Factorial design in the determination of differential limen values

This paper discusses the application of the principles of factorial design to an experiment in psychology. For the purpose of illustrating the principles, a simple experiment was designed dealing with the determination of the differential limen values of subjects for weights increasing at constant rates. The factorial design was of the type: 4 rates × 7 weights × 2 sexes × 2 sights × 2 dates. The appropriate statistical analysis for this type of design is the analysis of variance. The mathematical formulation of the problem was specified and the appropriate solution for the specific problem was obtained. Greater precision results from this type of design, in comparison with the traditional psychological experiment dealing with a single factor, in that measures are obtained of the effect of each of a number of factors together with their interactions.This is one of a number of studies on modern principles of experimental design. For the research grant to finance these studies, grateful acknowledgment is given to the Graduate School, the University of Minnesota.     

Factorial design and covariance in the study of individual educational development

This is the report of the application of the principles of factorial design to an investigation of individual educational development. The specific type of factorial design formulated was a 2 × 3 × 3 × 3 arrangement, that is, the effect of sex, grade location, scholastic standing, and individual order, singly and in all possible combinations was studied in relation to educational development as measured by theIowa Tests of Educational Development. An application of the covariance method was introduced which resulted in increased precision of this type of experimental design by significantly reducing experimental error. The two concomitant measures used to increase the sensitiveness of the experiment were initial status of individual development and mental age. Without these statistical controls all main effects and two first-order interactions would have been accepted as significant. With their use only sex (doubtful), scholastic standing, and individual order demonstrated significant effects. The chief beauty of the analysis of variance and covariance as an integral part of a self-contained experiment is demonstrated in the complete single analysis of the data. The statistical utilization of the experimental results has also been developed for purposes of estimation and prediction. The mathematical statistician is being continuously required to develop and analyze experimental designs of increasing complexity since the introduction of the analysis of variance and covariance. The mathematical formulation and solution of the problem of this investigation is carried out. The methods illustrated and explained in this study, and modifications and extensions of them are capable of very wide application. The general principles can be used to various degrees and in a number of ways.For the research grant to finance this study, grateful acknowledgment is given to the Graduate School, the University of Minnesota.     

A simple test for heterogeneity of variance in complex factorial designs

A simple procedure for testing heterogeneity of variance is developed which generalizes readily to complex, multi-factor experimental designs. Monte Carlo Studies indicate that the Z-variance test statistic presented here yields results equivalent to other familiar tests for heterogeneity of variance in simple one-way designs where comparisons are feasible. The primary advantage of the Z-variance test is in the analysis of factorial effects on sample variances in more complex designs. An example involving a three-way factorial design is presented.     

Multilevel factorial experiments for developing behavioral interventions: power, sample size, and resource considerations

Factorial experimental designs have many potential advantages for behavioral scientists. For example, such designs may be useful in building more potent interventions by helping investigators to screen several candidate intervention components simultaneously and to decide which are likely to offer greater benefit before evaluating the intervention as a whole. However, sample size and power considerations may challenge investigators attempting to apply such designs, especially when the population of interest is multilevel (e.g., when students are nested within schools, or when employees are nested within organizations). In this article, we examine the feasibility of factorial experimental designs with multiple factors in a multilevel, clustered setting (i.e., of multilevel, multifactor experiments). We conduct Monte Carlo simulations to demonstrate how design elements-such as the number of clusters, the number of lower-level units, and the intraclass correlation-affect power. Our results suggest that multilevel, multifactor experiments are feasible for factor-screening purposes because of the economical properties of complete and fractional factorial experimental designs. We also discuss resources for sample size planning and power estimation for multilevel factorial experiments. These results are discussed from a resource management perspective, in which the goal is to choose a design that maximizes the scientific benefit using the resources available for an investigation.     

Confusing developmental and individual differences

T. Globerson (1985, Developmental Review, 5, 261–273) compared the relative effects of an individual difference variable field dependency (FD/I) with a developmental variable (age) on a measure of M-capacity, in an analysis of variance (ANOVA) design. Age is a highly significant effect and the sum of squares associated with this factor accounts for 66% of the variance in M-capacity. FD/I is not significant and accounts for less than 1% of the variance in M-capacity. This is interpreted as indicating that M-capacity is a purely developmental phenomenon showing negligible individual differences. This paper argues that this result is an inevitable artifact of the experimental design. Further, the design is a common one in developmental psychology, and its dangers are largely unrecognized. The main problems concern (1) dichotomizing a continuous variable and treating the dichotomy as two levels of a factorial variable, (2) drawing inferences about the size of relationship between the continuous variable and the dependent variable on the basis of the variance accounted for by the factorial variable in the ANOVA, (3) differential reliability of measures of the independent variables, and (4) selective control of the sampling of the independent variables. These issues are explored in a simulation of Globerson's data and in an analogous analysis of data relating age and height, as independent variables, with weight as a dependent variable.     

Contribution of content, context, and process to understanding openness to organizational change: two experimental simulation studies

The authors examined the contribution of the content, context, and process of organizational transformation to employees' openness to change. The authors predicted that 5 factors would have a positive effect on openness to change: (a) threatening character of organizational change (content related), (b) trust in executive management (context related), (c) trust in the supervisor (context related), (d) history of change (context related), and (e) participation in the change effort (process related). The authors tested their hypotheses in 2 separate studies (N = 828 and N = 835) using an experimental simulation strategy. The first study crossed 4 variables in a completely randomized 2 x 2 x 2 x 2 factorial design. Results showed significant main effects for content, context, and process but no significant interaction effects. A second study, with a completely randomized 2 x 2 factorial design, crossed two context variables. Results showed a significant main and an interaction effect: Openness to change decreased dramatically only when history of change and trust in executive management were low.     

Investigations of perception and imagery using CAEs: The role of experimental design and psychophysical method

Orientation-contingent color aftereffects (CAEs) were measured using 174 observers ran-domly allocated to one of eight experimental conditions formed by the factorial combination of three two-level factors. These were experimental design (pretest and posttest measures taken vs. posttest measures taken alone), psychophysical method (forced-choice vs. magnitude estimation), and inspection procedure (inspection of real contours vs. inspection of imagined contours). In addition to color responses, written introspective reports of the strategies used in providing color responses were obtained. As a basis for the rejection of observers using inappropriate strategies, reported strategies were coded as appropriate or inappropriate. An analysis of the color reports of all observers was performed, as well as two analyses of the color reports of only those observers identified as using appropriate response strategies. In all cases, the results showed that CAEs were reported after inspection of real contours irrespective of the experimental design or psychophysical method used. There were no significant CAEs reported in any conditions that involved the imagination of contours. A separate analysis of the relationship between observers’ response strategies, the coding of these strategies, and observers’ actual color reports revealed some problems concerning the use of such introspective techniques. The implications of these findings are discussed in terms of theoretical assumptions about the nature of imagery and perception.     

Further evidence of the invalidity of the magnitude estimation method

IntroductionThe magnitude estimation method is widely applied in diverse areas of psychology. However, Weiss (1972) found that this method may produce judgments that inaccurately represent subjective magnitudes: constructing the stimuli according to a factorial design allowed the bias caused by magnitude estimation to be exposed as a spurious divergence of factorial curves.ObjectiveThe present study experimentally compared magnitude estimation with graphic rating and ratio estimation. The comparison of magnitude estimation with graphic rating retested Weiss's finding. The comparison of magnitude estimation with ratio estimation explored the hypothesis that magnitude estimation may cause a divergence of factorial curves due to the requirement of magnitude estimation to multiply and divide numbers—both ratio estimation and magnitude estimation required participants to judge ratios, but only magnitude estimation required mental multiplication and division.MethodUsing a 3 × 4 factorial experimental design, an achromatic disk was presented concentrically with a larger achromatic square. The brightness of the disk varied with the luminance of the disk and, due to simultaneous contrast, with the luminance of the square. Three different groups of participants judged the brightness of the disk: one group by graphic rating, another by ratio estimation, and the third by magnitude estimation.ResultsIn accord with Weiss's results, graphic rating yielded roughly parallel factorial curves whereas magnitude estimation yielded diverging factorial curves. Ratio estimation also yielded roughly parallel factorial curves, supporting the hypothesis that the magnitude estimation task itself may cause divergence of factorial curves due to the requirement of this method to mentally multiply and divide numbers.ConclusionCaution should be taken in using magnitude estimation, especially in factorial experiments, because this method is likely to cause the researcher to make incorrect inferences about the functional relations between the variables involved in mental information integration. Ratio estimation yields a pattern of factorial curves essentially equivalent to that obtained by graphic rating.     

Evaluation apprehension,hypothesis awareness,and the weapons effect

The present study was designed to extend Turner and Simons' (1974) investigation of possible experimental artifacts in aggression research. Employing a 2 × 2 × 2 factorial design, 60 male subjects were given either a low or high evaluation apprehension treatment, and then were either exposed or not exposed to weapons in an otherwise close replication of the seven shock-unassociated weapons condition of Berkowitz and LePage (1967). Based on an objectively worded postexperimental questionnaire, subjects were classified as being aware or not aware that the procedures were designed to make them give more shock to their partners. The findings, which were consistent with the analysis of Berkowitz (1974) and Turner and Simons (1974), indicated that the weapons-no weapons comparison was significant only for non-apprehensive, nonaware subjects. The results were interpreted as suggesting that evaluation apprehension, hypothesis awareness, and sophistication in aggression experiments may produce inhibitions which can modify the effects of experimental manipulations.     

Experimental components analysis of brief theory-based HIV/AIDS risk-reduction counseling for sexually transmitted infection patients.

An experimental components analysis of brief HIV risk-reduction counseling based on the information-motivation-behavioral skills (IMB) model was conducted with 432 men and 193 women receiving sexually transmitted infection (STI) clinic services. Following baseline assessments, participants were randomly assigned to 1 of 4 90-min risk-reduction counseling sessions that deconstructed the IMB model within a full factorial design. Participants were followed for 9 months, with STI diagnoses monitored over 12 months. Men who received the full IMB session evidenced relatively greater use of risk-reduction behavioral skills and relatively lower rates of unprotected intercourse over 6-months follow-up and had fewer new STIs. For women, however, the motivational counseling demonstrated the most positive outcomes. Results suggest that brief single-exposure HIV prevention counseling can reduce HIV transmission risks.     

Combining effect size estimates in meta-analysis with repeated measures and independent-groups designs

When a meta-analysis on results from experimental studies is conducted, differences in the study design must be taken into consideration. A method for combining results across independent-groups and repeated measures designs is described, and the conditions under which such an analysis is appropriate are discussed. Combining results across designs requires that (a) all effect sizes be transformed into a common metric, (b) effect sizes from each design estimate the same treatment effect, and (c) meta-analysis procedures use design-specific estimates of sampling variance to reflect the precision of the effect size estimates.     

Paradoxical effects of stimulation on normal neonates

To determine the effects of unimodal and multimodal stimulation on early development and maternal perception of newborn behavior, 81 full-term healthy newborns were randomly assigned to 1 of 3 experimental or control conditions. An expanded Solomon four-group design was used, resulting in a 2×4 factorial analysis. Unimodal stimulation was provided through daily performance of a stroking procedure, whereas multimodal stimulation was offered through use of a multisensory hammock during expected sleep cycles. At the end of a 1-month intervention period, the Brazelton Neonatal Behavioral Assessment Scale (NBAS) and the Neonatal Perception Inventory (NPI) were administered and weight gain was assessed. On the Brazelton Scale, treated infants showed less mature orientation, motor, and state regulation behaviors than the control group. Weight gain was not enhanced. A positive correlation was found between quantity of multimodal stimulation and favorable maternal perception of infant behavior.     

The effects of competition and social reinforcement upon perceptual motor performance

Junior high school boys (n=180) performed 40 trials on the ball roll-up task in order to determine whether competition, initial ability level, and social reinforcement interact to influence performance. A 3 x 2 x 3 x 8 factorial design, with repeated measures on the last factor (initial ability level x competition x social reinforcement x blocks of trials) was used. The analysis of the data demonstrated that: (a) competition, initial ability level, and social reinforcement factors did not interact with one another; (b) social reinforcement (reproof) facilitated performance, while competition had no effect; and (c) the competition group had a greater tonic heart rate (increase from basal) than the noncompetitive group. Although it was not entirely clear why changes in the performance of the experimental groups did not accompany the changes in the groups' tonic heart rate, it was concluded that competition was an incentive.     

The Effect Of Shifts In Fixed-Interval Schedules On Acquisition,Extinction, Spontaneous Recovery,And Reacquisition

Using both multivariate and univariate statistical tests, the effect of two opposite patterns of primary reinforcement schedules upon the acquisition, extinction, spontaneous recovery and reacquisition of bar pressing was determined. Two control and two experimental groups, each consisting of eight Ss, formed the standard 2x2 factorial design for the experiment. The main effects were (a) effect of primary reinforcement (experimental vs. control groups) and (b) direction of shift in the schedule values (ascending vs. descending groups). The extinction data showed that the experimental group exposed to the ascending-order shift pressed the bar more than the descending group. Therefore, persistence of nonreinforced responding is a function of the direction of shift in fixed-interval values. The recovery data for first day indicated that primary reinforcements following either pattern facilitated spontaneous recovery. However, on second day neither direction of shift nor primary reinforcement had any significant effect on it. Covariance analysis of recovery and reacquisition data for second day revealed that the experimental ascending group showed better recovery and reacquisition when the frequency of prior nonreinforced responding was held constant. The SL analyses suggest a higher acquisition level for the experimental ascending group than for the other groups. It is possible to explain the result in terms of Hull's habit- strength theory.     

Three-way metric unfolding via alternating weighted least squares

Three-way unfolding was developed by DeSarbo (1978) and reported in DeSarbo and Carroll (1980, 1981) as a new model to accommodate the analysis of two-mode three-way data (e.g., nonsymmetric proximities for stimulus objects collected over time) and three-mode, three-way data (e.g., subjects rendering preference judgments for various stimuli in different usage occasions or situations). This paper presents a revised objective function and new algorithm which attempt to prevent the common type of degenerate solutions encountered in typical unfolding analysis. We begin with an introduction of the problem and a review of three-way unfolding. The three-way unfolding model, weighted objective function, and new algorithm are presented. Monte Carlo work via a fractional factorial experimental design is described investigating the effect of several data and model factors on overall algorithm performance. Finally, three applications of the methodology are reported illustrating the flexibility and robustness of the procedure.We wish to thank the editor and reviewers for their insightful comments.     

An experimental test of stimulus estimation theory: danger and safety with snake phobic stimuli

The stimulus estimation model (Taylor & Rachman, 1994) asserts that fear overprediction stems from: (a) overprediction of the danger elements of a phobic stimulus, and (b) underprediction of existing safety resources. Using a 2x2 factorial design, with danger (high vs low) and safety (high vs low) as between-subjects variables, an experimental test of the model was conducted with 25 snake-fearful participants per condition. The four experimental conditions were matched on initial levels of snake fearfulness, as assessed by the Snake Questionnaire (SNAQ). For the 51 participants who demonstrated overprediction of fear, high danger led to reliably more fear overprediction than low danger; and low safety led to reliably more fear overprediction than high safety. The interaction between danger and safety was not statistically significant. The results offer the first convincing experimental support for the stimulus estimation model of fear overprediction.     

Reliability and validity of the Padua Inventory in an Iranian population

In order to study the reliability and validity of the Padua Inventory (PI), its items were translated into Persian and then back-translated into English as suggested by Brislin, Lonner, Thorndike, Cross-cultural research methods, Wiley, New York, 1973. Using the translated version of the PI, two studies were carried out. The first study included 219 healthy participants selected from Shiraz city in Iran to examine the reliability and factorial structure of the PI via test-retest, internal consistency and principal component analysis methods. The results of this study showed good reliability for the PI and confirmed its factorial structure reported by previous studies. The second study included four groups of participants as follows: patients with obsessive-compulsive disorder (OCD), patients with anxiety disorders, patients with depressive disorder and normal individuals. The PI scores were able to differentiate between OCD patients and normal individuals, but failed to differentiate between neurotic and OCD patients. Two subscales of PI-R version of the PI (precision and washing) differentiated depressive and OCD patients.     

Studentized maximum root procedures for coherent analyses of two-factor fixed-effects designs

The authors provide generalizations of R. J. Boik's (1993) studentized maximum root (SMR) procedure that allow for simultaneous inference on families of product contrasts including simple effect contrasts and differences among simple effect contrasts in coherent analyses of data from 2-factor fixed-effects designs. Unlike the F-based simultaneous test procedures (STPs) proposed by M. A. Betz and K. R. Gabriel (1978) for coherent analyses allowing for inferences on all factorial contrasts, SMR STPs are designed for analyses where each contrast of interest is a product contrast, which is usually the case in analyses of data from factorial experiments. When both factors have more than 2 levels, SMR STPs always provide more power and precision than F STPs for inferences on product contrasts.     

Indian college students' attitudes toward abortion in different contexts

Two experimental studies used the hypothetical situation of a pregnant woman (the stimulus person) considering the possibility of an abortion and obtained from Bombay college students estimates of the likelihood of an abortion and their own recommendations in favor of abortion. The first study had a 2 (subject's sex) × 2 (reason for abortion: unplanned child vs. female child) × 2 (person more in favor of abortion: stimulus person vs. her husband) × 3 (stimulus person's socioeconomic status: upper, middle, or lower class) factorial design with 15 subjects per cell. Male subjects indicated greater likelihood of abortion than female subjects. Recommendations were stronger for the unplanned rather than female child and for the lower class rather than upper and middle class stimulus persons. The second study had a 2 (subject's sex) × 3 (reason for abortion: unplanned marital pregnancy, premarital pregnancy, or rape pregnancy) × 3 (stimulus person's socioeconomic status: upper, middle, or lower class) factorial design with 15 subjects per cell. Perceived likelihood and strength of recommendation were highest for rape pregnancy followed by premarital pregnancy and marital pregnancy in that order. The interactions in this study suggested a greater vulnerability of the middle class as compared to the other two classes with respect to sexual morality.     

The High Cost of Complexity in Experimental Design and Data Analysis: Type I and Type II Error Rates in Multiway ANOVA

The availability of statistical software packages has led to a sharp increase in use of complex research designs and complex statistical analyses in communication research. An informal examination of studies from 2 leading communication journals suggests that the analysis of variance (ANOVA) is often the statistic of choice, and a substantial proportion of published research reports using ANOVA employ complex (k ≥ 3) factorial designs, often involving multiple dependent variables. This article reports a series of Monte Carlo simulations which demonstrate that this complexity may come at a heavier cost than many communication researchers realize. As frequently used, complex factorial ANOVA yield Type I and Type II error rates that many communication scholars would likely consider unacceptable. Consequently, quality of statistical inference in many studies is highly suspect. Communication researchers are warned about problems associated with design and statistical complexity and solutions are suggested.