How is this term "effect of treatment" justified? In Mathematical Statistics with Applications, 5th Ed., by Wackerly, Mendenhall, and Scheaffer, Section 13.5, the authors present the Statistical Model For a One-Way Layout (p. 587):
For $i=1,2,\dots,k$ and $j=1,2,\dots,n_i,$
\begin{align*}
Y_{ij}&=\mu+\tau_i+\varepsilon_{ij}\qquad\text{where}\\
Y_{ij}&=\text{the $j$th observation from population (treatment) $i$}\\
\mu&=\text{the overall mean}\\
\tau_i&=\text{the nonrandom effect of treatment $i,$ where } \sum_{i=1}^k\tau_i=0\\
\varepsilon_{ij}&=\text{random error terms such that $\varepsilon_{ij}$ are independent normally}\\
&\phantom{=}\;\;\text{distributed random variables with $E(\varepsilon_{ij})=0$ and
$V(\varepsilon_{ij})=\sigma^2.$}
\end{align*}
From the perspective of the New Causal Revolution, the terms "cause" and "effect" are defined much more carefully, and are understood more precisely than in older statistical literature (the 5th edition of this book dates to 1996.)
My question is this: how can $\tau_i$ be called the effect of treatment $i?$ Would the perspective of the New Causal Revolution call this sloppy terminology? Or can the New Causal Revolution confirm this terminology? Or would it be accurate in some cases and not in others?
 A: The model you've presented here looks a lot like an ANOVA.  If memory serves me right, ANOVA and many other methods were developed from applications in agriculture, in which plots of land would be given various treatments.  In these applications, the plots of land were given treatments randomly and so the estimated $\tau_i$ would be the effect of the treatment (due to the randomization, assuming the experiment was well controlled).
Contrast this with the way ANOVA and most other linear models can be seen being used at present.  More often than not, the data are observational (or at the very least, not from a controlled experiment with randomization) and so referring to estimated $\tau$'s as "effects" may not be appropriate (depending on whatever causal model you subscribe to).
The causal language which once was appropriate in the times of controlled experiments has persisted even though it is not appropriate all the time (even the phrase "we controlled for..." is not appropriate since you can't control someone's sex or age).  To answer your question, I suspect people doing causal inference would call $\tau$ the effect only if the associated DAG would justify it, else they might call it the "association".
