Correction

View correction statement:

Correction

This article refers to:

Correction

Article title: Complex sampling designs in large-scale education surveys: a two-level sample distribution approach

Authors: Ting Shen and Spyros Konstantopoulos

Journal: The Journal of Experimental Education

DOI: https://doi.org/10.1080/00220973.2021.1891007

When the article was first published online, the Equations (4), (5), (6), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), (19), (20), (21), (22), and (23) have been published incorrectly.

The original article has been amended for correct equations and republished online.

The correct equations as follows: (4) $$f_s\left(y_{ij}\right|{\mathit{X}}_{ij})=f_p(y_{ij}|{\mathit{X}}_{ij},\mathrm{ }{I}_{ij}=1)=\frac{\mathit{Prob}(I_{ij}=1|y_{ij},{\mathit{X}}_{ij})}{\mathit{Prob}\mathrm{ }\left(I_{ij}=1|{\mathit{X}}_{ij}\right)}{f}_p\left(y_{ij}|{\mathit{X}}_{ij}\right)$$(4) (5) $$f_s\left(y_j\right|{\mathit{X}}_j)=\frac{\mathit{Prob}(I_j=1|y_j,{\mathit{X}}_j)}{\mathit{Prob}\left(I_j=1|{\mathit{X}}_j\right)}{f}_p(y_j|{\mathit{X}}_j).$$(5) (6) $$\mathit{Prob}(I_{ij}=1|y_{ij},{\mathit{X}}_{ij})=E_p({\pi }_{ij}|y_{ij},{\mathit{X}}_{ij}\mathbf{)}\mathbf{,}$$(6) (9) $$\mathit{Prob}\left(I_j=1|{\mathit{X}}_j\right)=E_p\left({\pi }_j\right|{\mathit{X}}_j).$$(9) (10) $$f_s\left(y_{ij}\right|{\mathit{X}}_{ij})=\frac{E\left({\pi }_{ij}\right|y_{ij},{\mathit{X}}_{ij})}{E_p\left({\pi }_{ij}\right|{\mathit{X}}_{ij})}{f}_p(y_{ij}|{\mathit{X}}_{ij}),$$(10) (11) $$f_s\left(y_j\right|{\mathit{X}}_j)=\frac{E_p\left({\pi }_j\right|y_j,{\mathit{X}}_j)}{E_p\left({\pi }_j\right|{\mathit{X}}_j)}{f}_p(y_j|{\mathit{X}}_j).$$(11) (12) $$E_p\left({\pi }_{ij}\right|y_{ij},{\mathit{X}}_{ij})=\hspace{0.17em}\mathit{Exp}\hspace{0.17em}\left(d_0{y}_{ij}+{\mathit{X}}_{ij}{\mathit{D}}_1\right),$$(12) (13) $$E_p\left({\pi }_j\right|y_j,{\mathit{X}}_j)=\hspace{0.17em}\mathit{Exp}\hspace{0.17em}(b_0{y}_j+{\mathit{X}}_j{\mathit{B}}_1)$$(13) (14) $$E\left(w_{i|j}\right|y_{ij},{\mathit{X}}_{ij})=1/E_p({\pi }_{ij}|y_{ij},{\mathit{X}}_{ij})=Exp\left[-(d_0{y}_{ij}+{\mathit{X}}_{ij}{\mathit{D}}_1)\right],$$(14) (15) $$E\left(w_j\right|y_j,{\mathit{X}}_j)=1/E_p({\pi }_j|y_j,{\mathit{X}}_j)=\mathit{Exp}\hspace{0.17em}\left[-(b_0{y}_j+{\mathit{X}}_j{\mathit{B}}_{\mathbf{1}}\mathbf{)}\right].$$(15) (16) $$E_s\left(y_{ij}\right|{\mathit{X}}_{ij})=y_j+{\mathit{X}}_{ij}{\mathit{D}}_1+d_0{\sigma }_e^2;\mathrm{ }\mathit{Var}\mathrm{ }(y_{ij}\left|{\mathit{X}}_{ij}\right)={\sigma }_e^2,$$(16) (17) $$E_s\left(y_j\right|{\mathit{X}}_j)={\gamma }_{00}+{\mathit{X}}_j{\mathit{B}}_1+b_0{\sigma }_a^2;\mathrm{ }\mathit{Var}\mathrm{ }(y_j\left|{\mathit{X}}_j\right)={\sigma }_a^2.$$(17) (18) $$p\left(\theta \right|y)\propto p(\theta \left)f\right(y\left|\theta \right),$$(18) (19) $$p\left(\theta \right)=p{(\gamma }_{00},\mathrm{ }{\mathit{B}}_1,\mathrm{ }{\mathit{\Gamma }}_1,\mathrm{ }{\sigma }_a^2,{\sigma }_e^2)=p({\gamma }_{00}\left)p\right({\mathit{B}}_1\left)p\right({\mathit{\Gamma }}_1\left)p\right({\sigma }_a^2\left)p\right({\sigma }_e^2).$$(19) (20) $$f\left(\theta \right|y)=f({\gamma }_{00},{\mathit{B}}_1,{\mathit{\Gamma }}_1,{\sigma }_a^2,{\sigma }_e^2\left|y\right).$$(20) (21) $$p\left(\theta \right|y)\approx p({\gamma }_{00}\left)p\right({\mathit{B}}_1\left)p\right({\mathit{\Gamma }}_1\left)p\right({\sigma }_a^2\left)p\right({\sigma }_e^2\left)f\right({\gamma }_{00},{\mathit{B}}_1,{\mathit{\Gamma }}_1,{\sigma }_a^2,{\sigma }_e^2\left|y\right).$$(21) (22) $$P\left(\theta \right|y)=\int \prod _{j=1}^m\left[N\right({\theta }_j\left|{{\gamma }_{00},\sigma }_a^2\right)\left]p\right({{\gamma }_{00},\sigma }_a^2\left)d\right({{\gamma }_{00},\sigma }_a^2).$$(22) (23) $${P(\gamma }_{00}\left|y\right)=\int \int \int \int f({\gamma }_{00},{\mathit{B}}_1,{\mathit{\Gamma }}_1,{\sigma }_a^2,{\sigma }_e^2|y)d_{{\mathit{B}}_1}{d}_{{\mathbf{\Gamma }}_1}{d}_{{\sigma }_a^2}{d}_{{\sigma }_e^2}.$$(23)