背景
常規醫事人員專業認證筆試之目的在於評估考生是否具備執業能力之基礎，其測驗結果同時受到考生能力與題目難易程度所影響，因此，基於預判頻率理論，具備同時評估考生能力與題目難易程度之Rasch模式於筆試資料應用愈趨廣泛。 然而，具次序性之筆試資料並不適用Rasch模式，因此，於本論文將結合兩階段馬可夫鏈模式(two-state Markov chain)與Rasch模式，並以實證資料加以說明。

目的
本論文目的在於展示如何於Rasch模式中導入兩階段馬可夫鏈模式(two-state Markov chain)用以處理次序型筆試資料，並提出兩模式於個別考生能力與試題難易程度之相關應用理論。

方法
本論文以2007至2016年臺灣麻醉專科醫師筆試資料進行實證，兩階段馬可夫鏈模式將用於「對到錯」(Pcw)與「錯到對」(Pwc)兩階段之條件機率評估。其結果亦與Rasch模式所得結果進行比較，於兩階段馬可夫迴歸模式中拓展其隨機效應(random effect)，用以評估兩階段中共變數(如年齡、性別及受訓醫院所屬區域等)之影響。本研究將以貝氏馬可夫鏈蒙地卡羅方法(Bayesian Markov Cain Monte Carlo method)進行參數估計，其SAS程式碼(Proc MCMC)亦呈現於本論文中。

結果
本論文利用兩階段馬可夫鏈模式分析臺灣地區2007-2016十年間麻醉專科筆試資料，Pwc及Pcw兩個條件機率的估計結果分別為0.6957 (95% CI:0.6860-0.7044)及0.2326 (95% CI:0.2276-0.2374)，由此可得該考試考生長程答對機率為74.95% (95% CI:74.49%-75.42%)，接近Rasch模式得到的73.50%。考生能力指數估計結果為1.09 [=log (Pwc/Pcw)]，也與Rasch模式得到的1.02接近。此外，拓展兩階段馬可夫模式可進一步考慮考生特性與隨機效應，本資料顯示考生性別與年齡與考生能力有關：男性顯著低於女性、年齡長者分數較低。而混合效應模式證明考題難度的異質性(隨機效應在Pwc及Pcw的標準差分別為1.6781及 1.8416)高於考生的異質性(隨機效應在Pwc及Pcw的標準差分別為0.4399及 0.4875)。在未考慮隨機效益之下，二階馬可夫鏈(DIC=41935.14)的模式配適較一階馬可夫鏈(DIC=42872.54)佳，但在一階馬可夫鏈模式加入考生能力與考題難度的隨機效應後，其模式配適則優於二階馬可夫鏈模式。

結論
本論文利用兩階段馬可夫鏈模式評估麻醉專科筆試資料，並證明其分析結果與傳統Rasch模式分析方法之間的連結性。在未來數位化時代，兩階段馬可夫鏈模式可作為專業認證考試發展序列性適性考試之評估工具。

Background
Conventional written examination for accrediting health professionals is often based on pre-determined frequentist type with the Rasch model that models the corrected probability of answering the question as a function of individual ability and the difficulty of item. However, using the Rasch model may not be appropriate for sequential type written test. The two-state Markov chain (MC) method is therefore proposed. The link between it and the Rasch model is postulated and illustrated with one empirical data.
Aims
The specific aims of this thesis are to demonstrate how to apply two-state MC to modelling sequential type written test and to propose the theorem of linking two-state MC model with the Rasch model in terms of individual ability and the difficulty of item.
Methods
The empirical data used for illustration are derived from Taiwanese board certification examination in anaesthesiology from 2007 to 2016. The proposed two-state (wrong and correct) MC model was applied to estimating two parameters, i.e. two conditional probabilities (Pwc (wrongcorrect) and Pcw (correct wrong)). The results of two-state MC were compared with those based on the Rasch model. Two-state MC regression model with random effect was extended to assess the effect of relevant covariates (such as age, gender, and region) on two parameters of two-state MC model. Bayesian Markov Cain Monte Carlo (MCMC) method was used to estimate the parameters of interest for the Rasch model and two-state MC model. Computer algorithms with SAS Proc MCMC were also developed in this thesis.
Results
Based on the estimated 0.6957 (95% CI:0.6860-0.7044) of Pwc and 0.2326 (95% CI:0.2276-0.2374) of Pcw using the overall empirical data, the long-run corrected probability (74.95% (95% CI:74.49%-75.42%)) derived from the equilibrium distribution of two-state MC model was close to the estimated 73.50% based on the Rasch model. The overall estimated log (Pwc/Pcw) (1.09) was very close to the estimated individual ability (1.02 logit scale) based on the Rasch model. The application of two-state MC regression model to Taiwanese board certification examination in anaesthesiology from 2007 to 2016 based on sequential type test found the following relationships after adjustment for the relevant covariates with each other and also random effect of individual ability and item difficulty: age was inversely associated the corrected probability; females were more likely to have correct answer than males; the corrected probability varied by region. The heterogeneity of item difficulty (sigma=1.6781 and 1.8416 for Pwc and Pcw, respectively) was greater than that of individual ability (sigma=0.4399 and 0.4875 for Pwc and Pcw, respectively). Without considering covariates with random effects, the second-order Markov model (DIC=41935.14) had a better performance compared with the first-order Markov model (DIC=42872.54). However, the consideration of variation of individual ability and item difficulty with two random effect using the mixed Markov regression model outnumbered second-order Markov model.
Conclusions
This thesis demonstrates alternative use of two-state Markov chain model for the assessment of sequential type written test. The findings support the theorem of linking two-state MC model with the Rasch model. The proposed two-state MC model and its regression one may provide a new insight into the development of a prototype of adaptive sequential test that is very useful for certification of health professionals in era of digital age.

中文摘要 i
Abstract iii
Chapter 1 Introduction 1
1.1 The Rasch model for accrediting health professionals 1
1.2 Sequential type test 1
1.3 Link between the Rasch model and two-state MC 2
1.4 Empirical Data on written test for health professionals 3
1.5 Aims 3
Chapter 2 Literature review 4
Chapter 3 Two-state Markov Chain and the Item Response Theory model 10
3.1 The item response theory (IRT) model 10
3.2 Two-state Markov chain model 11
3.2.1 Basic model 11
3.2.2 Model incorporating random effects 12
3.2.3 Model incorporating covariates 13
3.2.4 Model with covariate and random effects 13
3.3 Theorem on the link between the Rasch model and two-state Markov chain model 14
Chapter 4 Empirical Data on Taiwanese board certification examinations in anesthesiology 17
Chapter 5 Computer algorithm for two-state MC and the Rasch Model with SAS program 19
5.1 The Rasch model 19
5.2 Two-state Markov Chain model 20
Chapter 6 Results 23
6.1 Descriptive results of score and transitions 23
6.2 Rasch model 24
6.3 Two-state Markov chain model 24
6.3.1 Basic model 24
6.3.2 Two-state Markov Chain Model with random effects 25
6.3.3 Univariate fixed effect Markov chain model 26
6.3.4 Univariate mixed effect Markov chain model 26
6.3.5 Multivariate fixed effect Markov chain model 27
6.3.6 Multivariate mixed effect Markov chain model 27
6.4 Second-order Two-state Markov chain model 28
Chapter 7 Discussion 29
7.1 Novelty and rationales for using two-stage Markov model 29
7.2 The comparison of findings between the Rasch model and two-stage Markov chain model 30
7.3 Methodological comparison between two-stage Markov chain and the Rasch model 31
7.4 Limitation 32
REFERENCES 34
Table 2.1 Literature review table of applications of Item Response Theory 37
Table 4.1 Basic characteristics of study subjects by year 41
Table 4.1 Basic characteristics of study subjects by year (continue). 42
Figure 4.1 Box plot of total score by examination year 43
Figure 6.1 Box plots showing the individual scores and the means by gender, training center, training center level and area of training center 44
Figure 6.2 Scatter plot of score by age of examinees 46
Figure 6.3 Transition of responses to questions given the status of answers to previous questions 46
Figure 6.4 Transition of response to questions given the status of answers to previous questions by examination year, gender, age group, level and area of training center, 47
Table 6.2.1 Rasch Model 50
Table 6.3.1 Estimated results of the two state Markov model. 51
Table 6.3.2 Estimated results of the two state Markov model by examination year. 52
Table 6.3.3 The ability index based on the estimated results of 2-state Markov chain model 54
Table 6.3.4 Estimated results of the two state Markov model with random effects of examinee and item. 55
Table 6.3.5 Estimated results of the two state Markov model with random effects of examinee, item, and examination year. 56
Table 6.3.6 Estimated results of the two-state Markov model using univariate analysis without random effect term. 57
Table 6.3.7 Estimated results of the two state Markov model using univariate analysis with random effect term. 58
Table 6.3.8 Estimated results of the two state Markov model using multivariate analysis without random effect term. 60
Table 6.3.9 Estimated results of the two state Markov model using multivariate analysis with random effect term. 61
Table 6.3.10 Estimated results of the second-order two-state Markov model without random effect. 62

1.Chang KY, Tsou MY, Chan KH, Chang SH, Tai JJ, Chen HH. Item analysis for the written test of Taiwanese board certification examination in anaesthesiology using the Rasch model. Br J Anaesth. 2010 Jun;104(6):717-22. doi: 10.1093/bja/aeq097. Epub 2010 Apr 28.
2.Cox DR, Miller HD. The theory of stochastic processes. London: Methuen, 1965.
3.De Ayala RJ. The Theory and Practice of Item Response Theory. New York: Guilford Press, 2009.
4.Fragoso TM, de Andrade M, Pereira AC, Rosa GJ, Soler JM. Bayesian Variable Selection in Multilevel Item Response Theory Models with Application in Genomics. Genet Epidemiol. 2016 Apr;40(3):253-63. doi: 10.1002/gepi.21960.
5.Hagquist C, Bruce M, Gustavsson JP. Using the Rasch model in nursing research: an introduction and illustrative example. Int J Nurs Stud. 2009 Mar;46(3):380-93. doi: 10.1016/j.ijnurstu.2008.10.007. Epub 2008 Dec 6.
6.Johansson S, Kottorp A, Lee KA, Gay CL, Lerdal A. Can the Fatigue Severity Scale 7-item version be used across different patient populations as a generic fatigue measure--a comparative study using a Rasch model approach. Health Qual Life Outcomes. 2014 Feb 22;12:24. doi: 10.1186/1477-7525-12-24.
7.Lee KYS, Lam JHS, Chan KTY, van Hasselt CA, Tong MCF. Applying Rasch model analysis in the development of the cantonese tone identification test (CANTIT). Int J Audiol. 2017 Mar 1:1-14.
8.MacDougall, M and Stone GE. Fortune-tellers or content specialists: challenging the standard setting paradigm in medical education programmes. Journal of Contemporary Medical Education. 2015; 3(3) ;134-142.
9.Nielsen JB, Kyvsgaard JN, Sildorf SM, Kreiner S, Svensson J. Item analysis using Rasch models confirms that the Danish versions of the DISABKIDS® chronic-generic and diabetes-specific modules are valid and reliable. Health Qual Life Outcomes. 2017 Mar 1;15(1):44. doi: 10.1186/s12955-017-0618-8.
10.O''Mara DA, Canny BJ, Rothnie IP, Wilson IG, Barnard J, Davies L. The Australian Medical Schools Assessment Collaboration: benchmarking the preclinical performance of medical students. Med J Aust. 2015 Feb 2;202(2):95-8.
11.O''Neill TR, Royal KD, Puffer JC. Performance on the American Board of Family Medicine (ABFM) certification examination: are superior test-taking skills alone sufficient to pass? J Am Board Fam Med. 2011 Mar-Apr;24(2):175-80. doi: 10.3122/jabfm.2011.02.100162.
12.Pallant JF, Tennant A. An introduction to the Rasch measurement model: an example using the Hospital Anxiety and Depression Scale (HADS).Br J Clin Psychol. 2007 Mar;46(Pt 1):1-18.
13.Smith AB, Rush R, Fallowfield LJ, Velikova G, Sharpe M. Rasch fit statistics and sample size considerations for polytomous data. BMC Med Res Methodol. 2008 May 29;8:33. doi: 10.1186/1471-2288-8-33.
14.Stephen W. Raudenbush, Christopher Johnson, Robert J. Sampson. A Multivariate, Multilevel Rasch Model with Application to Self-Reported Criminal Behavior. Sociological Methodology. 2004 ; 33(1); 169-211.
15.Stephen Slogoff. A History of the American Board of Anesthesiology Certifying Examinations. The Wondrous Story of Anesthesia. 2014; 459-470.
16.Strong DR, Lesieur HR, Breen RB, Stinchfield R, Lejuez CW. Using a Rasch model to examine the utility of the South Oaks Gambling Screen across clinical and community samples.Addict Behav. 2004 May;29(3):465-81.
17.Thellesen L, Bergholt T, Hedegaard M, Colov NP, Christensen KB, Andersen KS, Sorensen JL. Development of a written assessment for a national interprofessional cardiotocography education program. BMC Med Educ. 2017 May 18;17(1):88. doi: 10.1186/s12909-017-0915-2.
18.Tor E, Steketee C. Rasch analysis on OSCE data : An illustrative example. Australas Med J. 2011;4(6):339-45.
19.Yang SC , Tsou MY, Chen ET, Chan KH, Chang KY. Statistical item analysis of the examination in anesthesiology for medical students using the Rasch model. J Chin Med Assoc. 2011 Mar;74(3):125-9. doi: 10.1016/j.jcma.2011.01.027. Epub 2011 Feb 19.
20.Yoon SP, Cho SS. Outcome-based self-assessment on a team-teaching subject in the medical school. Anat Cell Biol. 2014 Dec;47(4):259-66. doi: 10.5115/acb.2014.47.4.259. Epub 2014 Dec 23.