spifa() can fit five model types that are determined
automatically from formula and the class of
data (see ?spifa):
eifa: Exploratory IFAcifa: Confirmatory IFAcifa_pred: CIFA with predictorsspifa: Spatial IFAspifa_pred: SPIFA with predictorsThe model type is determined based on the following:
eifa) is used when no
constraints are provided.cifa_pred/spifa_pred) are activated if terms
are included on the right-hand side of formula.spifa/spifa_pred) are only
activated when the data is geo-referenced as an
sf object, otherwise the non-spatial models are
activated.ngp = 0 forces the model to be non-spatial
(eifa/cifa/cifa_pred).We will showcase the five model types using the ipixuna
data.
The simplest case is the eifa model, the user does not
need to provide constraints neither a spatial
data. The constraints will be internally defined as a
lower-triangular matrix; and in case data is an
sf object, the spatial functionality can be dropped using
ngp = 0:
set.seed(123)
samples <- spifa(items ~ 1, data = ipixuna, nfactors = 2, ngp = 0, niter = 500)
samples#> Item factor analysis model: eifa
#> Formula: items ~ 1
#> Dimensions: 100 respondents, 10 items, 2 latent factors, 0 spatial processes
#> MCMC: 1 chain, iter = 500, thin = 1, samples = 500
#>
#> Item model parameters:
#> mean median sd q10 q90 ess_bulk rhat
#> c[1] -0.4102 -0.414 0.15 -0.590 -0.2106 121.8 1.0
#> c[2] -0.5215 -0.517 0.18 -0.754 -0.2975 57.4 1.0
#> c[3] -0.5718 -0.559 0.22 -0.851 -0.2868 61.6 1.0
#> c[4] -0.2970 -0.301 0.16 -0.499 -0.0884 118.0 1.0
#> c[5] -0.2675 -0.268 0.20 -0.514 0.0018 46.6 1.0
#> c[6] 0.1068 0.088 0.23 -0.148 0.3979 50.6 1.0
#> c[7] 0.2626 0.259 0.19 0.018 0.5172 45.4 1.0
#> c[8] 0.2074 0.199 0.21 -0.035 0.4479 45.9 1.0
#> c[9] 0.7201 0.684 0.26 0.427 1.0441 45.3 1.0
#> c[10] 0.5153 0.519 0.23 0.240 0.7878 41.7 1.1
#> A[1,1] 0.6013 0.589 0.23 0.326 0.9009 5.5 1.2
#> A[2,1] 0.8696 0.839 0.34 0.475 1.2985 13.9 1.1
#> A[3,1] 1.2459 1.219 0.39 0.787 1.7374 14.6 1.1
#> A[4,1] 0.6308 0.619 0.25 0.361 0.9279 36.6 1.1
#> A[5,1] -0.0097 0.078 0.41 -0.654 0.4757 12.0 1.1
#> A[6,1] -0.2310 -0.207 0.35 -0.692 0.2276 19.6 1.0
#> A[7,1] 0.5565 0.550 0.29 0.204 0.9338 3.8 1.2
#> A[8,1] 1.0440 0.900 0.59 0.566 1.8901 2.8 1.3
#> A[9,1] 1.3692 1.201 0.68 0.692 2.6265 11.3 1.1
#> A[10,1] 1.0474 1.070 0.39 0.506 1.5035 4.2 1.2
#> A[2,2] 0.4742 0.453 0.20 0.229 0.7346 38.6 1.0
#> A[3,2] 0.3524 0.345 0.27 0.016 0.7145 10.6 1.1
#> A[4,2] 0.0648 0.055 0.22 -0.203 0.3401 13.3 1.1
#> A[5,2] 1.6031 1.531 0.46 1.102 2.2714 13.9 1.0
#> A[6,2] 1.6781 1.601 0.63 0.923 2.5744 12.1 1.0
#> A[7,2] 1.0608 1.050 0.29 0.687 1.4535 13.3 1.1
#> A[8,2] 0.2790 0.273 0.27 -0.054 0.6172 7.2 1.1
#> A[9,2] 1.1263 1.104 0.37 0.684 1.6746 27.9 1.0
#> A[10,2] 1.5646 1.542 0.39 1.099 2.0739 17.6 1.0
#>
#> ess_bulk is the bulk effective sample size; rhat is the potential
#> scale reduction factor on split chains (Rhat = 1 at convergence).
#> Use summary() for the full set of statistics (incl. ess_tail).Notice how the constraints have been defined internally:
#> [,1] [,2]
#> [1,] 1 0
#> [2,] 1 1
#> [3,] 1 1
#> [4,] 1 1
#> [5,] 1 1
#> [6,] 1 1
#> [7,] 1 1
#> [8,] 1 1
#> [9,] 1 1
#> [10,] 1 1EIFA also never estimates the correlation between the latent factors, so only the easiness (c) and discrimination (A) parameters are estimated.
Supplying your own discrimination constraints leads to a
cifa model:
nfactors <- 3
A <- matrix(1, nitems, nfactors)
A[c(4, 8), 1] <- 0
A[c(4, 5, 6, 7, 8, 10), 2] <- 0
A[c(5, 6), 3] <- 0
samples <- spifa(
items ~ 1, data = ipixuna, nfactors = nfactors, ngp = 0, niter = 500,
constraints = list(discrimination = A)
)
samples#> Item factor analysis model: cifa
#> Formula: items ~ 1
#> Dimensions: 100 respondents, 10 items, 3 latent factors, 0 spatial processes
#> MCMC: 1 chain, iter = 500, thin = 1, samples = 500
#>
#> Item model parameters:
#> mean median sd q10 q90 ess_bulk rhat
#> c[1] -0.639 -0.624 0.25 -0.9412 -0.338 49 1.0
#> c[2] -0.509 -0.494 0.18 -0.7397 -0.312 86 1.0
#> c[3] -0.712 -0.709 0.27 -1.0354 -0.385 48 1.0
#> c[4] -0.348 -0.347 0.15 -0.5406 -0.161 72 1.0
#> c[5] -0.381 -0.374 0.24 -0.6974 -0.099 51 1.0
#> c[6] 0.026 0.025 0.17 -0.1868 0.242 102 1.0
#> c[7] 0.218 0.209 0.18 -0.0057 0.482 50 1.0
#> c[8] 0.143 0.148 0.20 -0.1105 0.397 48 1.0
#> c[9] 0.802 0.747 0.32 0.4527 1.215 23 1.0
#> c[10] 0.500 0.479 0.27 0.1580 0.838 47 1.0
#> A[1,1] 0.474 0.504 0.30 0.0713 0.840 18 1.0
#> A[2,1] 0.367 0.379 0.24 0.0567 0.668 33 1.0
#> A[3,1] 0.217 0.211 0.28 -0.1100 0.590 36 1.0
#> A[5,1] 1.832 1.808 0.45 1.3315 2.433 23 1.0
#> A[6,1] 1.140 1.143 0.28 0.8016 1.500 13 1.1
#> A[7,1] 1.061 1.045 0.32 0.6408 1.485 27 1.0
#> A[9,1] 1.383 1.372 0.42 0.8790 1.928 27 1.1
#> A[10,1] 1.577 1.588 0.44 1.0507 2.095 17 1.0
#> A[1,2] -1.642 -1.700 0.45 -2.1196 -1.124 32 1.0
#> A[2,2] -0.281 -0.296 0.25 -0.6011 0.044 30 1.0
#> A[3,2] -0.757 -0.756 0.35 -1.1897 -0.318 20 1.0
#> A[9,2] -1.252 -1.233 0.48 -1.8385 -0.682 18 1.1
#> A[1,3] -0.422 -0.463 0.45 -0.9941 0.166 9 1.1
#> A[2,3] 0.484 0.489 0.29 0.1108 0.829 33 1.0
#> A[3,3] 1.153 1.143 0.28 0.8127 1.518 19 1.1
#> A[4,3] 0.763 0.753 0.24 0.4767 1.072 35 1.0
#> A[7,3] 0.602 0.604 0.23 0.3013 0.907 36 1.0
#> A[8,3] 1.305 1.340 0.43 0.7765 1.869 29 1.0
#> A[9,3] 0.670 0.661 0.38 0.2168 1.213 14 1.0
#> A[10,3] 1.467 1.436 0.45 0.8870 2.072 13 1.0
#>
#> Factor model parameters:
#> mean median sd q10 q90 ess_bulk rhat
#> Corr[2,1] 0.015 0.0034 0.13 -0.182 0.186 9.6 1.1
#> Corr[3,1] 0.147 0.1048 0.16 -0.016 0.425 7.6 1.1
#> Corr[3,2] -0.233 -0.2421 0.18 -0.483 0.063 6.2 1.1
#>
#> ess_bulk is the bulk effective sample size; rhat is the potential
#> scale reduction factor on split chains (Rhat = 1 at convergence).
#> Use summary() for the full set of statistics (incl. ess_tail).Unlike eifa, the cifa model estimates the
correlation (Corr) between the latent factors.
Adding predictor terms to the right-hand side of formula
generates a cifa_pred, which models the latent factors in
terms of the predictor terms:
samples <- spifa(
items ~ poly(wealth, 2), data = ipixuna, nfactors = nfactors, ngp = 0, niter = 500,
constraints = list(discrimination = A)
)
samples#> Item factor analysis model: cifa_pred
#> Formula: items ~ poly(wealth, 2)
#> Dimensions: 100 respondents, 10 items, 3 latent factors, 0 spatial processes
#> MCMC: 1 chain, iter = 500, thin = 1, samples = 500
#>
#> Item model parameters:
#> mean median sd q10 q90 ess_bulk rhat
#> c[1] -0.419 -0.4127 0.15 -0.605 -0.243 127.1 1.0
#> c[2] -0.448 -0.4339 0.15 -0.653 -0.254 123.9 1.0
#> c[3] -0.412 -0.4142 0.14 -0.599 -0.222 131.9 1.0
#> c[4] -0.267 -0.2743 0.13 -0.435 -0.094 194.0 1.0
#> c[5] -0.249 -0.2525 0.16 -0.460 -0.041 105.8 1.0
#> c[6] 0.046 0.0459 0.15 -0.142 0.226 115.1 1.0
#> c[7] 0.255 0.2547 0.17 0.038 0.478 118.5 1.0
#> c[8] 0.157 0.1496 0.14 -0.022 0.335 180.6 1.0
#> c[9] 0.582 0.5792 0.18 0.354 0.814 36.8 1.0
#> c[10] 0.428 0.4284 0.20 0.157 0.689 59.9 1.0
#> A[1,1] 0.374 0.3834 0.18 0.147 0.586 47.9 1.0
#> A[2,1] 0.428 0.4196 0.27 0.102 0.796 41.4 1.0
#> A[3,1] 0.550 0.5758 0.32 0.140 0.915 16.5 1.1
#> A[5,1] 1.037 1.0138 0.29 0.739 1.390 16.7 1.1
#> A[6,1] 0.785 0.7982 0.23 0.544 1.059 55.6 1.0
#> A[7,1] 0.784 0.7731 0.27 0.460 1.118 56.8 1.0
#> A[9,1] 0.730 0.7208 0.28 0.377 1.069 56.5 1.0
#> A[10,1] 0.912 0.9268 0.26 0.574 1.248 32.7 1.0
#> A[1,2] 0.567 0.6136 0.41 0.079 1.025 13.6 1.1
#> A[2,2] 0.395 0.3975 0.18 0.160 0.621 8.8 1.1
#> A[3,2] 0.180 0.2980 0.51 -0.447 0.734 2.3 1.4
#> A[9,2] 0.206 0.2353 0.41 -0.351 0.704 4.7 1.2
#> A[1,3] -0.206 -0.1908 0.28 -0.609 0.142 17.2 1.1
#> A[2,3] -0.379 -0.3777 0.25 -0.713 -0.067 35.6 1.0
#> A[3,3] 0.019 0.0081 0.16 -0.165 0.237 133.0 1.0
#> A[4,3] -0.473 -0.4598 0.19 -0.729 -0.243 68.5 1.0
#> A[7,3] -0.444 -0.4405 0.22 -0.744 -0.153 69.0 1.0
#> A[8,3] -0.622 -0.6213 0.20 -0.881 -0.356 55.2 1.0
#> A[9,3] -0.623 -0.6187 0.27 -0.986 -0.281 36.5 1.0
#> A[10,3] -0.599 -0.5902 0.27 -0.949 -0.262 55.9 1.0
#>
#> Factor model parameters:
#> mean median sd q10 q90 ess_bulk rhat
#> B[1,1] -6.03 -6.04 1.56 -7.92 -4.20 15.7 1.1
#> B[2,1] 0.46 0.51 1.14 -1.04 1.80 66.1 1.0
#> B[1,2] 4.14 4.21 2.09 1.60 6.76 26.9 1.0
#> B[2,2] 0.75 0.65 1.51 -1.11 2.75 39.6 1.0
#> B[1,3] 3.16 2.98 1.88 0.80 5.48 25.4 1.0
#> B[2,3] 0.77 0.76 1.36 -0.97 2.47 61.3 1.0
#> Corr[2,1] -0.47 -0.49 0.18 -0.62 -0.19 5.2 1.1
#> Corr[3,1] -0.35 -0.34 0.14 -0.53 -0.18 18.5 1.1
#> Corr[3,2] 0.17 0.14 0.24 -0.10 0.51 2.9 1.3
#>
#> ess_bulk is the bulk effective sample size; rhat is the potential
#> scale reduction factor on split chains (Rhat = 1 at convergence).
#> Use summary() for the full set of statistics (incl. ess_tail).If the data is an sf object and no
predictor terms are included in formula, the model type is
spifa automatically. By default, it includes a spatial term
for each factor ngp = nfactors; but custom number of
ngp can be provided with a constraints loading
matrix (T) of size nfactors x ngp.
#> [,1] [,2]
#> [1,] 1 0
#> [2,] 1 0
#> [3,] 0 1This structure means that 1 GP is shared between factor 1 and 2, and another GP is introduced for factor 3:
samples <- spifa(
items ~ 1, data = ipixuna, nfactors = nfactors, ngp = 2, niter = 500,
constraints = list(discrimination = A, loading = T_loading)
)
samples#> Item factor analysis model: spifa
#> Formula: items ~ 1
#> Dimensions: 100 respondents, 10 items, 3 latent factors, 2 spatial processes
#> MCMC: 1 chain, iter = 500, thin = 1, samples = 500
#>
#> Item model parameters:
#> mean median sd q10 q90 ess_bulk rhat
#> c[1] -0.414 -0.410 0.16 -0.640 -0.21 124.5 1.0
#> c[2] -0.508 -0.504 0.17 -0.726 -0.29 71.9 1.0
#> c[3] -0.519 -0.527 0.17 -0.740 -0.29 88.4 1.0
#> c[4] -0.323 -0.323 0.17 -0.539 -0.12 87.0 1.0
#> c[5] -0.265 -0.261 0.19 -0.512 -0.02 64.7 1.0
#> c[6] 0.081 0.074 0.19 -0.159 0.34 69.7 1.0
#> c[7] 0.260 0.258 0.18 0.037 0.49 10.5 1.1
#> c[8] 0.177 0.154 0.20 -0.059 0.43 50.3 1.0
#> c[9] 0.739 0.726 0.25 0.443 1.08 7.9 1.1
#> c[10] 0.472 0.461 0.25 0.173 0.77 21.4 1.1
#> A[1,1] 0.512 0.531 0.36 0.065 0.96 134.2 1.0
#> A[2,1] -0.188 -0.179 0.35 -0.650 0.27 64.7 1.0
#> A[3,1] 0.108 0.106 0.56 -0.646 0.85 55.7 1.0
#> A[5,1] 1.402 1.388 0.29 1.069 1.77 33.7 1.0
#> A[6,1] 1.286 1.284 0.36 0.850 1.77 28.5 1.0
#> A[7,1] 1.066 1.051 0.30 0.716 1.47 35.6 1.0
#> A[9,1] 0.762 0.735 0.52 0.134 1.44 69.9 1.0
#> A[10,1] 1.429 1.387 0.39 0.991 1.90 20.1 1.1
#> A[1,2] -0.346 -0.334 0.43 -0.904 0.17 67.7 1.0
#> A[2,2] 0.682 0.676 0.33 0.241 1.12 81.0 1.0
#> A[3,2] 0.173 0.204 0.57 -0.605 0.87 40.6 1.0
#> A[9,2] 0.516 0.519 0.64 -0.328 1.25 35.5 1.1
#> A[1,3] 0.401 0.396 0.20 0.155 0.65 56.1 1.0
#> A[2,3] 0.650 0.647 0.23 0.369 0.93 46.0 1.0
#> A[3,3] 1.058 1.036 0.24 0.753 1.38 31.2 1.0
#> A[4,3] 0.763 0.709 0.31 0.417 1.16 25.6 1.0
#> A[7,3] 0.546 0.533 0.22 0.287 0.82 34.2 1.0
#> A[8,3] 1.100 1.008 0.45 0.655 1.67 12.2 1.1
#> A[9,3] 1.264 1.227 0.41 0.802 1.78 19.0 1.0
#> A[10,3] 1.085 1.071 0.31 0.709 1.46 3.8 1.2
#>
#> Factor model parameters:
#> mean median sd q10 q90 ess_bulk rhat
#> T[1,1] 0.919 0.915 0.0630 0.832 1.006 14.5 1.1
#> T[2,1] 0.912 0.908 0.0667 0.814 1.011 12.9 1.1
#> T[3,2] 0.963 0.951 0.0671 0.882 1.047 22.3 1.1
#> phi[1] 0.049 0.049 0.0032 0.045 0.052 4.4 1.2
#> phi[2] 0.055 0.055 0.0041 0.049 0.060 30.1 1.0
#> Corr[2,1] 0.115 0.144 0.1652 -0.069 0.326 16.0 1.0
#> Corr[3,1] 0.254 0.257 0.0768 0.134 0.336 24.3 1.1
#> Corr[3,2] 0.213 0.218 0.1049 0.030 0.331 31.5 1.0
#>
#> ess_bulk is the bulk effective sample size; rhat is the potential
#> scale reduction factor on split chains (Rhat = 1 at convergence).
#> Use summary() for the full set of statistics (incl. ess_tail).Notice that the loading parameter is also available in
the samples.
A spifa_pred model is simply defined by adding predictor
terms to formula.
samples <- spifa(
items ~ wealth, data = ipixuna, nfactors = nfactors, niter = 500,
constraints = list(discrimination = A)
)
samples#> Item factor analysis model: spifa_pred
#> Formula: items ~ wealth
#> Dimensions: 100 respondents, 10 items, 3 latent factors, 3 spatial processes
#> MCMC: 1 chain, iter = 500, thin = 1, samples = 500
#>
#> Item model parameters:
#> mean median sd q10 q90 ess_bulk rhat
#> c[1] -0.57260 -0.528 0.27 -0.93231 -0.29 33.7 1.0
#> c[2] -0.56493 -0.563 0.19 -0.80248 -0.32 79.9 1.0
#> c[3] -0.58779 -0.577 0.21 -0.87309 -0.32 65.5 1.0
#> c[4] -0.36991 -0.355 0.18 -0.62738 -0.15 48.5 1.0
#> c[5] -0.35614 -0.346 0.20 -0.61555 -0.11 45.6 1.0
#> c[6] 0.00062 0.015 0.19 -0.22678 0.22 54.9 1.0
#> c[7] 0.19205 0.197 0.22 -0.10039 0.47 35.1 1.0
#> c[8] 0.19514 0.187 0.21 -0.06398 0.46 52.3 1.0
#> c[9] 0.56597 0.559 0.24 0.26629 0.87 34.9 1.0
#> c[10] 0.39335 0.403 0.26 0.04838 0.72 37.9 1.0
#> A[1,1] 0.23607 0.234 0.19 0.00078 0.47 44.3 1.0
#> A[2,1] 0.34453 0.329 0.19 0.12458 0.58 71.8 1.0
#> A[3,1] 0.20850 0.212 0.20 -0.05182 0.46 25.2 1.1
#> A[5,1] 1.24924 1.227 0.35 0.82162 1.66 14.7 1.1
#> A[6,1] 1.07571 1.042 0.34 0.68615 1.48 5.8 1.1
#> A[7,1] 0.95125 0.944 0.24 0.63978 1.27 27.1 1.1
#> A[9,1] 0.80681 0.780 0.23 0.53003 1.13 29.0 1.0
#> A[10,1] 1.10507 1.093 0.28 0.78731 1.46 36.6 1.0
#> A[1,2] 1.11162 1.055 0.40 0.64652 1.68 13.9 1.2
#> A[2,2] 0.66447 0.651 0.22 0.38657 0.94 50.6 1.0
#> A[3,2] 0.48612 0.462 0.26 0.16289 0.84 38.4 1.0
#> A[9,2] 0.71997 0.715 0.26 0.39855 1.02 44.1 1.0
#> A[1,3] 0.18727 0.179 0.31 -0.19703 0.56 6.1 1.2
#> A[2,3] 0.55237 0.569 0.26 0.21271 0.89 36.2 1.0
#> A[3,3] 1.02677 1.023 0.26 0.70991 1.33 47.2 1.0
#> A[4,3] 0.84582 0.839 0.30 0.45570 1.26 37.1 1.0
#> A[7,3] 0.57838 0.580 0.21 0.32502 0.84 40.9 1.0
#> A[8,3] 1.17996 1.136 0.34 0.76787 1.64 23.0 1.0
#> A[9,3] 0.80017 0.767 0.33 0.38408 1.24 26.6 1.1
#> A[10,3] 0.98692 0.981 0.33 0.62181 1.43 40.1 1.0
#>
#> Factor model parameters:
#> mean median sd q10 q90 ess_bulk rhat
#> B[1,1] -0.489 -0.476 0.1617 -0.701 -0.312 7.5 1.1
#> B[1,2] 0.109 0.117 0.1271 -0.068 0.265 258.6 1.0
#> B[1,3] -0.251 -0.248 0.1320 -0.423 -0.100 167.8 1.0
#> T[1,1] 1.008 0.946 0.1597 0.862 1.228 1.4 2.0
#> T[2,2] 0.920 0.925 0.0663 0.833 1.011 10.6 1.0
#> T[3,3] 0.938 0.936 0.1048 0.823 1.069 16.3 1.0
#> phi[1] 0.054 0.055 0.0052 0.046 0.060 9.8 1.1
#> phi[2] 0.055 0.055 0.0102 0.040 0.069 2.4 1.4
#> phi[3] 0.054 0.052 0.0078 0.044 0.063 9.3 1.1
#> Corr[2,1] 0.055 0.043 0.1257 -0.097 0.220 12.2 1.0
#> Corr[3,1] 0.039 0.028 0.1606 -0.160 0.275 2.0 1.6
#> Corr[3,2] 0.025 0.080 0.3392 -0.484 0.469 1.8 1.6
#>
#> ess_bulk is the bulk effective sample size; rhat is the potential
#> scale reduction factor on split chains (Rhat = 1 at convergence).
#> Use summary() for the full set of statistics (incl. ess_tail).Notice that the predictor effects B is now included in
the samples.