Flag to indicate whether the batch used the new process (newprocess)

Processing time for each batch, in hours (time)

Temperature of the batch, in degrees Celsius (temp)

Categorical variable indicating the supplier (A,B, orC) of the chemical used in the batch (supplier)

Number of defects in the batch (defects)

$${\text{defects}}_{ij}$$is the number of defects observed in the batch produced by factory $$i$$during batch $$j$$.

$${\mu }_{ij}$$is the mean number of defects corresponding to factory $$i$$(where $$i=1,2,...,20$$) during batch $$j$$(where $$j=1,2,...,5$$).

$${\text{newprocess}}_{ij}$$, $${\text{time}\text{\_}\text{dev}}_{ij}$$, and $${\text{temp}\text{\_}\text{dev}}_{ij}$$are the measurements for each variable that correspond to factory $$i$$during batch $$j$$. For example, $${\text{newprocess}}_{ij}$$indicates whether the batch produced by factory $$i$$during batch $$j$$used the new process.

$${\text{supplier}\text{\_}\text{C}}_{ij}$$and $${\text{supplier}\text{\_}\text{B}}_{ij}$$are dummy variables that use effects (sum-to-zero) coding to indicate whether companyCorB, respectively, supplied the process chemicals for the batch produced by factory $$i$$during batch $$j$$.

$$b_i\sim N(0,{\sigma }_b^2)$$is a random-effects intercept for each factory $$i$$that accounts for factory-specific variation in quality.