= Homogeneity-study time-term identifiability
\b[Homogeneity-study time-term identifiability] asks whether a measurement schedule supplies enough independent information to distinguish <instrumental drift> from differences between containers. Adding time to a <regression analysis> does not guarantee that its coefficient can be estimated separately from the container effects.
In an original fixed-container <measurement model>,
$$
y_{ij}=a_i+\gamma t_{ij}+e_{ij},
$$
each container has an unknown intercept $a_i$ and all share a linear drift rate $\gamma$. If every observation of container $i$ occurs at one time $t_i$, any rate $\gamma'$ gives the same predictions after replacing $a_i$ by $a_i+(\gamma-\gamma')t_i$. The data cannot distinguish those parameter sets. Algebraically, the time column lies in the span of the container-indicator columns of the <design matrix>.
Repeated observations at different times within at least one container can break that exact ambiguity. Subtract each container's mean:
$$
y_{ij}-\overline y_i=\gamma(t_{ij}-\overline t_i)+(e_{ij}-\overline e_i).
$$
For independent, equal-variance errors with variance $\sigma^2$, the least-squares drift estimate has
$$
\operatorname{Var}(\widehat\gamma)=\frac{\sigma^2}{\sum_{i,j}(t_{ij}-\overline t_i)^2},
$$
provided the denominator is positive. The formula distinguishes <identifiability> from precision: a positive but tiny within-container time spread permits a formally unique yet noisy estimate.
For three containers each observed twice one time unit apart, the denominator is $3/2$. If each pair is instead separated by five time units, it is $75/2$. With the same residual variance and a valid linear-drift model, the latter schedule reduces drift-estimate variance by 25, or its standard deviation by five. These are design calculations, not a guarantee that a longer experiment remains linear or stable.
Randomized order and controls address additional risks. The primary IUPAC protocol requires randomized analysis of duplicate portions and inspection for suspect data features.<The International Harmonized Protocol for the proficiency testing of analytical chemistry laboratories by Michael Thompson and others>[Appendix 1, A1.1 and A1.3]{ref} NIST's comparison designs also use observation order to suppress specified drift contributions.<The Gauge Block Handbook by Ted Doiron and John S. Beers>[§5.6.1, drift-eliminating designs]{ref}
Record timestamps and check the <rank of a matrix> for the fitted design, but also assess residual time patterns, uncertainty and physical plausibility. A randomized schedule can still have poor numerical leverage; a full-rank model can still be wrong. <Drift-aware homogeneity analysis> must report those distinctions rather than equating a returned regression coefficient with established material homogeneity.
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