Bayesian Imputation of Missing Covariates

# Bayesian<br>Imputation of<br>Missing Covariates
### <br>Nicole S. Erler<br><span style="font-size: 20pt">June 12, 2019</span>

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# <span style="color:#593661">Bayesian<br>Imputation of</span><br>Missing Covariates

???

Thank you __Sir Rector__,
dear __committee members__, __colleagues__, __family__ and __friends__.

I'd like to **thank you** all for being here today and 
would like to **take this opportunity** to give you an **overview** of the
**research that forms my doctoral dissertation**.

Bayesian Imputation of Missing Covariates

What is that?

<hr>

Let's start with the __motivation of my research__: 
Missing Covariates

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# Missing Covariates

???

Say, we want to investigate if there is an <br>
**association** between

* the **amount of sugar-containing beverages** that a mother consumes during pregnancy
* and the **body composition of her child** after birth.

Since this will **typically be measured in an uncontrolled environment**,
many other **factors can influence** this association.

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# Missing Covariates

???

For example:

- whether the child plays __sports__ regularly, 
- how much time it spends watching __television__ or in front of a computer, or
- the __household income__ and the __educational level__ of the parents as proxies for their socio-economic status

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# Missing Covariates

???

In this example, child body composition is the __outcome__ and all other variables
are called __"covariates"__.

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# Missing Covariates

<div class="row">
  <div class="column">
<table style="color: #472050; font-size: 36px; border-spacing: 15px 40px;">
  <tr>
    <td style="text-align:center">
    <i class="far fa-list-alt"></i>
    </td>
    <td>questionnaire</td>
  </tr>
  <tr>
    <td style="text-align:center">
    <i class="fas fa-phone"></i>
    </td>
    <td> phone interview </td>
  </tr>
  <tr>
    <td style="text-align:center">
    <i class="far fa-hospital"></i>
    </td>
    <td>visit to the research centre</td>
  </tr>
</table>
</div>

</div>
</div>

???

In observational studies, variables are often measured using 
* **questionnaires** that participants have to fill in at home and return by mail, or
* by **interviews**, for example over the **phone**.
* In other cases, participants have to **visit the research center** where **physical measurements** are taken.

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# Missing Covariates

</div>

<div class="column">
<table style="color: #472050; font-size: 36px; border-spacing: 15px; padding:5px">
  <tr>
    <td style="text-align:center">
    <img src="index_files/skip_question.png" alt=""></img>
    </td>
    <td>skip a question</td>
  </tr>
  <tr>
    <td style="text-align:center">
    <img src="index_files/lost_in_mail.png" alt=""></img>
  </td>
  <td> lost in mail </td>
  </tr>
  <tr>
    <td style="text-align:center">
    <img src="index_files/refuse_measurement.png" alt=""></img>
    </td>
    <td>refuse measurement</td>
  </tr>
  <tr>
    <td style="text-align:center">
    <img src="index_files/miss_visit.png" alt=""></img>
    </td>
    <td>miss a visit</td>
  </tr>
</table>
</div>
</div>

???

It can happen that when filling in a questionnaire
* participants skip a question.
  * Maybe they __can not recall__, for instance, how much their __weight was before pregnancy__, or
  * they may __feel uncomfortable__ answering questions __about their income or alcohol consumption__.
* In other cases, the __whole questionnaire__ is not returned or may get lost in the mail.
* Sometimes, participants **do not give consent** for a particular measurement, 
for example, if blood has to be taken.
* It is also possible that participants __miss a scheduled appointment__.

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# Missing Covariates

???

The __consequence__ is that some **values are missing** from the resulting database.

The most common approach to work with such incomplete data is to
**fill in values for the missing observations**.

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# <span style="color:#593661">Bayesian<br></span>Imputation<span style="color:#593661"> of<br>Missing Covariates</span>

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This is called __imputation__.

An **important concept** in imputation is that we can **learn from the observed data** to be
able to make **good guesses about the missing values**.

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# Imputation

???

When, for instance, we want to __impute missing values in the variable income__,

we can **learn how income is associated with all the other variables** 
from those participants for whom income is observed.

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# Imputation

???

This allows us to **determine a distribution**<br>
that describes for any given set of values for the other variables<br>
which values of income are more or less likely.

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# Imputation

???

For example, for

* **smoking** mothers
* with a **university degree**
* who did not drink **alcohol** during pregnancy
* and had on average **three glasses** of sugar-containing beverages per day,

the **most likely values for income** might be around 3000 Euros.

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# Imputation

???

We can then **draw a value** from this distribution to **replace the missing observation**.

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# Multiple Imputation

???

To better represent the __uncertainty about the imputed values__ it is common
to **repeat** the imputation multiple times to obtain multiple sets of imputed values.

This is called **Multiple Imputation** and is the current gold standard
to deal with missing values in covariates.

The **idea**, that **missing values** have a **distribution** that depends on the other variables,<br>
**fits very naturally into the Bayesian framework**.

---

class: mychapter, animated, fadeIn
# Bayesian<br><span style="color:#593661">Imputation of<br>Missing Covariates</span>

???

The framework of "Bayesian" statistics is **named after Thomas Bayes**<br>
who was one of the first to write down his **thoughts** about how **observed data**
can be used to **gain information** about the underlying **cause of that data**.

Let's look at an example.

---

???

Say, I want to determine the **probability to obtain "heads"** in a coin toss.

**Before** starting the experiment, I already have an **expectation** of what the 
**result** might be.

---

???

For a **regular 1 Euro** coin, 
I expect the probability for "heads" to be **fairly close to 50%**<br>
and would expect values **below 40%** or **above 60%** as very **unlikely**.

---

???

If, however, my coin was a **very old coin**, my prior **assumption** about the
probability of "heads" might be **very different**:

I might expect that **anything can happen**, and would say **all values** between zero
and 100% are **equally likely**.

---

???

When I now **start the experiment**, I will **gain information** about the probability
for "heads" with every coin toss ...

---

???

... and my **belief about that probability will be adapted**.

---

???

The more often I toss the coin, the **more certain** I become about the 
probability of obtaining 'heads'.

---

???

The more often I toss the coin, the **more certain** I become about the 
probability of obtaining 'heads'.

---

???

The more often I toss the coin, the **more certain** I become about the 
probability of obtaining 'heads'.

---

???

The more often I toss the coin, the **more certain** I become about the 
probability of obtaining 'heads'.

---

# Bayesian Analysis

???

How does this **apply** to my initial question about the association between
sugar-containing beverages and body composition?

---

# Bayesian Analysis

???

When we **analyse our research question** in the **Bayesian framework**,<br>
all the associations, indicated by the **arrows in this figure**, <br>
are **estimated** the way I demonstrated it using the **coin experiment**.

Throughout this dissertation we **followed the example of the old coin** rather than the
1 Euro coin,<br>
meaning that **before having seen any data** 
we did **not have a strong belief** about what those **associations would be**.

The **observation** of each of the mother-child pairs allowed us to **learn more** about
the **strengths of each of the associations**.

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# Bayesian Analysis

???

This **concept of having distributions** for the strengths of the associations
is very **similar to** what we I've shown you earlier for the **missing covariate values**,
where we had a **distribution** telling us which values are **more likely than others**.

And this is why the **Bayesian framework lends itself very naturally to dealing 
with missing values**.

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# Bayesian Analysis and Imputation

<div class="wrapper2">

???

As with the association between variables<br>
we could have an **expectation** of what **likely values** for the missing
observations are **before having seen any data**.

The **observation of data** then allows us to **update our belief** about 
**which values are most likely**.

**Contrary to the concept of multiple imputation**,<br>
where the **imputation** of missing values is done **separately** from the **analysis** of the data,<br>
in the **Bayesian framework** **imputation** and **analysis**
are performed **simultaneously**.

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# Bayesian Analysis and Imputation

???

Specifically, in the **Bayesian approach** applied **in my research**,
besides the **model used to answer the research question**, ...

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# Bayesian Analysis and Imputation

???

... a **model** is specified **for each incomplete covariate**.

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# Bayesian Analysis and Imputation

---

# Bayesian Analysis and Imputation

---

# Bayesian Analysis and Imputation

---

# Bayesian Analysis and Imputation

???

The models are specified in a **sequential manner**...

... so that **all variables**, observed and unobserved, **and their relationships**
are **described**.

<hr>

I would now like to **highlight** the content of **some of the chapters of my dissertation**.

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class: mychapter, animated, fadeIn
# Chapter 2<br><br>Imputation in<br>Longitudinal Settings

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Chapter 2 deals with **imputation in longitudinal settings**.

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# Imputation in Longitudinal Settings

???

This research was motivated by the question if **diet** and **weight** during
pregnancy are associated.

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# Imputation in Longitudinal Settings

???

The **particular challenge** is that gestational weight was **measured repeatedly throughout pregnancy**.

---

# Imputation in Longitudinal Settings

<div class = "column">
<center>
<h3 style="text-color:#472050"> Bayesian approach<br>&nbsp;</h3>
</center>
</div>

<div class = "column">
<center>
<h3 style="text-color:#472050"> Multiple Imputation using<br>Fully Conditional Specification</h3>
</center>
</div>

</div>

???

In this chapter, we **compare** the **Bayesian approach**<br>
to the **current gold standard ** for multiple imputation:<br>
**Multiple Imputation using a Fully Conditional Specification**.

This approach is also called **multiple imputation using chained equations**, or short, **MICE**.

An **important difference** between the two approaches is **how** the **models** for 
incomplete variables **are specified**.

---

# Imputation in Longitudinal Settings

<div class = "column">
<center>
<h3 style="text-color:#472050"> Bayesian approach<br>&nbsp;</h3>

<div class = "column">
<center>
<h3 style="text-color:#472050"> Multiple Imputation using<br>Fully Conditional Specification</h3>
</center>
</div>
</div>

???

While in the Bayesian approach **models** for the **outcome** and **all incomplete
covariates** are specified **in the sequence** I've shown you before, ...

---

# Imputation in Longitudinal Settings

<div class = "column">
<center>
<h3 style="text-color:#472050"> Bayesian approach<br>&nbsp;</h3>

<div class = "column">
<center>
<h3 style="text-color:#472050"> Multiple Imputation using<br>Fully Conditional Specification</h3>

???

... in the **multiple imputation approach** models are specified 
**only for the incomplete covariates**...

...and **each of the models** uses information from **all other variables**, including the **outcome**.

This **works well** when the **outcome**, maternal weight in our example, is **measured
only once**.

---

# Imputation in Longitudinal Settings

<div class = "column">
<center>
<h3 style="text-color:#472050"> Bayesian approach<br>&nbsp;</h3>

<div class = "column">
<center>
<h3 style="text-color:#472050"> Multiple Imputation using<br>Fully Conditional Specification</h3>

???

It could even still work if we had **multiple measurements** of weight, as long as all women
had the **same number** of measurements and measured at the **same time**.

---

# Imputation in Longitudinal Settings

???

However, in our data, the **weight measurements** are **unbalanced**,<br>
meaning that

* they are obtained at **different time-points** and, 
* **because** some women **entered** the study **late** or **missed visits**,
* we do not have the **same number** of weight **measurements** for everybody.

---

# Imputation in Longitudinal Settings

<div class = "row">

<div class = "column">
<center>
<h3 style="text-color:#472050"> Bayesian approach<br>&nbsp;</h3>

<div class = "column">
<center>
<h3 style="text-color:#472050"> Multiple Imputation using<br>Fully Conditional Specification</h3>

???

In the **Bayesian approach**,<br>
weight is used **only on the left-hand side** of the **analysis model**,<br>
where such **unbalanced data** can be **handled quite easily**.

Using the **fully conditional specification**,<br>
however, maternal weight is used as a **covariate in the imputation models**,<br>
and there, **unbalanced measurements are a problem**.

**Summaries** or **simplifications** of the repeated measurements of weight are **needed**.

---

# Imputation in Longitudinal Settings

<div class = "row">
  <div class = "column">
   <center>
     <h3> Bayesian approach<br>&nbsp;</h3>
     <div class = 'mybox' style="display: inline-block;width:300px;">
     <div class = "row">
       <div class = "column" style="width:90px;position:relative; top:5px;">
       <i class="far fa-smile fa-2x"></i>
      </div>
      <div class = "column" style="width:200px;text-align:left;">
       unbiased results
       </div>
     </div>
     </div>
   </center>
  </div>

<div class = "column">
    <center>
      <h3> Multiple Imputation using<br>Fully Conditional Specification</h3>
     <div class = 'mybox' style="display:inline-block;">
     <div class = "row">
       <div class = "column" style="width:90px;position:relative; top:5px;">
        <i class="far fa-frown fa-2x"></i>
      </div>
      <div class = "column" style="width:250px;text-align:left;">
        potentially<br>biased results
      </div>
     </div>
     </div>
   </center>
  </div>
</div>

???

In summary, we found that

* the **Bayesian approach** can provide **unbiased results** in longitudinal settings
* Multiple Imputation using a fully conditional specification will produce **biased results** when
  * missingness is related to the outcome, and
  * **relevant information** gets lost when a **summary** of a longitudinal outcome **is used**

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class: mychapter, animated, fadeIn
# Chapter 5<br><br>Imputation with<br>Time-Varying Covariates

???

In Chapter 5 we **extend the Bayesian approach** to settings with **time-varying covariates**.

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# Time-Varying Covariates

???

This **extension was motivated** by two research questions:

- Is **maternal weight** associated with **maternal blood pressure**?, and
- Is **maternal weight** associated with **child BMI**?

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# Time-Varying Covariates

???

In the first question,

* **blood pressure** and **weight** were both **measured repeatedly** during pregnancy, and 
* both measurements were always obtained **at the same time points**.

---

# Time-Varying Covariates

???

The **difficulty** here is that

* there may **not only be an effect** of **weight on blood pressure**, ...

---

# Time-Varying Covariates

???

... but at the same time, **blood pressure may affect gestational weight**.

---

# Time-Varying Covariates

<div class = 'endobox'>
endogenous covariate
</div>

???

In this setting, weight is therefore called an **endogenous covariate**.

Since we had seen previously that<br>
multiple imputation using a **fully conditional specification** may have 
**problems in longitudinal data**,

in this chapter we compared the Bayesian approach to a **different way of creating multiple imputations**

called **Joint Model multiple imputation**.

---

# Time-Varying Covariates

<div class = 'row'>
<div class = 'column' style="width:45%">
<center>
<h3>Bayesian Approach<br>&nbsp;</h3>

<div class = 'column' style="width:55%">
<center>
<h3>Joint Model<br>Multiple Imputation</h3>

???

In joint model multiple imputation,

* a model is specified for the **outcome**, in our case **blood pressure**,
* the **time-varying covariates**, in our case that was **maternal weight**, 
* and all other **covariates that have missing values**.

In each of these models,<br>
only those **variables that do not have any missing values** are used as **covariates**.

---

# Time-Varying Covariates

<div class = 'row'>
<div class = 'column' style="width:45%">
<center>
<h3>Bayesian Approach<br>&nbsp;</h3>

<div class = 'column' style="width:55%">
<center>
<h3>Joint Model<br>Multiple Imputation</h3>

???

To take into account the<br>
**associations** between

* **incomplete covariates**,
* **time-varying covariate** and 
* the **outcome**

these models are then **connected**.

This connection does **not imply a direction** of the associations,<br>
which is why joint model multiple imputation **can be used for endogenous** covariates.

A particular **challenge that remains**, however, is the **analysis of the imputed data**<br>
because **standard analysis methods** cannot handle endogenous covariates which may cause
**bias in the results**.

---

# Time-Varying Covariates

<div class = 'row'>
<div class = 'column' style="width:45%">
<center>
<h3>Bayesian Approach<br>&nbsp;</h3>

<div class = 'column' style="width:55%">
<center>
<h3>Joint Model<br>Multiple Imputation</h3>

???

We **extended the Bayesian approach** to handle endogenous covariates by 
**introducing a similar connection** between

* the model for the **outcome** and
* the models for **time-varying covariates**.

Since in the Bayesian approach analysis and imputation are performed **simultaneously**
**endogeneity is taken into account** during **imputation as well as analysis**.

---

# Time-Varying Covariates

???

In the **second research question** about the association between **gestational weight**
and **child BMI**, again,

* **both variables** were measured **repeatedly**, however,
* **maternal weight** was measured **during pregnancy**, and 
* **child BMI after birth**.

It is therefore **not meaningful** to assume an association of maternal weight and
child BMI **at the same time points** as was the case with blood pressure.

---

# Time-Varying Covariates

<div class = 'row' style="width:100%; position:fixed; bottom:10%">
<div class = 'column' style="width:35%;">
<img src = "index_files/weight_plot1c.png" style="width:400px"></img>
</div>
<div class = 'column' style="width:20%;">
<div class = 'endobox' style="width:200px;margin-top:100px;text-align:center;">
non-linear<br>association
</div>
</div>
<div class = 'column' style="width:35%;">
<img src = "index_files/BMI_plot.png" style="width:400px"></img>
</div>
</div>

???

The **challenge of this research question** was to<br>
**allow for more complex, non-linear associations** between a time-varying
covariate and the outcome.

---

# Time-Varying Covariates

<div class = 'row' style="width:100%; position:fixed; bottom:10%">
<div class = 'column' style="width:35%;">
<img src = "index_files/weight_plot1d.png" style="width:400px"></img>
</div>
<div class = 'column' style="width:20%;">
<div class = 'endobox' style="width:200px;margin-top:100px;text-align:center;">
non-linear<br>association
</div>
</div>
<div class = 'column' style="width:35%;">
<img src = "index_files/BMI_plot.png" style="width:400px"></img>
</div>
</div>

???

And for **this particular application**,<br>
**trimester-specific weight gain** was of interest,
meaning, the amount of weight gained during each trimester.

---

# Time-Varying Covariates

<div class = 'row'>
<div class = 'column' style="width:45%">
<center>
<h3>Bayesian Approach<br>&nbsp;</h3>

<div class = 'column' style="width:55%">
<center>
<h3>Joint Model<br>Multiple Imputation</h3>

???

In the **Bayesian approach**, analysis and imputation are performed **simultaneously**.

This means, that

* the **non-linear** and potentially complex **association**
* which is **part of the analysis model** is
* **automatically taken into account** during imputation.

---

# Time-Varying Covariates

<div class = 'row'>
<div class = 'column' style="width:45%">
<center>
<h3>Bayesian Approach<br>&nbsp;</h3>

<div class = 'column' style="width:55%">
<center>
<h3>Joint Model<br>Multiple Imputation</h3>

???

In **joint model multiple imputation**<br>
the **only connection** between child BMI and maternal weight is **through the link** of the two models.

This **link**, however, **implies a linear association**.

Because

* **missing values** are **imputed under a wrong assumptions**
* the results may be **biased**, even when 
* the **correct, non-linear association** is used **during analysis** of the imputed data.

---

# Time-Varying Covariates

<div class = "row">
  <div class = "column" style="width:35%;">
   <center>
     <h3> Bayesian approach<br>&nbsp;</h3>
     <div class = 'mybox' style="display: inline-block;width:300px; margin-top: 15px;">
     <div class = "row">
       <div class = "column" style="width:90px;position:relative; top:5px;">
       <i class="far fa-smile fa-2x"></i>
      </div>
      <div class = "column" style="width:200px;text-align:left;">
       unbiased results
       </div>
     </div>
     </div>
   <div class = 'mybox' style="display: inline-block;width:300px;">
     <div class = "row">
       <div class = "column" style="width:90px;position:relative; top:5px;">
       <i class="far fa-smile fa-2x"></i>
      </div>
      <div class = "column" style="width:200px;text-align:left;">
       unbiased results
       </div>
     </div>
     </div>
   </center>
  </div>

<div class = 'mybox2' style="display:inline-block; margin-top: 15px;">
  endogeneity
  </div>
  <div class = 'mybox2' style="display:inline-block;">
  non-linearity
  </div>

</div>

<div class = "column" style="width:45%;">
    <center>
    <h3>Joint Model<br>Multiple Imputation</h3>
     <div class = 'mybox' style="display:inline-block; margin-top: 15px;">
     <div class = "row">
       <div class = "column" style="width:60px;position:relative; top:2px;">
        <i class="far fa-smile fa-1x"></i>
      </div>
      <div class = "column" style="width:320px;text-align:left;">
        imputation OK
      </div>
     </div>
     <div class = "row">
       <div class = "column" style="width:60px;position:relative; top:2px;">
        <i class="far fa-frown fa-1x"></i>
      </div>
      <div class = "column" style="width:345px;text-align:left;">
        analysis problematic
      </div>
     </div>
     </div>
     <div class = 'mybox' style="display:inline-block;">
     <div class = "row">
       <div class = "column" style="width:60px;position:relative; top:2px;">
        <i class="far fa-frown fa-1x"></i>
      </div>
      <div class = "column" style="width:345px;text-align:left;">
        imputation biased
      </div>
     </div>
     <div class = "row">
       <div class = "column" style="width:60px;position:relative; top:2px;">
        <i class="far fa-smile fa-1x"></i>
      </div>
      <div class = "column" style="width:345px;text-align:left;">
        analysis OK
      </div>
     </div>
     </div>
   </center>
  </div>
</div>

???

In summary:

* The Bayesian approach can handle
   * **endogenous** variables as well as 
   * complex, **non-linear associations**
   
* Joint Model multiple imputation can
   * handle **endogenous** covariates, but
   * **analysis** of the imputed data **remains difficult**.
   * And Joint model MI will **introduce bias** when associations are **non-linear**

---

class: mychapter, animated, fadeIn
# Chapter 6<br><br>Software Implementation<br>into an R Package

???

Chapter 6 describes the **implementation** of the Bayesian approach in the
**statistical software "R"**.

---

# Software Implementation

<center>
<div class="row" style="margin-top: -10px">
<div class="cardcol">
<div class="card">
<div class = 'card-header'>Longitudinal Outcome</div>
<img src = "index_files/weight_plot2b.png" style="width:90%">
</div>
</div>

<div class="cardcol" style="padding: 0 15px;">
<div class="card">
<div class = 'card-header'>Endogenous Covariate</div>
<img src = "index_files/BP_plot2b.png" style="width:90%">
</div>
</div>

<div class="cardcol">
<div class="card">
<div class = 'card-header'>Non-linear Association</div>
<img src = "index_files/weight_plot1e.png" style="width:90%">
</div>
</div>
</div>

<div class="row" style = 'padding: 10px 0 0 0;position:relative; left:16%;'>

</div>
</center>

???

In the **previous chapters** of my **dissertation**, we have shown that 
**standard imputation methods** may have difficulties in settings

* where the **outcome is longitudinal**,
* where covariates are **endogenous** or
* when the **association** between a covariate and the outcome is **complex**.

---

# Software Implementation

<div class="cardcol" style="padding: 0 15px;">
<div class="card">
<div class = 'card-header'>Endogenous Covariate</div>
<img src = "index_files/BP_plot2b.png" style="width:90%">
</div>
</div>

<div class="cardcol">
<div class="card">
<div class = 'card-header'>Non-linear Association</div>
<img src = "index_files/weight_plot1e.png" style="width:90%">
</div>
</div>
</div>

<div class="row" style = 'padding: 10px 0 0 0;position:relative; left:16%;'>
<div class="cardcol">
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<img src = "index_files/surv_plot.png" style="width:90%">
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<img src = "index_files/int_plot.png" style="width:90%">
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**Other studies** have shown that the same is the case for

* **time-to-event outcomes** and 
* models that involve **interaction terms**.

Based on

* **theory**,
* our **real data examples** and 
* **simulation studies**

we know that the **Bayesian approach can overcome these difficulties**
and  can provide correct results.

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# Software Implementation
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For it to be **useful in practice**, however, theoretical results and examples 
are **not sufficient**.

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# Software Implementation
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<div class="arrow_box" style= 'width:200px;'>JointAI</div>
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<span style="color:#472050">Joint A</span>nalysis and <span style="color:#472050">I</span>mputation
</h2>

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To **facilitate the step** from theory to practice, 
we implemented the Bayesian approach in the **R package JointAI**,
short for Joint Analysis and Imputation.

Its **use** follows the **use of very well-known functions** for
**analysis of complete data** so that

the **step from standard, complete data models** to 
**analysis and imputation under the Bayesian framework** requires **hardly any extra effort**.

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# Software Implementation
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<div class="arrow_box" style= 'width:200px;'>JointAI</div>
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<h2 style="font-weight:bold">
<span style="color:#472050">Joint A</span>nalysis and <span style="color:#472050">I</span>mputation
</h2>

???

The package is

* available for download from **The Comprehensive R Archive Network**. 
* Development versions can be obtained via **GitHub**, and
* the package has its own **website** with **detailed documentation and examples**.

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# Bayesian<br>Imputation of<br>Missing Covariates

<h3> <br>Nicole S. Erler<br><span style='font-size: 20pt'>June 12, 2019</span></h3>

???

Thank you for your attention.
I give the word back to you, Sir Rector.