Regression of total energy intake and total protein intake v. the dietary contribution of ultra-processed foods, evaluated by restricted cubic splines (, predicted values; , 95 % CI), among the US population aged ≥2 years (n 9042), National Health and Nutrition Examination Survey 2009–2010. (a) The values shown on the x-axis correspond to the 5th, 27·5th, 50th, 72·5th and 95th percentiles for percentage of total energy from ultra-processed foods (knots). Coefficient for linear term=0·024 (95 % CI 0·002, 0·046). There was little evidence of linearity in the restricted cubic spline model (Wald test for linear term P=0·035; Wald test for all non-linear terms P=0·049). (b) The values shown on the x-axis correspond to the 5th, 27·5th, 50th, 72·5th and 95th percentiles for percentage of total energy from ultra-processed foods (knots). Coefficient for linear term=−0·001 (95 % CI −0·007, 0·004). There was little evidence of linearity in the restricted cubic spline model (Wald test for linear term P=0·7; Wald test for all non-linear terms P=0·0009; Wald test for all terms P<0·001)