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<h1 class="title toc-ignore">Variance Structures</h1>
</div>
<p>Here you will find a list of variance structures (<em>a.k.a</em> covariance structures, variance-covariance-structures, correlation structures) with short explanations and possibly examples.</p>
<div id="iid---independent-identically-distributed" class="section level1">
<h1>IID - Independent & Identically Distributed</h1>
<p><strong>This covariance structure has homogeneous variances and zero correlation between elements.</strong></p>
<p>Number of parameters = 1 (<em>i.e.</em> <span class="math inline">\(\sigma\)</span>)</p>
<p><span class="math display">\[
\left(\begin{array}{cc}
\sigma^2 & 0 & 0 & 0\\
& \sigma^2 & 0 & 0\\
& & \sigma^2 & 0\\
& & & \sigma^2\\
\end{array}\right) =
\left(\begin{array}{cc}
1 & 0 & 0 & 0\\
& 1 & 0 & 0\\
& & 1 & 0\\
& & & 1\\
\end{array}\right) \sigma^2
\]</span></p>
<p>This is the simplest covariance structure and the default setting in most, if not all, linear modelling packages.</p>
</div>
<div id="diagonal" class="section level1">
<h1>Diagonal</h1>
<p><strong>This covariance structure has heterogenous variances and zero correlation between elements.</strong></p>
<p>Number of parameters: <span class="math inline">\(t\)</span> (<em>i.e.</em> <span class="math inline">\(\sigma_1\)</span>, <span class="math inline">\(\sigma_2\)</span>, …, <span class="math inline">\(\sigma_t\)</span>), which is the overall dimension of the covariance matrix (<em>e.g.</em> number of treatement levels).</p>
<p><span class="math display">\[
\left(\begin{array}{cc}
\sigma_1^2 & 0 & 0 & 0\\
& \sigma_2^2 & 0 & 0\\
& & \sigma_3^2 & 0\\
& & & \sigma_4^2\\
\end{array}\right) =
\left(\begin{array}{cc}
1 & 0 & 0 & 0\\
& k_1 & 0 & 0\\
& & k_2 & 0\\
& & & k_3\\
\end{array}\right) \sigma^2
\]</span></p>
<p>In our <a href="heterogeneous_error_variance.html" target="_blank">chapter on heterogeneous error variances</a>, we fit models in which we allow for different error variances for two of the treatments. Thus, the off-diagonals are all 0, but there are multiple variances on the diagonal. More specifically, in <code>mod5</code> we allows for 8 different error variances - one for each factor-level-combination of the respective factor effects <code>date</code> and <code>density</code>. This is visualized in the plot below as 8 different colors. Speaking in the <a href="https://cran.r-project.org/web/packages/nlme/nlme.pdf#Rfn.nlmeStruct.1" target="_blank">syntax of <code>nlme</code></a>, we obtain 8 parameter estimates: 1 estimate for the <a href="https://cran.r-project.org/web/packages/nlme/nlme.pdf#Rfn.nlmeObject.1" target="_blank">model-object’s <code>sigma</code></a> (= standard deviation for error term) and 7 estimates (that are different from 1) in the <a href="https://cran.r-project.org/web/packages/nlme/nlme.pdf#Rfn.nlmeStruct.1" target="_blank">model-object’s <code>varStruct</code></a> as can be seen on the y-axis:</p>
<p><img src="variance_structures_files/figure-html/unnamed-chunk-2-1.png" width="672" /></p>
<blockquote>
<p>In order to give a clearer picture, the variance matrix presented here was reduced to data of a single block in order to have dimensions 16x16. Since there were <a href="heterogeneous_error_variance.html" target="_blank">4 complete blocks in the dataset</a>, the entire variance matrix of the error term has dimensions 64x64. However, given that data/errors are sorted accordingly, our presented matrix is simply 1 out of 4 blocks in a <a href="https://www.wikiwand.com/en/Block_matrix#/Block_diagonal_matrices" target="_blank">block diagonal matrix</a>.</p>
</blockquote>
</div>
<div id="first-order-autoregressive-ar1" class="section level1">
<h1>First order autoregressive AR(1)</h1>
<p><strong>This covariance structure has homogeneous variances, while the correlation between any two elements gets smaller the further apart they are separated (e.g. in terms of time or space).</strong></p>
<p>Number of parameters: 2 (<em>i.e.</em> <span class="math inline">\(\sigma\)</span> and <span class="math inline">\(\rho\)</span>).</p>
<p><span class="math display">\[
\left(\begin{array}{cc}
\sigma^2 & \sigma^2\rho & \sigma^2\rho^2 & \sigma^2\rho^3\\
& \sigma^2 & \sigma^2\rho & \sigma^2\rho^2\\
& & \sigma^2 & \sigma^2\rho\\
& & & \sigma^2\\
\end{array}\right) =
\left(\begin{array}{cc}
1 & \rho & \rho^2 & \rho^3\\
& 1 & \rho & \rho^2\\
& & 1 & \rho\\
& & & 1\\
\end{array}\right) \sigma^2
\]</span></p>
<p>As can be seen, the correlation between any two elements can be described more speficically as: it is equal to <span class="math inline">\(\rho\)</span> for adjacent elements, <span class="math inline">\(\rho^2\)</span> for elements that are separated by a third, and so on. Note that since it is a correlation, we have -1 < <span class="math inline">\(\rho\)</span> < 1 and therefore <span class="math inline">\(\rho\)</span> indeed gets smaller when squared etc. This correlation model is useful, if all time points are equally spaced.</p>
<p>As an explicit example, take the correlation between errors of two adjacent time points, in this case weeks, to be <span class="math inline">\(\rho=0.9\)</span>. The correlation between errors that are two weeks apart is then <span class="math inline">\(\rho^2=0.9^2=0.81\)</span> and when they are three weeks apart it is <span class="math inline">\(\rho^3=0.9^3=0.729\)</span> and so on. Thus, the advantage here is that correlations become smaller over time and thus we have different correlation estimates, yet only a single correlation parameter <span class="math inline">\(\rho\)</span> is fitted in the model.</p>
</div>
<div id="multiplicative" class="section level1">
<h1>Multiplicative</h1>
<p><strong>It is possible to combine any two or more variance structures via direct multiplication <em>a.k.a.</em> the <a href="https://www.wikiwand.com/en/Kronecker_product" target="_blank">Kronecker product</a>.</strong></p>
<p><span class="math display">\[
\left(\begin{array}{cc}
1 & 0 \\
& k_1 \\
\end{array}\right)
\otimes
\left(\begin{array}{cc}
1 & 0 & 0 \\
& k_2 & 0 \\
& & k_3 \\
\end{array}\right) \sigma^2
=
\left(\begin{array}{cc}
1\cdot1 & 0 & 0 & 0 & 0 & 0 \\
& 1\cdot k_2 & 0 & 0 & 0 & 0 \\
& & 1\cdot k_3 & 0 & 0 & 0 \\
& & & k_1\cdot 1 & 0 & 0 \\
& & & & k_1\cdot k_2 & 0 \\
& & & & & k_1\cdot k_3 \\
\end{array}\right) \sigma^2
\]</span></p>
<p>This operation on two matrices of arbitrary size resulting in a block matrix is sometimes denoted by <span class="math inline">\(\otimes\)</span>. To give an example, we refer to <code>mod4</code> in the <a href="heterogeneous_error_variance.html" target="_blank">chapter on heterogeneous error variances</a>. Here, a multiplicative variance structure results from the kronecker product of two diagonal variance structures. The first diagonal variance structure allows for different variances for the 2 levels of <code>date</code>, while the second diagonal variance structure allows for different variances for the 4 levels of <code>density</code>. Their Kronecker product therefore results in 8 different variances, visualized in the plot below as 8 different colors.</p>
<p><img src="variance_structures_files/figure-html/unnamed-chunk-3-1.png" width="672" /></p>
<div id="advantage" class="section level2">
<h2>Advantage</h2>
<p>One may now ask where the difference lies between this multiplicative variance structure for <code>mod4</code> on the one hand, and the simple diagonal variance structure for all 8 <code>date</code>-<code>density</code>-combinations in <code>mod5</code> (see <a href="#Diagonal">diagonal section above</a>) on the other hand. The question comes intuitively, since both lead to obtaining 8 different variance estimates for the error term. However, while the combinations for which the 8 estimates are obtained are the same, the estimates themselves are different between <code>mod4</code> and <code>mod5</code>. In order to understand this, one must realize that fewer parameters need to be estimated here for <code>mod4</code> (= 6 parameters) compared to the simple diagonal variance structure for <a href="#Diagonal"><code>mod5</code></a> (= 8 parameters) - even though both result in 8 different variance estimates! One can retrace this manually by counting the number of <code>varStruct</code> values on the y-axes of the two plots. There should be 5 values for <code>mod4</code> and 7 values for <code>mod5</code> <strong>that are not equal to 1</strong> and in addition, <a href="https://cran.r-project.org/web/packages/nlme/nlme.pdf#Rfn.nlmeObject.1" target="_blank"><code>sigma</code></a> (= standard deviation for error term) itself is the <em>missing</em> parameter here.</p>
<p>Therefore, <strong>direct multiplication can lead to the desired structure with fewer parameters needing to be estimated</strong>. Notice that the number of parameters penalizes the AIC and therefore has a direct impact on model selection decisions. In the underlying <a href="heterogeneous_error_variance.html" target="_blank">chapter on heterogeneous error variances</a>, <code>mod4</code> (= multiplicative) is indeed chosen over <code>mod5</code> based on the AIC.</p>
</div>
</div>
<div id="summary" class="section level1">
<h1>Summary</h1>
<table class="table table-striped table-hover table-condensed table-responsive" style="width: auto !important; margin-left: auto; margin-right: auto;border-bottom: 0;">
<thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="1">
</th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3">
<div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">
number of parameters
</div>
</th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="5">
<div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">
name in package
</div>
</th>
</tr>
<tr>
<th style="text-align:left;">
Variance Structure
</th>
<th style="text-align:left;">
total
</th>
<th style="text-align:left;">
var
</th>
<th style="text-align:left;">
cor
</th>
<th style="text-align:left;">
nlme
</th>
<th style="text-align:left;">
lme4
</th>
<th style="text-align:left;">
glmmTMB
</th>
<th style="text-align:left;">
sommer
</th>
<th style="text-align:left;">
SAS
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
Identitiy
</td>
<td style="text-align:left;">
1
</td>
<td style="text-align:left;">
1
</td>
<td style="text-align:left;">
0
</td>
<td style="text-align:left;">
default
</td>
<td style="text-align:left;">
default
</td>
<td style="text-align:left;">
default
</td>
<td style="text-align:left;">
default
</td>
<td style="text-align:left;">
VC
</td>
</tr>
<tr>
<td style="text-align:left;">
Diagonal
</td>
<td style="text-align:left;">
t
</td>
<td style="text-align:left;">
t
</td>
<td style="text-align:left;">
0
</td>
<td style="text-align:left;">
varIdent
</td>
<td style="text-align:left;">
–
</td>
<td style="text-align:left;">
diag
</td>
<td style="text-align:left;">
ds
</td>
<td style="text-align:left;">
UN(1)
</td>
</tr>
<tr>
<td style="text-align:left;">
First order autoregressive
</td>
<td style="text-align:left;">
2
</td>
<td style="text-align:left;">
1
</td>
<td style="text-align:left;">
1
</td>
<td style="text-align:left;">
corAR1
</td>
<td style="text-align:left;">
–
</td>
<td style="text-align:left;">
ar1
</td>
<td style="text-align:left;">
AR1
</td>
<td style="text-align:left;">
AR(1)
</td>
</tr>
</tbody>
<tfoot>
<tr>
<td style="padding: 0; " colspan="100%">
<span style="font-style: italic;">Note: </span>
</td>
</tr>
<tr>
<td style="padding: 0; " colspan="100%">
<sup></sup> t = overall dimension of the covariance matrix (e.g. number of treatement levels).
</td>
</tr>
</tfoot>
</table>
</div>
<div id="more-on-this" class="section level1">
<h1>More on this</h1>
<p><a href="https://asreml.kb.vsni.co.uk/wp-content/uploads/sites/3/2018/02/ASReml-R-Reference-Manual-4.pdf#section.4.2" target="_blank">ASReml-R documentation</a></p>
<p><a href="https://documentation.sas.com/?docsetId=statug&docsetTarget=statug_mixed_syntax14.htm&docsetVersion=14.3&locale=en#statug.mixed.repeatedstmt_type" target="_blank">SAS documentation</a></p>
<p><a href="https://www.ibm.com/support/knowledgecenter/SSLVMB_23.0.0/spss/advanced/covariance_structures.html" target="_blank">SPSS documentation</a></p>
<p><span style="color:red">in progress</span></p>
</div>
<hr />
<p style="text-align: center;">Please feel free to contact us about any of this! </p>
<p style="text-align: center;"><span style="color: #003f75ff;"><em>[email protected]</em></span></p>
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