The nontruncated marginal of a truncated bivariate normal distribution

Inference is considered for the marginal distribution ofX, when (X, Y) has a truncated bivariate normal distribution. TheY variable is truncated, but only theX values are observed. The relationship of this distribution to Azzalini's skew-normal distribution is obtained. Method of moments and maximum likelihood estimation are compared for the three-parameter Azzalini distribution. Samples that are uniformative about the skewness of this distribution may occur, even for largen. Profile likelihood methods are employed to describe the uncertainty involved in parameter estimation. A sample of 87 Otis test scores is shown to be well-described by this model.     

Fisher transformation based confidence intervals of correlations in fixed- and random-effects meta-analysis

Meta-analyses of correlation coefficients are an important technique to integrate results from many cross-sectional and longitudinal research designs. Uncertainty in pooled estimates is typically assessed with the help of confidence intervals, which can double as hypothesis tests for two-sided hypotheses about the underlying correlation. A standard approach to construct confidence intervals for the main effect is the Hedges-Olkin-Vevea Fisher-z (HOVz) approach, which is based on the Fisher-z transformation. Results from previous studies (Field, 2005, Psychol. Meth., 10, 444; Hafdahl and Williams, 2009, Psychol. Meth., 14, 24), however, indicate that in random-effects models the performance of the HOVz confidence interval can be unsatisfactory. To this end, we propose improvements of the HOVz approach, which are based on enhanced variance estimators for the main effect estimate. In order to study the coverage of the new confidence intervals in both fixed- and random-effects meta-analysis models, we perform an extensive simulation study, comparing them to established approaches. Data were generated via a truncated normal and beta distribution model. The results show that our newly proposed confidence intervals based on a Knapp-Hartung-type variance estimator or robust heteroscedasticity consistent sandwich estimators in combination with the integral z-to-r transformation (Hafdahl, 2009, Br. J. Math. Stat. Psychol., 62, 233) provide more accurate coverage than existing approaches in most scenarios, especially in the more appropriate beta distribution simulation model.     

Bayesian estimation of semiparametric nonlinear dynamic factor analysis models using the Dirichlet process prior

Parameters in time series and other dynamic models often show complex range restrictions and their distributions may deviate substantially from multivariate normal or other standard parametric distributions. We use the truncated Dirichlet process (DP) as a non‐parametric prior for such dynamic parameters in a novel nonlinear Bayesian dynamic factor analysis model. This is equivalent to specifying the prior distribution to be a mixture distribution composed of an unknown number of discrete point masses (or clusters). The stick‐breaking prior and the blocked Gibbs sampler are used to enable efficient simulation of posterior samples. Using a series of empirical and simulation examples, we illustrate the flexibility of the proposed approach in approximating distributions of very diverse shapes.     

A short note on the maximal point‐biserial correlation under non‐normality

The aim of this paper is to derive the maximal point‐biserial correlation under non‐normality. Several widely used non‐normal distributions are considered, namely the uniform distribution, t‐distribution, exponential distribution, and a mixture of two normal distributions. Results show that the maximal point‐biserial correlation, depending on the non‐normal continuous variable underlying the binary manifest variable, may not be a function of p (the probability that the dichotomous variable takes the value 1), can be symmetric or non‐symmetric around p = .5, and may still lie in the range from ?1.0 to 1.0. Therefore researchers should exercise caution when they interpret their sample point‐biserial correlation coefficients based on popular beliefs that the maximal point‐biserial correlation is always smaller than 1, and that the size of the correlation is always further restricted as p deviates from .5.     

A first-order method for estimating correlation coefficients

A rapid method of estimating a correlation coefficient is given. The method expresses the correlation coefficient as the ratio between two differences in sums (or means) of the dependent variable computed only for extremes of the bivariate distribution. A trial shows that this method gives results similar to the product-moment correlation coefficient. Extensions of the method to qualitative data are also suggested.     

The structural processing of the truncated passive in children and adults

Three experiments were designed to test the hypothesis that the structural processing of truncated passives is dependent upon the degree to which the past participle syntactically resembles an adjective. Two types of truncated passive were distinguished, one predicted to be processed according to the standard transformational account, and the other analogously to the predicate adjective construction. In sentence-completion tasks, both children and adults showed differential rates of agentive phrase recovery to the two types of truncated passive. A third experiment showed that the linguistic intuitions of adult subjects differed for the two types of truncated passive.     

A note on the use of rank-ordered logit models for ordered response categories

Models for rankings have been shown to produce more efficient estimators than comparable models for first/top choices. The discussions and applications of these models typically only consider unordered alternatives. But these models can be usefully adapted to the case where a respondent ranks a set of ordered alternatives that are ordered response categories. This paper proposes eliciting a rank order that is consistent with the ordering of the response categories, and then modelling the observed rankings using a variant of the rank ordered logit model where the distribution of rankings has been truncated to the set of admissible rankings. This results in lower standard errors in comparison to when only a single top category is selected by the respondents. And the restrictions on the set of admissible rankings reduces the number of decisions needed to be made by respondents in comparison to ranking a set of unordered alternatives. Simulation studies and application examples featuring models based on a stereotype regression model and a rating scale item response model are provided to demonstrate the utility of this approach.     

Discrimination between solid forms in early infancy

This paper describes three experiments concerned with the ability of 3-month-old infants to discriminate between stationary solid forms. Experiment 1 confirmed, using a habituation/recovery technique, the finding of Cook, Field, and Griffiths (1978) that the infants do not distinguish between a cube and a truncated pyramid (and a range of other solid forms). Experiment 2 showed that infants can distinguish the square face of the cube from a trapezoidal face of the truncated pyramid when these shapes are presented in isolation. Experiment 3 showed that this discrimination is not made when the corresponding faces (square and trapezoid) of the solid cube and trapezoid are colored distinctively. These results are discussed in terms of limitations in the infants' capacity to resolve solid form.     

Are True Scores and Construct Scores the Same? A critical examination of their substitutability and the implications for research results

Relations between constructs are estimated based on correlations between measures of constructs corrected for measurement error. This process assumes that the true scores on the measure are linearly related to construct scores, an assumption that may not hold. We examined the extent to which differences in distribution shape reduce the correlation between true scores on a measure and scores on the underlying construct they are intended to measure. We found, via a series of Monte Carlo simulations, that when the actual construct distribution is normal, nonnormal distributions of true scores caused this correlation to drop by an average of only .02 across 15 conditions. When both construct and true score distributions assumed different combinations of nonnormal distributions, the average correlation was reduced by .05 across 375 conditions. We conclude that theory‐based scales intended to measure constructs usually correlate highly with the constructs they are constructed to measure. We show that, as a result, in most cases true score correlations only modestly underestimate correlations between different constructs. However, in cases in which the two constructs are redundant, this underestimation can lead to the false conclusion that the constructs are ‘correlated but distinct constructs,’ resulting in construct proliferation.     

The origins of levels-of-processing effects in a conceptual test: evidence for automatic influences of memory from the process-dissociation procedure

In three experiments, we explored automatic influences of memory in a conceptual memory task, as affected by a levels-of-processing (LoP) manipulation. We also explored the origins of the LoP effect by examining whether the effect emerged only when participants in the shallow condition truncated the perceptual processing (the lexical-processing hypothesis) or even when the entire word was encoded in this condition (the conceptual-processing hypothesis). Using the process-dissociation procedure and an implicit association-generation task, we found that the deep encoding condition yielded higher estimates of automatic influences than the shallow condition. In support of the conceptual processing hypothesis, the LoP effect was found even when the shallow task did not lead to truncated processing of the lexical units. We suggest that encoding for meaning is a prerequisite for automatic processing on conceptual tests of memory.     

Asymptotic confidence intervals for the Pearson correlation via skewness and kurtosis

When bivariate normality is violated, the default confidence interval of the Pearson correlation can be inaccurate. Two new methods were developed based on the asymptotic sampling distribution of Fisher's z′ under the general case where bivariate normality need not be assumed. In Monte Carlo simulations, the most successful of these methods relied on the (Vale & Maurelli, 1983, Psychometrika, 48, 465) family to approximate a distribution via the marginal skewness and kurtosis of the sample data. In Simulation 1, this method provided more accurate confidence intervals of the correlation in non-normal data, at least as compared to no adjustment of the Fisher z′ interval, or to adjustment via the sample joint moments. In Simulation 2, this approximate distribution method performed favourably relative to common non-parametric bootstrap methods, but its performance was mixed relative to an observed imposed bootstrap and two other robust methods (PM1 and HC4). No method was completely satisfactory. An advantage of the approximate distribution method, though, is that it can be implemented even without access to raw data if sample skewness and kurtosis are reported, making the method particularly useful for meta-analysis. Supporting information includes R code.     

Estimating correlations with missing data,a bayesian approach

The posterior distribution of the bivariate correlation is analytically derived given a data set wherex is completely observed buty is missing at random for a portion of the sample. Interval estimates of the correlation are then constructed from the posterior distribution in terms of highest density regions (HDRs). Various choices for the form of the prior distribution are explored. For each of these priors, the resulting Bayesian HDRs are compared with each other and with intervals derived from maximum likelihood theory.     

Population models and simulation methods: The case of the Spearman rank correlation

The purpose of this paper is to highlight the importance of a population model in guiding the design and interpretation of simulation studies used to investigate the Spearman rank correlation. The Spearman rank correlation has been known for over a hundred years to applied researchers and methodologists alike and is one of the most widely used non‐parametric statistics. Still, certain misconceptions can be found, either explicitly or implicitly, in the published literature because a population definition for this statistic is rarely discussed within the social and behavioural sciences. By relying on copula distribution theory, a population model is presented for the Spearman rank correlation, and its properties are explored both theoretically and in a simulation study. Through the use of the Iman–Conover algorithm (which allows the user to specify the rank correlation as a population parameter), simulation studies from previously published articles are explored, and it is found that many of the conclusions purported in them regarding the nature of the Spearman correlation would change if the data‐generation mechanism better matched the simulation design. More specifically, issues such as small sample bias and lack of power of the t‐test and r‐to‐z Fisher transformation disappear when the rank correlation is calculated from data sampled where the rank correlation is the population parameter. A proof for the consistency of the sample estimate of the rank correlation is shown as well as the flexibility of the copula model to encompass results previously published in the mathematical literature.     

Evidence of rapid correlation‐based perceptual category learning by 4‐month‐olds

Young infants are very sensitive to feature distribution information in the environment. However, existing work suggests that they do not make use of correlation information to form certain perceptual categories until at least 7 months of age. We suggest that the failure to use correlation information is a by‐product of familiarization procedures that encourage infants to over encode individual exemplars rather than relations across exemplars. By changing the exemplar presentation regime to one in which exemplars are rapidly (2 s durations) and repeatedly presented we find that 4‐month‐olds can form perceptual categories on the basis of feature correlation information. In addition, this ability emerges rapidly between 114 and 134 days. We argue that the ability to process correlation information is present very early on but that the demonstration of that ability in categorization tasks is mediated by the demands of the task the infant is tested with. Copyright © 2005 John Wiley & Sons, Ltd.     

Roll-over characteristics of human walking on inclined surfaces

Roll-over characteristics of able-bodied human subjects walking on ramped surfaces were examined in this study. Ten subjects walked at their normal self-selected speed on a level surface, a 5-deg ramp, and a 10-deg ramped surface. Ramps were designed such that ground reaction forces and center of pressure of the ground reaction forces could be measured on their surfaces. This set-up facilitated calculation of the effective rockers that the ankle-foot (AF) and knee-ankle-foot (KAF) systems conformed to during single-limb stance (contralateral toe off to contralateral heel contact). Since our original "roll-over shapes" were characterized between heel contact and opposite heel contact, we label the shapes found during single-limb stance as "truncated roll-over shapes". We hypothesized that the ankle-foot system would adapt to the various surfaces, creating a roll-over shape that would change in orientation with different levels of inclination. The truncated AF roll-over shapes supported this hypothesis for uphill walking but did not support the hypothesis for downhill walking. However, truncated roll-over shapes of the KAF system did adjust their orientation to match both the positive and negative levels of surface inclination. In general, the ankle appears to be the main adapting joint when walking up inclined surfaces while the knee becomes important for the overall adaptation in downhill walking. Knowledge of physiological lower-limb roll-over characteristics on ramped surfaces may help in the development of biomimetic prostheses and orthoses that will automatically adapt to changes in walking surface inclination.     

Range restrictions for product-moment correlation matrices

It is well-known that for a trivariate distribution if two correlations are fixed the remaining one is constrained. Indeed, if one correlation is fixed, then the remaining two are constrained. Both results are extended to the case of a multivariate distribution. The results are applied to some special patterned matrices.     

Generating multivariate data from nonnormal distributions: Mihal and Barrett revisited

An algorithm described by Graybill (1969) factors a population correlation matrix, R, into an upper and lower triangular matrix, T and T′, such that R=T′T. The matrix T is used to generate multivariate data sets from a multinormal distribution. When this algorithm is used to generate data for nonnormal distributions, however, the sample correlations are systematically biased downward. We describe an iterative technique that removes this bias by adjusting the initial correlation matrix. R, factored by the Graybill algorithm. The method is illustrated by simulating a multivariate study by Mihal and Barrett (1976). Large-N simulations indicate that the iterative technique works: multivariate data sets generated with this approach successfully model both the univariate distributions of the individual variables and their multivariate structure (as assessed by intercorrelation and regression analyses).