How to make predictions over time

Question

I need to build a statistical model to predict the time of assembly of a machine. Those predictions should be updated during the assembly on a daily basis. I have some variables which for each machine built do not change in time such as the machine model and others, and some variables time-dependent such as the number of missing parts required to make the machine, the number of days spent on the machine so far and so on.

So in the dataset for each machine made in the last years (about 100 of them) I have a number of rows equal to the number of days spent to build that machine, in which the response variable is always the same (known at the end of the assembly), while some of the predictor variables changes.

How can I model this kind of data? I think is not a repeated measures analysis because for each machine the response is always the same, only some predictors changes, I just make predictions several times over time because I expect to improve my forecast approaching the end of the assembly. I can change the response variable to be the number of days remaining to finish assembly instead of the total number of days of assembly, perhaps in this case I can fit a mixed model for repeated measures? Or maybe Time-Varying survival regression?

Answer 1

With this structure of the data, I would build a discrete time hazard model. The literature on those is quite raw on the web, since I last searched about it. But it boils down to logistic regression where each row is a period (day, minute, etc...).

In response variable,for each subject(machine in your case), you have a zero, indicating that the subject did not experience the event (the machine wasn't ready at that period) and in the last observation you have a 1 indicating the machine was ready in that period, and then no more rows for that observation.

Here is a quick artificial dataset I built in R with this structure.

library(reprex)
library(tidyverse)


set.seed(11)
tibble(time=(1:6)) %>% crossing(machine = LETTERS[0:10]) %>% 
  arrange(machine) %>% 
  mutate(time_varying_covar = runif(n())) %>%
  group_by(machine) %>% 
  mutate(finished_at = runif(1, 0, max(time))) %>% 
  mutate(non_time_varying_covar = rnorm(1, 10, 3)) %>% 
  mutate(not_finished = time < finished_at) %>% 
  mutate(response = cumsum(!not_finished)) %>% 
  filter(response <=1) %>% 
  select(time, machine, response, finished_at, time_varying_covar, non_time_varying_covar)
#> # A tibble: 29 x 6
#> # Groups:   machine [10]
#>     time machine response finished_at time_varying_covar non_time_varying_covar
#>    <int> <chr>      <int>       <dbl>              <dbl>                  <dbl>
#>  1     1 A              1       0.808             0.277                   11.5 
#>  2     1 B              0       4.51              0.0865                   9.44
#>  3     2 B              0       4.51              0.290                    9.44
#>  4     3 B              0       4.51              0.881                    9.44
#>  5     4 B              0       4.51              0.123                    9.44
#>  6     5 B              1       4.51              0.175                    9.44
#>  7     1 C              0       1.04              0.907                   14.6 
#>  8     2 C              1       1.04              0.851                   14.6 
#>  9     1 D              0       3.48              0.158                    8.17
#> 10     2 D              0       3.48              0.480                    8.17
#> # … with 19 more rows

^{Created on 2020-10-28 by the reprex package (v0.3.0)}

With this structure, you can model:

• the effect of time in how likely it is for a subject to experience the event at that period
• as well as the effect of time varying predictors and non time varying predictors
• model subject specific effects through mixed/hierarchical modeling

You can learn more about those models in Part II of Applied Longitudinal Data Analysis by Singer and Willett.