Intuitions about sample size: the empirical law of large numbers

According to Jacob Bernoulli, even the ‘stupidest man’ knows that the larger one's sample of observations, the more confidence one can have in being close to the truth about the phenomenon observed. Two-and-a-half centuries later, psychologists empirically tested people's intuitions about sample size. One group of such studies found participants attentive to sample size; another found participants ignoring it. We suggest an explanation for a substantial part of these inconsistent findings. We propose the hypothesis that human intuition conforms to the ‘empirical law of large numbers’ and distinguish between two kinds of tasks–one that can be solved by this intuition (frequency distributions) and one for which it is not sufficient (sampling distributions). A review of the literature reveals that this distinction can explain a substantial part of the apparently inconsistent results. © 1997 John Wiley & Sons, Ltd.     

A comparison of stimulus set size on tact training for children with autism spectrum disorder

Previous studies on skill acquisition have taught targets in stimulus sets composed of different numbers of stimuli. Although the rationale for selection of a stimulus set size is not clear, the number of target stimuli trained within a set is a treatment decision for which there is limited empirical support. The current investigation compared the efficiency of tact training in 4 stimulus set sizes, each of which included 12 stimuli grouped into (a) 4 sets of 3 stimuli, (b) 3 sets of 4 stimuli, (c) 2 sets of 6 stimuli, and (d) 1 set of 12 stimuli. Results of all 4 participants with autism spectrum disorder show tact training with larger (i.e., 6 and 12) stimulus set sizes was more efficient than training with smaller (i.e., 3 and 4) stimulus set sizes.     

Children learn words easier when they are interested in the category to which the word belongs

The overall pattern of vocabulary development is relatively similar across children learning different languages. However, there are considerable differences in the words known to individual children. Historically, this variability has been explained in terms of differences in the input. Here, we examine the alternate possibility that children's individual interest in specific natural categories shapes the words they are likely to learn – a child who is more interested in animals will learn a new animal name easier relative to a new vehicle name. Two‐year‐old German‐learning children (N = 39) were exposed to four novel word–object associations for objects from four different categories. Prior to the word learning task, we measured their interest in the categories that the objects belonged to. Our measure was pupillary change following exposure to familiar objects from these four categories, with increased pupillary change interpreted as increased interest in that category. Children showed more robust learning of word–object associations from categories they were more interested in relative to categories they were less interested in. We further found that interest in the novel objects themselves influenced learning, with distinct influences of both category interest and object interest on learning. These results suggest that children's interest in different natural categories shapes their word learning. This provides evidence for the strikingly intuitive possibility that a child who is more interested in animals will learn novel animal names easier than a child who is more interested in vehicles.     

Orthographic neighbors and visual word recognition

Two lexical decision experiments, using words that were selected and closely matched on several criteria associated with lexical access, provide evidence of facilitatory effects of orthographic neighborhood size and no significant evidence of inhibitory effects of orthographic neighborhood frequency on lexical access. The words used in Experiment 1 had few neighbors that were higher in frequency. In Experiment 2, the words employed had several neighbors that were higher in frequency. Both experiments showed that words possessing few neighbors evoked slower responses than those possessing many neighbors. Also, in both experiments, neighborhood size effects occurred even though words from large neighborhoods had more potentially interfering higher-frequency neighbors than words from small neighborhoods.     

Is the link between forebrain size and feeding innovations caused by confounding variables? A study of Australian and North American birds

The short notes of ornithology journals feature new and unusual feeding behaviours, which, when systematically collated, could provide a quantitative estimate of behavioural flexibility in different bird groups. Previous studies suggest that taxonomic variation in the frequency of new behaviours (innovations) is correlated with variation in relative forebrain size. Recent work on primates shows, however, that observer bias can affect innovation frequency. We assess this possibility in birds via three estimates in North America and Australia: the number of full-length papers in academic journals, the frequency of photographs in birding magazines and a questionnaire on reporting bias given to ornithologists at a meeting. We also look at sampling effects due to single journal sources by doing a split-half analysis of our North American database (The Wilson Bulletin vs. six other journals) and adding three new Australian journals to the one we had used previously. In multiple regressions that also included species number per taxon, none of the potential biases could account for the correlation between forebrain size and innovation frequency. Species number was the best predictor of full-length paper frequency, which was the best predictor of photograph numbers. Ornithologists are not preferentially interested in innovative, large-brained taxa, suggesting that the correlation between innovations and neural substrate size is not a spurious effect of the biases examined here. Accepted after revision: 27 July 2001 Electronic Publication     

Skull size and intelligence, and King Robert Bruce's IQ

An estimate of someone's IQ is a potentially informative personal datum. This study examines the association between external skull measurements and IQ scores, and uses the resulting regression equation to provide an estimate of the IQ of King Robert I of Scotland (Robert Bruce, 1274–1329). Participants were 48 relatively healthy Caucasian men (age 71–76 years) resident in Scotland. Using magnetic resonance imaging data, intracranial volume estimated from external skull length and width correlated greater than .5 (p < .001) with measured intracranial area, which correlates very highly with brain volume. IQ scores estimated from the National Adult Reading Test (NART) correlated .56 (p < .001) with measured intracranial area, and .49 (p < .01) with estimated intracranial volume based on external skull width and length. The partial correlation coefficient of this latter association was .25 (p = .09) after adjustment for measured intracranial area. Thus, actual intracranial area accounts for about 74% of the variance shared by NART and estimated intracranial volume. A cast of the skull of Robert Bruce was measured and its intracranial volume estimated. A regression equation between IQ and estimated intracranial volume, based on data from the 48 subjects, estimated the IQ of Robert Bruce at about 128 (95% confidence interval 106 to  130), i.e. almost two standard deviations above the mean. NART scores show a ceiling effect, so this estimated IQ might be an underestimate. Robert Bruce's estimated high IQ is congruent with his military, political and other intellectual achievements.     

A Unified Approach to Power Calculation and Sample Size Determination for Random Regression Models

The underlying statistical models for multiple regression analysis are typically attributed to two types of modeling: fixed and random. The procedures for calculating power and sample size under the fixed regression models are well known. However, the literature on random regression models is limited and has been confined to the case of all variables having a joint multivariate normal distribution. This paper presents a unified approach to determining power and sample size for random regression models with arbitrary distribution configurations for explanatory variables. Numerical examples are provided to illustrate the usefulness of the proposed method and Monte Carlo simulation studies are also conducted to assess the accuracy. The results show that the proposed method performs well for various model specifications and explanatory variable distributions. The author would like to thank the editor, the associate editor, and the referees for drawing attention to pertinent references that led to improved presentation. This research was partially supported by National Science Council grant NSC-94-2118-M-009-004.     

The measure of a nation: The USDA and the rise of survey methodology

Survey research has played a major role in American social science. An outgrowth of efforts by the United States Department of Agriculture in the 1930s, the Division of Program Surveys (DPS) played an important role in the development of survey methodology. The DPS was headed by the ambitious and entrepreneurial Rensis Likert, populated by young and talented social scientists getting their first practical experience, and fed by the needs of the US government fighting World War II. The DPS innovations included open-ended interviewing and area probability sampling methodology as illustrated in the War Bond studies and the Master Sample of Agriculture. This paper examines the creation of the DPS, its work, and its legacy.     

European starlings are capable of discriminating subtle size asymmetries in paired stimuli

Small deviations from bilateral symmetry (fluctuating asymmetries) are cues to fitness differences in some animals. Therefore, researchers have considered whether animals use these small asymmetries as visual cues to determine appropriate behavioral responses (e.g., mate preferences). However, there have been few systematic studies of animals' abilities to visually discriminate such minor asymmetries. If the asymmetries cannot be discriminated, fluctuating asymmetry can not be a visual cue. Here, we report an investigation of European starlings' (Sturnus vulgaris) abilities to discriminate small size asymmetries. We trained starlings, through operant conditioning in a free-flight aviary, to discriminate achromatic, symmetric paired stimuli from size-matched asymmetric stimuli. By starting the learning process with a large asymmetry and progressing through sequential trials of decreasing asymmetry, we elucidated a behavioral limit to asymmetry discrimination. We found that starlings are capable of discriminating a 10% size asymmetry. There was weaker evidence for discrimination of 5% asymmetry but no evidence for signal discrimination at 2.5% size asymmetry. This level of asymmetry discrimination suggests that many size asymmetry cues in nature can be discriminated by birds. At each level of asymmetry discrimination, we also tested whether starlings could generalize their learned symmetry preference to unreinforced novel images. Consistent with previous findings, we found that starlings could generalize their symmetry preferences.