Article Text
Abstract
Objective Selfreported recall data are often used in wireless phone epidemiological studies, which in turn are used to indicate relative risk of health outcomes from extended radiofrequency exposure. We sought to explain features commonly observed in wireless phone recall data and to improve analytical procedures.
Setting Wellington Region, New Zealand.
Participants Each of the 16 schools selected a year 7 and/or 8 class to participate, providing a representative regional sample based on socioeconomic school ratings, school type and urban/rural balance. There was an 85% participation rate (N=373).
Main outcome measures Planned: the distribution of participants’ estimated extent of SMStexting and cordless phone calls, and the extent of rounding to a final zero or five within the full set of recall data and within each order of magnitude. Unplanned: the distribution of the leading digits of these raw data, compared with that of billed data in each order of magnitude.
Results The nature and extent of numberrounding, and the distribution of data across each order in recall data indicated a logarithmic (ratiobased) mental process for assigning values. Responses became less specific as the leadingdigit increased from 1 to 9, and 69% of responses for weekly texts sent were rounded by participants to a single nonzero digit (eg, 2, 20 and 200).
Conclusions Adolescents’ estimation of their cellphone use indicated that it was performed on a mental logarithmic scale. This is the first time this phenomenon has been observed in the estimation of recalled, as opposed to observed, numerical quantities. Our findings provide empirical justification for logtransforming data for analysis. We recommend the use of the geometric rather than arithmetic mean when a recalled numerical range is provided. A point of calibration may improve recall.
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Article summary
Article focus:

Selfreported cellphone and cordless phone use is skewed and has a very large estimation error. This is important because estimated data are often all that is available for epidemiological studies.

We explored the logarithmic (ratiobased) mental process evident in the distribution of participants’ estimated data.
Key messages

We show for the first time that the mental process in recalling the extent of recently sent text messages mirrors in most respects that used when estimating a number of observed objects, as described in the literature on estimation magnitude.

This mental process is not linear (except, perhaps, for numbers <10), and indicates the use of a logarithmic mental numberline to recall numbers of events (in this case texts sent and cordless phone calls made).

We suggest the statistical methods to reduce misclassification and to improve estimation accuracy.
Strengths and limitations of this study

There could have been an influence on estimation by other participants, although they were asked not to ‘compare notes’; frank estimation was encouraged by assurance of confidentiality.

The findings and suggested ways of addressing them should be applicable to other medical studies using recall data of numbers of recent events.

The literature on estimation of numerosity of observed objects provides wellresearched support for the current findings.
Introduction
Using recalled cellphone data is problematic for case–control studies which are exploring a possible relationship between wireless phone radiation and health effects. This is because studies that have used this approach1–5 have routinely reported recall data as skewed and having a large estimation error. Rather than trying to explain this, there have been calls for caution in interpretation4 and doubt expressed about the usefulness of recall data.5
In 2009, we ran a survey of New Zealand adolescents’ wireless phone use. We also found recalled use to be positively skewed, with the distribution of recalled texts sent being log normal. We had asked participants to estimate various aspects of their cordless phone and cellphone use, including the number of SMS texts they sent daily, or weekly or monthly. They could estimate a range, if preferred. Many students chose to estimate the extent of their texting for all three periods. This led, serendipitously, to the analysis presented in this paper.
During data entry we noticed a common tendency for individuals’ weekly and monthly estimates to be very low in comparison with their daily texting estimates. For instance, one participant estimated 10 daily, 35 weekly and 150 monthly, and another recalled 20 daily, 50 weekly and 150 monthly. At first, we thought this might reflect poor arithmetic skills, but one teacher informed us that the class selected for participation was a topstream one: students’ science grades all exceeded 85%. Despite this, they showed the same estimation tendency. Consequently, we explored the literature on magnitude estimation.
Magnitude estimation is a basic area of research, currently considered in the field of neuroscience. In 1834, Weber observed what change in weight was needed for the person lifting it to notice. He realised that ‘the extent to which two stimuli can be discriminated is determined by their ratio’.6 Fechner developed this theory, ‘postulating that the external stimulus is scaled into a logarithmic internal representation of sensation’.7 These concepts came to be called the WeberFechner law whereby linear change in sensation (S) is proportional to the logarithm of the stimulus magnitude (m): S=k×log(m), where k is a constant. It has been shown to apply generally to the way our senses perceive environmental stimuli (eg, light intensity, volume and length). Over the last few decades, research has suggested that a logarithmic mental number line is also consistent with the estimation of observed numerical quantity (referred to as the numerosity).8 ,9
Here, we explore our data for indicators of the mental process behind estimating a number of past events—specifically, the extent of cellphone texting and cordless calls made weekly. We checked whether the consistent, but unexpected, tendencies we had observed in participants’ texting estimates were explained by the WeberFechner law. We sought to find explanations for commonly observed features of recall and use these to improve analytical procedure in epidemiological risk analyses which use numerical recall data. Results based on such data provide indications of public health risk from environmental exposures or medical interventions, therefore it is important to minimise bias in the analytical methods and resulting inferences.
Methods
The methodology evolved during examination of the data. The analysis was undertaken using data from our crosssectional survey of New Zealand adolescents’ wireless phone habits. The study population has been described previously.10 Briefly, it was representative of the region for school type and decile (socioeconomic ranking of schools by their area), and included the capital city through to rural areas. Years 7 and 8 students (N=373; 207 male, 165 female and 1 transgender) from around the region participated. The median age was 12.3 years. There was an 85% participation rate. Ethical approval was given by the Victoria University of Wellington Human Ethics Committee. Informed consent was obtained from principals of participating schools and parents of participating students. Students could choose to opt out.
Primaryindependent variables
We examined the following variables: recalled and billed weekly texts sent, pairs of recalled and billed weekly texts sent from those on 500 and 2000/month plans, and the estimated number of cordless phone calls made weekly.
Participants retrieved their remaining text balance on their prepaid monthly plan from their provider. This allowed us to calculate their daily actual use (‘billed’) pro rata by dividing the used portion by the number of days since billing, and multiplying this by 7 for the weekly rate.
Statistical analyses
Distribution of the estimation data
We considered two aspects of the distribution of the estimation data. First, that of the estimates themselves, overall and within each order of magnitude, which could reasonably be expected to reflect the distribution of actual use. Second, that of the leading digits, which we would expect to be randomly and uniformly distributed if the mental processes involved in recollection were linear. Analyses were undertaken using the statistical programmes SPSS V.17.0.1, Chicago, Illinois, 2008, and Microsoft Excel, 2010.
The distribution of estimated and billed weekly texts sent was examined with cumulative distribution plots using raw and logtransformed data. These raw text data were plotted on threedimensional column graphs (for the orders 1–9, 10–99 and 100–999; second order of each at figure 1). This was to enable us to examine the nature and extent of rounding within each order of magnitude, and the distribution of data across each order. We calculated the extent of rounding to fives/tens and fifties/hundreds in the second and third orders of magnitude, respectively. The percentage of datapoints in the lower 31.6% of each order of magnitude was calculated, 31.6% being the halfway point on a logarithmic scale for 10 (base 10; geometric mean (1,10)=√10=3.16). Regression plots were used to assess ‘daily’ versus ‘weekly’ and ‘billed’ versus ‘estimated’ texts sent. We checked whether there was a tendency towards overestimation or underestimation with increasing numerosity (in the texting data) by regressing the difference of the logrecalled and logbilled against the logbilled. The explanation for this variation to the Bland and Altman approach11 is given elsewhere.12
We assigned the geometric mean to responses given as a range (explanation below); these were included in overall distribution reporting but excluded from digit analysis in this paper as we focused on specific estimates when exploring the mental process of estimation.
Valid zeros were included in reporting the overall distributions, but not in the calculations of mean and SD of the logtransformed data.
Distribution of the first digits in estimation data
We assessed firstdigit distribution in estimates of weekly texts sent and cordless phone calls made. For comparison, we did the same for a set of random numbers drawn from the same distribution. We began by removing all data given as a range, and all estimates of zero. We sorted those remaining into nine groups, one for each digit from 1 to 9. Each was then allocated into two groups: those with only a single nonzero leading digit (eg, 2, 20 and 200) and the remaining estimates starting with that digit (in the example case 2). These were displayed as stacked columns, with each twopart column representing the percentage of estimates starting with that digit (figure 2).
Results
Descriptive statistics
At least one cellphone was owned by 285 (76%) of participants, while 331 (89%) currently used one. Most participants had a cordless phone at home which they used (341, 91%). We retrieved paired estimated and billed texting data from 108 participants (38% of cellphone owners). Other relevant descriptive statistics can be seen in table 1.
Overall distribution of estimation data
Recalled estimates of recent texts sent were right skewed. The variance of estimates increased by a fixed ratio with increasing estimated numerosity. Once the data were log transformed, the regression of estimated dailytoweekly texts became linear (Pearson's r 0.91 p<0.01; figure 3), showing a systematic tendency to underestimate use over a week compared to that estimated for a day. The log estimated to billed texts (Pearson's r 0.78 p<0.01) revealed a large, but homogeneous, variance of the residuals of logtolog regression (random error).
Logtransformed data from all those who sent texts followed a normal distribution (not shown), while the influence of a plan with a known prepaid quantity of texts monthly (500 or 2000) appeared to have a calibrating effect on daily and weekly estimates. This was evident in each plan's data, which had a distribution closer to exponential, that is, where μ was the population mean use, estimated by the sample mean. The mean of estimated texts sent weekly for the 2000/month plan fell within the 95% CI of four times that of those with the 500/month plan.
Two types of systematic error existed in recall. The first resulted in a trend significantly different from zero, moving from overestimation by those who sent few texts towards underestimation by those who sent many (figure 4). The second systematic error was apparent when comparing recalled texts sent over different periods (figure 3). The ratios of individual recall (daily:weekly and daily:monthly) were both only a little over half that expected (0.58 and 0.54), while that of weekly:monthly was 0.90. This applied, both between and within participants, in data which ranged from 0 to >1000.
Distribution of estimated and billed texts within each order
About half or more of participants’ estimates fell in the lower 31.6% of each order of magnitude (table 1). This represents the halfway point on a logarithmic scale. The billed data was homogeneously spread (illustrated for the second order of each at figure 1). The second order of magnitude (10–99) is most relevant for comparison as there were no outside influences on its distribution.
Distribution of digits and rounding effect
The leading digits of texting and cordless phonecall estimation data were distributed very unevenly, with proportions of each digit from 1 to 9 resembling those of the intervals on a log scale (figure 2A).
There were several rounding effects. Responses became less specific as numerosity increased within each order of magnitude, and as the leading digit increased from 1 to 9 (figure 2B,C). Distribution of leading digits showed that 123 (68%) of weekly texting responses were rounded by participants to a single nonzero digit, as were 158 (86%) of cordless call responses.
There was an additional rounding effect to final digits of 5 and 50 in the upper 68.5% of the second and third orders, respectively; these are visible as bright blue columns in figure 1. Only three texting responses (5%) greater than 35 or 350 in the second and third orders, respectively, were not rounded thus, being 68, 525 and 839; for cordless phone calls, only 2 (15%) were not rounded (being 53 and 59).
We can only hypothesise about the spike in estimates starting with five and seven (figure 2B). An excess of leading fives is probably related to the rounding effect and is shown in fewer leading fours. (The same applies to almost no nines in preference for rounding to a final zero.) However, the excess of leading sevens may reflect a more linear approach from a quarter of those estimating 1, 10 or 100 texts daily, whose weekly estimates were seven times greater. This explanation is supported by there being no excess of leading sevens for the estimated number of cordless phone calls weekly (figure 2C).
Although 50% of firstorder texting estimates fell in the lower 31.6% (table 1), we could not resolve whether the mental process for estimating very low numerosity is better described as linear or logarithmic, but speculate that the first order of magnitude is transitional towards the latter.
Discussion
We report for the first time that the way numerosity of recent events (specifically cellphone use) is recalled conforms to the WeberFechner 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 figures 1 and 2. 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 (figure 1).
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 appreciableimagined 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 nonzero digits were vastly overrepresented (figure 2). 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 (figure 1), 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 ratiobased estimations are logtransformed they become linear.14 This mental process reflects logarithmically compressed numberneurons operating like a sliderule 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 inborn 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 overreporting in this situation.2
Psychological studies of observed numerosityestimation have resulted in the hypothesis of a consistent variance of the residuals once the data are logtransformed6 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 ,18 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 quantilecuts occur at round numbers (particularly those starting with 1 and 5), there may be many samevalue digits. Forming cutpoints 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 ,22 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 logtransformed 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.
Acknowledgments
MR would like to thank the Dominion Post and Victoria University of Wellington for scholarship funding, and the principals and students for their willing and often enthusiastic participation. The authors would like to thank the reviewers for their overall impressions and specific suggestions which have helped with improving the paper considerably.
Footnotes

Contributors The study was designed by MR and MJA. MR conducted the survey, analysed the data and wrote the first draft of the paper. MR and ES analysed implications. All authors contributed to the paper and approved the final version for publication.

Competing interests MR and ES have no competing interests to declare. MA holds small parcels of shares in Telstra and SingTel which operate cell telephone networks in Australia.

Ethics approval Victoria University of Wellington Human Ethics Committee.

Provenance and peer review Not commissioned; externally peer reviewed.

Data sharing statement Further figures showing distribution of billed and estimated texting could be supplied on request, as could more details on ratios of daily:weekly estimations.
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