`DImodels`

The `DImodels`

package is designed to make fitting Diversity-Interactions models easier. Diversity-Interactions (DI) models (Kirwan et al 2009) are a set of tools for analysing and interpreting data from experiments that explore the effects of species diversity (from a pool of *S* species) on community-level responses. Data suitable for DI models will include (at least) for each experimental unit: a response recorded at a point in time, and a set of proportions of *S* species \(p_1\), \(p_2\), …, \(p_S\) from a point in time prior to the recording of the response. The proportions sum to 1 for each experimental unit.

`DImodels`

installation and loadThe `DImodels`

package is installed from CRAN and loaded in the typical way.

```
install.packages("DImodels")
library("DImodels")
```

It is recommended that users unfamiliar with Diversity-Interactions (DI) models read the introduction to `DImodels`

, before using the package. Run the following code to access the documentation.

```
?DImodels
```

There are seven example datasets included in the `DImodels`

package: `Bell`

, `sim1`

, `sim2`

, `sim3`

, `sim4`

, `sim5`

, `Switzerland`

. Details about each of these datasets is available in their associated help files, run this code, for example:

```
?sim3
```

In this vignette, we will describe the `sim3`

dataset and show a worked analysis of it.

The `sim3`

dataset was simulated from a functional group (FG) Diversity-Interactions model. There were nine species in the pool, and it was assumed that species 1 to 5 come from functional group 1, species 6 and 7 from functional group 2 and species 8 and 9 from functional group 3, where species in the same functional group are assumed to have similar traits. The following equation was used to simulate the data.

\[ y = \sum_{i=1}^{9}\beta_ip_i + \omega_{11}\sum_{\substack{i,j = 1 \ i

Parameter | Value | Parameter | Value | |
---|---|---|---|---|

\(\beta_1\) | 10 | \(\omega_{11}\) | 2 | |

\(\beta_2\) | 9 | \(\omega_{22}\) | 3 | |

\(\beta_3\) | 8 | \(\omega_{33}\) | 1 | |

\(\beta_4\) | 7 | \(\omega_{12}\) | 4 | |

\(\beta_5\) | 11 | \(\omega_{13}\) | 9 | |

\(\beta_6\) | 6 | \(\omega_{23}\) | 3 | |

\(\beta_7\) | 5 | \(\gamma_1\) | 3 | |

\(\beta_8\) | 8 | \(\gamma_2\) | 0 | |

\(\beta_9\) | 9 | \(\sigma\) | 1.2 |

Here, the non-linear parameter \(\theta\) that can be included as a power on each \(p_ip_j\) component of each interaction variable (Connolly et al 2013) was set equal to one and thus does not appear in the equation above.

The 206 rows of proportions contained in the dataset `design_a`

(supplied in the package) were used to simulate the `sim3`

dataset. Here is the first few rows from `design_a`

:

community | richness | p1 | p2 | p3 | p4 | p5 | p6 | p7 | p8 | p9 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |

1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |

2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |

2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |

3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

Where `community`

is an identifier for unique sets of proportions and `richness`

is the number of species in the community.

The proportions in `design_a`

were replicated over two treatment levels, giving a total of 412 rows in the simulated dataset. The `sim3`

data can be loaded and viewed in the usual way.

```
data("sim3")
knitr::kable(head(sim3, 10))
```

community | richness | treatment | p1 | p2 | p3 | p4 | p5 | p6 | p7 | p8 | p9 | response |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | A | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 10.265 |

1 | 1 | B | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 7.740 |

1 | 1 | A | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 12.173 |

1 | 1 | B | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 8.497 |

2 | 1 | A | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 10.763 |

2 | 1 | B | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 8.989 |

2 | 1 | A | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 10.161 |

2 | 1 | B | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 7.193 |

3 | 1 | A | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 10.171 |

3 | 1 | B | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 6.053 |

There are several graphical displays that will help to explore the data and it may also be useful to generate summary statistics.

```
hist(sim3$response, xlab = "Response", main = "")
```

```
# Similar graphs can also be generated for the other species proportions.
plot(sim3$p1, sim3$response, xlab = "Proportion of species 1", ylab = "Response")
```

```
summary(sim3$response)
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 4.134 9.327 10.961 10.994 12.604 17.323
```

`autoDI`

The function `autoDI`

in `DImodels`

provides a way to do an automated exploratory analysis to compare a range of DI models. It works through a set of automated steps (Steps 1 to 4) and will select the 'best' model from the range of models that have been explored and test for lack of fit in that model. The selection process is not exhaustive, but provides a useful starting point in analysis using DI models.

```
auto1 <- autoDI(y = "response", prop = 4:12, treat = "treatment",
FG = c("FG1","FG1","FG1","FG1","FG1","FG2","FG2","FG3","FG3"), data = sim3,
selection = "Ftest")
#> Warning in DI_data_prepare(y = y, block = block, density = density, prop = prop, : One or more rows have species proportions that sum to approximately 1, but not exactly 1. This is typically a rounding issue, and has been corrected internally prior to analysis.
#>
#> --------------------------------------------------------------------------------
#>
#> Step 1: Investigating the diversity effect
#> Selection using F tests
#> Description
#> DI Model 1 Structural 'STR' DImodel with treatment
#> DI Model 2 Species identity 'ID' DImodel with treatment
#> DI Model 3 Average interactions 'AV' DImodel with treatment
#> DI Model 4 Functional group effects 'FG' DImodel with treatment
#> DI Model 5 Separate pairwise interactions 'FULL' DImodel with treatment
#>
#> DI_model treat estimate_theta Resid. Df Resid. SSq Resid. MSq
#> DI Model 1 STR 'treatment' FALSE 410 1496.1645 3.6492
#> DI Model 2 ID 'treatment' FALSE 402 841.2740 2.0927
#> DI Model 3 AV 'treatment' FALSE 401 694.3095 1.7314
#> DI Model 4 FG 'treatment' FALSE 396 559.7110 1.4134
#> DI Model 5 FULL 'treatment' FALSE 366 522.9727 1.4289
#> Df SSq F Pr(>F)
#> DI Model 1
#> DI Model 2 8 654.8905 57.2903 <0.0001
#> DI Model 3 1 146.9645 102.8524 <0.0001
#> DI Model 4 5 134.5985 18.8396 <0.0001
#> DI Model 5 30 36.7383 0.857 0.686
#>
#> Selected model: Functional group effects 'FG' DImodel with treatment
#> Formula:
#> response = p1 + p2 + p3 + p4 + p5 + p6 + p7 + p8 + p9 + FG_bfg_FG1_FG2 + FG_bfg_FG1_FG3 + FG_bfg_FG2_FG3 + FG_wfg_FG1 + FG_wfg_FG2 + FG_wfg_FG3 + treatmentA
#>
#>
#> --------------------------------------------------------------------------------
#>
#> Step 2: Investigating the treatment effect
#> Selection using F tests
#> Description
#> DI Model 1 Functional group effects 'FG' DImodel
#> DI Model 2 Functional group effects 'FG' DImodel with treatment
#>
#> DI_model treat estimate_theta Resid. Df Resid. SSq Resid. MSq
#> DI Model 1 FG none FALSE 397 1550.682 3.9060
#> DI Model 2 FG 'treatment' FALSE 396 559.711 1.4134
#> Df SSq F Pr(>F)
#> DI Model 1
#> DI Model 2 1 990.9711 701.12 <0.0001
#>
#> Selected model: Functional group effects 'FG' DImodel with treatment
#> Formula:
#> response = p1 + p2 + p3 + p4 + p5 + p6 + p7 + p8 + p9 + FG_bfg_FG1_FG2 + FG_bfg_FG1_FG3 + FG_bfg_FG2_FG3 + FG_wfg_FG1 + FG_wfg_FG2 + FG_wfg_FG3 + treatmentA
#>
#>
#> --------------------------------------------------------------------------------
#>
#> Step 3: Investigating whether theta is equal to 1 or not
#>
#> Theta estimate: 0.968140012342718
#> Selection using F tests
#> Description
#> DI Model 1 Functional group effects 'FG' DImodel with treatment
#> DI Model 2 Functional group effects 'FG' DImodel with treatment, estimating theta
#>
#> DI_model treat estimate_theta Resid. Df Resid. SSq Resid. MSq
#> DI Model 1 FG 'treatment' FALSE 396 559.7110 1.4134
#> DI Model 2 FG 'treatment' TRUE 395 558.9274 1.4150
#> Df SSq F Pr(>F)
#> DI Model 1
#> DI Model 2 1 0.7836 0.5538 0.4572
#>
#> Selected model: Functional group effects 'FG' DImodel with treatment
#>
#> --------------------------------------------------------------------------------
#>
#> Step 4: Comparing the final selected model with the reference (community) model
#> 'community' is a factor with100levels, one for each unique set of proportions.
#>
#> model Resid. Df Resid. SSq Resid. MSq Df SSq F Pr(>F)
#> DI Model 1 Selected 396 559.7110 1.4134
#> DI Model 2 Reference 311 445.9889 1.4340 85 113.7222 0.933 0.6423
#>
#> --------------------------------------------------------------------------------
#> autoDI is limited in terms of model selection. Exercise caution when choosing your final model.
#>
#> --------------------------------------------------------------------------------
```

The output of `autoDI`

, works through the following process:

- Step 1 compared five DI models using F tests. The models each had a different form of species interactions and treatment was always included. The functional group model (FG) was the selected model. This assumes that pairs of species interact according to functional group membership.
- Step 2 provided a test for the treatment and indicated that the treatment, included as an additive factor, was significant and needed in the model (p < 0.0001).
- Step 3 uses profile likelihood to estimate the non-linear parameter \(\theta\) and tests whether or not it differs from one. (Each model in Step 1 was fitted assuming theta equalled one.) Theta was estimated to be 0.96814 and was not significantly different from one (p = 0.4572).
- Step 4 provides a lack of fit test, here there was no indication of lack of fit in the model selected in Step 3 (p = 0.6423).

Further details on each of these steps are available in the `autoDI`

help file. Run the following code to access the documentation.

```
?autoDI
```

All parameter estimates from the selected model can be viewed using `summary`

.

```
summary(auto1)
#>
#> Call:
#> glm(formula = new_fmla, family = family, data = new_data)
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -3.8425 -0.8141 0.0509 0.8048 3.5657
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> p1 9.7497 0.3666 26.595 < 2e-16 ***
#> p2 8.5380 0.3672 23.253 < 2e-16 ***
#> p3 8.2329 0.3666 22.459 < 2e-16 ***
#> p4 6.3644 0.3665 17.368 < 2e-16 ***
#> p5 10.8468 0.3669 29.561 < 2e-16 ***
#> p6 5.9621 0.4515 13.205 < 2e-16 ***
#> p7 5.4252 0.4516 12.015 < 2e-16 ***
#> p8 7.3204 0.4515 16.213 < 2e-16 ***
#> p9 8.2154 0.4515 18.196 < 2e-16 ***
#> FG_bfg_FG1_FG2 3.4395 0.8635 3.983 8.09e-05 ***
#> FG_bfg_FG1_FG3 11.5915 0.8654 13.395 < 2e-16 ***
#> FG_bfg_FG2_FG3 2.8711 1.2627 2.274 0.02351 *
#> FG_wfg_FG1 2.8486 0.9131 3.120 0.00194 **
#> FG_wfg_FG2 0.6793 2.3553 0.288 0.77319
#> FG_wfg_FG3 2.4168 2.3286 1.038 0.29997
#> treatmentA 3.1018 0.1171 26.479 < 2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for gaussian family taken to be 1.413412)
#>
#> Null deviance: 52280.33 on 412 degrees of freedom
#> Residual deviance: 559.71 on 396 degrees of freedom
#> AIC: 1329.4
#>
#> Number of Fisher Scoring iterations: 2
```

If the final model selected by autoDI includes a value of theta other than 1, then a 95% confidence interval for \(\theta\) can be generated using the `theta_CI`

function:

```
theta_CI(auto1$selected_model_obj, conf = .95)
```

Here, this code would not run, since the final model selected by `autoDI`

does not include theta estimated.

`DI`

functionFor some users, the selection process in `autoDI`

will be sufficient, however, most users will fit additional models using `DI`

. For example, while the treatment is included in `autoDI`

as an additive factor, interactions between treatment and other model terms are not considered. Here, we will first fit the model selected by `autoDI`

using `DI`

and then illustrate the capabilities of `DI`

to fit specialised models.

`autoDI`

using `DI`

```
##
m1 <- DI(y = "response", prop = 4:12,
FG = c("FG1","FG1","FG1","FG1","FG1","FG2","FG2","FG3","FG3"), treat = "treatment",
DImodel = "FG", data = sim3)
#> Warning in DI_data_prepare(y = y, block = block, density = density, prop = prop, : One or more rows have species proportions that sum to approximately 1, but not exactly 1. This is typically a rounding issue, and has been corrected internally prior to analysis.
#> Fitted model: Functional group effects 'FG' DImodel
summary(m1)
#>
#> Call:
#> glm(formula = new_fmla, family = family, data = new_data)
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -3.8425 -0.8141 0.0509 0.8048 3.5657
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> p1 9.7497 0.3666 26.595 < 2e-16 ***
#> p2 8.5380 0.3672 23.253 < 2e-16 ***
#> p3 8.2329 0.3666 22.459 < 2e-16 ***
#> p4 6.3644 0.3665 17.368 < 2e-16 ***
#> p5 10.8468 0.3669 29.561 < 2e-16 ***
#> p6 5.9621 0.4515 13.205 < 2e-16 ***
#> p7 5.4252 0.4516 12.015 < 2e-16 ***
#> p8 7.3204 0.4515 16.213 < 2e-16 ***
#> p9 8.2154 0.4515 18.196 < 2e-16 ***
#> FG_bfg_FG1_FG2 3.4395 0.8635 3.983 8.09e-05 ***
#> FG_bfg_FG1_FG3 11.5915 0.8654 13.395 < 2e-16 ***
#> FG_bfg_FG2_FG3 2.8711 1.2627 2.274 0.02351 *
#> FG_wfg_FG1 2.8486 0.9131 3.120 0.00194 **
#> FG_wfg_FG2 0.6793 2.3553 0.288 0.77319
#> FG_wfg_FG3 2.4168 2.3286 1.038 0.29997
#> treatmentA 3.1018 0.1171 26.479 < 2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for gaussian family taken to be 1.413412)
#>
#> Null deviance: 52280.33 on 412 degrees of freedom
#> Residual deviance: 559.71 on 396 degrees of freedom
#> AIC: 1329.4
#>
#> Number of Fisher Scoring iterations: 2
```

`DI`

functionThere are two ways to fit customised models using `DI`

; the first is by using the option `DImodel =`

in the `DI`

function and adding the argument `extra_formula =`

to it, and the second is to use the `custom_formula`

argument in the `DI`

function. If species interaction variables (e.g., the FG interactions or the average pairwise interaction) are included in either `extra_formula`

or `custom_formula`

, they must first be created and included in the dataset. The function `DI_data_prepare`

can be used to compute several types of species interaction variables.

`extra_formula`

```
m2 <- DI(y = "response", prop = 4:12,
FG = c("FG1","FG1","FG1","FG1","FG1","FG2","FG2","FG3","FG3"), treat = "treatment",
DImodel = "FG", extra_formula = ~ (p1 + p2 + p3 + p4):treatment,
data = sim3)
#> Warning in DI_data_prepare(y = y, block = block, density = density, prop = prop, : One or more rows have species proportions that sum to approximately 1, but not exactly 1. This is typically a rounding issue, and has been corrected internally prior to analysis.
#> Fitted model: Functional group effects 'FG' DImodel
summary(m2)
#>
#> Call:
#> glm(formula = new_fmla, family = family, data = new_data)
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -3.6892 -0.7859 0.0436 0.7781 3.6227
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> p1 9.391824 0.540485 17.377 < 2e-16 ***
#> p2 8.492825 0.540879 15.702 < 2e-16 ***
#> p3 8.406038 0.540471 15.553 < 2e-16 ***
#> p4 6.015296 0.540391 11.131 < 2e-16 ***
#> p5 10.802270 0.378776 28.519 < 2e-16 ***
#> p6 5.917565 0.461482 12.823 < 2e-16 ***
#> p7 5.380703 0.461535 11.658 < 2e-16 ***
#> p8 7.275881 0.461506 15.766 < 2e-16 ***
#> p9 8.170907 0.461471 17.706 < 2e-16 ***
#> FG_bfg_FG1_FG2 3.439508 0.865279 3.975 8.38e-05 ***
#> FG_bfg_FG1_FG3 11.591458 0.867140 13.367 < 2e-16 ***
#> FG_bfg_FG2_FG3 2.871063 1.265295 2.269 0.02381 *
#> FG_wfg_FG1 2.848612 0.915008 3.113 0.00199 **
#> FG_wfg_FG2 0.679285 2.360195 0.288 0.77365
#> FG_wfg_FG3 2.416774 2.333420 1.036 0.30097
#> treatmentA 3.190868 0.216493 14.739 < 2e-16 ***
#> `p1:treatmentB` 0.626667 0.668369 0.938 0.34902
#> `p2:treatmentB` 0.001213 0.668369 0.002 0.99855
#> `p3:treatmentB` -0.435322 0.668369 -0.651 0.51522
#> `p4:treatmentB` 0.609180 0.668369 0.911 0.36262
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for gaussian family taken to be 1.419257)
#>
#> Null deviance: 52280.33 on 412 degrees of freedom
#> Residual deviance: 556.35 on 392 degrees of freedom
#> AIC: 1335
#>
#> Number of Fisher Scoring iterations: 2
```

`extra_formula`

First, we create the FG pairwise interactions, the `DI_data_FG`

function.

```
newlist <- DI_data_FG(prop = 4:12, FG = c("FG1","FG1","FG1","FG1","FG1","FG2","FG2","FG3","FG3"),
data = sim3)
sim3a <- data.frame(sim3, newlist$FG)
```

Then we fit the model using `extra_formula`

.

```
m3 <- DI(y = "response", prop = 4:12,
FG = c("FG1","FG1","FG1","FG1","FG1","FG2","FG2","FG3","FG3"),
treat = "treatment", DImodel = "FG",
extra_formula = ~ (bfg_FG1_FG2 + bfg_FG1_FG3 + bfg_FG2_FG3 + wfg_FG1 + wfg_FG2
+ wfg_FG3):treatment, data = sim3a)
#> Warning in DI_data_prepare(y = y, block = block, density = density, prop = prop, : One or more rows have species proportions that sum to approximately 1, but not exactly 1. This is typically a rounding issue, and has been corrected internally prior to analysis.
#> Fitted model: Functional group effects 'FG' DImodel
summary(m3)
#>
#> Call:
#> glm(formula = new_fmla, family = family, data = new_data)
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -3.8251 -0.8208 0.0554 0.7982 3.4218
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> p1 9.68668 0.40000 24.217 < 2e-16 ***
#> p2 8.47495 0.40053 21.159 < 2e-16 ***
#> p3 8.16990 0.39998 20.426 < 2e-16 ***
#> p4 6.30140 0.39987 15.759 < 2e-16 ***
#> p5 10.78379 0.40031 26.938 < 2e-16 ***
#> p6 5.89908 0.47958 12.301 < 2e-16 ***
#> p7 5.36222 0.47963 11.180 < 2e-16 ***
#> p8 7.25740 0.47960 15.132 < 2e-16 ***
#> p9 8.15243 0.47957 17.000 < 2e-16 ***
#> FG_bfg_FG1_FG2 4.00191 1.12383 3.561 0.000415 ***
#> FG_bfg_FG1_FG3 11.77389 1.12973 10.422 < 2e-16 ***
#> FG_bfg_FG2_FG3 3.83681 1.64287 2.335 0.020027 *
#> FG_wfg_FG1 2.81860 1.16226 2.425 0.015757 *
#> FG_wfg_FG2 -1.58378 3.11717 -0.508 0.611682
#> FG_wfg_FG3 1.32358 3.07561 0.430 0.667181
#> treatmentA 3.22783 0.33480 9.641 < 2e-16 ***
#> `treatmentA:bfg_FG1_FG2` -1.12480 1.43053 -0.786 0.432178
#> `treatmentA:bfg_FG1_FG3` -0.36487 1.44450 -0.253 0.800717
#> `treatmentA:bfg_FG2_FG3` -1.93150 2.09024 -0.924 0.356029
#> `treatmentA:wfg_FG1` 0.06003 1.42911 0.042 0.966517
#> `treatmentA:wfg_FG2` 4.52613 4.06260 1.114 0.265924
#> `treatmentA:wfg_FG3` 2.18638 3.99748 0.547 0.584733
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for gaussian family taken to be 1.42436)
#>
#> Null deviance: 52280.3 on 412 degrees of freedom
#> Residual deviance: 555.5 on 390 degrees of freedom
#> AIC: 1338.3
#>
#> Number of Fisher Scoring iterations: 2
```

`custom_formula`

First, we create a dummy variable for level A of the treatment (this is required for the `glm`

engine that is used within `DI`

and because there is no intercept in the model).

```
sim3a$treatmentA <- as.numeric(sim3a$treatment=="A")
```

Then we fit the model using `custom_formula`

.

```
m3 <- DI(y = "response", custom_formula = response ~ 0 + p1 + p2 + p3 + p4 + p5 + p6 + p7 + p8 + p9
+ treatmentA + bfg_FG1_FG2 + bfg_FG1_FG3 + bfg_FG2_FG3, data = sim3a)
#> Fitted model: Custom DI model
summary(m3)
#>
#> Call:
#> glm(formula = custom_formula, family = family, data = data)
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -4.0272 -0.7831 0.0404 0.7570 3.7016
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> p1 10.3417 0.3138 32.957 < 2e-16 ***
#> p2 9.1766 0.3103 29.573 < 2e-16 ***
#> p3 8.8268 0.3134 28.164 < 2e-16 ***
#> p4 6.9742 0.3122 22.341 < 2e-16 ***
#> p5 11.4422 0.3141 36.426 < 2e-16 ***
#> p6 5.9177 0.3994 14.815 < 2e-16 ***
#> p7 5.3967 0.3999 13.496 < 2e-16 ***
#> p8 7.4468 0.3983 18.695 < 2e-16 ***
#> p9 8.3449 0.3984 20.945 < 2e-16 ***
#> treatmentA 3.1018 0.1184 26.198 < 2e-16 ***
#> bfg_FG1_FG2 2.9359 0.8042 3.651 0.000296 ***
#> bfg_FG1_FG3 10.8896 0.8343 13.053 < 2e-16 ***
#> bfg_FG2_FG3 2.9410 1.2233 2.404 0.016667 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for gaussian family taken to be 1.443887)
#>
#> Null deviance: 52280.33 on 412 degrees of freedom
#> Residual deviance: 576.11 on 399 degrees of freedom
#> AIC: 1335.3
#>
#> Number of Fisher Scoring iterations: 2
```

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