tips:models_with_or_without_intercept

# Differences

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tips:models_with_or_without_intercept [2020/10/31 08:46] – Wolfgang Viechtbauer | tips:models_with_or_without_intercept [2021/02/12 16:09] – Wolfgang Viechtbauer | ||
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& | & | ||

\end{align} | \end{align} | ||

- | But what about the contrast between random and systematic allocation? It turns out that we can obtain this from the model as the difference between the $\beta_1$ and $\beta_2$ coefficients. In particular, if we subtract $\beta_1$ from $\beta_2$, then | + | But what about the contrast between systematic and random allocation? It turns out that we can obtain this from the model as the difference between the $\beta_1$ and $\beta_2$ coefficients. In particular, if we subtract $\beta_1$ from $\beta_2$, then |

$$ | $$ | ||

\beta_2 - \beta_1 = (\mu_r - \mu_a) - (\mu_s - \mu_a) = \mu_r - \mu_s | \beta_2 - \beta_1 = (\mu_r - \mu_a) - (\mu_s - \mu_a) = \mu_r - \mu_s | ||

$$ | $$ | ||

- | so this difference reflects how different random allocation is compared to systematic allocation. Using the '' | + | so this contrast reflects how different systematic allocation is compared to random allocation. Using the '' |

<code rsplus> | <code rsplus> | ||

anova(res, L=c(0, | anova(res, L=c(0, | ||

Line 224: | Line 224: | ||

</ | </ | ||

- | It is important to realize that this does not test whether there are differences between the different forms of allocation (this is what we tested earlier in the model that included the intercept term). However, we can again use contrasts of the model coefficients to test differences between the levels. For example, let's test the difference between alternating and random allocation and the difference between systematic allocation and random allocation: | + | It is important to realize that this does not test whether there are differences between the different forms of allocation (this is what we tested earlier in the model that included the intercept term). However, we can again use contrasts of the model coefficients to test differences between the levels. Let's test all pairwise differences (i.e., between random and alternating allocation, between systematic and alternating allocation, and between systematic and random allocation): |

<code rsplus> | <code rsplus> | ||

- | anova(res, L=rbind(c(-1, | + | anova(res, L=rbind(c(-1,,1), c(0,-1,1))) |

</ | </ | ||

<code output> | <code output> | ||

- | Hypotheses: | + | Hypotheses: |

- | 1: | + | 1: |

- | 2: -factor(alloc)alternate + factor(alloc)systematic = 0 | + | 2: -factor(alloc)alternate + factor(alloc)systematic = 0 |

+ | 3: -factor(alloc)random + factor(alloc)systematic = 0 | ||

Results: | Results: | ||

- | | + | |

- | 1: -0.4478 0.5158 -0.8682 0.3853 | + | 1: -0.4478 0.5158 -0.8682 0.3853 |

- | 2: | + | 2: |

+ | 3: | ||

+ | </ | ||

+ | These are now the exact same results we obtained earlier for the model that included the intercept term. | ||

+ | Note that the output does not contain an omnibus test for the three contrasts because the matrix with the contrast coefficients ('' | ||

+ | <code rsplus> | ||

+ | anova(res, L=rbind(c(-1, | ||

+ | </ | ||

+ | <code output> | ||

Omnibus Test of Hypotheses: | Omnibus Test of Hypotheses: | ||

QM(df = 2) = 1.7675, p-val = 0.4132 | QM(df = 2) = 1.7675, p-val = 0.4132 | ||

</ | </ | ||

- | These are now the exact same results we obtained earlier for the model that included the intercept term. | ||

==== Parameterization ==== | ==== Parameterization ==== | ||

- | What the example above shows is that, whether we remove the intercept or not, we are essentially fitting the same model, but using a different [[wp> | + | What the example above shows is that, whether we remove the intercept or not, we are essentially fitting the same model, but using a different [[wp>(on regression models in general) is Christensen (2011). |

==== Models with Continuous Moderators ==== | ==== Models with Continuous Moderators ==== |

tips/models_with_or_without_intercept.txt · Last modified: 2022/08/03 11:34 by Wolfgang Viechtbauer