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This vignette shows how to specify and estimate a two-factor measurement system with factorana. The latent factors are identified from a set of noisy indicators (measurements) whose loadings the package estimates jointly with the factor variances.
The example below uses two latent factors (“cognitive” and “non-cognitive” skills) each measured by three continuous indicators, and is small enough (n = 300) to run quickly.
library(factorana)
set.seed(1)
n <- 300
# Latent factors (true values, unobserved in practice)
f_cog <- rnorm(n, mean = 0, sd = 1.0)
f_noncog <- rnorm(n, mean = 0, sd = 0.8)
# Cognitive indicators: loadings (1.0, 0.9, 0.7); error sd = 0.5
cog1 <- 0.0 + 1.0 * f_cog + rnorm(n, 0, 0.5)
cog2 <- 0.2 + 0.9 * f_cog + rnorm(n, 0, 0.5)
cog3 <- 0.1 + 0.7 * f_cog + rnorm(n, 0, 0.5)
# Non-cognitive indicators: loadings (1.0, 1.1, 0.8); error sd = 0.5
nc1 <- 0.0 + 1.0 * f_noncog + rnorm(n, 0, 0.5)
nc2 <- 0.1 + 1.1 * f_noncog + rnorm(n, 0, 0.5)
nc3 <- 0.0 + 0.8 * f_noncog + rnorm(n, 0, 0.5)
dat <- data.frame(
intercept = 1,
cog1 = cog1, cog2 = cog2, cog3 = cog3,
nc1 = nc1, nc2 = nc2, nc3 = nc3
)
head(dat)
#> intercept cog1 cog2 cog3 nc1 nc2 nc3
#> 1 1 -0.7969873 -1.1345097 -1.11703554 1.1399607 1.2434756 -0.0985743
#> 2 1 0.9348556 0.4624395 1.19013215 -1.3004950 -0.5309301 -0.6931244
#> 3 1 -0.5714748 -0.4198545 -1.41335484 2.0238605 1.7614149 2.3556515
#> 4 1 1.8663765 1.0763851 0.16363734 -0.7774106 0.5158946 0.4655233
#> 5 1 0.2611711 0.8220335 0.67947970 1.5927923 1.4158815 1.1502765
#> 6 1 -1.3888353 -1.0548717 -0.02060567 1.1187830 2.4446165 0.6413497Two independent latent factors. Loading normalizations are set on the component side: each factor has one indicator with loading fixed at 1 (to pin the scale) and two free loadings.
For each indicator we declare a linear equation, which factor(s) it
loads on, and any fixed loadings. loading_normalization
takes a vector of length n_factors:
1 = loading fixed at 1 (scale normalization)0 = loading fixed at 0 (indicator is unrelated to that
factor)NA_real_ = free parameter, to be estimated# Cognitive indicators: load on factor 1 only
mc_cog1 <- define_model_component(
name = "cog1", data = dat, outcome = "cog1", factor = fm,
covariates = "intercept", model_type = "linear",
loading_normalization = c(1, 0) # factor 1 loading = 1, factor 2 loading = 0
)
mc_cog2 <- define_model_component(
name = "cog2", data = dat, outcome = "cog2", factor = fm,
covariates = "intercept", model_type = "linear",
loading_normalization = c(NA_real_, 0) # factor 1 loading free, factor 2 loading = 0
)
mc_cog3 <- define_model_component(
name = "cog3", data = dat, outcome = "cog3", factor = fm,
covariates = "intercept", model_type = "linear",
loading_normalization = c(NA_real_, 0)
)
# Non-cognitive indicators: load on factor 2 only
mc_nc1 <- define_model_component(
name = "nc1", data = dat, outcome = "nc1", factor = fm,
covariates = "intercept", model_type = "linear",
loading_normalization = c(0, 1)
)
mc_nc2 <- define_model_component(
name = "nc2", data = dat, outcome = "nc2", factor = fm,
covariates = "intercept", model_type = "linear",
loading_normalization = c(0, NA_real_)
)
mc_nc3 <- define_model_component(
name = "nc3", data = dat, outcome = "nc3", factor = fm,
covariates = "intercept", model_type = "linear",
loading_normalization = c(0, NA_real_)
)Assemble the components into a system:
The estimator uses Gauss-Hermite quadrature to integrate over the
latent factors; we keep n_quad_points modest here for
speed.
# Tidy table of parameter estimates with standard errors
components_table(fit, digits = 3)
#>
#> Factor Model Results by Component
#> ========================================================================================================================
#>
#> Parameter factor cog1 cog2 cog3 nc1 nc2 nc3
#> ----------------------------------------------------------------------------------------------------------
#> Intercept 0.170** 0.353*** 0.264*** -0.030 0.089 0.025
#> (0.065) (0.057) (0.047) (0.059) (0.061) (0.049)
#> Sigma 0.608*** 0.535*** 0.483*** 0.545*** 0.580*** 0.531***
#> (0.037) (0.032) (0.027) (0.037) (0.043) (0.028)
#> factor_var_1 0.794***
#> (0.089)
#> factor_var_2 0.659***
#> (0.062)
#> cog2_loading_1 0.888***
#> (0.053)
#> cog3_loading_1 0.692***
#> (0.044)
#> nc2_loading_2 1.036***
#> (0.067)
#> nc3_loading_2 0.769***
#> (0.048)
#>
#> ----------------------------------------------------------------------------------------------------------
#> N 0 300 300 300 300 300 300
#> Log-likelihood: -2053.38
#>
#> Standard errors in parentheses
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1Factor variances are near the true values (1.0 for the cognitive factor and 0.64 = 0.8^2 for the non-cognitive factor); loadings recover the simulated values (cog2 ≈ 0.9, cog3 ≈ 0.7, nc2 ≈ 1.1, nc3 ≈ 0.8).
Posterior mean factor scores for each observation can be recovered from the estimated model:
fscores <- estimate_factorscores_rcpp(
fit, dat, control = ctrl, parallel = FALSE, verbose = FALSE
)
head(fscores[, c("obs_id", "factor_1", "factor_2",
"se_factor_1", "se_factor_2", "converged")])
#> Factor Score Estimates
#> ======================
#>
#> Observations: 6
#> Factors: 2
#> Converged: 6 (100.0%)
#>
#> Summary Statistics:
#> Factor 1: mean=-0.295, sd=0.944, mean_se=0.365
#> Factor 2: mean=0.702, sd=0.957, mean_se=0.340
#>
#> First 6 rows:
#> obs_id factor_1 factor_2 se_factor_1 se_factor_2 converged
#> 1 1 -1.2915708 0.7037167694 0.364723 0.3399574 TRUE
#> 2 2 0.5875902 -0.8011073548 0.364723 0.3399574 TRUE
#> 3 3 -1.0696673 1.8108452339 0.364723 0.3399574 TRUE
#> 4 4 0.7438436 -0.0003981072 0.364723 0.3399574 TRUE
#> 5 5 0.3346083 1.2404898768 0.364723 0.3399574 TRUE
#> 6 6 -1.0742353 1.2587562101 0.364723 0.3399574 TRUE
# Correlation of estimated factor scores with the true (unobserved) factors
cor(fscores$factor_1, f_cog)
#> [1] 0.936484
cor(fscores$factor_2, f_noncog)
#> [1] 0.9422921vignette("roy_model", package = "factorana") — an
applied example with a discrete sector-choice component and partially
observed potential outcomes.?define_model_component — supported model types
(linear, probit, logit, ordered probit) and options for factor
interactions and type intercepts.?estimate_model_rcpp — full list of estimator options,
including parallel estimation, checkpointing, and adaptive integration
for two-stage workflows.These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.