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<!DOCTYPE html>
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<title>Planejamento experimental e Análise de dados agronômicos</title>
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</head>
<body class="quarto-light">
<div class="reveal">
<div class="slides">
<section id="title-slide" class="quarto-title-block center">
<h1 class="title">Planejamento experimental <br> e Análise de dados agronômicos</h1>
<div class="quarto-title-authors">
<div class="quarto-title-author">
<div class="quarto-title-author-name">
José Tiago Barroso Chagas
</div>
</div>
</div>
</section>
<section id="apresentação" class="slide level2 smaller">
<h2>Apresentação</h2>
<img data-src="misc/linkedin.png" class="r-stretch"><p><em>Engenheiro agronômo (<strong>UFCA</strong>- 2018)</em></p>
<p><em>Mestre em genética e melhoramento de plantas (<strong>UENF</strong>-2020)</em></p>
<p><em>Doutor em genética e melhoramento (<strong>UFV</strong>- 2024)</em></p>
<p><em>Pesquisador de pós doutorado (<strong>UFV</strong>- Atual)</em></p>
</section>
<section id="sumário" class="slide level2">
<h2>Sumário</h2>
<ul>
<li class="fragment">Princípios da experimentação agrícola</li>
<li class="fragment">Planejamento de Experimentos agronômicos</li>
<li class="fragment">Equações de Modelos Mistos</li>
<li class="fragment">Análise de variância individual</li>
<li class="fragment">Análise de variância conjunta</li>
<li class="fragment">Predição de valores genotípicos</li>
<li class="fragment">Análise de Fatores</li>
<li class="fragment">Índices de seleção</li>
</ul>
</section>
<section id="princípios-da-experimentação-agrícola" class="slide level2 smaller">
<h2>Princípios da Experimentação agrícola</h2>
<div>
</div>
<div class="quarto-layout-panel" data-layout-ncol="2">
<div class="quarto-layout-row">
<div class="quarto-layout-cell" style="flex-basis: 50.0%;justify-content: flex-start;">
<ul>
<li class="fragment">Repetição</li>
<li class="fragment">Casualização</li>
<li class="fragment">Controle local</li>
</ul>
</div>
<div class="cell quarto-layout-cell" style="flex-basis: 50.0%;justify-content: flex-start;">
<div class="cell-output cell-output-stdout">
<pre><code>DIC - 5trat*3rep</code></pre>
</div>
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="Exp_Plan_-_data_analysis_files/figure-revealjs/unnamed-chunk-1-1.png" width="960"></p>
</figure>
</div>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>DBC - 5trat*3rep</code></pre>
</div>
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="Exp_Plan_-_data_analysis_files/figure-revealjs/unnamed-chunk-1-2.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div>
</div>
</section>
<section id="planejamento-de-experimentos-agronômicos" class="slide level2 smaller">
<h2>Planejamento de Experimentos agronômicos</h2>
<div>
</div>
<div class="quarto-layout-panel" data-layout-ncol="2">
<div class="quarto-layout-row">
<div class="quarto-layout-cell" style="flex-basis: 50.0%;justify-content: flex-start;">
<ul>
<li class="fragment">Experimentos com grande número de tratamentos para os blocos casualizados em DBC</li>
<li class="fragment"><strong>k</strong> Tratamentos divididos em <strong>k</strong> blocos (ktrat*kblocos)<br>
</li>
<li class="fragment">Látice quadrado (9trat/k3/rep3)</li>
</ul>
</div>
<div class="cell quarto-layout-cell" style="flex-basis: 50.0%;justify-content: flex-start;">
<div class="cell-output cell-output-stdout">
<pre><code>LÁTICE</code></pre>
</div>
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="Exp_Plan_-_data_analysis_files/figure-revealjs/unnamed-chunk-2-1.png" width="768"></p>
</figure>
</div>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> ID LOCATION PLOT REP IBLOCK UNIT ENTRY TREATMENT
1 1 COIMBRA 1001 1 1 1 7 G-7
2 2 COIMBRA 1002 1 1 2 3 G-3
3 3 COIMBRA 1003 1 1 3 4 G-4</code></pre>
</div>
</div>
</div>
</div>
</section>
<section id="planejamento-de-experimentos-agronômicos-1" class="slide level2 smaller">
<h2>Planejamento de Experimentos agronômicos</h2>
<div class="columns">
<div class="column" style="width:35%;">
<ul>
<li class="fragment">Experimento com restrição quanto à disponibilidade de sementes e/ou volume grande de tratamentos para a área experimental</li>
<li class="fragment">Delineamento em blocos aumentados</li>
</ul>
<p>(30gen/3test/k6)</p>
</div><div class="column" style="width:65%;">
<div class="cell">
<div class="cell-output cell-output-stdout">
<pre><code> ID EXPT LOCATION YEAR PLOT ROW COLUMN CHECKS BLOCK ENTRY TREATMENT
1 1 Expt1 COIMBRA 2024 1001 1 1 1 1 3 CH3
2 2 Expt1 COIMBRA 2024 1002 1 2 1 1 2 CH2
3 3 Expt1 COIMBRA 2024 1003 1 3 0 1 23 G23</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>DBA</code></pre>
</div>
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="Exp_Plan_-_data_analysis_files/figure-revealjs/unnamed-chunk-3-1.png" width="576"></p>
</figure>
</div>
</div>
</div>
</div></div>
<div style="width: 80%">
<pre><code></code></pre>
</div>
<p>::::</p>
</section>
<section id="equações-de-modelos-mistos" class="slide level2 smaller">
<h2>Equações de modelos mistos</h2>
<p><span class="citation" data-cites="henderson1953estimation">Charles R. Henderson (<a href="#/referências" role="doc-biblioref" onclick="">1953</a>)</span> (melhorista animal) propôs as equações de modelos mistos</p>
<p>As quais possibilitaram obter BLUPs dos efeitos aleatórios (Melhor preditor linear não viesado) e estimadores de máxima verossimilhança dos efeitos fixos de <span class="math inline">\(\beta\)</span>.</p>
<img data-src="https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQVMCAcy-db2L5NduVmnliFiasE3-6Qk2vCjg&s.png" class="r-stretch"></section>
<section id="equações-de-modelos-mistos-1" class="slide level2 smaller scrollable">
<h2>Equações de modelos mistos</h2>
<p><span class="math display">\[
\textbf{y} = \mathbf{X}\beta + \mathbf{Z}u+ \mathbf{e}
\]</span></p>
<p><span class="math display">\[
\begin{align*}
\begin{bmatrix}
\mathbf{X}'\mathbf{X} & \mathbf{X}'\mathbf{Z} \\
\mathbf{Z}'\mathbf{X} & \mathbf{Z}'\mathbf{Z} + \lambda \mathbf{K}^{-1}
\end{bmatrix}
\begin{bmatrix}
\mathbf{\hat{b}} \\
\mathbf{\hat{u}}
\end{bmatrix}
&=
\begin{bmatrix}
\mathbf{X}'\mathbf{y} \\
\mathbf{Z}'\mathbf{y}
\end{bmatrix}
\end{align*}
\]</span></p>
<h3 id="máxima-verossimilhança-restrita-reml-restricted-maximum-likelihood">Máxima verossimilhança restrita <br> (REML-Restricted maximum likelihood)</h3>
<p><span class="citation" data-cites="Patterson1971">Patterson and Thompson (<a href="#/referências" role="doc-biblioref" onclick="">1971</a>)</span></p>
<p>Função de densidade de probabilidade normal</p>
<p><span class="math display">\[
f(x)= \frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}
\]</span></p>
<p>A função de máxima verossimilhança buscar a otimização dos valores da média <span class="math inline">\(\mu\)</span> e variância <span class="math inline">\(\sigma\)</span>, dado as observações e considerando distribuição normal.</p>
<p>Neste exemplo consideramos 100 valores aleatórios gerados de uma distribuição normal com média 10 (<span class="math inline">\(\mu\)</span>) e desvio padrão <span class="math inline">\(\sqrt{10}\)</span> (<span class="math inline">\(\sigma\)</span>)</p>
<div class="cell">
<div class="cell-output cell-output-stdout">
<pre><code>head de valores amostrados</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 13.105260 11.482207 9.658565 9.326820 13.662229 14.086785</code></pre>
</div>
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="Exp_Plan_-_data_analysis_files/figure-revealjs/unnamed-chunk-4-1.png" width="960"></p>
</figure>
</div>
</div>
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="Exp_Plan_-_data_analysis_files/figure-revealjs/unnamed-chunk-4-2.png" width="960"></p>
</figure>
</div>
</div>
</div>
</section>
<section id="equações-de-modelos-mistos-2" class="slide level2 smaller scrollable">
<h2>Equações de modelos mistos</h2>
<p>BLUE (Melhor estimador linear não viesado - efeitos fixos)</p>
<p>BLUP (Melhor preditor linear não viesado - efeitos aleatórios)</p>
<p><span class="citation" data-cites="Henderson1975">C. R. Henderson (<a href="#/referências" role="doc-biblioref" onclick="">1975</a>)</span></p>
<h3 id="simulação-de-dados-fenotípicos---exemplo">Simulação de dados fenotípicos - Exemplo</h3>
<div>
</div>
<div class="quarto-layout-panel" data-layout-ncol="2">
<div class="quarto-layout-row">
<div class="quarto-layout-cell" style="flex-basis: 50.0%;justify-content: flex-start;">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="misc/gy.jpeg"></p>
<figcaption>Fonte: Bayer crop science</figcaption>
</figure>
</div>
</div>
<div class="quarto-layout-cell" style="flex-basis: 50.0%;justify-content: flex-start;">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="misc/ph.jpeg"></p>
<figcaption>Fonte: Bayer crop science</figcaption>
</figure>
</div>
</div>
</div>
</div>
<p><strong>2 características; 100 genótipos; 2 blocos</strong></p>
<div class="cell">
<div class="cell-output cell-output-stdout">
<pre><code>Variâncias residuais Trait per environment -
c(Yld:E1, Yld:E2, Yld:E3, Pht:E1, Pht:E2, Pht:E3</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0.20 0.28 0.14 15.10 8.50 11.70</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Correlação dos erros devido tendências espaciais</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> 1 2
1 1.00 0.49
2 0.49 1.00</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Correlação dos erros devido causas não conhecidas
(Caminhamento,máquinas e implementos)</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> 1 2
1 1.00 0.02
2 0.02 1.00</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Banco de dados simulado de Milho</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>'data.frame': 200 obs. of 8 variables:
$ env : Factor w/ 1 level "1": 1 1 1 1 1 1 1 1 1 1 ...
$ block: Factor w/ 2 levels "1","2": 1 1 1 1 1 1 1 1 1 1 ...
$ col : Factor w/ 10 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 1 1 ...
$ row : Factor w/ 20 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
$ id : Factor w/ 100 levels "1","2","3","4",..: 39 4 77 14 49 62 47 94 40 63 ...
$ rep : Factor w/ 2 levels "1","2": 1 1 1 1 1 1 1 1 1 1 ...
$ GY : num 5.51 5.7 5.8 5.17 5.32 ...
$ PH : num 230 230 230 232 239 ...</code></pre>
</div>
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="Exp_Plan_-_data_analysis_files/figure-revealjs/unnamed-chunk-5-1.png" width="960"></p>
</figure>
</div>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Solving mixed models with RHS</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 5.513532 5.703086 5.796593 5.168647 5.323202 5.033188</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>X matrix - Efeitos fixos</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> block1 block2
1 1 0
2 1 0
3 1 0
4 1 0
5 1 0
6 1 0</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Z matrix - Efeitos aleatórios</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> id1 id2 id3 id4 id5 id6 id7 id8 id9 id10
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 1 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0
7 0 0 0 0 0 0 0 0 0 0
8 0 0 0 0 0 0 0 0 0 0
9 0 0 0 0 0 0 0 0 0 0
10 0 0 0 0 0 0 0 0 0 0</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Phenotypic data</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> env block col row id rep GY PH
1 1 1 1 1 39 1 5.513532 230.2357
2 1 1 1 2 4 1 5.703086 230.2436
3 1 1 1 3 77 1 5.796593 230.4429
4 1 1 1 4 14 1 5.168647 231.6617
5 1 1 1 5 49 1 5.323202 238.9142
6 1 1 1 6 62 1 5.033188 233.4361
7 1 1 1 7 47 1 5.969180 227.9464
8 1 1 1 8 94 1 5.296897 233.5849
9 1 1 1 9 40 1 3.751499 226.4649
10 1 1 1 10 63 1 4.588704 235.6345</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Error variance</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0.2075</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Genotypic variance</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0.0948</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Lambda</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 2.188819</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>XpX</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> block1 block2
block1 100 0
block2 0 100</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>XpZ</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> id1 id2 id3 id4 id5
block1 1 1 1 1 1
block2 1 1 1 1 1</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>ZpX</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> block1 block2
id1 1 1
id2 1 1
id3 1 1
id4 1 1
id5 1 1
id6 1 1</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>ZpZ</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> id1 id2 id3 id4 id5
id1 2 0 0 0 0
id2 0 2 0 0 0
id3 0 0 2 0 0
id4 0 0 0 2 0
id5 0 0 0 0 2</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Xpy</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> [,1]
block1 498.0205
block2 509.2963</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Zpy</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> [,1]
id1 8.834306
id2 10.478144
id3 11.530348
id4 10.042302
id5 9.734658
id6 9.007352</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>LHS</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> block1 block2 id1 id2 id3
block1 100 0 1.000000 1.000000 1.000000
block2 0 100 1.000000 1.000000 1.000000
id1 1 1 4.188819 0.000000 0.000000
id2 1 1 0.000000 4.188819 0.000000
id3 1 1 0.000000 0.000000 4.188819</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> [,1]
block1 498.020468
block2 509.296324
id1 8.834306
id2 10.478144
id3 11.530348
id4 10.042302</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code> [,1]
block1 4.980204681
block2 5.092963242
id1 -0.295754501
id2 0.096680330
id3 0.347873860
id4 -0.007368719</code></pre>
</div>
</div>
<div class="cell">
<div class="cell-output cell-output-stdout">
<pre><code>Solving mixed models with package LME4</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Linear mixed model fit by REML ['lmerMod']
Formula: GY ~ rep + (1 | id)
Data: pheno_env1
REML criterion at convergence: 320.2
Scaled residuals:
Min 1Q Median 3Q Max
-2.15158 -0.56334 0.05882 0.58978 2.09491
Random effects:
Groups Name Variance Std.Dev.
id (Intercept) 0.1010 0.3178
Residual 0.1982 0.4451
Number of obs: 200, groups: id, 100
Fixed effects:
Estimate Std. Error t value
(Intercept) 4.98020 0.05469 91.058
rep2 0.11276 0.06295 1.791
Correlation of Fixed Effects:
(Intr)
rep2 -0.576</code></pre>
</div>
<div class="cell-output-display">
<table class="caption-top">
<caption>RHS</caption>
<thead>
<tr class="header">
<th style="text-align: right;">x</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">-0.2957545</td>
</tr>
<tr class="even">
<td style="text-align: right;">0.0966803</td>
</tr>
<tr class="odd">
<td style="text-align: right;">0.3478739</td>
</tr>
<tr class="even">
<td style="text-align: right;">-0.0073687</td>
</tr>
<tr class="odd">
<td style="text-align: right;">-0.0808127</td>
</tr>
<tr class="even">
<td style="text-align: right;">-0.2544430</td>
</tr>
</tbody>
</table>
</div>
<div class="cell-output-display">
<table class="caption-top">
<caption>LME4</caption>
<thead>
<tr class="header">
<th style="text-align: left;"></th>
<th style="text-align: right;">x</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">id.(Intercept)1</td>
<td style="text-align: right;">-0.3126610</td>
</tr>
<tr class="even">
<td style="text-align: left;">id.(Intercept)2</td>
<td style="text-align: right;">0.1022069</td>
</tr>
<tr class="odd">
<td style="text-align: left;">id.(Intercept)3</td>
<td style="text-align: right;">0.3677597</td>
</tr>
<tr class="even">
<td style="text-align: left;">id.(Intercept)4</td>
<td style="text-align: right;">-0.0077899</td>
</tr>
<tr class="odd">
<td style="text-align: left;">id.(Intercept)5</td>
<td style="text-align: right;">-0.0854323</td>
</tr>
<tr class="even">
<td style="text-align: left;">id.(Intercept)6</td>
<td style="text-align: right;">-0.2689879</td>
</tr>
</tbody>
</table>
</div>
</div>
</section>
<section id="análise-conjunta" class="slide level2 smaller scrollable">
<h2>Análise conjunta</h2>
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="Exp_Plan_-_data_analysis_files/figure-revealjs/unnamed-chunk-7-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
<p><span class="math display">\[
\mathbf{y} = \mathbf{X}_1\mathbf{e} + \mathbf{X}_2\mathbf{be} + \mathbf{Z}_1\mathbf{g} + \mathbf{Z}_2\mathbf{ge} + \mathbf{\epsilon}
\]</span></p>
<p><span class="math inline">\([\mathbf{g}\sim N(0,\sigma_g^2\mathbf{I}_n)]\)</span> e <span class="math inline">\([\mathbf{ge}\sim N(0,\sigma_{ge}^2\mathbf{I}_{np})]\)</span>, onde n é o número de genótipos e p é o número de ambientes.</p>
<p><strong>2 características; 100 genótipos; 2-3 blocos; 3 ambientes</strong></p>
<p>Testar os efeitos de genótipo e da interação genótipo x ambiente para as características avaliadas. <span class="math display">\[
LRT = -2(LogL_{cm}-LogL_{rm})
\]</span> onde <span class="math inline">\(LogL_{cm}\)</span> é o ponto de máxima da função de verossimilhança residual do modelo completo e <span class="math inline">\(LogL_{rm}\)</span> é o mesmo para o modelo reduzido.</p>
<div class="cell">
<div class="cell-output cell-output-stdout">
<pre><code>Análise de variância conjunta - GY</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Linear mixed model fit by REML ['lmerMod']
Formula: GY ~ env + rep:env + (1 | id) + (1 | id:env)
Data: pheno_df
REML criterion at convergence: 1063.2
Scaled residuals:
Min 1Q Median 3Q Max
-2.67400 -0.55510 0.01111 0.57905 2.30558
Random effects:
Groups Name Variance Std.Dev.
id:env (Intercept) 0.09360 0.3059
id (Intercept) 0.01043 0.1021
Residual 0.18065 0.4250
Number of obs: 700, groups: id:env, 300; id, 100
Fixed effects:
Estimate Std. Error t value
(Intercept) 4.98020 0.05336 93.340
env2 0.13322 0.07406 1.799
env3 0.48945 0.07406 6.609
env1:rep2 0.11276 0.06011 1.876
env2:rep2 -0.02553 0.06011 -0.425
env3:rep2 -0.12331 0.06011 -2.051
env3:rep3 -0.19016 0.06011 -3.164
Correlation of Fixed Effects:
(Intr) env2 env3 env1:2 env2:2 env3:2
env2 -0.694
env3 -0.694 0.500
env1:rep2 -0.563 0.406 0.406
env2:rep2 0.000 -0.406 0.000 0.000
env3:rep2 0.000 0.000 -0.406 0.000 0.000
env3:rep3 0.000 0.000 -0.406 0.000 0.000 0.500
fit warnings:
fixed-effect model matrix is rank deficient so dropping 2 columns / coefficients</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>LRT GxE effect - Trait GY</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 43.31988</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>Data: pheno_df
Models:
mmcr: GY ~ env + rep:env + (1 | id)
mmc: GY ~ env + rep:env + (1 | id) + (1 | id:env)
npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
mmcr 9 1096.1 1137.0 -539.03 1078.1
mmc 10 1054.3 1099.8 -517.15 1034.3 43.758 1 3.717e-11 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
</div>
<div class="cell-output-display">
<table class="caption-top">
<caption>LME4 - g x e</caption>
<thead>
<tr class="header">
<th style="text-align: left;"></th>
<th style="text-align: right;">x</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">id:env.(Intercept)1</td>
<td style="text-align: right;">-0.2994339</td>
</tr>
<tr class="even">
<td style="text-align: left;">id:env.(Intercept)2</td>
<td style="text-align: right;">-0.0883629</td>
</tr>
<tr class="odd">
<td style="text-align: left;">id:env.(Intercept)3</td>
<td style="text-align: right;">0.1092676</td>
</tr>
<tr class="even">
<td style="text-align: left;">id:env.(Intercept)4</td>
<td style="text-align: right;">0.0992936</td>
</tr>
<tr class="odd">
<td style="text-align: left;">id:env.(Intercept)5</td>
<td style="text-align: right;">-0.0491773</td>
</tr>
<tr class="even">
<td style="text-align: left;">id:env.(Intercept)6</td>
<td style="text-align: right;">0.0160696</td>
</tr>
</tbody>
</table>
</div>
<div class="cell-output-display">