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Suppose the response of three different treatments, $A$, $B$ and $C$ are measured in two different hospitals of a country. The data are given below.

enter image description here

My question is: so far I understand it is a randomised block design if I want to compare the treatments. But whatever I search on the internet, I see there is only one observation per each cell. And the model is :

$$Y_{ij}=\beta_0+\beta_i+\gamma_j+\epsilon_{ij},$$

where $\epsilon_{ij}\sim N(0,\sigma^2)$, $\beta_i$ is the effect of the $i$th treatment, and $\gamma_j$ is the effect of the $j$th block.

But in my particular example, there are multiple observations per cell. It makes me doubt if I am adopting the correct experimental design. Is it possible to have more than one observation per cell in a randomised block design? If so, will the model become

$$Y_{ijk}=\beta_0+\beta_{ik}+\gamma_{jk}+\epsilon_{ijk},\quad k=1,\ldots, n$$

???

Could you please give me any reference regarding this?

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    $\begingroup$ It should be something like $Y_{ijk}=\beta_0+\beta_i+\gamma_j+\varepsilon_{ijk}$, where $k=1,2,\ldots,n_{ij}$. $\endgroup$ Apr 28, 2020 at 7:20

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