On sample size calculation for 2×2 fixed‐effect ANOVA when variances are unknown and possibly unequal

The factorial 2 × 2 fixed‐effect ANOVA is a procedure used frequently in scientific research to test mean differences between‐subjects in all of the groups. But if the assumption of homogeneity is violated, the test for the row, column, and the interaction effect might be invalid or less powerful. Therefore, for planning research in the case of unknown and possibly unequal variances, it is worth developing a sample size formula to obtain the desired power. This article suggests a simple formula to determine the sample size for 2 × 2 fixed‐effect ANOVA for heterogeneous variances across groups. We use the approximate Welch t test and consider the variance ratio to derive the formula. The sample size determination requires two‐step iterations but the approximate sample sizes needed for the main effect and the interaction effect can be determined separately with the specified power. The present study also provides an example and a SAS program to facilitate the calculation process.     

New heterogeneous test statistics for the unbalanced fixed‐effect nested design

When the underlying variances are unknown or/and unequal, using the conventional F test is problematic in the two‐factor hierarchical data structure. Prompted by the approximate test statistics (Welch and Alexander–Govern methods), the authors develop four new heterogeneous test statistics to test factor A and factor B nested within A for the unbalanced fixed‐effect two‐stage nested design under variance heterogeneity. The actual significance levels and statistical power of the test statistics were compared in a simulation study. The results show that the proposed procedures maintain better Type I error rate control and have greater statistical power than those obtained by the conventional F test in various conditions. Therefore, the proposed test statistics are recommended in terms of robustness and easy implementation.     

Mini‐breaks: the use of escape on a fixed‐time schedule to reduce unauthorized breaks from vocational training sites for individuals with brain injury

This study evaluated mini‐breaks with three individuals who were traumatically brain injured and engaged in unauthorized breaks. The vocational instructors reported that the three participants were having problems staying at their scheduled vocational training sites (walking away without permission). Following baseline, participants were placed on mini‐breaks; that is, participants were given a ten‐minute mini‐break every hour. Thus, mini‐breaks were given without regard to unauthorized breaks. Unauthorized breaks decreased to zero levels for all three participants when mini‐breaks commenced. Following 1 month of the mini‐breaks, two of the participants were put back on a regular break schedule (one 15‐minute AM break and one 15‐minute PM break and a 30‐minute lunch break). The third participant was placed in supported work. The effects of the mini‐breaks were maintained. Copyright © 1999 John Wiley & Sons, Ltd.     

The relationship between confidence and self-concept — Towards a model of response confidence

According to Stankov [Stankov, L. (2000). Complexity, metacognition and fluid intelligence. Intelligence, 28, 121–143.] response confidence in cognitive tests reflects a trait on the boundary of personality and abilities. However, several studies failed in relating confidence scores to other known traits, including self-concept. A model of response confidence is proposed which predicts that confidence judgements do reflect self-concept, but only to the extent that they do not reflect a calibration process based on task-inherent cues. In the current study, 101 students completed various tests of cognitive abilities and skills as well as scales of the SDQ III measuring general academic and problem-solving self-concept. As expected, self-concept predicts a significant proportion of variance in the confidence factor that is over and above the influence of test scores.     

On the potential value and limitations of emphasis change and other exploration-enhancing training methods

Two experiments are presented that examine the efficiency of training methods that force trainees to explore the possible strategy space. Both experiments used a 2-dimensional search task. Experiment 1 studies a method that enforces exploration by preventing repetitions within short sequences. It shows that the effect of this "enhanced exploration" method in the abstract setting is similar to the effect of "emphasis change" training (see D. Gopher, 1993) in high cognitive workload tasks. This finding appeared only in strategic spaces where intuitive exploration converges to a local optimum. Experiment 2 compares the effect of exploration enhancement with the effect of guided instruction. The results of the experiments, which are captured well by a generalization of I. Erev and D. A. Gopher's (1999) model of the conditions for the success of emphasis change training, shed light on the limitations and potential of exploration-enhancing training methods.     

Modelling non‐ignorable missing‐data mechanisms with item response theory models

A model‐based procedure for assessing the extent to which missing data can be ignored and handling non‐ignorable missing data is presented. The procedure is based on item response theory modelling. As an example, the approach is worked out in detail in conjunction with item response data modelled using the partial credit and generalized partial credit models. Simulation studies are carried out to assess the extent to which the bias caused by ignoring the missing‐data mechanism can be reduced. Finally, the feasibility of the procedure is demonstrated using data from a study to calibrate a medical disability scale.     

Eyewitness recall: Regulation of grain size and the role of confidence

Eyewitness testimony plays a critical role in Western legal systems. Three experiments extended M. Goldsmith, A. Koriat, and A. Weinberg-Eliezer's (2002) framework of the regulation of grain size (precision vs. coarseness) of memory reports to eyewitness memory. In 2 experiments, the grain size of responses had a large impact on memory accuracy. Further, participants achieved a compromise between the accuracy and informativeness of their testimony by volunteering precise answers only when likely to be correct. The level of detail reported was strongly, positively related to confidence in the accuracy of the response. This highlights the importance of considering the level of detail, not just the accuracy, of eyewitness testimony.     

On the chi‐square approximation to the exact distribution of goodness‐of‐fit statistics in multinomial models with composite hypotheses

Multinomial models are increasingly being used in psychology, and this use always requires estimating model parameters and testing goodness of fit with a composite null hypothesis. Goodness of fit is customarily tested with recourse to the asymptotic approximation to the distribution of the statistics. An assessment of the quality of this approximation requires a comparison with the exact distribution, but how to compute this exact distribution when parameters are estimated from the data appears never to have been defined precisely. The main goal of this paper is to compare two different approaches to defining this exact distribution. One of the approaches uses the marginal distribution and is, therefore, independent of the data; the other approach uses the conditional distribution of the statistics given the estimated parameters and, therefore, is data—dependent. We carried out a thorough study involving various parameter estimation methods and goodness‐of‐fit statistics, all of them members of the general class of power‐divergence measures. Included in the study were multinomial models with three to five cells and up to three parameters. Our results indicate that the asymptotic distribution is rarely a good approximation to the exact marginal distribution of the statistics, whereas it is a good approximation to the exact conditional distribution only when the vector of expected frequencies is interior to the sample space of the multinomial distribution.     

Approximate sample size formulas for the two‐sample trimmed mean test with unequal variances

Yuen's two‐sample trimmed mean test statistic is one of the most robust methods to apply when variances are heterogeneous. The present study develops formulas for the sample size required for the test. The formulas are applicable for the cases of unequal variances, non‐normality and unequal sample sizes. Given the specified α and the power (1?β), the minimum sample size needed by the proposed formulas under various conditions is less than is given by the conventional formulas. Moreover, given a specified size of sample calculated by the proposed formulas, simulation results show that Yuen's test can achieve statistical power which is generally superior to that of the approximate t test. A numerical example is provided.     

On the Rationality of Vow‐making

I offer a philosophical account of vowing and the rationality of vow‐making. I argue that vows are most productively understood as exceptionless resolutions that do not have any excusing conditions. I then articulate an apparent problem for exceptionless vow‐making: how can it be rational to bind yourself unconditionally, when circumstances might change unexpectedly and make it the case that vow‐keeping no longer makes sense for you? As a solution, I propose that vows can be rational to make only if they are implicitly conditional on a personal identification or social role that is itself escapable.     

On the form of ROCs constructed from confidence ratings

A classical question for memory researchers is whether memories vary in an all-or-nothing, discrete manner (e.g., stored vs. not stored, recalled vs. not recalled), or whether they vary along a continuous dimension (e.g., strength, similarity, or familiarity). For yes-no classification tasks, continuous- and discrete-state models predict nonlinear and linear receiver operating characteristics (ROCs), respectively (D. M. Green & J. A. Swets, 1966; N. A. Macmillan & C. D. Creelman, 1991). Recently, several authors have assumed that these predictions are generalizable to confidence ratings tasks (J. Qin, C. L. Raye, M. K. Johnson, & K. J. Mitchell, 2001; S. D. Slotnick, S. A. Klein, C. S. Dodson, & A. P. Shimamura, 2000, and A. P. Yonelinas, 1999). This assumption is shown to be unwarranted by showing that discrete-state ratings models predict both linear and nonlinear ROCs. The critical factor determining the form of the discrete-state ROC is the response strategy adopted by the classifier.     

Self‐Handicapping: On the Self‐Perpetuating Nature of Defensive Behavior

The term self‐handicapping was introduced by Jones and Berglas (1978 ) to refer to the creation of barriers to successful performance for the purpose of controlling attributions about the self. In the event of failure, attributions to lack of ability are diminished or discounted because of the handicap and, in the event of success, attributions to ability are enhanced or augmented because of the handicap. This article reviews over 25 years of research on self‐handicapping. A process model is presented in which individual differences in goals and concerns dynamically interact with situational threats to elicit self‐handicapping behavior which produces consequences that perpetuate the use of the behavior in future situations.     

Inferring the size of a goal object from an actor's grasping movement in 6‐ and 9‐month‐old infants

The present study applied a preferential looking paradigm to test whether 6‐ and 9‐month old infants are able to infer the size of a goal object from an actor's grasping movement. The target object was a cup with the handle rotated either towards or away from the actor. In two experiments, infants saw the video of an actor's grasping movement towards an occluded target object. The aperture size of the actor's hand was varied as between‐subjects factor. Subsequently, two final states of the grasping movement were presented simultaneously with the occluder being removed. In Experiment 1, the expected final state showed the actor's hand holding a cup in a way that would be expected after the performed grasping movement. In the unexpected final state, the actor's hand held the cup at the side which would be unexpected after the performed grasping movement. Results show that 6‐ as well as 9‐month‐olds looked longer at the unexpected than at the expected final state. Experiment 2 excluded an alternative explanation of these findings, namely that the discrimination of the final states was due to geometrical familiarity or novelty of the final states. These findings provide evidence that infants are able to infer the size of a goal object from the aperture size of the actor's hand during the grasp.     

Limited‐information goodness‐of‐fit testing of item response theory models for sparse 2P tables

Bartholomew and Leung proposed a limited‐information goodness‐of‐fit test statistic (Y) for models fitted to sparse 2^P contingency tables. The null distribution of Y was approximated using a chi‐squared distribution by matching moments. The moments were derived under the assumption that the model parameters were known in advance and it was conjectured that the approximation would also be appropriate when the parameters were to be estimated. Using maximum likelihood estimation of the two‐parameter logistic item response theory model, we show that the effect of parameter estimation on the distribution of Y is too large to be ignored. Consequently, we derive the asymptotic moments of Y for maximum likelihood estimation. We show using a simulation study that when the null distribution of Y is approximated using moments that take into account the effect of estimation, Y becomes a very useful statistic to assess the overall goodness of fit of models fitted to sparse 2^P tables.     

On interpreting the relationship between remember–know judgments and confidence: The role of instructions

Two experiments were designed to test the hypothesis that the nature of the remember–know instructions given to participants influences whether these responses reflect different memory states or different degrees of memory confidence. Participants studied words and nonwords, a variable that has been shown to dissociate confidence from remember–know judgments and were given a set of published remember–know instructions that either emphasized know judgments as highly confident (Experiment 1) or as less confident (Experiment 2) states of recognition. Experiment 1 replicated the standard finding showing that remembering and knowing were differently influenced by the word–nonword variable, whereas confidence responses were not. By contrast, Experiment 2 showed a similar pattern of data for remember–know and sure–unsure responses, thus demonstrating the importance of the instructions for interpreting the relationship between remembering and knowing and confidence.     

The effect of on‐ or off‐line extinction of a first‐order conditioned stimulus on a second‐order conditioned response in rats

In a water‐licking experiment with rats, the effects of the extinction of a first‐order conditioned stimulus (CS1) on the suppression of licking established for a second‐order conditioned stimulus (CS2) were explored. Extinction of the first‐order conditioned response (CR1) attenuated the conditioned suppression induced by CS2 when the rats had been allowed to lick water during the CR1 extinction phase. However, if water had been unavailable during the CR1 extinction phase, suppression by CS2 was not affected. The latter result is consistent with other studies of rats' conditioned suppression and suggests that the underlying mechanism of second‐order conditioning in this experiment is a connection between CS2 and the response elicited by CS1 rather than a CS2‐CS1 connection. The former result was interpreted as the CR1 extinction phase encouraging the rats to lick water despite the fear elicited by CS1, and thus, in testing, they licked despite the fear elicited by CS2.     

Non‐zero‐sum collaboration,reciprocity, and the preference for similarity: Developing an adaptive model of close relational functioning

An evolutionary framework was used to develop a model of relational functioning among friends. The proposed model focuses on the collaborative nature of close friendships and attempts to highlight two adaptive mechanisms important in creating non‐zero‐sum outcomes among highly interdependent parties. The model emphasizes the importance of reciprocity in creating mutually beneficial outcomes through social exchange, and it articulates how the preference for similarity is useful when creating non‐zero‐sum rewards through synergistic coordination. In particular, the focus is on the unique role that shared interests and mutual knowledge play when individuals attempt to pursue common goals through joint activity. The implications of the model are discussed with respect to a host of issues ranging from deceptive communication to relational satisfaction and commitment.     

Measurement of body size and shape perception in eating‐disordered and control observers using body‐shape software

A disturbance in the perception of personal body size and shape is a key feature of both anorexia and bulimia nervosa, but it has proved difficult to quantify. Previous attempts have used methods like the distorting video technique (DVT), which alters an image by stretching the figure in either the X‐ or Y‐axis. This is a poor representation of the way fat is added to or lost from the body, and the pattern of distortion provides a host of cues to the degree to which the image has been altered. To overcome these problems we have used a specially designed software system that uses biometric data based on real body shapes, instead of simply stretching or compressing images of bodies. This technique also allows individual body parts to be altered separately, so we can determine whether a specific body part is overestimated relative to others. We can also calculate the apparent body mass index (BMI) of our modified pictures, using the perimeter‐area ratio (PAR). This allows us to compare an observer's actual BMI with that calculated for their estimated and ideal bodies. We tested 30 anorexic, 30 bulimic and 137 control observers. Our results show that all three observer groups tend to overestimate their body size, but not significantly so. Both the control and bulimic observers prefer an ideal body with a BMI of 20, which is at the lower end of the ‘normal’ BMI range. However, the anorexics ideal BMI is 15, which is on the border between the emaciated and underweight BMI categories.