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Model types

spifa() can fit five model types that are determined automatically from formula and the class of data (see ?spifa):

The model type is determined based on the following:

  1. Exploratory structure (eifa) is used when no constraints are provided.
  2. Models with predictors (cifa_pred/spifa_pred) are activated if terms are included on the right-hand side of formula.
  3. Spatial models (spifa/spifa_pred) are only activated when the data is geo-referenced as an sf object, otherwise the non-spatial models are activated.
  4. Setting ngp = 0 forces the model to be non-spatial (eifa/cifa/cifa_pred).

We will showcase the five model types using the ipixuna data.

library(spifa)

data(ipixuna)
nitems <- ncol(ipixuna$items)

Exploratory item factor analysis (EIFA)

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:

attr(samples, "fit_args")$constrain_L
#>       [,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    1

EIFA also never estimates the correlation between the latent factors, so only the easiness (c) and discrimination (A) parameters are estimated.

Confirmatory item factor analysis (CIFA)

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.

CIFA model with predictors

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).

Spatial item factor analysis (SPIFA)

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.

T_loading <- matrix(c(1, 1, 0, 0, 0, 1), nfactors, 2)
T_loading
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    1    0
#> [3,]    0    1

This 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.

SPIFA model with predictors

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.

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.