Encoding, maintenance, and retrieval processes in the lag effect: A multinomial processing tree analysis

Abstract

Short-term studies on repeated learning of verbatim material have typically revealed an overall benefit of long lags compared to short lags between repetitions. This has been referred to as the lag effect. On educationally relevant time scales, however, an inverted-U-shaped relation between lag and memory performance is often observed. Recently, Cepeda et al. (2009) showed that the optimal lag for relearning depends heavily on the time interval between the last learning session and the final memory test (i.e., the retention interval; RI). In order to explore the cognitive mechanisms underlying this result in more detail we independently manipulated both the lag and the RI in a 3×2 experimental design and analysed our data using a multinomial processing tree model for free-then-cued-recall data. Our results reveal that the lag effect trends are mainly driven by encoding and maintenance processes rather than by retrieval mechanisms. Our findings have important implications for theories of the lag effect.

Full text

Encoding, Maintenance, and Retrieval Processes 1
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Running head: ENCODING, MAINTENANCE, AND RETRIEVAL IN THE LAG EFFECT
Encoding, Maintenance, and Retrieval Processes in the Lag Effect:
A Multinomial Processing Tree Analysis
Carolina E. Küpper-Tetzel and Edgar Erdfelder
University of Mannheim, Germany
Corresponding author’s address:
Carolina E. Küpper-Tetzel
University of Mannheim
L9, 7, room 209
68131 Mannheim, Germany
Phone: +49-621-181 3569
Fax: +49-621-181 3997
Email: kuepper-tetzel@psychologie.uni-mannheim.de
PublishedinMemory:
http://www.tandfonline.com/doi/abs/10.1080/09658211.2
011.631550#.Ul1aAVD2hak

Encoding, Maintenance, and Retrieval Processes 2
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Abstract
Short-term studies on repeated learning of verbatim material have typically revealed an
overall benefit of long lags compared to short lags between repetitions. This has been referred
to as the lag effect. On educationally relevant time scales, however, an inverted-U-shaped
relation between lag and memory performance is often observed. Recently, Cepeda et al.
(2009) showed that the optimal lag for relearning heavily depends on the time interval
between the last learning session and the final memory test (i.e., the retention interval (RI)).
In order to explore the cognitive mechanisms underlying this result in more detail, we
independently manipulated both the lag and the RI in a 3 by 2 experimental design and
analysed our data using a multinomial processing tree model for free-then-cued-recall data.
Our results reveal that the lag effect trends are mainly driven by encoding and maintenance
processes rather than by retrieval mechanisms. Our findings have important implications for
theories of the lag effect.

Encoding, Maintenance, and Retrieval Processes 3
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Acknowledgements
The authors are grateful to Kerstin Mertz and Jana Scheible for their considerable help with
data collection and to Pia Sue Helferich for editing the Multinomial Processing Tree graph.
We thank David Riefer and an anonymous reviewer for helpful comments on this manuscript.

Encoding, Maintenance, and Retrieval Processes 4
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Encoding, Maintenance, and Retrieval Processes in the Lag Effect:
A Multinomial Processing Tree Analysis
The lag between initial learning (i.e., when new information is acquired for the first
time) and relearning (i.e., when this information is repeatedly studied) has a strong effect on
memory performance on a final test. The finding that memory performance benefits from
increasing lags between study episodes has been referred to as the lag effect. Past studies,
however, have profoundly challenged this simple lag effect finding (e.g., Ausubel, 1966;
Glenberg & Lehmann, 1980; Cepeda, Vul, Rohrer, Wixted, & Pashler, 2008). They have
shown that the retention interval (RI) (i.e., the time between the last study episode and the
final test) plays a crucial role in the modulation of the lag effect function. Glenberg and
Lehmann (1980), for instance, found that, given a 7-day RI, memory performance increased
between the massed practice condition and a 1-day lag condition, but decreased again for a
lag of 7 days between learning sessions. In other words, memory performance on the final
test administered after one week followed an inverted-U-shaped trend with increasing lag.
Recently, Cepeda et al. (2008) and Cepeda et al. (2009) investigated this effect at
educationally relevant time scales. Cepeda et al. (2009) conducted two experiments with RIs
of 10 and 168 days, respectively, combined with six different lags between initial learning
and relearning. They also found that memory performance on the final test followed an
inverted-U-shaped trend with increasing lag. The maximum of the lag effect function,
however, depended on the length of the RI. More precisely, the optimal lag between study
episodes increased as the RI increased. On the basis of an extensive web study, Cepeda et al.
(2008) formalised and reinforced this systematic relationship between lag and RI. Taken
together, these findings suggest that memory performance follows a nonmonotonic trend with
increasing lag, but that the optimal point in time for relearning increases with RI. We use the
term “lag effect” in a way that includes these recent findings.

Encoding, Maintenance, and Retrieval Processes 5
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An important aspect that has not been examined yet is the contribution of encoding,
maintenance, and retrieval processes to the lag effect trends. Basically, improved memory in
the optimal lag condition compared to other study conditions could emerge from any
combination of three different influences: (1) enhanced encoding during repeated practice
leading to a strengthening of the memory trace, (2) improved maintenance leading to
enduring memory traces and resistance to forgetting until the time of testing, and (3) better
retrieval during the final test phase. The answer to the question about the exact contributions
of these memory processes to the lag effect is – to date – not only unknown, but also has
fundamental impact on the evaluation of lag effect explanations. All theories of the lag effect
proposed so far aim at explaining the emergence of the lag effect by focussing on different
memory processes.
The contextual variability theory (Glenberg, 1976, 1979) states that with increasing
lag more different context components are stored along with the to-be-learned information.
This boosts the probability of successful retrieval at final test because more effective retrieval
cues are available due to increased overlap between context components at test and at study.
Hence, retrieval should benefit from the increase in context variability associated with longer
lags. However, Glenberg (1976) also points out that longer lags must not always translate to
increased memory performance. If the RI is short compared to the lag between study sessions
inverted-U-shaped memory functions may occur because the retrieval cues at test are biased
towards the second learning occurrence, and thus, share less contextual components with the
stored memory trace which also contains contextual features from the first learning session.
According to the study-phase retrieval theory (Thios & D’Agostino, 1976), memory
performance improves when during the second occurrence of an item its first occurrence is
retrieved from memory. The second occurrence serves as a cue initiating automatic study-
phase retrieval. Successful study-phase retrieval is assumed to strengthen the stored memory

Encoding, Maintenance, and Retrieval Processes 6
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trace. The benefit of successful retrieval during practice increases with lag because
successful study-phase retrieval is more effortful which, in turn, increases performance on the
test. However, lags may become too long and lead to a failure in study-phase retrieval
decreasing later memory performance.
The Multiscale Context Model (MCM) (Mozer, Pashler, Cepeda, Lindsey, & Vul,
2009) combines the Search of Associative Memory (SAM; Raaijmakers, 2003) and the
predictive utility theory (Staddon, Chelaru, & Higa, 2002). SAM encompasses assumptions
of the contextual variability and the study-phase retrieval theory. The novel aspect in MCM,
however, is the predictive utility assumption. It states that the time that elapses before the
reencounter of information (i.e., the lag) determines for how long this information will be
maintained in memory for the future. More precisely, if the to-be-learned material is
relearned after a long lag our memory system will store and, importantly, maintain the
material for a longer period of time. By contrast, if the lag is short the material will be
available for a short time only.
Taken together, the contextual variability theory, the study-phase retrieval theory, and
MCM offer plausible theoretical explanations for the lag effect. They can be distinguished,
however, with respect to the underlying memory processes they put forward to explain the
lag effect trends. The contextual variability theory emphasises the role of retrieval processes
during testing. The study-phase retrieval theory focuses on the importance of encoding
processes during repeated practice. MCM advances an adaptive feature of the memory
system and introduces the importance of maintenance processes. Thus, in order to evaluate
which theory makes the most plausible assumptions in terms of the underlying memory
processes it would be crucial to disentangle contributions of encoding, maintenance, and
retrieval processes to the lag effect.

Encoding, Maintenance, and Retrieval Processes 7
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A paradigm that has often been used in the past to determine the role of storage (i.e.,
encoding or maintenance) versus retrieval processes makes use of two memory tests: one test
that depends highly on retrieval processes (e.g., free recall) followed by one that depends less
on retrieval processes (e.g., cued recall) (see, e.g., Drachman & Leavitt, 1972; Hogan &
Kintsch, 1971; Thomson & Tulving, 1970). Consequently, finding a memory effect in free
recall, but not in cued recall, suggests that retrieval processes play a major role for the
phenomenon. In contrast, if the effect emerges in both memory tests alike, this hints at the
conclusion that the phenomenon is rather driven by storage processes (i.e., encoding,
maintenance, or both).
Although performance profiles in the free-then-cued-recall paradigm provide valuable
information, a more fine-grained analysis of the memory processes involved would be
preferable. Multinomial processing tree (MPT) models (Batchelder & Riefer, 1999; Erdfelder
et al., 2009) offer such an analysis by providing separate estimates of the cognitive processes
underlying performance scores. In the past, MPT models have been used successfully to
disentangle storage and retrieval contributions to well-known memory phenomena, for
example, the bizarreness effect (Riefer & Rouder, 1992), the recognition failure effect (Riefer
& Batchelder, 1995), retroactive inhibition (Bäuml, 1991, 1996), and, more recently, the
enactment effect (Steffens, Jelenec, Mecklenbräuker, & Thompson, 2006; Steffens, Jelenec,
& Mecklenbräuker, 2009).
Thus, in order to evaluate prevailing theories of the lag effect, we examined the role
of encoding, maintenance, and retrieval processes by assessing memory performance with
both free and cued recall tests and by analysing the data with an extended1 version of Rouder
and Batchelder’s (1998) storage-retrieval MPT model for a free-then-cued-recall paradigm.
In our study, we assessed joint effects of different lags and RIs on memory
performance. Critically to our approach, memory performance was assessed by a free recall

Encoding, Maintenance, and Retrieval Processes 8
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test on weakly associated cue-target word pairs immediately followed by a cued recall test for
the target word given the cue word. Based on the performance on the cued recall test at the
end of practice combined with the performances on the free and cued recall final tests
administered after the RI, 12 observable events can occur for each word pair (Table 1).
(Table 1 about here)
Based on the observed frequencies of these 12 events, MPT modelling allows
estimation of seven parameters representing underlying memory processes: one probability of
associative encoding (e), two probabilities of associative maintenance in memory until the
final test (ms and mu for maintenance following successful vs. unsuccessful initial cued recall,
respectively), two probabilities of successful retrieval in free and cued recall (rf and rc,
respectively), and, finally, two probabilities of single word retrieval in free recall in case of
successful vs. unsuccessful associative encoding or maintenance (s and u, respectively).
(Figure 1 about here)
To facilitate the understanding of this method, the extended MPT model is presented
as a processing tree diagram in Figure 1. It has 32 branches, each terminating in one of the 12
events summarized in Table 1. Each of these branches represents a possible sequence of
encoding, maintenance, and retrieval processes underlying performance in free and cued
recall. Specifically, a word pair is either encoded as an association with probability e or is not
encoded as an association with the complementary probability 1-e. In case of successful
associative encoding, cued recall at the end of practice may be successful with probability rc
or fail with probability 1-rc. Hence, parameter rc represents the probability of successful
associative retrieval in a cued recall test and is likely to be close to one whenever the cue-
target association is stored in memory at the time of testing. Associative maintenance to the
time of testing occurs upon encoding and successful initial cued recall with probability ms. If
associative maintenance of the word pair is successful (ms), associative retrieval during final

Encoding, Maintenance, and Retrieval Processes 9
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cued recall may be successful with probability rc or unsuccessful with the complementary
probability 1-rc. In either case, the intact word pair may be retrieved during free recall with
probability rf resulting in event E1 (in case of successful final cued recall) or E4 (in case of
unsuccessful final cued recall). Associative retrieval during free recall, however, may fail
with probability 1-rf, so that the intact word pair cannot be retrieved as association.
Nevertheless, each word of a pair may be independently retrieved during free recall with
probability s or not retrieved with probability 1-s, so that both words of a pair (E1 or E4,
depending on successful vs. unsuccessful final cued recall, respectively), exactly one word
(E2 or E5, depending on successful vs. unsuccessful final cued recall, respectively), or neither
word is recalled (E3 or E6, depending on successful vs. unsuccessful final cued recall,
respectively). In contrast, if maintenance of the word pair association in memory fails (1-ms),
the final cued recall will also fail. However, items may be retrieved (u) or not retrieved (1-u)
as nonassociated single words during free recall, so that both words of a pair (E4), exactly one
word (E5), or neither word is recalled (E6)2. Moreover, successful associative maintenance
may also occur following unsuccessful cued recall at the end of learning, albeit with
probability mu that may, in principle, differ from ms. From there, the MPT tree progresses as
described above except that the branches terminate in event categories E7 to E12 instead.
Finally, associative encoding can fail altogether with probability 1-e. This implies a failure of
both cued recall at the end of practice and final cued recall. However, items may be retrieved
(u) or not retrieved (1-u) independently as nonassociated single words during free recall, so
that both words of a pair (E10), exactly one word (E11), or neither word is recalled (E12).
Summing up the branch probabilities that terminate in the same observable event, we obtain
the following set of model equations for the 12 possible events:
p(E1) = e rc2 ms [rf +(1-rf) s2]
p(E2) = 2 e rc2 ms (1-rf) s (1-s)
p(E3) = e rc2 ms (1-rf) (1-s)2
p(E4) = e rc [ms (1-rc) (rf + (1-rf) s2) + (1-ms) u2]

Encoding, Maintenance, and Retrieval Processes 10
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p(E5) = 2 e rc [ms (1-rc) (1-rf) s (1-s) + (1-ms) u (1-u)]
p(E6) = e rc [ms (1-rc) (1-rf) (1-s)2 + (1-ms) (1-u)2]
p(E7) = e (1-rc) rc mu [rf + (1-rf) s2]
p(E8) = 2 e (1-rc) rc mu (1-rf) s (1-s)
p(E9) = e (1-rc) mu rc (1-rf) (1-s)2
p(E10) = e (1-rc) [mu (1-rc) (rf + (1-rf) s2) + (1-mu) u2] + (1-e) u2
p(E11) = 2 [e (1-rc)2 mu (1-rf) s (1-s) + e (1-rc) (1-mu) u (1-u) + (1-e) u (1-u)]
p(E12) = e (1-rc) [mu (1-rc) (1-rf) (1-s)2 + (1-mu) (1-u)2] + (1-e) (1-u)2
On the basis of the observed event frequencies, the seven model parameters (e, ms, mu,
rc, rf, s, and u) are then estimated using standard maximum likelihood techniques (Hu &
Batchelder, 1994) as implemented in freely available software for MPT models (e.g.,
Moshagen, 2010). The parameters of Rouder and Batchelder’s original free-then-cued-recall
MPT model have previously been validated experimentally by applying specific
manipulations that were assumed to influence one parameter (e.g., retrieval) while leaving
others (e.g., storage) unaffected (see Rouder & Batchelder, 1998). Thus, it has been
established that the model not only fits empirical data but also provides parameters that
capture the intended memory processes selectively. Our MPT model keeps all the basic
assumptions of Rouder and Batchelder’s (1998) model and represents a straightforward
extension by (1) differentiating between processes of encoding and processes of maintenance
to the time of testing and by (2) providing an estimate of associative retrieval during cued
recall in addition. Therefore, it seems appropriate to use our extended MPT model to measure
encoding, maintenance, and retrieval contributions to the trends in the lag effect and, more
importantly, to evaluate theoretical explanations of the lag effect.
Method
Participants
Sixty-two persons participated in this experiment. Two participants were excluded
from all analyses because they severely underperformed during the relearning session (cued

Encoding, Maintenance, and Retrieval Processes 11
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recall performance < 40%). The remaining 60 participants were current students or alumni of
the University of Mannheim. Thirty-seven were female, mean age 22.45 (range, 18-33).
Materials
The word material consisted of 30 weakly associated cue-target word pairs from 30
common categories. All words were concrete German nouns taken from German production
norms (Hager & Hasselhorn, 1994). Both words of a pair always came from the same
category (e.g., foods). The cue word was a weakly associated word of the category (e.g.,
candy, production index < 0.02) and the respective target word was one of the four most
frequently produced words for that category (e.g., bread). Weakly associated word pairs were
used to avoid inferential processes during recall. In order to ensure that significant effects
would not be due to word list characteristics, we constructed four word lists, each containing
30 target words with high associations to their respective category and to each other (e.g.,
category: foods, target words: bread, meat, butter, vegetables). These four target word lists
were counterbalanced across participants. Thus, the same cue word was combined with one
target word that varied depending on the active word list.
Design
The experiment consisted of two learning sessions separated by a lag and one final
test session occurring after the RI. The lag between learning sessions was either 0 days, 1
day, or 11 days and the RI was either 7 or 35 days. This resulted in a 3 x 2 between-subjects
design. These intervals were chosen on the basis of Cepeda et al. (2008), who suggest an
inverted-U-shaped memory trend in the 7-day RI group (i.e., peak at a 1-day lag) and a
positive linear trend in the 35-day RI group for lags increasing from 0 to 11 days. Participants
were randomly assigned to their experimental condition, subject to the constraint of their

Encoding, Maintenance, and Retrieval Processes 12
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availability for attending sessions on different days. Four experimental conditions contained
10 participants each and two conditions contained 9 and 11 participants, respectively.
Procedure
Participants attended two learning sessions and one test session. During the first
learning session, participants worked on two study-test trials. The 1-day lag and 11-day lag
group were dismissed after these trials and returned after the respective lag to their second
learning session. The 0-day lag group worked for five minutes on an unrelated distractor task
and continued with the second learning session on the same day. The second learning session
contained one more study-test trial. After the second learning session, all participants were
dismissed and returned after a RI of either 7 or 35 days to the final test session.
A study-test trial consisted of a study phase, a distractor task, and a test phase. During
the study phase, each word pair was presented for three seconds separated by a 750
milliseconds interstimulus interval. Word pairs were presented in a different random order in
each study phase. After the study phase, participants worked for two minutes on an unrelated
arithmetic task. Afterwards, participants were administered a free recall test immediately
followed by a cued recall test. Both memory tests were self-paced. For the free recall test,
they were instructed to write all word pairs they could remember on a lined recall sheet. It
was pointed out to them that if they could only remember one word of a pair, they should
nevertheless write it down. For the cued recall test, participants were presented with the cue
word and asked to recall and type the correct target word. They were asked to recall all 30
target words given the cue. The cues were presented sequentially in random order.
Participants could skip to the next cue word if they could not remember the target word. They
were not provided with feedback about their performance during the memory tests.
On the final test session, participants worked on a free recall test immediately
followed by a cued recall test. Afterwards, they were compensated for their participation and

Encoding, Maintenance, and Retrieval Processes 13
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debriefed. The whole experiment lasted about 1.5 hours divided over three sessions. All
participants were required to attend three sessions on three different days. Participants in the
0-day lag condition who completed the experiment within two sessions worked for 15
minutes on an unrelated experiment during their third session. This was done to establish
comparable motivational conditions for all participants.
Results
Learning performance
Participants showed good learning performances in both memory tests at the end of the
first learning session after two study-test trails, 52% free-recall accuracy and 83% cued-recall
accuracy. Moreover, as expected, free and cued recall performances assessed by the end of
the first learning session were not affected by lag, F(2,57) = 1.31, p = .279, p2 = .04 and
F(2,57) = 1.24, p = .297, p2 = .04, respectively.
Memory performance in the relearning session (i.e., after the lag) showed the expected
decline in free and cued recall as a consequence of increasing lag. We performed multiple
comparisons and adjusted the α-level of .05 using the Holm-Bonferroni correction method
(Holm, 1979). All significance tests are reported with p-values corresponding to two-tailed
tests even when directed hypotheses were imposed. The Welch test was applied whenever
Levene’s test indicated unequal variances at  = .05. Memory performance on both memory
tests was significantly lower during practice after an 11-day lag compared to a 1-day lag
(free-recall accuracy: 48% vs. 69%; cued-recall accuracy: 78% vs. 96%), t(32.76) = 3.51, p =
.001, 2 = 0.27 and t(20.80) = 3.74, p = .001, 2 = 0.40 for free and cued recall, respectively.
Also, the decrease between a 0-day lag and an 11-day lag was significant (free-recall
accuracy: 70% vs. 48%; cued-recall accuracy: 92% vs. 78%), t(38) = 3.52, p = .001, 2 = 0.25
and t(23.98) = 2.81, p = .010, 2 = 0.25 for free and cued recall, respectively. There was no

Encoding, Maintenance, and Retrieval Processes 14
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significant difference in memory performance between the 0-day and the 1-day lag condition,
all ts ≤ 1.69, ps ≥ .10.
Final test performance
An initial analysis of the final test performance revealed that there was no systematic
effect due to the counterbalance factor word list neither for free recall nor for cued recall,
F(3,56) = 1.16, p = .332, p2 = .06 and F(3,56) = 1.05, p = .376, p2 = .05, respectively. Thus,
for further analyses, data were collapsed across the four word lists. An α-level of .05 was
used for all statistical tests. Again, significance tests are reported as two-tailed tests even in
case of directed predictions.
Not surprisingly, participants recalled more word pairs on both memory tests after a
7-day RI (free recall: M = 41%, SD = 20; cued recall: M = 74%, SD = 23) than after a 35-day
RI (free recall: M = 19%, SD = 17; cued recall: M = 43%, SD = 23), t(58) = -4.43, p < .001,
2 = 0.25 and t(58) = -5.12, p < .001, 2 = 0.31 for free and cued recall, respectively.
Of greatest interest, however, were the different trends in the two RI conditions as a
function of increasing lag. To revisit, we expected that with increasing lag (0-day < 1-day <
11-day) memory performance would follow an inverted-U-shaped trend (i.e., negative
quadratic trend) in the 7-day RI group and a positive linear (or at least monotonically
increasing) trend in the 35-day RI group. Descriptively, the data fit our expectations nicely
(see Figure 2). In fact, a significant negative quadratic trend emerged in the 7-day RI
condition for cued recall, t(27) = 2.08, p = .048, 2 = 0.14, and a marginally significant
quadratic trend occurred for free recall, t(27) = 1.71, p = .099, 2 = 0.10. In contrast, the
linear trend was neither significant for free recall, t(27) = 1.07, p = .292, 2 = 0.04, nor for
cued recall, t(27) = -0.25, p = .807, 2 < 0.01. In the 35-day RI condition, however, a
significant positive linear trend was detected for both free recall, t(27) = 2.24, p = .033, 2 =

Encoding, Maintenance, and Retrieval Processes 15
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0.16, and cued recall, t(27) = 2.79, p = .010, 2 = 0.22. The quadratic trend was not
significant in this condition, neither for free recall, t(27) = 0.71, p = .482, 2 = 0.02, nor for
cued recall, t(27) = 0.52, p = .606, 2 = 0.01.
(Figure 2 about here)
Model-based analyses
Our extended MPT model for free-then-cued-recall was used to disentangle encoding,
maintenance, and retrieval contributions to free and cued recall in the final memory test. For
model-based analyses, the frequencies of the 12 event categories were calculated for each
participant and aggregated separately for each of the 3 x 2 = 6 experimental conditions,
resulting in N = 1,800 data points in total. The Type I error level was set to α = .05 for all
model-based analyses. A sensitivity analysis was performed using G*Power 3.1 (Faul,
Erdfelder, Buchner, & Lang, 2009). This analysis showed that with a sample size of N =
1,800 data points, a significance level of α = .05, and a desired power of 1-β = .95, the
detectable effect size for G2 goodness-of-fit tests based on df ≤ 30 is ω ≤ 0.14 (i.e., a small
effect; cf. Cohen, 1988). Thus, all G2 tests reported below allowed detecting already small
deviations from the model. The multiTree software (Moshagen, 2010) was used for all MPT
model analyses reported here. To revisit, our extended free-then-cued-recall MPT model
contains seven parameters (e, ms, mu, rc, rf, s, and u) per condition (see Figure 1), that is, 6 · 7
= 42 parameters across all six conditions. Hence, the overall goodness-of-fit test has 6 · (12-
1) – 42 = 24 degrees of freedom. This general model fitted the data well (G2(24) = 19.65, p =
.716). To specify our model as parsimoniously as possible and to increase the precision of
parameter estimates, we tested the additional restriction that the maintenance probabilities ms
and mu can be set equal in each condition. A priori, we hypothesised that ms > mu, because
success in associative cued recall at the end of practice is likely to have positive effects on
subsequent maintenance. However, as revealed by a G2 difference test, this effect was not

Encoding, Maintenance, and Retrieval Processes 16
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significant in our data, G2(6) = 1.44, p = .964. Consequently, we report results based on the
more parsimonious version of our extended model that contains a single maintenance
parameter m only.3 The overall goodness-of-fit test for this restricted model has 6 · (12-1) –
36 = 30 degrees of freedom and indicates an excellent fit to the data (G2(30) = 21.09, p =
.885).
Parameter Estimates. Of greatest interest for our research question are the probability
estimates for associative encoding e, associative maintenance m, and associative retrieval rf
presented in the upper, middle, and lower chart of Figure 3, respectively.
(Figure 3 about here)
The associative encoding parameter e followed an inverted-U-shaped trend with
increasing lag in both RI conditions (Figure 3, upper chart). More specifically, the probability
of associative encoding increased significantly between the 0-day and the 1-day lag
condition, ΔG2(1) = 6.60, p = .010 and ΔG2(1) = 5.90, p = .015, and decreased between the 1-
day lag and the 11-day lag condition, ΔG2(1) = 57.12, p < .001 and ΔG2(1) = 27.36, p < .001,
for the 7- and 35-day RI group, respectively. In line with our expectations, associative
encoding was not affected by the length of the RI, ΔG2(3) = 4.28, p = .233. This result is
important because it shows that the additional associative encoding parameter e in our
extended MPT model can be considered as a valid measure of encoding processes at practice
that is not affected by the length of the RI.
The parameter for associative maintenance m, however, was affected differently by
the length of the RI (Figure 3, middle chart). In the 7-day RI group, associative maintenance
increased between the 0-day lag and the 1-day lag as well as the 11-day lag, ΔG2(1) = 20.49,
p < .001 and ΔG2(1) = 17.25, p < .001, respectively. There was no difference in associative
maintenance between the 1-day and the 11-day lag, ΔG2(1) = 0.06, p = .813. In the 35-day RI
condition, associative maintenance increased significantly between the 0-day lag and the 1-

Encoding, Maintenance, and Retrieval Processes 17
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day lag, ΔG2(1) = 15.44, p < .001, and increased further between the 1-day and the 11-day
lag, ΔG2(1) = 18.20, p < .001.
In addition, we tested the difference in m between the two RI conditions for each lag.
As expected, better associative maintenance of the material emerged in the 7-day RI
condition than in the 35-day RI condition for all lag conditions, ΔG2(1) = 104.60, p < .001,
ΔG2(1) = 112.51, p < .001, and ΔG2(1) = 30.52, p < .001, for 0-, 1-, and 11-day lag,
respectively. Again, this validates the current MPT model because the length of the RI should
have a strong negative impact on maintenance of the to-be-learned material to the time of
testing.
The parameter estimates for associative retrieval rf during free recall are presented in
the lower chart of Figure 3. In the 7-day RI condition, associative retrieval increased
significantly between the 0-day lag and the 1-day lag as well as the 11-day lag, ΔG2(1) =
6.79, p < .009 and ΔG2(1) = 7.51, p = .006, respectively. We found the same results in the 35-
day RI group, for the comparison between 0-day and 1-day lag, ΔG2(1) = 4.45, p = .035, and
for the comparison between 0-day and 11-day lag, ΔG2(1) = 4.12, p = .043. Importantly, the
probability of associative retrieval did not differ significantly between the two distributed
learning conditions, neither for the 7-day RI, ΔG2(1) = 0.15, p = .695, nor for the 35-day RI,
ΔG2(1) = 0.04, p = .849.
In addition, we analysed the difference in associative retrieval between the two RI
groups for each lag condition. Not surprisingly, the probability of associative retrieval in free
recall was significantly smaller after a 35-day RI than after a 7-day RI for all lag conditions,
ΔG2(1) = 5.49, p = .019, ΔG2(1) = 4.91, p = .027, and ΔG2(1) = 7.46, p = .006, for a 0-, 1-,
and 11-day lag, respectively.
Last but not least, estimates of the associative cued recall probability rc (not shown in
Figure 3) differed significantly between conditions, G2(5) = 24.25, p < .001. However, as

Encoding, Maintenance, and Retrieval Processes 18

expected, all six rc parameter estimates were very close to 1, ranging between .96 and 1.00
with a mean of .99. This result is roughly in line with the original free-then-cued-recall MPT
model (Rouder & Batchelder, 1998) which is based on the simplifying approximation rc = 1.
Discussion
Our data convincingly show that different lags of relearning can affect memory
performance either in a linear or in a negatively accelerated quadratic manner depending on
the length of the RI. More precisely, in the 7-day lag condition, we revealed an inverted-U-
shaped trend with increasing lag. Memory performance in this condition peaked at a 1-day
lag and decreased for shorter or longer lags. In contrast, in the 35-day RI condition, memory
performance increased with lag, thereby suggesting that memory performance improves from
a 0-day to an 11-day lag. Thus, we successfully replicated the lag effect trends detected by
Cepeda et al. (2008).
The model-based analyses contribute to a better understanding of the underlying
cognitive processes. Our extended MPT model for free-then-cued-recall based on Rouder and
Batchelder (1998) fit the empirical data successfully. Not surprisingly, associative retrieval
decreased with the length of the RI. The lag effect trends, however, were particularly driven
by processes captured by the associative encoding parameter e and the associative
maintenance parameter m. We found a systematic interplay between encoding and
maintenance processes that influenced memory performance in the final test and that was
mediated by the length of the RI.
More precisely, associative encoding revealed an inverted-U-shaped trend in both the
7-day and the 35-day RI condition alike. This result represents the drop in learning
performance after a long lag (i.e., 11 days) compared to a short lag (i.e., 1 day) – a result that
is in line with earliest findings in memory research (Ebbinghaus, 1885/1964). Memory
performance after a short RI of 7 days was particularly affected by this inverted-U-shaped

Encoding, Maintenance, and Retrieval Processes 19

trend in associative encoding. In this condition, associative maintenance and associative
retrieval (captured by rf) both increased between the massed and the spaced conditions, but
there was no further significant difference between the two spaced conditions. In contrast, in
the long 35-day RI condition, the detrimental effect of poor encoding after a long lag of 11
days was outweighed by better associative maintenance processes. Thus, the positive effect of
increasing lag for memory performance after a RI of 35 days was most certainly due to
enhanced associative maintenance processes. Note that retrieval processes cannot explain this
effect because the associative retrieval parameter rf reflected the advantage of spaced practice
over massed practice only and was not sensitive to lags of different lengths. However, the
latter finding is consistent with previous research suggesting that spaced practice, in general,
leads to enhanced retrieval compared to massed practice (e.g., Batchelder & Riefer, 1980).
Taken together, encoding processes dictated the inverted-U-shaped memory
performance in the 7-day RI condition, whereas maintenance processes to the time of testing
were responsible for the linear increasing memory performance in the 35-day RI condition
with increasing lag. Thus, given a long RI, the detrimental effects of decreased encoding as a
consequence of a long lag are outweighed by better maintenance processes of the encoded
memory traces.
Consequently, a theory that attempts to explain the lag effect must emphasise the
crucial roles and interplay of encoding and maintenance processes for this learning effect.
Thus, the contextual variability theory, which conceives retrieval processes at test as most
important, cannot be considered as a potential candidate for explaining the processes
producing the curvilinear lag effect trend. Nevertheless, the MPT results show that the
contextual variability theory can account for the spacing effect since the advantage of spaced
over massed practice was reflected in the associative retrieval parameter.

Encoding, Maintenance, and Retrieval Processes 20

Our model findings suggest that both the study-phase retrieval theory and the
predictive utility assumption of MCM provide the most plausible explanations for the lag
effect trends. In accordance with the study-phase retrieval account, better encoding occurred
after a lag of 1 day compared to the massed condition. However, if the lag becomes too long
(e.g., 11-day lag), successful study-phase retrieval may fail, which leads to decreased
encoding. This, in turn, has negative effects on the later memory performance. The inverted-
U-shaped memory performance on the final test after a 7-day RI is consistent with this
explanation. MCM endorses the crucial role of forgetting and proposes that more enduring
memory traces are stored after a long lag compared to a short lag between study episodes.
Using MPT modelling, we were able to find that the linear increasing lag effect in the 35-day
RI condition was specifically affected by the model parameter that captures maintenance of
the material to the time of testing. Thus, consistent with MCM, the memory traces that were
encoded after an 11-day lag were maintained better than memory traces that were encoded
after a 1-day lag. This proved to be particularly beneficial if memory performance was
assessed after a long 35-day RI.
Taken together, our MPT analyses showed that the study-phase retrieval theory and
the predictive utility assumption of MCM offer the most plausible assumptions in regard to
the underlying cognitive mechanisms for the lag effect. The current findings suggest that the
shift of optimal lag with RI is not due to a single mechanism, but rather to a systematic
interplay of encoding and maintenance processes that is moderated by the length of the RI.
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Footnotes
1 Rouder and Batchelder’s (1998) original MPT model for free-then-cued-recall differs from
our extended model mainly by using a single storage parameter that combines encoding and
maintenance processes. For our research question, however, it was crucial to differentiate
between encoding and maintenance processes. Inclusion of cued recall performances at the
end of practice (i.e., in the relearning session after the lag) enabled us to introduce an
additional encoding parameter and to decompose Rouder and Batchelder’s associative storage
parameter a into associative encoding and maintenance parameters, e und m, respectively.
Keeping the basic assumptions of Rouder and Batchelder’s MPT model unchanged, our
extension provides separate estimates for associative encoding, maintenance, retrieval in cued
recall, and retrieval in the final free recall test.
2 One reviewer pointed out correctly that the MPT model does not distinguish between the
probability to recall the cue and the probability to recall the target. Rather, a single parameter
s or u is assigned to the independent retrieval of one word of the pair, irrespective of whether
it is a cue or target. This model assumption, most certainly, represents an approximation that
allows us to keep the model as simple and parsimonious as possible. Nevertheless, we
checked whether cues and targets were recalled equally often. This was indeed the case for all
but one experimental condition: Only in the 11-day lag group that was tested after a 35-day
RI, participants recalled slightly more targets (M = 11, SD = 5.9) than cues (M = 9, SD = 5.7)
in the final free recall test, t(9) = 3.35, p = .008. In all other five conditions differences were
even smaller and not significant (all ts ≤ 2.06, all ps ≥ .067). Thus, for the sake of simplicity
and model parsimony, since the difference in cue and target recall in our experiment was
small and the s and u parameters were not of major interest for the current research question,
we decided not to distinguish between cue and target retrieval.

Encoding, Maintenance, and Retrieval Processes 26

3 We would like to point out that the restriction ms = mu= m has negligible effects on
estimates of the other parameters. Estimates for m reported in the present paper resemble
those for ms in the unrestricted version of the extended model. In other words, substantive
conclusions are not affected by the version of the extended model used for data analyses.

Encoding, Maintenance, and Retrieval Processes 27

Figure Captions
Figure 1. Extended MPT model for a free-then-cued-recall paradigm to disentangle encoding,
maintenance, and retrieval processes based on Rouder and Batchelder (1998). The processing
tree presents the latent cognitive processes leading to 12 observable event categories:
Successful cued recall at the end of practice, successful final cued recall and free recall of the
complete word pair (E1), exactly one word of the pair (E2), or neither word of the pair (E3) or
successful cued recall at the end of practice, unsuccessful final cued recall and free recall of
the complete word pair (E4), exactly one word of the pair (E5), or neither word of the pair
(E6) or unsuccessful cued recall at the end of practice, successful final cued recall and free
recall of the complete word pair (E7), exactly one word of the pair (E8), or neither word of
the pair (E9) or unsuccessful cued recall at the end of practice, unsuccessful final cued recall
and free recall of the complete word pair (E10), exactly one word of the pair (E11), or neither
word of the pair (E12). The transition probabilities between cognitive states (rounded
rectangles) are represented by the model parameters (e = probability of associative encoding
during study, ms, mu = probability of associative maintenance to the time of testing upon
successful encoding (following successful or unsuccessful cued recall at the end of practice,
respectively), rc = probability of associative retrieval during cued recall, rf = probability of
associative retrieval during free recall, s = probability of associated single word retrieval
during free recall, u = probability of nonassociated single word retrieval during free recall).
Figure 2. Mean and standard errors of correctly recalled word pairs on the final free recall
(upper chart) and final cued recall test (lower chart) as a function of lag and retention
interval.
Figure 3. Parameter estimates and standard errors for the probability of associative encoding
e (upper chart), for the probability of associative maintenance m (middle chart), and for the
probability of associative retrieval rf (lower chart) as a function of lag and retention interval.

Encoding, Maintenance, and Retrieval Processes 28

Table 1. Twelve event categories for a memory paradigm that combines cued recall
performance at the end of practice with final free and cued recall performances.
Cued recall at
end of practice Final cued
recall Final free recall
Both words Exactly one word Neither word
Correct Correct E1 E
2 E
3
Incorrect E4 E
5 E
6
Incorrect Correct E7 E
8 E
9
Incorrect E10 E
11 E
12

Encoding, Maintenance, and Retrieval Processes 29

Figure 1

Encoding, Maintenance, and Retrieval Processes 30

0
10
20
30
40
50
60
70
80
90
100
0days 1day 11days
InterstudyInterval
0
10
20
30
40
50
60
70
80
90
100
0days 1day 11days
InterstudyInterval
RI=7days
RI=35days
Figure 2
Correctlyrecalledwordpairsin%
FreeRecall
CuedRecall

Encoding, Maintenance, and Retrieval Processes 31

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0days 1day 11days
Probability estimate for
associative encoding e
InterstudyInterval
RI=7days RI=35days
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0days 1day 11days
Probability estimate for
associative maintenance m
InterstudyInterval
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0days 1day 11days
Probability estimate for
associative retrieval rf
InterstudyInterval
Figure 3