Environmental exposures often show a time-lagged association with outcomes [1–3]. Distributed lag models have been used to capture such lag patterns by incorporating time-lagged values of exposures, with the corresponding of the lag structure approximated by polynomials or splines [1, 4]. These models require the correct input of cut-off time, or pre-specified window (hereafter termed lag length), after which the association diminishes to a constant level, typically zero [5, 6]. However, lag length is often unknown [5–7]. To fit distributed lag models without specifying lag length, we revisit transfer functions (TFs), a method to specify time-lagged associations commonly used in econometrics and introduced to epidemiology in 1991 [8–10]. We provide a case study to capture the time-lagged association between weekly purchasing outcome of sugar-sweetened drinkable yogurt and weekly-varying display promotion of these beverages, which is an obesogenic food environmental exposure in supermarkets.

TFs capture a time-lagged exposure-outcome association using a structural variable, denoted E _t , which summarizes the current association (at time t) and cumulative association (up to time t) between the outcome variable Y _t and time-lagged exposure variable X _t−1 + X _t−2 + X _t−3+... [8, 11] (Supplementary Appendix S1). We illustrate a simple form of TF to capture a commonly observed shape of lag pattern, a monotonically decreasing association of outcome and lagged exposure, often called the Koyck decay [12]. Using the decay coefficient of lagged association λ up to lag h, the decreasing associations are represented as E t = β X t + λ 1 β X t − 1 + λ 2 β X t − 2 + ⋯ + λ h β X t − h , which recursively reduces to E t = β X t + λ E t − 1 , E 0 = 0

The coefficient β captures the immediate association at time t, and the value of decay coefficient λ closer to 1 implies a more persistent association over time (i.e., slower decay), while a value closer to zero indicates a shorter lag [12, 13]. Constraining λ to be 0 < λ < 1 ensures the association monotonically decaying towards zero when the value of β is positive (Supplementary Figure S1A), and previous studies also imposed the decay towards zero [14, 15]. The variable E _t is added to a time-series regression for the outcome Y _t to estimate β and λ as Y _t = E _t + Z _t γ + ε _t , where Z _t represents a set of covariates and intercept with coefficients γ, and ε _t represents the error term [10, 13].

A visual interpretation of a lagged association combining these coefficients is provided by an impulse response function (IRF), representing the change of the outcome Y _t+0 + Y _t+1 + Y _t+2 + … + Y _t+h to an impulse (one-unit increase of x at time t only), while holding other variables constant [16]. The IRF of the Koyck decay is β + βλ ¹ + βλ ² + … + βλ ^h , visualized in Figure 1.

The general specification of the TF capturing various shapes of lag structure is E t = β 0 X t − 0 + β 1 X t − 1 … β p X t − p + λ 0 E t − 0 + λ 1 E t − 1 + … + λ q E t − q (1) where the Koyck decay is captured by p = 0, q = 1 in Eq. 1 above. More complex shapes are specified by higher values of p and q (Figure 2; Supplementary Appendix S2), allowing generalization to classical lag models, such as the Almon polynomial [10, 17].

Unlike commonly used distributed lag models, TF models obviates pre-specification of a lag length h, but require prior biological and epidemiological knowledge to help select plausible shapes of the lag (values of p and q). Deciding among candidate shapes is facilitated by model selection using fit metrics such as an information criterion [11].

The exposure is the weekly within-store display promotion of sugar-sweetened food items that potentially exhibits time-lagged association with the number of these items sold (outcome). Display promotion is the temporary placement of items in prominent locations to increase sales of (typically) ultra-processed food [18]. Our food of interest is sugar sweetened (not plain) drinkable yogurt, a hidden and important source of dietary sugar among children [19, 20]. A time series of weekly proportion of display-promoted sugar-sweetened drinkable yogurt items (continuous exposure) and weekly sum of the sales quantity of these items (continuous outcome) are recorded from a large supermarket in Montreal, Canada over T = 311 weeks (6 years). Supplementary Appendix S3 and Supplementary Figures S2, S3 elaborate the definition of the exposure and outcome.

The time-series regression used in this study is a dynamic linear model [21, 22]. We added the structural variable, E _t , covariates, a seasonal term, and an intercept. We selected the Koyck lag TF (p = 0, q = 1) for E _t , since the promotion exposure is likely to have a monotonically decaying association with purchasing [6]. The model was fit under the Bayesian framework as described in Supplementary Appendix S4.

The estimated immediate effect of the TF β was 0.68 (95% Posterior Credible Interval [CI]: 0.39–0.96), implying two-fold increase in sales at week t, if all yogurt items were display- promoted in the same week. The point estimate of the decay coefficient λ was moderately strong: 0.47 (95% CI 0.20–0.72), as shown by the distinct lag in the estimated IRF (Figure 3). Residual diagnostics indicate the absence of temporally autocorrelated residuals (Supplementary Figure S4).

Time-lagged exposure-outcome associations are of critical interest in time-series analysis. We described TF modeling to estimate lagged associations when lag length is unknown a priori. Previous applications of TFs include environmental time-series analysis to capture decaying associations between arbovirus incidence and temperature [23] and interrupted time-series analysis to capture the persistent effect of interventions [11, 24]. TF modeling requires pre-specification of the shape of a lag structure from investigators’ prior knowledge followed by their selection based on model fit. When such knowledge is lacking, existing distributed lag models such as those using splines allow data-driven estimation of the shape of lag. They require the specification of lag length by model selection applied to plausible lag lengths [25], by setting a long enough length to cover the unobserved true lag window with a potential sacrifice of precision [4], or alternatively estimating the lag length from data [26, 27]. Limitations of TFs include challenges in selecting the most appropriate shape of lag, when competing shapes show similar model fit. Finally, a comprehensive evaluation of TFs to capture lagged associations from simulated environmental health data is warranted, including their capacities to capture non-linear exposure-outcome associations by making β time-varying (dynamic) or imposing non-linear structure to E _t [17, 28].