混合時間模型─考慮輕率受訪者回答含有虛題的問卷行為─之應用
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(2) 致謝 一段經過短暫時間的漫長旅程，遇到很多很多的貴人。在此先感謝我的兩 位指導教授：蔡蓉青老師與呂翠珊老師。非常感謝她們的嚴格的要求與耐心、 細心地指導才得以完成這篇文章。完成的過程中也得到許多人建議與幫助。感 謝口試委員林定香老師給予建議，使文章內容更貼近初讀者並一目瞭然。淑貞 學姐與育瑋學長的熱情相助，在反覆的討論中建立對基礎反應模型的認識並給 予程式語言的指導。系上助教與彰化師範大學的老師們協助蒐集並提供實證資 料，讓這篇文章得以短時間完成。另外，家人的支持和朋友的鼓勵與協助，在 研究生活的咖啡中加點糖跟奶精。這篇文章絕非只是學術方面的結果，也是我 與這些生命中的貴人在這段旅程中的紀錄。如還有未提到而疏忽的貴人，一言 難盡，不勝感激。. i.
(3) 摘要 一份問卷調查的健全推論建立在有效率的估計之上，但是有效率的估計往 往受到不認真的受訪者的影響。問卷調查廣泛應用虛題 (bogus item) 找出不認 真的受訪者、發現那些不看題目的人。另一種方法是反應答題的時間，其使用 方式已由 Rasch 混合模型結合反應時間的方法確立，並能有效地提升估計的效 率。在這篇研究中，我們提出結合虛題與混合反應時間的模型。此模型是利用 虛題並考慮反應時間來區分「兩種不同類型的不認真受訪者的回答以及認真受 訪者的回覆」。分析時保留虛題做為共變數與時間做為潛在類別的應變數的優 點。在模擬實驗中，當樣本數 500 人以上，反應時間有降低估計值的標準誤的 趨勢。更進一步，我們應用該模型分析真實資料時，其中有高達 22% 的不認 真受訪者。結果顯示：我們未受限制的模型表現得比受限制的模型好。. 關鍵字：虛題、反應時間、潛在類別、Rasch 混合模型結合反應時間. ii.
(4) Abstract A valid inference for a survey is always based on efficient estimation. The efficient estimation is threatened and distorted by the careless response. Bogus item is commonly used in survey questionnaire to detect the careless respondents and powerfully unveils those responding without reading the items. Response time is another way to identify the careless responses and its use of enhancing item parameter estimation was verified by the mixture Rasch model with response time component (MRM-RT). Therefore, we propose a mixture response time model to analyze survey questionnaire with a bogus item. The goal of our proposed model is to classify two behaviors of the careless responding in addition to the attentive respondents, using the bogus item and taking account into response time. We attempt to converse the benefit of the bogus item used as a covariate and the response time used as the latent class indicators in analysis. In the simulation studies, we find that the standard errors of the model with response time decrease substantially as the sample size is equal or larger than 500. Further, we also apply our proposed model to a real data set and with a larger proportion of careless responding (22%), the results show that the unrestricted models perform better than the restricted models.. Keywords: bogus item, response time, latent class, MRM-RT. iii.
(5) Contents i. 致謝 中文摘要. ii. Abstract. iii. Contents. iv. List of Tables. vi. List of Figures. viii. 1 Introduction. 1. 2 Model. 6. 2.1. The Mixture Response Time Model . . . . . . . . . . . . . . . . .. 6. 2.2. A Restricted Condition . . . . . . . . . . . . . . . . . . . . . . . .. 11. 2.3. Estimation and Identification . . . . . . . . . . . . . . . . . . . .. 12. 3 Simulation 3.1. 3.2. 3.3. 14. Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14. 3.1.1. Class Size . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16. 3.1.2. Response and Response Time . . . . . . . . . . . . . . . .. 16. 3.1.3. Factors and Model Fittings . . . . . . . . . . . . . . . . .. 17. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. 3.2.1. Under The Equal Case . . . . . . . . . . . . . . . . . . . .. 19. 3.2.2. Under The Unequal Case . . . . . . . . . . . . . . . . . .. 39. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59. iv.
(6) 4 Data Analysis. 61. 4.1. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 62. 4.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 62. 5 Conclusion and Future Direction. 66. Reference. 68. Appendix. 71. v.
(7) List of Tables 1. The parameters transformed from logistic orgive to normal orgive. 15. 2. The estimates of the loading and thresholds for Item 1 (equal) . .. 21. 3. The estimates of the loading and thresholds for Item 2 (equal) . .. 22. 4. The estimates of the loading and thresholds for Item 3 (equal) . .. 23. 5. The estimates of the loading and thresholds for Item 4 (equal) . .. 24. 6. The estimates of the loading and thresholds for Item 5 (equal) . .. 25. 7. The estimates of the loading and thresholds for Item 6 (equal) . .. 26. 8. The estimates of the loading and thresholds for Item 7 (equal) . .. 27. 9. The estimates of the loading and thresholds for Item 8 (equal) . .. 28. 10. The estimates of the loading and thresholds for Item 9 (equal) . .. 29. 11. The estimates of the loading and thresholds for Item 10 (equal) .. 30. 12. The estimates of the loading and thresholds for Item 11 (equal) .. 31. 13. The estimates of the loading and thresholds for Item 12 (equal) .. 32. 14. The estimates of the loading and thresholds for Item 13 (equal) .. 33. 15. The estimates of the loading and thresholds for Item 14 (equal) .. 34. 16. The estimates of the loading and thresholds for Item 15 (equal) .. 35. 17. The estimates of the loading and thresholds for Item 16 (equal) .. 36. 18. The estimates of the loading and thresholds for Item 17 (equal) .. 37. 19. The estimates of the loading and thresholds for Item 18 (equal) .. 38. 20. The estimates of the loading and thresholds for Item 1 (unequal). 41. 21. The estimates of the loading and thresholds for Item 2 (unequal). 42. 22. The estimates of the loading and thresholds for Item 3 (unequal). 43. 23. The estimates of the loading and thresholds for Item 4 (unequal). 44. 24. The estimates of the loading and thresholds for Item 5 (unequal). 45. 25. The estimates of the loading and thresholds for Item 6 (unequal). 46. 26. The estimates of the loading and thresholds for Item 7 (unequal). 47. vi.
(8) 27. The estimates of the loading and thresholds for Item 8 (unequal). 48. 28. The estimates of the loading and thresholds for Item 9 (unequal). 49. 29. The estimates of the loading and thresholds for Item 10 (unequal). 50. 30. The estimates of the loading and thresholds for Item 11 (unequal). 51. 31. The estimates of the loading and thresholds for Item 12 (unequal). 52. 32. The estimates of the loading and thresholds for Item 13 (unequal). 53. 33. The estimates of the loading and thresholds for Item 14 (unequal). 54. 34. The estimates of the loading and thresholds for Item 15 (unequal). 55. 35. The estimates of the loading and thresholds for Item 16 (unequal). 56. 36. The estimates of the loading and thresholds for Item 17 (unequal). 57. 37. The estimates of the loading and thresholds for Item 18 (unequal). 58. 38. The model fit indices of each model setting . . . . . . . . . . . . .. 63. 39. The item estimates for Model D.1 and Model D.2 . . . . . . . . .. 64. 40. The thresholds of the careless responses estimated from D.1 and D.2 65. 41. The response time distribution estimated from D.2 (logarithm scale) 65. vii.
(9) List of Figures 1. The factor mixture model used in the Meade and Craig’s study .. 2. An exact example of the bogus item in Hargittai’s study without. 2. misunderstanding. . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 3. (a) Meade and Craig’s model; (b) Chan, Lu and Tsai’s model . .. 6. 4. A mixture model incorporating response time and involving a bogus item.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 5. The added bogus item in the questionnaire. . . . . . . . . . . . .. 61. 6. The response time of logarithm scale distribution . . . . . . . . .. 65. viii.
(10) 1 Introduction In a research based on survey responses, careless or random responses are often concerned and become increasingly vital. It had been concluded that the careless responses may distort inference of the survey. The distorted inference results from the erroneous estimation, and the careless or random responses lead to the loss in estimation efficiency. For example, Wood (2006) examined five indices: Comparative fit index (CFI), Tucker–Lewis incremental fit index (TLI), root mean square error of approximation (RMSEA), standardized root mean-square residual (SRMR), weighted root mean square residual (WRMR) and found that, with random responses, the two-factor model was often mistakenly concluded to fit better than the true one-factor model. That is, the researchers might reject a single-factor confirmatory factor model because the model with about 10 percent of the respondents carelessly responding to reverse-worded items. Credé (2010) found that even low base rates of random responding can significantly affect the observed correlations. With a 5 percent of random response rate, the observed squared correlation can be as little as. 1 44. of the squared correlation that would be. observed without random responding. Hence, to avoid the improper inference, a solid and efficient model which can classify the survey behavior and simultaneously improve the accuracy of estimation is needed. Learning from a similiar situation of the achivement test is random guessing. The random guessing also distorts inference due to the inefficient estimation. Wise and DeMar (2006) used an effort-moderated model to enhance item parameters by classifying response behavior. When excluding the random-behavior examinees, the bias and root mean-square error (RMSE) of their model are both smaller than the standard 3-parameter logistic (3PL) model. Meade and Craig (2012) described that there were at least 7 kinds of indicators for the careless respondents in a questionnaire. Meade and Craig had applied a factor mixture. 1.
(11) model to identify careless responding. The model is displayed in Figure 1. Both the effort-moderated model and the mixture factor model rely on the covariate (the position of the careless response indicators of Figure 1) to classify the respondents’ latent classes. In the following, we will review a useful covariate employed in a survey: Bogus Item.. Figure 1: The factor mixture model used in the Meade and Craig’s study The bogus item design is useful and common in a survey. An advantage of an unambiguous bogus item is that if a careless person responds incorrectly, there is little doubt that he or she is responding carelessly or dishonestly (Meade and Craig, 2012). Beach (1988) was aware of the careless responses and illustrated the bogus item like “I was born on Feburary 30th ” to detect careless respondents. This example was used in the “Ture or False” survey. In fact, Beach had a good explanation to the bogus item: such a scale is made up of the questions that the subjects who read and understand the question can only answer in one way. Hargittai (2008) gave an example of such a bogus item, as illustrated in Figure 2. With certain response to the bogus items, the careless respondents can be easily identified. Other careless response indicators may be used as well, but it is. 2.
(12) sometimes harder to directly link certain response pattern to careless or random responding. For example, the researchers so far have no consensus in the criteria for long string and Mahalanobis D. Consistency like antonyms and synonyms may create other negative factors (Schmitt and Stults, 1985). Wood’s finding (2006) agreed with their results. Therefore, comparing with other indicators, a bogus item design can be more straightforward and easily implemented in a study.. Figure 2: An exact example of the bogus item in Hargittai’s study without misunderstanding.. The length of response time (RT) can be another method to identify careless respondents in the survey. For instance, RT was treated as a continuous covariate and used in a personality test to detect those dissimulting or dishonest. Their model (Holden, et al., 1992) predicted that dissimulators are relatively faster in answering consistent with the dissimulation and relatively slower in responding inconsistently with the dissimulation. Meade and Craig (2012) also considered RT as indicator of careless responding and excluded the repondents whose RT exceeded 15 minutes for their analysis. In the achivement test, Wise and DeMar (2006) used RT to distinguish solution behavior and random guessing. Their effort-moderated model refined the item parameter estimates by classifying response behavior. However, the thresholds of the RT would be subjective when it was treated as a covariate, and these models relied on the empirical RT plot. There were other ways to treat RT as a function of the latent trait in the. 3.
(13) achivement test. One of the models that combined responses and RT is based on speed-distance hypothesis(Ferrando and Lorenzo-Seva, 2007). The hypothesis constructed a measure for the distance between the personal trait and item difficulty, and linked this measure to the logarithmized RT by regression. The measure had described that the RT decreased as the distance of personal trait and item difficulty increased. However, The shorter RT is merely corresponded to the larger distance between the trait and difficulty. It is insufficient to confirm whether the classes are guessing. Another model proposed by Meyer(2010) was based on the assumption that the response and the RT are independent given the latent class. According to Meyer, the item estimates from the mixture Rasch model with response time component (MRM-RT) could be recovered by a Bayesian method. The analysis of extant data showed that a two-class model fit the test data better than the one-class model when 15 percent of examinees engaged in rapid-guessing behavior. Chan, Lu and Tsai (2014) employed the MRM-RT, adjusted it into a structure equation model (SEM) and enhanced the item parameters. According to their study, the mixture SEM was extended to 3 classes which defined random guessing, well-skilled and the ordinary. The mixture SEM used maximum liklihood estimation and it takes much less time than the Bayesian approach. Upon these reaserch, a model treating RT as a latent class indicator can be more precise than a model treating RT as a function of the latent trait. Another consideration for the relation of the RT to a survey is not corresponded to the trait-difficulty distance hypothesis. The distance represent the degree of agreement in a survey. Therefore, we propose a model with RT as a latent class indicator for a survey to avoid the defect of the adjustment in an achievement test. We will review Chan, Lu and Tsai’s model (2014), Meade and Craig’s model (2012) and resultantly propose a mixture response time model with a bogus item. 4.
(14) for the quetionnaire. This model conserves both advantage of the bogus item and RT. In Section 2, we introduce the likelihood of the proposed model and list the identification constaints. In Section 3, the simulation studies are stated and the results are reported and discussed. In Section 4, we apply the model to analyze real data; we will then conclude and discuss our study in Section 5.. 5.
(15) 2 Model 2.1 The Mixture Response Time Model According to Meade and Craig (2012), their model structure can be described in Figure 3-(a), where X denotes as the covariates previously shown in Figure 2; g represents the latent classes of the careless and the attentive respondents; θ is the latent trait of interest and U is the response. Meyer (2010) proposed a Bayesian model to enhance the item estimates and later Chan, Lu and Tsai (2014) constructed a structure equation model corresponding to Meyer’s. The path diagram is displayed in Figure 3-(b), where T is response time, logarithmtransformed normally distributed and treated as an indicator of the latent classes g; U is the response as before. Note that θ is interpreted as the ability in achievement testing.. (b). (a). Figure 3: (a) Meade and Craig’s model; (b) Chan, Lu and Tsai’s model. 6.
(16) In Figure 3-(a), the careless respondents were viewed partially careless; therefore, the dotted line represented ”partially” and revealed that some careless responding did not depend on θ. That is, the factor loadings vary across the latent classes. Figures 3-(a) and 3-(b) attempt to distinguish the careless/random responding from attentive/standard behavior solution. The more accurate the sizes of the latent classes, the more reliable estimates one can obtain. Based on the models above, we propose a mixture model incorporating response time and involving a bogus item; the model is depicted in Figure 4.. Figure 4: A mixture model incorporating response time and involving a bogus item.. In Figure 4, the ellipse including X and g represents the joint probability. The ellipse substitutes g only since those careless respondents are apt to respond incorrectly on X. That is, we believe those careless without reading in g might still be identified by X. Therefore we suggest using (X, g) instead of g. According to Meade and Craig (2012), X = (1, X1 , X2 , · · · , X7 ) representing the covariate vactor in Figure 2 was linked with the latent classes by a logistic. 7.
(17) regression: P (gi |Xi ) =. exp(Xi β) , 1 + exp(Xi β). (1). where gi is the ith person’s latent class and Xi is the vector of his or her covariates. β = (β0 , · · · , β7 ) are the intercept and the coefficient parameters. In Meyer’s justification (2010), the probability of personal responses and response time are independent when the latent class is given. However this justification should be adjusted since now we treat X and g simultaneously. Similarly, the response and response time are independent given (Xi , gi ), so that we can write P (Uij , Tij |Xi , gi ) = P (Uij |Xi , gi )P (Tij |Xi , gi ) ,. (2). where Uij and Tij are respectively the ith person’s response and response time for item j. P (Uij |Xi , gi ) and P (Tij |Xi , gi ) will be further discussed next. To relate a personal latent trait of interest and fitting a polytomous item response model, the graded response model (GRM) (Samejima, 1973) was extended and developed from the continuous response model. Forero and Maydeu-Olivares (2009) not only had a summary of the approximation between logistic GRM and normal ogive GRM, but also derived normal ogive GRM from a factor analytic framework. That is, either logistic or normal GRM can be derived from a latent trait model, and moreover the parameter space can be identified via each model constriants. Based on the latent trait model, suppose that the ith person’s latent response to item j is Uij∗ : Uij∗ = λ(X,g)j θi + εij ,. (3). where λ(X,g)j is the factor loading of item j. θi is the ith person’s latent trait of interest and εij follows N (0, σ 2 ). As a result, the observation of the ith person responding to item j is: Uij = k if τ(X,g)jk ≤ Uij∗ ≤ τ(X,g)j(k+1) ,. 8. (4).
(18) where τ(X,g)j = (τ(X,g)j0 , · · · , τ(X,g)j(k+1) ), specific to the class (X, g), is a vector consisting of threshold parameters for the item j. Note that k = 0, · · · , K − 1, where K is the number of categories of item j; τ(X,g)j0 = −∞ and τ(X,g)jK = ∞. Hence the response model of Uij can be expressed in the following: P (Uij = k|Xi , gi , θi ; λ(X,g)j , τ(X,g)j ) τ(X,g)jk − λ(X,g)j θi τ(X,g)j(k+1) − λ(X,g)j θi ) − Φ( ). = Φ( σ σ. (5). It has been shown that lognormal-distributed response time has a better fit when compared with other distributions (Linden & Krimpen-Stoop, 2003). As a result, Tij will be transformed to logarithm scale in this study. Since Meyer (2010) assigned a distribution to each latent class, the response time model will be similarly designed to adapt to (X, g). The adjusted response time model now is P (Tij = tij |Xi , gi ; ν(X,g) , ξ(X,g) ) = √. −(tij − ν(X,g) )2 1 exp( ), 2 2ξ(X,g) 2πξ(X,g). (6). 2 where tij is the realization of Tij ; (ν(X,g) , ξ(X,g) ) are the mean and the variance of. response time for class (X, g). Under conditional independence of all hte response and response time as well, given the class membership (X, g), a mixture model. 9.
(19) incorporating response time and involving bogus items is P (U, T|X; Θ) =. N ∏. P (Ui , Ti |Xi ; Θ). i=1. =. =. =. N ∑ G ∫ ∏. ∞. i=1 gi =1 −∞ N ∑ G ∫ ∏. ∞. i=1 gi =1 −∞ N ∑ G ∫ ∏. ∞. P (Ui , Ti |Xi , gi , θi ; Θ)P (gi , θi |Xi ; Θ)dθi. P (Ui , Ti |Xi , gi , θi ; Θ)P (θi |Xi , gi ; Θ)P (gi |Xi ; Θ)dθi [∏ J. i=1 gi =1 −∞ j=1. ]. P (Uij |Xi , gi , θi ; Θ)P (Tij |Xi , gi , θi ; Θ) P (θi |Xi , gi ; Θ)P (gi |Xi ; Θ)dθi , (7). where Θ = (σ, η, δ, λ, τ , ν, ξ, β) , exp(Xi β) , 1 + exp(Xi β) 1 −(θi − η)2 P (θi |Xi , gi ; Θ) = √ exp( ), 2δ 2 2πδ −(tij − ν(X,g) )2 1 exp( ), P (Tij = tij |Xi , gi , θi ; Θ) = √ 2 2ξ(X,g) 2πξ(X,g) P (gi |Xi ; Θ) =. P (Uij = k|Xi , gi , θi ; Θ) = Φ(. τ(X,g)j(k+1) − λ(X,g)j θi τ(X,g)jk − λ(X,g)j θi ) − Φ( ), σ σ. and i = 1, · · · , N ; j = 1, · · · , J; g = 1, · · · , G, where N , J and G are the total numbers of the respondents, items and latent classes, respectively.. 10.
(20) 2.2 A Restricted Condition Without loss of generality, we consider Xi as a two-dimension vector Xi = (1, Xi ) .. (8). Xi = 1 if person i has an incorrect response on the bogus item, and 0 if correct. Abbreviating the conditional probability P (gi |Xi , Θ), we replace it with P (gi |Xi ) as follows. Denote that the latent class gi = 1 if person i is a careless respondent and gi = 2 if he/she is an attentive respondent. The conditional probability can be tabulated as,. Xi = 0 Xi = 1 P (gi = 1|Xi ). π1|0. π1|1. P (gi = 2|Xi ). π2|0. π2|1. where 1 , 1 + exp(β0 ) exp(β0 ) , = P (gi = 2|Xi = 0) = 1 + exp(β0 ) 1 = P (gi = 1|Xi = 1) = , 1 + exp(β0 + β1 ) exp(β0 + β1 ) = P (gi = 2|Xi = 1) = . 1 + exp(β0 + β1 ). π1|0 = P (gi = 1|Xi = 0) = π2|0 π1|1 π2|1. (9) (10) (11) (12). Meade and Craig did not restrict π2|1 to be zero when estimating (β0 , β1 ). Hence P (Xi = 1, gi = 2) would not equal 0 in their study. Instead, we believe that it barely happens when an attentive person responds incorrectly, and such that a model constraint will be β1 → −∞ ⇒ π1|1 = 1, π2|1 = 0 .. 11. (13).
(21) To account for the careless responding, the response distributions of these classes (Xi = 0, gi = 1) and (Xi = 1, gi = 1) should be specified. Since these careless respondents do not read the items thoroughly, θi of the respondents do not react on P (Uij = k|Xi , gi , θi ; Θ). In fact, θi of a certain careless person may follow a distribution in his or her class. But regardless of the value of θi , the response distribution can be determined once given the class (Xi , gi ). That is, the careless responding will be simplified to P (Uij = k|Xi , gi = 1, θi ; Θ) = P (Uij = k|Xi , gi = 1; Θ).. (14). To specify such a condition, we can then simply the factor loadings of the careless classes (Xi = 0, gi = 1) and (Xi = 1, gi = 1) to zero, λ(X,gi =1)j = 0. ∀ X, j .. (15). Later in the result, we will estimate τ(X,g)jk under the careless responsees. However it is not allowed to set loadings equal to zeroes in Mplus Version 5. We instead replace it with λ(X,gi =1)j = 1 and θi |gi =1 = 0. Moreover, we restrict all the response to follow the same distribution within (Xi = 0, gi = 1) and (Xi = 1, gi = 1).. 2.3 Estimation and Identification From (7), the likelihood function can be rewritten as:. L(Θ; U, T) =. N ∑ G ∫ ∏. ∞. [∏ J. i=1 gi =1 −∞ j=1. ]. P (Uij |Xi , gi , θi ; Θ)P (Tij |Xi , gi , θi ; Θ) P (θi |Xi , gi ; Θ)P (gi |Xi ; Θ)dθi . (16). 12.
(22) The estimates are obtained using the maximum likelihood method. In Mplus, the numerical integration is used to approximate the integral of latent response (Muthén and Muthén, 1998-2010). As we discussed previously, identification becomes an issue in estimating the parameters. For example, taking a look at (5), it is obvious that the same probabilities can be obtained from different response patterns. For example, {θ = 0, τ(X,g)j = τ ∗ , σ = 1} and {θ = 0, τ(X,g)j = 2τ ∗ , σ = 2} can produce the same probability. Hence, Forero and Maydeu-Olivares (2009) set constraints for item estimation. To confirm a unique solution set of parameters, the distribution of Uij∗ is prerequisite and the variance of εij has to be specified. The identification constraints used in our model are E(θi |Θ) = 0 ,. (17). Var(θi |Θ) = 1 ,. (18). Var(εij ) = 1 .. (19). That is, θi ∼ N (η = 0, δ 2 = 1) and εij ∼ N (0, σ 2 = 1).. 13.
(23) 3 Simulation In this simulation, we employ the mixture model discussed in Section 2 and compare performances of various model fittings. In Section 3.1, we exhibit the data generation, eight settings of model constraints and the criteria used for model comparison. In Section 3.2, the simulation results are presented and we will discuss the results and findings in Section 3.3.. 3.1 Data Generation To simulate survey items, we apply Fraley, Waller and Brennan’s analysis (2000) of self-report measures. There were 18 items and the item parameter estimates were obtained from GRM. Their result is displayed in Table 1-(a). Note that aj and bj referred to the slope and intercepts, respectively, in IRT terminology; in SEM terminology, aj is the factor loading and bj are the thresholds (Forero and Maydeu-Olivares, 2009). Here we use normal ogive in the simulation. According to Koch (1983), we can interchange these two ogives by using aj , 1.7 = λ(X,g)j bj .. λ(X,g)j =. (20). τ(X,g)j. (21). We further collapse these thresholds down to four by eliminating two extreme options in order to avoid extremely large standard errors. Therefore, the factor loadings and thresholds we used in the data generation are presented in Table 1-(b).. 14.
(24) Table 1: The parameters transformed from logistic orgive to normal orgive (a) The item estimates cited from Fraley, Waller and Brennan’s analysis j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18. aj 2.79 2.33 2.21 2.10 1.98 1.93 1.87 1.74 1.50 1.49 1.36 1.36 1.35 1.35 1.34 1.32 1.24 1.24. -1.12 -1.38 -1.07 -1.64 -1.32 -1.71 -1.36 -1.85 -1.86 -0.72 -1.69 -1.38 -1.31 -0.90 -0.97 -1.52 -1.91 -0.45. -0.39 -0.52 -0.21 -0.76 -0.57 -0.73 -0.45 -0.89 -0.67 0.35 -0.45 -0.29 -0.18 0.11 0.10 -0.45 -0.71 0.83. bj 0.00 -0.12 0.25 -0.39 -0.29 -0.20 0.04 -0.40 -0.06 0.87 0.11 0.16 0.30 0.50 0.52 0.03 -0.29 1.40. 0.45 0.41 0.82 0.19 0.28 0.36 0.50 0.16 0.60 1.68 0.73 1.07 1.02 1.09 1.00 0.79 0.33 2.17. 1.08 1.15 1.53 0.93 0.86 1.00 1.32 0.90 1.29 2.43 1.42 1.96 1.73 1.91 1.65 1.74 1.18 2.86. 1.70 1.85 2.11 1.80 1.58 1.75 2.05 1.80 2.26 3.62 2.22 2.99 2.59 2.81 2.61 2.69 2.25 3.53. (b) The item estimates from logistic ogive transforemed to normal ogive j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18. λ(X,g)j 1.64 1.37 1.30 1.24 1.16 1.14 1.10 1.02 0.88 0.88 0.80 0.80 0.79 0.79 0.79 0.78 0.73 0.73. -0.640 -0.713 -0.273 -0.939 -0.664 -0.829 -0.495 -0.911 -0.591 0.307 -0.360 -0.232 -0.143 -0.087 0.079 -0.349 -0.518 0.605. 15. τ(X,g)j 0.000 0.739 -0.164 0.562 0.325 1.066 -0.482 0.235 -0.338 0.326 -0.227 0.409 0.044 0.550 -0.409 0.164 -0.053 0.529 0.763 1.472 0.088 0.584 0.128 0.856 0.238 0.810 0.397 0.866 0.410 0.788 0.023 0.613 -0.212 0.241 1.021 1.583. 1.772 1.576 1.989 1.149 1.002 1.135 1.452 0.921 1.138 2.130 1.136 1.568 1.374 1.517 1.300 1.350 0.861 2.086.
(25) 3.1.1 Class Size In Section 2, the respondents were classified according to (X, g). For better understanding, we denote the classes of (Xi = 0, gi = 1), (Xi = 1, gi = 1) and (Xi = 0, gi = 2) as “the careless 1”, “the careless 2” and “the attentive,” respectively. Most literature suggested that there were 10 percent of careless responding in a survey, so that we choose 10 percent in our simulation. Furthermore, according to Johnson (2005), there was about 3.5 percent of careless responding without reading the items in the population. Hence we set 3.5, 6.5 and 90 percent for “the careless 1”, “the careless 2” and “the attentive,” respectively.. 3.1.2 Response and Response Time According to Table 1-(b), the response of “the attentive” can be generated via (5). For the response time of “the attentive,” we randomly sampled ten students to answer these questions and recorded their response time. Then the obtained emperical time distribution is lognormal(1.5, 0.35). This time distribution will be used to generate reponse time of “the attentive.” The response distributions of “the careless 1” and “the careless 2” depend on their tendency. We considered two conditions, the “equal” and “unequal” cases. In the “equal” case, since the response patterns of the “careless 1” and “careless 2” were set unifromly in most literature, the probability distribution of responses in these two classes is set to be (0.2, 0.2, 0.2, 0.2, 0.2). In the “unequal” case, the “careless 1” and “careless 2” have their own response styles. The response distributions were determined from another empirical data. In the empirical data, the response distribution of the “careless 1” was (0.05, 0.15, 0.25, 0.45, 0.1), and for the “careless 2,” it was (0.05, 0.15, 0.2, 0.4, 0.2). The response time distribution of the “careless 1” is lognormal(−0.8, 0.3) (Chan, Lu & Tsai, 2014). The response time of the “care-. 16.
(26) less 2” may be longer than “careless 1” but shorter than “the attentive,” so we assigned lognormal(1.37, 0.4) to the “careless 2.”. 3.1.3 Factors and Model Fittings To evaluate the performance of the proposed model, we consider the following eight fitting models: • A.1 : Only bogus item as covariate, without response time. From (9) to (12), we restrict P (gi |Xi ; Θ) with β1 → −∞ and β0 → ∞. Therefore the model switches into the miltiple-group analysis. From (16), the likelihood is rewritten as L(Θ; U) =. N ∫ ∏. ∞ [J=18 ∏. i=1 −∞. ]. P (Uij |Xi , θi ; Θ) P (θi |Xi ; Θ)dθi .. j=1. • A.2 : Based on A.1, add the factor of response time. The likelihood is L(Θ; U) =. N ∫ ∏. ∞ [J=18 ∏. i=1 −∞. ]. P (Uij |Xi , θi ; Θ)P (Tij |Xi , θi ; Θ) P (θi |Xi ; Θ)dθi .. j=1. • B.1 :Without response time, the response distributions of “careless 1” and “careless 2” are fixed at (0.2, 0.2, 0.2, 0.2, 0.2) when fitting model. That is, for k = {0, 1, 2, 3, 4}, P (Uij = k|Xi = (1, 0), gi = 1, θi ; Θ) = P (Uij = k|Xi = (1, 1), gi = 1, θi ; Θ) = 0.2. The likelihood is rewritten as L(Θ; U) =. N G=2 ∏ ∑∫. ∞ [J=18 ∏. i=1 gi =1 −∞. ]. P (Uij |Xi , gi , θi ; Θ) P (θi |Xi , gi ; Θ)P (gi |Xi ; Θ)dθi .. j=1. • B.2 : Based on B.1, add the factor of response time. • C.1 : Without reponse time, we let the thresholds of “careless 1” and “careless 2” be equal when fitting the model, and we relax the probabilities of the responses from being uniform. That is, k = {0, 1, 2, 3, 4}, P (Uij = k|Xi = (1, 0), gi = 1, θi ; Θ) = P (Uij = k|Xi = (1, 1), gi = 1, θi ; Θ).. 17.
(27) • C.2 : Based on C.1, add the factor of response time. • D.1 : Without reponse time, we relax the restriction in Model C.1 of “careless 1” and “careless 2” when fitting the model. “Careless 1” and “careless 2” can have their response distributions. • D.2 : Based on D.1, add the factor of response time. ˆ In this simulation, these eight models will be compared using the criteria, SE(Θ), ˆ is the estimate of Θ. Note that Θ is the parameter. where Θ We also consider different sample sizes, N = 250, 500, and 1000. The replication is 200 times for each condition setting.. 18.
(28) 3.2 Results Since we generate data from two different response distributions: “equal case” and “unequal case,” we will present their results seperately. In Section 3.2.1, we display the results of the eight fitted models under the “equal case”. In Section 3.2.2, we show the results for the “unequal case” as described in 3.1.2. In Section 3.2.3, we will discuss the results of both cases and make the conclusions.. 3.2.1 Under The Equal Case Tables 2 to 19 present the item estimates of “the attentive.” In Table 3, when N = 250, the models considering the latent classes (B.1, B.2, C.1, C.2 and D. 1, D.2) generally have smaller bias than the models without latent classes (A.1, A.2). The loadings of A.1 and A.2 have larger bias than the other models, and the thresholds are the same. Models B.1 and B.2 are the true models with their thresholds fixed to yeild discrete uniform distribution of the response. Comparing B.1 with B.2, we find that there is no substantial difference between the model with response time and the model without reponse time if just eyeballing the estimates and their standard errors. This finding can also be seen in C.1 versus C.2 and D.1 versus D.2. However the bias and the standard errors are not consistent. Comparing C.1 with D.1 and C.2 with D.2, we can not conclude which of C.1, C.2, D.1 and D.2 is better. When N = 500, the bias of the estimates of A.1 and A.2 are larger than the small sample size N = 250. However the other models (B.1, B.2, C.1, C.2 and D.1, D.2) have smaller bias than those at N = 250. The standard errors are decreasing if coming with the models for N = 250, which is as expected. The bias of the models (B.1, B.2, C.1, C.2 and D.1, D.2) do not have consistent performance, but the standard errors of the models with response time (B.2, C.. 19.
(29) 2 and D.2) have become smaller than the models without response time (B.1, C. 1, and D.1). For example, λ1 of Item 1, the estimates of B.2, C.2 and D.2 are smaller than the estimates of the models B.1, C.1 and D.1. It seem that the bias of the models with response time estimates do not have much difference from those without time estimates. When N = 1000, these phenomenon become more obviously.. Summary: there are some conclusions in this section as following, • The models considering the latent classes have smaller bias than those without the latent classes. • The models with response time have smaller standard errors than those without response time. • As the sample size increases, we obtain consistent results that both the bias and the standard errors decrease.. 20.
(30) Table 2: The estimates of the loading and thresholds for Item 1 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ1 1.640 1.556(0.162) 1.556(0.162) 1.660(0.179) 1.660(0.179) 1.662(0.179) 1.661(0.178) 1.661(0.178) 1.661(0.178). τ1 -0.640 -0.635(0.143) -0.635(0.143) -0.640(0.160) -0.641(0.158) -0.638(0.158) -0.641(0.160) -0.642(0.157) -0.638(0.158). τ2 0.000 -0.004(0.147) -0.004(0.147) -0.001(0.163) -0.001(0.161) 0.001(0.164) -0.002(0.165) -0.002(0.163) 0.001(0.164). τ3 0.739 0.706(0.153) 0.706(0.153) 0.744(0.171) 0.743(0.170) 0.745(0.171) 0.742(0.170) 0.741(0.168) 0.745(0.171). τ4 1.772 1.676(0.175) 1.676(0.175) 1.780(0.209) 1.780(0.209) 1.779(0.202) 1.775(0.203) 1.775(0.202) 1.779(0.202). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.543(0.104) 1.543(0.104) 1.644(0.121) 1.646(0.117) 1.644(0.122) 1.646(0.117) 1.643(0.123) 1.645(0.117). -0.640(0.116) -0.640(0.116) -0.647(0.123) -0.642(0.120) -0.646(0.123) -0.643(0.120) -0.647(0.123) -0.644(0.120). -0.003(0.110) -0.003(0.110) 0.001(0.119) 0.006(0.117) 0.003(0.120) 0.005(0.118) 0.001(0.120) 0.004(0.118). 0.703(0.116) 0.703(0.116) 0.737(0.130) 0.741(0.129) 0.739(0.130) 0.741(0.130) 0.737(0.129) 0.739(0.128). 1.673(0.132) 1.673(0.132) 1.773(0.151) 1.778(0.151) 1.775(0.151) 1.778(0.152) 1.772(0.154) 1.776(0.151). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.538(0.086) 1.538(0.086) 1.638(0.094) 1.641(0.094) 1.637(0.095) 1.641(0.094) 1.637(0.095) 1.640(0.093). -0.638(0.072) -0.638(0.072) -0.641(0.077) -0.640(0.077) -0.641(0.079) -0.639(0.078) -0.641(0.079) -0.639(0.078). -0.006(0.072) -0.006(0.072) -0.001(0.083) 0.001(0.079) -0.002(0.081) -0.001(0.079) -0.002(0.081) 0.001(0.079). 0.710(0.080) 0.710(0.080) 0.742(0.089) 0.744(0.088) 0.741(0.088) 0.742(0.088) 0.740(0.087) 0.742(0.087). 1.685(0.096) 1.685(0.096) 1.776(0.106) 1.778(0.104) 1.775(0.107) 1.777(0.103) 1.774(0.106) 1.778(0.103). 21.
(31) Table 3: The estimates of the loading and thresholds for Item 2 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ2 1.370 1.283(0.142) 1.283(0.142) 1.362(0.158) 1.364(0.159) 1.364(0.157) 1.365(0.158) 1.364(0.160) 1.364(0.160). τ1 -0.713 -0.710(0.127) -0.710(0.127) -0.720(0.142) -0.721(0.138) -0.720(0.143) -0.721(0.138) -0.720(0.141) -0.720(0.141). τ2 -0.164 -0.155(0.133) -0.155(0.133) -0.166(0.147) -0.169(0.145) -0.166(0.148) -0.169(0.145) -0.167(0.148) -0.167(0.148). τ3 0.562 0.545(0.122) 0.545(0.122) 0.566(0.138) 0.565(0.137) 0.567(0.140) 0.565(0.137) 0.567(0.139) 0.567(0.139). τ4 1.576 1.496(0.153) 1.496(0.153) 1.572(0.175) 1.573(0.176) 1.574(0.176) 1.572(0.176) 1.573(0.180) 1.573(0.180). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.291(0.095) 1.291(0.095) 1.369(0.108) 1.371(0.108) 1.369(0.108) 1.371(0.108) 1.370(0.109) 1.371(0.108). -0.707(0.090) -0.707(0.090) -0.718(0.102) -0.715(0.102) -0.717(0.102) -0.716(0.103) -0.718(0.102) -0.717(0.102). -0.153(0.094) -0.153(0.094) -0.163(0.103) -0.160(0.101) -0.161(0.103) -0.160(0.102) -0.163(0.102) -0.162(0.101). 0.540(0.095) 0.540(0.095) 0.559(0.103) 0.562(0.102) 0.561(0.103) 0.562(0.103) 0.560(0.102) 0.560(0.102). 1.508(0.111) 1.508(0.111) 1.585(0.129) 1.587(0.126) 1.587(0.129) 1.587(0.127) 1.586(0.129) 1.585(0.126). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.290(0.069) 1.290(0.069) 1.368(0.077) 1.367(0.076) 1.368(0.076) 1.366(0.074) 1.368(0.075) 1.366(0.074). -0.711(0.063) -0.711(0.063) -0.715(0.071) -0.713(0.070) -0.716(0.071) -0.715(0.070) -0.715(0.070) -0.715(0.069). -0.161(0.057) -0.161(0.057) -0.166(0.064) -0.164(0.063) -0.166(0.064) -0.165(0.063) -0.165(0.063) -0.164(0.062). 0.533(0.063) 0.533(0.063) 0.566(0.070) 0.565(0.069) 0.565(0.070) 0.565(0.068) 0.564(0.069) 0.565(0.067). 1.507(0.074) 1.507(0.074) 1.579(0.081) 1.579(0.080) 1.577(0.083) 1.576(0.082) 1.577(0.084) 1.575(0.082). 22.
(32) Table 4: The estimates of the loading and thresholds for Item 3 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ3 1.300 1.263(0.144) 1.263(0.144) 1.324(0.159) 1.324(0.158) 1.325(0.159) 1.325(0.158) 1.326(0.158) 1.326(0.158). τ1 -0.273 -0.311(0.127) -0.311(0.127) -0.277(0.136) -0.277(0.136) -0.277(0.136) -0.277(0.135) -0.276(0.137) -0.276(0.137). τ2 0.325 0.289(0.133) 0.289(0.133) 0.323(0.146) 0.322(0.142) 0.324(0.148) 0.322(0.141) 0.324(0.143) 0.324(0.143). τ3 1.066 1.028(0.140) 1.028(0.140) 1.093(0.157) 1.092(0.154) 1.093(0.159) 1.091(0.153) 1.094(0.156) 1.094(0.156). τ4 1.989 1.903(0.189) 1.903(0.189) 2.021(0.216) 2.020(0.213) 2.021(0.217) 2.019(0.213) 2.022(0.215) 2.022(0.215). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.246(0.109) 1.246(0.109) 1.304(0.117) 1.307(0.116) 1.304(0.118) 1.307(0.116) 1.304(0.118) 1.307(0.116). -0.315(0.095) -0.315(0.095) -0.283(0.106) -0.281(0.105) -0.281(0.106) -0.281(0.105) -0.283(0.105) -0.282(0.105). 0.281(0.097) 0.281(0.097) 0.316(0.110) 0.319(0.107) 0.318(0.110) 0.319(0.108) 0.316(0.109) 0.318(0.108). 1.000(0.107) 1.000(0.107) 1.066(0.121) 1.071(0.118) 1.069(0.123) 1.071(0.120) 1.067(0.122) 1.070(0.119). 1.885(0.130) 1.885(0.130) 2.003(0.148) 2.008(0.145) 2.006(0.150) 2.008(0.147) 2.005(0.150) 2.007(0.146). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.248(0.073) 1.248(0.073) 1.300(0.075) 1.303(0.075) 1.301(0.076) 1.303(0.076) 1.302(0.077) 1.304(0.075). -0.317(0.066) -0.317(0.066) -0.273(0.071) -0.272(0.071) -0.274(0.070) -0.272(0.070) -0.273(0.071) -0.273(0.070). 0.288(0.065) 0.288(0.065) 0.325(0.070) 0.327(0.070) 0.326(0.070) 0.327(0.070) 0.326(0.070) 0.327(0.070). 0.999(0.072) 0.999(0.072) 1.066(0.082) 1.068(0.080) 1.066(0.081) 1.068(0.080) 1.065(0.082) 1.067(0.081). 1.872(0.096) 1.872(0.096) 1.989(0.109) 1.990(0.110) 1.990(0.112) 1.992(0.112) 1.990(0.111) 1.991(0.111). 23.
(33) Table 5: The estimates of the loading and thresholds for Item 4 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ4 1.240 1.177(0.135) 1.177(0.135) 1.263(0.151) 1.262(0.149) 1.265(0.151) 1.263(0.149) 1.262(0.150) 1.262(0.150). τ1 -0.939 -0.919(0.138) -0.919(0.138) -0.955(0.152) -0.954(0.150) -0.956(0.154) -0.955(0.149) -0.953(0.151) -0.953(0.151). τ2 -0.482 -0.447(0.127) -0.447(0.127) -0.490(0.145) -0.489(0.142) -0.491(0.145) -0.490(0.140) -0.488(0.142) -0.488(0.142). τ3 0.235 0.245(0.114) 0.245(0.114) 0.237(0.127) 0.236(0.126) 0.236(0.125) 0.235(0.124) 0.238(0.125) 0.238(0.125). τ4 1.149 1.131(0.128) 1.131(0.128) 1.168(0.143) 1.166(0.142) 1.167(0.144) 1.164(0.142) 1.166(0.143) 1.166(0.143). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.159(0.089) 1.159(0.089) 1.243(0.098) 1.244(0.097) 1.243(0.099) 1.244(0.097) 1.243(0.099) 1.244(0.097). -0.916(0.100) -0.916(0.100) -0.952(0.107) -0.947(0.105) -0.951(0.107) -0.947(0.106) -0.952(0.107) -0.948(0.105). -0.453(0.093) -0.453(0.093) -0.495(0.100) -0.491(0.098) -0.494(0.101) -0.491(0.098) -0.495(0.100) -0.492(0.098). 0.234(0.087) 0.234(0.087) 0.225(0.096) 0.227(0.093) 0.226(0.096) 0.227(0.094) 0.225(0.094) 0.226(0.093). 1.106(0.091) 1.106(0.091) 1.142(0.101) 1.145(0.100) 1.144(0.102) 1.144(0.100) 1.143(0.102) 1.144(0.100). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.144(0.069) 1.144(0.069) 1.238(0.076) 1.239(0.076) 1.238(0.076) 1.238(0.076) 1.239(0.075) 1.241(0.074). -0.911(0.067) -0.911(0.067) -0.941(0.075) -0.940(0.073) -0.941(0.074) -0.940(0.073) -0.941(0.074) -0.939(0.073). -0.451(0.063) -0.451(0.063) -0.485(0.069) -0.486(0.069) -0.487(0.068) -0.486(0.067) -0.485(0.068) -0.486(0.067). 0.244(0.064) 0.244(0.064) 0.235(0.068) 0.235(0.068) 0.236(0.068) 0.236(0.067) 0.236(0.068) 0.237(0.067). 1.110(0.069) 1.110(0.069) 1.141(0.073) 1.144(0.072) 1.141(0.073) 1.143(0.072) 1.141(0.072) 1.144(0.072). 24.
(34) Table 6: The estimates of the loading and thresholds for Item 5 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ5 1.160 1.123(0.128) 1.123(0.128) 1.187(0.142) 1.187(0.141) 1.188(0.143) 1.188(0.141) 1.188(0.140) 1.188(0.140). τ1 -0.664 -0.679(0.121) -0.679(0.121) -0.684(0.133) -0.684(0.135) -0.684(0.134) -0.686(0.133) -0.684(0.133) -0.684(0.133). τ2 -0.338 -0.321(0.117) -0.321(0.117) -0.350(0.134) -0.349(0.132) -0.350(0.135) -0.350(0.132) -0.348(0.131) -0.348(0.131). τ3 0.326 0.321(0.121) 0.321(0.121) 0.316(0.136) 0.318(0.136) 0.316(0.137) 0.317(0.135) 0.319(0.135) 0.319(0.135). τ4 1.002 1.004(0.129) 1.004(0.129) 1.019(0.144) 1.019(0.144) 1.018(0.145) 1.019(0.143) 1.021(0.143) 1.021(0.143). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.101(0.094) 1.123(0.128) 1.162(0.105) 1.163(0.103) 1.162(0.105) 1.163(0.103) 1.162(0.105) 1.163(0.103). -0.674(0.098) -0.679(0.121) -0.680(0.108) -0.676(0.107) -0.678(0.107) -0.676(0.107) -0.679(0.107) -0.677(0.107). -0.318(0.090) -0.321(0.117) -0.346(0.101) -0.344(0.099) -0.344(0.100) -0.344(0.099) -0.345(0.100) -0.344(0.099). 0.326(0.088) 0.321(0.121) 0.321(0.096) 0.322(0.094) 0.323(0.096) 0.322(0.095) 0.321(0.096) 0.321(0.094). 0.990(0.098) 1.004(0.129) 1.000(0.107) 1.003(0.105) 1.002(0.107) 1.003(0.106) 1.001(0.107) 1.003(0.105). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.107(0.070) 1.107(0.070) 1.159(0.075) 1.158(0.074) 1.160(0.075) 1.159(0.075) 1.160(0.073) 1.159(0.072). -0.665(0.065) -0.665(0.065) -0.668(0.067) -0.666(0.067) -0.667(0.066) -0.667(0.066) -0.666(0.066) -0.664(0.065). -0.319(0.065) -0.319(0.065) -0.340(0.070) -0.339(0.069) -0.340(0.069) -0.340(0.069) -0.339(0.068) -0.338(0.068). 0.324(0.058) 0.324(0.058) 0.326(0.064) 0.326(0.062) 0.325(0.063) 0.325(0.063) 0.326(0.060) 0.325(0.060). 0.995(0.066) 0.995(0.066) 1.003(0.074) 1.003(0.074) 1.000(0.074) 1.001(0.073) 1.001(0.074) 1.002(0.073). 25.
(35) Table 7: The estimates of the loading and thresholds for Item 6 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ6 1.140 1.086(0.115) 1.086(0.115) 1.143(0.121) 1.145(0.122) 1.144(0.120) 1.146(0.121) 1.145(0.121) 1.145(0.121). τ1 -0.829 -0.829(0.115) -0.829(0.115) -0.839(0.130) -0.842(0.129) -0.840(0.131) -0.842(0.129) -0.840(0.130) -0.840(0.130). τ2 -0.227 -0.220(0.110) -0.220(0.110) -0.228(0.121) -0.231(0.121) -0.228(0.122) -0.231(0.120) -0.229(0.120) -0.229(0.120). τ3 0.409 0.408(0.118) 0.408(0.118) 0.415(0.130) 0.412(0.129) 0.415(0.131) 0.412(0.128) 0.413(0.130) 0.413(0.130). τ4 1.135 1.125(0.138) 1.125(0.138) 1.148(0.148) 1.147(0.148) 1.147(0.149) 1.146(0.147) 1.147(0.148) 1.147(0.148). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.082(0.082) 1.082(0.082) 1.141(0.089) 1.143(0.088) 1.141(0.089) 1.143(0.088) 1.141(0.089) 1.143(0.088). -0.816(0.095) -0.816(0.095) -0.829(0.103) -0.826(0.103) -0.827(0.104) -0.826(0.103) -0.828(0.102) -0.827(0.102). -0.216(0.083) -0.216(0.083) -0.228(0.093) -0.227(0.092) -0.226(0.094) -0.227(0.092) -0.228(0.093) -0.228(0.092). 0.408(0.084) 0.408(0.084) 0.414(0.092) 0.417(0.091) 0.415(0.093) 0.417(0.091) 0.414(0.093) 0.416(0.090). 1.111(0.095) 1.111(0.095) 1.135(0.103) 1.139(0.101) 1.137(0.104) 1.139(0.102) 1.136(0.104) 1.139(0.101). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.079(0.064) 1.079(0.064) 1.139(0.067) 1.138(0.067) 1.139(0.068) 1.139(0.068) 1.140(0.067) 1.139(0.067). -0.820(0.060) -0.820(0.060) -0.827(0.067) -0.827(0.067) -0.830(0.066) -0.829(0.065) -0.830(0.066) -0.829(0.065). -0.219(0.057) -0.219(0.057) -0.227(0.062) -0.226(0.063) -0.227(0.062) -0.225(0.062) -0.227(0.062) -0.226(0.062). 0.408(0.059) 0.408(0.059) 0.414(0.063) 0.416(0.063) 0.414(0.062) 0.414(0.061) 0.413(0.062) 0.415(0.061). 1.110(0.063) 1.110(0.063) 1.136(0.073) 1.138(0.073) 1.136(0.073) 1.137(0.073) 1.136(0.073) 1.138(0.072). 26.
(36) Table 8: The estimates of the loading and thresholds for Item 7 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ7 1.100 1.071(0.114) 1.071(0.114) 1.111(0.121) 1.112(0.120) 1.112(0.121) 1.113(0.120) 1.113(0.120) 1.113(0.120). τ1 -0.495 -0.514(0.119) -0.514(0.119) -0.494(0.129) -0.494(0.128) -0.493(0.130) -0.494(0.127) -0.493(0.128) -0.493(0.128). τ2 0.044 0.035(0.115) 0.035(0.115) 0.051(0.123) 0.049(0.123) 0.051(0.124) 0.049(0.122) 0.050(0.123) 0.050(0.123). τ3 0.550 0.549(0.109) 0.549(0.109) 0.562(0.119) 0.561(0.119) 0.562(0.119) 0.561(0.118) 0.563(0.120) 0.563(0.120). τ4 1.452 1.422(0.139) 1.422(0.139) 1.468(0.157) 1.469(0.154) 1.468(0.157) 1.469(0.153) 1.470(0.154) 1.470(0.154). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.062(0.086) 1.062(0.086) 1.102(0.093) 1.103(0.093) 1.102(0.094) 1.104(0.093) 1.103(0.094) 1.103(0.092). -0.514(0.091) -0.514(0.091) -0.496(0.096) -0.493(0.095) -0.495(0.095) -0.493(0.095) -0.497(0.095) -0.494(0.094). 0.035(0.088) 0.035(0.088) 0.049(0.096) 0.052(0.096) 0.050(0.095) 0.052(0.096) 0.049(0.095) 0.051(0.095). 0.540(0.089) 0.540(0.089) 0.555(0.094) 0.557(0.094) 0.557(0.094) 0.557(0.095) 0.555(0.095) 0.557(0.094). 1.412(0.109) 1.412(0.109) 1.459(0.118) 1.462(0.115) 1.460(0.118) 1.462(0.115) 1.460(0.118) 1.461(0.114). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.058(0.060) 1.058(0.060) 1.098(0.062) 1.097(0.062) 1.099(0.064) 1.098(0.063) 1.099(0.064) 1.098(0.063). -0.519(0.061) -0.519(0.061) -0.496(0.063) -0.496(0.063) -0.497(0.063) -0.496(0.063) -0.495(0.063) -0.496(0.062). 0.029(0.057) 0.029(0.057) 0.044(0.059) 0.043(0.059) 0.044(0.060) 0.044(0.060) 0.044(0.059) 0.043(0.059). 0.538(0.059) 0.538(0.059) 0.548(0.064) 0.550(0.063) 0.549(0.063) 0.549(0.063) 0.548(0.063) 0.550(0.063). 1.406(0.063) 1.406(0.063) 1.453(0.072) 1.453(0.070) 1.452(0.072) 1.453(0.070) 1.452(0.072) 1.453(0.070). 27.
(37) Table 9: The estimates of the loading and thresholds for Item 8 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ8 1.020 0.975(0.114) 0.975(0.114) 1.037(0.126) 1.038(0.123) 1.039(0.126) 1.039(0.122) 1.038(0.122) 1.038(0.122). τ1 -0.911 -0.912(0.122) -0.912(0.122) -0.937(0.134) -0.938(0.132) -0.939(0.135) -0.939(0.131) -0.938(0.133) -0.938(0.133). τ2 -0.409 -0.389(0.121) -0.389(0.121) -0.415(0.134) -0.417(0.133) -0.417(0.136) -0.418(0.132) -0.416(0.134) -0.416(0.134). τ3 0.164 0.184(0.112) 0.184(0.112) 0.170(0.125) 0.170(0.126) 0.169(0.127) 0.169(0.125) 0.171(0.127) 0.171(0.127). τ4 0.921 0.920(0.122) 0.920(0.122) 0.930(0.133) 0.930(0.132) 0.929(0.133) 0.929(0.131) 0.931(0.133) 0.931(0.133). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.961(0.079) 0.961(0.079) 1.020(0.087) 1.019(0.086) 1.020(0.087) 1.020(0.086) 1.020(0.087) 1.019(0.086). -0.896(0.089) -0.896(0.089) -0.919(0.095) -0.914(0.094) -0.918(0.094) -0.914(0.094) -0.919(0.094) -0.914(0.093). -0.387(0.080) -0.387(0.080) -0.414(0.089) -0.411(0.087) -0.413(0.088) -0.411(0.088) -0.414(0.088) -0.412(0.087). 0.170(0.084) 0.170(0.084) 0.156(0.093) 0.159(0.092) 0.157(0.093) 0.159(0.092) 0.157(0.093) 0.158(0.092). 0.919(0.090) 0.919(0.090) 0.930(0.099) 0.932(0.097) 0.931(0.099) 0.931(0.098) 0.931(0.099) 0.931(0.098). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.959(0.052) 0.959(0.052) 1.018(0.058) 1.017(0.058) 1.018(0.057) 1.018(0.057) 1.018(0.058) 1.017(0.057). -0.896(0.054) -0.896(0.054) -0.912(0.062) -0.912(0.062) -0.912(0.062) -0.912(0.062) -0.913(0.062) -0.912(0.062). -0.387(0.057) -0.387(0.057) -0.410(0.064) -0.409(0.064) -0.410(0.065) -0.409(0.065) -0.410(0.065) -0.409(0.064). 0.177(0.056) 0.177(0.056) 0.165(0.060) 0.165(0.060) 0.164(0.061) 0.164(0.060) 0.164(0.061) 0.164(0.060). 0.913(0.055) 0.913(0.055) 0.923(0.066) 0.925(0.066) 0.923(0.066) 0.924(0.065) 0.923(0.066) 0.924(0.065). 28.
(38) Table 10: The estimates of the loading and thresholds for Item 9 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ9 0.880 0.858(0.102) 0.858(0.102) 0.890(0.110) 0.890(0.109) 0.891(0.111) 0.891(0.109) 0.891(0.109) 0.891(0.109). τ1 -0.591 -0.602(0.111) -0.602(0.111) -0.590(0.120) -0.590(0.120) -0.590(0.120) -0.591(0.120) -0.590(0.123) -0.590(0.123). τ2 -0.053 -0.049(0.105) -0.049(0.105) -0.041(0.112) -0.042(0.112) -0.041(0.112) -0.042(0.111) -0.041(0.113) -0.041(0.113). τ3 0.529 0.523(0.114) 0.523(0.114) 0.540(0.123) 0.538(0.122) 0.539(0.123) 0.538(0.121) 0.539(0.123) 0.539(0.123). τ4 1.138 1.127(0.115) 1.127(0.115) 1.149(0.125) 1.148(0.123) 1.149(0.126) 1.147(0.123) 1.148(0.123) 1.148(0.123). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.847(0.074) 0.847(0.074) 0.875(0.076) 0.876(0.076) 0.875(0.076) 0.876(0.076) 0.875(0.076) 0.876(0.076). -0.618(0.086) -0.618(0.086) -0.605(0.090) -0.602(0.089) -0.604(0.090) -0.603(0.089) -0.605(0.091) -0.603(0.090). -0.071(0.075) -0.071(0.075) -0.062(0.079) -0.060(0.078) -0.061(0.079) -0.061(0.079) -0.062(0.079) -0.061(0.078). 0.513(0.082) 0.513(0.082) 0.528(0.088) 0.530(0.088) 0.529(0.088) 0.529(0.088) 0.528(0.088) 0.529(0.087). 1.119(0.091) 1.119(0.091) 1.136(0.095) 1.139(0.094) 1.137(0.095) 1.139(0.094) 1.137(0.095) 1.138(0.093). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.849(0.053) 0.849(0.053) 0.878(0.059) 0.878(0.058) 0.879(0.059) 0.879(0.058) 0.878(0.059) 0.879(0.057). -0.604(0.055) -0.604(0.055) -0.591(0.060) -0.591(0.059) -0.591(0.060) -0.591(0.059) -0.591(0.060) -0.591(0.059). -0.062(0.052) -0.062(0.052) -0.054(0.056) -0.053(0.056) -0.054(0.055) -0.053(0.055) -0.054(0.055) -0.053(0.055). 0.513(0.056) 0.513(0.056) 0.529(0.061) 0.529(0.060) 0.528(0.060) 0.529(0.059) 0.529(0.061) 0.529(0.059). 1.119(0.062) 1.119(0.062) 1.137(0.067) 1.138(0.066) 1.137(0.066) 1.138(0.066) 1.136(0.068) 1.137(0.066). 29.
(39) Table 11: The estimates of the loading and thresholds for Item 10 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ10 0.880 0.867(0.109) 0.867(0.109) 0.904(0.120) 0.904(0.119) 0.905(0.120) 0.905(0.118) 0.904(0.118) 0.904(0.118). τ1 0.307 0.231(0.103) 0.231(0.103) 0.313(0.118) 0.311(0.115) 0.314(0.118) 0.312(0.115) 0.313(0.117) 0.313(0.117). τ2 0.763 0.687(0.110) 0.687(0.110) 0.772(0.130) 0.770(0.128) 0.773(0.130) 0.771(0.127) 0.771(0.128) 0.771(0.128). τ3 1.472 1.370(0.143) 1.370(0.143) 1.505(0.178) 1.500(0.174) 1.506(0.179) 1.501(0.175) 1.501(0.175) 1.501(0.175). τ4 2.130 2.007(0.178) 2.007(0.178) 2.178(0.218) 2.174(0.210) 2.179(0.219) 2.175(0.211) 2.173(0.212) 2.173(0.212). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.848(0.080) 0.848(0.080) 0.882(0.088) 0.883(0.087) 0.881(0.088) 0.883(0.087) 0.882(0.088) 0.882(0.087). 0.223(0.081) 0.223(0.081) 0.303(0.089) 0.304(0.089) 0.304(0.089) 0.304(0.089) 0.303(0.089) 0.303(0.088). 0.683(0.086) 0.683(0.086) 0.764(0.097) 0.766(0.096) 0.766(0.098) 0.766(0.096) 0.764(0.099) 0.764(0.096). 1.350(0.100) 1.350(0.100) 1.477(0.116) 1.480(0.116) 1.479(0.118) 1.480(0.117) 1.476(0.118) 1.478(0.117). 1.985(0.135) 1.985(0.135) 2.151(0.166) 2.154(0.162) 2.153(0.169) 2.154(0.164) 2.152(0.171) 2.153(0.164). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.848(0.063) 0.848(0.063) 0.879(0.067) 0.880(0.067) 0.878(0.067) 0.880(0.066) 0.879(0.067) 0.880(0.066). 0.226(0.055) 0.226(0.055) 0.307(0.057) 0.308(0.057) 0.307(0.057) 0.308(0.056) 0.307(0.057) 0.309(0.056). 0.683(0.054) 0.683(0.054) 0.763(0.062) 0.764(0.060) 0.764(0.060) 0.764(0.060) 0.763(0.060) 0.764(0.060). 1.349(0.073) 1.349(0.073) 1.472(0.086) 1.472(0.085) 1.472(0.085) 1.473(0.084) 1.473(0.086) 1.474(0.084). 1.990(0.090) 1.990(0.090) 2.136(0.111) 2.138(0.109) 2.136(0.111) 2.137(0.110) 2.137(0.110) 2.139(0.110). 30.
(40) Table 12: The estimates of the loading and thresholds for Item 11 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ11 0.800 0.788(0.103) 0.788(0.103) 0.806(0.106) 0.806(0.105) 0.807(0.106) 0.806(0.105) 0.806(0.105) 0.806(0.105). τ1 -0.360 -0.396(0.100) -0.396(0.100) -0.365(0.107) -0.364(0.107) -0.365(0.106) -0.364(0.107) -0.363(0.108) -0.363(0.108). τ2 0.088 0.067(0.106) 0.067(0.106) 0.089(0.114) 0.089(0.114) 0.088(0.114) 0.089(0.113) 0.090(0.115) 0.090(0.115). τ3 0.584 0.573(0.105) 0.573(0.105) 0.593(0.118) 0.593(0.117) 0.593(0.118) 0.593(0.117) 0.595(0.118) 0.595(0.118). τ4 1.136 1.134(0.126) 1.134(0.126) 1.149(0.137) 1.149(0.135) 1.149(0.137) 1.149(0.134) 1.150(0.134) 1.150(0.134). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.775(0.071) 0.775(0.071) 0.792(0.074) 0.793(0.074) 0.791(0.074) 0.793(0.074) 0.792(0.074) 0.793(0.074). -0.396(0.068) -0.396(0.068) -0.366(0.073) -0.365(0.072) -0.366(0.073) -0.365(0.073) -0.366(0.073) -0.365(0.073). 0.064(0.070) 0.064(0.070) 0.083(0.074) 0.085(0.073) 0.085(0.074) 0.085(0.074) 0.083(0.074) 0.084(0.074). 0.558(0.073) 0.558(0.073) 0.577(0.078) 0.579(0.078) 0.578(0.078) 0.579(0.078) 0.577(0.078) 0.578(0.078). 1.114(0.085) 1.114(0.085) 1.131(0.090) 1.132(0.090) 1.132(0.090) 1.132(0.091) 1.131(0.090) 1.131(0.090). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.774(0.055) 0.774(0.055) 0.796(0.055) 0.797(0.054) 0.798(0.055) 0.798(0.053) 0.797(0.055) 0.797(0.054). -0.396(0.044) -0.396(0.044) -0.362(0.051) -0.361(0.050) -0.362(0.051) -0.361(0.050) -0.362(0.052) -0.360(0.050). 0.068(0.053) 0.068(0.053) 0.088(0.056) 0.087(0.055) 0.088(0.057) 0.087(0.056) 0.087(0.057) 0.086(0.056). 0.563(0.052) 0.563(0.052) 0.581(0.058) 0.584(0.057) 0.582(0.057) 0.584(0.056) 0.582(0.057) 0.584(0.056). 1.121(0.059) 1.121(0.059) 1.135(0.063) 1.136(0.062) 1.135(0.063) 1.136(0.063) 1.135(0.062) 1.137(0.062). 31.
(41) Table 13: The estimates of the loading and thresholds for Item 12 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ12 0.800 0.792(0.101) 0.792(0.101) 0.813(0.105) 0.814(0.103) 0.814(0.104) 0.815(0.103) 0.814(0.102) 0.814(0.102). τ1 -0.232 -0.284(0.101) -0.284(0.101) -0.247(0.108) -0.248(0.105) -0.246(0.107) -0.248(0.105) -0.247(0.106) -0.247(0.106). τ2 0.128 0.102(0.105) 0.102(0.105) 0.124(0.113) 0.124(0.110) 0.124(0.113) 0.124(0.110) 0.126(0.112) 0.126(0.112). τ3 0.856 0.821(0.121) 0.821(0.121) 0.866(0.131) 0.868(0.128) 0.866(0.132) 0.868(0.129) 0.869(0.131) 0.869(0.131). τ4 1.568 1.518(0.142) 1.518(0.142) 1.577(0.154) 1.579(0.155) 1.577(0.154) 1.580(0.156) 1.581(0.159) 1.581(0.159). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.775(0.064) 0.775(0.064) 0.795(0.067) 0.796(0.067) 0.794(0.067) 0.796(0.067) 0.795(0.066) 0.795(0.066). -0.267(0.071) -0.267(0.071) -0.230(0.077) -0.228(0.076) -0.229(0.077) -0.228(0.076) -0.230(0.078) -0.229(0.076). 0.114(0.074) 0.114(0.074) 0.134(0.080) 0.135(0.079) 0.135(0.080) 0.135(0.079) 0.134(0.080) 0.134(0.079). 0.817(0.076) 0.817(0.076) 0.857(0.082) 0.858(0.081) 0.858(0.083) 0.858(0.081) 0.857(0.082) 0.857(0.081). 1.525(0.096) 1.525(0.096) 1.585(0.106) 1.585(0.105) 1.586(0.107) 1.584(0.105) 1.585(0.107) 1.584(0.105). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.774(0.052) 0.774(0.052) 0.799(0.055) 0.798(0.055) 0.798(0.056) 0.798(0.055) 0.799(0.055) 0.798(0.054). -0.276(0.051) -0.276(0.051) -0.232(0.052) -0.234(0.051) -0.236(0.052) -0.234(0.051) -0.236(0.053) -0.235(0.052). 0.104(0.047) 0.104(0.047) 0.127(0.050) 0.128(0.050) 0.128(0.051) 0.128(0.049) 0.127(0.050) 0.128(0.049). 0.804(0.052) 0.804(0.052) 0.853(0.055) 0.854(0.055) 0.854(0.054) 0.854(0.054) 0.855(0.055) 0.854(0.053). 1.506(0.071) 1.506(0.071) 1.567(0.078) 1.568(0.077) 1.568(0.077) 1.568(0.076) 1.567(0.077) 1.567(0.076). 32.
(42) Table 14: The estimates of the loading and thresholds for Item 13 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ13 0.790 0.785(0.095) 0.785(0.095) 0.800(0.099) 0.800(0.099) 0.801(0.099) 0.800(0.099) 0.800(0.100) 0.800(0.100). τ1 -0.143 -0.190(0.099) -0.190(0.099) -0.147(0.108) -0.147(0.107) -0.147(0.109) -0.147(0.106) -0.146(0.107) -0.146(0.107). τ2 0.238 0.208(0.092) 0.208(0.092) 0.238(0.101) 0.238(0.102) 0.238(0.103) 0.238(0.101) 0.238(0.101) 0.238(0.101). τ3 0.810 0.774(0.103) 0.774(0.103) 0.812(0.114) 0.811(0.115) 0.812(0.116) 0.811(0.116) 0.812(0.116) 0.812(0.116). τ4 1.374 1.357(0.120) 1.357(0.120) 1.395(0.133) 1.393(0.131) 1.394(0.135) 1.392(0.132) 1.393(0.132) 1.393(0.132). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.775(0.072) 0.775(0.072) 0.788(0.075) 0.789(0.075) 0.787(0.074) 0.789(0.074) 0.788(0.075) 0.788(0.074). -0.187(0.072) -0.187(0.072) -0.143(0.079) -0.141(0.077) -0.142(0.079) -0.141(0.078) -0.142(0.079) -0.141(0.077). 0.210(0.071) 0.210(0.071) 0.240(0.077) 0.242(0.076) 0.241(0.077) 0.242(0.076) 0.240(0.077) 0.242(0.076). 0.774(0.081) 0.774(0.081) 0.811(0.089) 0.813(0.087) 0.812(0.089) 0.813(0.087) 0.811(0.090) 0.812(0.087). 1.343(0.092) 1.343(0.092) 1.377(0.098) 1.378(0.098) 1.378(0.099) 1.378(0.097) 1.378(0.099) 1.377(0.098). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.776(0.050) 0.776(0.050) 0.790(0.054) 0.791(0.054) 0.790(0.054) 0.791(0.054) 0.791(0.054) 0.792(0.053). -0.192(0.051) -0.192(0.051) -0.143(0.055) -0.144(0.054) -0.146(0.056) -0.144(0.054) -0.145(0.054) -0.144(0.052). 0.206(0.052) 0.206(0.052) 0.237(0.055) 0.238(0.054) 0.236(0.055) 0.238(0.053) 0.236(0.055) 0.238(0.053). 0.770(0.052) 0.770(0.052) 0.809(0.059) 0.810(0.058) 0.808(0.059) 0.811(0.057) 0.808(0.058) 0.810(0.057). 1.338(0.060) 1.338(0.060) 1.374(0.067) 1.374(0.066) 1.372(0.067) 1.374(0.066) 1.371(0.066) 1.374(0.065). 33.
(43) Table 15: The estimates of the loading and thresholds for Item 14 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ14 0.790 0.780(0.106) 0.780(0.106) 0.792(0.110) 0.792(0.110) 0.793(0.111) 0.792(0.110) 0.792(0.110) 0.792(0.110). τ1 -0.087 0.032(0.097) 0.032(0.097) 0.092(0.104) 0.092(0.103) 0.093(0.104) 0.093(0.103) 0.094(0.105) 0.094(0.105). τ2 0.397 0.362(0.103) 0.362(0.103) 0.404(0.112) 0.403(0.110) 0.405(0.113) 0.403(0.110) 0.405(0.111) 0.405(0.111). τ3 0.866 0.829(0.121) 0.829(0.121) 0.870(0.133) 0.869(0.131) 0.871(0.134) 0.869(0.131) 0.870(0.132) 0.870(0.132). τ4 1.517 1.472(0.136) 1.472(0.136) 1.524(0.161) 1.523(0.158) 1.524(0.162) 1.522(0.158) 1.523(0.159) 1.523(0.159). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.779(0.070) 0.779(0.070) 0.790(0.072) 0.791(0.072) 0.790(0.073) 0.791(0.072) 0.790(0.073) 0.791(0.072). 0.017(0.074) 0.017(0.074) 0.076(0.081) 0.078(0.079) 0.077(0.081) 0.078(0.079) 0.075(0.081) 0.077(0.079). 0.342(0.075) 0.342(0.075) 0.383(0.081) 0.385(0.078) 0.383(0.081) 0.385(0.079) 0.382(0.081) 0.384(0.078). 0.817(0.080) 0.817(0.080) 0.856(0.085) 0.858(0.083) 0.858(0.086) 0.858(0.083) 0.856(0.086) 0.857(0.083). 1.471(0.107) 1.471(0.107) 1.517(0.115) 1.517(0.109) 1.518(0.115) 1.516(0.110) 1.517(0.115) 1.516(0.109). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.775(0.051) 0.775(0.051) 0.789(0.054) 0.789(0.052) 0.789(0.053) 0.789(0.052) 0.788(0.054) 0.789(0.053). 0.027(0.053) 0.027(0.053) 0.088(0.056) 0.087(0.055) 0.088(0.056) 0.087(0.054) 0.087(0.054) 0.087(0.054). 0.356(0.055) 0.356(0.055) 0.397(0.058) 0.397(0.058) 0.397(0.058) 0.397(0.057) 0.397(0.058) 0.397(0.057). 0.830(0.057) 0.830(0.057) 0.870(0.060) 0.872(0.059) 0.870(0.060) 0.870(0.059) 0.871(0.059) 0.872(0.059). 1.478(0.066) 1.478(0.066) 1.520(0.073) 1.520(0.073) 1.519(0.073) 1.517(0.072) 1.519(0.073) 1.517(0.072). 34.
(44) Table 16: The estimates of the loading and thresholds for Item 15 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ15 0.790 0.794(0.117) 0.794(0.117) 0.804(0.121) 0.805(0.122) 0.804(0.121) 0.805(0.121) 0.804(0.122) 0.804(0.122). τ1 0.079 0.015(0.103) 0.015(0.103) 0.073(0.114) 0.072(0.110) 0.074(0.115) 0.073(0.110) 0.073(0.111) 0.073(0.111). τ2 0.410 0.367(0.104) 0.367(0.104) 0.410(0.114) 0.409(0.112) 0.411(0.114) 0.410(0.112) 0.410(0.113) 0.410(0.113). τ3 0.788 0.759(0.116) 0.759(0.116) 0.790(0.125) 0.789(0.123) 0.790(0.124) 0.789(0.122) 0.790(0.123) 0.790(0.123). τ4 1.300 1.293(0.126) 1.293(0.126) 1.316(0.135) 1.315(0.135) 1.316(0.135) 1.314(0.134) 1.315(0.135) 1.315(0.135). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.783(0.076) 0.783(0.076) 0.790(0.079) 0.792(0.078) 0.790(0.079) 0.792(0.078) 0.790(0.079) 0.791(0.078). 0.018(0.078) 0.018(0.078) 0.076(0.085) 0.078(0.083) 0.077(0.085) 0.077(0.083) 0.076(0.086) 0.077(0.083). 0.369(0.078) 0.369(0.078) 0.412(0.086) 0.413(0.085) 0.413(0.086) 0.413(0.085) 0.412(0.086) 0.413(0.085). 0.754(0.078) 0.754(0.078) 0.784(0.086) 0.786(0.084) 0.785(0.085) 0.786(0.084) 0.784(0.085) 0.785(0.084). 1.278(0.091) 1.278(0.091) 1.300(0.098) 1.302(0.099) 1.301(0.098) 1.302(0.098) 1.300(0.098) 1.301(0.098). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.783(0.050) 0.783(0.050) 0.788(0.051) 0.790(0.050) 0.787(0.051) 0.790(0.050) 0.788(0.051) 0.790(0.051). 0.016(0.050) 0.016(0.050) 0.077(0.054) 0.077(0.054) 0.078(0.052) 0.077(0.052) 0.078(0.053) 0.078(0.052). 0.363(0.050) 0.363(0.050) 0.406(0.054) 0.406(0.053) 0.408(0.054) 0.407(0.054) 0.408(0.054) 0.407(0.053). 0.757(0.053) 0.757(0.053) 0.788(0.059) 0.789(0.059) 0.788(0.059) 0.789(0.058) 0.788(0.059) 0.788(0.058). 1.280(0.063) 1.280(0.063) 1.301(0.067) 1.301(0.066) 1.302(0.068) 1.301(0.066) 1.301(0.066) 1.302(0.064). 35.
(45) Table 17: The estimates of the loading and thresholds for Item 16 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ16 0.780 0.762(0.108) 0.762(0.108) 0.782(0.111) 0.783(0.112) 0.782(0.111) 0.783(0.112) 0.783(0.112) 0.783(0.112). τ1 -0.349 -0.381(0.105) -0.381(0.105) -0.352(0.114) -0.353(0.112) -0.351(0.114) -0.353(0.112) -0.353(0.113) -0.353(0.113). τ2 0.023 0.009(0.108) 0.009(0.108) 0.024(0.113) 0.022(0.112) 0.024(0.113) 0.022(0.111) 0.023(0.112) 0.023(0.112). τ3 0.613 0.598(0.103) 0.598(0.103) 0.622(0.108) 0.620(0.108) 0.621(0.107) 0.620(0.108) 0.621(0.109) 0.621(0.109). τ4 1.350 1.331(0.125) 1.331(0.125) 1.372(0.140) 1.369(0.138) 1.371(0.139) 1.369(0.138) 1.369(0.139) 1.369(0.139). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.753(0.074) 0.753(0.074) 0.772(0.076) 0.773(0.076) 0.772(0.076) 0.773(0.075) 0.772(0.075) 0.773(0.075). -0.390(0.078) -0.390(0.078) -0.361(0.084) -0.358(0.083) -0.360(0.084) -0.359(0.083) -0.361(0.084) -0.359(0.082). 0.003(0.077) 0.003(0.077) 0.015(0.084) 0.017(0.082) 0.016(0.084) 0.017(0.083) 0.015(0.083) 0.016(0.081). 0.593(0.089) 0.593(0.089) 0.613(0.096) 0.615(0.094) 0.614(0.096) 0.615(0.094) 0.613(0.095) 0.614(0.093). 1.311(0.097) 1.311(0.097) 1.345(0.109) 1.347(0.106) 1.346(0.110) 1.347(0.106) 1.345(0.109) 1.346(0.106). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.766(0.049) 0.766(0.049) 0.780(0.053) 0.782(0.053) 0.780(0.052) 0.782(0.052) 0.781(0.053) 0.781(0.052). -0.383(0.052) -0.383(0.052) -0.350(0.055) -0.349(0.054) -0.348(0.054) -0.349(0.055) -0.350(0.054) -0.350(0.053). 0.005(0.054) 0.005(0.054) 0.024(0.056) 0.021(0.056) 0.022(0.055) 0.021(0.053) 0.022(0.055) 0.020(0.054). 0.585(0.049) 0.585(0.049) 0.610(0.053) 0.609(0.053) 0.611(0.053) 0.611(0.052) 0.610(0.053) 0.611(0.052). 1.318(0.060) 1.318(0.060) 1.351(0.067) 1.351(0.066) 1.353(0.066) 1.352(0.065) 1.353(0.066) 1.351(0.064). 36.
(46) Table 18: The estimates of the loading and thresholds for Item 17 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ17 0.730 0.709(0.098) 0.709(0.098) 0.733(0.103) 0.734(0.103) 0.733(0.103) 0.734(0.103) 0.733(0.103) 0.733(0.103). τ1 -0.518 -0.540(0.112) -0.540(0.112) -0.523(0.121) -0.524(0.119) -0.523(0.121) -0.524(0.119) -0.523(0.119) -0.523(0.119). τ2 -0.212 -0.207(0.103) -0.207(0.103) -0.210(0.110) -0.211(0.110) -0.211(0.111) -0.212(0.110) -0.211(0.110) -0.211(0.110). τ3 0.241 0.248(0.103) 0.248(0.103) 0.241(0.110) 0.241(0.109) 0.241(0.111) 0.240(0.110) 0.241(0.111) 0.241(0.111). τ4 0.861 0.867(0.113) 0.867(0.113) 0.866(0.119) 0.865(0.117) 0.865(0.118) 0.864(0.116) 0.865(0.117) 0.865(0.117). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.713(0.068) 0.713(0.068) 0.736(0.072) 0.737(0.073) 0.736(0.072) 0.737(0.073) 0.736(0.072) 0.736(0.072). -0.540(0.073) -0.540(0.073) -0.523(0.079) -0.521(0.078) -0.522(0.078) -0.521(0.078) -0.523(0.078) -0.521(0.078). -0.212(0.073) -0.212(0.073) -0.215(0.078) -0.213(0.077) -0.214(0.079) -0.213(0.077) -0.214(0.078) -0.214(0.077). 0.244(0.073) 0.244(0.073) 0.237(0.079) 0.239(0.078) 0.238(0.080) 0.239(0.078) 0.238(0.080) 0.239(0.078). 0.866(0.080) 0.866(0.080) 0.864(0.087) 0.867(0.087) 0.865(0.087) 0.866(0.087) 0.865(0.087) 0.866(0.087). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.710(0.042) 0.710(0.042) 0.729(0.047) 0.730(0.048) 0.728(0.048) 0.730(0.047) 0.730(0.048) 0.730(0.047). -0.539(0.044) -0.539(0.044) -0.518(0.052) -0.519(0.053) -0.518(0.052) -0.519(0.053) -0.519(0.052) -0.518(0.051). -0.213(0.046) -0.213(0.046) -0.213(0.050) -0.213(0.050) -0.212(0.049) -0.212(0.048) -0.213(0.050) -0.212(0.049). 0.239(0.042) 0.239(0.042) 0.238(0.048) 0.238(0.048) 0.237(0.048) 0.237(0.048) 0.238(0.048) 0.237(0.047). 0.862(0.050) 0.862(0.050) 0.861(0.055) 0.860(0.056) 0.858(0.055) 0.858(0.056) 0.860(0.055) 0.860(0.055). 37.
(47) Table 19: The estimates of the loading and thresholds for Item 18 (equal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ18 0.730 0.707(0.105) 0.707(0.105) 0.735(0.114) 0.735(0.114) 0.737(0.114) 0.736(0.113) 0.736(0.114) 0.736(0.114). τ1 0.605 0.498(0.108) 0.498(0.108) 0.608(0.131) 0.607(0.123) 0.611(0.132) 0.608(0.124) 0.609(0.126) 0.609(0.126). τ2 1.021 0.918(0.119) 0.918(0.119) 1.036(0.145) 1.034(0.137) 1.041(0.148) 1.036(0.137) 1.037(0.139) 1.037(0.139). τ3 1.583 1.457(0.142) 1.457(0.142) 1.615(0.178) 1.611(0.163) 1.620(0.185) 1.613(0.165) 1.615(0.171) 1.615(0.171). τ4 2.086 1.956(0.170) 1.956(0.170) 2.138(0.230) 2.130(0.221) 2.142(0.234) 2.131(0.221) 2.130(0.223) 2.130(0.223). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.696(0.074) 0.696(0.074) 0.721(0.081) 0.722(0.082) 0.721(0.082) 0.722(0.082) 0.722(0.082) 0.722(0.082). 0.495(0.073) 0.495(0.073) 0.601(0.083) 0.603(0.080) 0.602(0.083) 0.603(0.080) 0.601(0.082) 0.602(0.080). 0.900(0.076) 0.900(0.076) 1.013(0.088) 1.016(0.088) 1.014(0.089) 1.015(0.088) 1.014(0.089) 1.015(0.088). 1.431(0.093) 1.431(0.093) 1.581(0.109) 1.583(0.108) 1.583(0.112) 1.583(0.109) 1.582(0.112) 1.582(0.109). 1.922(0.110) 1.922(0.110) 2.084(0.141) 2.088(0.139) 2.085(0.143) 2.088(0.139) 2.085(0.143) 2.087(0.140). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 0.709(0.050) 0.709(0.050) 0.731(0.057) 0.730(0.057) 0.731(0.057) 0.730(0.057) 0.730(0.057) 0.730(0.056). 0.495(0.051) 0.495(0.051) 0.604(0.054) 0.605(0.054) 0.605(0.054) 0.605(0.053) 0.605(0.054) 0.605(0.052). 0.920(0.053) 0.920(0.053) 1.022(0.060) 1.023(0.058) 1.022(0.059) 1.023(0.058) 1.023(0.060) 1.023(0.058). 1.437(0.060) 1.437(0.060) 1.586(0.069) 1.586(0.070) 1.586(0.070) 1.587(0.070) 1.586(0.071) 1.587(0.069). 1.936(0.082) 1.936(0.082) 2.089(0.099) 2.090(0.097) 2.091(0.099) 2.091(0.096) 2.091(0.097) 2.090(0.095). 38.
(48) 3.2.2 Under The Unequal Case Under the “unequal case,” the results are presented form Tables 20 to 37. In Table 20, when N = 250, we have the same conclusion about the models considering the latent classes as described in 3.2.1. The models with latent classes (B.1, B. 2, C.1, C.2 and D.1, D.2) have smaller bias than the models without latentent classes (A.1 and A.2). An unrestricted model is better than a restricted model. For instance, the bias of the estimates of B.2 are smaller than B.1. Both of B.1 and B.2 response distribution are not exact, but the cooperating with response time model (B.2) can minimize the damage of improper response distribution model (B.1). Therefore, the models C.1, C.2 and D.1, D.2 have smaller bias of the estimates because their response distributions are relaxed. For example, the τ1 of Item 1, the bias of C.1 and C.2 are smaller than B.1 and B.2. It is similiar when we compare the models B.1 nad B.2 versus D.1 and D.2. However, the bias are not consistent as described in Section 3.2.1. The bias of the models (C. 1, C.2 and D.1, D.2) have larger bias than B.1 and B.2. But it becomes better while the sample size is large enough. The comparison of the models C.1 and C. 2 versus D.1 and D.2 are stated: the bias of D.1 and D.2 are larger than C.1 and C.2. Though D.1 and D.2 are the true models, the models C.1 and C.2 which constrain the carelss 1 and careless 2 to have the similiar response distribution with smaller bias than models D.1 and D.2. When N = 500, there are two same findings as Section 3.2.1. First, the bias and the standard errors become smaller than the size 250 as expected. Second, the effect of the response time to the standard errors has become obviously. The models with response time (B.2, C.2 and D.2) have smaller standard errors than the models without time (B.1, C.1 and D.1).. 39.
(49) Summary: there are some conclusions in this section as following, • The unrestricted models (B.2, C.1, C.2, and D.1, D.2) have smaller bias than the restricted model (B.1). • The models with response time have smaller standard errors than those without response time, but there are no consistency upon the bias.. 40.
(50) Table 20: The estimates of the loading and thresholds for Item 1 (unequal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ1 1.640 1.611(0.167) 1.611(0.167) 1.629(0.175) 1.649(0.178) 1.668(0.174) 1.673(0.175) 1.675(0.175) 1.675(0.174). τ1 -0.640 -0.724(0.154) -0.724(0.154) -0.726(0.158) -0.712(0.160) -0.659(0.159) -0.662(0.160) -0.651(0.159) -0.653(0.162). τ2 0.000 -0.074(0.159) -0.074(0.159) -0.075(0.162) -0.059(0.162) -0.006(0.162) -0.008(0.163) 0.002(0.164) 0.002(0.167). τ3 0.739 0.681(0.158) 0.681(0.158) 0.686(0.162) 0.701(0.164) 0.748(0.167) 0.744(0.169) 0.755(0.170) 0.753(0.173). τ4 1.772 1.736(0.196) 1.736(0.196) 1.752(0.204) 1.773(0.210) 1.794(0.213) 1.793(0.212) 1.807(0.218) 1.805(0.215). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.587(0.122) 1.587(0.122) 1.602(0.126) 1.623(0.128) 1.641(0.130) 1.646(0.130) 1.648(0.132) 1.649(0.130). -0.710(0.117) -0.710(0.117) -0.710(0.117) -0.692(0.117) -0.639(0.120) -0.638(0.119) -0.632(0.121) -0.634(0.120). -0.074(0.105) -0.074(0.105) -0.073(0.105) -0.054(0.107) 0.001(0.109) 0.002(0.109) 0.010(0.110) 0.006(0.109). 0.669(0.100) 0.669(0.100) 0.675(0.101) 0.696(0.103) 0.741(0.104) 0.740(0.105) 0.748(0.106) 0.744(0.105). 1.720(0.124) 1.720(0.124) 1.735(0.125) 1.763(0.131) 1.786(0.133) 1.788(0.133) 1.800(0.135) 1.795(0.134). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.584(0.086) 1.584(0.086) 1.593(0.085) 1.620(0.083) 1.642(0.088) 1.641(0.086) 1.643(0.088) 1.642(0.086). -0.710(0.074) -0.710(0.074) -0.711(0.076) -0.690(0.079) -0.639(0.080) -0.638(0.080) -0.637(0.080) -0.638(0.081). -0.070(0.072) -0.070(0.072) -0.064(0.074) -0.044(0.076) 0.002(0.077) 0.002(0.078) 0.001(0.079) 0.000(0.078). 0.673(0.081) 0.673(0.081) 0.677(0.085) 0.692(0.088) 0.743(0.091) 0.741(0.090) 0.744(0.091) 0.742(0.090). 1.720(0.099) 1.720(0.099) 1.730(0.102) 1.758(0.105) 1.779(0.108) 1.781(0.107) 1.785(0.108) 1.783(0.107). 41.
(51) Table 21: The estimates of the loading and thresholds for Item 2 (unequal) N Sample Size Model A.1 A.2 B.1 B.2 250 C.1 C.2 D.1 D.2. 500. 1000. λ2 1.370 1.325(0.138) 1.325(0.138) 1.338(0.141) 1.354(0.144) 1.367(0.146) 1.373(0.149) 1.373(0.149) 1.374(0.151). τ1 -0.713 -0.771(0.143) -0.771(0.143) -0.774(0.147) -0.762(0.147) -0.719(0.149) -0.722(0.151) -0.712(0.152) -0.714(0.152). τ2 -0.164 -0.229(0.132) -0.229(0.132) -0.232(0.136) -0.220(0.135) -0.177(0.138) -0.181(0.138) -0.171(0.139) -0.173(0.139). τ3 0.562 0.507(0.141) 0.507(0.141) 0.509(0.144) 0.519(0.145) 0.554(0.150) 0.550(0.149) 0.560(0.151) 0.557(0.150). τ4 1.576 1.535(0.168) 1.535(0.168) 1.546(0.171) 1.562(0.176) 1.576(0.178) 1.574(0.179) 1.586(0.180) 1.584(0.181). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.325(0.100) 1.325(0.100) 1.335(0.102) 1.351(0.104) 1.364(0.107) 1.368(0.108) 1.368(0.109) 1.370(0.108). -0.775(0.093) -0.775(0.093) -0.774(0.094) -0.759(0.095) -0.714(0.097) -0.713(0.098) -0.707(0.098) -0.710(0.098). -0.222(0.092) -0.222(0.092) -0.222(0.093) -0.208(0.094) -0.163(0.095) -0.162(0.095) -0.156(0.096) -0.159(0.095). 0.508(0.090) 0.508(0.090) 0.511(0.092) 0.527(0.094) 0.563(0.098) 0.563(0.097) 0.569(0.099) 0.565(0.097). 1.533(0.116) 1.533(0.116) 1.543(0.119) 1.562(0.121) 1.579(0.125) 1.579(0.125) 1.588(0.127) 1.584(0.123). A.1 A.2 B.1 B.2 C.1 C.2 D.1 D.2. 1.332(0.070) 1.332(0.070) 1.349(0.073) 1.361(0.076) 1.370(0.078) 1.372(0.076) 1.374(0.078) 1.373(0.077). -0.781(0.075) -0.781(0.075) -0.781(0.078) -0.766(0.078) -0.714(0.079) -0.714(0.078) -0.713(0.079) -0.712(0.078). -0.222(0.068) -0.222(0.068) -0.219(0.069) -0.182(0.069) -0.167(0.077) -0.166(0.076) -0.165(0.077) -0.166(0.076). 0.505(0.069) 0.505(0.069) 0.514(0.071) 0.526(0.070) 0.560(0.073) 0.560(0.072) 0.561(0.070) 0.562(0.070). 1.543(0.082) 1.543(0.082) 1.547(0.086) 1.570(0.087) 1.576(0.090) 1.578(0.090) 1.576(0.090) 1.575(0.091). 42.
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