Incorporating scenarios and generating projections

The imputed individual-level dataset was denoted by I = (I¹, …, I^t) where waves 1, …, t = 7 corresponded the years 2000–2012 biennially, and the sequential projection step was indexed by s. The projection steps and dataflow are presented in Fig. 1.

The dataflow of generating projections and datasets used in in each projection step. Projections are generated sequentially by imputing the values in the dataset marked with grey color in each step

At the first projection step (s = 1) the dataset R₁ was the combination of the matrices IT,A1TT defining A1:=I2,⋯,It+1i:di,t=1,· the subset of I, where an individual i was alive at wave t, where values of I^t + 1 were set to missing values, except age and sex, which remained constant or changed deterministically.

Then a copy of R₁ was generated for each scenario and the scenarios are incorporated as follows:

Scenario 0:

The missing values of I^t + 1 in A₁ were imputed with no modification of IADL or ADL scores.

Scenario 1:

First, the missing values of I^t + 1 were imputed (except outcomes and mortality status). Then, the SMD estimate 1.12 was transformed to the absolute scale of IADL score by multiplying the SMD estimate with the study-specific standard deviation (SD) of the IADL score in 2000 [SD(y₂₀₀₀)]. IADL score values for 2014 (y₂₀₁₄) of individuals with no vigorous physical activity were replaced with a random number from a normal distribution

Finally, outcomes and mortality status in I^t + 1 were imputed.

Scenario 2:

Scenario 2 followed the same procedure as in scenario 1, but in addition to the modification of IADL scores, also the ADL score values for 2014 (z₂₀₁₄) of individuals with no vigorous physical activity were replaced, using the study-specific SD of the IADL score at 2000 [SD(z₂₀₀₀)], with a random number from a normal distribution

In scenarios 1 and 2, the modified IADL and ADL scores below 0 were coded as 0, and scores over 6 were coded as 6.

In all scenarios I^t + 1 was imputed using R₁ with predictors in the imputation model. The assumption was that the survey variables will change between 2002 and 2014 with the same transition probabilities as between 2000 and 2012.

At the second projection step (s = 2) the dataset (R₂) is the combination of the matrices A1T,A2TT defining A2:=I3,⋯,It+2i:di,t+1=1,· the subset of A₁, where an individual i was alive at wave t + 1 and the missing values of I^t + 2 were imputed for all scenarios. Projection steps were repeated until we had imputed all the necessary values of I^t + s of the desired projection. The theory behind generating projections is described in Additional file 1.