Conclude Resin

Missing data are a rule rather than an exception in quantitative research. Enders (
2003 (link)
) stated that a missing rate of 15% to 20% was common in educational and psychological studies. Peng et al. (
2006
) surveyed quantitative studies published from 1998 to 2004 in 11 education and psychology journals. They found that 36% of studies had no missing data, 48% had missing data, and about 16% cannot be determined. Among studies that showed evidence of missing data, 97% used the listwise deletion (LD) or the pairwise deletion (PD) method to deal with missing data. These two methods are ad hoc and notorious for biased and/or inefficient estimates in most situations (
Rubin 1987
;
Schafer 1997
). The APA Task Force on Statistical Inference explicitly warned against their use (
Wilkinson and the Task Force on Statistical Inference 1999 (link)
p. 598). Newer and principled methods, such as the multiple-imputation (MI) method, the full information maximum likelihood (FIML) method, and the expectation-maximization (EM) method, take into consideration the conditions under which missing data occurred and provide better estimates for parameters than either LD or PD. Principled missing data methods do not replace a missing value directly; they combine available information from the observed data with statistical assumptions in order to estimate the population parameters and/or the missing data mechanism statistically.
A review of the quantitative studies published in Journal of Educational Psychology (JEP) between 2009 and 2010 revealed that, out of 68 articles that met our criteria for quantitative research, 46 (or 67.6%) articles explicitly acknowledged missing data, or were suspected to have some due to discrepancies between sample sizes and degrees of freedom. Eleven (or 16.2%) did not have missing data and the remaining 11 did not provide sufficient information to help us determine if missing data occurred. Of the 46 articles with missing data, 17 (or 37%) did not apply any method to deal with the missing data, 13 (or 28.3%) used LD or PD, 12 (or 26.1%) used FIML, four (or 8.7%) used EM, three (or 6.5%) used MI, and one (or 2.2%) used both the EM and the LD methods. Of the 29 articles that dealt with missing data, only two explained their rationale for using FIML and LD, respectively. One article misinterpreted FIML as an imputation method. Another was suspected to have used either LD or an imputation method to deal with attrition in a PISA data set (
OECD 2009
;
Williams and Williams 2010 (link)
).
Compared with missing data treatments by articles published in JEP between 1998 and 2004 (
Table 3.1 in Peng et al. 2006
), there has been improvement in the decreased use of LD (from 80.7% down to 21.7%) and PD (from 17.3% down to 6.5%), and an increased use of FIML (from 0% up to 26.1%), EM (from 1.0% up to 8.7%), or MI (from 0% up to 6.5%). Yet several research practices still prevailed from a decade ago, namely, not explicitly acknowledging the presence of missing data, not describing the particular approach used in dealing with missing data, and not testing assumptions associated with missing data methods. These findings suggest that researchers in educational psychology have not fully embraced principled missing data methods in research.
Although treating missing data is usually not the focus of a substantive study, failing to do so properly causes serious problems. First, missing data can introduce potential bias in parameter estimation and weaken the generalizability of the results (
Rubin 1987
;
Schafer 1997
). Second, ignoring cases with missing data leads to the loss of information which in turn decreases statistical power and increases standard errors(
Peng et al. 2006
). Finally, most statistical procedures are designed for complete data (
Schafer and Graham 2002 (link)
). Before a data set with missing values can be analyzed by these statistical procedures, it needs to be edited in some way into a “complete” data set. Failing to edit the data properly can make the data unsuitable for a statistical procedure and the statistical analyses vulnerable to violations of assumptions.
Because of the prevalence of the missing data problem and the threats it poses to statistical inferences, this paper is interested in promoting three principled methods, namely, MI, FIML, and EM, by illustrating these methods with an empirical data set and discussing issues surrounding their applications. Each method is demonstrated using SAS 9.3. Results are contrasted with those obtained from the complete data set and the LD method. The relative merits of each method are noted, along with common features they share. The paper concludes with an emphasis on assumptions associated with these principled methods and recommendations for researchers. The remainder of this paper is divided into the following sections: (1) Terminology, (2) Multiple Imputation (MI), (3) Full Information Maximum-Likelihood (FIML), (4) Expectation-Maximization (EM) Algorithm, (5) Demonstration, (6) Results, and (6) Discussion.

To use the FDR technique to detect outliers, we compute a P value for each residual testing the null hypothesis that that residual comes from a Gaussian distribution. Additionally, we restrict the maximum number of outliers that we will detect to equal 30% of N, so only compute P values for the 30% of the residuals that are furthest from the curve.
Follow these steps:
1. Fit the model using robust regression. Compute the robust standard deviation of the residuals (RSDR, defined in Equation 1).
2. Decide on a value for Q. We recommend setting Q to 1%.
3. Rank the absolute value of the residuals from low to high, so Residual_N corresponds to the point furthest from the curve.
4. Loop from i = int(0.70*N) to N (we only test the 30% of the points furthest from the curve).
a. Compute
αi=Q(N−(i−1))N17 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFXoqydaWgaaWcbaGaemyAaKgabeaakiabg2da9maalaaabaGaemyuae1aaeWaaeaacqWGobGtcqGHsislcqGGOaakcqWGPbqAcqGHsislcqaIXaqmcqGGPaqkaiaawIcacaGLPaaaaeaacqWGobGtaaGaaCzcaiaaxMaacqaIXaqmcqaI3aWnaaa@3EFE@
b. Compute
t=|Residuali|RSDR18 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG0baDcqGH9aqpdaWcaaqaamaaemaabaacbiGae8NuaiLae8xzauMaem4CamNaemyAaKMaemizaqMaemyDauNaemyyaeMaemiBaW2aaSbaaSqaaiabdMgaPbqabaaakiaawEa7caGLiWoaaeaacqWGsbGucqWGtbWucqWGebarcqWGsbGuaaGaaCzcaiaaxMaacqaIXaqmcqaI4aaoaaa@466F@
c. Compute the two-tailed P value from the t distribution with N-K degrees of freedom (N is number of data points; K is number of parameters fit by nonlinear regression).
d. Test whether this P value (calculated in 4c) is less than α_i. (calculated in 4a).
• If yes, define the data point that corresponds to Residual_i, and all data points further from the curve, to be outliers (and stop the outlier hunt).
• If no, and if i = N, then conclude there are no outliers.
• If no, and i5. Delete the outliers, and run least-squares regression on the remaining points.

Two separate Markov decision models were developed to compare the long-term costs and health benefits of the IraPEN program (primary CVD prevention) with the status quo (no prevention) in two distinct scenarios. In the base case scenario, individuals without diabetes were included, while patients with diabetes were included in the alternative scenario. Each Markov model has four health states with transitions between the states according to age, sex, and the CVD risk characteristics of participants (Figure 1). In contrast to the usual Markov models, which are structured based on cohorts with average profiles, we decided to categorize the individuals based on their CVD risks. As the intervention (treatment) varied according to CVD risk level, it is logical to model them separately. In this way, we can take into account their specific characteristics. Therefore, based on WHO/ISH CVD risk prediction charts for EMR B, four index cohorts were constructed (5 ). These hypothetical cohorts were used as a representative for individuals with low, moderate, high, and very high CVD risk profiles. The CVD risk state represents the starting point for all people who are 40 years old. It was assumed that people in this state may either remain in the same health state, move to the stroke state, or CHD (coronary heart disease) state, or die. As long as they are event-free, these individuals can stay in a healthy state, but after the first event, they move to the CHD or stroke state and stay there until their death.
In WHO/ISH CVD risk prediction charts, the CVD risk is calculated based on individuals' age and risk factors such as blood pressure, lipid profile, diabetes, and smoking status and categorized into the following five groups: below 10% (low-risk group), between 10 and 19% (moderate-risk group), between 20 and 29% (high-risk group), between 30 and 39%, and above 40% (very high-risk group). As the individuals in the two latter groups are treated the same, in the IraPEN program, whoever has a CVD risk above 30% is categorized as the very high-risk group.
Therefore, considering what was mentioned earlier, all the Iranians aged older than 40 years who did not have CHD or stroke events before were eligible for this program. According to the recent census (2016), 31.16% of Iranians were older than 40 years (6 ). By adding individuals aged older than 30 years with the aforementioned risk factors, we can conclude that this program is going to screen at least 25 million people yearly.
The healthcare perspective and a 40-year time horizon were adopted for this analysis. As the analysis is a comparison between IraPEN (intervention) and status quo (no intervention) which both have the same Markov structure and transition probabilities, it is not expected that half cycle correction (HCC) approach makes any difference in ICER results; therefore, HCC was not applied to this analysis (7 (link)).
The hypothetical cohorts were used as a representative for individuals with low, moderate, high, and very high CVD risk profiles (Table 1). Progressively, a proportion of the cohort can go to the CHD state, who are the survivors of the first CHD event, or to the stroke state who are the survivors of the first stroke event. Those CHD and stroke events that were fatal moved to the death state. In general, the people in these two states are at a higher risk of dying from CHD or stroke, but they may die from any other causes like the normal population. Table 2 summarizes the assumptions of this analysis.

The data were exported from an online Kobo collect server to STATA version 14.1 for analysis. Numerical descriptive statistics were expressed by using median with Interquartile Range (IQR) after checking the data distribution with histogram and Shapiro–Wilk test for continuous variables whereas categorical variables were expressed by the frequency with percentage. The outcome of each participant was dichotomized into censored or event. The incomplete data were managed with the assumption of multiple imputations after ascertaining the missing data were completely at random. The Incidence Density Rate (IDR) of mortality was calculated for the entire follow-up period. Kaplan–Meier (KM) failure curve was used to estimate the median survival time and cumulative probability of death and the KM survival curve and the log-rank test were considered to test the presence of difference in the probability of death among the groups. Proportional hazard assumptions were checked both graphically and statistically using log (-log) plot and Schoenfeld residual test, respectively, and it was satisfied (p = 0.993). Multicollinearity was checked with variance inflation factor (VIF) and the mean was 1.49 which indicated that there was no significant multicollinearity. In addition, shared frailty was checked to see unobserved heterogeneity between hospitals and the p-value for the likelihood ratio test for theta was non-significant at p = 0.375 which showed the classical Cox regression model was the best-fitted model over the Cox frailty model for the sake of model parsimony. The Cox proportional hazard regression was used to explore the association between each independent variable with the outcome variable. The model's fitness was checked by using Cox–Snell residuals test and the hazard function follows 45˚ close to the baseline hazard which indicated that the model was well fitted. For the residual test, it was possible to conclude that the final model fit for data well. Both bivariable and multivariable Cox proportional hazard regression were used to identify predictor variables. Variables having a p-value < 0.2 in the bivariable analysis were candidates for the multivariable analysis and Adjusted Hazard Ratio (AHR) with 95% Confidence Intervals (CI) was computed to evaluate the strength of association and variables with a p-a value less than 0.05 were considered as statistically significant with the incidence of mortality among trauma patients.