Replication and Analysis of Ebbinghaus’ Forgetting Curve

We believe that we may conclude that our attempt to replicate Ebbinghaus’ classic forgetting was successful. We were able to follow his method quite closely and the resulting curve is very similar to both that of Ebbinghaus and that of the two subjects in an earlier German replication, with the exception of the savings value at 31 days, which in our case is much lower than the others. The latter difference remains even if we correct for increased learning time over the course of the experiment. It is possible that with Ebbinghaus and Heller et al. there were far fewer intermediary lists learned between learning and relearning and hence much less interference. Unfortunately, this information is not available, so this must remain speculation.

Effects of serial position on forgetting

Ebbinghaus does not say anything about serial position curves or indeed about the order in which he acquired the syllables. Our data allow us to say a little more about this. When interpreting the serial position scores in Fig 6, one has to bear in mind the nature of the savings methods with lists of nonsense syllables. With Ebbinghaus-type relearning, a row is always first studied before it is relearned. Savings experiments are very different from normal memory retention experiments where the subject learns something at after some time interval is tested for retention. With savings, the retention measurement itself consists of relearning the original material in repeated recall trials each of which is preceded by prior exposure to the stimulus materials. In theory it is, therefore, possible in extreme cases that there are subjects who can learn a row of 13 syllables in a single learn-recall trial but who subsequently always forget the learned materials in say 10 min. By conventional (non-savings) retention tests, these subjects would be reported as have 0% retention past 10 min, but these same subjects would show 100% savings for all retention intervals past 10 min because when testing their savings performance they would go through the exposure-learn sequence, always (re)learning the materials immediately: at t = 0, they take 1 trial to learn the material and at t = 1 day they also take 1 trial, showing perfect savings (i.e., seemingly no forgetting). Thus, when stimuli are very easy to learn true forgetting becomes impossible to measure with the savings method, because original learning will be at ceiling (one-trial learning) as will be relearning after a time-interval (again: one-trial learning), no matter how long the interval.

Here, it seems that the first two and the last three syllables were very easily learned (and relearned), probably because of primacy and recency effects. This effect was also noticed by other early researchers who adapted Ebbinghaus’ method. Relearning the harder parts of a list, in particular the middle syllables, benefits most from recently having learned these sometime before (i.e., before the relearning phase of the savings experiment), as is evident from the savings scores. The discrepancy between savings and recall or recognition has also been found by other authors and appears here as a function of serial position.

The fairly evident primacy and recency effects also suggest that the despite Ebbinghaus’ efforts to construct equivalent stimuli, there is a great variation in how well they eventually were learned: During the first phase (t = 0), stimuli in the first and last positions were very easily and hence very well learned, while those in the middle were learned much less. The effect seems quite constant, however, so that it need not affect the validity of the shape of the forgetting curve, but we should be aware that its shape is based on a combination of very well learned items and just barely learned items. In that sense, the forgetting curve by Ebbinghaus is an average over different forgetting curves of items in various serial positions, which have been learned to varying degrees. This in itself may explain part of the characteristic shape of the curve, which we will explore in the next section.

Curve fitting

Hermann Ebbinghaus was the first to try to find a mathematical equation that describes the shape of forgetting. Many researchers who used his method have followed suit, also trying to summarize the forgetting curve in a concise equation. In his first manuscript, from 1880, Ebbinghaus proposes the equation x = [1 − (2/t)0.099]0.51, where x equals 1 minus savings at time t (in min). It is of some interest that Ebbinghaus (p. 57–63) puts the entire section in which he fits this equation to the data between square brackets, making it an aside: something that is also interesting but not belonging to the main text. Nonetheless, he gives a well-motivated derivation based on a differential equation of a gradually slowing forgetting process. Interesting is that in the write-up of the experiment in 1885, the equation has been changed to the very different one, which has been become generally known as the ‘Ebbinghaus Forgetting Equation’, rather than the first one, namely Q(t) = 1.84 / ((log t)1.25 + 1.84), where the log is taken with base 10. This equation is lacking a derivation and Ebbinghaus remarks on it: “Of course this statement and the formula upon which it rests have here no other value than that of a shorthand statement of the above results which have been found but once and under the circumstances described. Whether they possess a more general significance so that, under other circumstances or with other individuals, they might find expression in other constants I cannot at the present time say.”

We are now in a better position to verify Ebbinghaus’ question about the general significance of his equation by fitting his equations to the other data. The results are given in Table 4 and were obtained with a nonlinear fitting procedure in Mathematica 9 (the Mathematica code is available from the author). We calculated several goodness-of-fit measures including the Akaike Information Criterion or AIC, which contrary to, for example, variance explained (R ²) or sum of squared differences (SSD), takes into account (and penalizes for) the number of free parameters. It also allows a comparison of the goodness-of-fit of different models, even if they have different numbers of parameters. Lower values indicate better fit, where a difference of more than 2 is seen as a meaningful difference in goodness-of-fit.

In the table, we see that in the case of his power function from 1880, Ebbinghaus' calculations, carried out by hand, were quite close the computer-optimized parameter values: he found values 0.51 and 0.099 for the parameters, whereas we found 0.523 and 0.101, respectively. For the logarithmic function from 1885, we also found similar parameters to those parameters Ebbinghaus reported: 1.8 and 1.21 for his values of 1.85 and 1.25 respectively.

The goodness-of-fit of his functions is quite good, in both cases explaining 98.8% of the variance (R ²) for his own data. The equation from 1885 has a slightly smaller SSD value (i.e., fits better), which in fact is the lowest value for an individually fitted curve we obtained (also see below and Table 5). Though the equations found by Ebbinghaus fit his own data very well, they do not always fit the other curves well, with especially Mack and Dros showing a relatively bad fit on these ‘classic’ equations. This is perhaps not concluded from the variance explained (R ²), which is very high for all studies, but if we base our judgment on the AIC we observe large differences where the AIC for the Ebbinghaus data is almost twice as low as on the Dros data. This suggests that the general applicability of Ebbinghaus equations may be lacking. We further investigate this by comparing Ebbinghaus’ functions with some other functions that have been proposed in the literature.

Ebbinghaus function from 1880 is a type of double power function. A normal power function is described by the equation Q(t)=(1+μ1t)−a1, where Q(t) is savings at time t and μ 1 and a 1 are parameters. The latter equation has been proposed by several authors to describe the time-course of forgetting. The forgetting mechanism typically associated with a power function is a constant slowing down of the forgetting rate with time (cf. Ebbinghaus' account from 1880 mentioned above). Whereas this is certainly a viable mechanism of forgetting, it can be proven mathematically that (spurious) power functions may emerge from averaging over different subjects or items. This has also been shown in simulations considering a wide range of circumstances. As argued in the previous section, the forgetting curve (also) averages over items that have been learned to various degrees, due to their serial position. There are, therefore, several reasons to expect and consider the power function.

The goodness-of-fit of the simple power function to our data is given in Table 5. As can be seen, the fit to Ebbinghaus’ data is still impressive, though somewhat less good than either of his own equations. The goodness-of-fit of the power function, as expressed by the SSD, averaged over all four subjects’ is comparable to the Ebbinghaus 1885 ‘logarithmic’ function and it is somewhat worse than his 1880 ‘power’ function. The AIC is slightly worse, but probably not meaningfully so as the difference in AIC measures is only 1.

Heller et al. also fitted the Ebbinghaus 1885 ‘logarithmic’ equation to the Mack and Seitz data and noticed that it did not fit the Mack and Seitz data well. They therefore proposed a different equation, the sum of two exponentials: Q(t)=μ1e−a1+μ2e−a2 A similar function has independently been proposed by Rubin, Hinton and Wenzel to successfully fit a forgetting curve with very large numbers of observations per data point, which could not be fitted satisfactorily with any of the more than hundred functions studied in Rubin and Wenzel. None of these authors gave a forgetting mechanism associated with these functions.

Though providing us with a superior fit, a disadvantage of the summed exponential is that there are no memory models that explain why forgetting might have this shape. As remarked by several authors investigating the shape of learning and forgetting, simply fitting sums of exponentials is expected to yield progressively better fits for the simple reasons that any function may be approximated by such a sum, which is related to the Laplace transformation.

A model of forgetting and amnesia developed by our group also yields a summed exponential function, but with a different parameterization. This so called Memory Chain Model assumes that a memory passes through several neural processes or stores, from short-term to very long-term memory. While a memory is (exponentially) declining in intensity in Store 1 (e.g., the hippocampus), its contents is steadily transferred to a Store 2 (e.g., the neocortex) from which it will decline at a lower rate. We still have two exponentially declining stores, as in the summed exponential function above, but they are linked by a memory consolidation process. The decay rates in Store 1 and Store 2 are given by a 1 and a 2, respectively. The initial strength of the memory traces in Store 1 are given by μ 1 and the rate of consolidating the contents of Store 1 to Store 2 is given by μ 2. In experiments with dementia patients and experimental animals, Store 1 may typically be identified with the hippocampus and Store 2 with the neocortex. Lesioning Store 1, will produce a retrograde amnesia gradient that can be modeled by the Memory Chain Model simultaneously with the forgetting gradient of healthy controls.

The Memory Chain Model (MCM) equation for type of savings studied here is given by The MCM function has the same number of parameters but they are arranged differently. The proof that this equation is a mathematical formalization of the memory consolidation process can be found elsewhere. As can be seen in Table 5, the summed exponential and the MCM function give exactly the same fits, though the parameters differ. The gain of using the MCM function lies primarily in the fact that its parameters can be interpreted more clearly, that it is associated with a type of consolidation mechanism, and that also explains other types of data than the savings function. The MCM function assumes a neural system consolidation mechanism that has been dubbed the 'Standard Consolidation Theory', where the latter authors propose a different theory, the so called Multiple Trace Theory of consolidation. It is here not our goal to evaluate the merits of these theories; we have reviewed these and other theories of consolidation elsewhere. We merely want to apply the MCM equation to these four savings curves and evaluate the goodness-of-fit, viewing it as a conceptual improvement of the summed exponential.

If we compare the MCM equation or summed exponential function to the other functions, this only makes sense if we rely on the AIC, which takes into account the varying number of parameters. The MCM function (or double exponential function) fits two of the four curves (Mack and Dros) better than Ebbinghaus’ own equations from 1880 and 1885, it gives about the same fit on the Seitz data, and it does much worse on Ebbinghaus' own data. The average AIC is 0.5 less than the average AIC for the classic Ebbinghaus functions, which—though it indicates a better fit—may not be considered a meaningful difference; a difference of 2 is considered 'meaningful'. We also fitted a single exponential with only two parameters but this fared far much worse on all data sets, including Ebbinghaus’ data (also see).

Summarizing, Ebbinghaus' data fit his own equations and the power function best. The AIC indicates that on average the MCM equation (or summed exponential function) is on average better than all equations considered thus far, where the difference with the power function is 1.5. The difference with Ebbinghaus' own equations is only 0.5 but this is partially because his own data have an exceptionally good fit on his own equations, with a very low AIC of about -30.5 (and an extremely high 99.8% variance explained). It is likely, however, that Ebbinghaus actively searched for an equation that achieves such an exceptional fit, which in his eyes was no more than a 'summary' of the forgetting curve (see citation above). This also explains why he has no problems substituting a 'logarithmic' equations for the earlier 'power' equation: it shows a slightly better fit.

Fitting data is always done with a purpose. Ebbinghaus achieved a concise summary of his forgetting data, the power function is a parsimonious description of the forgetting function that shows a good or at least adequate fit in many types of forgetting experiments, and the MCM equation attempts to capture the shape of a hypothetical consolidation process in the brain albeit at the expense of additional parameters. Taking into account these extra parameters, however, does not give a worse fit on the AIC and approaches a meaningful improvement over the power function.

The 24-hour point in Ebbinghaus' forgetting curve

When looking at the shapes of the four curves in Fig 2, savings after 1 day (or 2 days) seems higher than expected. Ebbinghaus notices this as well but merely writes it off as a discrepancy from his fitted curve (see above) that still falls within the error bars (p. 62). He clearly did not trust this data point because in his text from 1885 he reports that he later had replicated this 24 hour data point. The replicated data for this point gave a very similar score, so we must consider it a valid measurement. Jenkins and Dallenbach, however, interpreted the discrepancy as an effect of sleep, which motivated them to investigate this closer in an experiment on the effect of sleep on forgetting. They also refer to the forgetting curve by Radossawljewitsch, who also found higher savings after both 1 and 2 days (0.689 and 0.609, resp.) compared with after 8 hours (0.474). To them, this is suggestive of a very strong effect of sleep, but Finkenbinder points out that Radossawljewitsch's 8-hour data point may not be reliable, because these lists were all relearned during the afternoon, when there was less rapid learning resulting in fewer savings. He, therefore, suggests using a corrected savings score at 8 hours of 0.66, which is not unreasonable given that Ebbinghaus also corrected his savings scores for time-of-day effects, in some cases up to 13%. Even if savings would be 0.66 at 8 hours, however, the 1 day savings score is still higher than the 8 hour score and the 2 day savings is still higher than what one would expect.

Using free recall and retention up to 8 hours, the seminal study by Jenkins and Dallenbach yielded a positive effect of sleep on recall. This effect has since been replicated many times, for example in recent studies on the effects of different sleep stages on both procedural and declarative memory. Whereas the older studies from the 1970s and before typically confound the sleep manipulation with time-of-day effects or fatigue, this is no longer the case in the recent studies, so that there is now very strong evidence that sleep does indeed have an effect on memory independent of the effects of, say, rest or lack of interference. In some of the sleep-memory experiments cited above, we even see a temporary increase in the forgetting curve, where subjects score better than after learning in the days following sleep, but not if they skipped the night of sleep after learning. This result—and other studies—suggests that the first night of sleep after learning has a particularly important effect on memory that may continue to evolve for several days afterwards. Such an effect may also be observed in savings curve by Mack and to some extent in the Seitz curve, both show a tendency to increase in savings score for two days following learning.

Given that we can trace the history of research on the effects of sleep on memory to the 24 hour point of Ebbinghaus' forgetting curve, we think it is interesting to evaluate this data point more formally. If we can establish the jump in the curve more formally, it will make a stronger case that the 'true shape' of the long-term forgetting curve has a jump in at 24 hours (or perhaps right after the subject has slept), although we may not conclude from this that the local increase is due to sleep per se, which would require more research is necessary for that.

If we first informally inspect the data points shown in Fig 2 and compare them with the fitted power function, we see relatively less forgetting at either day 1, day 2, or both. In all four panels, at least one of these points is above the fitted power function curve at a distance of at least one standard error. The same is true for the fitted curves of the summed exponential, Ebbinghaus’ 1880 ‘power’ function and his 1885 ‘logarithmic’ function (not shown here). We also see this effect for the Memory Chain Model curve in Fig 3, though somewhat less pronounced (in the Seitz panel, the fitted curve crosses the error bars at 1 and 2 days). The reason for this is that the Memory Chain Model already incorporates the effects of a hypothetical consolidation process. In short, we observe that there is seems to be a memory ‘boost’ in the classic Ebbinghaus’ forgetting curve and its replications. The current body of research on sleep and memory would predict such a boost after one or two nights and attribute it to sleep, though for this particular type of experiments this has to established more firmly in further experiments. Whatever its cause, we can better quantify the visually observed boost by including it in the equations fitted. We, therefore, made a variant of the power function that differs only in the addition of a constant boost factor to the savings of the retention intervals of 1 day and higher. This power function with boost is also plotted in Fig 2.

The results are mixed, though on average they suggest a trend towards improvement with a boost parameter. The original power function had a average AIC of -22, a sum-of-squared-differences (SSD) of 0.00932 and explained 98.7% of the variance (see Table 5), whereas adding the boost reduces the AIC to -23.6, the SSD to 0.00628 and increases the average variance explained to 99.1%, putting its goodness-of-fit on a par with the MCM (or summed exponential) function. A difference in average AIC of 1.6 may perhaps be called a trend towards a meaningful difference, though there were large differences between the individual subjects. The boost parameter in Table 5 shows the size of the upward jump after 24 hours. We see that for Ebbinghaus, this jump is small (0.030), whereas for Mack it is quite substantial (0.131). The Dros data show no evidence for a boost but these fits are probably influenced strongly by the very low 31 day data point.