Reply to Farrell & Lewandowsky: Recency-contiguity interactions predicted by the temporal context model

Abstract

Farrell & Lewandowsky (2008) argue that the temporal context model (TCM; Howard & Kahana, 2002) cannot explain non-monotonicities in the contiguity effect seen at extreme lags. However TCM actually predicts that these non-monotonicities to the extent that end-of-list context persists as a retrieval cue during recall, and to the extent that end-of-list context generates a recency effect. We show that the observed non-monotonicity in the contiguity effect interacts with the recency effect as predicted by TCM. In conditions that exhibit strong recency, such as immediate and continual distractor free recall, one observes more prominent non-monotonicities in the contiguity effect than in conditions that attenuate recency, such as delayed free recall. Rather than posing a challenge to the model, the non-monotonicities in the contiguity effect at extreme lags and the interactions between recency and contiguity result from the role of end-of-list context as a retrieval cue in TCM.

In free recall, participants are presented with a list of words and then instructed to recall them in the order they come to mind. Because the order of recall is unconstrained by the experimenter, regularities in the transition probabilities reflect properties of the organization of memory. Perhaps the most important of these regularities in constraining models of episodic memory retrieval is the conditional response probability as a function of lag, or lag-CRP (Kahana, 1996). Given that a participant has just recalled an item from serial position i, the lag-CRP estimates the probability that the next item recalled will be from serial position i + lag, attempting to control for the availability of potential recalls in a number of ways. The lag-CRP has a canonical shape, exhibiting a strong contiguity effect favoring adjacent transitions over more remote transitions and an asymmetry favoring forward transitions over backward transitions (see Kahana, Howard, & Polyn, 2008, for a review).

According to the temporal context model (Howard & Kahana, 2002; Howard, Fotedar, Datey, & Hasselmo, 2005; Sederberg, Howard, & Kahana, 2008), the contiguity effect arises because items from neighboring serial positions are associated with similar contexts. In TCM, recall of an item retrieves its associated contexts which in turn cue items studied in nearby list positions. Because context in TCM is driven by items (rather than fluctuating randomly) the model naturally predicts a forward bias in the contiguity effect (see Howard & Kahana, 2002 for details). Because context drifts due to experience (e.g., the presentation and retrieval of items) the context at the start of an immediate recall test will be more similar to the contexts associated with recently studied items. In this way, TCM accounts for the recency effect as well as the contiguity effect. Since Howard & Kahana's original formulation of TCM, the model has been extended to address the effects of hippocampal lesion on transitive associations (Howard et al., 2005), the effect of aging on the shape of lag-CRP curves (Howard, Kahana, & Wingfield, 2006) and the dynamics of immediate, delayed, and continual-distractor free recall (Sederberg et al., 2008). All of these applications of TCM have preserved the model's basic assumptions concerning how contextual dynamics gives rise to recency and contiguity.

Farrell and Lewandowsky (2008) showed that in immediate recall, TCM predicts that in addition to the contiguity effect, transitions also tend to be made to items near the end of the list. This tendency can be seen as a non-monotonicity in the lag-CRP. If we follow the forward lag-CRP outward from zero, eventually the tendency to make recalls to the end of the list overcomes the advantage conferred by contiguity to the just-recalled word. This can result in an increase in the lag-CRP at extreme values of lag. The non-monotonicity in the lag-CRP reflects an excess of transitions from extreme serial positions to other extreme serial positions. A persistent primacy effect would manifest as an increase in the lag-CRP in the backward direction—extreme negative lags—whereas a persistent recency effect would manifest as an increase at extreme positive lags.

The persistence of the primacy effect in recall transitions has been known for some time. Laming (unpublished manuscript) observed that there was an excess of transitions to the first serial position—we noted this persistent primacy effect early on in describing the lag-CRP (p. 939, Howard & Kahana, 1999). A moment's reflection reveals that the existence of the primacy effect in the serial position curve obtained in immediate free recall, coupled with the tendency to initiate free recall from the recency portion of the list necessitates an excess of remote transitions to the early part of the list across subsequent retrieval attempts. That is, to the extent that the primacy effect in the serial position curve is not solely attributable to the tendency to initiate recall with the first item in the list, then there must have been an excess of remote transitions to the first item.

Farrell & Lewandowsky's primary empirical contribution is to suggest that there is a non-monotonicity in the forward direction consistent with what would be expected from a persistent recency effect. Unfortunately the empirical status of this effect in any individual experiment is ambiguous because of conceptual flaws in the quantitative analyses they used. Farrell and Lewandowsky (2008) evaluated non-monotonicity in the lag-CRP by comparing the fits of descriptive models to the observed lag-CRPs. Two of these models, the quadratic and complementary exponential are capable of exhibiting non-monotonicity and two, the linear and power function, are not (Table 2, Farrell & Lewandowsky, 2008). However, the superior fit of the non-monotonic functions, in the subset of experiments where they show a better fit, is not necessarily attributable to their non-monotonicity. Farrell and Lewandowsky (2008) evaluated non-monotonicity by comparing three-parameter models that can exhibit non-monotonicity with one parameter models that cannot. However, the ability to exhibit non-monotonicity is not the only way in which these classes of models differ. Most notably, the three-parameter models are able to exhibit a non-zero asymptotic value whereas the one-parameter models cannot. Consider Farrell & Lewandowsky's Eq. 4:

$$P(l)=c\text{exp}(-al)+(1-c)exp(-bl).$$

This complementary exponential function is nested with a two-parameter model that is not monotonic yet gives rise to a non-zero asymptote that is achieved by setting b = 0. If b = 0, then the resulting two-parameter model describes an exponential decay to an asymptote. It does not have the ability to increase at long lags, yet exhibits a degree of flexibility that the “monotonic” functions considered by Farrell & Lewandowsky (in press) cannot. It remains possible that the (monotonic) exponential-to-asymptote model provides a better fit than the (non-monotonic) complementary exponential function.

Although the statistics reported by Farrell & Lewandowsky (in press) for individual experiments are not appropriate to determine whether a particular experiment demonstrates a non-monotonicity, there are three additional sources of evidence that convince us that lag-CRPs can exhibit non-monotonicity at extreme positive values of lag, at least under some circumstances. These sources of evidence are Farrell and Lewandowsky's (2008) analysis collapsing across experiments (their Figure 2), their observation that the version of TCM they refer to as TCM_evo provides a better fit than the model they refer to as TCM_pub across a wide variety of experiments, and our own secondary analyses of additional data sets that are reported here.

In this reply we explore the variables that affect the change of shape of the lag-CRP by examining lag-CRPs from a set of experiments with largely similar methods but differing delay schedules. Qualitative modeling will assess the degree to which this pattern of results is consistent with the predictions of TCM. We start by describing in more detail the source of the non-monotonicity and skew in the lag-CRP predicted by TCM.

Skew and non-monotonicity in the lag-CRP predicted by TCM

TCM proposes that the cue for episodic recall is the current state of a gradually-changing temporal context vector. Potential recalls are cued by a state of context to the extent that it overlaps with the context that obtained when they were studied. The current state of temporal context is driven by presented items, which can also recover their study context. This enables the model to account for contiguity effects—when a studied item is recalled, the input it causes to the temporal context vector resembles the encoding context of neighboring list items, resulting in an increased tendency to recall neighbors of the recalled item. These basic ideas are common to all of the studies that have applied TCM to a variety of topics, although these treatments have varied in a number of details (see Howard & Kahana, 2002; Howard et al., 2005, 2006; Rao & Howard, 2008; Sederberg et al., 2008).

TCM predicts a change in the shape of the lag-CRP evident in extreme lags to the extent that end-of-list context persists as part of the retrieval cue and to the extent that this end-of-list context supports a recency effect.¹ From the earliest work on TCM (Howard, 1999), we have considered the implications of assuming that context changes gradually during retrieval. In TCM, the degree of contextual drift at any given time step is a function of the amount of information that is provided as input to temporal context. This leads to the interesting property that when no input is provided, there is no change in the state of temporal context, predicting that recency can remain intact in response to an unfilled delay (Baddeley & Hitch, 1977; Murdock, 1963). It is straightforward to ask whether the amount of information provided as input following successful retrieval of a memory probe is different than the amount of input caused by study of an item. Thus, the rate of contextual drift during study, ρ_study, may differ from the rate of contextual drift during retrieval, ρ_test. Farrell and Lewandowsky (2008) consider two special cases, which they treat as dichotomous. They refer to the case in which ρ_test = 0 as TCM_pub and to the case in which ρ_test = ρ_study as TCM_evo. This approach can lead to confusion on at least two counts. First, there are actually a continuity of models rather than just these two alternatives. Second, the designation TCM_pub is somewhat misleading in that the assumption that ρ_study = ρ_test has been considered since the first work on TCM (e.g., Howard, 1999, see also Sederberg et al., 2008; Polyn, Norman, & Kahana, in press).

Figure 1a illustrates the predictions of TCM for the shape of the lag-CRP across the entire range of lags for immediate free recall for a variety of values of ρ_test. The rate of drift during test controls how much end-of-list context contributes to the retrieval cue at subsequent retrieval attempts. As can be seen from Figure 1a, TCM predicts that the shape of the lag-CRP changes to the extent end-of-list context persists during retrieval. This is evidenced not only by the non-monotonicity at extreme forward lags, but also by a skew between forward and backward retrievals. Put another way, the model predicts that when end-of-list context does not contribute to retrieval, the asymmetry between forward and backward transitions is approximately constant as the absolute value of lag increases. In contrast, when end-of-list context persists and contributes to the retrieval cue, the difference between forward and backward retrievals grows as the absolute value of lag increases. In the backward direction, retrieved context and end-of-list context both favor recall of an item contiguous to a just-recalled item from the middle of the list. In contrast, for retrievals in the forward direction, these cues are in conflict, resulting in a noticeable skew and even a non-monotonicity.

Figure 1.

TCM can predict non-monotonicities and skew in the lag-CRP. a. Continuity between with (Howard, et al., 2006; Howard, 2004) and the version with retrieval of end-of-list context (Howard & Kahana, 2002). CRP curves were generated at the first output position with ρ_test = ρ_study (black line, referred to as TCM_evo by Farrell & Lewandowsky, in press) and gradually decreasing values of ρ_test, ending with ρ_test = 0 (referred to as TCM_pub by Farrell & Lewandowsky, in press), for the lightest shaded lines. b. According to TCM, non-monotonicity depends on recency. Even without allowing ρ_test to vary from ρ_study, TCM can predict a variety of non-monotonicities if the recency effect is altered. The black curves shows the CRP observed at the first output position in immediate free recall with ρ = .85, γ = 0.8, and τ = 0.3. The value of the retention interval was gradually increased from zero to an infinite delay (successively lighter lines).

TCM predicts forward non-monotonicities only to the extent that end-of-list context is an effective cue for recall of items from the end of the list. Figure 1b shows the lag-CRP predicted by TCM for immediate free recall (black curve) and increasingly long retention intervals (successively lighter curves) when ρ_test = ρ_study, i.e., the extreme case referred to as TCM_evo by Farrell and Lewandowsky (2008). It is well-known that increasing the retention interval between study of the last item and test results in a decrease of the recency effect in free recall (e.g. Postman & Phillips, 1965). As can be seen from Figure 1b, skew and non-monotonicity are present when the test is immediate and gradually decrease with the increase in the retention interval.

Note that the results of Figure 1b falsify one of the claims of Farrell and Lewandowsky (2008) regarding the predictions of TCM:

• An initial examination of the model revealed a striking non-monotonicity of the forward lag-CRP functions in TCM_evo. Irrespective of whether recall was immediate or delayed or involved a continuous distractor task, lags greater than 5 attracted nearly as many—or indeed more—transitions than lags +1.

The light grey lines in Figure 1b, generated with ρ_test = ρ_study, which Farrell and Lewandowsky (2008) refer to as TCM_evo, show predictions for delayed free recall from TCM. The light grey curves are monotonically decreasing. At extreme lags they do not in any way approach—let alone exceed—the much higher values observed at lags near zero. Farrell and Lewandowsky's (2008) conclusion is a consequence of fixing the effective delay of the retention interval at a particular value and/or insufficiently exploring the range of values ρ can assume.

It is important to note that non-monotonicity at extreme lags is not the only issue in differentiating the predictions of TCM when end-of-list context is allowed to persist as a cue from TCM when end-of-list context is not allowed to persist as a cue. This is particularly relevant from an empirical perspective because only the most extreme lags exhibit non-monotonicity (see Figure 2 in Farrell & Lewandowsky, 2008). These lags are infrequently observed. For instance, a lag of +11 in a 12-item list, which reflects a transition from the very first item in the list to the very last item in the list, can only be observed if two conditions are met. The very first item in the list must have been recalled and the very last item in the list must be available as a newly-recalled item. In immediate free recall, in which very strong recency effects obtain, these conditions are infrequently met, especially early in output, resulting in a paucity of observations. If limiting one's attention to the first recall transition, these extreme transitions are only observed to the extent that subjects initiate recall with the very first item, which may reflect a serial recall strategy (Bhatarah, Ward, & Tan, 2008).

A qualitative exploration of the shape of the lag-CRP across delay conditions

Given that appropriate statistical tools have not yet been developed to fully characterize the skew and non-monotonicities in the lag-CRP in any individual experiment, we must rely on qualitative analyses. Indeed, Farrell and Lewandowsky (2008) use a qualitative approach to good effect in their Figure 2, which calculates an end-adjusted lag-CRP across a broad variety of experiments that vary in delay schedule, list length, modality of presentation, presence of orienting task, presentation rate, level of practice, individual vs group testing, and method of recall (verbal vs written). Here we will examine the effects of different experimental manipulations on the non-monotonicity and skew in the lag-CRP. In order to control as many variables as possible, we will restrict our attention to published experiments from our labs in which relatively short lists of words were presented visually under conditions designed to minimize rehearsal and verbal free recall was collected. These analyses utilize a subset of the experiments examined by Farrell and Lewandowsky (2008), plus final free recall data (reported in Howard, Youker, & Venkatadass, 2008) from the Howard, Venkatadass, Norman, and Kahana (2007) immediate free recall study. In an attempt to maximize the amount of available data, we will examine lag-CRPs collapsed across output positions rather than only considering the first output position. Our analyses suggest that skew and non-monotonicity in the lag-CRP are observed to the extent that there is a recency effect, regardless of whether recall is immediate or delayed. In order to establish this, we first summarize the recency effect observed in these studies.

Recency across delay conditions

Figure 2 shows the recency effect observed in the experimental data we will consider here. Figure 2a shows the probability of first recall (PFR) curves from Experiments 1 and 2 of Howard and Kahana (1999), exhibiting immediate, delayed and continuous-distractor free recall. In immediate free recall, the list is presented and test immediately follows presentation of the last item. A strong recency effect is observed in the PFR. In delayed free recall, a delay intervenes between study of the last item and the test. In delayed free recall, the recency effect is attenuated relative to immediate free recall. In continuous-distractor free recall, a delay intervenes between each list item and also at the end of the list. The recency effect in the PFR is larger in continuous-distractor free recall than in delayed free recall. The long-term recency effect in continuous-distractor free recall is also observed when examining final free recall across lists (Tzeng, 1973; Glenberg et al., 1980). Howard et al. (2007) examined immediate free recall of 48 lists. A recency was observed in immediate free recall testing. At the end of the session, they tested final free recall of all the items from all the lists. Howard et al. (2008) reported the final free recall results from this study and observed a recency effect across lists in the PFR (Figure 2b). Notably, there was no evidence for a recency effect relative to within-list serial position in final free recall (not shown). This makes sense in that the delay between study of a particular (10 item) list and the final free recall period was on the average several tens of minutes.

Figure 2.

The recency effect across delay schedules. a. The recency effect, as illustrated by the probability of first recall (PFR), from immediate, delayed and continuous-distractor free recall from Experiments 1 and 2 from Howard & Kahana (1999). While the recency effect is attenuated in delayed free recall, it is amplified in continuous distractor free recall. b. The recency effect in the PFR across lists from Howard, Youker, & Venkatadass (2008). In both panels, error bars reflect the standard error of the mean.

The lag-CRP at extreme values across delay conditions

Figure 3 illustrates lag-CRPs from immediate, delayed and continuous-distractor free recall, as well as final free recall across lists. While prior work has focused on documenting the existence of a contiguity effect by focusing on lags around zero (e.g. Howard & Kahana, 1999; Howard et al., 2008), here we examine all possible lags, as suggested by Farrell and Lewandowsky (2008). Figure 3a compares immediate to delayed free recall from Experiment 1 of Howard and Kahana (1999). There is a boost in the contiguity effect and appears to be a larger non-monotonicity in immediate free recall compared to delayed free recall.

Figure 3.

Non-montonicities in the lag-CRP appears to co-occur with the recency effect. In all panels, lag-CRPs are calculated across all output positions. a. Immediate condition and delayed condition of Experiment 1 of Howard & Kahana (1999). b. Comparison of the continuous-distractor condition with the longest interpresentation interval (IPI) (labeled “Continuous-distractor”) with the delayed free recall condition of Experiment 2 of Howard & Kahana (1999). c. Lag-CRP from the immediate free recall test of the control lists of Howard, Venkatadass, Norman & Kahana (2007) contrasted with the within-list CRP calculated from a final free recall session (“FFR”). Error bars in panels a-c reflect one standard error. d. Across-list CRP (Howard, Youker, & Venkatadass, 2008).

Figure 3b compares the continuous-distractor condition of Experiment 2 of Howard and Kahana (1999) with the longest interpresentation interval (IPI), labeled “continuous-distractor,” to the delayed free recall (IPI = 0s) condition from the same experiment. Recall that continuous-distractor free recall shows a larger recency effect than delayed free recall (Figure 2a). Although the contiguity effect is similar in magnitude across the conditions, the non-monotonicity exhibited at extreme positive lags appears stronger in continuous-distractor free recall than in delayed free recall.

Figure 3c compares the lag-CRP from immediate free recall in the Howard et al. (2007) data with the within-list lag-CRP observed in final free recall of the same items. The final free recall data is comparable to delayed free recall with an extremely long delay and, not surprisingly, does not demonstrate a recency effect within-list. The results appear to be consistent with Figure 3a. Again the contiguity effect is larger in magnitude in immediate recall. In addition a non-monotonicity in the forward direction is observed in immediate free recall. With the delay, leaving aside the primacy effect observed at extreme backward lags, the lag-CRP has a consistent degree of asymmetry in contrast to the skew observed in the immediate free recall data. Figure 3d illustrates the across-list lag-CRP observed by Howard et al. (2008). There appears to be a steep non-monotonicity observed at extreme positive lags presumably corresponding to the strong across-list recency effect observed in the same data (Howard et al., 2008).

Qualitative predictions of TCM

Here we consider whether the properties of the the lag-CRP across the entire range of possible lags are consistent with the predictions of TCM. Our strategy is to use a common set of parameters that illustrate the qualitative behavior of the model across conditions, which we will compare to the pattern of observed results across experiments. The goal of this approach is to provide insight as to whether the source of the skew and non-monotonicity in the lag-CRP are consistent with the origin predicted by TCM.

Farrell and Lewandowsky (2008) evaluated fits of a two-parameter TCM model and found that the predictions of the model deviated from the observed results to an extent significantly different from chance. This is not a surprising result; there are many sources of variability that are not included in this two-parameter description. For instance, it is known that free recall is strongly affected by the degree of proactive interference the items are subject to (Goodwin, 1976), the duration of the delay interval (Postman & Phillips, 1965), and the semantic organization of the list (Romney, Brewer, & Batchelder, 1993; Glanzer, Koppenaal, & Nelson, 1972). Free recall models have typically used a great many more than two parameters and been content to describe the qualitative pattern of results across experiments (Raaijmakers & Shiffrin, 1980; Davelaar, Goshen-Gottstein, Ashkenazi, Haarmann, & Usher, 2005; Sederberg et al., 2008; Sirotin, Kimball, & Kahana, 2005; Kimball, Smith, & Kahana, 2007).²

In evaluating the qualitative predictions of TCM, we informally searched for a set of parameters that would exhibit the basic properties of the recency effect and lag-CRP effects exhibited by the data across conditions from the single-list free recall experiments, i.e., the data in Figures 2a and 3a-c. A relatively broad range of parameters exhibit the same basic properties. These parameters, with ρ = .85, γ = .8, τ = .3 and ρ_D = 0.4 were used in the predictions generated in Figures 1a and b, Figure 2a and Figures 3a-c (see the Appendix for details of the parameterization). The parameter ρ controls the rate of contextual drift; γ controls the degree to which recalled items recover their temporal context; τ controls the sensitivity of the Luce choice retrieval rule that maps activations onto probability of recall and ρ_D controls the effective length of the distractor interval where appropriate (see Appendix for details). Because we treated the lists as single items in the across-list FFR simulations, a separate set of parameters were chosen for the across-list free recall data, with the effective rate of contextual drift across lists set to .8, γ = .8, τ for the PFR was set to .6 and τ for the CRP set to .9.³ In all of these predictions, ρ_study was equal to ρ_test, equivalent to the formulation Farrell and Lewandowsky (2008) referred to as TCM_evo.

Figure 4a shows the predictions of TCM for the PFR from immediate, delayed and continuous-distractor free recall. As can be seen from Figure 4a, the model correctly predicts a strong recency effect in immediate free recall, an attenuation of the recency effect in delayed free recall and a strong recency effect in continuous-distractor free recall (compare to the empirical results shown in Figure 2a, see also Howard & Kahana, 2002). In modeling the recency effect in final free recall across lists, TCM can also predict a recency effect that extends across multiple lists.

Figure 4.

TCM with the Luce choice retrieval rule produces recency effects that correspond to the qualitative pattern observed. Compare to Figure 2. a. Probability of first recall functions for immediate (black), delayed (light grey) and continuous-distractor (dark grey) free recall. The same parameters were used in Figure 5a-c. b. Across-list PFR. The same parameters were used in Figure 5d.

To avoid some of the complexities modeling multiple free recalls, we generated predictions for the lag-CRP at the first output position from TCM. Some caution should be exercised in comparing the predictions of the model with the empirical results in Figure 3, which collapse across output positions. In order to successfully model lag-CRPs across more than the first output transition, one must have a mechanism for dealing with resampling and termination of recall, neither of which depend on the structural properties of TCM. While the comparison of the qualitative shape of the lag-CRP is not dramatically altered by collapsing across output position, the magnitude of the contiguity effect in immediate free recall is affected by output position (Howard & Kahana, 1999; Kahana, Howard, Zaromb, & Wingfield, 2002). As has been well-known for some time, TCM under-predicts the magnitude of the lag-CRP effect at early output positions in immediate free recall, underestimating the dramatic boost to the lag-CRP at early stages of output in immediate free recall (Howard & Kahana, 2002).⁴ The use of lag-CRPs from only one output position in the modeling and the use of lag-CRPs collapsed across output positions tends to obscure this difference in immediate free recall.

Figure 5 shows the lag-CRPs from the first output position from the same experimental settings that generated the corresponding PFR curves in Figure 4. As can be seen from Figure 5a, TCM predicts that the lag-CRP from immediate free recall should have a larger contiguity effect, somewhat greater skew and a larger non-monotonicity in the forward direction than the lag-CRP from delayed free recall. In continuous-distractor free recall (Figure 5b), the model correctly predicts that although the contiguity effect should be of similar magnitude across conditions, the difference between the lag-CRP from continuous-distractor free recall and delayed free recall should be manifest as a large and sharp non-monotonicity at extreme positive lags in continuous-distractor free recall relative to delayed free recall.⁵

Figure 5.

Simulations of lag-CRPs from the first output position using TCM, with the Luce choice retrieval rule. Compare to Figure 3. a. Immediate free recall (black) and delayed free recall. b. Continuous-distractor free recall (black) and delayed free recall (grey). c. Immediate free recall compared to delayed free recall with an infinite delay. d. Across-list CRP.

Figure 5c shows predictions comparing the lag-CRP from the immediate free recall within-list lag-CRP to the lag-CRP from final free recall of the same lists. To simulate the very long delay between study of a typical list from the experiment and the final free recall session, we simply set the effective length of the retention interval to be infinite in generating the grey lag-CRP curve in Figure 5c, rather than modeling delayed free recall as reflecting a small residual recency effect (Figures 2a, 4a). Indeed, there is no evidence for a within-list recency effect or primacy effect in these final free recall data (Howard et al., 2008). The correspondence between the predictions (Figure 5c) and the empirical observations (Figure 3c) in this case are particularly strong. The model not only correctly predicts a non-monotonicity in the immediate lag-CRP that is larger than in final free recall, but also describes a continuously-increasing difference between immediate free recall and delayed final free recall for increasing values of lag. Moreover, the model also correctly predicts a benefit for backward transitions in delayed final free recall. It is perhaps worth noting that this was a very large study, with almost three hundred participants performing more than 7,000 trials of immediate free recall.

Comparing Figure 5d, which shows predicted values of across-list lag-CRPs, with Figure 3d, which shows empirically-observed across-list lag-CRPs, we see that the model has successfully captured several aspects of the data. First, the model predicts a contiguity effect across several lists that is also exhibited in the data.⁶ Second, there is a large non-monotonicity in the forward direction, such that extremely large across-list lags are better recalled than adjacent across-list lags. Third, there is a persistent skew to the entire curve that appears to be reflected in the data.

From these analyses, the model appears to correctly predict the range of shapes of lag-CRP curves that are observed, and the variation in the skew and non-monotonicity observed across experiments. In particular, these seem to co-occur with the recency effect, as predicted by TCM. Rather than ruling out TCM as a description of recency and contiguity across scales, the data from examining the entire range of lag-CRPs across conditions are qualitatively quite consistent with the predictions of TCM. Finally, we note that we used the assumption that ρ_test = ρ_study, which is equivalent to the version of TCM that Farrell and Lewandowsky (2008) referred to as TCM_evo and achieved a good description of the data, these results do not uniquely support TCM_evo. It remains possible that other nonzero values of ρ_test provide a superior description of the data.

Contiguity effects predicted by TCM in immediate free recall are not an averaging artifact

Farrell and Lewandowsky (2008) claimed that TCM incorrectly predicts an artifactual lag-CRP due to averaging across serial positions with a strong recency effect. In their Figure 7, Farrell and Lewandowsky (2008) examined the lag-CRP conditionalized on the serial position of the previously-recalled word. Although noisy, the experimental data from the Howard et al. (2007) study show a contiguity effect in the forward direction for each previously-recalled serial position with a non-monotonicity at extreme positive lags for some items. In contrast, the best-fitting parameter values of their two-parameter implementation of TCM showed a pure recency effect in the lag-CRP for the forward direction when conditionalized on serial position of the just-recalled item. If this were a general property of the model, it would clearly rule out TCM as a description of the contiguity effect. We will show, however, that Farrell & Lewandowsky's (2008) finding is an artifact of the particular choice of parameters used in generating the predictions and does not reflect a general property of the model.

Figures 6a and b show the predictions from TCM with the Luce choice retrieval rule using the same parameters as Figures 4 and 5. The model clearly shows a contiguity effect at each serial position in addition to a recency effect that appears as a non-monotonicity at extreme lags. To determine if these results were due to some detail of implementation of TCM used here, we examined a number of alternative implementations, including one that closely approximates the version of the model described in Howard and Kahana (2002). All exhibited the same qualitative behavior.⁷ As further evidence that this does not depend on any implementational details of the models used here, we also examined predictions generated by TCM-A, a variant of TCM in which the Luce choice rule is replaced with a set of leaky competing accumulators (Sederberg et al., 2008). TCM-A uses the convention that ρ_study = ρ_test, referred to as TCM_evo by Farrell and Lewandowsky (2008). Using the same set of parameters used by Sederberg et al. (2008), figures 6c and d show the lag-CRP segregated by serial position of the just-recalled word generated by a simulation of TCM-A with 50,000 simulated trials using the same parameters and methods described in Sederberg et al. (2008). As can be clearly seen from Figure 6c, TCM-A predicts a contiguity effect, as well as a non-monotonicity in the forward direction, even when the lag-CRP is conditionalized on the serial position of the just-recalled item. We conclude that although it is possible to find parameters for TCM with the Luce choice rule—and probably TCM-A as well—that generate an artifactual contiguity effect, this is not a weakness of the model per se so much as a weakness of the specific choice of parameters used by Farrell and Lewandowsky (2008).

Figure 6.

The CRP predicted by TCM in immediate free recall need not be an artifact of the recency effect. Compare to Figure 7, Farrell & Lewandowsky (in press). Panels on the left (a and c) correspond to immediate free recall. Panels on the right (b and d) correspond to delayed free recall (note change of scale). a,b. Simulations from the Howard & Kahana (2002) version of TCM with the Luce choice retrieval rule. The same parameters were used as in Figure 4a and Figure 5a-c. c,d. Simulations from the TCM-A model of Sederberg, Howard & Kahana (2008). CRP curves broken down by serial position of the just-recalled item are shown. Parameter values are as reported in Sederberg et al. (2008).

General Discussion

Farrell and Lewandowsky (2008) pointed out that TCM makes a prediction about the shape of the lag-CRP in free recall when the lag-CRP is considered across all possible lags—even those with very few observations. Their meta-analysis that aggregates lag-CRP curves collected under a wide variety of experimental conditions (Figure 2 of Farrell & Lewandowsky, 2008), their finding that the variant of TCM they refer to as TCM_evo provided a superior fit than the variant of TCM they refer to as TCM_pub, and our own analyses (Figure 3) all suggest that these changes in the shape of the lag-CRP at extreme values of lag are observed—at least under some circumstances. Our own secondary analyses (Figure 3) suggest that the non-monotonicity in the extreme values of the lag-CRP and skew are driven by persistent serial position effects. We focused our attention on the non-monotonicity in the forward direction and the persistent skew in the lag-CRP which are a consequence of a persistent recency effect. TCM successfully describes the conditions under which the recency effect is observed (Figure 4) and, as a consequence, also the qualitative pattern of lag-CRPs observed across conditions (Figure 5).

Much of our previous work has focused on varying a small number of parameters of TCM to provide a minimal description of canonical free recall effects—the behavior of recency and contiguity effects across conditions. This approach necessarily leaves a number of gaps about the details of the processes underlying free recall. These gaps often reflect gaps in our understanding of the empirical situation. Although the empirical status of persistent recency in free recall remains somewhat ambiguous, the primary contribution of this exchange has been to place some constraint on the relationship between ρ_study and ρ_test.

Farrell and Lewandowsky (2008) have provided a service by pointing out that the entirety of the lag-CRP curve can yield important constraints on models of free recall. We know that setting ρ_test = 0, the model referred to as TCM_pub by Farrell and Lewandowsky (2008), yields a poor description of the lag-CRP when there is a recency effect. However, the constraints appear on further examination to be quite consistent with the qualitative predictions of the model. Moreover, the effect of the value of ρ_test is only apparent to the extent that there is a recency effect in the serial position curve, so that the error induced by setting ρ_test = 0 (or equivalently assuming an infinite delay prior to presentation of the cue) is minimal if one is examining delayed free recall (Howard et al., 2006).

Two detailed free recall models based on TCM have been recently developed. The TCM-A model of Sederberg et al. (2008) provides a description of dissociations between immediate recency and long-term recency. The CMR model of Polyn et al. (in press) uses a framework very similar to TCM to account for associative effects observed in free recall of words encoded in different task contexts. Although we would be the first to agree that TCM is not a complete description of free recall (and there are probably limitations of TCM-A and CMR as well), our conviction that the ideas at the core of TCM—that a gradually-changing state of temporal context is the cue for episodic recall and that some items can recover the state of temporal context in which they were encoded—have only been reinforced by this exchange.

Appendix: Modeling details and mathematical description of TCM

Here we describe the details of the implementation of TCM used here. In the last decade or so, there have been a number of works using TCM that differ in some details of the implementation. We note similarities and differences between this implementation and prior studies where appropriate. However, we wish to emphasize that we have confirmed that none of the qualitative patterns illustrated in this paper depend on any of these specific choices. That is, for all implementations of TCM that we have explored, a long-term recency effect and long-term contiguity effect are observed, skew and non-monotonicity in the lag-CRP cooccur with the recency effect across delay schedules and, for some range of parameters there is a non-artifactual contiguity effect in immediate free recall even when the lag-CRP is conditionalized on the serial position of the first-recalled item.

In TCM, the current state of context t_i is generated from the previous state of context t_i−1 and the current input ${\mathbf{\text{t}}}_i^{\text{IN}}$ according to

$${\mathbf{\text{t}}}_i={\rho }_i{\mathbf{\text{t}}}_{i-1}+\beta {\mathbf{\text{t}}}_i^{\text{IN}}$$

where β is a free parameter and ρ_i is chosen such that the length of t_i is unity. Note that this implies that the rate of contextual drift depends on the amount and nature of the input vector ${\mathbf{\text{t}}}_i^{\text{IN}}$. We treated the asymptotic rate of drift $\rho :=\sqrt{1-{\beta }^2}$ the parameter. Here the Euclidean norm is used for t, rather than the ¹ norm used in some recent papers (Shankar, Jagadisan, & Howard, submitted; Rao & Howard, 2008).

In TCM items are encoded in their temporal contexts. This is accomplished by means of a Hebbian outer product matrix M which is updated using

$$\mathrm{\Delta }\mathbf{\text{M}}={\mathbf{\text{f}}}_i{\mathbf{\text{t}}}_{i-1}^{\prime },$$

where f_i is the item representation of item i and the prime denotes the transpose. That is, we associated item i to the state of context that preceded it, t_i−1. The use of t_i−1 in this equation, rather than t_i, is consistent with recent treatments of TCM (e.g. Howard et al., 2006; Rao & Howard, 2008; Sederberg et al., 2008; Howard, Jing, Rao, Provyn, & Datey, in press) but different from earlier work (Howard & Kahana, 2002; Howard, 1999; Howard et al., 2005).

In TCM, context cues for retrieval of items. Each item is activated to some extent by the state of context used as a cue and then the items compete to be retrieved. Given a contextual cue t, item i is activated by

$$a_i\equiv {\mathbf{\text{f}}}_i^{\prime }\mathbf{\text{Mt}}.$$

The distractors for the retention interval in the case of delayed free recall and continuous-distractor free recall were implemented by adding a vector orthogonal to all preceding states of context weighted by β and multiplying the preceding context vector by ρ_D. This is conceptually identical to all previous implementations of TCM, although the parameterization is a bit different.

In TCM, repetition of a study item during retrieval causes a change of the contextual state, which in turn leads to a contiguity effect. This contextual state recovered by the remembered item contains two components. As in all previous treatments of TCM, these two components are differentially responsible for the asymmetry observed in the lag-CRP. Here the parameter γ was used to weight the degree of contextual retrieval upon repetition of an item. Allowing the mixing of the two parameters to vary freely was not done in Howard and Kahana (2002), but has been a consistent feature of more recent treatments of TCM (Howard et al., 2006, 2005; Sederberg et al., 2008). The parameterization of the weighting here is somewhat different from previous work. Here, the consistent part of the retrieved context vector is an orthogonal vector c chosen separately for each item in the list. The c vector for each item remains fixed throughout. Recovery of c leads to a strong forward association and is closely analogous to the pre-experimental component of previous implementations of TCM (Howard & Kahana, 2002; Howard et al., 2006, 2005). In addition to the fixed component, each item is also associated with an h vector. The h vector recovers a contextual state during list presentation and thus serves the same role as the “newly-learned” context in previous treatments of TCM (Howard & Kahana, 2002; Howard et al., 2005, 2006). The h vector provides a symmetric retrieval cue for the neighbors of the recalled item. Following recent treatments of TCM (Rao & Howard, 2008; Howard et al., in press), if item A is presented (or recalled) at time step i, then the input pattern at time step i is given by

$${\mathbf{\text{t}}}_i^{\text{IN}}\propto (1-\gamma ){\mathbf{\text{c}}}_A+\gamma {\mathbf{\text{h}}}_A,$$

where the proportionality symbol indicates that ${\mathbf{\text{t}}}_i^{\text{IN}}$ is normalized to be of unit length before entering into the evolution equation above. Each item's h vector is initialized to zero, and then updated according to

$$\mathrm{\Delta }{\mathbf{\text{h}}}_A={\mathbf{\text{t}}}_{i-1}.$$

In TCM, contiguity effects are a consequence of the similarity of the c and h vectors caused by repeated (recalled) items have with the encoding contexts of items presented at similar times.

In TCM, after a word is recalled, it provides input to the context vector and the resulting context vector is used as a cue for recall of the other items. Each item is activated to some extent. Once the activations have been calculated, it is necessary to translate this number into a probability of recall in order to compare to behavioral data. As in early treatments of TCM (Howard & Kahana, 2002; Howard et al., 2006) but not more recent studies that utilize accumulators to model the retrieval process (Sederberg et al., 2008; Polyn et al., in press), we used the Luce choice rule to map the activation of an item a_i onto the probability of recalling that item:

$$P_R(i)=\frac{\text{exp}{a}_i/\tau }{{\sum }_j\text{exp}{a}_j/\tau }.$$

The parameter τ controls the sensitivity of this mapping.

The predictions shown in the paper were generated by using a computer program implemented in the R programming language (available on request). The predicted lag-CRP value was generated by first calculating the probability of first recall for each serial position. The probability of second recall for each other serial position conditionalized on the first recall was then calculated for each item. These values were aggregated by lag, weighted by the probability of first recall of the first item. The result is thus an analytic evaluation of the lag-CRP.