Table of Contents

### Study design

This is a retrospective time series study examining the association between weekly discounts and the sales of five SSB categories at a single supermarket in Metropolitan Montreal, Canada. The study period represents the period covered by our beverage transaction data, which is between January 2008 and December 2013, so 311 weeks or 6 years. The unit of analysis is weekly sales transactions for each beverage category. Note that this is not a longitudinal data analysis that uses measurements from multiple stores, as shown in our previous studies [16, 17]ie these are not panel data. Instead, we performed a time series analysis (ie a single store), which allowed us to examine time-bound effects while taking into account the temporal correlation of sales.

### transaction data

The transaction data was generated by a large supermarket owned by a major Canadian retail chain (the chain’s identity has been anonymized) and was purchased from a marketing company, Nielsen [25].

The data consists of the weekly sales quantity of individual beverage items, as uniquely defined by the universal product code and item name, the weekly price of items sold in Canadian cents, flyer promotion, and retail display promotion (described below). We classified these items into the five non-alcoholic SSB categories based on the product name of each beverage item and the corresponding food category assigned by Nielsen. For example, soda items were categorized by the company as “carbonated soda”, but we manually excluded diet soda, i.e. items with artificial sweeteners based on terms such as “diet”, “zero”, “non-sugar”.

### Result

The weekly sales quantities of each beverage item were standardized on the Food and Drug Administration’s single 240 ml serving for beverage (about 1 cup). The outcome variable was the aggregated sum of the sales of items in each category in each week, with the category-specific mean number of individual items over the entire 6-year period in our store being 109 (soft drinks), 152 (fruit drinks), 36. (sports and energy drinks), 22 (coffee and tea) and 29 (drinking yoghurts). The category-specific sales were, of course, transformed to reduce skew. We did not analyze the disaggregated, individual item-level association between sales and discounts because such an analysis required us to consider product dependence on sales. Since the change in sales at the category level is of primary importance to the diet of the population rather than to the sales of individual foods or brands, our unit of analysis for exposure, outcome, and covariates was defined at the beverage category level.

### Exposure

The exposure variable is category-specific discounting each week. Specifically, it is a continuous variable calculated as the weighted average of weekly price discounts of individual items in each category, with the weights representing each item’s market share (share of standardized sales) within the category it belongs to. Price discount of an individual item is a continuous measure and was calculated as a percentage decrease in the standardized serving price sold (net price) from the base price (i.e. unpromoted) price [16, 26]. Detailed calculation of portion standardized discounts for each item and subsequent aggregation by category can be found in Appendix S1 and Supplementary Image. S1 in the supplementary information file.

### Statistical analysis: regression variables to capture lagging association of price discounts and SSB sales

A delayed association between time-varying outcome (log-transformed sales quantity) and exposure (discounting) is usually captured by a distributed lag model, which is a regression model containing multiple time-delayed values of an exposure. Regression coefficients for these time-delayed variables have functional limitations (i.e., the value of the coefficients is constrained to change smoothly over delay), as is often seen in the epidemiology and econometrics of environmental time series [27, 28]. One of those limitations is the Koyck lag decay [29]which captures the monotonous decay of the effect of an exposure over time with two regression coefficients: as the immediate effect (at zero retardation) and λ as the retardation coefficient quantifying the decay rate. The functional form of the Koyck decay is represented by a polynomial of form:

$$\beta {\lambda}^0+\beta {\lambda}^1+\beta {\lambda}^2+\beta {\lambda}^3+\dots +\beta {\lambda}^h, $$

Where *h* indicates lag, and ^{}= is the immediate effect. An estimated value of the retardation coefficient closer to 0 indicates the absence of a delay, while a value closer to 1 indicates a stronger delayed effect. The visual interpretation of the delayed effect represented by this polynomial function is given in Supplementary Figures. S2 a and b (appendix S2). We pre-specified the range of the estimated value of be 0 < 1 so that the effect of discounting decreased monotonically to zero during the delay, recording a decreasing effect.

### Statistical analysis: time series regression model to include the Koyck lag model

The Koyck lag variables have been added to a linear time series regression, dynamic linear model [30, 31]. The details of the model, including the interception and the lag coefficients, can be found in Appendix S3. We accounted for seasonal trends in sales by adding the sine- and cosine-transformed harmonic wave of a weekly variable, as described in Appendix S3.

Covariates were weekly varying variables likely to temporarily correlate with price discounts and sales. These include non-discount promotion: weekly changing display promotion and flyers, which often come with price discounts (though not always) and associated with higher sales [3]. Display promotion is the temporary placement of items in a prominent location in stores, such as the front of a store. We calculated the value of these variables at the SSB category level each week by aggregating the binary promotion status for items. Specifically, display promotion was coded as 1 if an item was temporarily placed in one of the prominent retail locations of the original shelf space, such as the end of the aisle, the store entrance, or by the cashier. Flyer promotion was coded as 1 if an item was in the flyer and 0 otherwise. These item-level binary variables were aggregated to the category-level proportion as the weighted proportion of items promoted in each category in a given week, with the weights represented an item’s standardized market share, as in the discount variable. In addition, an indicator variable has been added for whether or not to include national and provincial statutory holidays. Other covariates were the regular (base) price of each drink category, the average daily temperature in each week and the residual value of the sale itself (autoregressive of order 1).

We independently modeled each of the five food categories under the Bayesian framework. We have therefore specified previous distributions for regression parameters (Appendix S3). Interpretation of regression coefficients is based on point estimates (posterior mean or median) and uncertainty (95% credible interval [CI]) as summarized from the posterior distribution of the parameters approximated by Markov Chain Monte Carlo methods. We used the Stan software, which uses Hamiltonian Monte Carlo methods and is accessible through the Rstan package in R software [32]. Model selection, specifically selecting a subset of variables from the covariates described above, was guided by the value of the Watanabe-Akaike Information Criterion (WAIC) indicator of model fit [33]. Since the sales of many food categories are expected to have a priori seasonal trends, we have not carried out a selection of the seasonal conditions and therefore they have been retained in all models. A lower WAIC value indicates a better fit model. Codes are publicly available in an online repository [34].

As a sensitivity analysis, we tested an alternative form of promotion decay by changing the constraint of the lag parameterfrom 0< 1 to −1<0. The last specification implies that, instead of assuming monotonous decay in supplementary figures. S2 a and b, we let the model capture a so-called 'post-promotion dip' (Supplementary Figures S3), a sharp drop in sales below the pre-discount period immediately after discounting [3]. Theoretical explanations for the post-promotion dip are given elsewhere [3, 35, 36].

The study was approved by the Institutional Review Board, Faculty of Medicine, McGill University (IRB Approval#: A07-E45-16B), which did not require written or verbal consent from subjects as the study was used in aggregated (store level) secondary data. All methods followed institutional guidelines and regulations.