# Question on this study's design

I am struggling to convince myself that my answer to this question for my stats assignment is correct as the way the question was asked is making me doubt my own answer. In brief, we have a supposed n=8 number of babies, where the study is interested in observing the weight growth (in ounces) of these babies over the span on 4 days.

In perspective, the sample table for the data is:-

Baby: Day1, Day2, Day3, Day4

1. 116, 116, 117, 118
2. 91, 92, 93, 95
3. 125, 127, 127, 128

.....

1. 89, 90, 91, 93

The question I'm having issues with is:-

1) 'What is the design of the study? What are the fixed factors and random factors?'

This looks to me like a one-way repeated measures ANOVA is applicable. The day groupings are definitely the fixed factor here. What I am confused about, is that this study is not quite interested in the between subjects interaction but rather, the within subjects growth rate. As such, even though the random assignment of 8 babies seem like a random factor, it isn't relevant in a one-way repeated measure ANOVA, is it? (Ie. why would the question ask for multiple factors when this is a one-way design? Or can you actually have more than one random/fixed factor even though it is one-way? Or.. is this actually a 2-way mixed design?)

I would analyze this dataset with a random intercept ($b_0$) and a random slope ($b_1$) for time for every baby, together with a fixed intercept ($\beta_0$) and slope ($\beta_1$) for time as follows:

$$E(W_{ij}) = \beta_0 + \beta_1t_i + b_{0j} + b_{1j}t_i$$

with $t_i$ the time indicator of day $i$, $W_{ij}$ the weight of baby $j$ at time $i$. Hereby $b_0 \sim N(0, \sigma^2_0)$ and $b_1 \sim N(0, \sigma^2_1)$. The day groupings have thus a fixed effect and a random effect part. The variance of the random slopes $\sigma^2_1$ represents the within-subjects variability in growth rates. I would agree that this is indeed a mixed design.