在各種調查研究中,
評估多個項目的喜好所產生的 排序資料 (rank data) 是一種很常見的資料型態,
但在實務上, 常常會因為一些因素而無法取得所有的排序,
形成不完全排序資料. 一般對於排序資料分析,
多數研究者會估計重要的共變數對排序結果的影響,
使用各種統計模型分析, 但現有的統計方法,
多未考慮同時會有排序值與共變數值缺失的情況,
且未考慮如何分析具有缺失值的不完全排序資料.
對於不完全排序資料,
目前有一些多重插補方法僅針對不完全排序的缺失值進行插補,
但未考慮共變數缺失的情形, 且沒有相對應的套裝統計軟體可用.

本研究利用統計模擬,
探討排序值與共變數值同時缺失的一般情況,
針對不同 缺失機制 (missing mechanism) 下,
利用現有多重插補的套裝統計軟體進行多重插補,
本研究比較插補前與插補後的結果差異.
現有多重插補的套裝統計軟體僅考慮排序資料為連續變數或類別變數,
本研究比較不同變數性質的插補差異與不同缺失機制的差異.

本研究發現進行多重插補之後,
插補資料排序值與完全排序資料的排序值差距減小,
插補資料有助於排序資料的分析.
研究也發現部分平衡不完全區集設計的插補資料與完全排序資料差距最大,
但在市場調查實務上, 部分平衡不完全區集設計卻是最常用的設計.
研究結果也顯示考慮排序資料為連續變數或類別變數的插補方法會對分析有不同程度的影響,
且影響因子 (共變數) 的缺失值在不同的缺失機制下,
也會對分析有不同程度的影響.
在實例分析中發現插補缺失排序值後會改變對品牌的喜好排序.
排序資料與常態連續變數, 順序類別變數, 名目類別變數有所不同,
建議進行敏感度分析多重插補對排序資料使用不同分析模型的影響.

Ranking multiple items is common in survey research and forms complete rank data.
Researchers also use models to quantify the covariates' impact on the ranks.
Because of various reasons, raters often can not evaluate all items
resulting incomplete rank data.
Many statistical softwares provide multiple imputation for missing values to analyzed an imputed data as a full data.
However, these statistical softwares only consider imputing continuous or categorical variables.
And few studies reported how to analyze incomplete rank data with missing covariates.
Rank data is different from continuous and categorical data.
Therefore, this study aim to
explore the effects of multiple imputations on
analyzing incomplete rank data with missing covariates by using the available statistical softwares.

We used a complete rank data set to generated 1,000 full data sets by using bootstrap sampling.
We constructed incomplete rank data sets with missing covariates from 3 different missing mechanisms.
We generated imputed rank data sets by using multiple imputation that treats
rank value as continuous, ordinal and nominal value.
We assessed the effects of missing mechanisms and multiple imputations on ranks.

We found that the imputed rank data had smaller difference
in terms of the original rank values of the complete rank data
than those of missing rank data.
The differences were small among treating rank value as continuous, ordinal and nominal value.
We also found that partial balanced incomplete block design
had the worst performance among different missing mechanisms after imputation.
We also found that imputing missing values changed the rank of brand preferences
with a sequential conditional logit model.
When using multiple imputation for incomplete rank data with the available softwares,
we should conduct sensitivity analysis based on type of variables, missing mechanisms and
statistical models.

目錄
1 緒論.............................................1
2 文獻探討..........................................3
2.1 排序資料(Rank Data).............................3
2.2 缺失資料(Missing Data)..........................5
2.3 多重插補(Multiple Imputation)...................6
2.4 變數型態........................................7
3 研究方法與資料模擬..................................9
3.1 研究背景及目的...................................9
3.2 研究資料.......................................10
3.3 模擬完全排序資料(Complete Rank Data).............10
3.4 模擬不完全排序資料(Incomplete Rank Data).........10
3.5 模擬插補排序資料(Imputed Rank Data)..............12
3.6 比較分析.......................................12
3.7 模擬結果.......................................13
4 實例分析.........................................23
4.1 資料說明.......................................23
4.2 分析方法.......................................24
4.3 分析結果比較....................................25
5 討論與建議........................................29
參考文獻............................................31


圖目錄
3.1 插補後之排序差距盒鬚圖.............................15
3.2 常態連續變數插補之排序差距盒鬚圖.....................16
3.3 順序類別變數插補之排序差距盒鬚圖.....................16
3.4 名目類別變數插補之排序差距盒鬚圖.....................17

表目錄
2.1 平衡不完全排序資料..................................4
2.2 部分平衡不完全排序資料...............................4
2.3 非平衡不完全排序資料.................................5
2.4 缺失機制...........................................6
2.5 插補方法...........................................8
3.1 以常態連續變數插補之排序差距摘要表.....................18
3.2 以順序類別變數插補之排序差距摘要表.....................18
3.3 以名目類別變數插補之排序差距摘要表.....................18
3.4 MCAR 1 排序差距摘要表..............................19
3.5 MCAR 2 排序差距摘要表..............................19
3.6 MCAR 3 排序差距摘要表..............................19
3.7 MCAR 4 排序差距摘要表..............................20
3.8 MAR 1 排序差距摘要表...............................20
3.9 MAR 2 排序差距摘要表...............................20
3.10 MAR 3 排序差距摘要表............................. 21
3.11 MNAR 1 排序差距摘要表.............................21
3.12 MNAR 2 排序差距摘要表.............................21
3.13 用常態連續變數插補 1 個檔案所需時間(秒)...............22
3.14 用順序類別變數插補 1 個檔案所需時間(秒)...............22
3.15 用名目類別變數插補 1 個檔案所需時間(秒)...............22
4.1 咖啡實例原始資料....................................24
4.2 未調整主作用估計................................... 27
4.3 未調整主作用變異數估計...............................27
4.4 未調整主作用的 95% 信賴區間..........................27
4.5 調整主作用與交互作用估計..............................28
4.6 調整主作用與交互作用變異數估計.........................28
4.7 調整主作用與交互作用的 95% 信賴區間....................28

[1] William H Greene and David A Hensher. Modeling ordered choices: A
primer. Cambridge University Press, 2010.
[2] Paul D Allison and Nicholas A Christakis. Logit models for sets of
ranked items. Sociological methodology, 24(1994):199–228, 1994.
[3] Paul H Lee and LH Philip. An r package for analyzing and modeling
ranking data. BMC medical research methodology, 13(1):65, 2013.
[4] TUEH-LI LEE. Compare four different methods for incomplete rank
data (in chinese ). 2004.
[5] TING-MEI HSU. Missing data analysis of ranking experiments. 2003.
[6] CHIH-HSUEH WU. Incomplete rank data analysis - likelihood ap-
proach. 2005.
[7] HSIANG-TING PENG. Bayesian analysis for incomplete data (in chi-
nese ). 2007.
[8] John I Marden. Analyzing and modeling rank data, volume 64. CRC
Press, 1996.
[9] Roderick JA Little and Donald B Rubin. Statistical analysis with miss-
ing data. 2002.
[10] DB Rubin. Multiple imputation for nonresponse in surveys. New York,
USA: John Willey & Sons, 1987.
[11] Joseph L Schafer. Analysis of incomplete multivariate data. CRC press,
1997.
[12] Stef Buuren and Karin Groothuis-Oudshoorn. Mice: Multivariate impu-
tation by chained equations in r. Journal of statistical software, 45(3),
2011.
[13] Bradley Efron and Robert J Tibshirani. An introduction to the bootstrap,
volume 57. CRC press, 1994.
[14] David R Cox et al. Regression models and life tables. JR stat soc B,
34(2):187–220, 1972.