Multivariable regression analysis

We estimated multivariate regression models of the observed association between our two outcome measures of diet quality (dependent variable) and diet cost, namely an individual’s daily expenditure and time cost they incurred on cooking. We also estimated whether socio-economic status is associated with diet quality, and included a large set of other explanatory variables that are likely to be correlated with observed food expenditures and time costs as well as being associated with diet quality. We use both 24h dietary recall assessments for each individual as our unit of observation, hence we have more than 4000 potential observations. In economic terms, the model is considered a health (diet quality) production function with diet expenditures and time costs the amount of inputs consumed by individuals in order to produce a desired level of diet quality. Socio-economic status and other food knowledge variables reflect their ability to choose and transform these inputs efficiently i.e. to achieve the healthiest diet possible for the least amount of diet cost. Other variables in the model are included to capture preferences for healthy eating, cultural or demographic influences on diet quality.

We estimated the impact of diet cost on diet quality using a Poisson regression model for count dependent variables for the Pyramid Score. We also estimated a random effects or mixed-effects Poisson regression that allows for unobserved variation between individuals and included sampling weights. For the HEI-2015, we estimated an ordinary least squares (OLS) regression and a random-effects model. All models included sampling weights and allowed for cluster/robust standard errors for the models without random-effect between surveyed individuals for whom we have repeated observations.

Our baseline model specification included all covariates in a linear specification, The linear model assumes a monotonic (constantly increasing or decreasing) association between diet cost and diet quality. The linear base model is presented as follows:

𝐷𝑖𝑒𝑡 𝑄𝑢𝑎𝑙𝑖𝑡𝑦 = 𝛽 +𝑙𝑜𝑔 𝛽 𝑙𝑜𝑔 (𝐸𝑥𝑝) + 𝛽 𝑇 + 𝛽 𝑆𝐸𝑆 + 𝛽 𝐻 + 𝛽 𝐷𝐸𝑀 + 𝛽 𝐹 + 𝛽 𝑋 + 𝜀

The dependent variable is diet quality (as measured by the Swiss Food Pyramid score or the HEI-2015), and the independent variables are the logarithm of daily expenditure on food 𝑙𝑜𝑔 𝑙𝑜𝑔 (𝐸𝑥𝑝) and time cost (𝑇).

Logarithmic transformation of the food expenditure variable was a convenient means of transforming a highly skewed variable into one that is more approximately normal. It also implies that the estimated association between diet quality and expenditure is a proportionate one, so that the underlying relationship between increases in actual expenditure and diet quality is a non-linear relationship (diminishing absolute marginal effects with increasing expenditures), while still preserving the log-linear specification of the model.

Apart from the two measures of diet cost, the model included the socioeconomic status variables (SES) net household income, education level and work status to test for significant inequalities in diet quality across SES groups. Income is likely to determine expenditure on food, but in the model that already included food expenditure it indicates if there are stronger preferences for healthier diets between richer and poorer households. Education is a proxy for individuals health literacy, cognitive ability, and, along with work status, is a measure of their degree of social capital.[187] We included smoking status, perceived health status and physical activity as proxies for healthy behaviors and preferences (H). We included as demographic factors (DEM) gender, ge and language region, which is a proxy for cultural differences within Switzerland. Further explanatory variables (F) closely related to dietary choices and nutritional knowledge, specifically being on a diet or being a vegetarian as well as awareness of either the food pyramid or the 5-a-day recommendation or both. Summary statistics of all the covariates used in the regression please are listed in Table 6.3. Finally, we controlled for the methods used for survey data collection (face-to-face or telephone interviews) and for seasonality using monthly indicator variables. Moreover, in the random effects model only, there is an individual specific error term 𝜀 to control for unobserved heterogeneity that is persistent within individuals over time and is assumed independent of the other explanatory variables. The random error term is represented by 𝜀 .

We tested if the log-linear specification for the relationship between diet cost and diet quality holds by using both a visual approach, namely the augmented residuals plot with a non-parametric kernel density line plot, as well as a regression model with a quadratic specification for food expenditures and time costs with the Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) to test for goodness-of-fit.[188]

The intuition for the quadratic specification is that the association between diet quality and diet costs is non-linear, in the sense that increases in diet expenditure may not be associated with the same degree of improvement in diet quality at all levels of expenditure. Instead, it could be hypothesized that there is initially a larger positive association at lower levels of expenditure, but that at higher levels of expenditure additional expenditure may be associated with a smaller increase or even a decrease in diet quality.

For instance, at low levels of diet expenditure or time costs, individuals may face significant financial or time constraints to eating healthier, so that any additional increase in diet costs will improve diet quality.

However, as the diet cost increases the additional benefits to diet quality diminish and, beyond a certain level of expenditure, diet quality could decline. This is because, either individuals overconsume relative to

We specified the following model with quadratic terms in the logarithm of food expenditures and the level values of time costs to examine the existence of non‐linearity between diet cost and diet quality:

𝐷𝑖𝑒𝑡 𝑄𝑢𝑎𝑙𝑖𝑡𝑦 = 𝛽 +𝑙𝑜𝑔 𝛽 𝑙𝑜𝑔 (𝐸𝑥𝑝) +𝑙𝑜𝑔 𝛽 𝑙𝑜𝑔 (𝐸𝑥𝑝) + 𝛽 𝑇 + 𝛽 𝑇 + 𝛽 𝑆𝐸𝑆 + 𝛽 𝐻 + 𝛽 𝐷𝐸𝑀 + 𝛽 𝐹 + 𝛽 𝑋 + 𝜀

Where all variables except 𝑙𝑜𝑔 𝑙𝑜𝑔 (𝐸𝑥𝑝) and 𝑇 are similar to the variables used in the previous model specifications and the random individual effect 𝜀 is estimated only in the random/mixed effects models.

The non‐linear relationship is reflected by the coefficients 𝛽 , 𝛽 and 𝛽 , 𝛽 for daily expenditure and time cost, respectively. As a robustness check, we estimated the linear model with quintiles of 𝑙𝑜𝑔 𝑙𝑜𝑔 (𝐸𝑥𝑝) and quintiles of 𝑇. AIC and BIC were used to determine model goodness-of-fit between the alternative linear and non-linear specifications. Statistical significance was set to 5%.

2.7 Results

2.7.1 Descriptive data analysis

The mean values for our diet quality scores were 2.17 (upper bound 7) for the Food Pyramid Score and 47.5 (upper bound 100) for the HEI. Average daily food expenditure was 20.07 CHF and average expected daily time cost from preparing meals at home was 20.87 CHF.

Figure 2.1 and Figure 2.2 show the distributions of the Food Pyramid Score and HEI. There is less variation in the Food Pyramid score as it can only take on 7 values, and most individuals achieve scores of 2 and 3 with few individuals achieving scores above 4. There is more variation in the HEI, which is quite symmetrically distributed around a score of 50. Few individuals achieve scores above 75 or below 25.

The distribution of daily expenditure is heavily rightly skewed (Figure 2.3) with a small but non-negligible number of individuals with estimated daily expenditures above 50 CHF and even over 100 CHF (over 5 times the mean). After having applied the natural logarithm to the transform the variable (see

Figure 2.4) the distribution is now close to that of the normal distribution. Taking the natural logarithm of daily expenditures reduces the influence of the unusually large daily expenditures while dispersing the concentration of expenditures at the lower end of the scale.