We report for the first time that the way numerosity of recent events (specifically cellphone use) is recalled conforms to the Weber-Fechner law. In other words, there appears to be a mental logarithmic scale consistent with that found in the estimation of observed numerosity. This provides a new direction for understanding human magnitude estimation, as, rather than a mental representation of an environmental stimulus, it is the outcome of an internally generated (ie, recalled) stimulus.
Let us examine the evidence for this. Texting estimation data were very unevenly distributed, but with strong similarities in each order. First, the majority of estimates fell in the lower 31.6% of each order, possibly related to a mental logarithmic scale, as well as consistent with the estimations accurately representing the log normal or exponential distribution of the billed data. Second, there was a strong rounding effect; data were almost exclusively rounded in the upper 68.5% of each order, reflecting a logarithmic mental estimation scale. This is clearly visible in and . Further, the pattern of leading digits in the estimation data did not match that of the leading digits of random numbers drawn from this distribution. This only occurred (in the first digit after the decimal) after log transformation.
If estimation were carried out linearly for data which, overall, formed a log normal distribution, then we might expect more than half of all estimates to be evenly distributed through the lower 31.6% of the full range 1–1000, with the balance being evenly distributed through the remainder. This is what we saw in the billed data ().
The neuroscience literature describes a numerical magnitude effect: ‘discrimination of two numerosities of a given numerical distance becomes more difficult as the absolute values of the two sets get higher’ (ref.13
, p.4). Our data show that this applies within each order
. There needed to be an appreciable-imagined difference (stimulus) in numerosity on a log scale for it to be acknowledged in the resulting estimate. This is evidenced in the rounding effect within orders. Testing of visual estimation of numerosity has generally been limited to the first two orders of magnitude (1–9 and 10–99) so rounding appears only to have been commented upon by Krueger14
who reported 89% of estimates being rounded to a last digit of 5 or 0 when participants were shown arrays of Xs numbering 25–300.
If the mental estimation process were linear we would expect all leading digits to be equally represented, but their distribution closely resembled the intervals of a logarithmic scale. This also applied to the distribution of leading digits in recalled cordless call data. Integers with single non-zero digits were vastly over-represented (). Looking to the remaining digits in estimations, these were also far from being evenly distributed. The rounding effect was so strong that estimates in the top 65% of the second order were almost exclusively rounded to tens or fives (), and in the third order to hundreds or fifties. These effects are all consistent with estimation on a logarithmic mental scale.
Most of the phenomena we have reported are consistent with the estimation of observed
numerosity, but estimation of recalled
numbers of events over recent months has not previously been reported in the cognitive science literature. Estimation of observed numerosity is one of the several foci of magnitude estimation. When these ratio-based estimations are log-transformed they become linear.14
This mental process reflects logarithmically compressed number-neurons operating like a slide-rule by ensuring accuracy proportional to the size of the numbers being processed,7
thus maximising neuronal efficiency. In humans, this neuronal activity has been traced to the horizontal segment of the intraparietal sulcus.15
It has been suggested that this logarithmic method of weighing the comparative value to ascribe to a large numerosity may be ‘deeply embedded’ as the default method in humans,8
a prelinguistic in-born approach to number.16
The logarithmic mental process has been shown to result in increasing numerosity progressively being assigned proportionally lower comparative values, with high numbers commonly underestimated.6
This applied to our data, that of Inyang et al4
and to the CEFALO study.17
Hollingsworth et al9
reported the same tendency in a psychological test resulting in mean overestimation of an array of <130 dots and underestimation of large arrays up to 650 dots. Several cellphone studies have found the opposite tendency, with high values overestimated. Since much of the literature on magnitude estimation has adult participants, we doubt that this ‘reverse’ trend is a feature of age, but suggest that it may result from the elapsed period since that being recalled, as cellphone studies often ask participants to recall their phone use over periods up to 10 years. The Interphone study reported greater over-reporting in this situation.2
Psychological studies of observed numerosity-estimation have resulted in the hypothesis of a consistent variance of the residuals once the data are log-transformed6
thus providing a common probabilistic range at any given point on the line. We found this applied to recalled numerosity, as has been reported in other cellphone studies.1
However, recalled estimation has an important difference from the visual estimation process as the variance of the residuals in recall estimation reported in this and other cellphone studies is routinely much wider than when numerosity is observed. It appears that this is a function of recall, introducing greater random error.
Implications for epidemiology
Our findings have implications for other cellphone studies and other epidemiological studies involving recalled numbers of events. A high proportion of rounded estimates could affect categorisation. Specifically, if quantile-cuts occur at round numbers (particularly those starting with 1 and 5), there may be many same-value digits. Forming cut-points before or after these would form irregularly sized quantiles. Arbitrarily allotting same values to different quantiles is not viable as it would return different results when analysed against other variables depending on how the dataset was ordered prior to categorisation. This would be true independent of sample size.
The mental process of estimation affects how given ranges of data should be averaged. The geometric rather than arithmetic mean is likely to align better with single value estimates as this is equivalent to averaging the logarithms of the values and back transforming. It would thus avoid introducing bias which would occur by mixing specific estimates made on a logarithmic scale with the arithmetic mean of a range, which is appropriate for a linear process. The geometric mean would also be better when imputing missing central data between two provided estimates. Typically in cellphone research, these situations have been allocated the arithmetic mean or median.19–21
An example from our study of the possible outcome being strongly affected is when the range is wide and starts at a low number, for instance, 1–70. Here the arithmetic mean is 35.5, while the geometric mean is 8.37. A quarter of all weekly text estimates were provided as a range. Recording their geometric means instead of arithmetic means resulted in the mean of all the data being 10% lower.
There is some evidence from the cognitive neuroscience literature that it may be possible to reduce recall inaccuracy by providing a calibration point.6
Variability in our study was smaller where participants knew the monthly maximum available on their account compared to those with no account. This is also applied to two Interphone studies1
where location questions may have acted as contextual prompts.18
Variability was considerably broader in the MoRPhEUS study4
where no prompts were given, and in the UK Interphone validation study3
that was conducted by postal questionnaire. The possible beneficial influence of a calibration point suggests that supplying participants in case–control studies with an accurate record of their recent cellphone use may allow them to better judge their earlier levels of use. This could be tested in further research.
In summary, recalled numerosity of recent events appears to be processed in the brain in a very similar way as is observed numerosity. This finding extends the cognitive science literature on estimation of numerical quantity, and lends some predictability to epidemiological studies involving recalled numerosity: Numerical recall estimated on a logarithmic mental scale means that as numerosity increases, estimations reduce comparatively. This trend from overestimation to better estimation or underestimation in recall of the extent of recent events is of great importance for epidemiology, as is the large variance in the residuals of recalled data. If these aspects are not allowed for during analysis, it may introduce error or bias, leading to overestimation or underestimation of relative risk for those with extremes of cellphone use. Bias or error may also be introduced as the high incidence of rounding could affect categorisation.
We offer some solutions. First, the rounding effect and a logarithmic mental process imply that recalled numbers should be log-transformed prior to analysis. This is usual, but our study provides empirical justification. Second, recalled number ranges and imputed missed data between given estimates are better represented by the geometric rather than arithmetic mean. Third, informing study participants of their correct current level of use over a short period may improve estimation of use over somewhat longer periods. These steps should help reduce random and systematic bias in cellphone studies, but we anticipate that they will also be applicable to other research which relies on recalled estimations of recent numbers of events.