Philosophiae Doctor - PhD (Statistics and Population Studies) / The aim of this study is to look at the methods and processes involved in imputing missing data and more specifically, complete missing blocks of data. A further aim of this study is to look at the effect that the imputed data has on the accuracy of various predictive models constructed on the imputed data and hence determine if the imputation method involved is suitable. The identification of the missingness mechanism present in the data should be the first process to follow in order to identify a possible imputation method. The identification of a suitable imputation method is easier if the mechanism can be identified as one of the following; missing completely at random (MCAR), missing at random (MAR) or not missing at random (NMAR). Predictive models constructed on the complete imputed data sets are shown to be less accurate for those models constructed on data sets which employed a hot-deck imputation method. The data sets which employed either a single or multiple Monte Carlo Markov Chain (MCMC) or the Fully Conditional Specification (FCS) imputation methods are shown to result in predictive models that are more accurate. The addition of an iterative bagging technique in the modelling procedure is shown to produce highly accurate prediction estimates. The bagging technique is applied to variants of the neural network, a decision tree and a multiple linear regression (MLR) modelling procedure. A stochastic gradient boosted decision tree (SGBT) is also constructed as a comparison to the bagged decision tree. Final models are constructed from 200 iterations of the various modelling procedures using a 60% sampling ratio in the bagging procedure. It is further shown that the addition of the bagging technique in the MLR modelling procedure can produce a MLR model that is more accurate than that of the other more advanced modelling procedures under certain conditions. The evaluation of the predictive models constructed on imputed data is shown to vary based on the type of fit statistic used. It is shown that the average squared error reports little difference in the accuracy levels when compared to the results of the Mean Absolute Prediction Error (MAPE). The MAPE fit statistic is able to magnify the difference in the prediction errors reported. The Normalized Mean Bias Error (NMBE) results show that all predictive models constructed produced estimates that were an over-prediction, although these did vary depending on the data set and modelling procedure used. The Nash Sutcliffe efficiency (NSE) was used as a comparison statistic to compare the accuracy of the predictive models in the context of imputed data. The NSE statistic showed that the estimates of the models constructed on the imputed data sets employing a multiple imputation method were highly accurate. The NSE statistic results reported that the estimates from the predictive models constructed on the hot-deck imputed data were inaccurate and that a mean substitution of the fully observed data would have been a better method of imputation. The conclusion reached in this study shows that the choice of imputation method as well as that of the predictive model is dependent on the data used. Four unique combinations of imputation methods and modelling procedures were concluded for the data considered in this study.

Šiame darbe išnagrinėjome kelis praleistų reikšmių įrašymo metodus, kuriuos taikėme išvykstamojo turizmo statistinio tyrimo 2.6. klausimo pirmiems dviem punktams: paslaugų paketo ir transporto išlaidoms. Įrašymo metodų efektyvumo analizę atlikome su pilnais duomenimis, juose fiktyviai padarydamos praleistas reikšmes ir į jas įrašydamos reikšmes keliais praleistų reikšmių įrašymo metodais. Tuomet turėdamos tikras ir įrašytas reikšmes galėjome palyginti parametrų įverčius. Kadangi praleistos reikšmės gali atsirasti atsitiktinai ir neatsitiktinai, todėl mes praleistų reikšmių įrašymo metodus taikėme trims atvejams: kai praleistos reikšmės atsiranda atsitiktinai, kai praleistos reikšmės atsiranda tada, kai neatsako respondentai turėję didžiausias ar mažiausias išlaidas kelionėje. Praleistų reikšmių įrašymui taikėme skirstiniu pagrįstą, vidurkio, atsitiktinio pakartojimo, santykiu pagrįstą ir daugiareikšmio įrašymo metodus, nesudarydamos įrašymo klasių ir sudarydamos įrašymo klases. Taigi, siūlome tokį pat praleistų reikšmių įrašymo metodų efektyvumo tyrimą atlikti ir likusiems 2.6. klausimo punktams, nusistatyti tinkamiausią įrašymo metodą ir tada jį taikyti jau tikroms praleistoms reikšmėms įrašyti. Be to, reikėtų atsižvelgti ir į dėl įrašymo atsirandančios dispersijos įvertinį, nes jos indėlis į bendrą dispersijos įvertinį yra nemažas. Atlikus praleistų reikšmių įrašymą, bus galima taikyti kompiuterinius įverčių skaičiavimo metodus ir nebus prarasta kita informacija, kurią... [toliau žr. visą tekstą] / In this work, we examined some missing data imputation methods in the survey on outbound tourism for the package tour and transport expenses. We performed an analysis of the efficiency of missing data imputation methods using full data sets with fictitious missing data applying various missing data imputation methods to fill in the missing data. Thus, we had real values and imputed values and could compare the estimated parameters. The missing data can appear randomly and non-randomly, so we applied missing data imputation methods in three cases: when missing data appear randomly and when missing data appear in case of non-response of respondents who had the highest or the lowest travel expenses. We applied distribution, average, random, ratio and multiple imputation methods for missing data imputation without using imputation classes and using imputation classes. We propose to perform the same efficiency survey of missing data imputation methods for the remaining items of expenses in the outbound tourism questionnaire in order to find out a convenient missing data imputation method and apply it for the real missing data (the current analysis was performed applying fictitious missing data). After the missing data imputation, we can apply the procedures of parameter estimation and we will not lose other information as it would be the case with the elimination of questionnaires having missing data.

Šiame darbe išnagrinėjome kelis praleistų reikšmių įrašymo metodus, kuriuos taikėme išvykstamojo turizmo statistinio tyrimo 2.6. klausimo pirmiems dviem punktams: paslaugų paketo ir transporto išlaidoms. Įrašymo metodų efektyvumo analizę atlikome su pilnais duomenimis, juose fiktyviai padarydamos praleistas reikšmes ir į jas įrašydamos reikšmes keliais praleistų reikšmių įrašymo metodais. Tuomet turėdamos tikras ir įrašytas reikšmes galėjome palyginti parametrų įverčius. Kadangi praleistos reikšmės gali atsirasti atsitiktinai ir neatsitiktinai, todėl mes praleistų reikšmių įrašymo metodus taikėme trims atvejams: kai praleistos reikšmės atsiranda atsitiktinai, kai praleistos reikšmės atsiranda tada, kai neatsako respondentai turėję didžiausias ar mažiausias išlaidas kelionėje. Praleistų reikšmių įrašymui taikėme skirstiniu pagrįstą, vidurkio, atsitiktinio pakartojimo, santykiu pagrįstą ir daugiareikšmio įrašymo metodus, nesudarydamos įrašymo klasių ir sudarydamos įrašymo klases. Taigi, siūlome tokį pat praleistų reikšmių įrašymo metodų efektyvumo tyrimą atlikti ir likusiems 2.6. klausimo punktams, nusistatyti tinkamiausią įrašymo metodą ir tada jį taikyti jau tikroms praleistoms reikšmėms įrašyti. Be to, reikėtų atsižvelgti ir į dėl įrašymo atsirandančios dispersijos įvertinį, nes jos indėlis į bendrą dispersijos įvertinį yra nemažas. Atlikus praleistų reikšmių įrašymą, bus galima taikyti kompiuterinius įverčių skaičiavimo metodus ir nebus prarasta kita informacija, kurią... [toliau žr. visą tekstą] / In this work, we examined some missing data imputation methods in the survey on outbound tourism for the package tour and transport expenses. We performed an analysis of the efficiency of missing data imputation methods using full data sets with fictitious missing data applying various missing data imputation methods to fill in the missing data. Thus, we had real values and imputed values and could compare the estimated parameters. The missing data can appear randomly and non-randomly, so we applied missing data imputation methods in three cases: when missing data appear randomly and when missing data appear in case of non-response of respondents who had the highest or the lowest travel expenses. We applied distribution, average, random, ratio and multiple imputation methods for missing data imputation without using imputation classes and using imputation classes. We propose to perform the same efficiency survey of missing data imputation methods for the remaining items of expenses in the outbound tourism questionnaire in order to find out a convenient missing data imputation method and apply it for the real missing data (the current analysis was performed applying fictitious missing data). After the missing data imputation, we can apply the procedures of parameter estimation and we will not lose other information as it would be the case with the elimination of questionnaires having missing data.

We compare different multiple imputation methods for categorical variables using the MICE package in R. We take a complete data set and remove different levels of missingness and evaluate the imputation methods for each level of missingness. Logistic regression imputation and linear discriminant analysis (LDA) are used for binary variables. Multinomial logit imputation and LDA are used for nominal variables while ordered logit imputation and LDA are used for ordinal variables. After imputation, the regression coefficients, percent deviation index (PDI) values, and relative frequency tables were found for each imputed data set for each level of missingness and compared to the complete corresponding data set. It was found that logistic regression outperformed LDA for binary variables, and LDA outperformed both multinomial logit imputation and ordered logit imputation for nominal and ordered variables. Simulations were ran to confirm the validity of the results.

Missing data continues to be one of the main problems in data analysis as it reduces sample representativeness and consequently, causes biased estimates. Multiple imputation methods have been established as an effective method of handling missing data. In this study, we examined multiple imputation methods for quantitative variables on twelve data sets with varied sizes and variability that were pseudo generated from an original data. The multiple imputation methods examined are the predictive mean matching, Bayesian linear regression and linear regression, non-Bayesian in the MICE (Multiple Imputation Chain Equation) package in the statistical software, R. The parameter estimates generated from the linear regression on the imputed data were compared to the closest parameter estimates from the complete data across all twelve data sets.

Despite considerable advances in missing data imputation methods over the last three decades, the problem of missing data remains largely unsolved. Many techniques have emerged in the literature as candidate solutions. These techniques can be categorised into two classes: statistical methods of data imputation and computational intelligence methods of data imputation. Due to the longstanding use of statistical methods in handling missing data problems, it takes quite some time for computational intelligence methods to gain profound attention even though these methods have analogous accuracy, in comparison to other approaches. The merits of both these classes have been discussed at length in the literature, but only limited studies make significant comparison to these classes. This thesis contributes to knowledge by firstly, conducting a comprehensive comparison of standard statistical methods of data imputation, namely, mean substitution (MS), regression imputation (RI), expectation maximization (EM), tree imputation (TI) and multiple imputation (MI) on missing completely at random (MCAR) data sets. Secondly, this study also compares the efficacy of these methods with a computational intelligence method of data imputation, ii namely, a neural network (NN) on missing not at random (MNAR) data sets. The significance difference in performance of the methods is presented. Thirdly, a novel procedure for handling missing data is presented. A hybrid combination of each of these statistical methods with a NN, known here as the post-processing procedure, was adopted to approximate MNAR data sets. Simulation studies for each of these imputation approaches have been conducted to assess the impact of missing values on partial least squares structural equation modelling (PLS-SEM) based on the estimated accuracy of both structural and measurement parameters. The best method to deal with particular missing data mechanisms is highly recognized. Several significant insights were deduced from the simulation results. It was figured that for the problem of MCAR by using statistical methods of data imputation, MI performs better than the other methods for all percentages of missing data. Another unique contribution is found when comparing the results before and after the NN post-processing procedure. This improvement in accuracy may be resulted from the neural network's ability to derive meaning from the imputed data set found by the statistical methods. Based on these results, the NN post-processing procedure is capable to assist MS in producing significant improvement in accuracy of the approximated values. This is a promising result, as MS is the weakest method in this study. This evidence is also informative as MS is often used as the default method available to users of PLS-SEM software.

Despite considerable advances in missing data imputation methods over the last three decades, the problem of missing data remains largely unsolved. Many techniques have emerged in the literature as candidate solutions. These techniques can be categorised into two classes: statistical methods of data imputation and computational intelligence methods of data imputation. Due to the longstanding use of statistical methods in handling missing data problems, it takes quite some time for computational intelligence methods to gain profound attention even though these methods have analogous accuracy, in comparison to other approaches. The merits of both these classes have been discussed at length in the literature, but only limited studies make significant comparison to these classes. This thesis contributes to knowledge by firstly, conducting a comprehensive comparison of standard statistical methods of data imputation, namely, mean substitution (MS), regression imputation (RI), expectation maximization (EM), tree imputation (TI) and multiple imputation (MI) on missing completely at random (MCAR) data sets. Secondly, this study also compares the efficacy of these methods with a computational intelligence method of data imputation, ii namely, a neural network (NN) on missing not at random (MNAR) data sets. The significance difference in performance of the methods is presented. Thirdly, a novel procedure for handling missing data is presented. A hybrid combination of each of these statistical methods with a NN, known here as the post-processing procedure, was adopted to approximate MNAR data sets. Simulation studies for each of these imputation approaches have been conducted to assess the impact of missing values on partial least squares structural equation modelling (PLS-SEM) based on the estimated accuracy of both structural and measurement parameters. The best method to deal with particular missing data mechanisms is highly recognized. Several significant insights were deduced from the simulation results. It was figured that for the problem of MCAR by using statistical methods of data imputation, MI performs better than the other methods for all percentages of missing data. Another unique contribution is found when comparing the results before and after the NN post-processing procedure. This improvement in accuracy may be resulted from the neural network¿s ability to derive meaning from the imputed data set found by the statistical methods. Based on these results, the NN post-processing procedure is capable to assist MS in producing significant improvement in accuracy of the approximated values. This is a promising result, as MS is the weakest method in this study. This evidence is also informative as MS is often used as the default method available to users of PLS-SEM software. / Minister of Higher Education Malaysia and University Utara Malaysia

Em pesquisas da área da saúde é muito comum que o pesquisador defronte-se com o problema de dados faltantes. Nessa situação, é freqüente que a decisão do pesquisador seja desconsiderar os sujeitos que tenham não-resposta em alguma ou algumas das variáveis, pois muitas das técnicas estatísticas foram desenvolvidas para analisar dados completos. Entretanto, essa exclusão de sujeitos pode gerar inferências que não são válidas, principalmente se os indivíduos que permanecem na análise são diferentes daqueles que foram excluídos. Nas duas últimas décadas, métodos de imputação de dados foram desenvolvidos com a intenção de se encontrar solução para esse problema. Esses métodos usam como base a idéia de preencher os dados faltantes com valores plausíveis. O método mais complexo de imputação é a chamada imputação múltipla. Essa tese tem por objetivo divulgar o método de imputação múltipla e através de dois artigos procura atingir esse objetivo. O primeiro artigo descreve duas técnicas de imputação múltipla e as aplica a um conjunto de dados reais. O segundo artigo faz a comparação do método de imputação múltipla com duas técnicas de imputação única através de uma aplicação a um modelo de risco para mortalidade cirúrgica. Para as aplicações foram usados dados secundários já utilizados por Klück (2004). / Missing data in health research is a very common problem. The most direct way of dealing with missing data is to exclude observations with missing data, probably because the traditional statistical methods have been developed for complete data sets. However, this decision may give biased results, mainly if the subjects considered in the analysis are different of those who have been excluded. In the last two decades, imputation methods were developed to solve this problem. The idea of the imputation is to fill in the missing data with reasonable values. The multiple imputation is the most complex method. The objective of this dissertation is to divulge the multiple imputation method through two papers. The first one describes two different types of multiple imputation and it shows an application to real data. The second paper shows a comparison among the multiple imputation and two single imputations applied to a risk model for surgical mortality. The used data sets were secondary data used by Klück (2004).

Em pesquisas da área da saúde é muito comum que o pesquisador defronte-se com o problema de dados faltantes. Nessa situação, é freqüente que a decisão do pesquisador seja desconsiderar os sujeitos que tenham não-resposta em alguma ou algumas das variáveis, pois muitas das técnicas estatísticas foram desenvolvidas para analisar dados completos. Entretanto, essa exclusão de sujeitos pode gerar inferências que não são válidas, principalmente se os indivíduos que permanecem na análise são diferentes daqueles que foram excluídos. Nas duas últimas décadas, métodos de imputação de dados foram desenvolvidos com a intenção de se encontrar solução para esse problema. Esses métodos usam como base a idéia de preencher os dados faltantes com valores plausíveis. O método mais complexo de imputação é a chamada imputação múltipla. Essa tese tem por objetivo divulgar o método de imputação múltipla e através de dois artigos procura atingir esse objetivo. O primeiro artigo descreve duas técnicas de imputação múltipla e as aplica a um conjunto de dados reais. O segundo artigo faz a comparação do método de imputação múltipla com duas técnicas de imputação única através de uma aplicação a um modelo de risco para mortalidade cirúrgica. Para as aplicações foram usados dados secundários já utilizados por Klück (2004). / Missing data in health research is a very common problem. The most direct way of dealing with missing data is to exclude observations with missing data, probably because the traditional statistical methods have been developed for complete data sets. However, this decision may give biased results, mainly if the subjects considered in the analysis are different of those who have been excluded. In the last two decades, imputation methods were developed to solve this problem. The idea of the imputation is to fill in the missing data with reasonable values. The multiple imputation is the most complex method. The objective of this dissertation is to divulge the multiple imputation method through two papers. The first one describes two different types of multiple imputation and it shows an application to real data. The second paper shows a comparison among the multiple imputation and two single imputations applied to a risk model for surgical mortality. The used data sets were secondary data used by Klück (2004).

Em pesquisas da área da saúde é muito comum que o pesquisador defronte-se com o problema de dados faltantes. Nessa situação, é freqüente que a decisão do pesquisador seja desconsiderar os sujeitos que tenham não-resposta em alguma ou algumas das variáveis, pois muitas das técnicas estatísticas foram desenvolvidas para analisar dados completos. Entretanto, essa exclusão de sujeitos pode gerar inferências que não são válidas, principalmente se os indivíduos que permanecem na análise são diferentes daqueles que foram excluídos. Nas duas últimas décadas, métodos de imputação de dados foram desenvolvidos com a intenção de se encontrar solução para esse problema. Esses métodos usam como base a idéia de preencher os dados faltantes com valores plausíveis. O método mais complexo de imputação é a chamada imputação múltipla. Essa tese tem por objetivo divulgar o método de imputação múltipla e através de dois artigos procura atingir esse objetivo. O primeiro artigo descreve duas técnicas de imputação múltipla e as aplica a um conjunto de dados reais. O segundo artigo faz a comparação do método de imputação múltipla com duas técnicas de imputação única através de uma aplicação a um modelo de risco para mortalidade cirúrgica. Para as aplicações foram usados dados secundários já utilizados por Klück (2004). / Missing data in health research is a very common problem. The most direct way of dealing with missing data is to exclude observations with missing data, probably because the traditional statistical methods have been developed for complete data sets. However, this decision may give biased results, mainly if the subjects considered in the analysis are different of those who have been excluded. In the last two decades, imputation methods were developed to solve this problem. The idea of the imputation is to fill in the missing data with reasonable values. The multiple imputation is the most complex method. The objective of this dissertation is to divulge the multiple imputation method through two papers. The first one describes two different types of multiple imputation and it shows an application to real data. The second paper shows a comparison among the multiple imputation and two single imputations applied to a risk model for surgical mortality. The used data sets were secondary data used by Klück (2004).