2.11. Analysis
All analyses were conducted using R software (version 3.53, R Core Team, Vienna, Austria). All participants were included based on the intention to treat (ITT) principle. We imputed missing values (m = 20) using all variables of cognitive function performance, age, and sex. To impute missing dates, we used the “mice” function in the mice package [Moreover, to investigate change of the blood glucose levels among groups, we conducted GLMM with “lmer” in the lme4 package (For the main analysis, we analyzed cognitive functions using a GLMM. In the GLMM, we used drinks (15-g, 10-g, and 0-g milk proteins), time (before the drink intake, 15 min after the drink intake, and 60 min after the drink intake), age, and sex as fixed effects. Participants were used as a random effect. In these GLMM analyses, we were interested in the interaction between the factors of drinks and time. Significance was inferred for multiple comparison methods We also conducted an additional analysis for mood state measurements. We used a similar GLMM model to cognitive functions’ analysis. We used the false discovery rate correction to adjust 