# Computational Scaling

    library(galamm)

This vignette describes the computational scaling properties of `galamm` for selected example models. We focus on the relationship between the size of the input data and the speed of the algorithm.

We will fit models to different random subsets of the input data, and here specify the number of such random subsets:

    n_replicates <- 20

## Linear Mixed Models with Factor Structures

We start with the linear mixed model with factor structures described in another vignette. Our goal here is to study how the computing time scales with the number of observations.

    data(KYPSsim, package = "PLmixed")

We start by defining the necessary arguments to provide to galamm.

    loading_matrix <- rbind(
      c(1, 0),
      c(NA, 0),
      c(NA, 1),
      c(NA, NA)
    )
    factors <- c("ms", "hs")
    load.var <- "time"
    form <- esteem ~ time + (0 + ms | mid) + (0 + hs | hid) + (1 | sid)

The KYPSsim dataset has 11,494 rows.

    nrow(KYPSsim)
    #> [1] 11494

There are 2,924 unique student IDs, and we will fit models by extracting subsets of these students of increasing size.

    all_sids <- unique(KYPSsim$sid)
    length(all_sids)
    #> [1] 2924

We set up a simulation grid in `sim_params`, with five different samples sizes, and specifying that we want ten random replicates for each sample size.

    sim_params <- expand.grid(
      n_students = c(100, 500, 1000, 2000, 2924),
      replicate = seq(from = 1, to = n_replicates, by = 1)
    )

    set.seed(11)
    times_to_convergence <- Map(function(n, r) {
      sids <- sample(all_sids, size = n)
      dat <- KYPSsim[KYPSsim$sid %in% sids, ]
      t1 <- Sys.time()
      mod <- galamm(
        formula = form,
        data = dat,
        factor = factors,
        load.var = load.var,
        lambda = loading_matrix
      )
      t2 <- Sys.time()

      data.frame(
        n = n,
        nrow = nrow(dat),
        time = t2 - t1
      )
      },
      n = sim_params$n_students,
      r = sim_params$replicate
      )

The plot below suggests that for this model, the algorithm scales linearly in the number of IDs.

    plot_dat <- do.call(rbind, times_to_convergence)
    fit <- lm(time ~ n, data = plot_dat)
    plot(plot_dat$n, plot_dat$time, xlab = "Number of IDs",
         ylab = "Time until convergence")
    abline(fit)

Time until convergence for different model sizes of linear mixed model with factor structures. Black line shows linear model fit to the data.

## Model with Group-Wise Heteroscedasticity

We now consider the model with group-wise heteroscedasticity consider in the vignette on heteroscedastic linear mixed models.

There 1200 rows and 200 unique IDs in the dataset. We take subset of IDs, while keeping the remaining model structure unchanged.

    nrow(hsced)
    #> [1] 1200
    all_ids <- unique(hsced$id)
    length(all_ids)
    #> [1] 200

    sim_params <- expand.grid(
      n_ids = c(100, 150, 200),
      replicate = seq(from = 1, to = n_replicates, by = 1)
    )

    set.seed(11)
    times_to_convergence <- Map(function(n, r) {
      ids <- sample(all_ids, size = n)
      dat <- hsced[hsced$id %in% ids, ]
      t1 <- Sys.time()
      mod <- galamm(
        formula = y ~ x + (1 | id),
        weights = ~ (1 | item),
        data = dat
      )
      t2 <- Sys.time()

      data.frame(
        n = n,
        nrow = nrow(dat),
        time = t2 - t1
      )
      },
      n = sim_params$n_ids,
      r = sim_params$replicate
      )

A plot of the time until convergence suggests that also in this case the algorithm scales linearly with the unput size.

    plot_dat <- do.call(rbind, times_to_convergence)
    fit <- lm(time ~ n, data = plot_dat)
    plot(plot_dat$n, plot_dat$time, xlab = "Number of IDs",
         ylab = "Time until convergence")
    abline(fit)

Time until convergence as a function of input data size for heteroscedastic linear mixed model. Black line shows linear model fit to the data.

## Generalized Linear Mixed Models with Factor Structures

We now consider an example described in the vignette on generalized linear mixed models with factor structures.

    data(IRTsim, package = "PLmixed")

This dataset has 2,500 rows and 500 student IDs.

    nrow(IRTsim)
    #> [1] 2500
    all_sids <- unique(IRTsim$sid)
    length(all_sids)
    #> [1] 500

We again sample subsets of student IDs.

    sim_params <- expand.grid(
      n_students = seq(from = 100, to = 500, by = 100),
      replicate = seq(from = 1, to = n_replicates, by = 1)
    )

    set.seed(11)
    times_to_convergence <- Map(function(n, r) {
      sids <- sample(all_sids, size = n)
      dat <- IRTsim[IRTsim$sid %in% sids, ]
      t1 <- Sys.time()
      mod <- galamm(
        formula = y ~ item + (0 + ability | school / sid),
        data = IRTsim,
        family = binomial,
        load.var = "item",
        factor = "ability",
        lambda = matrix(c(1, NA, NA, NA, NA), ncol = 1)
      )
      t2 <- Sys.time()

      data.frame(
        n = n,
        nrow = nrow(dat),
        time = t2 - t1
      )
      },
      n = sim_params$n_students,
      r = sim_params$replicate
      )

The plot below suggests that for this model and for the range of IDs considered here, there is almost no relationship between the input data size and the time until convergence.

    plot_dat <- do.call(rbind, times_to_convergence)
    fit <- lm(time ~ n, data = plot_dat)
    plot(plot_dat$n, plot_dat$time, xlab = "Number of IDs",
         ylab = "Time until convergence")
    abline(fit)

Time until convergence for different model sizes of generalized linear mixed model with factor structures. Black line shows linear model fit to the data.

We speculate that this is caused by the fact that for non-Gaussian models, a penalized iteratively reweighted least squares algorithm has to be run to find the posterior modes of the random effect for each candidate value of the model parameters, whereas for models with Gaussian responses, the linear system is solved exactly in a single iteration. There might hence be a trade-off: by increasing the number of observations, fewer iterations are required to reach convergence, since uncertainty is reduced, while on the other hand, the time taken to solve a single system increases with the sample size.

## Semiparametric Model with Gaussian Responses

We can study the scaling issue further in semiparametric models with factor structures, which are described in a separate vignette.

The data for domain 1 has 4,800 rows and 200 unique IDs.

    dat <- subset(cognition, domain == 1)
    nrow(dat)
    #> [1] 4800
    all_ids <- unique(dat$id)
    length(all_ids)
    #> [1] 200

We define the simulation parameters as before.

    sim_params <- expand.grid(
      n_ids = seq(from = 50, to = 200, by = 50),
      replicate = seq(from = 1, to = n_replicates, by = 1)
    )

Then we fit the model for different subsets of the data.

    set.seed(11)
    times_to_convergence <- Map(function(n, r) {
      ids <- sample(all_ids, size = n)
      dat <- dat[dat$id %in% ids, ]
      t1 <- Sys.time()
      mod <- galamm(
        formula = y ~ 0 + item + sl(x, factor = "loading") +
          (0 + loading | id / timepoint),
        data = dat,
        load.var = "item",
        lambda = matrix(c(1, NA, NA), ncol = 1),
        factor = "loading"
      )
      t2 <- Sys.time()

      data.frame(
        n = n,
        nrow = nrow(dat),
        time = t2 - t1
      )
      },
      n = sim_params$n_ids,
      r = sim_params$replicate
      )

The plot below suggests that for this model, the algorithm scales linearly in the number of IDs.

    plot_dat <- do.call(rbind, times_to_convergence)
    fit <- lm(time ~ n, data = plot_dat)
    plot(plot_dat$n, plot_dat$time, xlab = "Number of IDs",
         ylab = "Time until convergence")
    abline(fit)

Time until convergence for different model sizes of semiparametric model with factor structures. Black line shows linear model fit to the data.

## Semiparametric Model with Binomial Responses

We now consider domain 2, which has binomial responses.

The data for domain 2 has 3,200 rows and 200 unique IDs.

    dat <- subset(cognition, domain == 2)
    nrow(dat)
    #> [1] 3200
    all_ids <- unique(dat$id)
    length(all_ids)
    #> [1] 200

We define the simulation parameters as before.

    sim_params <- expand.grid(
      n_ids = seq(from = 50, to = 200, by = 50),
      replicate = seq(from = 1, to = n_replicates, by = 1)
    )

Then we fit the model for different subsets of the data.

    set.seed(11)
    times_to_convergence <- Map(function(n, r) {
      ids <- sample(all_ids, size = n)
      dat <- dat[dat$id %in% ids, ]

      dt <- tryCatch({
        t1 <- Sys.time()
        mod <- galamm(
          formula = y ~ 0 + item + sl(x, factor = "loading") +
            (0 + loading | id / timepoint),
          data = dat,
          family = binomial,
          load.var = "item",
          lambda = matrix(c(1, NA), ncol = 1),
          factor = "loading"
        )
        t2 <- Sys.time()
        t2 - t1
      },
      error = function(e) {
        NA
      })

      data.frame(
        n = n,
        nrow = nrow(dat),
        time = dt
      )
      },
      n = sim_params$n_ids,
      r = sim_params$replicate
      )

    plot_dat <- do.call(rbind, times_to_convergence)

We check how many models failed to converge, and then remove those rows:

    sum(is.na(plot_dat))
    #> [1] 1
    plot_dat <- na.omit(plot_dat)

The plot below suggests that for this model, the algorithm scales linearly in the number of IDs.

    fit <- lm(time ~ n, data = plot_dat)
    plot(plot_dat$n, plot_dat$time, xlab = "Number of IDs",
         ylab = "Time until convergence")
    abline(fit)

Time until convergence for different model sizes of semiparametric model with factor structures. Black line shows linear model fit to the data.

## Conclusion

The algorithm used by galamm, which is described in detail in Sørensen, Fjell, and Walhovd (2023), seems to scale linearly with the input data size in the tests considered in this vignette, with the exception of the the generalized linear mixed model with factor structures, for which the relationship was almost flat.

# References

Sørensen, Øystein, Anders M. Fjell, and Kristine B. Walhovd. 2023. “Longitudinal Modeling of Age-Dependent Latent Traits with Generalized Additive Latent and Mixed Models.” Psychometrika 88 (2): 456–86. https://doi.org/10.1007/s11336-023-09910-z.