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B. Zumbo, A. Rupp (2004)
Responsible Modeling of Measurement Data for Appropriate Inferences: Important Advances in Reliability and Validity Theory
L. Guttman (1945)
A basis for analyzing test-retest reliabilityPsychometrika, 10
S. Green, S. Hershberger (2000)
Correlated Errors in True Score Models and Their Effect on Coefficient AlphaStructural Equation Modeling: A Multidisciplinary Journal, 7
W. Kristof (1963)
The statistical theory of stepped-up reliability coefficients when a test has been divided into several equivalent partsPsychometrika, 28
S. Green, Yanyun Yang (2009)
Commentary on Coefficient Alpha: A Cautionary TalePsychometrika, 74
K. Barchard, A. Hakstian (1997)
The Effects of Sampling Model on Inference with Coefficient AlphaEducational and Psychological Measurement, 57
A. Duhachek, D. Iacobucci (2004)
Alpha's Standard Error (ASE): An Accurate and Precise Confidence Interval Estimate
M. Novick, Charles Lewis (1967)
Coefficient alpha and the reliability of composite measurements.Psychometrika, 32 1
K. Barchard, A. Hakstian (1997)
The Robustness of Confidence Intervals for Coefficient Alpha Under Violation of the Assumption of Essential Parallelism.Multivariate behavioral research, 32 2
D. Zimmerman, B. Zumbo, Coralie Lalonde (1993)
Coefficient Alpha as an Estimate of Test Reliability Under Violation of Two AssumptionsEducational and Psychological Measurement, 53
K. Sijtsma (2008)
On the Use, the Misuse, and the Very Limited Usefulness of Cronbach’s AlphaPsychometrika, 74
B. Zumbo (1999)
A Glance at Coefficient Alpha with an Eye Towards Robustness Studies : Some Mathematical Notes and a Simulation Model
Michael Miller (1995)
Coefficient alpha: A basic introduction from the perspectives of classical test theory and structural equation modelingStructural Equation Modeling, 2
(1994)
Psychometric Theory, 3rd Edn
Todd Headrick (2002)
Fast fifth-order polynomial transforms for generating univariate and multivariate nonnormal distributionsComputational Statistics & Data Analysis, 40
Sharon Shultz (1993)
A Monte Carlo study of the robustness of coefficient alpha.
J. Zyl, H. Neudecker, D. Nel (2000)
On the distribution of the maximum likelihood estimator of Cronbach's alphaPsychometrika, 65
Allen Fleishman (1978)
A method for simulating non-normal distributionsPsychometrika, 43
R. Caplan, R. Naidu, R. Tripathi (1984)
Coping and defense: constellations vs. components.Journal of health and social behavior, 25 3
G. Johanson, G. Brooks (2010)
Initial Scale Development: Sample Size for Pilot StudiesEducational and Psychological Measurement, 70
K. Bay (1973)
THE EFFECT OF NON‐NORMALITY ON THE SAMPLING DISTRIBUTION AND STANDARD ERROR OF RELIABILITY COEFFICIENT ESTIMATES UNDER AN ANALYSIS OF VARIANCE MODELBritish Journal of Mathematical and Statistical Psychology, 26
(2010)
MATLAB (Version 7.11) [Computer software
J. Graham (2006)
Congeneric and (Essentially) Tau-Equivalent Estimates of Score ReliabilityEducational and Psychological Measurement, 66
M. Tatsuoka, F. Lord, M. Novick, A. Birnbaum (1971)
Statistical Theories of Mental Test Scores.Journal of the American Statistical Association, 66
Todd Headrick (2009)
Statistical Simulation: Power Method Polynomials and Other Transformations
J. Fleiss (1987)
The design and analysis of clinical experiments
T. Raykov (1998)
Coefficient Alpha and Composite Reliability With Interrelated Nonhomogeneous ItemsApplied Psychological Measurement, 22
L. Feldt (1965)
The approximate sampling distribution of Kuder-Richardson reliability coefficient twentyPsychometrika, 30
Jeff Lin (2006)
Design and Analysis of Clinical Trials
L. Cronbach (1951)
Coefficient alpha and the internal structure of testsPsychometrika, 16
ORIGINAL RESEARCH ARTICLE published: 15 February 2012 doi: 10.3389/fpsyg.2012.00034 Is coefﬁcient alpha robust to non-normal data? 1 2 Yanyan Sheng * and Zhaohui Sheng Department of Educational Psychology and Special Education, Southern Illinois University, Carbondale, IL, USA Department of Educational Leadership, Western Illinois University, Macomb, IL, USA Edited by: Coefﬁcient alpha has been a widely used measure by which internal consistency reliabil- Jason W. Osborne, Old Dominion ity is assessed. In addition to essential tau-equivalence and uncorrelated errors, normality University, USA has been noted as another important assumption for alpha. Earlier work on evaluating this Reviewed by: assumption considered either exclusively non-normal error score distributions, or limited Pamela Kaliski, College Board, USA conditions. In view of this and the availability of advanced methods for generating univariate James Stamey, Baylor University, USA non-normal data, Monte Carlo simulations were conducted to show that non-normal distri- *Correspondence: butions for true or error scores do create problems for using alpha to estimate the internal Yanyan Sheng, Department of consistency reliability. The sample coefﬁcient alpha is affected by leptokurtic true score Educational Psychology and Special distributions, or skewed and/or kurtotic error score distributions. Increased sample sizes, Education, Southern Illinois not test lengths, help improve the accuracy, bias, or precision of using it with non-normal University, Wham 223, MC 4618, Carbondale, IL 62901-4618, USA. data. e-mail: ysheng@siu.edu Keywords: coefﬁcient alpha, true score distribution, error score distribution, non-normality, skew, kurtosis, Monte Carlo, power method polynomials INTRODUCTION 2004; Graham, 2006; Green and Yang, 2009), which have been Coefﬁcient alpha (Guttman, 1945; Cronbach, 1951) has been one considered as two major assumptions for alpha. The normality of the most commonly used measures today to assess internal assumption, however, has received little attention. This could be consistency reliability despite criticisms of its use (e.g., Raykov, a concern in typical applications where the population coefﬁ- 1998; Green and Hershberger, 2000; Green and Yang, 2009; Sijtsma, cient is an unknown parameter and has to be estimated using the sample coefﬁcient. When data are normally distributed, sam- 2009). The derivation of the coefﬁcient is based on classical test theory (CTT; Lord and Novick, 1968), which posits that a person’s ple coefﬁcient alpha has been shown to be an unbiased estimate of the population coefﬁcient alpha (Kristof, 1963; van Zyl et al., observed score is a linear function of his/her unobserved true score (or underlying construct) and error score. In the theory, measures 2000); however, less is known about situations when data are can be parallel (essential) tau-equivalent, or congeneric, depend- non-normal. ing on the assumptions on the units of measurement, degrees of Over the past decades, the effect of departure from normality precision, and/or error variances. When two tests are designed to on the sample coefﬁcient alpha has been evaluated by Bay (1973), measure the same latent construct, they are parallel if they mea- Shultz (1993), and Zimmerman et al. (1993) using Monte Carlo sure it with identical units of measurement, the same precision, simulations. They reached different conclusions on the effect of and the same amounts of error; tau-equivalent if they measure it non-normal data. In particular, Bay (1973) concluded that a lep- with the same units, the same precision, but have possibly differ- tokurtic true score distribution could cause coefﬁcient alpha to seriously underestimate internal consistency reliability. Zimmer- ent error variance; essentially tau-equivalent if they assess it using the same units, but with possibly different precision and differ- man et al. (1993) and Shultz (1993), on the other hand, found that the sample coefﬁcient alpha was fairly robust to departure ent amounts of error; or congeneric if they assess it with possibly different units of measurement, precision, and amounts of error from the normality assumption. The three studies differed in the (Lord and Novick, 1968; Graham, 2006). From parallel to con- design, in the factors manipulated and in the non-normal distribu- generic, tests are requiring less strict assumptions and hence are tions considered, but each is limited in certain ways. For example, becoming more general. Studies (Lord and Novick, 1968, pp. 87– Zimmerman et al. (1993) and Shultz (1993) only evaluated the 91; see also Novick and Lewis, 1967, pp. 6–7) have shown formally effect of non-normal error score distributions. Bay (1973), while that the population coefﬁcient alpha equals internal consistency looked at the effect of non-normal true score or error score dis- reliability for tests that are tau-equivalent or at least essential tau- tributions, only studied conditions of 30 subjects and 8 test items. equivalent. It underestimates the actual reliability for the more Moreover, these studies have considered only two or three scenar- ios when it comes to non-normal distributions. Speciﬁcally, Bay general congeneric test. Apart from essential tau-equivalence, coef- ﬁcient alpha requires two additional assumptions: uncorrelated (1973) employed uniform (symmetric platykurtic) and exponen- errors (Guttman, 1945; Novick and Lewis, 1967) and normality tial (non-symmetric leptokurtic with positive skew) distributions (e.g., Zumbo, 1999). Over the past decades, studies have well doc- for both true and error scores. Zimmerman et al. (1993) generated umented the effects of violations of essential tau-equivalence and error scores from uniform, exponential, and mixed normal (sym- uncorrelated errors (e.g., Zimmerman et al., 1993; Miller, 1995; metric leptokurtic) distributions, while Shultz (1993) generated Raykov, 1998; Green and Hershberger, 2000; Zumbo and Rupp, them using exponential, mixed normal, and negative exponential www.frontiersin.org February 2012 | Volume 3 | Article 34 | 1 Sheng and Sheng Effect of non-normality on coefﬁcient alpha (non-symmetric leptokurtic with negative skew) distributions. for tau-equivalence, and Since the presence of skew and/or kurtosis determines whether and how a distribution departs from the normal pattern, it is X = t + υ + e , (3) ij i j ij desirable to consider distributions with varying levels of skew and kurtosis so that a set of guidelines can be provided. Gen- where Σ υ = 0, for essential tau-equivalence. j j erating univariate non-normal data with speciﬁed moments can Summing across k items, we obtain a composite score (X ) i+ be achieved via the use of power method polynomials (Fleish- and a scale error score (e ). The variance of the composite scores i+ man, 1978), and its current developments (e.g., Headrick, 2010) is then the summation of true score and scale error score variances: make it possible to consider more combinations of skew and 2 2 2 kurtosis. σ = σ + σ . (4) X t e + + Further, in the actual design of a reliability study, sample size determination is frequently an important and difﬁcult aspect. The The reliability coefﬁcient, ρ , is deﬁned as the proportion of XX literature offers widely different recommendations, ranging from composite score variance that is due to true score variance: 15 to 20 (Fleiss, 1986), a minimum of 30 (Johanson and Brooks, 2010) to a minimum of 300 (Nunnally and Bernstein, 1994). ρ = . (5) XX Although Bay (1973) has used analytical derivations to suggest that coefﬁcient alpha shall be robust against the violation of the normality assumption if sample size is large, or the number of Under (essential) tau-equivalence, that is, for models in (2) and items is large and the true score kurtosis is close to zero, it is (3), the population coefﬁcient alpha, deﬁned as never clear how many subjects and/or items are desirable in such situations. j=j X X k j In view of the above, the purpose of this study is to investigate α = , k − 1 σ the effect of non-normality (especially the presence of skew and/or + kurtosis) on reliability estimation and how sample sizes and test or lengths affect the estimation with non-normal data. It is believed ⎛ ⎞ that the results will not only shed insights on how non-normality k 2 j=1 j affects coefﬁcient alpha, but also provide a set of guidelines for ⎝ ⎠ α = 1 − , (6) k − 1 researchers when specifying the numbers of subjects and items in a reliability study. is equal to the reliability as deﬁned in (5). As was noted, ρ and XX MATERIALS AND METHODS α focus on the amount of random error and do not evaluate error This section starts with a brief review of the CTT model for coef- that may be systematic. ﬁcient alpha. Then the procedures for simulating observed scores Although the derivation of coefﬁcient alpha based on Lord and used in the Monte Carlo study are described, followed by measures Novick (1968) does not require distributional assumptions for t that were used to evaluate the performance of the sample alpha in and e , its estimation does (see Shultz, 1993; Zumbo, 1999), as the ij each simulated situation. sample coefﬁcient alpha estimated using sample variances s , PRELIMINARIES ⎛ ⎞ Coefﬁcient alpha is typically associated with true score the- 2 j=1 j ⎝ ⎠ ory (Guttman, 1945; Cronbach, 1951; Lord and Novick, 1968), α ˆ = 1 − ,(7) k − 1 s where the test score for person i on item j, denoted as X ,is + ij assumed to be a linear function of a true score (t ) and an error ij score (e ): is shown to be the maximum likelihood estimator of the popu- ij lation alpha assuming normal distributions (Kristof, 1963; van X = t + e , (1) Zyl et al., 2000). Typically, we assume t ∼ N (μ , σ ) and ij ij ij i t 2 2 e ∼ N (0, σ ),where σ has to be differentiated from the scale ij e e i = 1, ..., n and j = 1, ..., k,where E(e ) = 0, ρ = 0, and ij te error score variance σ deﬁned in (4). ρ = 0. Here, e denotes random error that reﬂects unpre- e ,e ij ij ij dictable trial-by-trial ﬂuctuations. It has to be differentiated from STUDY DESIGN systematic error that reﬂects situational or individual effects that To evaluate the performance of the sample alpha as deﬁned in may be speciﬁed. In the theory, items are usually assumed to be tau- (7) in situations where true score or error score distributions equivalent, where true scores are restricted to be the same across depart from normality, a Monte Carlo simulation study was car- items, or essentially tau-equivalent, where they are allowed to dif- ried out, where test scores of n persons (n = 30, 50, 100, 1000) for k fer from item to item by a constant (υ ). Under these conditions items (k = 5, 10, 30) were generated assuming tau-equivalence and (1) becomes where the population reliability coefﬁcient (ρ ) was speciﬁed to XX be 0.3, 0.6, or 0.8 to correspond to unacceptable, acceptable, or X = t + e (2) very good reliability (Caplan et al., 1984, p. 306; DeVellis, 1991, ij i ij Frontiers in Psychology | Quantitative Psychology and Measurement February 2012 | Volume 3 | Article 34 | 2 Sheng and Sheng Effect of non-normality on coefﬁcient alpha p. 85; Nunnally, 1967, p. 226). These are referred to as small, 4. c =−0.446924, c = 1.242521, c = 0.500764, c =−0.184710, 0 1 2 3 moderate, and high reliabilities in subsequent discussions. Speciﬁ- c =−0.017947, c = 0.003159; 4 5 cally,truescores(t )and errorscores(e ) were simulated from their 5. c =−0.276330, c = 1.506715, c = 0.311114, c =−0.274078, i ij 0 1 2 3 σ ρ c =−0.011595, c = 0.007683; 2 2 e XX 4 5 respective distributions with σ = 1, μ = 5 and σ = . e t (1−ρ )k XX 6. c =−0.304852, c = 0.381063, c = 0.356941, c = 0.132688, 0 1 2 3 The observed scores (X ) were subsequently obtained using ij c =−0.017363, c = 0.003570. 4 5 Eq. (2). In addition, true score or error score distributions were manip- It is noted that the effect of the true score or error score distrib- ulated to be symmetric (so that skew, γ , is 0) or non-symmetric ution was investigated independently, holding the other constant (γ > 0) with kurtosis (γ ) being 0, negative or positive. It is 1 2 by assuming it to be normal. noted that only positively skewed distributions were considered Hence, a total of 4 (sample sizes) × 3 (test lengths) × 3 (lev- in the study because due to the symmetric property, negative els of population reliability) × 6 (distributions) × 2 (true or error skew should have the same effect as positive skew. Generating score) = 432 conditions were considered in the simulation study. non-normal distributions in this study involves the use of power Each condition involved 100,000 replications, where coefﬁcient method polynomials. Fleishman (1978) introduced this popu- alpha was estimated using Eq. (7) for simulated test scores (X ). ij lar moment matching technique for generating univariate non- The 100,000 estimates of α can be considered as random samples normal distributions. Headrick (2002, 2010) further extended from the sampling distribution of α ˆ , and its summary statistics from third-order to ﬁfth-order polynomials to lower the skew including the observed mean, SD, and 95% interval provide infor- and kurtosis boundary. As is pointed out by Headrick (2010, p. mation about this distribution. In particular, the observed mean 26), for distributions with a mean of 0 and a variance of 1, the indicates whether the sample coefﬁcient is biased. If it equals α, α ˆ skew and kurtosis have to satisfy γ γ − 2, and hence it is is unbiased; otherwise, it is biased either positively or negatively not plausible to consider all possible combinations of skew and depending on whether it is larger or smaller than α. The SD of kurtosis using power method polynomials. Given this, six distrib- the sampling distribution is what we usually call the SE. It reﬂects utions with the following combinations of skew and kurtosis were the uncertainty in estimating α, with a smaller SE suggesting more considered: precision and hence less uncertainty in the estimation. The SE is directly related to the 95% observed interval, as the larger it is, the 1. γ = 0, γ = 0 (normal distribution); 1 2 more spread the distribution is and the wider the interval will be. 2. γ = 0, γ =− 1.385 (symmetric platykurtic distribution); 1 2 With respect to the observed interval, it contains about 95% of 3. γ = 0, γ = 25 (symmetric leptokurtic distribution); 1 2 α ˆ around its center location from its empirical sampling distrib- 4. γ = 0.96, γ = 0.13 (non-symmetric distribution); 1 2 ution. If α falls inside the interval, α ˆ is not signiﬁcantly different 5. γ = 0.48, γ =− 0.92 (non-symmetric platykurtic distribu- 1 2 from α even though it is not unbiased. On the other hand, if α tion); falls outside of the interval, which means that 95% of the esti- 6. γ = 2.5, γ = 25 (non-symmetric leptokurtic distribution). 1 2 mates differ from α, we can consider α ˆ to be signiﬁcantly different from α. A normal distribution was included so that it could be used as In addition to these summary statistics, the accuracy of the a baseline against which the non-normal distributions could be estimate was evaluated by the root mean square error (RMSE) and compared. To actually generate univariate distributions using the bias, which are deﬁned as ﬁfth-order polynomial transformation, a random variate Z is ﬁrst generated from a standard normal distribution, Z∼ N (0,1). Then α ˆ − α the following polynomial, RMSE = , (9) 100, 000 2 3 4 5 Y = c + c Z + c Z + c Z + c Z + c Z (8) 0 1 2 3 4 5 and is used to obtain Y. With appropriate coefﬁcients (c , ..., c ), Y 0 5 α ˆ − α bias = , (10) would follow a distribution with a mean of 0, a variance of 1, and 100, 000 the desired levels of skew and kurtosis (see Headrick, 2002, for a detailed description of the procedure). A subsequent linear trans- respectively. The larger the RMSE is, the less accurate the sample formation would rescale the distribution to have a desired location coefﬁcient is in estimating the population coefﬁcient. Similarly, or scale parameter. In this study, Y could be the true score (t )or i the larger the absolute value of the bias is, the more bias the sam- the error score (e ). For the six distributions considered for t or ple coefﬁcient involves. As the equations suggest, RMSE is always ij i e herein, the corresponding coefﬁcients are: ij positive, with values close to zero reﬂecting less error in estimating the actual reliability. On the other hand, bias can be negative or 1. c = 0, c = 1, c = 0, c = 0, c = 0, c = 0; positive. A positive bias suggests that the sample coefﬁcient tends 0 1 2 3 4 5 2. c = 0, c = 1.643377, c = 0, c =−0.319988, c = 0, c = to overestimate the reliability, and a negative bias suggests that it 0 1 2 3 4 5 0.011344; tends to underestimate the reliability. In effect, bias provides simi- 3. c = 0, c = 0.262543, c = 0, c = 0.201036, c = 0, c = lar information as the observed mean of the sampling distribution 0 1 2 3 4 5 0.000162; of α ˆ . www.frontiersin.org February 2012 | Volume 3 | Article 34 | 3 Sheng and Sheng Effect of non-normality on coefﬁcient alpha RESULTS distribution to determine if α ˆ was affected by non-normality in The simulations were carried out using MATLAB (MathWorks, true scores. Take the condition where a test of 5 items with the 2010), with the source code being provided in the Section “Appen- actual reliability being 0.3 was given to 30 persons as an example. dix.” Simulation results are summarized in Tables 1–3 for condi- A normal distribution resulted in an observed mean of 0.230 and tions where true scores follow one of the six distributions speciﬁed a SE of 0.241 for the sampling distribution of α ˆ (see Table 1). in the previous section. Here, results from the ﬁve non-normal Compared with it, a symmetric platykurtic distribution, with an distributions were mainly compared with those from the normal observed mean of 0.234 and a SE of 0.235, did not differ much. Table 1 | Observed mean and SD of the sample alpha (α ˆ ) for the simulated situations where the true score (t ) distribution is normal or non-normal. nk Mean (α ˆ ) SD (α ˆ ) dist1 dist2 dist3 dist4 dist5 dist6 dist1 dist2 dist3 dist4 dist5 dist6 ρ = 0.3 XX 30 5 0.230 0.234 0.198 0.230 0.231 0.201 0.241 0.235 0.290 0.242 0.237 0.288 10 0.231 0.234 0.199 0.229 0.233 0.202 0.229 0.223 0.278 0.230 0.224 0.276 30 0.231 0.233 0.199 0.230 0.233 0.200 0.221 0.215 0.269 0.222 0.216 0.270 50 5 0.252 0.253 0.233 0.253 0.254 0.232 0.176 0.172 0.214 0.177 0.172 0.214 10 0.252 0.256 0.232 0.252 0.254 0.233 0.166 0.161 0.205 0.166 0.162 0.204 30 0.254 0.254 0.231 0.252 0.254 0.233 0.160 0.156 0.202 0.160 0.157 0.199 100 5 0.269 0.269 0.258 0.268 0.269 0.258 0.118 0.116 0.148 0.119 0.117 0.148 10 0.268 0.269 0.257 0.269 0.270 0.258 0.112 0.109 0.143 0.112 0.110 0.142 30 0.269 0.270 0.257 0.268 0.269 0.256 0.108 0.105 0.141 0.108 0.106 0.140 1000 5 0.282 0.282 0.281 0.282 0.282 0.281 0.036 0.035 0.048 0.036 0.035 0.048 10 0.282 0.282 0.281 0.282 0.282 0.281 0.034 0.033 0.046 0.034 0.033 0.046 30 0.282 0.282 0.281 0.282 0.282 0.281 0.033 0.032 0.045 0.033 0.032 0.045 ρ = 0.6 XX 30 5 0.549 0.556 0.479 0.549 0.554 0.482 0.142 0.125 0.239 0.142 0.131 0.238 10 0.551 0.557 0.481 0.549 0.554 0.480 0.133 0.117 0.232 0.136 0.122 0.232 30 0.550 0.557 0.480 0.550 0.555 0.481 0.129 0.112 0.230 0.131 0.118 0.229 50 5 0.563 0.567 0.517 0.563 0.566 0.517 0.103 0.092 0.179 0.104 0.095 0.180 10 0.563 0.567 0.516 0.563 0.566 0.517 0.097 0.086 0.176 0.098 0.089 0.174 30 0.563 0.567 0.516 0.563 0.566 0.518 0.093 0.082 0.174 0.094 0.086 0.172 100 5 0.572 0.574 0.545 0.572 0.573 0.546 0.069 0.062 0.128 0.070 0.065 0.126 10 0.572 0.574 0.545 0.572 0.573 0.547 0.066 0.057 0.126 0.066 0.060 0.124 30 0.572 0.574 0.545 0.572 0.573 0.546 0.063 0.055 0.124 0.064 0.058 0.122 1000 5 0.580 0.580 0.576 0.580 0.580 0.577 0.021 0.019 0.043 0.021 0.020 0.042 10 0.580 0.580 0.577 0.580 0.580 0.576 0.020 0.018 0.042 0.020 0.018 0.042 30 0.580 0.580 0.576 0.580 0.580 0.576 0.019 0.017 0.042 0.019 0.018 0.041 ρ = 0.8 XX 30 5 0.771 0.778 0.701 0.770 0.776 0.703 0.072 0.056 0.171 0.075 0.062 0.172 10 0.771 0.778 0.702 0.770 0.776 0.702 0.068 0.052 0.167 0.070 0.057 0.169 30 0.771 0.778 0.701 0.771 0.776 0.702 0.066 0.049 0.166 0.068 0.055 0.167 50 5 0.778 0.782 0.733 0.778 0.780 0.733 0.052 0.041 0.125 0.053 0.045 0.125 10 0.778 0.782 0.733 0.778 0.781 0.733 0.049 0.038 0.123 0.050 0.042 0.123 30 0.778 0.782 0.732 0.778 0.781 0.733 0.048 0.036 0.122 0.049 0.040 0.122 100 5 0.782 0.784 0.757 0.782 0.784 0.757 0.035 0.028 0.086 0.036 0.031 0.085 10 0.783 0.784 0.757 0.782 0.784 0.757 0.033 0.026 0.085 0.034 0.028 0.084 30 0.783 0.784 0.757 0.782 0.784 0.757 0.032 0.024 0.084 0.033 0.027 0.084 1000 5 0.786 0.787 0.783 0.786 0.787 0.783 0.011 0.009 0.028 0.011 0.009 0.027 10 0.786 0.787 0.783 0.787 0.787 0.783 0.010 0.008 0.028 0.010 0.009 0.027 30 0.787 0.787 0.783 0.786 0.787 0.784 0.010 0.007 0.027 0.010 0.008 0.027 dist1, Normal distribution for t ; dist2, distribution with negative kurtosis for t ; dist3, distribution with positive kurtosis for t ; dist4, skewed distribution for t ; dist5, i i i i skewed distribution with negative kurtosis for t ; dist6, skewed distribution with positive kurtosis for t . i i Frontiers in Psychology | Quantitative Psychology and Measurement February 2012 | Volume 3 | Article 34 | 4 Sheng and Sheng Effect of non-normality on coefﬁcient alpha Table 2 | Root mean square error and bias for estimating α for the simulated situations where the true score (t ) distribution is normal or non-normal. nk RMSE bias dist1 dist2 dist3 dist4 dist5 dist6 dist1 dist2 dist3 dist4 dist5 dist6 ρ = 0.3 XX 30 5 0.251 0.244 0.308 0.252 0.247 0.305 −0.070 −0.066 −0.102 −0.070 −0.069 −0.100 10 0.240 0.233 0.296 0.241 0.234 0.292 −0.069 −0.067 −0.101 −0.071 −0.067 −0.098 30 0.232 0.226 0.287 0.232 0.226 0.288 −0.069 −0.067 −0.101 −0.070 −0.067 −0.101 50 5 0.182 0.178 0.224 0.183 0.178 0.224 −0.048 −0.047 −0.067 −0.047 −0.046 −0.068 10 0.173 0.167 0.216 0.173 0.169 0.215 −0.048 −0.044 −0.068 −0.048 −0.046 −0.067 30 0.166 0.162 0.213 0.167 0.164 0.210 −0.046 −0.046 −0.069 −0.048 −0.046 −0.067 100 5 0.122 0.120 0.154 0.123 0.121 0.154 −0.031 −0.031 −0.042 −0.032 −0.031 −0.042 10 0.116 0.114 0.149 0.116 0.114 0.148 −0.032 −0.031 −0.043 −0.031 −0.031 −0.042 30 0.112 0.109 0.147 0.113 0.110 0.147 −0.031 −0.030 −0.043 −0.032 −0.031 −0.044 1000 5 0.040 0.040 0.052 0.041 0.040 0.051 −0.018 −0.018 −0.019 −0.018 −0.018 −0.019 10 0.038 0.038 0.050 0.039 0.038 0.050 −0.018 −0.018 −0.019 −0.018 −0.018 −0.020 30 0.038 0.037 0.049 0.038 0.037 0.049 −0.018 −0.018 −0.019 −0.018 −0.018 −0.019 ρ = 0.6 XX 30 5 0.151 0.132 0.268 0.151 0.139 0.266 −0.051 −0.044 −0.121 −0.051 −0.046 −0.118 10 0.142 0.125 0.261 0.145 0.131 0.261 −0.050 −0.043 −0.120 −0.051 −0.046 −0.120 30 0.139 0.120 0.260 0.140 0.126 0.258 −0.050 −0.043 −0.120 −0.051 −0.045 −0.119 50 5 0.109 0.097 0.198 0.110 0.101 0.198 −0.037 −0.033 −0.083 −0.037 −0.035 −0.083 10 0.104 0.092 0.195 0.105 0.096 0.193 −0.037 −0.033 −0.084 −0.037 −0.034 −0.083 30 0.100 0.088 0.193 0.102 0.092 0.191 −0.037 −0.033 −0.084 −0.037 −0.034 −0.083 100 5 0.075 0.067 0.139 0.076 0.070 0.137 −0.028 −0.026 −0.055 −0.028 −0.027 −0.054 10 0.071 0.063 0.137 0.072 0.066 0.135 −0.028 −0.026 −0.056 −0.028 −0.027 −0.053 30 0.069 0.061 0.135 0.070 0.064 0.133 −0.028 −0.026 −0.055 −0.028 −0.027 −0.054 1000 5 0.029 0.028 0.049 0.029 0.028 0.049 −0.020 −0.020 −0.024 −0.020 −0.020 −0.024 10 0.028 0.027 0.048 0.029 0.027 0.048 −0.020 −0.020 −0.023 −0.020 −0.020 −0.024 30 0.028 0.026 0.048 0.028 0.027 0.048 −0.020 −0.020 −0.024 −0.020 −0.020 −0.024 ρ = 0.8 XX 30 5 0.078 0.060 0.197 0.080 0.066 0.198 −0.030 −0.022 −0.099 −0.030 −0.024 −0.097 10 0.074 0.056 0.194 0.076 0.062 0.196 −0.029 −0.023 −0.098 −0.030 −0.024 −0.099 30 0.072 0.053 0.193 0.074 0.060 0.193 −0.029 −0.022 −0.099 −0.029 −0.024 −0.098 50 5 0.057 0.045 0.142 0.058 0.049 0.142 −0.022 −0.018 −0.067 −0.023 −0.020 −0.067 10 0.054 0.042 0.140 0.055 0.046 0.140 −0.022 −0.018 −0.067 −0.022 −0.020 −0.067 30 0.052 0.040 0.140 0.054 0.044 0.139 −0.022 −0.018 −0.068 −0.022 −0.019 −0.067 100 5 0.039 0.032 0.096 0.040 0.035 0.095 −0.018 −0.016 −0.043 −0.018 −0.016 −0.043 10 0.038 0.030 0.095 0.038 0.033 0.094 −0.018 −0.016 −0.043 −0.018 −0.016 −0.043 30 0.037 0.029 0.094 0.037 0.032 0.094 −0.018 −0.016 −0.043 −0.018 −0.016 −0.043 1000 5 0.017 0.016 0.033 0.017 0.016 0.032 −0.014 −0.013 −0.017 −0.014 −0.013 −0.017 10 0.017 0.016 0.032 0.017 0.016 0.032 −0.014 −0.013 −0.017 −0.014 −0.013 −0.017 30 0.017 0.015 0.032 0.017 0.016 0.032 −0.014 −0.013 −0.017 −0.014 −0.013 −0.017 dist1, Normal distribution for t ; dist2, distribution with negative kurtosis for t ; dist3, distribution with positive kurtosis for t ; dist4, skewed distribution for t ; dist5, i i i i skewed distribution with negative kurtosis for t ; dist6, skewed distribution with positive kurtosis for t . i i On the other hand, a symmetric leptokurtic distribution resulted −0.066 for bias, whereas the leptokurtic distribution had a rela- in a much smaller mean (0.198) and a larger SE (0.290), indicat- tively larger RMSE value (0.308) and a smaller bias value (−0.102), ing that the center location of the sampling distribution of α ˆ was indicating that it involved more error and negative bias in estimat- further away from the actual value (0.3) and more uncertainty ing α. Hence, under this condition, positive kurtosis affected (the was involved in estimating α. With respect to the accuracy of the location and scale of) the sampling distribution of α ˆ as well as the estimate, Table 2 shows that the normal distribution had a RMSE accuracy of using it to estimate α whereas negative kurtosis did of 0.251 and a bias value of −0.070. The platykurtic distribution not. Similar interpretations are used for the 95% interval shown gave rise to smaller but very similar values: 0.244 for RMSE and in Table 3, except that one can also use the intervals to determine www.frontiersin.org February 2012 | Volume 3 | Article 34 | 5 Sheng and Sheng Effect of non-normality on coefﬁcient alpha Table 3 | Observed 95% interval of the sample alpha (α ˆ ) for the simulated situations where the true score (t ) distribution is normal or non-normal. nk dist1 dist2 dist3 dist4 dist5 dist6 LB UB LB UB LB UB LB UB LB UB LB UB ρ = 0.3 XX 30 5 −0.351 0.580 −0.329 0.577 −0.490 0.635 −0.356 0.580 −0.342 0.576 −0.481 0.637 10 −0.323 0.563 −0.305 0.556 −0.457 0.630 −0.328 0.561 −0.308 0.558 −0.450 0.624 30 −0.303 0.550 −0.285 0.545 −0.435 0.618 −0.303 0.551 −0.286 0.547 −0.435 0.616 50 5 −0.155 0.528 −0.143 0.524 −0.252 0.587 −0.155 0.529 −0.147 0.524 −0.255 0.583 10 −0.136 0.512 −0.115 0.508 −0.233 0.576 −0.134 0.514 −0.123 0.510 −0.229 0.573 30 −0.116 0.505 −0.106 0.500 −0.219 0.571 −0.119 0.504 −0.109 0.501 −0.216 0.568 100 5 0.005 0.469 0.013 0.465 −0.062 0.522 0.004 0.469 0.010 0.466 −0.062 0.521 10 0.020 0.457 0.027 0.454 −0.050 0.515 0.021 0.458 0.025 0.455 −0.046 0.512 30 0.030 0.452 0.039 0.447 −0.044 0.512 0.028 0.451 0.035 0.447 −0.040 0.508 1000 5 0.208 0.350 0.211 0.349 0.186 0.374 0.208 0.350 0.210 0.348 0.186 0.373 10 0.213 0.346 0.215 0.344 0.189 0.371 0.213 0.346 0.214 0.345 0.190 0.371 30 0.215 0.343 0.217 0.342 0.192 0.369 0.215 0.344 0.217 0.343 0.192 0.369 ρ = 0.6 XX 30 5 0.212 0.754 0.258 0.742 −0.088 0.836 0.206 0.754 0.239 0.746 −0.086 0.834 10 0.231 0.743 0.277 0.730 −0.067 0.833 0.219 0.744 0.261 0.734 −0.071 0.832 30 0.239 0.737 0.289 0.723 −0.063 0.831 0.235 0.737 0.273 0.727 −0.059 0.828 50 5 0.325 0.723 0.357 0.713 0.111 0.809 0.322 0.724 0.348 0.718 0.105 0.807 10 0.338 0.716 0.371 0.704 0.118 0.807 0.335 0.716 0.358 0.707 0.122 0.801 30 0.349 0.711 0.377 0.697 0.127 0.806 0.343 0.710 0.370 0.702 0.130 0.801 100 5 0.417 0.689 0.439 0.680 0.270 0.770 0.416 0.689 0.430 0.683 0.274 0.768 10 0.426 0.682 0.448 0.672 0.277 0.768 0.426 0.684 0.440 0.676 0.280 0.768 30 0.432 0.678 0.452 0.668 0.283 0.767 0.430 0.679 0.446 0.672 0.286 0.764 1000 5 0.537 0.619 0.541 0.616 0.492 0.660 0.537 0.620 0.539 0.617 0.493 0.659 10 0.539 0.617 0.544 0.613 0.494 0.660 0.539 0.617 0.543 0.615 0.495 0.658 30 0.541 0.616 0.546 0.612 0.494 0.658 0.540 0.616 0.544 0.613 0.495 0.657 ρ = 0.8 XX 30 5 0.596 0.875 0.646 0.864 0.281 0.930 0.590 0.875 0.630 0.868 0.274 0.928 10 0.607 0.869 0.655 0.857 0.292 0.929 0.598 0.869 0.641 0.861 0.283 0.926 30 0.612 0.866 0.663 0.852 0.300 0.927 0.604 0.867 0.645 0.858 0.291 0.926 50 5 0.656 0.860 0.688 0.849 0.436 0.917 0.653 0.860 0.677 0.853 0.433 0.914 10 0.664 0.855 0.696 0.844 0.444 0.916 0.660 0.856 0.686 0.848 0.439 0.913 30 0.667 0.853 0.700 0.840 0.444 0.915 0.664 0.853 0.690 0.845 0.443 0.913 100 5 0.704 0.842 0.723 0.833 0.562 0.896 0.703 0.842 0.717 0.837 0.564 0.896 10 0.708 0.838 0.728 0.829 0.567 0.896 0.706 0.840 0.722 0.833 0.568 0.895 30 0.711 0.836 0.731 0.827 0.569 0.896 0.710 0.837 0.725 0.830 0.568 0.894 1000 5 0.764 0.807 0.769 0.803 0.726 0.837 0.764 0.807 0.768 0.804 0.728 0.836 10 0.766 0.805 0.771 0.802 0.728 0.836 0.766 0.806 0.769 0.803 0.728 0.836 30 0.766 0.805 0.772 0.801 0.728 0.836 0.766 0.805 0.770 0.802 0.729 0.836 dist1, Normal distribution for t ; dist2, distribution with negative kurtosis for t ; dist3, distribution with positive kurtosis for t ; dist4, skewed distribution for t ; dist5, i i i i skewed distribution with negative kurtosis for t ; dist6, skewed distribution with positive kurtosis for t ; LB, lower bound; UB, upper bound. i i whether the sample coefﬁcient was signiﬁcantly different from α or bias in estimating α, either (see Table 2). On the other as described in the previous section. hand, symmetric or non-symmetric distributions with posi- Guided by these interpretations, one can make the following tive kurtosis tend to result in a much smaller average of α ˆ observations: with a larger SE (see Table 1), which in turn makes the 95% observed interval wider compared with the normal distribu- 1. Among the ﬁve non-normal distributions considered for t , tion (see Table 3). In addition, positive kurtosis tends to involve skewed or platykurtic distributions do not affect the mean or more bias in underestimating α with a reduced accuracy (see the SE for α ˆ (see Table 1). They do not affect the accuracy Table 2). Frontiers in Psychology | Quantitative Psychology and Measurement February 2012 | Volume 3 | Article 34 | 6 Sheng and Sheng Effect of non-normality on coefﬁcient alpha 2. Sample size (n) and test length (k) play important roles for 2. Sample size (n) and test length (k) have different effects on α ˆ α ˆ and its sampling distribution, as increased n or k tends to and its sampling distribution. Increased n consistently results result in the mean of α ˆ that is closer to the speciﬁed population in a larger mean of α ˆ with a reduced SE. However, increased reliability (ρ ) with a smaller SE. We note that n has a larger k may result in a reduced SE, but it has a negative effect on XX and more apparent effect than k. Sample size further helps off- the mean in pushing it away from the speciﬁed population reli- set the effect of non-normality on the sampling distribution ability (ρ ), especially when ρ is not large. In particular, XX XX of α ˆ . In particular, when sample size gets large, e.g., n = 1000, with larger k, the mean of α ˆ decreases to be much smaller for the departure from normal distributions (due to positive kurto- non-normal distributions that are leptokurtic, non-symmetric, sis) does not result in much different mean of α ˆ although the or non-symmetric platykurtic; but it increases to exceed ρ XX SE is still slightly larger compared with normal distributions for symmetric platykurtic or non-symmetric leptokurtic dis- (see Table 1). tributions. It is further observed that with increased n, the 3. Increased n or k tends to increase the accuracy in estimating difference between non-normal and normal distributions of α while reducing bias. However, the effect of non-normality e on the mean and SE of α ˆ reduces. This is, however, not ij (due to positive kurtosis) on resulting in a larger estimating observed for increased k (see Table 4). error and bias remains even with increased n and/or k (see 3. The RMSE and bias values presented in Table 5 indicate that Table 2). It is also noted that for all the conditions considered, non-normal distributions for e , especially leptokurtic, non- ij α ˆ has a consistently negative bias regardless of the shape of the symmetric, or non-symmetric platykurtic distributions tend distribution for t . to involve larger error, if not bias, in estimating α. In addition, 4. The 95% observed interval shown in Table 3 agrees with the when k increases, RMSE or bias does not necessarily reduce. corresponding mean and SE shown in Table 1. It is noted that On the other hand, when n increases, RMSE decreases while regardless of the population distribution for t , when n or k gets bias increases. Hence, with larger sample sizes, there is more larger, α ˆ has a smaller SE, and hence a narrower 95% interval, accuracy in estimating α, but bias is not necessarily reduced for as the precision in estimating α increases. Given this, and that symmetric platykurtic or non-symmetric leptokurtic distribu- all intervals in the table, especially those for n = 1000, cover tions, as some of the negative bias values increase to become the speciﬁed population reliability (ρ ), one should note that positive and non-negligible. XX although departure from normality affects the accuracy, bias, 4. The effect of test length on the sample coefﬁcient is more and precision in estimating α, it does not result in systematically apparent in Table 6. From the 95% observed intervals for α ˆ , different α ˆ . In addition, when the actual reliability is small (i.e., and particularly those obtained when the actual reliability is ρ = 0.3), the use of large n is suggested, as when n < 1000, small to moderate (i.e., ρ ≤ 0.6) with large sample sizes (i.e., XX XX the 95% interval covers negative values of α ˆ . This is especially n = 1000), one can see that when test length gets larger (e.g., the case for the (symmetric or non-symmetric) distributions k = 30), the intervals start to fail to cover the speciﬁed popula- with positive kurtosis. For these distributions, at least 100 sub- tion reliability (ρ ) regardless of the degree of the departure XX jects are needed for α ˆ to avoid relatively large estimation error from the normality for e . Given the fact that larger sample sizes ij when the actual reliability is moderate to large. For the other result in less dispersion (i.e., smaller SE) in the sampling dis- distributions, including the normal distribution, a minimum tribution of α ˆ and hence a narrower 95% interval, and the fact of 50 subjects is suggested for tests with a moderate reliability that increased k pushes the mean of α ˆ away from the speciﬁed (i.e., ρ = 0.6), and 30 or more subjects are needed for tests reliability, this ﬁnding suggests that larger k ampliﬁes the effect XX with a high reliability (i.e., ρ = 0.8; see Table 2). of non-normality of e on α ˆ in resulting in systematically biased XX ij estimates of α, and hence has to be avoided when the actual reli- In addition, results for conditions where error scores depart from ability is not large. With respect to sample sizes, similar patterns normal distributions are summarized in Tables 4–6. Given the arise. That is, the use of large n is suggested when the actual design of the study, the results for the condition where e followed ij reliability is small (i.e., ρ = 0.3), especially for tests with 30 XX a normal distribution are the same as those for the condition where items, whereas for tests with a high reliability (i.e., ρ = 0.8), a XX the distribution for t was normal. For the purpose of comparisons, sample size of 30 may be sufﬁcient. In addition, when the actual they are displayed in the tables again. Inspections of these tables reliability is moderate, a minimum of 50 subjects is needed for result in the following ﬁndings, some of which are quite different α ˆ to be fairly accurate for short tests (k ≤ 10), and at least 100 from what are observed from Tables 1–3: are suggested for longer tests (k = 30; see Table 5). 1. Symmetric platykurtic distributions or non-symmetric lep- Given the above results, we see that non-normal distributions tokurtic distributions consistently resulted in a larger mean but for true or error scores do create problems for using coefﬁcient not a larger SE of α ˆ than normal distributions (see Table 4). alpha to estimate the internal consistency reliability. In particu- Some of the means, and especially those for non-symmetric lar, leptokurtic true score distributions that are either symmetric leptokurtic distributions, are larger than the speciﬁed popula- or skewed result in larger error and negative bias in estimat- tion reliability (ρ ). This is consistent with the positive bias XX ing population α with less precision. This is similar to Bay’s values in Table 5. On the other hand, symmetric leptokurtic, (1973) ﬁnding, and we see in this study that the problem remains non-symmetric, or non-symmetric platykurtic distributions even after increasing sample size to 1000 or test length to 30, tend to have larger SE of α ˆ than the normal distribution (see although the effect is getting smaller. With respect to error score Table 4). www.frontiersin.org February 2012 | Volume 3 | Article 34 | 7 Sheng and Sheng Effect of non-normality on coefﬁcient alpha Table 4 | Observed mean and SD of the sample alpha (α ˆ ) for the simulated situations where the error score (e ) distribution is normal or ij non−normal. nk Mean (α ˆ ) SD (α ˆ ) dist1 dist2 dist3 dist4 dist5 dist6 dist1 dist2 dist3 dist4 dist5 dist6 ρ = 0.3 XX 30 5 0.230 0.255 0.215 0.206 0.213 0.313 0.241 0.233 0.257 0.252 0.250 0.223 10 0.231 0.295 0.158 0.174 0.185 0.367 0.229 0.207 0.256 0.248 0.248 0.195 30 0.231 0.371 0.103 0.155 0.139 0.460 0.221 0.177 0.258 0.244 0.249 0.160 50 5 0.252 0.279 0.232 0.231 0.237 0.324 0.176 0.169 0.191 0.183 0.181 0.166 10 0.252 0.316 0.180 0.198 0.213 0.380 0.166 0.150 0.187 0.180 0.178 0.143 30 0.254 0.390 0.128 0.181 0.164 0.474 0.160 0.128 0.188 0.176 0.181 0.116 100 5 0.269 0.295 0.245 0.247 0.255 0.332 0.118 0.113 0.130 0.124 0.122 0.115 10 0.268 0.331 0.196 0.216 0.229 0.390 0.112 0.101 0.128 0.121 0.120 0.098 30 0.269 0.403 0.146 0.198 0.182 0.484 0.108 0.086 0.127 0.119 0.122 0.078 1000 5 0.282 0.308 0.254 0.261 0.269 0.338 0.036 0.035 0.040 0.038 0.037 0.036 10 0.282 0.343 0.208 0.231 0.244 0.398 0.034 0.031 0.039 0.037 0.037 0.030 30 0.282 0.414 0.161 0.213 0.197 0.493 0.033 0.026 0.039 0.036 0.037 0.024 ρ = 0.6 XX 30 5 0.549 0.550 0.565 0.552 0.551 0.571 0.142 0.140 0.163 0.141 0.140 0.159 10 0.551 0.560 0.547 0.543 0.545 0.586 0.133 0.127 0.157 0.141 0.139 0.132 30 0.550 0.615 0.452 0.482 0.500 0.669 0.129 0.102 0.180 0.160 0.156 0.093 50 5 0.563 0.564 0.574 0.565 0.564 0.579 0.103 0.100 0.121 0.103 0.101 0.118 10 0.563 0.573 0.559 0.557 0.560 0.595 0.097 0.092 0.115 0.102 0.100 0.099 30 0.563 0.625 0.472 0.499 0.518 0.676 0.093 0.074 0.131 0.115 0.112 0.068 100 5 0.572 0.573 0.579 0.574 0.574 0.584 0.069 0.068 0.084 0.069 0.068 0.082 10 0.572 0.582 0.567 0.567 0.570 0.600 0.066 0.062 0.078 0.069 0.067 0.068 30 0.572 0.633 0.484 0.511 0.530 0.681 0.063 0.050 0.088 0.078 0.075 0.046 1000 5 0.580 0.581 0.583 0.582 0.582 0.588 0.021 0.021 0.026 0.021 0.021 0.026 10 0.580 0.589 0.574 0.576 0.578 0.605 0.020 0.019 0.024 0.021 0.020 0.021 30 0.580 0.638 0.496 0.522 0.540 0.686 0.019 0.015 0.027 0.024 0.023 0.014 ρ = 0.8 XX 30 5 0.771 0.771 0.777 0.772 0.772 0.779 0.072 0.070 0.094 0.072 0.070 0.092 10 0.771 0.771 0.776 0.773 0.772 0.777 0.068 0.067 0.081 0.068 0.067 0.080 30 0.771 0.782 0.760 0.763 0.766 0.798 0.066 0.058 0.085 0.076 0.073 0.057 50 5 0.778 0.778 0.782 0.779 0.779 0.783 0.052 0.051 0.070 0.052 0.051 0.070 10 0.778 0.778 0.781 0.779 0.779 0.784 0.049 0.048 0.060 0.049 0.048 0.059 30 0.778 0.788 0.768 0.771 0.774 0.802 0.048 0.042 0.060 0.054 0.052 0.042 100 5 0.782 0.783 0.785 0.783 0.783 0.787 0.035 0.034 0.049 0.035 0.034 0.049 10 0.783 0.783 0.785 0.784 0.784 0.787 0.033 0.033 0.041 0.033 0.033 0.041 30 0.783 0.792 0.774 0.777 0.779 0.805 0.032 0.028 0.040 0.036 0.034 0.029 1000 5 0.786 0.787 0.788 0.787 0.787 0.789 0.011 0.010 0.015 0.011 0.011 0.015 10 0.786 0.787 0.788 0.788 0.788 0.791 0.010 0.010 0.013 0.010 0.010 0.013 30 0.787 0.795 0.779 0.781 0.784 0.807 0.010 0.009 0.012 0.011 0.010 0.009 dist1, Normal distribution for e ; dist2, distribution with negative kurtosis for e ; dist3, distribution with positive kurtosis for e ; dist4, skewed distribution for e ; dist5, ij ij ij ij skewed distribution with negative kurtosis for e ; dist6, skewed distribution with positive kurtosis for e . ij ij distributions, unlike conclusions from previous studies, depar- (1973) and Shultz (1993), an increase in test length does have an ture from normality does create problems in the sample coefﬁ- effect on the accuracy and bias in estimating reliability with the cient alpha and its sampling distribution. Speciﬁcally, leptokurtic, sample coefﬁcient alpha when error scores are not normal, but skewed, or non-symmetric platykurtic error score distributions it is in an undesirable manner. In particular, as is noted earlier, tend to result in larger error and negative bias in estimating popu- increased test length pushes the mean of α ˆ away from the actual lation α with less precision, whereas platykurtic or non-symmetric reliability, and hence causes the sample coefﬁcient alpha to be sig- leptokurtic error score distributions tend to have increased posi- niﬁcantly different from the population coefﬁcient when the actual tive bias when sample size, test length, and/or the actual reliability reliability is not high (e.g., ρ ≤ 0.6) and the sample size is large XX increases. In addition, different from conclusions made by Bay (e.g., n = 1000). This could be due to the fact that e is involved ij Frontiers in Psychology | Quantitative Psychology and Measurement February 2012 | Volume 3 | Article 34 | 8 Sheng and Sheng Effect of non-normality on coefﬁcient alpha Table 5 | Root mean square error and bias for estimating α for the simulated situations where the error score (e ) distribution is normal or ij non−normal. nk RMSE bias dist1 dist2 dist3 dist4 dist5 dist6 dist1 dist2 dist3 dist4 dist5 dist6 ρ = 0.3 XX 30 5 0.251 0.237 0.271 0.269 0.264 0.224 −0.070 −0.045 −0.085 −0.094 −0.087 0.013 10 0.240 0.208 0.293 0.278 0.273 0.206 −0.069 −0.005 −0.142 −0.126 −0.115 0.067 30 0.232 0.191 0.325 0.284 0.297 0.226 −0.069 0.071 −0.197 −0.145 −0.162 0.160 50 5 0.182 0.170 0.202 0.195 0.192 0.167 −0.048 −0.022 −0.068 −0.069 −0.063 0.024 10 0.173 0.151 0.222 0.207 0.198 0.164 −0.048 0.016 −0.120 −0.103 −0.088 0.080 30 0.166 0.157 0.255 0.212 0.226 0.209 −0.046 0.090 −0.172 −0.119 −0.136 0.174 100 5 0.122 0.113 0.141 0.135 0.130 0.119 −0.031 −0.006 −0.055 −0.053 −0.045 0.032 10 0.116 0.106 0.165 0.148 0.140 0.133 −0.032 0.031 −0.105 −0.084 −0.071 0.090 30 0.112 0.134 0.200 0.157 0.170 0.200 −0.031 0.103 −0.154 −0.102 −0.118 0.184 1000 5 0.040 0.035 0.061 0.054 0.048 0.052 −0.018 0.008 −0.046 −0.039 −0.031 0.038 10 0.038 0.053 0.100 0.079 0.067 0.102 −0.018 0.043 −0.092 −0.070 −0.056 0.098 30 0.038 0.117 0.144 0.095 0.109 0.194 −0.018 0.114 −0.139 −0.087 −0.103 0.193 ρ = 0.6 XX 30 5 0.151 0.148 0.166 0.149 0.149 0.161 −0.051 −0.050 −0.035 −0.048 −0.050 −0.029 10 0.142 0.133 0.165 0.152 0.149 0.133 −0.050 −0.040 −0.053 −0.057 −0.055 −0.014 30 0.139 0.103 0.233 0.199 0.185 0.116 −0.050 0.015 −0.148 −0.118 −0.100 0.069 50 5 0.109 0.106 0.124 0.108 0.107 0.120 −0.037 −0.036 −0.026 −0.035 −0.036 −0.021 10 0.104 0.096 0.123 0.111 0.107 0.099 −0.037 −0.027 −0.041 −0.043 −0.040 −0.005 30 0.100 0.078 0.183 0.153 0.139 0.102 −0.037 0.025 −0.128 −0.101 −0.082 0.076 100 5 0.075 0.073 0.087 0.074 0.073 0.084 −0.028 −0.027 −0.022 −0.026 −0.026 −0.016 10 0.071 0.065 0.085 0.076 0.074 0.068 −0.028 −0.018 −0.033 −0.033 −0.031 0.000 30 0.069 0.060 0.146 0.118 0.103 0.093 −0.028 0.033 −0.116 −0.089 −0.070 0.081 1000 5 0.029 0.028 0.032 0.028 0.028 0.029 −0.020 −0.019 −0.017 −0.018 −0.018 −0.012 10 0.028 0.022 0.036 0.032 0.030 0.022 −0.020 −0.011 −0.026 −0.024 −0.022 0.005 30 0.028 0.041 0.108 0.082 0.064 0.087 −0.020 0.038 −0.104 −0.078 −0.060 0.086 ρ = 0.8 XX 30 5 0.078 0.076 0.097 0.077 0.076 0.095 −0.030 −0.029 −0.023 −0.028 −0.029 −0.021 10 0.074 0.073 0.085 0.073 0.073 0.084 −0.029 −0.029 −0.024 −0.027 −0.028 −0.023 30 0.072 0.060 0.094 0.085 0.080 0.057 −0.029 −0.018 −0.040 −0.037 −0.034 −0.003 50 5 0.057 0.055 0.073 0.057 0.055 0.072 −0.022 −0.022 −0.018 −0.021 −0.021 −0.017 10 0.054 0.053 0.063 0.053 0.053 0.062 −0.022 −0.022 −0.019 −0.021 −0.021 −0.016 30 0.052 0.044 0.068 0.061 0.058 0.042 −0.022 −0.012 −0.032 −0.029 −0.026 0.002 100 5 0.039 0.038 0.051 0.039 0.038 0.050 −0.018 −0.017 −0.015 −0.017 −0.017 −0.013 10 0.038 0.037 0.044 0.037 0.037 0.043 −0.018 −0.017 −0.015 −0.016 −0.016 −0.013 30 0.037 0.030 0.048 0.043 0.040 0.029 −0.018 −0.008 −0.026 −0.024 −0.021 0.005 1000 5 0.017 0.017 0.020 0.017 0.017 0.019 −0.014 −0.013 −0.012 −0.013 −0.013 −0.011 10 0.017 0.016 0.017 0.016 0.016 0.016 −0.014 −0.013 −0.012 −0.012 −0.012 −0.010 30 0.017 0.010 0.024 0.022 0.019 0.012 −0.014 −0.005 −0.021 −0.019 −0.016 0.007 dist1, Normal distribution for e ; dist2, distribution with negative kurtosis for e ; dist3, distribution with positive kurtosis for e ; dist4, skewed distribution for e ; dist5, ij ij ij ij skewed distribution with negative kurtosis for e ; dist6, skewed distribution with positive kurtosis for e . ij ij in each item, and hence an increase in the number of items would and this situation is much worse when it comes to measurement add up the effect of non-normality on the sample coefﬁcient. issues such as reliability. In actual applications, it is vital to not only evaluate the assumptions for coefﬁcient alpha, but also understand DISCUSSION them and the consequences of any violations. Normality is not commonly considered as a major assump- In practice, coefﬁcient alpha is often used to estimate reliabil- tion for coefﬁcient alpha and hence has not been well investigated. ity with little consideration of the assumptions required for the sample coefﬁcient to be accurate. As noted by Graham (2006, p. This study takes the advantage of recently developed techniques in generating univariate non-normal data to suggest that different 942), students and researchers in education and psychology are often unaware of many assumptions for a statistical procedure, from conclusions made by Bay (1973), Zimmerman et al. (1993), www.frontiersin.org February 2012 | Volume 3 | Article 34 | 9 Sheng and Sheng Effect of non-normality on coefﬁcient alpha Table 6 | Observed 95% interval of the sample alpha (α ˆ ) for the simulated situations where the error score (e ) distribution is normal or ij non-normal. nk dist1 dist2 dist3 dist4 dist5 dist6 LB UB LB UB LB UB LB UB LB UB LB UB ρ = 0.3 XX 30 5 −0.351 0.580 −0.305 0.592 −0.404 0.587 −0.403 0.572 −0.392 0.574 −0.220 0.638 10 −0.323 0.563 −0.203 0.595 −0.459 0.529 −0.421 0.536 −0.416 0.543 −0.104 0.647 30 −0.303 0.550 −0.058 0.629 −0.522 0.478 −0.437 0.508 −0.462 0.501 0.070 0.688 50 5 −0.155 0.528 −0.114 0.544 −0.212 0.531 −0.193 0.516 −0.179 0.519 −0.057 0.585 10 −0.136 0.512 −0.031 0.551 −0.256 0.475 −0.223 0.481 −0.199 0.493 0.050 0.604 30 −0.116 0.505 0.094 0.591 −0.307 0.423 −0.229 0.458 −0.257 0.447 0.203 0.653 100 5 0.005 0.469 0.042 0.486 −0.045 0.464 −0.031 0.456 −0.016 0.461 0.077 0.525 10 0.020 0.457 0.107 0.501 −0.087 0.411 −0.052 0.421 −0.039 0.431 0.172 0.554 30 0.030 0.452 0.211 0.549 −0.136 0.361 −0.067 0.399 −0.089 0.387 0.310 0.616 1000 5 0.208 0.350 0.237 0.372 0.172 0.330 0.184 0.332 0.193 0.338 0.264 0.405 10 0.213 0.346 0.280 0.400 0.128 0.282 0.155 0.300 0.170 0.313 0.336 0.454 30 0.215 0.343 0.360 0.463 0.082 0.234 0.139 0.280 0.122 0.267 0.444 0.537 ρ = 0.6 XX 30 5 0.212 0.754 0.212 0.752 0.171 0.794 0.212 0.756 0.210 0.752 0.186 0.796 10 0.231 0.743 0.253 0.745 0.167 0.768 0.197 0.744 0.210 0.743 0.267 0.778 30 0.239 0.737 0.370 0.766 0.015 0.711 0.099 0.713 0.121 0.721 0.444 0.803 50 5 0.325 0.723 0.331 0.722 0.289 0.758 0.326 0.725 0.329 0.722 0.302 0.762 10 0.338 0.716 0.360 0.717 0.286 0.736 0.319 0.715 0.328 0.714 0.363 0.749 30 0.349 0.711 0.455 0.743 0.167 0.675 0.229 0.680 0.256 0.691 0.520 0.783 100 5 0.417 0.689 0.422 0.688 0.389 0.716 0.420 0.691 0.422 0.689 0.398 0.721 10 0.426 0.682 0.444 0.687 0.392 0.697 0.415 0.682 0.420 0.682 0.448 0.714 30 0.432 0.678 0.521 0.718 0.287 0.632 0.338 0.643 0.363 0.656 0.579 0.759 1000 5 0.537 0.619 0.539 0.620 0.528 0.631 0.539 0.621 0.539 0.620 0.534 0.636 10 0.539 0.617 0.551 0.624 0.524 0.619 0.533 0.614 0.537 0.616 0.562 0.644 30 0.541 0.616 0.607 0.667 0.441 0.546 0.473 0.566 0.494 0.582 0.657 0.712 ρ = 0.8 XX 30 5 0.596 0.875 0.602 0.872 0.543 0.903 0.598 0.876 0.602 0.874 0.549 0.904 10 0.607 0.869 0.611 0.868 0.575 0.889 0.608 0.870 0.611 0.869 0.581 0.890 30 0.612 0.866 0.646 0.868 0.550 0.875 0.577 0.867 0.589 0.867 0.661 0.882 50 5 0.656 0.860 0.661 0.858 0.613 0.885 0.657 0.861 0.661 0.858 0.617 0.885 10 0.664 0.855 0.667 0.854 0.637 0.872 0.665 0.856 0.667 0.855 0.644 0.873 30 0.667 0.853 0.692 0.855 0.624 0.858 0.643 0.853 0.653 0.853 0.705 0.869 100 5 0.704 0.842 0.707 0.840 0.672 0.863 0.705 0.842 0.707 0.841 0.675 0.863 10 0.708 0.838 0.711 0.838 0.692 0.853 0.711 0.840 0.711 0.839 0.696 0.854 30 0.711 0.836 0.729 0.841 0.683 0.840 0.695 0.836 0.702 0.836 0.741 0.854 1000 5 0.764 0.807 0.766 0.806 0.756 0.816 0.766 0.807 0.766 0.807 0.757 0.817 10 0.766 0.805 0.767 0.806 0.762 0.812 0.767 0.807 0.767 0.806 0.765 0.814 30 0.766 0.805 0.778 0.812 0.754 0.802 0.759 0.802 0.762 0.803 0.789 0.824 dist1, Normal distribution for e ; dist2, distribution with negative kurtosis for e ; dist3, distribution with positive kurtosis for e ; dist4, skewed distribution for e ; dist5, ij ij ij ij skewed distribution with negative kurtosis for e ; dist6, skewed distribution with positive kurtosis for e ; LB, lower bound; UB, upper bound. ij ij and Shultz (1993), coefﬁcient alpha is not robust to the viola- bias. Neither case is desired in a reliability study, as the sample tion of the normal assumption (for either true or error scores). coefﬁcient would paint an incorrect picture of the test’s internal Non-normal data tend to result in additional error or bias in consistency by either estimating it with a larger value or a much estimating internal consistency reliability. A larger error makes smaller value and hence is not a valid indicator. For example, for a the sample coefﬁcient less accurate, whereas more bias causes it test with reliability being 0.6, one may calculate the sample alpha to further under- or overestimate the actual reliability. We note to be 0.4 because the true score distribution has a positive kurtosis, that compared with normal data, leptokurtic true or error score and conclude that the test is not reliable at all. On the other hand, distributions tend to result in additional negative bias, whereas one may have a test with actual reliability being 0.4. But because platykurtic error score distributions tend to result in a positive the error score distribution has a negative kurtosis, the sample Frontiers in Psychology | Quantitative Psychology and Measurement February 2012 | Volume 3 | Article 34 | 10 Sheng and Sheng Effect of non-normality on coefﬁcient alpha coefﬁcient is calculated to be 0.7 and hence the test is concluded to normality is assumed (see Table 2). However, the degree of bias be reliable. In either scenario, the conclusion on the test reliability becomes negligible when sample size increases to 1000 or beyond. is completely the opposite of the true situation, which may lead to In the study, we considered tests of 5, 10, or 30 items admin- an overlook of a reliable measure or an adoption of an unreliable istered to 30, 50, 100, or 1000 persons with the actual reliability instrument. Consequently, coefﬁcient alpha is not suggested for being 0.3, 0.6, or 0.8. These values were selected to reﬂect levels estimating internal consistency reliability with non-normal data. ranging from small to large in the sample size, test length, and Given this, it is important to make sure that in addition to satisfying population reliability considerations. When using the results, one the assumptions of (essential) tau-equivalence and uncorrelated should note that they pertain to these simulated conditions and errors, the sample data conform to normal distributions before may not generalize to other conditions. In addition, we evaluated one uses alpha in a reliability study. the assumption of normality alone. That is, in the simulations, Further, it is generally said that increased data sizes help approx- data were generated assuming the other assumptions, namely imate non-normal distributions to be normal. This is the case with (essential) tau-equivalence and uncorrelated error terms, were sat- sample sizes, not necessarily test lengths, in helping improve the isﬁed. In practice, it is common for observed data to violate more accuracy, bias and/or precision of using the sample coefﬁcient in than one assumption. Hence, it would also be interesting to see reliability studies with non-normal data. Given the results of the how non-normal data affect the sample coefﬁcient when other study, we suggest that in order for the sample coefﬁcient alpha to violations are present. Further, this study looked at the sample be fairly accurate and in a reasonable range, a minimum of 1000 coefﬁcient alpha and its empirical sampling distribution without subjects is needed for a small reliability, and a minimum of 100 considering its sampling theory (e.g., Kristof, 1963; Feldt, 1965). is needed for a moderate reliability when the sample data depart One may focus on its theoretical SE (e.g., Bay, 1973; Barchard and Hakstian, 1997a,b; Duhachek and Iacobucci, 2004) and compare from normality. It has to be noted that for the four sample size conditions considered in the study, the sample coefﬁcient alpha them with the empirical ones to evaluate the robustness of an consistently underestimates the population reliability even when interval estimation of the reliability for non-normal data. REFERENCES Experiment.New York: Miller, M. B. (1995). Coefﬁcient alpha: George, BC: Edgeworth Labora- Barchard, K. A., and Hakstian, R. Wiley. a basic introduction from the per- tory for Quantitative Behavioral Sci- (1997a). The effects of sampling Graham, J. M. (2006). Congeneric and spectives of classical test theory ence, University of Northern British model on inference with coefﬁ- (essential) tau-equivalent estimates and structural equation modeling. Columbia. cient alpha. Educ. Psychol. Meas. 57, of score reliability. Educ. Psychol. Struct. Equation Model. 2, 255–273. Zumbo, B. D., and Rupp, A. A. (2004). 893–905. Meas. 66, 930–944. Novick, M. R., and Lewis, C. (1967). “Responsible modeling of measure- Barchard, K. A., and Hakstian, R. Green, S. B., and Hershberger, S. L. Coefﬁcient alpha and the reliabil- ment data for appropriate infer- (1997b). The robustness of conﬁ- (2000). Correlated errors in true ity of composite measurements. Psy- ences: important advances in reli- dence intervals for coefﬁcient alpha score models and their effect on chometrika 32, 1–13. ability and validity theory,” in The under violation of the assumption coefﬁcient alpha. Struct. Equation Nunnally, J. C. (1967). Psychometric SAGE Handbook of Quantitative of essential parallelism. Multivariate Model. 7, 251–270. Theory. New York: McGraw-Hill. Methodology for the Social Sciences, Behav. Res. 32, 169–191. Green, S. B., and Yang, Y. (2009). Nunnally, J. C., and Bernstein, I. H. ed. D. Kaplan (Thousand Oaks: Bay, K. S. (1973). The effect of non- Commentary on coefﬁcient alpha: (1994). Psychometric Theory,3rd Sage), 73–92. normality on the sampling distribu- a cautionary tale. Psychometrika 74, Edn. New York: McGraw-Hill. tion and standard error of reliability 121–135. Raykov, T. (1998). Coefﬁcient alpha and coefﬁcient estimates under an analy- Guttman, L. A. (1945). A basis for composite reliability with interre- Conﬂict of Interest Statement: The sis of variance model. Br. J. Math. analyzing test-retest reliability. Psy- lated nonhomogeneous items. Appl. authors declare that the research was Stat. Psychol. 26, 45–57. chometrika 10, 255–282. Psychol. Meas. 22, 69–76. conducted in the absence of any com- Caplan, R. D., Naidu, R. K., and Tripathi, Headrick, T. C. (2002). Fast ﬁfth-order Shultz, G. S. (1993). A Monte Carlo study mercial or ﬁnancial relationships that R. C. (1984). Coping and defense: polynomial transforms for generat- of the robustness of coefﬁcient alpha. could be construed as a potential con- constellations vs. components. J. ing univariate and multivariate non- Masters thesis, University of Ottawa, ﬂict of interest. Health Soc. Behav. 25, 303–320. normal distributions. Comput. Stat. Ottawa. Cronbach, L. J. (1951). Coefﬁcient alpha Data Anal. 40, 685–711. Sijtsma, K. (2009). On the use, the mis- Received: 29 October 2011; paper pending and the internal structure of tests. Headrick, T. C. (2010). Statistical Sim- use, and the very limited usefulness published: 22 November 2011; accepted: Psychometrika 16, 297–334. of Cronbach’s alpha. Psychometrika ulation: Power Method Polynomi- 30 January 2012; published online: 15 DeVellis, R. F. (1991). Scale Develop- als and Other Transformations. Boca 74, 107–120. February 2012. ment. Newbury Park, NJ: Sage Pub- Raton, FL: Chapman & Hall. van Zyl, J. M., Neudecker, H., and Nel, Citation: Sheng Y and Sheng Z (2012) lications. Johanson, G. A., and Brooks, G. (2010). D. G. (2000). On the distribution of Is coefﬁcient alpha robust to non- Duhachek, A., and Iacobucci, D. (2004). Initial scale development: sample the maximum likelihood estimator normal data? Front. Psychology 3:34. doi: Alpha’s standard error (ASE): an size for pilot studies. Educ. Psychol. for Cronbach’s alpha. Psychometrika 10.3389/fpsyg.2012.00034 accurate and precise conﬁdence Meas. 70, 394–400. 65, 271–280. This article was submitted to Frontiers interval estimate. J. Appl. Psychol. 89, Kristof, W. (1963). The statistical theory Zimmerman, D. W., Zumbo, B. D., in Quantitative Psychology and Measure- 792–808. of stepped-up reliability coefﬁcients and Lalonde, C. (1993). Coefﬁcient ment, a specialty of Frontiers in Psychol- Feldt, L. S. (1965). The approximate when a test has been divided into alpha as an estimate of test reliabil- ogy. sampling distribution of Kuder- several equivalent parts. Psychome- ity under violation of two assump- Copyright © 2012 Sheng and Sheng . This Richardson reliability coefﬁcient trika 28, 221–238. tions. Educ. Psychol. Meas. 53, is an open-access article distributed under twenty. Psychometrika 30, 357–370. Lord, F. M., and Novick, M. R. (1968). 33–49. the terms of the Creative Commons Attri- Fleishman, A. I. (1978). A method Statistical Theories of Mental Test Zumbo, B. D. (1999). A Glance at Coef- bution Non Commercial License, which for simulating non-normal distrib- Scores. Reading: Addison-Wesley. ﬁcient Alpha With an Eye Towards permits non-commercial use, distribu- utions. Psychometrika 43, 521–532. MathWorks. (2010). MATLAB (Version Robustness Studies: Some Mathemat- tion, and reproduction in other forums, Fleiss, J. L. (1986). The Design 7.11) [Computer software]. Natick, ical Notes and a Simulation Model provided the original authors and source and Analysis of Clinical MA: MathWorks. (Paper No. ESQBS-99-1). Prince are credited. www.frontiersin.org February 2012 | Volume 3 | Article 34 | 11 Sheng and Sheng Effect of non-normality on coefﬁcient alpha APPENDIX CODE IN MATLAB function result=mcalpha(n,k,evar,rho,rep) % mcalpha - obtain summary statistics for sample alphas % result=mcalpha(n,k,evar,rho,rep) % returns the observed mean, standard deviation, and 95% interval (qtalpha) % for sample alphas as well as the root mean square error (rmse) and bias for % estimating the population alpha. % The INPUT arguments: % n - sample size % k - test length % evar - error variance % rho - population reliability % rep - number of replications alphav=zeros(rep,1); tbcd=[0,1,0,0,0,0]; ebcd=[0,1,0,0,0,0]; % note: tbcd and ebcd are vectors containing the six coefficients, c ,…,c , 0 5 % used in equation (8) for true scores and error scores, respectively. Each % of them can be set as: % 1. [0,1,0,0,0,0] (normal) % 2. [0,1.643377,0,-.319988,0,.011344] (platykurtic) % 3. [0,0.262543,0,.201036,0,.000162] (leptokurtic) % 4. [-0.446924 1.242521 0.500764 -0.184710 -0.017947,0.003159] (skewed) % 5. [-.276330,1.506715,.311114,-.274078,-.011595,.007683] (skewed % platykurtic) % 6. [-.304852,.381063,.356941,.132688,-.017363,.003570] (skewed leptokurtic) for i=1:rep alphav(i)=alpha(n,k,evar,rho,tbcd,ebcd); end rmse=sqrt(mean((alphav-rho).ˆ2)); bias=mean(alphav-rho); qtalpha=quantile(alphav,[.025,.975]); result=[mean(alphav),std(alphav),qtalpha,rmse,bias]; function A=alpha(n,k,evar,rho,tbcd,ebcd) % alpha - calculate sample alpha % alp=alpha(n,k,evar,rho,tbcd,ebcd) % returns the sample alpha. % The INPUT arguments: % n - sample size % k - test length % evar - error variance % rho - population reliability Frontiers in Psychology | Quantitative Psychology and Measurement February 2012 | Volume 3 | Article 34 | 12 Sheng and Sheng Effect of non-normality on coefﬁcient alpha % rep - number of replications % tbcd - coefficients for generating normal/nonnormal true score % distributions using power method polynomials % ebcd - coefficients for generating normal/nonnormal error score % distributions using power method polynomials tvar=evar*rho/((1-rho)*k); t=rfsimu(tbcd,n,1,5,tvar); e=rfsimu(ebcd,n,k,0,evar); xn=t*ones(1,k)+e; x=round(xn); alp=k/(k-1)*(1-sum(var(x,1))/var(sum(x,2),1)); function X=rfsimu(bcd,n,k,mean,var) % rfsimu - generate normal/nonnormal distributions using 5-th order power % method polynomials % X=rfsimu(bcd,n,k,mean,var) % returns samples of size n by k drawn from a distribution with the desired % moments. % The INPUT arguments: % bcd - coefficients for generating normal/nonnormal distributions using % the 5-th order polynomials % k - test length % evar - error variance % rho - population reliability % rep - number of replications % tbcd - coefficients for generating normal/nonnormal true score % distributions using power method polynomials % ebcd - coefficients for generating normal/nonnormal error score % distributions using power method polynomials Z=randn(n,k); Y=bcd(1)+bcd(2)*Z+bcd(3)*Z.ˆ2+bcd(4)*Z.ˆ3+bcd(5)*Z.ˆ4+bcd(6)*Z.ˆ5; X=mean+sqrt(var)*Y; www.frontiersin.org February 2012 | Volume 3 | Article 34 | 13
Frontiers in Psychology – Pubmed Central
Published: Feb 15, 2012
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