Package: mclogit 0.9.17.1

$$\ln\frac | References

$$\ln\frac | References

Application of the technique to conditional and baseline logit models | In case of the models supported by the mclogit package, the difference betweenFirth's PML technique and conventional ML estimation is that it solves for eachcoefficient $\alpha_q$ the modified gradient equation$$0 | U^*(\alpha_q) | \frac{\partial\ell}{\partial\alpha_q}+\frac12\mathrm{tr}\left((\boldsymbol{X}'\boldsymbol{W}\boldsymbol{X})^{-1}\boldsymbol{X}'\frac{\partial\boldsymbol{W}}{\partial\alpha_q}\boldsymbol{X}\right)$$instead of$$0 | U(\alpha_q) | Like the conventional MLE, the mclogit package uses an IWLS-algorithm,however, using a modified working response vector with elements$$y_{ij}^* | \boldsymbol{x}{ij}'\boldsymbol{\alpha} +\frac{y{ij}-\pi_{ij}}{\pi_{ij}}+\frac12\sum_r\sum_s\zeta_{ij,rs}I^{(r,s)}$$where $I^{(r,s)}$ is the $r,s$ element of$(\boldsymbol{X}'\boldsymbol{W}\boldsymbol{X})^{-1}$and$$\begin{aligned}\zeta_{ij,rs}&=(x_{ijr}-\sum_kx_{ikr}\pi_{ik})(x_{ijs}-\sum_l\pi_{il}x_{ils}) | A comparison of results obtained with other packages | References

$$\frac | \sum_ | \sum_{i,j}\boldsymbol{x}{ij}(n{ij}-n_{i+}\pi_{ij}) | \sum_{i,j}\boldsymbol{x}{ij}n{i+}(y_{ij}-\pi_{ij}) | $$\frac | \sum_ | \sum_{i,j,k}\boldsymbol{x}{ij}n{i+}(\delta_{jk}-\pi_{ij}\pi_{ik})\boldsymbol{x}_{ij}' | $$\boldsymbol | \boldsymbol | \left(\frac | $$\boldsymbol | \boldsymbol{X}'\boldsymbol{W}\boldsymbol{X}\boldsymbol{\alpha}^{(s)}+\boldsymbol{X}'\boldsymbol{N}(\boldsymbol{y}-\boldsymbol{\pi}) | \boldsymbol{X}'\boldsymbol{W}\left(\boldsymbol{X}\boldsymbol{\alpha}^{(s)}+\boldsymbol{W}^-\boldsymbol{N}(\boldsymbol{y}-\boldsymbol{\pi})\right) | $$\boldsymbol | $$y_ | References

The problem | $$\mathcal{L}_{\text{cpl}}(\boldsymbol{y},\boldsymbol{b}) | $$\mathcal{L}_{\text{obs}}(\boldsymbol{y}) | \int\exp\left(\ell_ | The Laplace approximation and PQL | Laplace approximation | $$\ell_{\text{cpl}}(\boldsymbol{y},\boldsymbol{b})\approx\ell(\boldsymbol{y}|\tilde{\boldsymbol{b}};\boldsymbol{\alpha}) | $$\ell^*_ | \ln\int \exp(\ell_ | \ell(\boldsymbol{y}|\tilde{\boldsymbol{b}};\boldsymbol{\alpha}) | \frac12\tilde | \frac12\ln\det(\boldsymbol{\Sigma}) | Penalized quasi-likelihood (PQL) | $$\begin | The Solomon-Cox approximation and MQL | The Solomon-Cox approximation | $$\ell_ | $$\ln\int \exp(\ell_ | \ell(\boldsymbol{y}|\boldsymbol{0};\boldsymbol{\alpha}) | \frac12\boldsymbol{g}_0'\left(\boldsymbol{H}_0+\boldsymbol{\Sigma}^{-1}\right)^{-1}\boldsymbol{g}_0 | Marginal quasi-likelhood (MQL) | References

References