An Introduction to Applied Multivariate Analysis

“…resulting in underestimation of scale reliability (Graham, 2006; Vaske, Beaman, & Sponarski, 2017), we tested the τ equivalence hypothesis for Cronbach’s α using factor analysis with restriction on each TPB variable in comparison with the model with no restriction (Raykov & Marcoulides, 2008).”

“Recommended value is calculated using the formula “p(p + 2)” where p = total number of observed indicators (Raykov & Marcoulides, 2008).”

“Table 2 Goodness-of-fit indices and acceptable limits Indices Acceptable limits χ2/df ≤5 acceptable fit, ≤3 perfect fit RMSEA ≤0.10 weak fit, ≤0.08 good fit, ≤0.05 perfect fit GFI .85–.89 acceptable fit, ≥ .90 good fit AGFI .85–.89 acceptable fit, ≥ .90 good fit CFI ≥ .90 acceptable fit, ≥ .95 good fit, ≥ .97 perfect…”

“The Mardia’s coefficient for the data in this study was 142.89, which is lower than the value of 440 computed, based on the formula p(p + 2), where p equals the number of observed variables in the model (Raykov & Marcoulides, 2008).”

“Naturally, it is desirable that p is significant [21].”

“If multivariate normal distribution is not met, under which Mardia’s coefficient is larger than p( p + 2), where p is the total number of observed indicators (Raykov and Marcoulides 2008), multivariate bootstrap will be used.”

“Sebelum diuji menggunakan anakova, data terlebih dahulu harus memenuhi beberapa asumsi yakni data harus berdistribusi normal, memiliki varian homogen dan berpola linier (Raykov & Marcoulides, 2008).”

“The calculated value of multivariate kurtosis was 15.629, less than the computed expected value of 168 (Raykov and Marcoulides 2008).”

“Then, cases outside the range of ± 3.0 were deleted from the data (Raykov & Marcoulides, 2008).”

“In the CFA model, all standardized estimated loadings on the latent construct were higher than the threshold value of 0.50 (Raykov and Marcoulides, 2008).”

“…administered q times to a sample of independent individuals from a studied population, as in a conventional longitudinal study (p, q > 1; e.g., Raykov & Marcoulides, 2008, 2016).1 Hence, at each assessment occasion, the single-factor model can be advanced: y ¼ μþ ληþ ε; (1) where y = (Y1,…”

“…is the k × qmatrix of factor loadings in that population, αg is the k × 1 vector of intercepts (mean intercepts) in it, and eg is the k × 1 vector of residual terms (unique factors) that there are with zero means and assumed uncorrelated with fgðg ¼ 1; . . . ;G; e.g., Raykov & Marcoulides, 2008).”

“Alternatively, one could consider univariate statistics as a special case of multivariate statistics (Raykov and Marcoulides 2008).”

“However, whereas univariate normality is not testable, as is well known it is implied from higher dimensional normality, in particular from bivariate normality (e.g., Raykov & Marcoulides, 2008).”

“In this study, Mardia’s coefficient (Mardia 1970) was 242.227, which was lower than the value of 1680 computed by the formula p(p + 2), where p equals the number of observed variables in the model (Raykov and Marcoulides 2008).”

“For more details on the exploratory factor analysis see for instance Raykov and Marcoulides (2008).”

“Because Mardia’s coefficient was 361.35, which is smaller than the value of 783 (i.e. p*[p + 2], where p represents the number of observed variables or items in this study), the assumption of multivariate normality was not violated (Raykov and Marcoulides 2008).”

“This result indicates that the model data conforms to multivariate normality (Murcia et al., 2009; Raykov & Marcoulides, 2008).”

“In particular, it is important to check that data are not multi-variate kurtotic, where the multivariate distribution of the observed variables has both tails and peaks that are non-characteristic of a normal distribution (Raykov & Marcoulides, 2008).”

“The relevant literature indicated that this value should be less than the recommended value calculated using the formula “p(p + 2)” where p = total number of observed indicators (Raykov & Marcoulides, 2008).”

“Specifically, Equation (1) could be viewed—at least conceptually—as a regression model with unobserved predictors that are the factors in the above vector h (e.g., Raykov & Marcoulides, 2008, 2011).”

“Within the scope of the multivariate normality assumption, the Mardia coefficient was lower than 1.96, the critical ratio (c.r) was lower than 5, and the multivariate normality assumption was met (Raykov & Marcoulides, 2008).”

“Mardia’s coefficient was used to examine the multivariate normality of the measurement model (Raykov & Marcoulides, 2008).”

“To check for violation of multivariate normality, we used Mardia’s (1970) coefficient, which we computed with the formula, p (p+2), where p refers to the number of observed variables in themodel (Raykov & Marcoulides, 2008, p. 18).”

“Mardia’s coefficient obtained from the combined sample (112.13) was less than the recommended value of p(p + 2) where p is the total number of observed indicators (Raykov & Marcoulides, 2008).”

“Responses to the 37 items were submitted to two factor analytic procedures (due to differing questionnaire stems) to identify any subscales using principal component extraction method with oblimin rotation and Kaiser normalisation [29].”

“…of Δ*, one might suggest that the population discrepancy between the criterion validity coefficients of the OLC Z and the overall scale score Y is at least Δ*, as then all plausible population values for Δv (at the considered confidence level) are above Δ* (cf., e.g., Raykov & Marcoulides, 2008).”

“Kullanılan bir diğer uyum indeksi RMSEA’nın .05 ve daha düşük olması mükemmel uyuma işaret etmektedir (Brown, 2006; Jöreskog & Sörbom, 1993; Raykov & Marcoulides, 2008; Schumacker & Lomax, 1996).”

“Given that the multivariate kurtosis value of 158.081 is less than the p(p + 2) = 685 and the number of observed variables (p) is 25, the data are reasonably assumed to be multivariate normal (Raykov & Marcoulides, 2008).”

“Thus, the data in this investigation were assumed to be multi-variate normal (Raykov & Marcoulides, 2008).”

“Consequently, multi-variate normality was assumed for the statistical data in this investigation (Raykov & Marcoulides, 2008).”

“First, an outlier analysis was performed, and 13 subjects were excluded from the dataset according to their standardized z-scores (Raykov & Marcoulides, 2008).”

“This command file differs from the preceding input file merely by the addition of the symbol ‘Q’ in the first line of the MODEL section (e.g., Raykov & Marcoulides, 2008).”

“The goal of MANCOVA cannot be achieved through a series of ANCOVAs (Raykov & Marcoulides, 2008, p. 209).”

“…analysis (DA) is a method that can be used in a multigroup setting to determine how a set of dependent variables or explanatory variables are related to group membership and, in particular, how they may be combined to enhance one’s understanding of group differences (Raykov and Marcoulides 2008).”

“In this study, the multivariate normality was examined by comparing the Mardia’s coefficient for the data under study to a critical value calculated from the Raykov and Marcoulides’s (2012) formula p(p+2) where p is the number of observed variables in the model.”

“Shulman (1987) proposed that teachers not only need content knowledge, but they also need pedagogical knowledge to teach effectively.”

“Multiple imputation is an acceptable solution and recommended by statisticians and researchers for handling missing data (Allison, 2002; Raykov & Marcoulides, 2008).”

“To do this, the likelihood ratio test was used (Raykov & Marcoulides, 2008, p. 430) to test the null hypothesis of no text differences in the five human-LSA reliabilities (Golden Summary Table 1.”

“As is well known (e.g., Raykov & Marcoulides, 2008), this interval could also be used for testing (point or simple) null hypotheses about D, if need be, in particular that of these two reliability coefficients’ equality, that is, H0: D = 0, or H * 0: rY = rZ.”

“Confirmatory factor analysis In this study, the multivariate kurtosis was 100.52 and the value computed based on the formula p(p + 2) where p represents the number of observed variables in the model (Raykov and Marcoulides 2008) was 624.”

“…we used Bartlett's test of sphericity to test the null hypothesis that the correlations among variables were zero or that the correlation matrix was an identity matrix (Raykov & Marcoulides, 2008) for all the physico-chemical properties and the total number of nodulated units per gram of soils.”

“The correlations between the LS factors respectively for math achievement simply reflect their degree of overlap (Raykov & Marcoulides, 2006, 2008).”

“…to +3 ranges, respectively, thus univariate normality was there (hair et al., 2019). the Mardia’s coefficient (103.4) was below the p(p + 2) value (624) that is recommended, where p is the number of observed variables (p = 24) (Raycov & Malcoulides, 2008). thus, multivariate normality was present.”

“To refine the model, we removed items 45, 33, 37, 46, and 49, as their standardized estimate loadings from observed variables to unobserved variables were below 0.45 (Raykov & Marcoulides, 2008).”

“To test now for no between-class differences in Cronbach’s alpha coefficients, that is, the null hypothesis H0: a(1) = a(2) (in this example), we can make use of the so-called ‘‘duality principle’’ (Raykov & Marcoulides, 2008).”

“(Brown 2006; Cokluk et al. 2012; Hu and Bentler 1999; Kelloway 2015; Kline 2011; Raykov and Marcoulides 2008; Meydan and Şeşen 2011; Schumacker and Lomax 2004; Schermelleh-Engel et al. 2003; Sümer 2000; Şimşek 2007; Tabachnick and Fidell 2013; Thompson 2004) (as cited in Savci and Aysan 2019).”

“…analysis to predict with the two outcome variables (career-dysfunctional thoughts and motivation to take the steps necessary to obtain employment) simultaneously, since the two outcome variables may be related, thus minimizing inflation of type 1 error rate (Raykov & Marcoulides, 2008).”

“041, lower than the value of 168 computed based on the formula p (p + 2) where p equals the number of observed variables in the model (Raykov and Marcoulides, 2008).”

“The critical value for multivariate normality is calculated according to the equation of p(p + 2) which is suggested by Raykov and Marcoulides (2008) and found to be 342.”

“The Kaiser-Meyer-Olkin (KMO) and Bartlett’s test of sphericity were computed to test adequacy of sample size and inter-item correlations for factor analysis (Raykov & Marcoulides, 2008).”

“……, Yp and are assumed to comprise a unidimensional scale evaluating a construct of interest, which is repeatedly administered q times to a sample of independent individuals from a population under investigation as in conventional longitudinal studies (p, q > 1; e.g., Raykov & Marcoulides, 2008).1”

“Canonical correlation analysis which is among the exploratory parametric statistical methods aiming to create theoretical knowledge, aims to examine the relationships between the variables in two different sets (Thompson, 2005; Tabachnick & Fidell, 2008; Ekşi, 2010; Raykov & Marcoulides, 2008).”

“Preliminary analyses were run to test the presence of multivariate normality in the data set (Raykov & Marcoulides, 2008).”