The variance of d′ estimates obtained in yes—no and two-interval forced choice procedures

In studies of detection and discrimination, data are often obtained in the form of a 2 × 2 matrix and then converted to an estimate of d′, based on the assumptions that the underlying decision distributions are Gaussian and equal in variance. The statistical properties of the estimate of d′, $\hat d'$ , are well understood for data obtained using the yes—no procedure, but less effort has been devoted to the more commonly used two-interval forced choice (2IFC) procedure. The variance associated with $\hat d'$ is a function of trued′ in both procedures, but for small values of trued′, the variance of $\hat d'$ obtained using the 2IFC procedure is predicted to be less than the variance of $\hat d'$ obtained using yes—no; for large values of trued′, the variance of $\hat d'$ obtained using the 2IFC procedure is predicted to be greater than the variance of $\hat d'$ from yes—no. These results follow from standard assumptions about the relationship between the two procedures. The present paper reviews the statistical properties of $\hat d'$ obtained using the two standard procedures and compares estimates of the variance of $\hat d'$ as a function of trued′ with the variance observed in values of $\hat d'$ obtained with a 2IFC procedure.     

Discriminability, loudness, and masking in the rat (Rattus norvegicus): a confirmation and extension

In Experiment 1, rats discriminated between two sound pressure levels (SPL) of a pure tone: standard (STD) SPLs of 84 and 74 dB and comparison (CO) SPLs 4, 14, and 24 dB below STD were tested in quiet and 60 dB noise at 4 and 12.5 kHz (24 conditions). The decibel difference between STD and CO accounted for only 43.52% of the variance in the signal detection measure of sensitivity, d', across conditions, whereas the loudness difference (LD = STD0.35 - CO0.35) accounted for 89.82% of the variance in d'. These results confirm and extend previous observations that: (a) equal decibel differences are not equally discriminable; (b) loudness for the rat increases as a power function of SPL with an exponent of 0.35: and (c) masked loudness is a linear function of loudness in quiet. In Experiment 2, the assumptions of normal distribution and equal variance implicit in the use of the d' measure were examined. Receiver operating characteristic curves that were well approximated by straight lines of unit slope in normal-normal coordinates were obtained and thereby validated the use of d' in Experiment 1.     

Stimulus discriminability in free-operant and discrete-trial detection procedures.

Six pigeons were trained to discriminate different light intensities in four experimental procedures. Experiment 1 compared stimulus discriminability in a yes-no signal-detection task with discriminability measures obtained from two free-operant procedures. Discriminability estimates were significantly lower in the detection procedure. Experiment 2 showed this lowered discriminability to be a function of the delay between stimulus presentation and the availability of the choice-response keys in the standard detection task. In addition, reinforcement sensitivity was lowest when correct choice responses were intermittently, rather than continuously, reinforced.     

Measures of sensitivity based on a single hit rate and false alarm rate: the accuracy, precision, and robustness of d', Az, and A'

Signal detection theory offers several indexes of sensitivity (d', Az, and A') that are appropriate for two-choice discrimination when data consist of one hit rate and one false alarm rate per condition. These measures require simplifying assumptions about how target and lure evidence is distributed. We examine three statistical properties of these indexes: accuracy (good agreement between the parameter and the sampling distribution mean), precision (small variance of the sampling distribution), and robustness (small influence of violated assumptions on accuracy). We draw several conclusions from the results. First, a variety of parameters (sample size, degree of discriminability, and magnitude of hits and false alarms) influence statistical bias in these indexes. Comparing conditions that differ in these parameters entails discrepancies that can be reduced by increasing N. Second, unequal variance of the evidence distributions produces significant bias that cannot be reduced by increasing N-a serious drawback to the use of these sensitivity indexes when variance is unknown. Finally, their relative statistical performances suggest that Az is preferable to A'.     

Testing for robustness in Monte Carlo studies

Monte Carlo studies provide the information needed to help researchers select appropriate analytical procedures under design conditions in which the underlying assumptions of the procedures are not met. In Monte Carlo studies, the 2 errors that one could commit involve (a) concluding that a statistical procedure is robust when it is not or (b) concluding that it is not robust when it is. In previous attempts to apply standard statistical design principles to Monte Carlo studies, the less severe of these errors has been wrongly designated the Type I error. In this article, a method is presented for controlling the appropriate Type I error rate; the determination of the number of iterations required in a Monte Carlo study to achieve desired power is described; and a confidence interval for a test's true Type I error rate is derived. A robustness criterion is also proposed that is a compromise between W. G. Cochran's (1952) and J. V. Bradley's (1978) criteria.     

Intensity discrilnination: Performance in three paradigms

Performance in the following auditory intensity-discrimination paradigms was compared: (1) 2IFC, (2) single-interval rating with an intensity cue preceding each observation interval, and (3) single-interval rating with no cue. ROCs obtained in the single-interval conditions could be well approximated by assuming that the underlying distributions were both normal and had equal variance. No large difference was observed between performance in the cue and the no-cue conditions. The normalized separation of the means for 2IFC was approximately twice the corresponding estimate for the single-interval conditions. i.e., d₂ = 1.91 d₁. This is consistent with the hypothesis that the O in intensity-discrimination tasks bases his decisions on the difference between the observed input and that of a noisy stored reference. The familiar result that d₂ = (2)^1/2 d₁ can be predicted for simple detection.     

The single interval adjustment matrix (SIAM) yes–no task: an empirical assessment using auditory and gustatory stimuli

The SIAM yes-no task is an efficient bias-free adaptive procedure for estimating absolute thresholds, though it arguably requires further evaluation prior to its adoption into mainstream psychological research. We report two experiments undertaken in the auditory and gustatory modalities designed to assess the accuracy and efficiency of the SIAM method. In the first experiment, estimates of absolute thresholds for 1000-Hz tones obtained using a two-alternative forced choice adaptive procedure were compared to those obtained using both the SIAM yes-no task and a modification of the SIAM task incorporating the method of free response, the SIAM-rapid. In Experiment 2, we compared absolute thresholds for sucrose in solution obtained with either a two-alternative forced choice adaptive procedure or the SIAM yes-no task. Both experiments demonstrated the efficiency and validity of the SIAM approach, with SIAM thresholds proving to be equivalent to those obtained in the two-alternative forced choice tasks and to those reported in the literature.     

Statistical properties of staircase estimates

The bias and variability of staircase estimators were studied by means of repeated computer simulations of staircase runs. Both forced-choice and yes-no staircases were simulated. The influence of the shape of the psychometric function, the location and spacing of the stimuli, the number of trials in a run, and the method of deriving the estimate from the data are discussed. The forced-choice staircase is compared to the yes-no staircase, and the limitations of the simulation procedure are outlined.     

Speed-accuracy tradeoff functions in choice reaction time: Experimental designs and computational procedures

The existence of tradeoffs between speed and accuracy is an important interpretative problem in choice reaction time (RT) experiments. A recently suggested solution to this problem is the use of complete speed-accuracy tradeoff functions as the primary dependent variable in choice RT ,experiments instead of a single mean RT and error rate. This paper reviews and compares existing procedures for generating empirical speed-accuracy tradeoff, functions for use as dependent variables in choice RT experiments. Two major types of tradeoff function are identified, and their experimental designs and computational procedures are discussed and evaluated. Systematic disparities are demonstrated between the two tradeoff functions in both empirical and computer-simulated data. Although all existing procedures for generating speed-accuracy tradeoff functions involve empirically untested assumptions, one procedure requires less stringent assumptions and is less sensitive to sources of experimental and statistical error. This procedure involves plotting accuracy against RT over a set of experimental conditions in which subjects’ criteria for speed vs. accuracy are systematically varied.     

A computer program for Spearman-K?rber and probit analysis of psychometric function data.

PMETRIC is a computer program for the analysis of observed psychometric functions. It can estimate the parameters of these functions, using either probit analysis (a parametric technique) or the Spearman-K?rber method (a nonparametric one). For probit analysis, either a maximum likelihood or a minimum chi 2 criterion may be used for parameter estimation. In addition, standard errors of parameter estimates can be estimated via bootstrapping. The program can be used to analyze data obtained from either yes-no or m-alternative forced-choice tasks. To facilitate the use of PMETRIC in simulation work, an associated program, PMETGEN, is provided for the generation of simulated psychometric function data. Use of PMETRIC is illustrated with data from a duration discrimination task.     

Measures of response bias at minimum-detectable luminance levels in the pigeon

Using an operant analogue of the yes-no detection task, six pigeons were trained to detect luminance changes under two different reinforcement-scheduling procedures. In the first, an uncontrolled reinforcement-ratio procedure, the relative frequency of food reinforcers obtained for correct detections was free to vary with the birds' behavior as luminance levels were changed. In the second, a controlled reinforcement-ratio procedure, changes in preference could not alter the relative distribution of food reinforcers between the two response alternatives. Extreme response biases developed as luminance was decreased to threshold in the uncontrolled procedure. No progressive changes in response bias as a function of decreasing luminance were seen in the controlled procedure. Absolute thresholds for light intensity were lower in the controlled than in the uncontrolled procedure.     

非正态分布下概化理论方差分量变异量估计

方差分量估计是概化理论的必用技术,但受限于抽样,需要对其变异量进行探讨。采用Monte Carlo数据模拟技术,探讨非正态数据分布对四种方法估计概化理论方差分量变异量的影响。结果表明：（1）不同非正态数据分布下,各种估计方法的“性能”表现出差异性;（2）数据分布对方差分量变异量估计有影响,适合于非正态分布数据的方差分量变异量估计方法不一定适合于正态分布数据。     

Characterizing sentence intonation in a right hemisphere-damaged population

Current production studies present a mixed view of right hemisphere-damaged (RHD) patients' ability to produce normal sentence intonation. The present study characterized the sentence intonation of RHD patients, focusing on a greater number of acoustic parameters than past works, and relying on more naturally elicited speech samples through use of a story completion task. Eight RHD speakers and seven nonneurological control subjects produced declarative and imperative sentences as well as yes-no and wh-questions. Slope of F0 change, linearity of pitch contour, and variance of F0 points were calculated for each utterance as a whole, as well as for the preterminal and the terminal contour separately. RHD contours were less linear and flatter in F0 decline than normal controls for the declarative sentences. The patients' yes-no questions also differed from normal productions, displaying smaller F0 dispersion around a mean F0. Preterminal range values were more restricted for patients' utterances of yes-no questions, while terminal properties between groups differed for three of the four sentence types examined. The present results suggest some disturbance in the patients' ability to manipulate fundamental frequency across sentential domains. These data are discussed in terms of current theories of a general dysprosody in RHD patients.     

Statistical simulation on microcomputers

The statistical simulation program DATASIM is designed to conduct large-scale sampling experiments on microcomputers. Monte Carlo procedures are used to investigate the Type I and Type II error rates for statistical tests when one or more assumptions are systematically violated-assumptions, for example, regarding normality, homogeneity of variance or covariance, mini-mum expected cell frequencies, and the like. In the present paper, we report several initial tests of the data-generating algorithms employed by DATASIM. The results indicate that the uniform and standard normal deviate generators perform satisfactorily. Furthermore, Kolmogorov-Smirnov tests show that the sampling distributions ofz, t, F, χ², andr generated by DATASIM simulations follow the appropriate theoretical distributions. Finally, estimates of Type I error rates obtained by DATASIM under various patterns of violations of assumptions are in close agreement with the results of previous analytical and empirical studies; These converging lines of evidence suggest that DATASIM may well prove to be a reliable and productive tool for conducting statistical simulation research.     

A computer program for Spearman-Kärber and probit analysis of psychometric function data

PMETRIC is a computer program for the analysis of observed psychometric functions. It can estimate the parameters of these functions, using either probit analysis (a parametric technique) or the Spearman-Kärber method (a nonparametric one). For probit analysis, either a maximum likelihood or a minimum χ² criterion may be used for parameter estimation. In addition, standard errors of parameter estimates can be estimated via bootstrapping. The program can be used to analyze data obtained from either yes-no orm-alternative forced-choice tasks. To facilitate the use of PMETRIC in simulation work, an associated program, PMETGEN, is provided for the generation of simulated psychometric function data. Use of PMETRIC is illustrated with data from a duration discrimination task.     

ESTIMATING CONTENT-ACQUIESCENCE CORRELATION BY COVARIANCE STRUCTURE ANALYSIS

A topic of continuing interest in the measurement area is response acquiescence. A recent study has demonstrated the feasibiliy of studying acquiescence or, more importantly, content/acquiescence correlation in the MMPI. Utilizing the components of variance approach, this study found that the variance due to acquiescence in scores on the Pt and Hg scales was small relative to content variance, but that the correlation between acquiescence and content may be substantial for the Pt scale. The present paper describes a general statistical procedure for investigating content variance, variance due to non-content characteristics of items, and the covariances of content and various item characteristics. The data from a previous paper are reanalyzed, using alternative covariance structure models. Maximum likelihood procedures which allow for a statistical test for parameters of interest are used. The results point to the significance of the content- acquiescence correlation in the Pt scale, but not in the Hy scale. The previous findings are verified statistically, and procedures which hold promise for other investigation into the properties of behavioral tests are described.     

Auditory phase and frequency discrimination: a comparison of nine procedures

Two auditory discrimination tasks were thoroughly investigated: discrimination of frequency differences from a sinusoidal signal of 200 Hz and discrimination of differences in relative phase of mixed sinusoids of 200 Hz and 400 Hz. For each task psychometric functions were constructed for three observers, using nine different psychophysical measurement procedures. These procedures included yes-no, two-interval forced-choice, and various fixed- and variable-standard designs that investigators have used in recent years. The data showed wide ranges of apparent sensitivity. For frequency discrimination, models derived from signal detection theory for each psychophysical procedure seem to account for the performance differences. For phase discrimination the models do not account for the data. We conclude that for some discriminative continua the assumptions of signal detection theory are appropriate, and underlying sensitivity may be derived from raw data by appropriate transformations. For other continua the models of signal detection theory are probably inappropriate; we speculate that phase might be discriminable only on the basis of comparison or change and suggest some tests of our hypothesis.     

Multiple Comparisons For Repeated Measures Means

The paper shows how multiple comparison procedures for repeated measures means employing a pooled estimate of error variance must conform to the sphericity assumptions of the design in order to provide a valid test. Since it is highly unlikely that behavioral science data will satisfy this condition the paper presents a test statistic that, depending upon the design, will provide either an exact or robust test and is generalizable to designs containing any number of repeated factors. Finally, various critical values are enumerated to limit the joint level of significance at α.     

An analysis of the strength-latency relationship

To account for latency in recognition memory, strength theory assumes that latency decreases symmetrically on both sides of the yes-no criterion. Some of the standard criticisms of the theory are presented. To evaluate these criticisms, explicit expressions for the latency distribution and its mean and variance were obtained. On analysis, the criticisms seem to be unwarranted. However, a satisfactory version of strength theory is not simple, and as many as a dozen or so parameters may be required to account for the data.     

Variability of magnitude estimates: A timing theory analysis

Three procedures for magnitude estimation were investigated, and a sufficient number of responses were obtained to make reasonable estimates of both the mean and variance of the responses. The conventional magnitude estimate procedure, without a standard signal, appeared to produce the most sensible data. The best method of establishing the central tendency of the data appears to be the plot of the mean ratio of successive responses against the intensity ratio of the corresponding signal intensities. When this is done, the average response ratio increases roughly as a power function of the signal ratios. The coefficient of variation, σ/m, varies from about 0.1 for small signal ratios and increases to 0.3 at about 20 dB and greater signal separations. The distribution of response ratios appears to be reasonably well approximated by a beta distribution. The change in σ/m with signal ratio is suggestive of an attention mechanism in which the sample size depends on the location of the attention band. The ratio estimation procedure suffers badly from discrete number tendencies.