Represent Competing Scientific Accounts With Candidate Models

From Scientific Explanations to Candidate Models

A regression formula is more than a list of variables. It encodes an account of how an outcome is expected to vary with measured characteristics. If two scientific explanations differ, their candidate models should make that difference visible. Writing one large formula containing every proposed term can conceal the disagreement instead of representing it.

The task here is to translate substantive accounts into candidate specifications. Each candidate uses the same outcome definition and makes its own claims about which predictors matter and how they relate to the outcome. These formulas are proposals to examine, not conclusions that one account is correct. Specifying candidates is also different from comparing their fit or choosing among them.

Definition: A candidate model is a prespecified mathematical description of how the conditional mean of an outcome depends on one or more predictors. Competing candidate models represent distinct substantive accounts of that dependence. They may include different predictors, different functional forms, or different terms describing how predictors work together.

For a continuous outcome \(Y_i\) observed for unit \(i\), a basic linear regression specification can be written as \(Y_i=\mu_i+\varepsilon_i\), where \(\mu_i=E(Y_i\mid\text{the predictors for unit }i)\) is the conditional mean and \(\varepsilon_i\) is the deviation from that mean. The basic condition is \(E(\varepsilon_i\mid\text{the predictors for unit }i)=0\). In the examples, we will treat units as independent and assume that the measured predictors and outcome refer to the same units. These conditions let us describe the candidate mean models clearly; they do not establish that a proposed account is true or that a regression association is causal.

Before writing formulas, state what the outcome and predictors mean, their units, and the scientific claim each account makes. Then translate the claim into terms. A coefficient is interpretable only in light of the other terms in its model, so the formula and its explanation must be considered together.

Distinguishing Additive Effects from an Interaction

Suppose two researchers propose different accounts of plant growth. One says that daily light and weekly water supply each contribute to growth by a fixed amount, regardless of the level of the other input. The other says that the growth response to water depends on how much light the plant receives. These are not just two ways of describing the same formula: they imply different changes in predicted growth.

Worked Example: Water and Light as Competing Accounts

In a hypothetical greenhouse, let \(G_i\) be plant growth in centimeters over a fixed period, \(L_i\) be average daily light in hours, and \(W_i\) be water supplied in liters per week. Assume the same growth period and measurement procedures for every plant. Consider two scientific accounts.

Account A: additive contributions. Light and water each have a constant association with mean growth, with no claim that the effect of either input changes with the other:

E(G_i\mid L_i,W_i)=\beta_0+\beta_1L_i+\beta_2W_i.

Account B: light modifies the water response. The association between water and mean growth may differ at different light levels:

E(G_i\mid L_i,W_i)=\gamma_0+\gamma_1L_i+\gamma_2W_i+\gamma_3L_iW_i.

For Account B, suppose purely for illustration that \(\gamma_0=2\), \(\gamma_1=0.5\), \(\gamma_2=1\), and \(\gamma_3=0.2\). These are hypothetical coefficient values, not estimates from data. At \(L=2\) hours per day and \(W=1\) liter per week, the predicted mean growth is

2+0.5(2)+1(1)+0.2(2)(1)=4.4\text{ cm}.

At those two light levels, increasing water from 1 to 2 liters per week changes the predicted mean by

\gamma_2+\gamma_3L=1+0.2(2)=1.4\text{ cm}.

The change is 1.4 centimeters, not 0.4 centimeters: the 0.4 is only the interaction contribution, and the water main-effect contribution of 1 centimeter must also be included. At \(L=0\), the same increase in water changes predicted mean growth by \(1+0.2(0)=1\) centimeter. Account A, in contrast, says the increase for one additional liter is \(\beta_2\) at every light level.

The interaction model includes both main effects as well as the product term. That structure allows it to represent the stated modification account while keeping light and water contributions in the formula. The two models are distinct because Account A requires a constant water change across light levels, while Account B allows that change to vary.

Why the Main Effects Stay in an Interaction Model

The product \(LW\) represents a change in the association of one predictor with the outcome across values of the other predictor. In a model containing \(L\), \(W\), and \(LW\), the coefficient of \(W\) alone is not the water effect at every light level. It is the water effect when \(L=0\); at another light level the product term also contributes.

The result follows directly by comparing predicted means with water values \(W\) and \(W+1\), holding light fixed:

\begin{aligned} \mu(L,W+1)-\mu(L,W) &=\{\gamma_0+\gamma_1L+\gamma_2(W+1)+\gamma_3L(W+1)\}\\ &\quad-\{\gamma_0+\gamma_1L+\gamma_2W+\gamma_3LW\}\\ &=\gamma_2+\gamma_3L. \end{aligned}

This is a finite-change derivation of the conditional effect for a one-unit increase in water. It also shows why interpreting \(\gamma_2\) as the water effect at all light levels is incorrect. With a product term in the model, a one-unit change in light has predicted effect \(\gamma_1+\gamma_3W\), so its interpretation likewise depends on water.

Keeping both main effects in the interaction specification is important: the term \(LW\) describes how the joint pattern varies, while the main effects describe the model’s component contributions at the reference value zero of the other predictor. If zero is not a meaningful or observed value, the main-effect interpretation may be inconvenient. Centering a predictor can change the reference point for that interpretation, but it does not change the substantive distinction between an additive account and an interaction account.

Competing Accounts Need Not Differ by an Interaction

Scientific accounts can also disagree about which predictors belong in the model. One account may emphasize a single driver; another may propose a different driver or an additional mechanism. The correct candidate set depends on the claims being represented, not on an automatic rule to include every proposed variable in every formula.

Worked Example: Soil Carbon and Habitat Connectivity

Suppose a hypothetical investigation records soil carbon concentration, in grams per kilogram, in forest habitat patches. Let \(A_i\) be patch area in hectares and \(C_i\) be a connectivity index ranging from 0 to 1. The outcome, units, and measurement period are held fixed across two accounts.

Account A: patch area is the relevant predictor. Larger patches are expected to have higher mean soil carbon, and connectivity is not part of this account:

E(Y_i\mid A_i)=\beta_0+\beta_1A_i.

Account B: area and connectivity both contribute. In addition to area, the account proposes a separate association between connectivity and mean soil carbon:

E(Y_i\mid A_i,C_i)=\gamma_0+\gamma_1A_i+\gamma_2C_i.

For an arithmetic illustration, take \(\gamma_0=7\), \(\gamma_1=0.035\) grams per kilogram per hectare, and \(\gamma_2=3\) grams per kilogram per unit of the connectivity index. For a patch with \(A=50\) hectares and \(C=0.6\), Account B gives

7+0.035(50)+3(0.6)=7+1.75+1.8=10.55\text{ grams per kilogram}.

At the same area, changing the connectivity index from 0.2 to 0.6 changes the predicted mean by \(3(0.6-0.2)=1.2\) grams per kilogram under Account B. Account A makes no such connectivity-based change: its formula contains no \(C_i\). The coefficients here are illustrative, not fitted estimates, and the formulas alone do not tell us which account is better supported.

A formula containing both predictors is appropriate for Account B because the account explicitly proposes both contributions. It should not be added merely because connectivity was suggested by someone. Conversely, omitting connectivity from Account A is part of what makes that account distinct, rather than an accidental oversight.

Representing Alternative Explanations

Two accounts can propose different primary predictors for the same outcome. Such models may be non-nested: neither formula is simply the other formula with an extra term. Write each account separately, then check what each one predicts for a unit described by the relevant measurements.

Worked Example: Two Accounts of Stream Oxygen

Consider a hypothetical set of stream measurements. Let \(Y_i\) be dissolved oxygen concentration in milligrams per liter, \(T_i\) be water temperature in degrees Celsius, and \(N_i\) be nutrient concentration in milligrams per liter. One scientific account attributes differences in oxygen chiefly to temperature; another emphasizes nutrient concentration. The outcome and measurement setting are the same, but the proposed predictors differ.

The temperature account is

E(Y_i\mid T_i)=\beta_0+\beta_1T_i.

The nutrient account is

E(Y_i\mid N_i)=\gamma_0+\gamma_1N_i.

For illustration, suppose the temperature account has \(\beta_0=10\) milligrams per liter and \(\beta_1=-0.25\) milligrams per liter per degree Celsius. At \(T=20\) degrees Celsius, its predicted mean is \(10-0.25(20)=5\) milligrams per liter. Suppose the nutrient account has \(\gamma_0=9\) milligrams per liter and \(\gamma_1=-1.5\) milligrams per liter per milligram per liter. At \(N=2\) milligrams per liter, its predicted mean is \(9-1.5(2)=6\) milligrams per liter.

For a stream with both of those measurements, the two candidate formulas therefore give different illustrative predictions: 5 and 6 milligrams per liter. Neither coefficient set is an empirical estimate, and the difference between these predictions does not establish which scientific account is correct. A model containing both \(T_i\) and \(N_i\) would express a third account—that both have distinct contributions conditional on each other—and should be included only if that account is scientifically intended.

A Practical Specification Check

Before treating a set of formulas as meaningful competing candidates, check that their differences are deliberate and interpretable. This is a specification task, not a way to decide which candidate wins.

Fix the outcome and units.
Write down exactly what \(Y_i\) measures, for whom or what, and over what period. Use the same outcome definition across the candidate models.

State each account in words.
Identify the proposed mechanism or substantive claim before translating it into a formula. Distinguish, for example, “both predictors contribute independently” from “one predictor changes the response to the other.”

Translate claims into terms.
Use main effects for proposed additive contributions, a product term for a specified interaction, or different predictor sets for different proposed drivers. Include the terms required to make the chosen relationship interpretable.

Check meaning and distinguishability.
Confirm that units are clear, that the model’s predictions make sense over the intended range, and that the proposed terms can in principle be distinguished in the data. If two predictors always take exactly proportional values, their separate linear contributions cannot be identified from those data.

Keep the rationale visible.
Record why a term is included or omitted in each candidate. Do not silently turn alternatives into a single formula that no longer represents either account.

Why Combining Every Proposed Term Can Mislead

A formula with many terms may look comprehensive, but it may answer a different scientific question from any of the proposed accounts. If one account says that light and water act additively and another says that light modifies the water response, fitting only a model with an interaction does not preserve the additive account as a separate candidate. Likewise, automatically adding a predictor that one account explicitly excludes changes that account’s claim.

There is also a practical issue: a model’s terms need information in the data to be distinguished. If two predictors are nearly identical in the observed sample, their separate contributions may be hard to estimate precisely; if they are exact multiples, the coefficients cannot be uniquely separated. This does not tell us which scientific explanation is right. It warns that the observed measurements may not distinguish the proposed terms well.

The purpose of candidate specification is therefore to make the scientific alternatives explicit before judging them. Later questions about model fit, complexity, and fair comparisons require candidates that have first been written clearly. Do not confuse a well-stated candidate with a selected or validated model.

Common Mistakes

Putting every suggested predictor into one formula. That creates a combined account, which may be scientifically different from the alternatives. Write separate formulas when the claims differ.
Calling an interaction coefficient the full effect. In an interaction model, the effect of one predictor includes its main-effect coefficient and the product-term contribution at the other predictor’s value.
Leaving out a main effect when the account includes an interaction. For the usual linear interaction specification, include both component predictors as well as their product; otherwise the model imposes additional restrictions that may not match the stated account.
Interpreting a candidate as a causal conclusion. A formula represents a proposed conditional relationship. Writing it does not establish that changing a predictor would cause a change in the outcome.
Treating different formulas as evidence that one is better. Different predictions make the scientific disagreement clear; they do not, by themselves, resolve it.

Key takeaway: A candidate model should express a specific scientific account, not merely collect available variables. Keep the outcome definition fixed, encode the substantive differences in the formulas, and interpret each term in the context of the other terms in its model.

Check Your Understanding

Try these questions using the distinctions in this tutorial.

In a model with light, water, and their product, what does the coefficient of water alone describe?
Write an additive candidate model for an outcome \(Y_i\) with predictors \(X_i\) and \(Z_i\). What scientific claim does it encode about their contributions?
Why might a model containing two predictors represent a different account from a model containing only one, rather than simply being a more complete version of it?
For \(E(Y_i\mid X_i,Z_i)=\gamma_0+\gamma_1X_i+\gamma_2Z_i+\gamma_3X_iZ_i\), derive the predicted change for a one-unit increase in \(Z_i\) at fixed \(X_i\).
Two proposed models use different predictors to explain the same outcome. What should be held constant when writing them as competing candidates, and what does their difference alone fail to establish?