**Transcript of Cyberseminar**

**VA Statisticians Association**

**Introduction on Statistical Methods for Analysis of Missing Data**

**Presented by Andrew Zhou, PhD**

**January 28, 2014**

*This is an unedited transcript of this session. As such, it may contain omissions or errors due to sound quality or misinterpretation. For clarification or verification of any points in the transcript, please refer to the audio version posted at www.hsrd.research.va.gov/cyberseminars/catalog-archive.cfm or contact andrew.zhou@va.gov*

Moderator:At this time I would like to introduce our speaker. We have Dr. Andrew Zhou joining us today. He is a professor in the Department of Biostatistics at The School of Public Health, adjunct professor in Department of Psychiatry and Behavioral Sciences at The School of Medicine at The University of Washington. He is also the Director in BH Research Center for the Biostatistics Unit at The HSR&D Center of Excellence at the VA Puget Sound Healthcare System. We are much honored to have Andrew joining us today and I will turn it over to you now.

Dr. Zhou:Thank you Molly. Welcome everyone to this cyber seminar organized by VASA, an introduction on how to analyze the data with Missing data. I just want to say a few words about VASA, which stands for VA Statisticians Association. 

On the screen the mission of VASA is:

· To promote and disseminate statistical methodological research relevant to VA studies;

· Facilitate communication and collaboration among VA-affiliated statisticians;

· Promote good statistical practice;

· Increase participation and visibility of VA statisticians;

· Increase participation of statisticians in VA merit review.

Moderator:Andrew, you can advance the slides using the downward facing arrow in the lower left hand corner of your screen.

Dr. Zhou:Thank you. For me to understand a little bit more about our audience I just have a very quick poll so you can click on what type of research you doing...applied statistic research work or health services work or clinical work and others. 

Moderator:Thank you. We do have answers streaming in. It looks like we are getting a good response rate from our audience. We will go ahead and go with some estimates now. It looks like we have around 19% that do applied statistics, around 42% that do health services research, around 26% doing clinical research and 11% identify as other. Thank you for those. I believe we have a couple more poll questions real quick.

Dr. Zhou:Also, I tried to do a very basic introduction for this course so I want to make sure our audience is suitable for this kind of introduction so just a very quick poll about the level of statistical knowledge you have.

Moderator:Thank you. Again, we are getting great response rates. We appreciate that from our audience. That helps us guide the level of talk. It looks like we have around 20% identifying as a beginner level of knowledge, around 60% saying intermediate and about 21% saying advanced so thank you for those responses.

Dr. Zhou:Thank you very much. I think for the audience who is advanced you may find this is easy but I hope you still learn from this introduction seminar. 

Moderator:Alright, a couple more questions before we get started.

Dr. Zhou:To help VASA to plan future cyber seminars, a quick poll is in future cyber seminars what type of category of statistical cyber seminar do you like to see or are you interested in attending. I just have very brief categories like study design, analyze causal inference, analyze observational data, analyze longitudinal data or correlate data or multilevel analysis and others.

Moderator:I understand people would like to select more than one of these. I am sorry; I should have made it where you can select all that apply but for now I guess just go with your top priority topic that you would like to see more on. It looks like we have around 8% requesting study design, about 22% asking for analysis observational data, about 26% analysis of longitudinal data, about 40% on multilevel analysis and about 3% on others. For those of you that selected others we will have a feedback survey at the end of the presentation in which you can write in topics that you would like to learn more about. We just have one more poll question and then we will get to the presentation.

Dr. Zhou:I think also to help VASA to plan our future work, we have a very quick poll to find out about the audience, in your own research, how many of them actually need statistical help and how many think you have adequate statistical support already?

Moderator:Thank you. It looks like it has been hovering right around 50/50. We are at 52% saying yes and about 47%-48% saying no. Thank you again to our attendees. We will go ahead and get back to the presentation.

Dr. Zhou:Thank you very much. Why do we care about the missing data? What is the problem in analysis when we have a missing value for some variable? I think has something to with how traditionally the statistic method is developed. Traditionally, the standard system has been developed to analyze rectangular data sets which means the row observation unit, columns of variables, and the entries of real values. The concern arises of what happens when some of those values are not observed? Most statistic software creates special codes for the missing value such as N/A or 999 and some statistic software excludes subjects with missing values with so called complete case analysis. Complete case analysis only valid in a very limited way and we are going to talk about that.

Which method we to analyze missing data depends on missing data patterns as well as missing mechanisms. I am going to first talk about missing data pattern of the data and then I am going to talk about mission data mechanism, which both are important for us to choose appropriate statistical methods. I will start with a simple missing data pattern called a univariate missing data pattern in this picture. In this table we have variable A and B and as you can see in this data set we have 5 observations and complete information on all 3 variables but we have 3 subjects who have missing information on the variable C which is noted by N/A. This is, of course, a simple missing data pattern. Let’s look at more complicated missing data patterns. One of the other commonly occurring missing data is in longitudinal data. It is called attrition or dropout. Attrition or dropout in longitudinal studies is very common because sometimes our longitudinal data involves long-term follow up or sometimes involves some difficulty with populations such as children. They may not stay on the study all the way through. The definition of attrition is as follows: if in a study the subject drops out before the end of the study or does not return then this situation will lead to so called volatile missing data pattern or the dropout. 

Let’s give a quick example of attrition. This is a multiclinical observation study on suspected cohort of primary care patients with clinical depression. This study invited in it depressive symptoms- mental and physical health for about 966 clinically depressed people from 6 large US clinics or cities. Those persons responded to several questionnaires that measured their physical mental health at the baseline, 6 weeks, 3 months and 9 months after baseline. After 9 months after baseline each patient was interviewed by a psychiatrist who determined whether the patient still suffered from clinical depression after 9 months. Here is the data. In this data you can see the majority of the patients have information on all 4 time points based on 6 months, 3 months and 9 months. However, there are 92 subjects who are missing 9 months data. They stayed in the study until 3 months and then dropped out. There were 27 subjects who stayed in the study until 6 weeks and after 6 weeks they dropped out of the study. There were 2 subjects who dropped out right after baseline. This is one example in which we have dropouts in the study. 

Another missing data pattern which is also patterned is called file matching missing data pattern. Suppose we are interested in 3 variables- A, B and C. However, in real data we have we do not have any subjects with observation of all the variables under study. In this case we do not have any subjects who have observation with all 3 variables but we do have information on paired or individual variables in the study. You can see that on this table there is no subject who has information on all 3 variables. They only had information on either paired variables or single variables. This is called a file matching missing data pattern. This has implications if we have this kind of data. For this kind of data, one of the implications is that if you have this kind of data then we may not be able to estimate the parameters relating to association between all 3 variables. For example, in the above data there is no information in the data about the partial correlation of the B and the C given A. Similarly, we do not have information about joint distribution of A, B and C. If you are interested in those kind of parameter associations then you have to make strong assumptions about the data in order to make estimations. If you have that kind of missing data you have to be very careful when you analyze your data.  You want to make sure the parameters you are interested in can be estimated from the data you have or you have to make some additional assumptions beyond the data to make inferences. That is very crucial. That is why it is important to know what type of missing data pattern you have. It will then help you determine what kind of parameters you have to estimate from the data you have or you have to make additional assumptions or collect additional data.  Then of course, the last one is the general missing pattern where there is non-monotone missing data that can be missing from anywhere. 

The next thing I want to talk about are missing data mechanisms. This is crucial in determining which methods we want to use. What is a missing data mechanism? Missing data mechanism is the reason for missing data. Why are we missing data? The missing data mechanism is supposed to tell us why we are missing data. That is why when we are conducting studies it is very important to record reasons for missing data even though those reasons may not be directly related to the outcome of our interested. You have to record it so it can help us determine what are the appropriate statistical methods. 

In order to define the missing data mechanism, I have to introduce a few simple notations. I want to introduce the vector of indicator variable R which identifies which variables are missing and which variables are observed. Suppose we have 2 variables, variable A and variable B and then for each subject we have an R vector which has 2 components. The first component indicates whether the first variable is missing or observed and the second component indicates whether the second variable is missing or observed. In this particular example for that subject R = (0,1). What that means is for that subject the first variable is missing and the second variable is observed. The R is just indicator to indicate which variable is missing and which variable is observed for individual patients. This R may have nothing to do with the outcome but this is additional information which will help us to determine the missing data mechanism. 

Before I discuss missing data mechanisms I want to introduce this important function which is sometimes ignored in the literature. What the assumption says is the missing indicator should hide 2 variables that are meaningful for analysis. I want to emphasize meaningful here. Sometimes we have missing values. It does not mean we have missing data problems. It sounds like a contradiction but you are going to see in the example some things have missing variables but it does not mean we have a missing data problem.

Let’s look at some examples to get this concept clear because this is a very important concept. This will help us if we use statistical methods to deal with missing data to handle your missing value in your study. Let’s look at the example on death. As you know, if you have long term follow up some patients will die in the study before you are able to collect information. The question is, how do you deal with those patients. Can those patients be considered missing data or not?  Suppose we have a randomized trial with 2 treatments and suppose our primary outcome of study is quality of life with health score Y measured 1 year after randomization to the treatment. Then suppose in this particular study some participants would die before you are able to collect the Y at one year. They died before 1 year during the study. The Y data is not available for those subjects because they died so we cannot interview them to ask those people who died about the quality of life of those patients. The issue here is do we consider those subjects missing data or not? Let’s do a simple poll here. Do you consider the patient who died before the end of the study as missing quality of life score?  You have three answers- yes, no or not sure. 

Moderator:Thank you. To our attendees, just click the circle next to your answer and that will submit your vote. It looks like we have about 40% of our audience responding so far. We will give people just a little bit more time to get those answers in. It looks like about 23% of our audience report yes, about 55% say no and about 21% are not sure. Thank you.

Dr. Zhou:Okay, that is very good. I think the answers will depend on who you talk to. This is just my personal opinion and I am going to tell you what I think. I think that I am glad to see that the majority of them agree with me. For those subjects who died before we were able to collect the information, it is not treated as missing data. Remember our assumption says you only consider your missing data when the missing data indicated to hide some true value which we do not observe and those values, even though we do not observe them are meaningful in analysis. If you die before we are able to collect information...suppose you die at 6 months. At one year it is hard to talk about quality of life for the people who already died. Maybe people can argue with me but I think from a statistic point of view it is hard to talk about the quality of life for dead people. I would not consider those people who died as missing data. I have to do it differently. I cannot just impugn them. I cannot use the methods we are going to talk about later on, the imputation method, to impugn the quality of life of the people who died. I think people have done that and they might have other options on this. From my point of view, if you die before we are able to measure that dying is special outcome. If he dies there is no time involved anymore after you die so it is not useful to consider those people as missing data.

Let’s look at another example. This is a little tricky. This example is looking at the long response to an opinion poll. Suppose we are interested in polling individuals about how they are going to vote in the future referendum on some education or an increased tax, for example, where available responses are yes, no or missing.  Let’s suppose we go out to ask people what they are going to vote in a future referendum and the answer we got from the questionnaire is yes, no or missing because some people did not fill out the questionnaire. I would like to hear your opinion. If you have this kind of data do you consider patients missing? Are those a true missing data problem? I would like to do a quick poll here.

Moderator:Thank you. Let me open that up real quick. It looks like the answers are streaming in. Great response rate. Thank you to everybody who is participating in our polls today. It looks like we have about 50% response rate. We will give people a little bit more time to get those answers in. We are at about 75% response rate. I am going to go ahead and close the poll now. It looks like 71% report yes, 17% say no and about 10% are not sure. Thank you to those who replied.

Dr. Zhou:Thank you for your part. As I said, this is a tricky question. Actually, the answer will be not sure. It depends on what are those people who did not give response. The individual who failed to respond the question might be due to 2 reasons. One is they just refused to reveal real answers or they might be the people who have no interested in voting. There are 2 types of people who have missing. One type is they are going to vote in the future referendum but they do not want to tell you what they want to vote and the other people are not going to the poll. They are not voting. You will see that the answer will depend. If you have those subjects who will be voting in the future referendums but do not give you answers then those are definitely a missing data problem. For the other subjects who will never vote in the future referendum are not considered missing data because it is not applicable for those subjections. In data analysis I think you have to distinguish the people who will not give you answers into the group who are going to vote in the future referendums and the group who will not vote. If you have that information then in your data analysis you should get rid of the people who are not voting in the future referendums because that is not a missing data problem. That is not relevant to the study of interest. 

You can see that even though the definition is very easy to say the missing data problem means that we have some true value which is meaningful we do not observe that is missing, this exists in the real world and we just do not observe it. In practice you have to think very hard when you do your own study whether the people who do not have answers are really missing data and if they are not you have to think of a different way to deal with that.  You cannot just use a missing data technique to impugn it. For example, the people that will never vote in the future referendums- I do not think it is a good idea to impugn what their response would be because they are not voting.

Now let’s talk about the missing data mechanisms. There are 3 missing data mechanisms depending on how the missing data indicator relates to the missing data and observed data. In order to define our 3 missing mechanisms I need to divide the data into 2 parts. Suppose Y contains all the data in our study sample and some of them might be missing. Since some of those study samples might be missing data I am going to devise a Y which will complete data if there is no missing data problem into 2 parts. One is Y obs represents all the observed data in the study sample and Y mis represents all the missing data in my study sample. I divide the information in my study sample Y into 2 parts: Y obs and Y mis and that determines what type of missing data mechanism we have. 

We are going to define 3 different missing data mechanisms. One is called missing completely at random (MCAR), the second is called missing at random (MAR) and the third one is missing not at random (MNAR). Let’s look at the definition of those three missing mechanisms.

First let’s talk about missing completely at random (MCAR). Data are called missing completely at random if missingness does not depend on the value of data Y regardless of the Y missing or not. Statistically speaking that means the distribution of R given Y only related to the distribution of R itself. That is a mathematical notation for all of Y. In other words, the distribution of missingness indicator has nothing to do with the data under study. This can happen but in most cases it may not happen. Consequences of this relationship means that the missingness indicator is conditionally independent of the study variable. Data on study variable regardless of missing or observed has nothing to do with mechanisms that generate the missing data. 

Let’s look at an example. Since the variable in the experiment involves random sampling, let’s say we do a random sampling 2 stage. We first draw a study sample randomly from the population and then I draw another random sample from my study population. If we do that then we have some missing data because not everybody in the study sample is selected into my other study sample. For that kind of missing data it is easy to deal with because since the complete data sample really is a random and representative sample of the study population we can just use complete data sample to do analysis. Only limitation or weakness of this approach is we may lose some efficiency because we are not using all study samples but you still get a valid statistical inference. If you are missing complete at random you do not have to worry too much about the missing data except efficiently. 

The next one is called missing at random. Missing at random is less restricted function of missing data mechanism than MCAR. Data is called missing at random when missingness depends only on observed components of your data, not on missing components Y mis. Statistically speaking the distribution of the missing indicator given all the data depends only on observed part of your data. The distribution of R depends only on Y obs which is observed data. I want to emphasize even though we call this missing at random it does not mean missing data is a simple random sample of all data values. The missing data does less than missing completely at random in a sense that the missing value can behave like a random sample of all of the values within subgroups defined by observed data rather than the entire data volume. 

Let’s look at an example. As we know in the VA, one of the variables is race and it is a very important variable in many studies. We know the race variable is missing substantially in the VA data particularly if you go to the VA administrative data site. Let’s suppose if we are able to assume your study is only related to the age of your veteran and we happen to have an age for all of the study subjects in my study. If that is the case then our missing data is missing at random. 

The third mechanism is called missing not at random. The mechanism is called missing not at random if the distribution of R depends on the missing value in Y. Strictly speaking, the missing not at random has two forms: One is missingness depends on the missing value itself so that is one possibility and the second is the missingness depends on the missing value itself but depends on some unobserved variables which we do not have. If you have those kinds of situations then you have not missing at random.

Let’s look at an example. Let’s continue with our VA example on race variables. Suppose the reason for missing race is related to race itself. Maybe due to his race sometimes they do not respond to the questionnaire or related to some unmajored covariates such as the socioeconomic variable which we do not have. Then the missing at random does not hold; we have a missing not at random problem. 

If you have a not missing at random problem it is much harder to deal with than missing at random problems. In this introduction talk I mostly focus on missing at random situations but if you have not missing at random there are still statistical methods available but more complicated and more beyond this introduction course.

Let’s look at some simulated data to show you what are the consequences if you have different types of missing data mechanism. Let’s consider a simple linear regression between two variables x and y. In this simulated data the x is always observed and y is missing for some subjects and they have different missing data mechanisms. Let’s assume that x and y follow this simple linear regression here. You can see that the beta coefficient in front of x is 1 so that is a true beta so we try to estimate that true beta. Here assuming both x and AR distribution has normal distribution I generate some data on x and y with sample size 300. This is my full data set, 300. Then I purposely get rid of y for some subjects depending on 3 different mechanisms. One is missing completely at random, one is missing at random and one is not missing at random. What happens? That is just simple linear regression that you can do. The first column are the results from analyzing 4 data sets, so that is the right data set. You can see the estimates are very close to truth and estimate the beta is 1.02. The true beta is 1 so it is very close. In the second column are the results I obtained by analyzing missing data when missing data follows missing completely at random. I simulated the data so I know what the true missing data mechanism is. For that data, you can see that the estimate on the beta is also very good. It is almost on bias but increased standard of error because in my data analysis I only have 148 subjects. The original subjects were 300. If you have missing at random that means the missing data mechanism actually depends on the missing data itself. You can that you have a little bit biased result so the estimated beta is not close to the true beta. Similarly, for the not missing at random you also have bias. This is the simplest example to show if you have missing at random or not missing at random, if you just get rid of the subject with missing value you might end up with biased results.

Next I want to talk about some of the methods. What are the methods available for us to deal with missing data? The first is complete data analysis. I already mentioned that is the simplest way. You get rid of all of the subjects who have missing variables and you use only subjects who have complete information on all the variables and then do analysis. The complete data analysis is actually a default on many statistical software’s. I already mentioned this is not a good way to do analysis. You can get biased results. As I mentioned before, if the missing completely at random assumption does not hold so if you are missing at random or not missing at random I already showed in the previous slide you might be biased results if you do a complete data analysis so this is not recommended. 

Another way you can modify the complete data analysis is called weighted complete data analysis. This strategy reweights samples to make it more representative. For example, if we know that the response rate is twice as likely among men as women then the data from each man in the sample could receive the weight of 2 in order to make the data more representative. This strategy is used a lot in sample surveys. This method usually requires the missing at random assumption.

The next approach is called the maximum likelihood approach. The fundamental idea behind the maximum likelihood approach is conveyed by its name: maximum likelihood. It means you find the values of the parameters that are most probable or most likely for the data that has actually been observed. Let’s have a formal definition here. Dente D obs as observed. The maximum likelihood I talk about here is under missing at random assumption. If you have not missing at random assumption you have to modify the maximum likelihood I talk about here. Let’s assume D obs is observed data which has probability density p D obs given theta. Theta is the unknown parameter we have to estimate. The likelihood function is equivalent or proportional to this density function but different from the density function. The density function considers the theta as your variable but the likelihood function considers the parameter as your variable so your theta is fixed. In the likelihood function the theta is unknown function which you want to maximize.

The maximum likelihood approach is to find value for theta which maximizes our likelihood function. The maximum principle says we should choose a value for theta that maximizes the likelihood function. But how you maximize likelihood function is tricky and sometimes not straightforward. 

So really there are three approaches to calculate maximum likelihood value. One is just to take the log of the like function and then you take derivative of this log like function and set the derivative equal to 0 and solve that equation. That is the first approach.

The second approach is maybe it is not very easy to solve that equation and then you use analytic approach like the Newton-Raphson or quasi-Newton algorithms. The third approach is the so called EM algorithm. I do not have time to get into detail. EM algorithm means expectation maximization algorithm. Sometimes they can help us to speed up the computation time.

Next I am going to talk about imputation based approach. There are two types of imputation based approaches: one is based on implicit modeling approach and the second is based on model based approach. Depending on what type of approach you take you might get different results. Implicit modeling approach means you do not have modeled the data explicitly just implicitly. 

For example, we have heard of the hot deck imputation and the cold deck imputation. The hot deck imputation means you replace the missing value from a value for similar responding units in the sample. The cold deck imputation means you replace missing data by something beyond your study sample, so from external sources. That can happen if you say I am going to use the data from other studies to do imputations. Model based approaches use a model for observed data and basing imputation from the draw for posterior distribution of missing data conditional on observed data under that particular model.

Let’s first talk about single imputation and then we are going to talk about multiple imputation briefly and then finally I am going to talk about software you can use to do imputations. Single imputation, as the name suggests, you are going to replace the missing value by some reasonably guessed value one time. Then you perform the analysis as if there is no missed data. After you replace the missing value with some value you guess, then you say I do not have missing data I am going to do the analysis.

 There are several approaches. One that is common is the mean imputation. You just replace each missing value with unconditional mean of the observed value for that particular variable. What that means is suppose you have age missing for one subject and then you have age variable for other subjects. You just calculate the value of age for those subjects who have the value and then use the mean of the subject with age variable to replace the subject with missing age variable. Of course, it has a lot of problems. You can argue that not everybody is the same in my study samples and how can you do that? 

The improved one is actually called a conditional mean. Conditional mean is that instead of replacing the missing variable by unconditional mean of all of the subjects with observed variables for that particular variable you do a conditional mean. You condition the subject. Let’s suppose you are missing age and the subject who has missing age has a socioeconomic status and other variables available, then you can do some similar subjects with that missing age subject and those observed variables and then do the conditional mean. 

Or you can do the regression imputation. You replace the missing value by predicting value from a regression of the missing item observed for the unit. That is called regression. 

Or, you can use a stochastic regression approach and instead of just using predicted value from your data you add some random noise to it. 

Another approach is called the near neighbor hot deck imputation. 

One of the limitations of regression is your predicted value may not be in the range of the value you have. To avoid that approach you use the hot deck imputing. That approach is called the nearest neighbor hot deck imputation. Let’s suppose we have a missing outcome for some subjects and we want to impute the missing outcome. We assume here all of the covariates observed. In order to do that we first have to define a matrix to measure the distance between the two subjects based on the value of observed covariates. Covariates could be multidimensional. They do have a mathematical matrix you can use to define the distance between the two vectors. Then we are going to choose imputed values that come from responding units close to the unit with the missing value. 

Let’s define that approach. Let x be the k covariate which are observable for all of the units. Here we assume all of the covariates are observed for the subject. Then I have a missing Y. Let d( i,j ) be the matrix between two covariates xi and xj of unit i and j. For unit i it is assumed that particular subject has the missing Y. I am going to impute the missing Y from that subject. From those subjects who had observed Y in addition their covariant is very close to the covariates of missing unit i. You can see the distance between i and j have to be less than some constant. That is called an imputation. 

Now I will give general comments about imputation. When you do the imputation you have to be very careful. First you have to conditionally observe variable to reduce bias due to response. Secondly, when you have more than one variable missing the best way to do is to impute the missing variables simultaneously, not individually because of multivariate imputations. Then we are going to talk about that it is better to do imputations from predictive distribution rather than means.

One of the problems with single imputations is are not counted for the fact that imputed value is not true value. There is uncertainty. How do you account for imputation uncertainty? That is called the multiple imputation that we are going to talk about.

What is a multiple imputation? Multiple imputation means you create multiple imputed value for each missing value. Let’s just say Y is missing.  You imputed the missing value multiple times. Then you are going to analyze and you end up with D complete data now. For each missing value you can have D imputed value and D complete data. Then you can analyze each of D complete data sets in the same way. It is very important you analyze the same way for each complete data. Then you combine the D set of estimates and standard deviation using Rubin’s rules. Suppose we have missing data here. The question marks are missing data. Based on imputation method whatever you choose you are going to impute each missing data M times. You can end up with M complete data, so 1, 2, 3 M complete data. Then you are going to analyze each data separately but you have to pay attention. When you analyze data you have to use the same methods to analyze data. You cannot say first I am going to use linear regression for the first data and then I am going to use nonparametric regression for the second data set. That is no good. You have to use the same technique for all the complete data.

After you analyze the data M times then you have estimates. You have M estimated beta, M estimated standard elevation and then you need to combine those estimates into one estimate using Rubin’s rule. 

Finally, I want to end my talk and leave some time for questions about software. There are many, many softwares you can use for multiple imputations so I start with SAS. If you use SAS there is a procedure in SAS called a PROC MI. MI stands for multiple imputations. It allows you to impute the data multiple times. There is a limitation for this. It can only handle normal diffusion. It can only handle continuous data. If you have not normal diffusion they may have problem. There is a lot of software in SAS which is still used commonly called IVEware in SAS. That will give you more options for multiple imputations. If you do not use that but use STATA, also in STATA there is a package for multiple imputations. That package is called ice. ICE stands for imputation chained equations. This is software that allows you to perform multiple imputations by changing equations. You can also use SPSS. It is a command for multiple imputations. That procedure is part of the missing values module which allows you to do multiple imputations by chained equations if you use SPSS. 

Finally there is R and R has many, many choices. I will just list a few of them. Many, many R packages allow you to perform multiple imputations. Amelia allows you to perform multiple imputations based on multivariate model or you can have BaBooN which is the multiple imputation by chained equations. You have cat which is multiple imputation for categorical data or you have multiple imputation procedures to perform Kaplan-Meier in survival analysis. If you have survival data and you want to do multiple imputations there is a software available in ours but maybe not available in other software. And you have mice, which is also multiple imputations by R which uses chained equations to perform multiple imputations.

That is actually the end of my talk. I will leave some time to answer some questions you have from the audience. This is only an introduction on analyzing missing data. I have not gotten into details about each approach. How do you actually perform maximum likelihood as an explanation but u think this will give you some idea about the available methods for performing multiple imputations. So we are going to open up for questions.

Moderator:Thank you so much. We do have some good questions that have come in. The first one: How does one join VASA?

Dr. Zhou:They can send me an email. My email is Andrew.zhou@va.gov. 

Unidentified Female: Great. Thank you. The next question that came in...oh we have a comment here. “Analysis of data with missing data requires the ability to handle values that are not real values. For example, nominal or ordinal.” Thank you for that comment.

Dr. Zhou:Yes, thank you for that comment.

Moderator:The next one is, “What if the death was related to the item being studied? Like taking a drug or not taking a drug?”

Dr. Zhou:Okay. What is the question?

Moderator:They are asking about when you were talking about if somebody died should that be counted as missing data and they are wondering if that had to do with taking the medication or not taking a medication related to the study does that count?

Dr. Zhou:No. Actually in most situations whether you die or not will depend on if you are taking medication or not. Even if that is the case, if you die before you are able to collect the quality of life I would still consider that not missing data. Of course, it depends on your interest of focus. If your focus is on death itself for quality of life variable.

Moderator:Great. Thank you for that reply. This is a question that came regarding slide #19. Let’s see if we can get back up to there. “For future reference it would be better to allow voters to choose “depends” as an answer for slide #19 as opposed to just yes or no.”

Dr. Zhou:Yes, I agree. That actually is a very interesting question. When we designed the study maybe we should also consider missing data when we design the study. That is exactly right. When you design a study when you think about potential for missing data and the potential for those people who never vote to have missing data, then maybe you should have a category to say are you going to vote or not first and then from there say, “If you are going to vote then we are going to ask if you are going to vote for the referendum or not.” That is actually a very good point. When we design the study the factors we consider when we design a questionnaire, we not only consider the study itself but also consider how we can analyze data to avoid missing data when we design the study. Just like this simple example that said if you design the study maybe an option choice to give to participants is not just yes or no; maybe it should have other questions asking whether they are going to vote or not first and then if they say yes you ask them whether they are going to vote for the referendum yes or no.

Moderator:Thank you for that reply. The next question, I believe, came in during this slide. Does the Y represent outcome only? For example, dependent variables?

Dr. Zhou:No. The Y represents all of the data you are studying which includes both outcome and covariant. I just denoted Y. I cannot misuse the notation here.

Moderator:Thank you. I just want to apologize to our attendees. We did lose the audio for just a second but it should be back up and running now. The next question is, “How to combine modeling results of multiple imputed categorical variables?”

Dr. Zhou:How to combine multiple imputed data?

Moderator:Yes.

Dr. Zhou:Well, the same way. Regardless if you have categorical missing data or continuing missing data to impute it use the method such as mice that you have completed. After you have complete data you are going to use a complete data analysis like regression or the T test or the Chi squared test. Suppose your goal is linear regression. After you have complete data you perform linear regression. Then you have the estimator for the beta in the linear regression.  You have multiple estimated beta and also multiple estimated standard divisions. Now you are going to use Rubin’s Rule which I did not talk about here but if you are interested I will be happy to give you reference. You use Rubin’s Rule to allow you to combine multiple estimated point estimated and standard deviation into a single estimator with a single standard deviation. After you have a single estimator and single standard deviation you can perform the P value and [inaud.]

Moderator:Thank you. The next question is, “Can you kindly explain missing value and missing data?”

Dr. Zhou:Okay. Missing value means you do not observe something in your study. Missing data means you have a missing value but the missing value has a meaning and I call that missing data. For the death example, for people who died before you are able to measure the quality of life, even though in the study quality of life for the dead people has missing value because you do not have that value collected they may not consider it missing data because the underlying value does not have any meaning because the people already died. Similarly for the voting example, for those people who never vote they are going to give you missing value in the questionnaire because they will not fill it so when you code it you are going to code a missing value there and say, “I do not have any information on those people.” I will give you N/A for example. For the missing value for the voting example it has no meaning because those people never vote. There is no clear meaning for us to consider the missing data. This is the definition I use in this talk to distinguish from missing value and missing data.

Moderator:Thank you for that reply. We have lots of great pending questions. “What are some good and recent references on your basic techniques?”

Dr. Zhou:Basic technique...We have a book in press. Hopefully it will come out in a few months by Whiting & Sons. The title of the book is called *Analysis Applied Statistical Methods for Missing Data.*There are several books available already. If you send an email I can send you some of the names of the textbooks for the basic techniques to deal with missing data.

Moderator:Thank you. The next question, “Should we first impute the missing covariates and then input missing outcomes?”

Dr. Zhou:No, I think you should impute simultaneously. Just like my Y here. Y could go for both missing outcome and missing covariate. I think that if you can, you want to impute in the same model but you can give different models for the outcome and the covariate. If you can, I think it is better to impute simultaneously so you can preserve the possible correlation between outcome and covariant because if you impute separately you may lose that correlation between outcome and covariant.

Moderator:Thank you for that reply. “In a longitudinal study where follow up is a sample of the baseline, do you weight Y based on baseline results or are non observations due to not being a follow up sample treated as missing data?”

Dr. Zhou:The baseline covariants are variable, yes. I would use those variables. Could you repeat the question?

Moderator:Yes, one second. “In a longitudinal study where follow up is a sample of the baseline, do you weight Y based on baseline results or are non observations treated as missing?”

Dr. Zhou:I would weight it using baseline observation. It would also depend on the missing data mechanism. If the missing data mechanism only depends on baseline then you can use baseline variable to do the weighting but if the missing data mechanism depends on other variables beyond the baseline then you cannot just use the baseline variable to do your weighting. 

Moderator:Thank you for that reply. The next question, “When using multiple imputation, are there any rules of thumb for determining the number of imputations per data set?”

Dr. Zhou:That is a good question. I think in the textbook they say 5 but I do not think that is a Golden Rule. The number of multiple imputations depends not on only the missing data proportion but also missing data information which I do not have time to talk about. There are a lot of concepts for missing data information. In other words, you can have a lot of missing data on some variables but maybe those missing data people have not much information provided to the primary interest you are interested in. There is a concept called missing data information. The missing information rate depends on the proportion missing as well as the primary interest what model you use. I think I do see in the literature they recommend 5. However, I think you have to be careful. If you believe the variable you are missing is a very, very important variable in your study then you may want to impute more than 5.

Moderator:Thank you for that reply. The next question is also regarding slide #19. “Does it matter why they are missing? They are still missing, especially if we have other data on them?”

Dr. Zhou:It does matter. Depending on how they are missing we have different statistical methods. Maybe the reason for that is in statistical literature we develop methods according to the missing data mechanism. If you have a missing at random, then we have a method. If you don’t have a missing at random methods, you have to use different methods and that is the reason to determine the missing data mechanism. In statistical literature, a statistical method is classified according to the missing data mechanism.

Moderator:Thank you. We do have some more pending questions. Are you able to stay on Andrew?

Dr. Zhou:Yes, for a few minutes.

Moderator:Okay, thank you. “What percent of data needs to be missing before consideration imputation?”

Dr. Zhou:That is a good question but I do not think there is one answer to it. I saw the literature say 5%. Like I said, once again, I think the safe way to do it if you are missing data is always do imputation. Then I think what you are supposed to do in this case is two analyses. Suppose you have only a very small percent missing like 1% or 2%. First do a complete data analysis and then you do an imputation approach and if the two analyses give you the same results you are probably confidant that complete data analysis makes some sense. It is that easy to explain. I would recommend if do have missing data, unless the missing data is very very small like just 1 or 2, I would recommend trying to do imputation approach. 

Moderator:Thank you. “What method or software would you recommend for multiple imputation of repeated measures data?”

Dr. Zhou:I would recommend software in R called pan, meaning panel data. It deals with panel data and allows you to deal with longitudinal data.

Moderator:Thank you. When you answered the questions for slide #19 we had actually lost audio so the person is asking again, “For future reference would it be better to allow voters to choose “depends” as an answer for the slide?”

Dr. Zhou:For that one I mentioned that I would suggest you add one more question in the questionnaire to ask whether the participant will vote in the referendum. That will solve this problem. If they say no they are not going to vote they will not have to answer any questions anymore. If they say yes they are going to vote in the future referendums you can ask their opinion about it.

Unknown Female:Thank you for that clarification. The next question, “Does including an outcome variable Y in a multiple imputation procedure in which you are trying to estimate missing values of a predictor variable X present a circularity problem if you then try to run an analysis to evaluate the relation between the predictors and outcome?”

Dr. Zhou:No, because you are doing two different analyses. Remember, really you put Y in and try to predict the missing data mechanism. At the beginning I introduced one additional variable R. R is different from Y. Think about you have multivaried analysis in this case. You have 2 outcome variable Y and you also have R. R is also your outcome. It is okay to use Y to predict missing data mechanism. From missing data mechanism then you do analysis on Y so it is not circular. Of course, if you estimate missing data mechanism and use missing data mechanism in your analysis when you calculate the variants you have to take that into consideration.

Unknown Female:Thank you. We just have one question remaining, “Some consider ML superior to MI approach. In what situations would you prefer ML or MI?”

Dr. Zhou:I think that ML is parametric approach. MI can be a non-parametric. It depends on whether you believe the parametric model you propose for the data is correct or not. If they do, I think ML definitely is the preferred approach and most efficient. They have to have some parametric models. The [inaud.] mutations is more flexible. You could have some semiparametric or non-parametric approach to use. 

Unidentified Female: Thank you. I want to thank you very much Dr. Zhou for presenting for us today. It was an excellent presentation. I want to also thank your attendees for joining us. I am going to ask our attendees to hold on for just one second. I am going to put up our feedback survey and ask for your feedback for this presentation. Once again, thank you very much Andrew and have a very nice day.

Dr. Zhou:Thank you.

 [End of audio]

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