllable in the word were also considered – left-edge, right-edge, closer-edge, and center. For both CM and LSS, the best-performing syllabic schemes were those using both-edges syllable position coding, and hence we report results for these schemes.
Within each syllable the position of each letter was defined by its role as an onset, nucleus, or coda letter. In the syllable DRANT, for instance, D and R make up the onset, A the nucleus, and N and T the coda. Finally, letters were differentiated by their positions within onsets, nuclei, and codas. The possible within-role positions were first, second, and third onset positions (Onset 1 - Onset 3), first and second nucleus positions (Nucleus 1 and Nucleus 2), and first through fourth coda positions (Coda 1 - Coda 4). Onset and nucleus letters were assigned to positions in a left-justified manner whereas coda letters were right-justified. In DRANT the D and R appear in the Onset 1 and Onset 2 positions, respectively, the A is in the Nucleus 1 position, and the N and T occupy, respectively, the Coda 3 and Coda 4 positions. The syllabic scheme as a whole therefore assigns the following two position representations to the D in DRANT: Syllable L+2: Onset 1 and Syllable R-1: Onset 1.
We considered both discrete and graded syllabic schemes. In the discrete scheme the coding of position within syllable role (e.g., Onset 1, Onset 2) was assumed to be exact, and hence a position match was tallied only when the position of the intruded letter in the perseveration response exactly matched the position of that letter in the source response. Consider the pair LAMP-BLOCK, in which the perseveration response LAMP is a misspelling of DAMP and so contains an intruded L. In LAMP the L is in an Onset 1 position, and so has syllabic position representations Syllable L+1: Onset 1 and Syllable R-1: Onset 1. In the potential source response BLOCK, however, the L occupies an Onset 2 position, and so is coded as Syllable L+1: Onset 2 and Syllable R-1: Onset 2. Onset 1 and Onset 2 positions are not the same, and consequently no position match would be tallied for this pair.
For the graded syllabic scheme we assumed that coding of position within syllable role was graded. In this scheme, the L in BLOCK might be represented by activating the L unit strongly in the Onset 2 pool for the appropriate syllable, and somewhat less strongly in the Onset 1 and Onset 3 pools. Accordingly, we tallied a position match for the graded syllabic scheme whenever the position of the intruded letter in the perseveration response matched its position in the source response with respect to syllable (e.g., Syllable L+1) and syllable role (e.g., Onset), regardless of whether the position-within-role matched exactly. Therefore, the graded syllabic scheme, unlike the discrete scheme, scored a position match for LAMP-BLOCK, even though the two responses do not exactly match with respect to the L's position within syllable role.
We excluded from Analysis 4 all responses that could not be parsed into orthographically acceptable syllables (e.g., DRNDY). For CM and LSS, respectively, the syllabic analyses included 2868 and 1641 potential perseveration-source pairs, with 1191 and 1309 of these estimated to be true perseveration-source pairs.
Analysis 4a: Discrete vs. Graded Syllabic Schemes
In Analysis 4a we first estimated the proportion of position matches for true perseverations for the discrete and graded syllabic schemes. As shown in the first two rows of , the graded scheme yielded substantially higher pPMforTrPS values for both CM and LSS (p < .0001). However, no firm conclusions can be drawn from this result, because the discrete syllabic scheme is a subcomponent of the graded scheme. To choose between the schemes we need to know whether the additional source predictions made by the graded scheme contribute systematically to its success. Accordingly, we compared the graded syllabic scheme with a discrete syllabic+random scheme that made the same number of source predictions. The results for this scheme are presented in the third row of . For both CM and LSS the estimated pPMforTrPS value was significantly higher for the graded scheme than for the discrete+random scheme, p < .0001. These results demonstrate that extending source predictions from exactly matching within-syllable-role position (e.g., Onset 1) to the entire syllable role (e.g., the entire onset) systematically improves the ability of a syllabic scheme to predict the source locations of perseverated letters.
Estimated pPMforTrPS Values for Discrete Syllabic, Graded Syllabic, and Discrete Syllabic Plus Random Schemes
The high pPMforTrPS values for the graded syllabic scheme are perhaps not too surprising, given that this scheme is well-correlated with the graded both-edges scheme (e.g., onsets are usually near the left edge of a word, and codas are usually near the right edge). In Analysis 4b we attempted to tease apart these two schemes.
Analysis 4b: Graded Syllabic vs. Graded Both-Edges Schemes
We first compared the pPMforTrPS values for the graded syllabic scheme with values obtained by applying the graded both-edges scheme to the data set used in the syllabic-scheme analyses. As shown in the estimated pPMforTrPS for both participants was higher for the graded both-edges scheme: This scheme successfully predicted the source location of the perseverated letter in 94% and 98% of the true perseveration-source pairs for CM and LSS, respectively, whereas the graded syllabic scheme succeeded only 91% of the time for each participant. The differences between both-edges and syllabic schemes were highly reliable (p < .0001 for both participants). Given that the graded syllabic scheme is not a subcomponent of the graded both-edges scheme, this result implies that the latter is a closer approximation than the former to the brain's scheme for coding letter position in graphemic spelling representations.
Estimated pPMforTrPS Values for Graded Syllabic and Graded Both-Edges Schemes
However, the graded syllabic scheme might nevertheless play some role in addition to that of the graded both-edges scheme in coding letter position. To evaluate this possibility we defined a graded both-edges+graded syllabic scheme, and compared this scheme to a graded both-edges+ random scheme. The results are presented in . For LSS the pPMforTrPS for the graded both-edges plus graded syllabic scheme (.97) was indistinguishable from that for the graded both-edges plus random scheme (.98), and in fact from that of the graded both-edges scheme alone (.98), ps > .25. These results indicate that the graded syllabic scheme made no systematic contribution beyond that of the graded both-edges scheme to the success of the graded both-edges+graded syllabic scheme.
Estimated pPMforTrPS Values for Graded Both-Edges, Graded Both-Edges Plus Graded Syllabic, and Graded Both-Edges Plus Random Schemes
For CM the results were not quite as clear. Adding the graded syllabic scheme to the graded both-edges scheme increased the pPMforTrPS from .94 to .97, whereas adding random source predictions yielded a slightly smaller increase (from .94 to .95). The difference between the .97 value for the both-edges plus syllabic scheme and the .95 value for the both-edges plus random scheme was significant only at the .05 level (as opposed to the .0001 level observed for almost all of our significant effects). Thus, for CM (but not for LSS) the graded syllabic scheme may make a small systematic contribution to the success of the graded both-edges+graded syllabic scheme.
What conclusions should we draw from these results? Given that both CM and LSS showed higher pPMforTrPS values for the graded both-edges scheme than for the graded syllabic scheme, the both-edges scheme should clearly be preferred to the syllabic scheme as a candidate for the scheme that underlies encoding of letter position in graphemic spelling representations. Furthermore, given that (a) the syllabic scheme clearly made no contribution beyond that of the both-edges scheme for LSS, (b) the syllabic scheme's contribution was small and uncertain for CM, and (c) with the sole exception of the both-edges plus syllabic versus both-edges plus random comparison CM and LSS presented with identical patterns of significant effects across all of our analyses, the most parsimonious conclusion is that syllabic position representations do not contribute to letter position coding in graphemic spelling representations. This conclusion should, however, be considered tentative.