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Bayesian evaluation of response styles in polytomous data with multiple group factor analysis model

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(1)國立臺灣師範大學數學系碩士班碩士論文. 指導教授: 蔡 蓉 青 博士. Bayesian evaluation of response styles in polytomous data with multiple group factor analysis model. 研 究 生: 賴 驥 緯 中 華 民 國 一零五 年 六 月.

(2) 致謝 非常感謝我的指導教授—蔡蓉青老師,在我求學的過程中給予我非常多的 指導。並且在我做論文的時候,提供我許多資源,並且給予我許多的協助,使我 能順利完成論文,謹此致上最高的感謝。. 同時也非常感謝兩位口試委員蔡恆修老師與張少同老師,對本論文提出指 正,並且針對研究的方法及論文的內容給予許多寶貴的意見,使我獲益良多,並 讓本論文能更完整。. 感謝與我一起合作論文的好夥伴–陳冠宏同學,在編寫程式碼上給予我很大 的幫助,以及林炯伊學長,一起討論,一起找資料,很高興我們能一起順利完成 論文且通過口試。. 另外,也要感謝師大數學所的其他同學們,讓我有個快樂的碩士生活,希 望畢業後能再相聚。. 最後,還要感謝我最親愛的家人和朋友們,在我就讀研究所及論文寫作期 間給予我經濟及精神上的支持,這篇論文是因為有他們才得以順利完成。 賴驥緯 謹識於 國立臺灣師範大學 數學所 統計組 中華民國 一零五年 七月.

(3) 摘要 本研究之主要目的在於檢定態度量表中,作答者是否受到作答風格而影響 問卷之作答,即兩群人是否有作答風格之差異。所謂作答風格乃是當潛在態度相 同的兩位作答者可能會因為作答風格的差異,而做出不同的回答。本研究利用了 多群組離散型驗證性因素分析模型來分析多群組的有序分類數據,其中利用貝 氏估計在最小限制式的條件下,來估計模型中的閾值、潛在因子的平均與變異 數以及因素負荷量等結構參數,並且使用Gibbs sampling來估計這些參數的聯合 分配。再利用貝氏因子來檢驗李克氏五點量表在兩群組間是否存在作答風格差 異。於本研究中,透過模擬所得之結果顯示,貝氏因子可用來檢定極端、默認肯 定及默認否定等作答風格。本研究中分析「1998 年國際資訊科技教育應用研究 (SITES 1998)」之跨國五點量表問卷,選取其中來自法國及義大利的作答者,分 析對於資訊融入教學的學習成效認同程度。其中,義大利的作答者相對於法國之 作答者具有默認肯定之作答風格。. 關 鍵 字 :作答風格、態度量表、多群組離散型因素分析模型、貝氏估計、貝氏因 子.

(4) Bayesian evaluation of response styles in polytomous data with multiple group factor analysis model Chi-Wei Lai July 28, 2016 Department of mathematics, National Taiwan Normal University Abstract The main purpose of this study is to use Bayesian estimation and Bayes factor to test for response styles in polytomous data using multiple group categorical confirmatory factor analysis model. Joint Bayesian estimates of the thresholds, the factor means and variances, as well as the factor loadings using Gibbs sampling are proposed subjected to some minimal identifiability constraints. Bayes factor is used to test hypotheses of different types of response styles with their corresponding inequality constraints among the thresholds. Our simulation studies show that Bayes factor is effective in testing for different types of response styles. Analysis of an international comparative research suggests that Italy, despite having similar mean attitude, exhibits the acquiescent response style on how much they think information and communication technology improve the student’s achievement or ability compared to France. Keyword: Likert scale, response style, Bayesian estimation, Bayes factor, inequality constrained hypotheses. 1.

(5) Contents 1 Introduction. 5. 2 Multiple group categorical CFA model. 7. 3 Bayesian estimation 3.1 Conditional distributions of parameters 3.1.1 Y ’s conditional distribution . . . . . . 3.1.2 µ’s conditional distribution . . . . . . 3.1.3 F ’s conditional distribution . . . . . . 3.1.4 Λ’s conditional distribution . . . . . . 3.1.5 Φ’s conditional distribution . . . . . . 3.1.6 α’s conditional distribution . . . . . . 3.2 Convergence criteria . . . . . . . . . . . . . . 4 Bayes factors 4.1 Definition . . . . . . . . . . . . . . . . 4.2 Extreme response style (ERS) . . . . . 4.3 Mid-point response style (MRS) . . . . 4.4 Acquiescent response style (ARS) . . . 4.5 Disacquiescent response style (DARS) .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . . . . .. . . . . .. . . . . . . . .. . . . . .. . . . . . . . .. . . . . .. . . . . . . . .. . . . . .. . . . . . . . .. . . . . .. . . . . . . . .. . . . . .. . . . . . . . .. . . . . .. . . . . . . . .. . . . . .. . . . . . . . .. . . . . .. . . . . . . . .. . . . . .. . . . . . . . .. . . . . .. . . . . . . . .. . . . . .. . . . . . . . .. 9 11 11 12 12 13 14 15 16. . . . . .. 17 17 19 19 19 20. 5 Simulation study 20 5.1 Parameters setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2 starting value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6 Real data 23 6.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7 Discussion. 28. 8 Conclusion. 34. References. 36. 2.

(6) List of Figures 1 2. 3. 4 5 6 7 8 9. Potential scale reduction factor. . . . . . . . . . . . . . . . . . . . . Convergence assessment using shrink factor of thresholds of item 1 for France with 1=threshold 1, 2=threshold 2, 3=threshold 3, 4=thresholds 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence assessment using shrink factor of factor mean and variance with 2=factor mean for Italy, 3=factor variance for France, and 4=factor variance for Italy. . . . . . . . . . . . . . . . . . . . . . . . Posterior distributions of thresholds for France. . . . . . . . . . . . Posterior distribution of factor loadings for both France (Group 1) and Italy (Group 2). . . . . . . . . . . . . . . . . . . . . . . . . . . Posterior distribution of factor mean µ(2) for Italy. . . . . . . . . . . Posterior distribution of φ for France and Italy. . . . . . . . . . . . Questionnaire of SITES1998 part1 . . . . . . . . . . . . . . . . . . . Questionnaire of SITES1998 part2 . . . . . . . . . . . . . . . . . . .. 3. . 17. . 26. . 27 . 30 . . . . .. 31 32 33 39 40.

(7) List of Tables 1 2 3 4 5 6 7 8 9 10 11 12. Choice of constrained thresholds for testing different types of response styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interpretation of the Bayes factor . . . . . . . . . . . . . . . . . . . . The parameter values of α(1) for group 1. . . . . . . . . . . . . . . . . The parameter values of α(2) for group 2 under various response styles. The Bayes factor under different response style settings. . . . . . . . . The seven questions chosen from the original survey on “Information and Communication Technology” from SITES. . . . . . . . . . . . . . The posterior means and standard deviations (SD) of the thresholds parameters, α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The posterior means and standard deviations (SD) of Λ, φ, and µ. . . The Bayes factor for testing various response styles of Italy versus France. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i , for different numbers of quesThe inverse of complexity’s odds, 1−c ci tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Bayes factor with 1 satisfying observation under different response styles for different number of items. . . . . . . . . . . . . . . . fi in the simulations and the real data . . . . . . . . . . . . . . . . .. 4. 16 18 21 21 24 25 28 29 29 32 34 35.

(8) 1. Introduction. Attitude scales are widely used by researchers to measure people’s attitudes towards issue they care about such as products, services, social events or government policies (Friedman, Herskovitz, & Pollack, 1993). The Likert scale is easy to use (Boone & Boone, 2012), merely asking subjects to express to what extent they agree or disagree with a certain statement, by endorsing only one of the categories “Strongly disagree”, “Disagree”, “Neutral”, “Agree”, and “Strongly agree”. When studies of attitudes across groups generally rely on these Likert-scale questions, it becomes important to assure that people have the same interpretations of the substantive meanings of the categories they respond to. Usually, researchers are expected to design the questionnaires so that respondents with the same level of attitude or equal standings on the latent trait should have the same expected observed responses (Drasgow, 1984). However, people’s responses are also influenced by their response styles. Response style is that for two respondents coming from different backgrounds, they respond systematically differently in spite of having the same levels of latent attitude. For example, consider two respondents with the same latent attitude towards foreign cultures, one responds “Neutral” for all the questions but the other responds to all the questions with ”Agree” due to expressiveness linked to their background or culture. That is, the probabilities of endorsing such response categories have the relations that for p = 1, . . . , P P (zp = 2 | ξ, g = 1) > P (zp = 2 | ξ, g = 2), P (zp = 3 | ξ, g = 1) < P (zp = 3 | ξ, g = 2), where zp = 2 and zp = 3 represent respectively giving the responses of “Neutral” and ”Agree” on the p-th item, ξ represents the latent attitude of the respondent, and g indicates his or her group membership. In other words, response styles might cause this type of systematic difference in response probabilities due to factors other than the latent attitude or trait the questionnaire intended to measure. Hence, ignorance of the existence of response style may cause bias on estimating or comparing mean attitudes or latent traits across different groups. The most commonly seen examples of response styles are acquiescent response style (ARS) of which respondents have the tendency to agree, disacquiescent response style(DARS) of which respondents have the tendency to disagree, extreme response style (ERS) of which respondents have the tendency to endorse extreme categories such as strongly disagree or strongly agree, and mid-point response style (MRS) of which respondents prefer to use the middle response such as neutral. Past researches have shown that response styles may affect people’s responses, particularly on rating scales. Van Herk, Poortinga and Verhallen (2004) conducted three different marketing studies on cooking behavior, washing clothes, and shaving and found ARS and ERS to be present more in the Mediterranean such as Greece, Italy, and Spain, than in the Northwestern Europe such as German, France, and the United Kingdom. Moreover, Harzing (2006) found that, while investigating for undergraduate students the impact of the language of the questionnaire on their 5.

(9) responses of learning situations, Spanish-speaking countries show higher ERS and ARS, while East Asian (Japanese & Chinese) respondents show a relatively high level of MRS, German respondents reflect higher ARS than British respondents, and within Europe the Greeks stand out as having the highest level of ARS and ERS. In Harzing’s study, respondents in Hong Kong and China exhibit low DARS and ERS but high MRS, Japanese show the highest MRS in the 26 countries whereas Taiwan locates in a middle position between these extremes. Song and Lee (2001) developed a Bayesian approach for the multiple groups factor analysis model with continuous and polytomous variables with data augmentation (Tanner & Wong, 1987) via the Gibbs sampling (Geman & Geman, 1984). We only consider the multiple groups factor analysis model with polytomous variables to analyze the Likert scales data. Different from Song and Lee (2001) setting the thresholds as the nuisance parameters, we focus on testing particular inequality relations among the threshold parameters associated with different types of response styles. Due to the ordinal nature of the thresholds parameters, we also improve on the choice of prior for the thresholds parameters using the order statistics of the uniform distribution rather than the non-informative prior. In addition, we use the minimal identifiability constraints to ensure model identifiability and convergence of the posterior distributions (Chang, Hsu, & Tsai, 2016; Millsap & Tein, 2004). The Bayes factor commonly used in Bayesian hypothesis testing was first discussed by Jeffreys (1935,1961) and integrated by Kass and Raftery (1995). It has also be applied to test for the hypotheses involving constraints among the structural parameters of the factor analysis models across different groups (Song & Lee, 2001). To test for response styles, we focus on testing for the hypotheses involving inequality constraints among the threshold parameters of the factor analysis model between different groups. Klugkist and Hoijtink (2007) and Hoijtink (2013) suggested that the Bayes factor could be used to test for inequality constrained hypotheses by choosing prior distributions such that one of the terms in the Bayes factor is the quantification of the complexity of the hypothesis of interest, and the other term in the Bayes factor represents the fit measurement of the hypothesis. The rest of this thesis is organized as follows. In Section 2, we first state the formulation for the Multiple group categorical CFA (MCCFA) model, followed by a detailed correspondence between the particular parameters patterns and different types of response styles. Secondly, the Bayesian estimation of the MCCFA model parameters using Gibbs sampling is described in Section 3. More importantly, Section 4 introduces the use of Bayes factor in examining the presence of response styles through testing inequality constrained hypotheses on the thresholds. Simulations studies are conducted to assess the validity of Bayes factor in assessing different types of response styles shown in Section 5. Furthermore, the empirical performance of Bayes factor in analyzing an international comparative research data for response styles are reported. At last, a brief discussion is given, followed by some concluding remarks.. 6.

(10) 2. Multiple group categorical CFA model (g). For the response data on the P polytomous items for person i in group g, z i = (g) (g) (g) (zi1 zi2 · · · ziP )T , the multiple group categorical confirmatory factor analysis (CFA) model is formulated as (g). (g). (g). y i = Λ(g) ξ i + i , (g). (g). (g). i = 1, . . . , ng ,. g = 1, . . . , G,. (1). (g). where y i = (yi1 yi2 · · · yip )T is the underlying latent responses on the P items for (g) person i in group g, Λ(g) is a P ×q factor loading matrix, and ξ i is a q×1 vector of latent factor scores distributed as a multivariate normal distribution with a q ×1 mean (g) (g) vector µ(g) , and a q × q covariance matrix Φ(g) , i.e., ξ i ∼ MVN(µ(g) , Φ(g) ). i is a P × 1 vector of error measurements distributed as MVN(0, Ψ(g) ) with the diagonal (g) (g) (g) (g) matrix Ψ(g) = Diag(ψ11 , . . . , ψP P ), and i is assumed to be independent of ξ i . As T (g) a result, the covariance structure of y i is given by Σ(g) = Λ(g) Φ(g) Λ(g) + Ψ(g) for (g) g = 1, . . . , G. It is further assumed that y i ’s are all independent with i = 1, . . . , ng and g = 1, . . . , G. (g) The unobservable random vector y i is assumed to give rise to the observed (g) polytomous random vector z i through discretization. The relationship between (g) (g) y i and z i is given by (g). (g). (g). (g). zip = k, if αp,k < yip ≤ αp,k+1 ,. k = 0, 1, . . . , bp , p = 1, . . . , P,. (g). where zip is the ordinal response for item p with bp + 1 categories {0, 1, . . . , bp } for (g) (g) (g) p = 1, . . . , P , and αp,0 = −∞ and αp,bp +1 = ∞ for all items. The elements αp,k and (g). αp,k+1 are the unknown thresholds that define the response categories k = 0, 1, . . . , bp (g). (g). (g). and αp = (αp,1 · · · αp,bp )T . Naturally, we assume that the numbers of polytomous variables and thresholds are the same across groups. Due to the five-point Likert scale, we set bp = 4 for p = 1, . . . , P . Response style is said to exist when some particular patterns occur on the differences in response probabilities of some categories between respondents of the same latent attitude but belonging to different groups. The four most commonly seen responses styles of group 2 in comparison to group 1 are as follows: • Acquiescent response style (ARS): tendency to agree and choose “Agree” or “Strongly agree” (1). (2). (1). (2). αp,3 > αp,3 and αp,4 > αp,4 for p = 1, . . . , P.. (2). • Disacquiescent response style (DARS): tendency to disagree and choose “Disagree” or “Strongly disagree” (1). (2). (1). (2). αp,1 < αp,1 and αp,2 < αp,2 for p = 1, . . . , P.. 7. (3).

(11) • Extreme response style (ERS): tendency to choose extreme categories such as “Strongly disagree” or “Strongly agree” (1). (2). (1). (2). αp,1 < αp,1 and αp,4 > αp,4 for p = 1, . . . , P.. (4). • Mid-point response style (MRS): tendency to choose middle category such as “Neutral” (1). (2). (1). (2). αp,2 > αp,2 and αp,3 < αp,3 for p = 1, . . . , P.. 8. (5).

(12) 3. Bayesian estimation (g). (g). For the g-th group, let Z (g) = (z 1 , . . . , z ng ) be the matrix of the observed poly(g) (g) (g) tomous data, Y (g) = (y 1 , . . . , y ng ) and F (g) = (ξ 1 , . . . , ξ (g) ng ) be respectively the (g) T. (g) T. matrix of the latent response and factor score data, and α(g) = (α1 , . . . , αP )T (g) be the group-specific threshold matrix with αp being the vector of thresholds associated with the p-th row of Z (g) , Z (g) p . In other words, respondents of the same group share the same threshold matrix α(g) . Moreover, let Z = (Z (1) , . . P . , Z (G) ), Y = (Y (1) , . . . , Y (G) ), F = (F (1) , . . . , F (G) ), α = (α(1) , . . . , α(G) ), and n = G i=1 ng . (g) Let Λ be the parameter matrix consisting of all the Λ ’s for g = 1, . . . , G, and µ, Φ, and Ψ are all similarly defined. For the multiple group categorical CFA model, the parameters need to be estimated are the thresholds α, the factor loadings Λ, the means µ, the covariance matrix of latent factors Φ, and the variance matrix of the error measurement Ψ. However, not all the above-mentioned parameters are identifiable due to the categorical nature of the polytomous responses. To ensure identifiability of the parameters, we constrain Ψ = I, the identity matrix. That is, the variance of ip is fixed at one for each of all the items. Thus, the parameters need to be estimated become α, Λ, Φ, and µ. To obtain the posterior distribution p(α, Λ, Φ, µ | Z), we need to solve complicated multiple integrals induced by the polytomous variables. Therefore, it would be difficult to obtain the Bayesian estimates of the parameters (α, Λ, Φ, µ) via direct derivation of p(α, Λ, Φ, µ | Z). Tanner and Wong(1987) introduced an iterative method for the computation of the posterior distributions with augmented data, which facilitate the simulations of the parameters of interest. Therefore, we consider the complete data as (Z, Y , F ) by augmenting the observed data Z with the missing data (Y , F ). Random draws will be obtained from the joint conditional distribution of [Y , µ, F , Λ, Φ, α | Z] through the Gibbs sampling (Geman & Geman, 1984). The iterative process of the Gibbs sampling is given as following: At the t-th iteration with current values of Y (t) ,µ(t) ,F (t) ,Λ(t) ,Φ(t) and α(t) , we successively draw from the following conditional distributions that • generate Y (t+1) from p(Y | Z, µ(t) F (t) , Λ(t) , Φ(t) , α(t) ); • generate µ(t+1) from p(µ | Z, Y (t+1) , F (t) , Λ(t) , Φ(t) , α(t) ); • generate F (t+1) from p(F | Z, Y (t+1) , µ(t+1) , Λ(t) , Φ(t) , α(t) ); • generate Λ(t+1) from p(Λ | Z, Y (t+1) , µ(t+1) , F (t+1) , Φ(t) , α(t) ); • generate Φ(t+1) from p(Φ | Z, Y (t+1) , µ(t+1) , F (t+1) , Λ(t+1) , α(t) ); • generate α(t+1) from p(α | Z, Y (t+1) , µ(t+1) , F (t+1) , Λ(t+1) , Φ(t+1) ). To derive the conditional distributions for the Gibbs sampling process, we need to specify the joint prior distribution of µ, Λ, Φ and α. Similar to Arminger and Muth´en (1998) who assumed the priors of µ, Λ and Φ to be independent to each 9.

(13) other, we assume µ,Λ and Φ as well as α are all independent. As a result, their joint prior distribution is given by p(µ, Λ, Φ, α) = p(µ)p(Λ)p(Φ)p(α). We first consider the prior distribution of the threshold parameters α. For g1 6= g2 , it is natural to assume that the prior distributions of α(g1 ) and α(g2 ) are independent. (g) And for the p-th item, αp is a random vector with a series of inequality relationships such that (g). (g). (g). (g). (g). (g). T α(g) p = (αp,1 αp,2 . . . αp,bp ) with αp,1 < αp,2 < · · · < αp,bp . (g). Here we use the order statistics of the a uniform distribution that αp,j = U(j) where U(j) is the j-th order statistic of U1 , . . . , Ubp ∼ Uniform(a, b) for j = 1, . . . , bp . Furthermore, we choose a to be small enough and b to be large enough to create a (g) near non-informative improper prior p(αp,j ) ∝ 1, for j = 1, . . . , bp and g = 1, . . . , G. (g) As a result, the prior distribution of αp becomes (g). (g). (g). (g). (g). (g). p(α(g) p ) ∝ p(αp,1 , αp,2 , · · · , αp,bp )I(αp,1 < αp,2 < · · · < αp,bp ) (g). (g). (g). (g). (g). (g). (g). (g). (g). (g). (g). (g). (g). (g). (g). ∝ p(αp,1 )p(αp,2 | αp,1 )p(αp,3 | αp,1 , αp,2 ) · · · p(αp,bp | αp,1 , αp,2 , . . . , αp,bp −1 ) (g). (g). ∝ p(αp,1 )p(αp,2 | αp,1 )p(αp,3 | αp,2 ) · · · p(αp,bp | αp,bp −1 ),. (6). where I is an indicator function. Since the marginal distributions of the order statistics of the uniform distribution on the unit interval are Beta distributions, we use some processes with change of variables to get the following (conditional) densities that (g). (g) p(αp,1 ). (g). (g). (g). αp,1 − a 0 αp,1 − a bp −1 1 bp ! = ( ) (1 − ) , 0!(bp − 1)! b − a b−a b−a (g). p(αp,j+1 | αp,j ) =. (g). p(αp,j , αp,j+1 ) (g). p(αp,j ) (g). =. (g). α −a j−1 αp,j+1 −a bp ! ( p,j ) (( b−a ) (j−1)!(j+1−j−1)!(bp −j−1)! b−a. (g). −. αp,j −a j+1−j−1 ) (1 b−a. (g). α −a bp ! ( p,j )j−1 (1 (j−1)!(bp −j)! b−a. (g). −. αp,j+1 −a b −j−1 1 2 )p ( b−a ) b−a. (g). −. αp,j −a 1 )bp −j b−a b−a. Let ΛTp denote the q × 1 vector as the transpose of Λ’s p-th row. For p1 6= p2 , we assume that the prior distributions of ΛTp1 and ΛTp2 are independent. Moreover, the following conjugate prior distributions are used for Λ, Φ, and µ such that (Broemeling, 1985; Lee, 1981; Lee & Zhu, 2000; Lindley & Smith, 1972, Gelman, Carlin, Stern, & Rubin, 2004): ΛTp ∼ MVN(ΛT0p , H 0p ), Φ(g). −1. (g). (g). (g). (g). ∼ Wq [R0 , ρ0 ], µ(g) ∼ MVN(µ0 , φ0 ). (7). where g = 1, . . . , G, Wq denotes a Wishart distribution, and the values of the hy(g) (g) (g) (g) perparameters ΛT0p , H 0p , R0 , ρ0 ,µ0 , and φ0 are to be given.. 10. ..

(14) Song and Lee (2001) derived the conditional distributions of Λ, Φ, and α necessary for the Gibbs sampling for the multiple group categorical CFA model with the restricted case of µ = 0. Here we relax such a restriction to allow for more generality. In the following sections, we detail all the conditional distributions of Y , µ, F , Λ, Φ and α used in our Gibbs sampling algorithm.. 3.1. Conditional distributions of parameters. Here we will show the conditional distributions in the Gibbs sampling as previously described. 3.1.1. p(Y. Y ’s conditional distribution (g). |F. (g). (g). (g). ,Λ ,α ,Z. (g). )=. =. ng Y. (g). (g). (g). p(y i | ξ i , Λ(g) , α(g) , z i ). i=1 ng P Y Y. (g). (g). (g). (g). (8). (g). (g) p(yip | ξ i , Λ(g) p , α , zip ). i=1 p=1. =. ng P Y Y. p(yip | ξ i , Λp(g) ) · I(α(g). (g) p,z ip. i=1 p=1. (g). ,α. (g) (g) p,z +1 ip. ). (yip ),. where (g). (g). (g). (g) (g) yip | ξ i , Λ(g) p ∼ MVN(Λp ξ i , ψpp = 1),. and I(α(g) p,z. (g). (g). ,α (g) ip. (g) p,z. (g) +1 ip. ). (yip ) is the indicator function of yip such that (. I(α(g). (9). ,α (g). p,z ip. (g) (g) p,z +1 ip. (g) (y ) ) ip. =. 1, α. (g) (g) p,zip. (g). < yip ≤ α. 0, otherwise.. 11. (g) (g). p,zip +1. ,. (10).

(15) 3.1.2. µ’s conditional distribution (g). (g) (g) p(µ(g) | F (g) , Φ(g) ) ∝ p(µ(g) )p(ξ 1 , . . . , ξ (g) ng | µ , Φ ) (g). ∝ p(µ ). ng Y. (g). p(ξ i | µ(g) , Φ(g) ). i=1. ∝|. 1 (g) (g) −1 (g) exp{− (µ(g) − µ0 )T φ0 (µ(g) − µ0 )} 2 ng n 1 X (g) − 2g −1 (g) | (ξ i − µ(g) )T Φ(g) (ξ i − µ(g) )} exp{− 2 i=1. 1 (g) − φ0 | 2. × | Φ(g). 1 (g) (g) −1 (g) ∝ exp{− [(µ(g) − µ0 )T φ0 (µ(g) − µ0 ) 2 ng X −1 (g) (g) + (ξ i − µ(g) )T Φ(g) (ξ i − µ(g) )},. (11). i=1. which is an exponential quadratic form in µ(g) . Completing the quadratic form and pulling out constant factors gives : µ(g) | F (g) , Φ(g) ∼ MVN(µng , φng ),. (12). where (g) −1. µng = (φ0. (g) −1. −1. + ng Φ(g) )−1 (φ0. −1 (g). (g). µ0 + ng Φ(g) ξ ). −1 −1 φng = (φ−1 0 + ng Φ ) ,. where ξ 3.1.3. (g). (g). is the average of ξ i , for i = 1, . . . , ng .. F ’s conditional distribution. Due to the previous section on the introduction of the MCCFA model, it is easy (g) (g) (g) (g) to see that y i = Λ(g) ξ i + i , where ξ i is a q × 1 vector with distribution (g) MVN(µ(g) , Φ(g) ) and i is a P × 1 vector with distribution MVN(0, Ψ(g) ). Thus, (g) y i is a P × 1 vector with distribution MVN(Λ(g) µ(g) , Λ(g) Φ(g) Λ(g)T + Ψ(g) ). Hence (g) (g) the covariance matrix of y i and ξ i becomes (g). (g). (g). (g). (g). (g). (g). Cov(y i , ξ i ) = Cov(Λ(g) ξ i + i , ξ i ) = Λ(g) Cov(ξ i , ξ i ) = Λ(g) Φ(g) . " # (g) ξ That is, i(g) is a (P + q) × 1 vector satisfying that yi ". #     (g) µ(g) ξi Φ(g) (Λ(g) Φ(g) )T (g) (g) (g) ∼ MVN , (g) | µ , Λ , Φ Λ(g) µ(g) Λ(g) Φ(g) Λ(g) Φ(g) Λ(g)T + Ψ(g) yi 12.

(16) According to the property of conditional multivariate normal distribution, the conditional distributions of F (g) becomes p(F. (g). |Y. (g). (g). (g). (g). ,µ ,Λ ,Φ ) =. ng Y. (g). (g). p(ξ i | y i , µ(g) , Λ(g) , Φ(g) ),. (13). i=1. where (g). (g). (g). ξ i | y i , µ(g) , Λ(g) , Φ(g) ∼ MVN(δ i , ∆(g) ),. (14). with (g). (g). δ i = µ(g) + (Λ(g) Φ(g) )T [Λ(g) Φ(g) Λ(g)T + Ψ(g) ]−1 (y i − Λ(g) µ(g) ), ∆(g) = Φ(g) − (Λ(g) Φ(g) )T [Λ(g) Φ(g) Λ(g)T + Ψ(g) ]−1 Λ(g) Φ(g) . Note that Ψ(g) has been fixed as the identity matrix I for identification purpose and therefore the above mean vector and variance-covariance matrix could be further simplified to (g). (g). δ i = µ(g) + (Λ(g) Φ(g) )T [Λ(g) Φ(g) Λ(g)T + I]−1 (y i − Λ(g) µ(g) ), and ∆(g) = Φ(g) − (Λ(g) Φ(g) )T [Λ(g) Φ(g) Λ(g)T + I]−1 Λ(g) Φ(g) . 3.1.4. Λ’s conditional distribution. p(Λ | Z, Y , F ) =. P Y. p(Λp | Y p , F ) ∝. p=1. ∝. P Y. p(Λp )p(Y k | Λp , F ). p=1. P Y. 1 T T exp{− (ΛTp − ΛT0p )T H −1 0p (Λp − Λ0p )} 2 p=1 G. ng. 1 XX (g) (g) × exp{− ψpp −1 (yip − Λp ξ i )2 } 2 g=1 i=1 =. P Y. 1 T −1 T −1 T −1 T exp{− [Λp H −1 0p Λp − Λp H 0p Λ0p − Λ0p H 0p Λp + Λ0p H 0p Λ0p ]} 2 p=1 G. ng. 1 XX (g) 2 (g) (g) (g) × exp{− ψpp −1 [yip − 2yip Λp ξ i + (Λp ξ i )2 ]} 2 g=1 i=1 =. P Y. 1 T −1 T −1 T −1 T exp{− [Λp H −1 0p Λp − Λp H 0p Λ0p − Λ0p H 0p Λp + Λ0p H 0p Λ0p ]} (15) 2 p=1 G. 1X T T (g) T × exp{− ψpp −1 [Y (g) − 2Λp F (g) Y (g) + Λp F (g) F (g) ΛTp ]}, p Yp p 2 g=1 13.

(17) T where ψpp denotes the p-th diagonal element of Ψ, Λ0p H −1 0p Λ0p can be regarded as T a constant, Λp H −1 0p Λ0p is of size 1 × 1, and H 0p is symmetric. Therefore, we have T T −1 T −1 T T −1 (Λp H −1 = H −1 0p Λ0p ) = Λ0p H 0p Λp , and {H 0p } = {H 0p } 0p .. Thus, (15) can be further simplified to P. p(Λ | Z, Y , F ) ∝ exp{−. −. T 2Λp [H −1 0p Λ0p. +. G. X 1X T {Λp [H −1 + ψpp −1 F (g) F (g) ]Λp T 0p 2 p=1 g=1. G X. −1 (g) (g) F Y p T] ψpp. +. G X. T. (g) ψpp −1 Y (g) }}. p Yk. (16). g=1. g=1. Thus, we obtain Λp ’s conditional distribution such that Λp | Z, Y p , F ∼ MVN(ω p , Ωp ),. (17). with ωp =. T Ωp [H −1 0p Λ0p. +. G X. T. −1 (g) (g) ψpp F Y p ],. g=1. Ωp = [H −1 0p +. G X. T. −1 (g) (g) −1 ψpp F F ] .. g=1. Because Ψ is fixed at I for identification purpose, ω p and Ωp are further simplified G G P P T T T to ω p = Ωp [H −1 F (g) Y p(g) ] and Ωp = [H −1 F (g) F (g) ]−1 . 0p Λ0p + 0p + g=1. 3.1.5. g=1. Φ’s conditional distribution −1. p(Φ. −1. −1. −1. −1. | F , µ) ∝ p(Φ )p(µ | Φ )p(F | µ, Φ ) ∝ p(Φ ). ng Y. p(ξ i | µ, Φ). i=1. =. | Φ−1 |. ρ0 −q−1 2. 2. ρ0 q 2. exp{− 12 tr(R0 −1 Φ−1 )} ρ0. | R0 | 2 Γq ( ρ20 ). ng Y. 1 1 exp{− (ξ i − µ)T Φ−1 (ξ i − µ)} √ 1 q 2 2 i=1 ( 2π) | Φ | ng ng +ρ0 −q−1 X 1 −1 −1 −1 2 ∝|Φ | exp{− [tr(R0 Φ ) + (ξ i − µ)T Φ−1 (ξ i − µ)]}. 2 i=1. ×. We can see that ng ng X X T −1 (ξ i − µ) Φ (ξ i − µ) = tr[Φ−1 (ξ i − µ)(ξ i − µ)T ] i=1. i=1 ng. =. X i=1. 14. tr[(ξ i − µ)(ξ i − µ)T Φ−1 ].. (18).

(18) Therefore, we have p(Φ. −1. −1. | F , µ) ∝ | Φ. |. ng +ρ0 −q−1 2. ng. X 1 exp{− tr[(R−1 + (ξ i − µ)(ξ i − µ)T )Φ−1 ]}. 0 2 i=1. We then obtain Φ−1 ’s conditional distribution that Φ−1 | F , µ ∼ Wq (R, ng + ρ0 ),. (19). with R=. (R−1 0. +. ng X. (ξ i − µ)(ξ i − µ)T )−1 .. i=1. 3.1.6. α’s conditional distribution. In this study, we will focus on the MCCFA model for the five-point Likert scale data with bp = 4 for all of the P items. The threshold α(g) ’s conditional distribution is given by (g). (g). (g). (g). (g) p(α(g) , Z (g) ) ∝ p(αp,1 , αp,2 , αp,3 , αp,4 ) p | Y (g). (g). (g). (g). × I(˜y(g) ,y(g) ) (αp,1 ) · I(˜y(g) ,y(g) ) (αp,2 ) · I(˜y(g) ,y(g) ) (αp,3 ) · I(˜y(g) ,y(g) ) (αp,4 ) 0. 1. 1. 2. 2. 3. 3. e e (g) e (g) (g) (g) (g) (g) (g) ∝ p(αp,1 )p(αp,2 | αp,1 )p(αp,3 | αp,2 )p(αp,4 | αp,3 ) (g). (g). 4. e. (g). (g). × I(˜y(g) ,y(g) ) (αp,1 ) · I(˜y(g) ,y(g) ) (αp,2 ) · I(˜y(g) ,y(g) ) (αp,3 ) · I(˜y(g) ,y(g) ) (αp,4 ), 0. 1. e. (g). 1. 2. 2. e. (g). (g). 3. e. 3. (g). (20). 4. e. (g). where y˜j−1 is the maximum of yip satisfying zip = j − 1, y j is the minimum of yip e (g) (g) (g) satisfying zip = j, and I(˜y(g) ,y(g) ) (αp,t ) is the indicator function of αp,j such that j−1 j e ( (g) (g) (g) 1, y˜j−1 < αp,j ≤ y j (g) (21) I(˜y(g) ,y(g) ) (αp,j ) = e j−1 j 0, otherwise. e To guarantee the identifiability of the Bayesian estimation, Bayesians have two choices between adding such constraints and placing proper prior distributions on the unidentified parameters. Here we consider the first approach of setting the minimum identifiability constraints such that (Chang, Hsu, & Tsai, 2016; Chang, Huang, & Tsai, 2015; Millsap & Tein, 2004) • µ(1) = 0, (1). (2). • λ1 = λ1 , • Ψ(1) = Ψ(2) = I P ×P , where I P ×P be the identity matrix with size P × P , (1). (2). • α1,c = α1,c , for some c ∈ {1, . . . , bk }. 15.

(19) With the above identifiability constraints, all the other parameters are just identified for the MCCFA model with two groups and P ≥ 3. Since we are interested to testing for response style in the five-point Likert scale, our bp = 4, which means that we have four thresholds for each item. But for each type of response styles, we have to choose different c to be constrained. Because, for example, when testing for ERS, we will focus on comparing the first and the last thresholds of each item between different groups. Thus, we need to avoid constraining the thresholds most relevant to the response style of interest. Besides, If we (1) (2) constrain α1,1 = α1,1 , we would then be unable to assess the inequality hypothesis on the first threshold of item 1 for ERS between the two groups. Therefore, we have to choose different c for each type of response styles by the criterion shown in Table 1. Table 1: Choice of constrained thresholds for testing different types of response styles Response style ERS 2 MRS 1 ARS 1 DARS 3. 3.2. c or or or or. 3 4 2 4. Convergence criteria. Since we use Gibbs sampling to obtain random draws from the joint posterior distribution of (µ, Λ, Φ, α | Z). It is necessary to assure that the MCMC runs have reached convergence to regard the random draws as from the joint posterior distribution. Robert and Casella (2010) suggested the use the package coda in R written by Plummer, Best, Cowles, and Vines (2006), which was primarily intended for processing the output of a BUGS run (Lunn, Thomas, Best, & Spiegelhalter, 2000) can also be utilized directly to handle an arbitrary output from the Gibbs sampling. In order to use coda in R to check for convergence, we need to set up several chains with different and reasonable starting values since the processes normally take a long time to reach convergence. Robert and Casella (2010) considered the central convergence criterion of Gelman & Rubin (1992) such that the scale value, shrink factor of coda being very far from 1 indicates that the multiple chains have not yet converged to the same region. Convergence is claimed to be achieved as all the shrink factor values are less than 1.2 (Song & Lee, 2001). The examplary results of convergence assessment using the shrink factor are given in Figure 1.. 16.

(20) Figure 1: Potential scale reduction factor.. 4. Bayes factors. Sir Harold Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory in a 1935 paper, and in his book Theory of Probability which played an important role in the revival of the Bayesian view of probability. Kass and Raftery (1995) gave a contemporary overview of the state of Bayes factors.. 4.1. Definition. Consider data Z arise under one of two hypotheses H1 and H2 according to a probability density p(Z | H1 ) or p(Z | H2 ). Given the prior probabilities p(H1 ) and p(H2 ) = 1 − p(H1 ), the data Z produce the posterior probabilities p(H1 | Z) and p(H2 | Z) = 1 − p(H1 | Z), respectively. Because of the Bayesian process, any prior opinions on the competing hypotheses will be updated after taking into account the observed data Z to form the posterior opinions on the hypotheses. Once we convert to these probabilities in terms of odds instead, the posterior odds is of a simple form. More specifically, we obtain based on Baye’s theorem that p(Hk | Z) =. p(Z | Hk )p(Hk ) , p(Z | H1 )p(H1 ) + p(Z | H2 )p(H2 ). where k = 1, 2. Thus, the posterior odds becomes p(H1 | Z) p(Z | H1 ) p(H1 ) = , p(H2 | Z) p(Z | H2 ) p(H2 ) 17. (22).

(21) Table 2: Interpretation of the Bayes factor BF12 2 log BF12 <1 <0 1−3 0−2 3 − 20 2−6 20 − 150 6 − 10 > 150 > 10. Evidence against H2 Negative(supports H2 ) Barely worth mentioning Postive(supports H1 ) Strong Decisive. and BF12 =. p(Z | H1 ) p(Z | H2 ). (23). which is the Bayes factor. In other words, posterior odds = Bayes factor × prior odds (see also Kass & Raftery, 1995). Our main purpose is to test response style between two groups, and for each type of response styles, it has different inequality constraints on the thresholds parameter α(g) for g = 1, . . . , G. Hoijtink (2013) suggested replacing H0 and H1 which were used in exploratory statistical analysis into Hi , Hc , and Hu while testing inequality constrained hypotheses. Hi is an hypothesis that reflects our expectations about what is going on in the population of interest by means of inequality constraints among the threshold parameters α; Hc is the complement of Hi ; and Hu is an hypothesis without any constraints on the threshold parameters α. Hoijtink (2013) also clarify that we can use complexity and fit to compute the Bayes factor, where complexity is the proportion of the prior distribution of Hu in agreement with Hi , denoted by ci ; and fit is the proportion of the posterior distribution of Hu in agreement with Hi , denoted by fi . With ci and fi , we can get BFiu =. fi . ci. (24). Since Hc is the complement of Hi , the proportion of the prior distribution of Hu in agreement with Hc is 1 − ci . And the proportion of the posterior distribution of Hu in agreement with Hc is 1 − fi . Hence we can obtain the Bayes factor BFic by the following: BFic =. BFiu fi 1 − fi f i 1 − ci = / = . BFcu ci 1 − ci 1 − f i ci. (25). Although there is no established rule for the interpretation of Bayes factor values, we report in Table 2 the guidelines provided by Kass and Raftery (1995) which is in line with Jeffreys(1961). Since we are interested in testing for response style in two groups, we compute the complexity for each type of response style using the order statistics of the uniform distribution, U(a, b) as the joint prior distribution of α as follow: 18.

(22) 4.2. Extreme response style (ERS) (1). (2). (1). (2). Hi : αp,1 < αp,1 and αp,4 > αp,4 for p = 1, . . . , P (1). (1). (2). (2). (1). (1). (2). (2). p(αp1 , αp4 , αp1 , αp4 ) = p(αp1 , αp4 )p(αp1 , αp4 ) (1). (1). (2). (2). 4! αp4 − a αp1 − a 2 1 2 4! αp4 − a αp1 − a 2 1 2 = [ − ]( ) [ − ]( ) 0!2!0! b − a b−a b − a 0!2!0! b − a b−a b−a 1 8 (1) 4! (1) (2) (2) ) (αp4 − αp1 )2 (αp4 − αp1 ) (26) = ( )2 ( 2! b − a Due to the independence between different items, we have P Z b Z α(1) Z α(2) Z α(2) Y p4 p4 p1 (1) (1) (2) (2) (1) (2) (2) (1) ci = P (Hi ) = p(αp1 , αp4 , αp1 , αp4 )dαp1 dαp1 dαp4 dαp4 p=1. =. 4.3. a. a. P Y 144. a. a P. P. Y 144 Y 3 1 8 3 ) (a − b)8 = = = ( )P . 672 b − a 672 p=1 14 14 p=1 p=1 (. (27). Mid-point response style (MRS) (1). (2). (1). (2). Hi : αp,2 > αp,2 and αp,3 < αp,3 for p = 1, . . . , P (1). (1). (2). (2). (1). (1). (2). (2). p(αp2 , αp3 , αp2 , αp3 ) = p(αp2 , αp3 )p(αp2 , αp3 ) (1). (1). (2). (2). αp3 − a αp3 − a 4! αp2 − a 1 2 4! αp2 − a 1 2 ( )(1 − )( ) ( )(1 − )( ) 1!0!1! b − a b−a b − a 1!0!1! b − a b−a b−a 1 8 (1) (1) (2) (2) = (4!)2 ( ) (αp2 − a)(b − αp3 )(αp2 − a)(b − αp3 ) (28) b−a =. Due to the independence between different items, we have P Z b Z α(2) Z α(1) Z α(1) Y p3 p3 p2 (1) (1) (2) (2) (2) (1) (1) (2) ci = P (Hi ) = p(αp2 , αp3 , αp2 , αp3 )dαp2 dαp2 dαp3 dαp3 p=1. =. 4.4. a. a. a. a. P P Y Y 576 1 8 9 9 ( ) (a − b)8 = = ( )P . 4480 b − a 70 70 p=1 p=1. (29). Acquiescent response style (ARS) (1). (2). (1). (2). Hi : αp,3 > αp,3 and αp,4 > αp,4 for p = 1, . . . , P (1). (1). (2). (2). (1). (1). (2). (2). p(αp3 , αp4 , αp3 , αp4 ) = p(αp3 , αp4 )p(αp3 , αp4 ) (1). (2). 4! αp3 − a 2 1 2 4! αp3 − a 2 1 2 ( )( ) ( )( ) 2!0!0! b − a b − a 2!0!0! b − a b−a 4! 1 8 (1) (2) = ( )2 ( ) (αp3 − a)2 (αp3 − a)2 2! b − a =. 19. (30).

(23) Due to the independence between different items, we have (1). P Z bZ Y ci = P (Hi ) = ( (1) αp4. Z bZ. (2). αp4. Z. (1). αp3. Z. (1). (1). (2). (2). (2). (1). (2). a. a (1) αp3. Z. (2) αp4. Z. (1). (1). (2). (2). (2). (2). (1). (1). p(αp3 , αp4 , αp3 , αp4 )dαp3 dαp4 dαp3 dαp4 ). + a P Y. a. a. a. P Y 4! 2 1 8 (a − b)8 (a − b)8 5 5 = [( ) ( )( + )] = = ( )P . 2! b − a 1008 672 14 14 p=1 p=1. 4.5. (1). p(αp3 , αp4 , αp3 , αp4 )dαp3 dαp3 dαp4 dαp4. a. a. p=1. αp4. (31). Disacquiescent response style (DARS) (1). (2). (1). (2). Hi : αp,1 < αp,1 and αp,2 < αp,2 for p = 1, . . . , P (1). (1). (2). (2). (1). (1). (2). (2). p(αp1 , αp2 , αp1 , αp2 ) = p(αp1 , αp2 )p(αp1 , αp2 ) (1). (2). αp2 − a 2 1 2 4! αp2 − a 2 1 2 4! (1 − )( ) (1 − )( ) = 0!0!2! b−a b − a 0!0!2! b−a b−a 1 8 4! (1) (2) ) (b − αp2 )2 (b − αp2 )2 (32) = ( )2 ( 2! b − a Due to the independence between different items, we have (2). P Z bZ Y ( ci = P (Hi ) =. αp2. a. Z bZ. αp2. (2). (2). αp1. Z. (1). (1). (2). (2). (1). (2). (1). (2). p(αp1 , αp2 , αp1 , αp2 )dαp1 dαp1 dαp2 dαp2. a. p=1. (1). αp2. Z a. Z. (2). αp1. a. Z. (1). αp2. (1). (1). (2). (2). (1). (1). (2). (2). p(αp1 , αp2 , αp1 , αp2 )dαp1 dαp2 dαp1 dαp2 ). + a. a. a. a. P P Y Y 4! 2 1 8 (a − b)8 (a − b)8 5 5 = [( ) ( )( + )] = = ( )P . 2! b − a 1008 672 14 14 p=1 p=1. (33). Once we have random draws from the posterior distribution of all the parameters via the Gibbs sampling, for T iterations after the burn-in, we can compute fi as PT (t) ∈ Hi ) t=1 I(α , (34) fi = T where I(α(t) ∈ Hi ) is the indicator function of whether the thresholds α(t) conform to the inequality constrained hypothesis Hi associated with a particular type of response style.. 5. Simulation study. We use R (R Core Team, 2015) to generate data Z from MCCFA model and create random sample draws from the posterior distributions by the Gibbs sampling to 20.

(24) test for response style. The Bayes factors of inequality constrained hypotheses with corresponding thresholds parameter α among different groups are computed to test for different types of response styles.. 5.1. Parameters setting. First of all, we consider the setting of two groups g = 1, 2 with ten items p = 1, . . . , 10 and one latent factor q = 1. The parameters of the MCCFA model for data generation are set as µ(1) = 0, µ(2) = −1, Φ(g) = φ(g) = 1 for both groups, and Λ and Ψ are also the same for the two groups such that Λ(g) =. . 1.0 0.9 0.7 0.88 0.72 0.86 0.74 0.84 0.76 0.82. T. , and Ψ(g) . = I. We report in Table 3 the parameter values α(1) used for group 1. Table 3: The parameter values of α(1) for group 1. p 1 2 3 4 5 6 7 8 9 10. (1). αp,1 -2.78 -1.89 -2.74 -1.82 -2.44 -2.35 -2.12 -1.97 -2.98 -2.80. (1). αp,2 -1.31 -0.67 -1.38 -0.65 -1.29 -1.15 -1.15 -0.70 -1.48 -1.35. (1). αp,3 -0.60 -0.08 -0.45 -0.20 -0.58 -0.46 -0.59 -0.12 -0.57 -0.43. (1). αp,4 0.96 1.48 1.33 0.87 0.85 0.95 1.00 1.28 1.35 1.31. Moreover, depending on the type of response style under consideration, a different set of parameters α would be used to generate data for group 2 with that particular type of response style. The sets of parameters α(g) used to generate data for group 2 with different types of response styles are listed in Table 4. Table 4: The parameter values of α(2) for group 2 under various response styles. Response style ERS MRS ARS DARS. α(2) (1) αp,1 + 0.4 and (1) αp,2 − 0.2 and (1) αp,3 − 0.2 and (1) αp,1 + 0.4 and. 21. (1). αp,4 − 0.4 (1) αp,3 + 0.2 (1) αp,4 − 0.4 (1) αp,2 + 0.2.

(25) We set all the hyperparameters of prior distributions of Λ(g) , Φ(g) , µ(g) and α(g) to be the same of all groups (g = 1, 2) as follows:     (g) (g) Λ0 (g) = 0.8 10×1 , H 0 (g) = 10I 10×10 , R0 = 18 , ρ0 = 20, (g). (g). µ0 = 0, φ0 = 10000, and a = −10000, b = 10000. (35). For each of the above parameter setting, we use a sample size of ng = 500 for each group and for each type of response style.. 5.2. starting value. Under the ERS setting, the starting values of the MCMC runs for Λ(g) , α(g) and Φ(g) are set to be the same for all groups and the setting for Chain 1 is given by.   Λ(g) = 1 10×1 , α(g).  T −2 −1     (2)  , Φ(g) = 2 , µ(1) = 1 10×1  0 = 0, µ0 = −2. 1 2. For the other two chains, we only change the setting of α(2) as :. α(g).  T  T −1.5 −2.5  −1   −1       for Chain 2, and α(g) = 1   for Chain 3. = 1 10×1   1  10×1  1  1.5 2.5 (36). In addition, for all three chains under ERS, we constrain equality on the second threshold (i.e., c = 2) between the two groups for identification purpose. Under the ARS setting, the starting values of the MCMC runs for Λ(g) , α(g) and Φ(g) are set to be the same as in the previous ERS setting for Chain 1. For the other two chains, we only change the setting of α(2) as : T  T −2 −1.6 −1 −0.8      for Chain 2, and α(g) = 1   for Chain 3. (37) = 1 10×1  0.8 10×1  0.8  1.6 1.6 . α(g). Different from equating the second threshold in ERS, we constrain the three chains for ARS the equality on the first threshold (i.e., c = 1) between the two groups for identification purpose. Under the DARS setting, the starting values of the MCMC runs for Λ(g) , α(g) and Φ(g) are set to be the same as in the previous ERS setting for Chain 1. For the. 22.

(26) other two chains, we only change the setting of α(2) as :. α(g). T  T  −1.6 −2.4 −0.8 −1.2      for Chain 2, and α(g) = 1   for Chain 3. = 1 10×1   1  10×1  1  2 2 (38). Different from equating the second threshold in ERS and the first threshold in ARS, we constrain the three chains for DARS the equality on the last threshold (i.e., c = 4) between the two groups for identification purpose. Under the MRS setting, the starting values of the MCMC runs for Λ(g) , α(g) and Φ(g) are set to be the same as in the previous ERS setting for Chain 1. For the other two chains, we only change the setting of α(2) as : T  T −2 −2.5 −1.2 −1.5      for Chain 2, and α(g) = 1   for Chain 3. = 1 10×1   1.2  10×1  0.5  2 1.5 (39) . α(g). In MRS case, we constrain the three chains for MRS the equality on the last threshold (i.e., c = 4) between the two groups for identification purpose.. 5.3. Results. We set the number of MCMC iterations to be 250000 for each chain. Due to the criteria of convergence, we choose to burn in the first 50000 iterations to ensure convergence and the thinning is set to be 10 to avoid high correlations between successive draws. Table 5 reports the Bayes factor values obtained under different response style setting. According to Table 5, the Bayes factors obtained in the ARS case show very decisively ARS pattern whereas they are 0 for the other three types of response styles. In the ERS case, the Bayes factors show both the ERS and ARS patterns. In the DARS case, the Bayes factors also show very decisively DARS pattern whereas they are 0 for the other three types of response styles. In the MRS case, the Bayes factors also show very decisively MRS pattern in Chain 1 and 3, but not in Chain 3.. 6. Real data. In 1998, the International Association for the Evaluation of Educational Achievement (IEA) established an international comparative research project of Information and Communication Technology (ICT) infusion in different countries educational systems which is called Second Information Technology in Education Study 23.

(27) Table 5: The Bayes factor under different response style settings. α. µ(2). ARS. -1. DARS. -1. ERS. -1. MRS. -1. Chain 1 2 3 1 2 3 1 2 3 1 2 3. ARS 39038 56632 36003 0 0 0 5671 4021 6530 0 0 0. DARS 0 0 0 24998 38905 28797 0 0 0 0 0 0. ERS 0 0 0 0 0 0 1297832 1920619 1488919 0 0 0. MRS 0 0 0 0 0 0 0 0 0 202583 0 121537. (SITES). The main purpose of the project was to provide policy-makers and educational practitioners with information on how ICT can to contribute to those system reforms to meet the needs of the Information Society. In this project, IEA made a survey in 26 countries. The survey collect samples of at least 200 computer-using schools from at least one of primary, lower secondary, and upper secondary. Van Herk, Poortinga & Verhallen (2004) found out that acquiescence and extreme response style present more in the Mediterranean such as Greece,Italy and Spain, than in the Northwestern Europe such as German, France and the United Kingdom. Fortunately, both Italy and France were included in the survey, so we will test for acquiescence and extreme response style between Italy and France. There are 768 respondents from France and 425 from Italy. The original questionnaire contains a total of 24 questions and for each question there are five response categories which are strongly disagree, slightly disagree, uncertain, slightly agree and strongly agree, i.e., in the form of a five-point Likert scale. Here, we only select seven questions from the survey which are presumably measuring the single factor that to what extent one thinks ICT can improve the student’s achievement or ability. The questions are listed in Table 6. We set France as group 1 and Italy as group 2, and all the hyperparameters of prior distributions will be the same as in (35). And the starting values for the MCMC runs are as follows:  T −2      −1 (1) (2) (g)  Λ(g) = 1 7×1 , α(g) = 1 7×1   1  , φ = 3, µ0 = 0, µ0 = −2. 2. 24.

(28) Table 6: The seven questions chosen from the original survey on “Information and Communication Technology” from SITES. item 1. 8 11 15 18 19 23. item description Students are more attentive when computers are used in class. ICT can effectively enhance problem solving and critical thinking skills of students. ICT-based learning enables students to take more responsibility for their own learning. ICT improves the monitoring of student’s learning progress. The achievement of students can be increased when using computers for teaching. The use of e-mail increases the motivation of students. Using computers in class leads to more productivity of students.. For the other two chains, we only change the setting of α(2) as :. α(g).  T  T −1.5 −2.5    −1     −1   for Chain 2, and α(g) = 1   for Chain 3. (40) = 1 7×1   1  7×1  1  1.5 2.5. Since we are interested in testing for ARS and ERS, we constrain the second threshold (i.e., c = 2) on the first item between Italy and France according to Table 1. (1) (2) That is, we set α1,2 = α1,2 .. 6.1. Results. We ran each chain for 150000 iterations. After the first 50000 burn-in, we used the thinning of 10 to collect a total of 10000 posterior draws. The convergence is assured with a shrink factor values 1, 1.01, 1.01, and 1 of the four thresholds of item 1 as shown in Figure 2. Moreover, Figure 3 has the shrink factors for µ(2) for Italy, and φ(1) and φ(2) for France and Italy respectively. Based on Figure 3 with shrink factor values of 1.05, 1, and 1, their MCMC are thought to have reached convergence after 50000 iterations. Thus, we only report the following results from Chain 1. Tables 7 and 8 report the posterior means and standard deviations of all the model parameter (1) (2) for France and Italy. Note that we do not discuss the posteriors of λ1 , λ1 and µ(1) because they are all fixed at 1 to ensure identifiability without estimating. Furthermore, we depict the posterior distributions for the model parameters. Figure 4 contains the posterior distributions of the thresholds of the first item for France. Figure 5 gives the posterior distributions of the factor loadings for both France and Italy. Note that we do not show the factor loading of the first item because they are both set to be equal to 1 for identification.The distributions with respect to the same threshold are paired for an easy comparison. For example, the loadings of items 2 to 7 for Italy appear to be larger than those for France, while constraining the loadings of item 1 to be equal to 1 for both countries. 25.

(29) Figure 2: Convergence assessment using shrink factor of thresholds of item 1 for France with 1=threshold 1, 2=threshold 2, 3=threshold 3, 4=thresholds 4. Moreover, since we set the latent factor mean of France to be equal to 0, we only show in Figure 6 the posterior distribution of the latent factor mean of Italy. Based on Figure 6, we can see that the posterior distribution of the factor mean of Italy have mean close to 0 in comparison to its standard deviation. That is, the mean attitude towards the benefit of using ICT do not seem to differ much between France and Italy. Moreover, we also show in the Figure 7 the factor variances for both France and Italy and they appear to have very similar factor variabilities.. 26.

(30) Figure 3: Convergence assessment using shrink factor of factor mean and variance with 2=factor mean for Italy, 3=factor variance for France, and 4=factor variance for Italy. Finally, we compute the Bayes factors to testing for different types of response styles. Table 9 reports the Bayes factor obtained from all the three chains. The large magnitudes of Bayes factor for ARS and 0 for the rest imply that Italy has ARS in comparison to France in responding to these ICT questions.. 27.

(31) Table 7: The posterior means and standard deviations (SD) of the thresholds parameters, α.. Country. France. Italy. 7. Parameter α1 α2 α3 α4 α5 α6 α7 α1 α2 α3 α4 α5 α6 α7. threshold 1 mean SD -2.322 0.116 -3.136 0.201 -3.988 0.343 -3.495 0.298 -3.559 0.230 -2.237 0.118 -3.125 0.187 -2.987 0.257 -3.137 0.312 -3.146 0.330 -3.135 0.319 -4.230 0.468 -2.416 0.248 -3.785 0.425. thershold 2 mean SD -1.861 0.078 -2.115 0.110 -2.509 0.131 -2.145 0.108 -2.906 0.171 -1.702 0.082 -2.154 0.122 -1.861 0.078 -2.131 0.265 -2.134 0.297 -2.002 0.262 -2.914 0.388 -1.717 0.220 -2.656 0.378. threshold 3 mean SD -0.534 0.050 -0.197 0.060 -0.859 0.069 -0.488 0.055 -0.715 0.076 0.027 0.051 0.106 0.067 -0.798 0.107 -1.088 0.241 -0.629 0.273 -0.954 0.240 -1.670 0.345 -0.580 0.200 -1.127 0.338. threshold 4 mean SD 1.135 0.066 1.523 0.083 1.258 0.083 1.154 0.067 1.554 0.093 1.282 0.067 2.059 0.125 -0.283 0.113 -0.021 0.234 0.624 0.284 0.106 0.227 -0.503 0.331 0.259 0.194 -0.100 0.332. Discussion. In the simulations, the Bayes factor showed for data with ARS very decisively ARS pattern and no the other three types of response styles with values of 0. For data with ERS, large values of Bayes values are also obtained as support for the presence of the ARS and ERS patterns. In the DARS case, the Bayes factors also show very decisively DARS pattern whereas they are 0 for the other three types of response styles. In the MRS case, the Bayes factors show very decisively MRS pattern in Chain 1 and 3, but not in Chain 2. In analyzing the real data from SITES1998, the Bayes factor suggested that Italy showed ARS pattern compared to France. However, it brought to our attention to suspect that the commonly used guideline for the interpretation of Bayes factor shown in Table 2 may not that appropriate for our proposed method in testing response style since those Bayes factor values we obtained from both the simulations and the analysis of real data were considerably much larger than 150, for criterion for a “Decisive” support, when response styles were present. For the example of ARS case shown in Table 5, a Bayes factor of 39038 was obtained in Chain 1 while testing for the ARS pattern. The reason for getting such extreme values of the Bayes factor is due to the small magnitude of complexity induced by our definition of having any type of response style in the MCCFA model. More specifically, when testing for the response styles, we expect that the respondents’ answer will reflect their response style patterns on all the questions in the questionnaire. For the questionnaire with P items, the. 28.

(32) Table 8: The posterior means and standard deviations (SD) of Λ, φ, and µ. Country France Italy Parameter mean SD mean SD (g) λ2 1.667 0.237 2.373 0.45 (g) λ3 2.004 0.270 2.836 0.515 (g) λ4 1.380 0.195 2.247 0.424 (g) λ5 2.266 0.306 3.492 0.645 (g) λ6 1.009 0.155 2.023 0.379 (g) λ7 2.335 0.318 3.407 0.628 (g) φ 0.270 0.060 0.245 0.078 µ(g) 0* 0* -0.027 0.101 µ(g) = 0∗ for France for identification. Table 9: The Bayes factor for testing various response styles of Italy versus France. Chain 1 2 3. ARS 428 517 635. DARS 0 0 0. ERS 0 0 0. MRS 0 0 0. calculation of the complexity of each response style becomes some smaller-than-one constant to the power of P . Therefore, ci will decrease as the number of items P in the questionnaires increases. As ci is such a small number, the value of (1 − ci )/ci will become very large, as shown in Table 10.. 29.

(33) Figure 4: Posterior distributions of thresholds for France. As ci is the proportion of the prior distribution of Hu in agreement with Hi , an extremely small ci implies that it is very unlikely to observe the corresponding response style pattern. If we do not obtain from any posterior draw with the parameters consistent with the response style pattern, i.e,, fi = 0, the Bayes factor will be 0. On the other hand, once we obtain even only one posterior draw with its threshold parameters satisfying the inequality constrained hypothesis Hi for the response style, the Bayes factor would become extremely large. For illustration, consider we have observed one single posterior draw, in the 20000 posterior draws, satisfying the inequality constrained hypotheses for particular response styles the resulting Bayes factors according to (25) are shown in Table 11. Based on the results in Table 11, we see that only one single posterior draw could make such a big impact on the magnitude of the Bayes factor. Therefore, it might be necessary to check the value of fi while computing the Bayes factor values as a seemingly decisive value of the Bayes factor could also result from a small fi but large ci odds inverse, (1 − ci )/ci . Thus, we went back to examine the fi values for out simulations and analysis of the real data, the fi , as reported in Table 12. In the ARS and ERS cases, the fi of ARS pattern were over 50% for the former, and fi ’s for ARS and ERS are both over 10%, for the DARS case, the fi of DARS 30.

(34) Figure 5: Posterior distribution of factor loadings for both France (Group 1) and Italy (Group 2). pattern were over 45%. Thus, we believed that the obtained decisively values of the Bayes factor in these cases were not caused simply by the incredibly large value of (1−ci )/ci , but the considerable amount of posterior draws satisfying the corresponding inequality constrained hypothesis. In other words, ARS response style pattern is correctly supported for data with ARS through using the Bayes factor. DARS response style pattern is also correctly supported for data with DARS through using the Bayes factor. However, both ERS and ARS patterns are supported for data with ERS. To further investigate whether this simply happens by chance, we generated a different dataset with nq = 1000 from the same setting of ERS, and found a Bayes factor of 0 for ARS and an extremely large value for ERS. More investigation is necessary to see whether the large Bayes factor value for ARS under the ERS data is simply due to chance. However, we would conclude that both ARS and ERS response style pattern could be correctly supported for data with ARS and ERS through using the Bayes factor for inequality constrained hypotheses. In MRS case, the Bayes factor of three chains show different results and the fi of MRS pattern were very close to 0.00005 in Chain 1 and 3. In other words, MRS case didn’t fit well in this study. However, it is obvious that the commonly used guideline in Table 31.

(35) Figure 6: Posterior distribution of factor mean µ(2) for Italy. Table 10: The inverse of complexity’s odds, number of items ARS 7 1348 8 3777 9 10577 10 29618 11 82934 12 232217. 1−ci , ci. DARS 1348 3777 10577 29618 82934 232217. for different numbers of questions.. ERS 48199 224932 1049688 4898551 22859910 106679600. MRS 1721822 13391962 104159712 810131101 6301020000 49007930000. 2 should not be directly applied to define or interpret the strength of a Bayes factor value. To build a guideline for the interpretation of Bayes factor for hypothesis that rarely occurs under the prior distributions is much needed for future research. In this paper, for the real data, we analyze response style by grouping respondents with their nationality because of the outcomes from the questionnaires. But the respondents with the same nationality may also have different response styles, in other words, respondents may group by their response styles. Therefore, the concept of latent group would be another direction for the future research. As for the running time of the proposed Bayes factor method using Gibbs sampling, in the ARS case, Chain 1 took 55.25 hours, Chain 2 with 54.48, and Chain 3 with 52.38 hours; in the ERS case, Chain 1 took 49.65 hours, Chain 2 with 50.52, and Chain 3 with 55.21 hours; in the DARS case, Chain 1 took 48.45 hours, Chain 2 with 49.41, and Chain 3 with 48.53 hours; in MRS case, Chain 1 took 49.88 hours, Chain 2 with 50.01, and Chain 3 with 51.31 hours; in analyzing real data of Italy and France, Chain 1 took 48.64 hours, Chain 2 with 49.52, and Chain 3 with 50.20 hours. Running times of obtaining the joint posterior distributions via the Gibbs. 32.

(36) Figure 7: Posterior distribution of φ for France and Italy. sampling depend on how many times of iterations we set, sample size, how many items in the questionnaire and what kind of settings on starting value. Since we can only test for hypotheses after the Gibbs sampling converge, if we choose the starting value with too far away, we need more iterations to reach convergence, which therefore cause more running time. Although the Gibbs sampling will spend longer time to obtain the Bayesian estimations rather than others estimation methods, its main benefit is that we can test for inequality constrained hypotheses on the corresponding parameter’s posterior distributions. To reduce the running time of the Gibbs sampling, we could try to use some starting values which were estimated by other estimation methods such as Mplus (Muth´en & Muth´en, 1998-2015), or using such values that have already been used by the previous researchers. Once the Gibbs sampling converges early, we could reduce the number of iterations necessary for obtaining posterior draws and therefore the running time could be directly reduced.. 33.

(37) Table 11: The Bayes factor with 1 satisfying observation under different response styles for different number of items. number of items 7 8 9 10 11 12. 8. ARS 0.067 0.188 0.528 1.481 4.146 11.61. DARS 0.067 0.188 0.528 1.481 4.146 11.61. ERS MRS 2.410 86.09 11.24 669.6 52.48 5208 244.9 40508 1143 315066 5334 2450519. Conclusion. In this study, we discussed a Bayesian approach for MCCFA model under minimal identifiability constraints for polytomous data via the Gibbs sampling, and used the Bayes factor to test for response styles with inequality constraints among the corresponding thresholds parameters. We conclude that the Bayes factor is effective in testing for different types of response styles but MRS case didn’t fit well in our simulations. We also conclude that the commonly used guideline for the interpretation of Bayes factor values in Table 2 may not be appropriate while using our proposed method to test for response styles.. 34.

(38) Table 12: fi in the simulations and the real data Setting ARS. DARS. ERS. MRS. Real data. Chain 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3. ARS DARS 0.56860 0 0.65660 0 0.54865 0 0 0.45770 0 0.56776 0 0.49296 0.1607 0 0.11955 0 0.18065 0 0 0 0 0 0 0 0.2411 0 0.2773 0 0.3204 0. 35. ERS MRS 0 0 0 0 0 0 0 0 0 0 0 0 0.20945 0 0.28165 0 0.23310 0 0 0.00025 0 0 0 0.00015 0 0 0 0 0 0.

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(42) Appendix. Figure 8: Questionnaire of SITES1998 part1. 39.

(43) Figure 9: Questionnaire of SITES1998 part2. 40.

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