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] -0.396 -0.393 0.15 -0.5764 -0.2043 75.9 1.0
#> c[2] -0.518 -0.506 0.20 -0.7757 -0.2745 38.8 1.0
#> c[3] -0.554 -0.553 0.24 -0.8527 -0.2580 31.0 1.0
#> c[4] -0.316 -0.316 0.16 -0.5271 -0.1102 94.1 1.0
#> c[5] -0.311 -0.292 0.24 -0.6408 -0.0086 7.3 1.1
#> c[6] 0.072 0.069 0.26 -0.2506 0.3731 25.4 1.0
#> c[7] 0.273 0.268 0.18 0.0454 0.4960 68.0 1.0
#> c[8] 0.201 0.202 0.17 0.0082 0.4032 77.7 1.0
#> c[9] 0.681 0.671 0.24 0.3968 0.9696 42.8 1.0
#> c[10] 0.496 0.486 0.24 0.1993 0.7943 25.1 1.0
#> A[1,1] 0.548 0.516 0.24 0.2597 0.8658 6.0 1.2
#> A[2,1] 0.767 0.716 0.35 0.3762 1.1924 14.7 1.1
#> A[3,1] 1.159 1.168 0.32 0.7743 1.5596 27.1 1.0
#> A[4,1] 0.731 0.713 0.30 0.4213 1.1538 29.7 1.0
#> A[5,1] -0.325 -0.343 0.44 -0.9079 0.2494 5.0 1.2
#> A[6,1] -0.438 -0.424 0.43 -1.0187 0.1151 4.5 1.2
#> A[7,1] 0.361 0.339 0.33 -0.0285 0.7999 9.8 1.1
#> A[8,1] 0.887 0.883 0.30 0.5286 1.2671 13.0 1.0
#> A[9,1] 0.918 0.925 0.40 0.3898 1.4541 13.6 1.1
#> A[10,1] 0.704 0.761 0.35 0.1872 1.1177 20.1 1.0
#> A[2,2] 0.563 0.559 0.24 0.2788 0.8568 5.7 1.1
#> A[3,2] 0.509 0.509 0.28 0.1369 0.8838 5.4 1.1
#> A[4,2] 0.127 0.137 0.25 -0.1728 0.4225 14.7 1.0
#> A[5,2] 1.708 1.683 0.47 1.1479 2.3277 17.7 1.1
#> A[6,2] 1.544 1.543 0.53 0.8674 2.2147 11.2 1.2
#> A[7,2] 1.174 1.164 0.33 0.7694 1.5600 14.7 1.1
#> A[8,2] 0.368 0.349 0.30 0.0546 0.7562 9.7 1.1
#> A[9,2] 1.222 1.193 0.36 0.7797 1.7185 5.8 1.2
#> A[10,2] 1.578 1.608 0.42 1.0197 2.0818 7.9 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).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] -0.741 -0.664 0.34 -1.130 -0.407 21.8 1.0
#> c[2] -0.546 -0.552 0.18 -0.784 -0.307 49.7 1.0
#> c[3] -0.696 -0.680 0.28 -1.104 -0.347 21.1 1.0
#> c[4] -0.335 -0.327 0.16 -0.555 -0.131 128.4 1.0
#> c[5] -0.370 -0.360 0.22 -0.661 -0.094 20.8 1.0
#> c[6] 0.031 0.026 0.20 -0.223 0.290 16.7 1.0
#> c[7] 0.192 0.206 0.21 -0.098 0.426 29.2 1.0
#> c[8] 0.183 0.173 0.21 -0.066 0.462 34.7 1.0
#> c[9] 0.754 0.756 0.24 0.442 1.078 37.3 1.0
#> c[10] 0.465 0.506 0.33 0.016 0.847 14.3 1.0
#> A[1,1] 0.568 0.568 0.27 0.234 0.888 29.2 1.0
#> A[2,1] 0.525 0.512 0.22 0.228 0.814 6.1 1.2
#> A[3,1] 0.322 0.317 0.26 -0.010 0.673 22.0 1.0
#> A[5,1] 1.595 1.564 0.40 1.141 2.097 18.1 1.0
#> A[6,1] 1.196 1.174 0.29 0.853 1.573 26.8 1.0
#> A[7,1] 1.083 1.035 0.36 0.672 1.564 10.9 1.1
#> A[9,1] 1.414 1.413 0.36 0.952 1.873 2.6 1.4
#> A[10,1] 1.713 1.635 0.43 1.256 2.318 5.4 1.2
#> A[1,2] 1.945 1.943 0.68 1.049 2.884 15.7 1.0
#> A[2,2] 0.770 0.745 0.25 0.478 1.139 52.7 1.0
#> A[3,2] 0.784 0.745 0.34 0.377 1.243 26.0 1.0
#> A[9,2] 1.218 1.247 0.40 0.671 1.700 9.3 1.1
#> A[1,3] -0.275 -0.215 0.47 -0.988 0.314 1.9 1.5
#> A[2,3] 0.395 0.416 0.27 0.010 0.749 4.6 1.2
#> A[3,3] 1.134 1.130 0.36 0.669 1.619 17.9 1.0
#> A[4,3] 0.949 0.945 0.31 0.569 1.367 41.1 1.0
#> A[7,3] 0.575 0.569 0.23 0.302 0.852 26.8 1.1
#> A[8,3] 1.489 1.479 0.56 0.817 2.262 2.9 1.3
#> A[9,3] 0.580 0.628 0.44 -0.021 1.104 3.2 1.3
#> A[10,3] 1.366 1.377 0.48 0.771 1.981 5.3 1.2
#>
#> Factor model parameters:
#> mean median sd q10 q90 ess_bulk rhat
#> Corr[2,1] -0.199 -0.205 0.15 -0.393 -0.026 2.6 1.4
#> Corr[3,1] 0.046 0.055 0.11 -0.081 0.171 24.1 1.0
#> Corr[3,2] 0.204 0.181 0.22 -0.072 0.551 1.5 1.9
#>
#> 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] -0.4405 -0.441 0.16 -0.652 -0.234 94.8 1.0
#> c[2] -0.4609 -0.458 0.16 -0.656 -0.257 102.8 1.0
#> c[3] -0.4569 -0.454 0.16 -0.656 -0.255 137.3 1.0
#> c[4] -0.2846 -0.283 0.14 -0.469 -0.106 168.7 1.0
#> c[5] -0.2574 -0.249 0.17 -0.484 -0.041 98.2 1.0
#> c[6] 0.0428 0.032 0.15 -0.143 0.229 123.4 1.0
#> c[7] 0.2192 0.208 0.17 0.006 0.442 116.9 1.0
#> c[8] 0.1636 0.161 0.15 -0.015 0.359 132.5 1.0
#> c[9] 0.5276 0.524 0.17 0.315 0.767 89.7 1.0
#> c[10] 0.3613 0.378 0.19 0.115 0.609 86.6 1.0
#> A[1,1] 0.3554 0.348 0.18 0.140 0.603 74.1 1.0
#> A[2,1] 0.2742 0.267 0.27 -0.067 0.652 36.1 1.1
#> A[3,1] 0.2701 0.266 0.26 -0.052 0.566 63.5 1.0
#> A[5,1] 1.0655 1.067 0.24 0.749 1.355 66.0 1.0
#> A[6,1] 0.9376 0.922 0.23 0.652 1.253 47.7 1.0
#> A[7,1] 0.8031 0.782 0.25 0.507 1.129 71.3 1.0
#> A[9,1] 0.6692 0.658 0.28 0.314 1.040 35.9 1.0
#> A[10,1] 0.9313 0.928 0.24 0.602 1.236 50.9 1.0
#> A[1,2] 0.1571 0.350 0.68 -0.840 0.919 1.5 1.9
#> A[2,2] 0.2982 0.324 0.23 -0.036 0.559 2.1 1.4
#> A[3,2] 0.0082 0.019 0.38 -0.491 0.523 1.9 1.5
#> A[9,2] 0.0218 0.024 0.39 -0.500 0.525 2.0 1.5
#> A[1,3] 0.2885 0.304 0.24 -0.023 0.584 34.8 1.0
#> A[2,3] 0.5672 0.570 0.28 0.184 0.942 38.9 1.0
#> A[3,3] 0.5483 0.551 0.15 0.349 0.740 139.8 1.0
#> A[4,3] 0.5220 0.507 0.18 0.298 0.755 53.0 1.0
#> A[7,3] 0.4387 0.422 0.21 0.181 0.698 81.6 1.0
#> A[8,3] 0.7184 0.701 0.23 0.437 1.039 19.7 1.1
#> A[9,3] 0.6897 0.694 0.24 0.391 0.967 37.4 1.0
#> A[10,3] 0.6648 0.658 0.20 0.420 0.934 75.3 1.0
#>
#> Factor model parameters:
#> mean median sd q10 q90 ess_bulk rhat
#> B[1,1] -5.64 -5.605 1.12 -7.062 -4.19 67.0 1.0
#> B[2,1] 0.18 0.088 1.29 -1.429 1.92 38.4 1.0
#> B[1,2] 1.50 3.038 5.02 -5.749 7.08 1.8 1.5
#> B[2,2] 0.37 0.508 1.85 -2.223 2.57 8.8 1.1
#> B[1,3] -3.43 -3.258 1.61 -5.368 -1.58 36.1 1.0
#> B[2,3] -0.63 -0.599 1.26 -2.334 1.03 99.9 1.0
#> Corr[2,1] -0.14 -0.185 0.45 -0.700 0.59 4.5 1.4
#> Corr[3,1] 0.30 0.278 0.16 0.082 0.49 4.3 1.2
#> Corr[3,2] -0.30 -0.263 0.16 -0.513 -0.13 4.9 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).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] -0.407 -0.410 0.14 -0.587 -0.222 150.8 1.0
#> c[2] -0.575 -0.566 0.19 -0.816 -0.355 36.5 1.0
#> c[3] -0.584 -0.585 0.20 -0.834 -0.332 67.0 1.0
#> c[4] -0.315 -0.304 0.14 -0.498 -0.140 109.0 1.0
#> c[5] -0.376 -0.360 0.27 -0.736 -0.051 29.0 1.0
#> c[6] 0.028 0.017 0.19 -0.220 0.269 67.0 1.0
#> c[7] 0.154 0.154 0.18 -0.087 0.384 13.5 1.1
#> c[8] 0.134 0.142 0.17 -0.087 0.351 76.2 1.0
#> c[9] 0.564 0.560 0.20 0.314 0.819 37.4 1.0
#> c[10] 0.363 0.371 0.23 0.071 0.654 6.0 1.2
#> A[1,1] 0.481 0.464 0.28 0.137 0.836 127.7 1.0
#> A[2,1] -0.190 -0.196 0.33 -0.596 0.245 96.8 1.0
#> A[3,1] 0.136 0.114 0.45 -0.381 0.758 70.1 1.0
#> A[5,1] 1.598 1.558 0.53 1.085 2.231 5.4 1.1
#> A[6,1] 1.115 1.109 0.30 0.746 1.509 17.7 1.0
#> A[7,1] 0.978 0.968 0.25 0.667 1.313 44.1 1.0
#> A[9,1] 0.697 0.703 0.49 0.041 1.293 69.8 1.0
#> A[10,1] 1.237 1.221 0.35 0.801 1.705 7.9 1.1
#> A[1,2] -0.346 -0.342 0.43 -0.898 0.240 56.0 1.0
#> A[2,2] 0.716 0.715 0.33 0.319 1.161 147.4 1.0
#> A[3,2] 0.125 0.132 0.50 -0.514 0.804 62.3 1.0
#> A[9,2] 0.309 0.326 0.59 -0.394 1.091 54.6 1.0
#> A[1,3] 0.460 0.441 0.20 0.217 0.726 50.1 1.0
#> A[2,3] 0.788 0.753 0.27 0.468 1.182 31.3 1.0
#> A[3,3] 1.106 1.113 0.23 0.804 1.396 69.7 1.0
#> A[4,3] 0.673 0.657 0.23 0.391 0.986 30.8 1.0
#> A[7,3] 0.579 0.572 0.21 0.323 0.829 20.2 1.1
#> A[8,3] 1.058 1.042 0.33 0.631 1.484 32.6 1.1
#> A[9,3] 1.097 1.107 0.25 0.775 1.396 58.1 1.1
#> A[10,3] 1.080 1.077 0.28 0.752 1.407 52.5 1.0
#>
#> Factor model parameters:
#> mean median sd q10 q90 ess_bulk rhat
#> T[1,1] 0.941 0.926 0.0690 0.860 1.0549 8.7 1.1
#> T[2,1] 0.882 0.900 0.0988 0.745 0.9917 2.9 1.3
#> T[3,2] 0.909 0.906 0.0777 0.800 1.0115 7.7 1.1
#> phi[1] 0.050 0.051 0.0047 0.044 0.0562 9.9 1.1
#> phi[2] 0.054 0.055 0.0080 0.043 0.0653 10.4 1.0
#> Corr[2,1] 0.157 0.145 0.2073 -0.075 0.4520 6.3 1.1
#> Corr[3,1] -0.128 -0.085 0.1066 -0.294 -0.0140 5.8 1.2
#> Corr[3,2] -0.257 -0.279 0.1541 -0.416 0.0044 2.5 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).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] -0.532 -0.531 0.22 -0.8073 -0.249 48.4 1.0
#> c[2] -0.528 -0.516 0.19 -0.7766 -0.302 72.7 1.0
#> c[3] -0.496 -0.495 0.19 -0.7363 -0.248 100.9 1.0
#> c[4] -0.343 -0.339 0.16 -0.5365 -0.136 103.7 1.0
#> c[5] -0.289 -0.284 0.19 -0.5286 -0.038 49.7 1.0
#> c[6] 0.036 0.037 0.18 -0.2012 0.274 38.5 1.1
#> c[7] 0.274 0.271 0.20 0.0273 0.530 49.0 1.0
#> c[8] 0.216 0.214 0.17 0.0051 0.427 105.4 1.0
#> c[9] 0.760 0.741 0.26 0.4419 1.125 16.1 1.1
#> c[10] 0.475 0.472 0.20 0.2096 0.710 51.9 1.0
#> A[1,1] 0.401 0.375 0.22 0.1312 0.720 45.9 1.0
#> A[2,1] 0.452 0.441 0.24 0.1634 0.764 7.8 1.1
#> A[3,1] 0.301 0.291 0.22 0.0288 0.599 79.6 1.0
#> A[5,1] 1.522 1.503 0.38 1.1259 2.071 21.2 1.0
#> A[6,1] 1.205 1.209 0.31 0.8209 1.586 46.2 1.0
#> A[7,1] 1.216 1.171 0.32 0.8575 1.642 20.5 1.0
#> A[9,1] 1.116 1.115 0.29 0.7787 1.475 32.4 1.0
#> A[10,1] 1.315 1.264 0.40 0.8853 1.824 16.4 1.0
#> A[1,2] 1.139 1.144 0.37 0.6788 1.626 21.6 1.1
#> A[2,2] 0.801 0.773 0.23 0.5203 1.111 43.3 1.1
#> A[3,2] 0.594 0.575 0.28 0.2576 0.967 53.4 1.0
#> A[9,2] 0.905 0.905 0.31 0.5326 1.293 7.9 1.1
#> A[1,3] 0.025 0.027 0.27 -0.3180 0.372 26.7 1.0
#> A[2,3] 0.532 0.539 0.24 0.2423 0.833 47.5 1.0
#> A[3,3] 0.892 0.893 0.20 0.6174 1.142 65.8 1.0
#> A[4,3] 1.006 0.991 0.32 0.6154 1.440 6.4 1.1
#> A[7,3] 0.693 0.697 0.24 0.3936 1.017 28.5 1.0
#> A[8,3] 1.258 1.216 0.38 0.8021 1.773 20.0 1.0
#> A[9,3] 0.839 0.851 0.36 0.3619 1.317 7.2 1.1
#> A[10,3] 1.053 1.059 0.29 0.6868 1.418 39.7 1.0
#>
#> Factor model parameters:
#> mean median sd q10 q90 ess_bulk rhat
#> B[1,1] -0.433 -0.437 0.1266 -0.582 -0.286 64.3 1.0
#> B[1,2] 0.096 0.094 0.1341 -0.067 0.269 206.3 1.0
#> B[1,3] -0.244 -0.238 0.1147 -0.389 -0.113 230.7 1.0
#> T[1,1] 0.889 0.881 0.0969 0.783 1.029 3.7 1.2
#> T[2,2] 0.920 0.879 0.1378 0.777 1.109 15.7 1.0
#> T[3,3] 0.895 0.869 0.0844 0.812 1.033 6.2 1.2
#> phi[1] 0.050 0.050 0.0082 0.039 0.065 12.6 1.2
#> phi[2] 0.046 0.046 0.0055 0.039 0.053 19.0 1.1
#> phi[3] 0.054 0.052 0.0078 0.044 0.064 3.8 1.3
#> Corr[2,1] 0.087 0.104 0.1250 -0.045 0.201 34.4 1.0
#> Corr[3,1] 0.164 0.133 0.1818 -0.023 0.472 13.9 1.2
#> Corr[3,2] 0.124 0.086 0.1968 -0.122 0.404 7.4 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).Notice that the predictor effects B is now included in
the samples.