Bayesian analysis of ordinal survey data using the Dirichlet process ...

More documents

Recommendations

Info

for extreme responses of 0’s and 5’s. This case highlights a distinction between our notionof extremism and variability.As another example of the adjustment made for personalities, the smallest posteriormean for the disposition parameter corresponds to a i = −0.30 which was recorded for astudent with an average observed response ¯X i = 2.06 over all m = 15 questions. Thisstudent has a corresponding extremism parameter b = 1.01. The question arises asto whether 2.06 is a measurement that should be taken at face value when the averageresponse and standard deviation over all students are 3.89 and 1.02 respectively. It appearsthat 2.06 is an extreme score lying 1.8 standard deviations from the mean. However, whenwe adjust for the personality of the student via (4), we obtain Y i = 1.01(Z i − 0.301 −31) + 31 ≈ Z i − 0.31. This implies that the standardized but latent response Z i is largerthan Y i . The student has a negative disposition, and when we account for the negativedisposition, the de-noised score Z i is not as extreme as the raw data X i .It is also instructive to look at the posterior mean of the variance-covariance matrixΣ which describes the relationships amongst the m = 15 survey questions. The largestcorrelation 0.63 occurred between survey questions 14 and 15. This is consistent withour intuition and personal teaching experience whereby students think highly of theirinstructors when they believe that their instructors care about them. The second highestcorrelation 0.57 occurred between survey questions 6 and 7 which is also believable fromthe view that learning is best achieved when material is clearly presented. However, weemphasize that the elimination of questions on the basis of redundancy should not bedone solely on the basis of high correlations. In addition to high correlations, we shouldalso have similar posterior means. With the estimated posterior means µ 14 = 4.57 andµ 15 = 4.52, SFU may feel comfortable in dropping either question 14 or question 15 fromthe survey. Furthermore, we note that there were no negative posterior correlations and14
Table 1: Posterior means and posterior standard deviations for the SFU survey data.Parameter Posterior Mean Posterior SDµ 1 3.69 0.19µ 2 3.53 0.15µ 3 4.04 0.18µ 4 3.45 0.17µ 5 3.85 0.17µ 6 4.54 0.19µ 7 4.33 0.18µ 8 4.41 0.17µ 9 4.78 0.15µ 10 3.23 0.17µ 11 4.51 0.19µ 12 4.01 0.18µ 13 4.11 0.17µ 14 4.57 0.19µ 15 4.52 0.18¯µ 4.10the minimum correlation 0.11 occurred between question 1 and question 13. Our intuitionaccordingly suggests that these two questions are independent. For comparison, we havealso calculated the sample correlation matrix based on the raw scores X. The values alignwith the posterior mean of Σ. For example, the smallest sample correlation is 0.10 andthis is observed between question 1 and question 13. The largest sample correlation is0.78 and this occurs between questions 14 and 15.It is good statistical practice to look at various plots related to the MCMC simulation.Trace plots for the parameters appear to stabilize immediately and hence provide no15
Page 1 and 2: Bayesian Analysis of Ordinal Survey
Page 3 and 4: survey where there is a respondent
Page 5 and 6: validity of the model. The proposed
Page 7 and 8: and 4’s. We instead consider a st
Page 9 and 10: our rationale for the Uniform(0, 6)
Page 11 and 12: For programming in WinBUGS, the Set
Page 13: Markov chain simulations. The resul
Page 17 and 18: our model gave D = 891. In both the
Page 19 and 20: Table 2: Posterior means and poster
Page 21 and 22: To provide a more stringent test, w
Page 23 and 24: Brooks, S. P. and Gelman, A. (1997)
Page 25 and 26: Parducci, A. (1965). “Category ju
Page 27 and 28: proportion0.00 0.02 0.04 0.06 0.08
Page 29 and 30: elative frequency0.0 0.1 0.2 0.3●

Bayesian analysis of ordinal survey data using the Dirichlet process ...

Create successful ePaper yourself

Delete template?

Save as template?