Choose observations that distinguish competing explanations

Try

You want to estimate an intercept and slope in y=a+bt. Two measurements taken at the same time give repeated information about one combination of a and b. Two distinct times can identify both in the noiseless case. The number of observations is therefore not enough to judge the design.

Write each observation as a row (1,t) multiplying the parameter vector. This turns the design question into a rank question before any data values are collected.

Predict

Compare designs at times (1,1), (0,1) and (0,3). Predict which is singular and which gives a more precise slope under equal independent measurement variance. Then consider whether the linear model is credible over the widest time span.

Experiment

For two distinct times, b̂=(y₂−y₁)/(t₂−t₁). If both errors have variance σ² and are independent, its variance is 2σ²/(t₂−t₁)².

Times	Separation	Slope variance
1,1	0	Not identifiable from these rows
0,1	1	2σ²
0,3	3	2σ²/9

The larger separation improves precision within the linear model. If curvature appears over that interval, the precision calculation does not remove the resulting model bias. Design must balance information with validity of the assumed regime.

Understand

For linear observations with equal independent noise, XᵀX summarizes parameter information. A zero determinant signals rank loss; a small eigenvalue signals a weakly measured direction. Maximizing a determinant can be a useful design criterion under specified assumptions, but it is not a universal measure of scientific value. The output of interest, parameter scales and feasible observation conditions also matter.

Replication at a fixed condition estimates noise and reduces the variance of its mean. With n independent equal-variance replicates, the mean variance is σ²/n. Correlation changes that formula. Replication does not create a missing parameter direction when all design rows remain identical.

Intervention questions require causal assumptions in addition to algebraic rank. Observing different inputs can be confounded by other variables. Random assignment, when appropriate and permitted, or another justified identification strategy can support a causal comparison. An invertible regression matrix alone does not prove that changing an input will cause the fitted response.

A small measurement budget should assign a role to every observation. Some conditions distinguish parameters, some replicate for noise checks, and some remain untouched for evaluation. If all observations are used repeatedly to select the model, there is no independent final test left. In a teaching investigation, synthetic pilots can reveal design weaknesses before any real collection, but they do not establish real-world precision.

Apply

Allocate eight observations across at least two conditions, replication and validation. Compute a simple rank or slope-variance comparison for two candidate designs. State assumptions about noise and model validity, and explain how the design would change if measurements were expensive at extreme inputs. Include a plan for an observation that contradicts your preferred mechanism.

Continue

Verification and reproducibility next ensure that the proposed computation can be checked independently. A well-designed experiment can still be undermined by a solver error or an undocumented processing step.