BIS Seminar: A General Framework for Quantile Estimation with Incomplete Data

October 14, 2020

Linglong Kong

Department of Mathematical and Statistical Sciences

University of Alberta

October 13, 2020

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ID: 5741
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00:00- Hi, everyone.
00:02Welcome to the departmental seminar of
00:04the Departmental Biostatistics, Yale University.
00:09I'm pleased to introduce you Linglong Kong.
00:12He was associate professor of the Department of Mathematical
00:16and Statistical Sciences at the University of Alberta.
00:20He's research interests are on, and correct me if I'm wrong,
00:24on functional and neuro imaging data analysis,
00:27statistical machine learning,
00:29and robost statistics and quantile regression.
00:32So today, he is gonna talk about his work on
00:35general framework for quantile estimation
00:38with incomplete data.
00:40Thank you, Linglong. And whenever you're ready.
00:44- Thank you Laura for the introduction.
00:47And also thanks Professor John for the invitation.
00:52I'm very happy to be here, although it's way too early.
00:57So today I'm going to talk about general framework for
01:01quantile estimation with incomplete data.
01:13So, this is a joint work with Peisong from
01:20University of Michigan and Jiwei from
01:23University of Wisconsin-Madison, and Xingeai.
01:27And we started this work when at the second year
01:33when I started my position at the University of Alberta.
01:37I know Peisong a long time ago before he was a student,
01:44and at that time he just started his position as
01:48assistant professor at the University of Waterloo.
01:51And I invited him to visit me and afterwards,
01:56he invited me to visit him.
01:58And we feel like we visited each other already,
02:02we should get something done.
02:04But I remember that I've known where he stayed in his office
02:11at the University of Waterloo and thinking about
02:15what do we have to do together.
02:17And eventually we thought, "Okay, what I'm good at
02:21and while all my research area is quantile regression.
02:24And what is Peisong good at?
02:27One of the research area of Peisong is missing the data."
02:31So we said maybe we can put them together,
02:34then we are write a couple of formula on the paper.
02:41Then we feel like, "Okay, we get a copy already."
02:45Then we went to have a dinner.
02:48And then one year later Peisong send me like
02:52two pages to trap, said maybe we should continue it.
02:57And that's the first scenario in this topic,
03:03I'm gonna talk about.
03:04And then another half year, I sent him my feedback.
03:12I said, "Why don't we make it more general,
03:15make it a framework?"
03:17So this semester we're going to be able to apply
03:20to honor other scenarios.
03:22And then we we both feel it's good idea,
03:26then we started working on it.
03:27At that time, Jiwei was posed to at a University of Waterloo
03:33and Xingeai where my post are.
03:35So, we thought together and started a project.
03:38Eventually, I wound a project that I'm kind of proud of.
03:47So, what's missing data?
03:49The missing data arise in almost all
03:52serious statistical analysis.
03:56Missing on values are representative of the
04:03messiness of real world.
04:05Why we would have missing a missing value,
04:08it could be all kinds of reason.
04:12For example, it may be due to social or natural process.
04:17Like for example, a student get a graduate,
04:20get a job out in clinical trial, people get died, and so on.
04:26And also could happen that you survey.
04:29For example, in certain question asked,
04:32only asked respondent answer yes,
04:35to continue to answer certain questions.
04:38Or maybe it's the intention missing
04:41as a part of a data collection process.
04:45Or some other scenario including random data collection
04:48issues respondent refusal or non-response.
04:56So, mathematically how we categorize these kind of missing,
05:01and here is the three scenario.
05:05Now, first scenario we call it missing completely at random.
05:10What does that mean?
05:11That means the missingness is nothing to do with the
05:15person being studied.
05:17They're just completely got missing,
05:19it's nothing related to any feature of this person.
05:23The second scenario is missing at random.
05:26Missing is to do with the person, but can be predicted
05:30from other information about the person.
05:34Like either a certain scenario need these project,
05:39the missingness maybe predictive from some
05:43auxiliary verbals auxiliary information.
05:48The third one is a very hard one, is missing not at random.
05:55The missingness depends on observed the information
05:59and sometime even the response itself.
06:05So, the missingness is specifically related to
06:08what is missing.
06:09For example, a person to not attend a drug test
06:13because the person took drugs the night before.
06:17And therefore the second day,
06:18he couldn't make to the drug test.
06:20Couldn't get to that drug test result.
06:23These are three missing mechanism.
06:30How do we handle those missing data?
06:33There are many strategies.
06:35For example, the first one would be,
06:37well, let's try to get the meeting data.
06:40That would be great.
06:42But in reality, that's usually impossible.
06:48But the second is, well, as we have incomplete cases,
06:52let's just discard.
06:57Just analyze the complete case, right?
07:02But these could cause some other problems.
07:05We will talk about it.
07:07And the third one is we replace missing data
07:12by some conservative estimation.
07:14For example, using sample mean, sample median, and so on.
07:20The first one is we are trying to estimate the missing data
07:25from other data on the person.
07:27We use on sort of more sophisticated method to impute.
07:37Now in particular, mathematically speaking,
07:43the strategy we are using today do to deal
07:46with missing data,
07:48the first one is a complete case analysis.
07:51These are very simple, okay?
07:52We just analyze compete case, okay?
07:56And we only analyze in consideration that individuals with
08:01no missing data.
08:05Sometimes it can provide good result,
08:07but the estimation obtained from this complete case analysis
08:12maybe biased if they excluded individuals are systematically
08:18different from those included.
08:20So hence, if the complete case would be a good
08:24representation of those missing case,
08:28then this method would it be fine.
08:34Otherwise, if the complete case is quite different from
08:38those we miss, then all result can be biased.
08:44And then there's inverse probability weighting method IPW.
08:50This is a commonly use method to correct the bias from a
08:53complete case analysis.
08:56What does that mean?
08:57It means, okay, we give each complete case a weight.
09:03This weight is the inverse of the probability of
09:07being a complete case.
09:12Well, this can also cause some bias
09:16if this IPW relies on the data distribution.
09:25The first strategy is more sophisticated to do
09:29these multiple imputation.
09:31It's quite common method,
09:32especially nowadays in genetic study.
09:35How do we do multiple imputation?
09:39We create multiple sets of imputation for
09:44the missing values, using imputation process
09:48with a random component.
09:51Now, we have an full data set.
09:54Then we analyze each data set.
09:59Those full data set can be a little bit different.
10:02Can be slightly different because the randomness of
10:08the imputation process.
10:11Anyway, analyze those data set, complete the data set,
10:14and then we get all set of parameter estimates.
10:17Then we can combine those result.
10:20We can combine this result,
10:22and we hopefully we get a better result.
10:26The multiple imputation sometimes works quite well,
10:31but only if the missing data can be ignored.
10:36And also, we have a good imputation models.
10:39And while it depends on the nature of the data,
10:41the auto mind depends on what kind of imputation model
10:45we are going to use.
10:51Now, that's how we deal with missing data,
10:56the strategy we happen to use to deal with missing data.
11:01But let's matched them together in terms of missing data.
11:06How we use these meeting dates age to deal with
11:11different missing mechanism.
11:14For example, if the data is missing complete at random,
11:19now in this case, the complete case analysis is quite good.
11:25Multiple imputation or any other imputation methods
11:29is also okay.
11:31Is also valid.
11:32So, this missing complete at random is
11:36the easiest case to deal with.
11:40What if data is missing at random?
11:43Then in this case, some complete case analysis are valid
11:51and multiple imputation nearly is okay too,
11:56if the bias is negligible.
12:00Now in a certain case,
12:02if the data is missing not at random,
12:05then we have to model the missingness explicitly.
12:11We need jointly modeling the response.
12:15We need jointly model the response,
12:17and also the missingness.
12:22In practice of course,
12:23we try to assume missing and random whenever it's possible
12:28and try to avoid to deal with
12:32missing not at a random situation.
12:34But the reality, it's not anything that we can control.
12:41Sometime we have data always missing not either random.
12:45Think in that case center or there is one special issue
12:53dedicated to missing data, not at a random situation.
13:02Now, we have different strategies.
13:04And that they state different strategies
13:07have different advantage and disadvantage.
13:12For example, multiple imputation is generally more efficient
13:17than IPW, but it's more complex.
13:23And the imputation and IPW approach
13:28require to model the data distribution
13:32and the missingness probability, respectively.
13:35Imputation, we need to model data distribution.
13:39IPW, we need model the missingness probability.
13:45And also, for all kinds of strategy,
13:48we would have have good property,
13:52only if the corresponding model is correctly specified.
13:59Most existing method are vulnerable to
14:03these model misspecifications.
14:06Of course can use nonparametric method to reduce the risk
14:11of model misspecification, but it's often impractical
14:16due to the curse of dimensionality.
14:21So now, how do we deal with this model misspecification?
14:27We have some method available.
14:30For example, we can use a double robust method.
14:37In particular, in double robust method,
14:40we have this augmented IPW.
14:44We are not only model the missingness probability,
14:49but also the distribution.
14:52Why is double robust?
14:54Because the result would be confusing
14:58if the model is correct.
15:02If the way we model missingness probability
15:07or the way we model the distribution is correct,
15:12then we would get consistent result.
15:14And that's why it's called double robust.
15:18Well, now that we are not satisfied with double robust,
15:22what about we can a multiple guarantee?
15:25So, we have these multiple robust.
15:27This is a proposal by Peisong.
15:33And they multiple robust method is proposed to account for
15:38multiple models for missingness probability
15:42and the distribution.
15:45In double robust, we can only one model for missingness
15:48probability and one model for data distribution.
15:51Well, for multiple robust,
15:54we get multiple models to model missingness probability,
15:59and we can have multiple models to model distribution.
16:05The good thing is the estimation result will be consistent
16:11if either one or the model is correct.
16:19Now, let's look at those crushing mathematically.
16:26So, we are looking at missing at random.
16:29We assume on the observed data are ID.
16:34So we have data R, RY XT.
16:38R, we use it to missingness, and the IPW estimator,
16:48essentially we are trying to solve these equation.
16:52And here, these π is the probability,
16:58although makes complete case.
17:01And IPW is consistent,
17:03only if this πX is correctly specified.
17:08And then, then from the equation,
17:10we can get consistent estimate of those
17:13permit we are interested in.
17:17This is IPW. The other one is imputation.
17:23For imputation, we need model that take distribution.
17:28And here we have on the model of a f(Y|X)
17:36And as you can see,
17:37we have our imputation for those missing data.
17:44This imputation is consistent,
17:47only if this state distribution is correctly modeled,
17:52this f(Y|X) is correctly modeled.
17:58Now for these augmented inverse probability waited method,
18:05we actually combined these two together.
18:11We had the first part from IPW,
18:14second part from implication.
18:17So the estimation result would be consistent
18:23if either this model for missingness probability
18:28or the model for data distribution is correctly specified.
18:35Well, for multiple robust method,
18:38they have a serious model for missingness probability
18:44and a serious model for data distribution.
18:49And all result would be consistent,
18:53if any one model is correctly specified.
19:01Well, this is something
19:03I just get a quick review about this missing data.
19:07Like I said, this is the part Peisong is
19:12one of the Peisong research area.
19:14For me, my research area is quantile regression.
19:18So, internal quantile regression at that time
19:23we were thinking, "Okay, those methods,
19:26these IPW, AIPW or double robust method,
19:32multiple robust method, had been quite well studied
19:35for when we model the conditional mean.
19:39Therefore, condition of quantile, there are not
19:41a lot of methods available.
19:44Why we care about the quantile?
19:46A quantile not only provide a central feature
19:49of the distribution, but also care about the tail behavior.
19:57And also under very mild conditions,
20:01the quantile function can uniquely determine
20:05the underlying distribution.
20:07So, there are a lot of advantages to model the quantiles.
20:13Then, we decided to study these missingness
20:18in quantile estimation.
20:21In particular, we proposed a general framework
20:23for quantile estimation with missing data.
20:30So, our proposed model, these kind of framework,
20:35can do a lot of estimation for
20:38missingness in quantile estimation.
20:43But in this paper,
20:46we particularly applied all proposed method,
20:50these three scenario.
20:52Okay, three commonly encountered situation.
20:56The first one we trying to estimate
21:01the marginal quantile of response.
21:04This response get some missingness.
21:09Well, there are fully observed covariates.
21:13That's the first scenario, response gets some missingness
21:16while the corresponding covariates get fully observed.
21:20The second scenario, we are looking at
21:23the conditional quantile of a fully observed response.
21:28In this scenario, we look at
21:31there are some covariates are partialy available.
21:36So, we have some missingness for covariates.
21:39And then the third scenario, we are still looking at
21:43the conditional quantile of a response.
21:47And in this case, the response gets some missingness
21:52and we have fully observed covariates
21:55and also extra auxiliary variable.
22:02Now, let's look at the first situation.
22:07We want to estimate the marginal quantile.
22:10In this scenario, we have the response gets some missingness
22:18and we have the covariates fully observed.
22:22Now, let m to be the number of subjects with
22:26data completely observed.
22:30Then our method consists of the following five steps.
22:38The first step, we calculate this α or estimate to this α.
22:45This α isn't related to the missingness probability, okay?
22:52The way we estimate this, is by maximizing
22:57the binomial likelihood.
23:01So, the first step we estimate the α,
23:03and then we get estimate of the missingness probability.
23:10Okay?
23:11The second step, we calculate gamma.
23:16This gamma is related to this data distribution.
23:21So, we maximize this data distribution.
23:25This gamma is a parameter related to the distribution.
23:33And then the third step is we can
23:39sort of preliminary estimate of the quantile
23:43or the marginal quantile through these imputation process,
23:51by solving this equation.
23:55And as you can see this is quite close to the AIPW scenario.
24:05Okay?
24:06And in this equation, this five is the score function
24:13of quantile lost function.
24:17This prosaic is r - i(r<0).
24:23This is the generalized derivative
24:28of quantile lost function, okay?
24:34Here, this one can not be exact zero.
24:39The reason this phosaica is a non-smooth function.
24:46and it sometime it won't be exact here.
24:53Basically the first step, okay?
24:57Now, we have a preliminary estimator
25:01of the marginal quantile.
25:03The first step is the case that of method
25:08is where the multiple robustness is coming from.
25:14Now, we calculates weights for the complete case.
25:19In total, do we have m complete case.
25:21For each case, we calculate the weight.
25:24As you can see, the weight is determined by three parts.
25:32The first part is related to this alpha,
25:36which is related to the missing probability, okay?
25:40Missing probability.
25:43The second part is related to this gamma.
25:47This is related to the data distribution.
25:52The third part is related to this cube.
25:56This preliminary estimate of these marginal quantile,
26:02which is related to this self step.
26:07As you can see from the first three step,
26:10we are trying to get ready for this,
26:14to get the estimate for the weight for the complete case,
26:18for this complete case.
26:21And also, we have our parameter,
26:23though is obtained through
26:27minimizing these equation, through minimizing this equation.
26:33Now, after we calculate the weight
26:36we get off final estimate of our multiple robust estimate
26:42by solving the following with estimated equation.
26:50This wi is the width.
26:52We estimate it from the first four steps.
26:58And this posy is a score function of quantile loss, okay?
27:06Now, you may get wondering on what's going on
27:10with these five steps.
27:14And let me try to explain it one by one, okay?
27:20In the first step, we get the estimate of alpha, okay?
27:24We get the estimate of alpha.
27:28In sense trying to model they missingness probability, okay?
27:33Missingness probability.
27:35And of course, this missingness probability is consistent
27:41only if this model is correctly specified, okay?
27:45So in the first step, we actually have multiple models
27:49to model the missingness probability.
27:52And you need a hope at least a one model is correct.
27:57Now, in the other case, the missingness probability
28:00will not be correctly specified.
28:05Well, in the second step, we only estimate gamma.
28:09We are trying to model the data distribution
28:14and we have models for the data distribution.
28:20And then the third step,
28:22we are sort of doing some imputation as made
28:26of these marginal quantile.
28:32And these marginal quantile will be correctly estimated,
28:42if those data distribution is correctly specified.
28:50Now for the key staff,
28:53(coughs)
28:54Excuse me.
28:55The step four is typical formulation of
28:59an empirical likelihood program.
29:03I will getting back to this in the next slide,
29:08why it's a empirical likelihood program.
29:12And this is a key contribution of methodology.
29:18Now, in step five, we have the structure of IPW, okay?
29:23For complete case, we have weight to correctify, okay?
29:32And do this weight actually, is coming from two parts.
29:35And one part is from the missingness probability.
29:41The other part is from the data distribution.
29:45Now, the weight actually does not distinguish
29:48the missingness probability and the data distribution.
29:54The way it treats them equally.
29:59And another note I want to say is step two and four
30:03are based on the complete case only.
30:12Now, let's look at step four.
30:15Okay? Let's look at step four.
30:18In step four, we saw assumption are missing at random.
30:26It's easy to verify this, okay?
30:29Like wx, which is the inverse of the missingness probability
30:34times b(X) - E{b(X)}| R-1 = 0, okay?
30:43And in thus case, we can let b(X) to be the score function
30:48of quantile lost function.
30:52And these probability are conditional estimation
30:56and the conditional probability under this density.
31:01And because of this, okay?
31:06We can easily write a sample case, a sample scenario.
31:14So, the scenario is like this.
31:16All the weight is inactive.
31:19Some weight is one,
31:22and this is the estimating equation part,
31:25estimation equation part.
31:29As you can see,
31:30this is a typical empirical likelihood scenario.
31:40So, this is a typical formulation for empirical likelihood.
31:47And the solution actually can be even as in all formula,
31:55our previous, can be given by this one, okay?
32:02The weight can be determined by this.
32:05And though hard, can be estimated by solving this equation.
32:16Okay?
32:19So, that's all key steps for this methodology, okay?
32:29This actually, is the formula we first written down
32:35on the paper.
32:36And then we thought, "Okay, this might also be able
32:40to be applied to the other scenario."
32:44Indeed it can be applied in other scenarios.
32:48For example, in this quantile regression
32:52with missing covariates.
32:55In this scenario, all parameter of interest is β0.
33:01This β0 is coming from these linear regression.
33:05We want to estimate this β0.
33:10And all covariates had two paths, X1 and X2.
33:17This X1 path is always observed,
33:22while this X2 may have some missing.
33:27So, the observed data.
33:31And I need copies of this format.
33:33This missingness response completely observed covariates
33:43and some covariates are missing,
33:45some covariates are observed, okay?
33:49So, in this setting, we want to estimate β0,
33:55as in previous scenario.
33:59We have two sets of models, okay?
34:02One set model is for π, the missing probability.
34:08And the other set of model is for data distribution.
34:15Here the distribution is related to X2,
34:19given the condition of the response
34:21and completely of the X1.
34:27Now, as previous, we have five steps.
34:35Step one and step two are same as in case one.
34:40And in step one, we estimate in the missing probability.
34:45In step two, we estimate the data distribution.
34:53And then in step three,
34:55we get preliminary imputation estimate pf β0
35:03by solving this seemed a very complicated equation.
35:09And here there's Xl, which had two parts,
35:17the complete the case and on the missing part.
35:21The missing part is random drawn
35:24from this data distribution.
35:28We estimate from step two.
35:32And then the step four, okay?
35:35The key is that the empirical likelihood part
35:39where we used to compute to the weight.
35:46And these weights that I had, is for complete case.
35:50And at previous, this weight depends on three parts.
35:59One is missing probability, α1 is the distribution.
36:05Gamma previous, it depend on the preliminary as estimate
36:09of margin quantile.
36:11Now, it's related to the preliminary estimate of
36:18linear quantile coefficient β.
36:22Okay?
36:23After we estimate these weight WI,
36:27then we can go to the estimating equation part, okay?
36:35Let's say five steps. Let's say five steps.
36:40As you can see you, step one, step two, step three,
36:44is all preexisting method we adapt trying to estimate
36:56the missing probability, the data distribution,
37:02and also impute to get a preliminary estimate
37:05of the parameter we are increasing.
37:08And then from all these,
37:10we pull all this information together to get
37:12a good weight for the compete case.
37:18And then the using this empirical likelihood method
37:25and then we adjust this complete case with the
37:31estimated weight to get a final estimate,
37:34to get the final multiple robust estimate.
37:41Now the case three, okay?
37:45In the case three, the parameter we are interested
37:49is still β0.
37:51This linear quantile regression are here.
37:55The scenario is the full-data vector is (Y, X).
38:02In this scenario, Y is missing and random, okay?
38:07Of course the simple complete a case analysis
38:10where lead to a consistent estimate,
38:14but it doesn't mean it will be optimal.
38:18Here we are trying to get a more complete educated
38:21but still very practical method.
38:30We are having some auxiliary variable.
38:33As this auxiliary variable,
38:36usually not the main study interest,
38:40and thus do not enter the quantile regression model.
38:43However, we can use it to help us to explain
38:48the missingness mechanism
38:51and to help us to build a more plausible model
38:55for the conditional distribution of Y.
39:00Now, here is the observed data.
39:06So, we now have an ID copies of these R, RY,
39:12this Y gets a missing, X is completely observed,
39:19and we have got auxiliary variable S.
39:23We have this missing and random scenario.
39:27We use π(X, S) to denote the probability,
39:34and we use f(Y| X, S) to denote conditional density.
39:40As previous, we have multiple models
39:43for missing probability,
39:46and we have multiple models for data distribution.
39:56And then once again, we have the all five steps.
40:00The first step, we modeled the missing probability.
40:05And here we have this additional auxiliary variable.
40:10The second step, we model the data distribution.
40:14Again, we have this auxiliary variable.
40:17And then step three,
40:18we get a preliminary estimate on
40:21using this imputation method.
40:24We have our preliminary estimate of the parameter
40:28we are interested in,
40:30which is a linear regression coefficient here.
40:36And then after the preparation of step one,
40:39step two, and step three,
40:41we finally be able to estimate our weight, okay?
40:46Our weight is for complete case.
40:50And from the formula here,
40:52you can tell why I put this scenario as scenario three
40:55because it got more and more complicated.
40:59Although the weight still depends on three parts,
41:02related to the first three step.
41:05The missing probability related to this alpha,
41:08the data distribution related to this gamma,
41:12and the preliminary estimate made by using the imputation
41:19in step three.
41:25And once we get the weight through
41:28this empirical likelihood method,
41:30we then put it into this estimating equation.
41:34Adjusted by this weight, we can get our proposed estimator
41:39as multiple robust estimator of
41:41the linear regression coefficient.
41:48Okay.
41:50(coughs)
41:51Our method all framework in general,
41:55these five sets, the key thing is step four
41:58is empirical likelihood method to estimate the weight.
42:03I'll estimate his probability
42:06and we will estimate our framework in these three scenarios.
42:13Of course there are some other scenarios,
42:15and you can easily adapt to these five steps.
42:20Now, let's look at some theoretical proprietary.
42:23Why we propose these seemingly complicated five steps.
42:30We first look at the case one. There are two parts.
42:36The first theorem is about this consistence.
42:40The second theorem is about asymptotic normality, okay?
42:46So, under certain conditions, if...
42:51Remember we have two sets of models.
42:53One set of model, we modeled the probability.
42:57The other set of model, we modeled the data distribution.
43:02So if either one from the model
43:07of modeling missingness probability
43:12or the model set model the data distribution,
43:15if either one is correctly specified, Okay?
43:21Then, our estimate will be consistent.
43:26Our estimate it well be consistent.
43:28So, all proposed method allow you to make mistakes, okay?
43:37But you at least make one good right decision,
43:44then you get a consistent result, okay?
43:49Of course if you make all the bad decisions,
43:52you didn't choose any track modeling,
43:55these two sets of model, then you probably won't be able
43:59to get that consistent result.
44:01Right?
44:04And then the second theorem is about
44:07the asymptotic normality.
44:09Under certain conditions, the model estimate
44:17some multiple robust estimate on the marginal quantile
44:20where I have asymptotic normal distribution
44:23with mean zero and variates here
44:28is related to this variable.
44:30Variates is related to this data one random variable.
44:38And as you can see these variates of data one
44:46actually coming from these three parts,
44:50the estimate of the missingness probability,
44:53the estimate of these data distribution,
44:56and also the imputation process, okay?
45:00That's for case one.
45:02Similarly for case two, we have these two theorem.
45:08Y is consistent.
45:11And as long as the one model is correctly specified,
45:14we would have this consistency.
45:17And then this asymptotic normality,
45:21we would have asymptotic normal distribution.
45:23And also the variates, they're two, as you can see.
45:28The two is ready to first three step
45:32to estimate the different component, okay?
45:38And then case three, two theorem.
45:43Consistency, we need at least one model.
45:47As long as one model is correctly specified,
45:50we have a consistent result.
45:53And we have this asymptotic normalcy
45:56and the variates come from their three part. Okay?
46:02As you can see, this is a very complicated formula.
46:07It's a model getting more and more complicated.
46:10And also, if you see that you can compound the variates
46:15of the three to the situation with complete case analysis.
46:21Because for complete case analysis,
46:23we also get the consistent result, but like I said,
46:28it doesn't mean the variates would be optimal.
46:30And here, we actually can verify the variates of the three
46:34will be smaller if our model are correctly specified, okay?
46:43Let's say theoretical propriety.
46:49Now, let's look at some simulation, okay?
46:54We did simulation for each scenario,
46:58but due to the timely meet, I will only present two.
47:03Let's look at the second scenario.
47:05In the second scenario, we have four here.
47:09We have X1 follow exponential distribution X2
47:13is a normal distribution.
47:16And so Y is discrete, one is continuous, okay?
47:20The model is the simple linear model
47:24and the error distribution Y,
47:28as you can see, is heteroscedastic.
47:32Because of these error distribution, it's reduced to X1.
47:38The missing mechanism for X2,
47:42in the second scenario, we have a part of X2 is missing is
47:47through this logistic regression, okay?
47:50Now, missingness rate is about 38%.
47:57Eventually, they have this conditional quantile regression,
48:00linear regression, they have those coefficient excess.
48:04This is our simulation setup is in the second scenario.
48:13Now, we consider two working models for π, okay?
48:19The fist one is correct. The second one is incorrect.
48:24We can see there are two models for the distribution, okay?
48:32All right.
48:33This is the incorrect one
48:35and for the ordinary least squares regression.
48:38And this is correct one with title 0.25 0.75.
48:48We have replication, 1,000 times.
48:51We have some π equals 500, L is 10.
48:55This L is really related to the first step
48:59of the imputation.
49:03Okay.
49:03Now, here is all our simulation result, okay?
49:09Although the result has to be multiplied by 100,
49:14as you can see Y is very large.
49:15And also we denote our mass as 0000, okay?
49:25The fist two digit represent
49:28the missing probability model.
49:31The last two is data distribution.
49:35For example, for IPW 1000,
49:40that means we only use inverse probability method.
49:44And the weight is estimating is based on
49:49this correct weight, okay?
49:52And for the imputation,
49:56that means we only use this data distribution.
50:00And for this IM 0010, that means we use our first model,
50:08which is to model the data distribution.
50:14This is the second model for data distribution.
50:18And in either case,
50:20is always the first one is correct model.
50:23The first one is correct model.
50:24The second one is not, okay?
50:26That's just from notation.
50:28As you can see here using IPW
50:31if the model is correctly specified,
50:34the bias is quite small
50:35and everything is quite good.
50:38However, if you miss specify the missingness probability,
50:42we see the estimate is quite out of control, okay?
50:47Let's say for IM imputation, if you specify correctly
50:53the data distribution, the result is good.
50:56If not, then it's not.
50:58Okay.
50:59Then there's multiple robust method.
51:03In the multiple robust method,
51:08we look at, for example, this one,
51:12we get a missing probability correctly specified,
51:15then we get a good result.
51:17If not, we get bad result as the IPW, okay?
51:22But anyway, if we can choose to use all these four models,
51:29as you can see, the result is quite good, okay?
51:33The taking home method for these simulation study is,
51:39if you have some ideas about missingness probability
51:47about the state of this data distribution,
51:50and you think, "Okay, maybe this one is right
51:53or maybe this one is also right, okay?
51:56So on my side, just tell you,
51:58"Okay, I don't have to just put all these
52:04potential candidate potential model into all framework.
52:11Then we look at the recount.
52:16This one of the simulation is scenario two.
52:22We also have a simulation in a scenario three,
52:27but I will skip it here and go directly to the
52:35real data analysis.
52:38So, in this real data analysis, we look at this
52:43AIDS clinical Trials Group Protocol 175 or ACTG 175 data.
52:52In this research, we evaluate treatment with either a single
53:01nucleosides or through HIV-infected subject
53:05whose CD4 cells count
53:08and are from 200 to 500 per cubic millimeters.
53:14So, we consider to arms or treatment.
53:17One is standardized,
53:19and the other one is with three newer treatments.
53:24The two arms respectively,
53:29have about 500 and 1,600 subjects.
53:34Now, model we are looking at is
53:36the linear quantile regression model
53:39and with those kind of covariates inside.
53:43The data can be found in this package.
53:51Now for the data, the average subject is 35 years old,
53:57standard variation is about nine,
54:01and the variable CD4 96 is missing for approximate 37%.
54:10It's quite similar to simulation scenario.
54:16Each athlete is part of set up of simulations scenario.
54:22However, at baseline during the followup,
54:25full measurements on additional variable are correlated
54:28with CD4 96 are obtained.
54:30So this would be the missing part. We get the missing part.
54:39Here we assumed this CD4 96 is the missing and random.
54:46And we also have other baseline, for example,
54:50CD4 80 and CD4 20, and so on.
54:56we will use these as auxiliary variables.
55:01So, we have our third scenario
55:07in this real data analysis.
55:12And why we choose this data?
55:16If we look at this CD4 96, the histogram of this, okay?
55:24The left one is before we do it's original skill.
55:32The right one is after we do log transformation.
55:39So, as you can see, the left one is kind of truncated,
55:46and the right one also truncated.
55:49So you may debate,
55:50"Okay, which one I should use?
55:52Do I take log transformation or not?
55:59Or to be, or not to be."
56:03So that's no apparent reason to favor one of them
56:10for the imputation method.
56:13Now, what do we do?
56:17In our proposed method,
56:19we can put all these two models in our framework, okay?
56:26We don't need to make the choice.
56:29And because no apparent reason,
56:31we take a log, or not take log.
56:33Now, let's put the two together into our model, okay?
56:38So we can simultaneously accommodate both simulation.
56:44And then we have a eight covariates and auxiliary variable.
56:49Then we have this probability is modeled by
56:54a logistic regression containing all main effect of X and S.
57:02So, here is the result. Here is the result.
57:04This is a big table, but let me summarize these table.
57:10Okay.
57:11They three newer treatment, significantly slow the progress.
57:16Our proposed method and the IPW method,
57:19produce very similar results, okay
57:23And the incubation estimate,
57:27one failed to catch difference in the treatment
57:31and treatment arm effect for different quantile.
57:38The amputation estimator 2 gives
57:40an increasing estimation effect and covariance.
57:44In addition, the two imputation estimates
57:48are quite sensitive to the selection of the working models.
58:04Okay?
58:05And also, from these real data,
58:07we can help complete case analysis
58:11overestimate the treatment arm effects once again,
58:16so that even sometimes the compete case analysis is valid
58:23but there are also advantage to use our proposed method.
58:34All right, so here's the summary of my talk.
58:40We proposed a general framework for
58:44quantile estimation with missing data.
58:48And we actually applied these framework
58:52in different scenario.
58:55Now, the taking home message is,
59:00our proposed method or whatever robust against
59:04possible model misspecification.
59:08So, as we have two sets of model,
59:10one for missing probability
59:12and one is for data distribution.
59:14As long as one model is correct,
59:17then we will get good result.
59:19And also our method can be easily to be generalized
59:23to many other scenario.
59:26And I think that's all of my talk,
59:32and thank you.
59:36- All right.
59:37Thank you, Linglong. This was very interesting.
59:39I think we're almost out of time, so if there's
59:43we have time probably for one question.
59:45So if there's any, if not
59:48Let's see if there are any questions.
59:52Feel free to write in the chat box or on cells.
01:00:12Okay.
01:00:13Just gonna ask one question
01:00:14and then I think I'm gonna ask all the questions
01:00:17when we meet.
01:00:19Just a quick question.
01:00:20Do you know why the complete case analysis have
01:00:24overestimation rather than underestimation?
01:00:27Like, do you have a feeling why that's the case and what?
01:00:33- Well, I don't know. No.
01:00:39- Yeah.
01:00:40I believe it will be interesting to see what cases,
01:00:42like what are the conditions for overestimation
01:00:45or underestimation for complete case analysis, I guess.
01:00:48I guess, it must depend on the data distribution
01:00:52and the missingness mechanism that's been made.
01:00:56But I'm not sure one.
01:00:59- I agree with you.
01:01:01The reason I would answer I don't know,
01:01:05because it's really hard to know how the data is miss.
01:01:11Although we assume it's missing at runtime.
01:01:13- Yeah.
01:01:14- But, who knows the reality?
01:01:17- Right. Yeah, right.
01:01:19I guess, under your assumption of missing at random,
01:01:22then I guess there could be conditions for underestimation
01:01:27or overestimation under the assumption of where MI.
01:01:31But, I don't know.
01:01:32I was wondering if people have derived those or not.
01:01:36(laughs)
01:01:37They could be future work, right?
01:01:40(laughs)
01:01:42All right.
01:01:43Linglong, thank you.
01:01:44I'll see you in an hour for a one on one meetings,
01:01:47and I know other students and maybe faculty have
01:01:51signed up for it to meet with you.
01:01:53So, thank you very much.
01:01:55And I'll see you later. All right.
01:01:57- Thank you.
01:01:57- Bye-bye. Thank you everyone for joining.
01:01:58Bye.
01:01:59- Bye.
01:02:00- Bye.