Table 1 Estimators of treatment effect and variance estimators in a homogeneous study population

ANOVA-Post

\( {\hat{\beta}}_{1, ols}^{(1)}={\overline{y}}_{.1{t}_1}-{\overline{y}}_{.0{t}_1} \)

U

\( \mathit{\operatorname{var}}\left({\hat{\beta}}_{1, ols}^{(1)}\right)=\frac{\sigma_1^2}{n_0}+\frac{\sigma_1^2}{n_1} \)

\( {\hat{\mathit{\operatorname{var}}}}_{ols}\left({\hat{\beta}}_{1, ols}^{(1)}\right)=\frac{{\hat{\sigma}}_1^2}{\sum_{j=0}^1{\sum}_{i=1}^{n_j}{\left({G}_{ij}-{G}_{..}\right)}^2} \)

\( {\hat{\sigma}}_1^2=\frac{\sum_{j=0}^1{\sum}_{i=1}^{n_j}{\left({y}_{ij{t}_1}-{\hat{y}}_{ij{t}_1}\right)}^2}{\left({n}_0+{n}_1-2\right)} \)

ANCOVA-Post I

\( {\hat{\beta}}_{1, ols}^{(2)}=\left({\overline{y}}_{.1{t}_1}-{\overline{y}}_{.0{t}_1}\right)-{\hat{\beta}}_{2, ols}^{(2)}\left({\overline{y}}_{.1{t}_0}-{\overline{y}}_{.0{t}_0}\right) \)

C

\( \mathit{\operatorname{var}}\left({\hat{\beta}}_{1, ols}^{(2)}|{Y}_{ij{t}_0}\right)=\left(\frac{1}{n_0}+\frac{1}{n_1}+\frac{{\left({\overline{y}}_{.1{t}_0}-{\overline{y}}_{.0{t}_0}\right)}^2}{\sum_{j=0}^1{\sum}_{i=1}^{n_j}{\left({y}_{ij{t}_0}-{\overline{y}}_{.j{t}_0}\right)}^2}\right){\sigma}_{\epsilon^{(2)}}^2 \),

\( {\sigma}_{\epsilon^{(2)}}^2=\left(1-{\rho}^2\right){\sigma}_1^2 \)

\( {\hat{\mathit{\operatorname{var}}}}_{ols}\Big({\hat{\beta}}_{1, ols}^{(2)}\left|{Y}_{ij{t}_0}\right)=\left(\frac{1}{n_0}+\frac{1}{n_1}+\frac{{\left({\overline{y}}_{.1{t}_0}-{\overline{y}}_{.0{t}_0}\right)}^2}{\sum_{j=0}^1{\sum}_{i=1}^{n_j}{\left({y}_{ij{t}_0}-{\overline{y}}_{.j{t}_0}\right)}^2}\right){\hat{\sigma}}_{e_{ij}^{(2)}}^2 \),

\( {\hat{\sigma}}_{e_{ij}^{(2)}}^2=\frac{\sum_{j=0}^1{\sum}_{i=1}^{n_j}{\left({y}_{ij{t}_1}-{\hat{y}}_{ij{t}_1}\right)}^2}{\left({n}_0+{n}_1-4\right)} \)

U

\( \mathit{\operatorname{var}}\left({\hat{\beta}}_{1, ols}^{(2)}\right)=\left(\frac{1}{n_0}+\frac{1}{n_1}\right)\left(1-{\rho}^2\right){\sigma}_1^2 \)

RM

\( {\hat{\gamma}}_{3,\kern0.5em gls}^{(3)}=\left({\overline{y}}_{.1{t}_1}-{\overline{y}}_{.1{t}_0}\right)-\left({\overline{y}}_{.0{t}_1}-{\overline{y}}_{.0{t}_0}\right) \)

U

\( \mathit{\operatorname{var}}\left({\hat{\gamma}}_{3,\kern0.5em gls}^{(3)}\right)=\left(\frac{1}{n_0}+\frac{1}{n_1}\right)\left({\sigma}_1^2+{\sigma}_0^2-2\rho {\sigma}_0{\sigma}_1\right) \)

cRM

\( {\hat{\gamma}}_{3,\kern0.5em gls}^{(4)}=\left({\overline{y}}_{.1{t}_1}-{\overline{y}}_{.0{t}_1}\right)-\frac{\rho {\sigma}_0{\sigma}_1}{\sigma_0^2}\left({\overline{y}}_{.1{t}_0}-{\overline{y}}_{.0{t}_0}\right) \)

U

\( \mathit{\operatorname{var}}\left({\hat{\gamma}}_{3, gls}^{(4)}\right)=\left(\frac{1}{n_0}+\frac{1}{n_1}\right)\left(1-{\rho}^2\right){\sigma}_1^2 \)

ANOVA-Change

\( {\hat{\beta}}_{1, ols}^{(5)}=\left({\overline{y}}_{.1{t}_1}-{\overline{y}}_{.1{t}_0}\right)-\left({\overline{y}}_{.0{t}_1}-{\overline{y}}_{.0{t}_0}\right) \)

U

\( \mathit{\operatorname{var}}\left({\hat{\beta}}_{1, ols}^{(5)}\right)=\left(\frac{1}{n_0}+\frac{1}{n_1}\right)\left({\sigma}_1^2+{\sigma}_0^2-2\rho {\sigma}_0{\sigma}_1\right) \)

\( {\hat{\mathit{\operatorname{var}}}}_{ols}\left({\hat{\beta}}_{1, ols}^{(5)}\right)=\frac{{\hat{\sigma}}_{\epsilon^{(5)}}^2}{\sum_{j=0}^1{\sum}_{i=1}^{n_j}{\left({G}_{ij}-{G}_{..}\right)}^2}, \)

\( {\hat{\sigma}}_{\epsilon^{(5)}}^2=\frac{\sum_{j=0}^1{\sum}_{i=1}^{n_j}{\left({\Delta }_{ij}-{\hat{\Delta }}_{ij}^{(5)}\right)}^2}{\left({n}_0+{n}_1-2\right)} \)