Decision policies minimizing risk in a multistage betting game

Considered in this paper is a decision task which has been employed to study multistage betting behavior. When the task commences, a decision maker (DM) is provided with some capital x (x > 0) which he is required to allocate over m (m > 1) mutually exclusive and collectively exhaustive alternatives, each of which occurs with probability p_i (p_i > 0, i = 1,…, m; Σ_i=1^mp_i = 1). If the amount y_i is allocated to alternative i (y_i ≥ 0, Σ_i=1^my_i = x) and alternative i obtains, DM's capital for the next stage of the game becomes y_ir_i, where r_i (r_i > 0) is the return per unit allocated to alternative i. The task consists of N stages.Defining risk in terms of the mean and variance of DM's bets, and assuming that the minimization of risk is DM's objective, decision policies satisfying this objective are derived in closed form and their testable properties are briefly discussed.     

Some counterexamples in multidimensional scaling

Given a set X with elements x, y,… which has a partial order < on the pairs of the Cartesian product X², one may seek a distance function ? on such pairs (x, y) which satisfies ?(x₁, y₁) < ?(x₂, y₂) precisely when (x₁, y₁) < (x₂, y₂), and even demand a metric space (X, ?) with some such compatible ? which has an isometric imbedding into a finite-dimensional Euclidean space or a separable Hilbert space. We exhibit here systems (X, <) which cannot meet the latter demand. The space of real m-tuples (ξ₁,…,ξ_m) with either the “city-block” norm Σ_i ∥ξ_i∥ or the “dominance” norm max_i, ∥ξ_i∥ cannot possibly become a subset of any finite-dimensional Euclidean space. The set of real sequences (ξ₁, ξ₂,…) with finitely many nonzero elements and the supremum norm sup_i, ∥ξ_i∥ cannot even become a subset of any separable Hilbert space.     

Generalized procrustes analysis

SupposeP _i ⁽ⁱ⁾ (i = 1, 2, ...,m, j = 1, 2, ...,n) give the locations ofmn points inp-dimensional space. Collectively these may be regarded asm configurations, or scalings, each ofn points inp-dimensions. The problem is investigated of translating, rotating, reflecting and scaling them configurations to minimize the goodness-of-fit criterion Σ _i=1 ^m Σ _i=1 ⁿ Δ²(P _j ⁽ⁱ⁾ G _i ), whereG _i is the centroid of them pointsP _i ⁽ⁱ⁾ (i = 1, 2, ...,m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special casem = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.     

The Geometry of Enhancement in Multiple Regression

In linear multiple regression, “enhancement” is said to occur when R ²=b′r>r′r, where b is a p×1 vector of standardized regression coefficients and r is a p×1 vector of correlations between a criterion y and a set of standardized regressors, x. When p=1 then b≡r and enhancement cannot occur. When p=2, for all full-rank R _xx≠I, R _xx=E[xx′]=V Λ V′ (where V Λ V′ denotes the eigen decomposition of R _xx; λ ₁>λ ₂), the set B₁:={b_i:R²=b_i￠r_i=r_i￠r_i;0 < R² ￡ 1}\boldsymbol{B}_{1}:=\{\boldsymbol{b}_{i}:R^{2}=\boldsymbol{b}_{i}'\boldsymbol{r}_{i}=\boldsymbol{r}_{i}'\boldsymbol{r}_{i};0R² ￡ 1;R²l_p ￡ r_i￠r_i < R²}0p≥3 (and λ ₁>λ ₂>⋯>λ _p ), both sets contain an uncountably infinite number of vectors. Geometrical arguments demonstrate that B ₁ occurs at the intersection of two hyper-ellipsoids in ℝ ^p . Equations are provided for populating the sets B ₁ and B ₂ and for demonstrating that maximum enhancement occurs when b is collinear with the eigenvector that is associated with λ _p (the smallest eigenvalue of the predictor correlation matrix). These equations are used to illustrate the logic and the underlying geometry of enhancement in population, multiple-regression models. R code for simulating population regression models that exhibit enhancement of any degree and any number of predictors is included in Appendices A and B.     

Specifically objective stochastic latency mechanisms

A two parameter additive model for psychological response latencies f_vi(t) = (θ_v + ?_i) exp(?(θ_v + ?_i)t) is suggested. Conditional inference procedures are derived for the psychometric latency model. If the model is considered as a theory for speed test construction the unrestricted comparability of measurements obtained under “speed” and “power” conditions is an attractive feature of the model. An extension of the parametric structure gives the “linear exponential model” f_vi(t) = (θ_v + Σ_ι^ma_iιη_ι) exp(?(θ_v + Σ_ι^ma_iιη_ι)t), where the time constants of the latencies are conceived as linear combinations of “basic parameters” corresponding to the experimental conditions of the latency situations. The linear exponential model includes as special cases some well known series latency mechanisms. Some generalizations of the distributional assumptions are discussed, especially the Weibull distributions. The estimation of parameters by conditional inference procedures and the evaluation of the fit of the model by conditional likelihood ratio tests is illustrated by means of the numerical data from a free recall experiment.     

Monotonic measures of agreement for ranked data

The τ_b and y statistics are interpreted as rank-monotonic coefficients of partial agreement. Using a method of transposition employed by Pearson's r_i intraclass correlation coefficient, the τ_bi and y_i intraclass coefficients of total monotonic agreement are created. Transpositional measures of agreement like τ_bi and τ_i measure the combined effects of cell and marginal disagreement which make them particularly suitable for reliability studies. The coefficients are also made applicable to K > 2 sets of ranks.     

Masking-level differences and the form of the psychometric function

Functions showing the relation between-the detectability and the energy of a signal were determined for various interaural phase conditions. The empirical relation between d′ and signal energy E is approximately d′=m(E/N₀)^k where m and k are constants for a particular function. The data show that k is fairly constant for a particular O and that m depends upon the interaural condition. That is, the various psychometrie functions for a given O all had the same form, independent of location. Therefore, the magnitude of the MLD=(10/k)log(m_i/m_r), where m_i is the constant for condition i, and m_r corresponds to a reference condition. Consequently, an MLD is relatively independent of the level of performance that is chosen for the determination of that MLD.     

An approximation of theK outN reliability of a test,and a scoring procedure for determining which items an examinee knows

Consider any scoring procedure for determining whether an examinee knows the answer to a test item. Letx _i = 1 if a correct decision is made about whether the examinee knows the ith item; otherwisex _i = 0. Thek out ofn reliability of a test isρ _k = Pr (Σx _i ≥k). That is,ρ _k is the probability of making at leastk correct decisions for a typical (randomly sampled) examinee. This paper proposes an approximation ofρ _k that can be estimated with an answer-until-correct test. The paper also suggests a scoring procedure that might be used whenρ _k is judged to be too small under a conventional scoring rule where it is decided an examinee knows if and only if the correct response is given.     

On linear classifications under varying choice probabilities

The paper deals with certain problems connected with the assumption that choice probabilities p_s(x, y) depend on the subject s. A set of postulates is given, which implies the existence of sequences of “classification standards”, i.e., sequences {z_j} such that whenever we have 0 < p_s0(x, z_i) < 1 for some s₀ and i, then p_s(z_i+k, x) = p_s(x, z_i?k) = 1 for all s, and k ≥ 1. Elements of any such sequence {z_j} can serve as boundaries between successive categories of classification based on the following rule: Assign x to jth category if you feel it is “to the right” of z_j and “to the left” of z_j+1. Under the condition stated above this rule is unambiguous, and the resulting classification has the property that every element is assigned to one of the two neighboring categories, regardless who performs the classification.Next, the postulates are enriched so as to imply the existence of “tightest” among such sequences {z_j}, hence leading to a classification with largest number of categories.     

Use of absolute and comparative performance feedback in absolute and comparative judgments and decisions

Which matters more—beliefs about absolute ability or ability relative to others? This study set out to compare the effects of such beliefs on satisfaction with performance, self-evaluations, and bets on future performance. In Experiment 1, undergraduate participants were told they had answered 20% correct, 80% correct, or were not given their scores on a practice test. Orthogonal to this manipulation, participants learned that their performance placed them in the 23rd percentile or 77th percentile, or they did not receive comparative feedback. Participants were then given a chance to place bets on two games—one in which they needed to get more than 50% right to double their money (absolute bet), and one in which they needed to beat more than 50% of other test-takers (comparative bet). Absolute feedback influenced comparative betting, particularly when no comparative feedback was available. Comparative feedback exerted weaker and inconsistent effects on absolute bets. Absolute feedback also had stronger (and more consistent) effects on satisfaction with performance and state self-esteem. Experiment 2 replicated these effects in a different university sample, and demonstrated that the effects emerge even when bets are placed after participants rate their satisfaction with their performance (although these ratings do not mediate the effect of feedback on bets). These findings suggest that information about one’s absolute standing on a dimension may be more influential than information about comparative standing, partially supporting a key tenet of Festinger’s [Festinger, L. (1954). A theory of social comparison processes. Human Relations, 7, 117–140.] theory of social comparison.     

An independent examination of the equivalence of the standard and digital administration formats of the Wechsler Intelligence Scale for Children-5th Edition

A new administration format for the Wechsler Intelligence Scale for Children-Fifth Edition (WISC-V; Wechsler, 2014) was introduced in 2016 on Q-interactive, Pearson's digital platform for test administration and scoring. The current study examined the measurement unit equivalence of the WISC-V standard and digital administration formats using counter-balanced administration of the 10 primary subtests to measure intellectual ability. The results indicated that correlations (r) between standard scores on subtests and composites administered in each format were generally moderate, with mean rs of 0.64 for subtests and 0.71 for composites after correction for attenuation, with the lowest rs for processing speed. Split-plot ANOVAs were conducted to examine within-subjects main effects for administration format and order and their interaction. The results of these analyses revealed significant main effects for format for the Full Scale IQ and Processing Speed composite scores, with small to medium effect sizes (ds > 0.40). These format effects largely stemmed from the non-equivalence of the Coding subtest, which is used to derive both composites. For Coding, the main effect for format was statistically significant, with a large effect size (d = 0.69). Statistically significant administration order by format interaction effects were also observed for a number of composites and subtests, with medium to large effect sizes (ƞ_ps² > 0.20). In each case, higher mean scores were observed when the WISC-V was administered first in digital format. Implications of these results for research and practice are discussed.     

The role of motion platform on postural instability and head vibration exposure at driving simulators

This paper explains the effect of a motion platform for driving simulators on postural instability and head vibration exposure. The sensed head level-vehicle (visual cues) level longitudinal and lateral accelerations (a_x,sensed = a_{x_head} and a_y,sensed = a_{y_head}, a_yv = a_{y_veh} and a_yv = a_{y_veh}) were saved by using a motion tracking sensor and a simulation software respectively. Then, associated vibration dose values (VDVs) were computed at head level during the driving sessions. Furthermore, the postural instabilities of the participants were measured as longitudinal and lateral subject body centre of pressure (X_CP and Y_CP, respectively) displacements just after each driving session via a balance platform. The results revealed that the optic-head inertial level longitudinal accelerations indicated a negative non-significant correlation (r = −.203, p = .154 > .05) for the static case, whereas the optic-head inertial longitudinal accelerations depicted a so small negative non-significant correlation (r = −.066, p = .643 > .05) that can be negligible for the dynamic condition. The X_CP for the dynamic case indicated a significant higher value than the static situation (t(47), p < .0001). The VDV_x for the dynamic case yielded a significant higher value than the static situation (U(47), p < .0001). The optic-head inertial lateral accelerations resulted a negative significant correlation (r = −.376, p = .007 < .05) for the static platform, whereas the optic-head inertial lateral accelerations showed a positive significant correlation (r = .418, p = .002 < .05) at dynamic platform condition. The VDV_y for the static case indicated a significant higher value rather than the dynamic situation (U(47), p < .0001). The Y_CP for the static case yielded significantly higher than the dynamic situation (t(47), p = .001 < 0.05).     

Visual evoked potential augmenting/reducing slopes in cats—1. Reliability as a function of flash intensity range

A cat model for the study of individual differences in augmenting/reducing of the visual evoked potential (VEP) is proposed. Multiple VEP sessions over an extended time period of 20 months permitted analysis of the stability of VEP amplitudes and augmenting/reducing slopes. The VEP was found to have a stable shape at a given intensity and served as a unique individual signature for each cat. Three intensity ranges (low, 0.2–15 lumens/m²/sterrad. sec; medium, 1–1501/m²/sr. s; and high, 20–1500 1/m²/sr.s), and four early components of the VEP (P1, N1, P2 and N2) were studied. A cortical inhibitory threshold was reached for most cats in the medium range of intensities. However, VEP amplitudes were variable, and only the N1 and P1 − N1 components at the medium range of intensities yielded reliable augmenting/reducing slopes. Averaging slopes of individual runs improved reliability, and the average N1 amplitude slopes for each of 3 sessions at the medium range of intensities correlated at r#62;0.9.     

A study of the spin-Hamiltonian parameters for Cr5+ at the rhombically-distorted tetrahedral P5+ site of Ca2PO4Cl crystal using a two-mechanism model

The complete high-order perturbation formulae of spin-Hamiltonian (SH) parameters (g factors g_i and hyperfine structure constants A_i , where i = x, y, z) containing contributions from both the crystal-field (CF) and charge-transfer (CT) mechanisms (the latter mechanism is neglected in the widely-used CF theory) are established for d¹ ions in rhombic tetrahedra. From these formulae, the SH parameters of Cr⁵⁺ ion at the rhombically-distorted tetrahedral P⁵⁺ site of Ca₂PO₄Cl crystal are calculated. The CF and CT energy levels used in the calculation are obtained from the optical spectra of the studied Ca₂PO₄Cl : Cr⁵⁺ crystal. The calculated results showed reasonable agreement with the experimental values. The signs of the hyperfine structure constants A_i and the relative importance of the CT mechanism to SH parameters are acquired from the calculations.     

Pseudocontingencies derived from categorically organized memory representations

Pseudocontingencies (PCs) allow for inferences about the contingency between two variables X and Y when the conditions for genuine contingency assessment are not met. Even when joint observations X _i and Y _i about the same reference objects i are not available or are detached in time or space, the correlation r(X _i ,Y _i ) is readily inferred from base rates. Inferred correlations are positive (negative) if X and Y base rates are skewed in the same (different) directions. Such PC inferences afford useful proxies for actually existing contingencies. While previous studies have focused on PCs due to environmental base rates, the present research highlights memory organization as a natural source of PC effects. When information about two attributes X and Y is represented in a hierarchically organized categorical memory code, as category-wise base rates p(X) and p(Y), the reconstruction of item-level information from category base rates will naturally produce PC effects. Three experiments support this contention. When the yes base rates of two respondents in four questionnaire subscales (categories) were correlated, recalled and predicted item-level responses were correlated in the same direction, even when the original responses to specific items within categories were correlated in the opposite direction.     

Monotone regression: Continuity and differentiability properties

Least-squares monotone regression has received considerable discussion and use. Consider the residual sum of squaresQ obtained from the least-squares monotone regression ofy _i onx _i . TreatingQ as a function of they _i , we prove that the gradient Q exists and is continuous everywhere, and is given by a simple formula. (We also discuss the gradient ofd=Q ^1/2.) These facts, which can be questioned (Louis Guttman, private communication), are important for the iterative numerical solution of models, such as some kinds of multidimensional scaling, in which monotone regression occurs as a subsidiary element, so that they _i and hence indirectlyQ are functions of other variables.     

Individual differences in spatial configuration learning predict the occurrence of intrusive memories

The dual-representation model of posttraumatic stress disorder (PTSD; Brewin, Gregory, Lipton, & Burgess, Psychological Review, 117, 210-232 2010) argues that intrusions occur when people fail to construct context-based representations during adverse experiences. The present study tested a specific prediction flowing from this model. In particular, we investigated whether the efficiency of temporal-lobe-based spatial configuration learning would account for individual differences in intrusive experiences and physiological reactivity in the laboratory. Participants (N = 82) completed the contextual cuing paradigm, which assesses spatial configuration learning that is believed to depend on associative encoding in the parahippocampus. They were then shown a trauma film. Afterward, startle responses were quantified during presentation of trauma reminder pictures versus unrelated neutral and emotional pictures. PTSD symptoms were recorded in the week following participation. Better configuration learning performance was associated with fewer perceptual intrusions, r = ?.33, p < .01, but was unrelated to physiological responses to trauma reminder images (ps > .46) and had no direct effect on intrusion-related distress and overall PTSD symptoms, rs > ?.12, ps > .29. However, configuration learning performance tended to be associated with reduced physiological responses to unrelated negative images, r = ?.20, p = .07. Thus, while spatial configuration learning appears to be unrelated to affective responding to trauma reminders, our overall findings support the idea that the context-based memory system helps to reduce intrusions.     

A strong developmental theory of field dependence-independence

Assume an X-linked gene in two alleles mediates performance on field dependent-independent tests such as the rod-and-frame test. Only the recessive gene with relative frequency q facilitates field independence. Other genotypes lead to field dependence. Under a simple genetic model, field dependence-independence may be viewed as outcomes of a discrete random variable B with field independent and dependent probabilities π_iq and 1 ? π_iq for men, and π_iq² and 1 ? π_iq² for women, respectively. The parameter π_i is a maturational ageindexed parameter, 0 < π_i ≤ 1, monotonically increasing with development until maturity when π_k = 1. Observations of performance are made on a random variable W of the form W = B + N, where N is normal in distribution independent of B; N represents a composite of influences including error. The model implies testable age-related between- and within-sex predictions regarding E(W) and Var(W), predictions which appear to coincide with major empirical findings; it also generates novel predictions. For instance, W is a mixture of normals distribution. The model is briefly evaluates in two data sets.     

On the scales of measurement

Let $\text{X}$ = 〈X, ≧, R₁, R₂…〉 be a relational structure, 〈$\text{X, ≧}$〉 be a Dedekind complete, totally ordered set, and n be a nonnegative integer. $\text{X}$ is said to satisfy n-point homogeneity if and only if for each x₁,…, x_n, y₁,…, y_n such that x₁ ? x₂ ? … ? x_n and y₁ ? y₂ … ? y_n, there exists an automorphism α of $\text{X}$ such that α(x₁) = y_i. $\text{X}$ is said to satisfy n-point uniqueness if and only if for all automorphisms β and γ of $\text{X}$, if β and γ agree at n distinct points of $\text{X}$, then β and γ are identical. It is shown that if $\text{X}$ satisfies n-point homogeneity and n-point uniqueness, then n ≦ 2, and for the case n = 1, $\text{X}$ is ratio scalable, and for the case n = 2, interval scalable. This result is very general and may in part provide an explanation of why so few scale types have arisen in science. The cases of 0-point homogeneity and infinite point homogeneity are also discussed.     

Effect of psychostimulant medications on static balance performance in adults with attention deficit hyperactivity disorder: Within-subjects repeated-measure study

ObjectiveThis study examined the effect of psychostimulant medications nPS) on balance and functional motor performance in adults with attention-deficit/hyperactivity disorder (ADHD).MethodsParticipants completed two sessions (off-medication and on-medication) in a within-subjects repeated-measure study design. There was a minimum of seven days between the two sessions. During both sessions, participants stood for 30 s per condition on a force platform. The conditions were: feet-apart with 1) eyes-open and 2) eyes-closed; feet-together with 3) eyes-open and 4) eyes-closed. Participants performed three trials of timed up and go (TUG) and lateral step-up test (LSUT) during both sessions. Outcome measures were sway area (SA [cm²]), average sway velocity (SV [cm/s]), TUG average time (s), and average number of LSUT repetitions. Data were analyzed using multivariate repeated measures analysis of variance and paired t-tests for examining PS effects on balance (SA and SV) and functional motor performance (TUG and LSUT), respectively.ResultsThe sample included 45 adults (35 females; mean age = 28.4 ± 6.3 years). The repeated-measures MANOVA indicated that PS was associated with better SA [F(1,44) = 9.6; p = 0.003;η_p² = 0.18] but not with SV [F(1,44) = 1.0; p = 0.319;η_p² = 0.02]. PS was associated with significantly better SA with decreasing base-of-support [F(1,44) = 9.9; p = 0.003;η_p² = 0.18]. Additionally, PS use was associated with better TUG [t(1,44) = 2.65; p = 0.014;Cohen's d = 0.39] but not LSUT performances [t(1,44) = −0.68; p = 0.499;Cohen's d = −0.10].ConclusionsPS was associated with better SA and TUG in adults with ADHD. Further studies are needed to investigate the effects of PS on balance performance using rigorous designs in this population.ImpactHealthcare providers should screen for PS status and balance when treating adults with ADHD to enhance safe motor performance.