-
Notifications
You must be signed in to change notification settings - Fork 17
/
cfa.Rmd
303 lines (209 loc) · 6.71 KB
/
cfa.Rmd
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
# Confirmatory Factor Analysis
This mostly follows Bollen (1989) for maximum likelihood estimation of a confirmatory factor analysis. In the following example we will examine a situation where there are two underlying (correlated) latent variables for 8 observed responses. The code as is will only work with this toy data set. Setup uses the <span class="pack" style = "">psych</span> and <span class="pack" style = "">mvtnorm</span> packages, and results are checked against the <span class="pack" style = "">lavaan</span> package.
## Data Setup
For the data we will simulate observed variables with specific loadings on two latent constructs (factors).
```{r cfa-setup}
library(tidyverse)
set.seed(123)
# loading matrix
lambda = matrix(
c(1.0, 0.5, 0.8, 0.6, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0, 1.0, 0.7, 0.6, 0.8),
nrow = 2,
byrow = TRUE
)
# correlation of factors
phi = matrix(c(1, .25, .25, 1), nrow = 2, byrow = TRUE)
# factors and some noise
factors = mvtnorm::rmvnorm(1000, mean = rep(0, 2), sigma = phi, "chol")
e = mvtnorm::rmvnorm(1000, sigma = diag(8))
# observed responses
y = 0 + factors%*%lambda + e
# Examine
#dim(y)
psych::describe(y)
# round(cor(y), 2)
# see the factor structure
psych::cor.plot(cor(y))
# example exploratory fa
#psych::fa(y, nfactors=2, rotate="oblimin")
```
## Functions
We will have two separate estimation functions, one for the covariance matrix,
and another for the correlation matrix.
```{r cfa-cov-func}
# measurement model, covariance approach
# trace function, strangely absent from base R
tr <- function(mat) {
sum(diag(mat), na.rm = TRUE)
}
cfa_cov <- function (parms, data) {
# Arguments-
# parms: initial values (named)
# data: raw data
# Extract parameters by name
l1 = c(1, parms[grep('l1', names(parms))]) # loadings for factor 1
l2 = c(1, parms[grep('l2', names(parms))]) # loadings for factor 2
cov0 = parms[grep('cov', names(parms))] # factor covariance, variances
# Covariance matrix
S = cov(data)*((nrow(data)-1)/nrow(data)) # ML covariance div by N rather than N-1, the multiplier adjusts
# loading estimates
lambda = cbind(
c(l1, rep(0,length(l2))),
c(rep(0,length(l1)), l2)
)
# disturbances
dist_init = parms[grep('dist', names(parms))]
disturbs = diag(dist_init)
# factor correlation
phi_init = matrix(c(cov0[1], cov0[2], cov0[2], cov0[3]), 2, 2) #factor cov/correlation matrix
# other calculations and log likelihood
sigtheta = lambda%*%phi_init%*%t(lambda) + disturbs
# in Bollen p + q (but for the purposes of this just p) = tr(data)
pq = dim(data)[2]
# a reduced version; Bollen 1989 p.107
# ll = -(log(det(sigtheta)) + tr(S%*%solve(sigtheta)) - log(det(S)) - pq)
# this should be the same as Mplus H0 log likelihood
ll = ( (-nrow(data)*pq/2) * log(2*pi) ) -
(nrow(data)/2) * ( log(det(sigtheta)) + tr(S%*%solve(sigtheta)) )
-ll
}
```
We can use the correlation matrix for standardized results. Lines correspond to those in `cfa_cov`.
```{r cfa-cor-func}
cfa_cor <- function (parms, data) {
l1 = parms[grep('l1', names(parms))] # loadings for factor 1
l2 = parms[grep('l2', names(parms))] # loadings for factor 2
cor0 = parms[grep('cor', names(parms))] # factor correlation
S = cor(data)
lambda = cbind(
c(l1, rep(0,length(l2))),
c(rep(0,length(l1)), l2)
)
dist_init = parms[grep('dist', names(parms))]
disturbs = diag(dist_init)
phi_init = matrix(c(1, cor0, cor0, 1), ncol=2)
sigtheta = lambda%*%phi_init%*%t(lambda) + disturbs
pq = dim(data)[2]
#ll = ( log(det(sigtheta)) + tr(S%*%solve(sigtheta)) - log(det(S)) - pq )
ll = ( (-nrow(data)*pq/2) * log(2*pi) ) -
(nrow(data)/2) * ( log(det(sigtheta)) + tr(S%*%solve(sigtheta)) )
-ll
}
```
## Estimation
Corresponding to the functions, we will get results based on the covariance and
correlation matrix respectively.
### Raw
Set initial values.
```{r cfa-cov-init}
par_init_cov = c(rep(1, 6), rep(.05, 8), rep(.5, 3))
names(par_init_cov) = rep(c('l1','l2', 'dist', 'cov'), c(3, 3, 8, 3))
```
Estimate and extract.
```{r cfa-cov-est}
fit_cov = optim(
par = par_init_cov,
fn = cfa_cov,
data = y,
method = "L-BFGS-B",
lower = 0
)
loadings_cov = data.frame(
f1 = c(1, fit_cov$par[1:3], rep(0, 4)),
f2 = c(rep(0, 4), 1, fit_cov$par[4:6])
)
disturbances_cov = fit_cov$par[7:14]
```
### Standardized
```{r cfa-cor-init}
par_init_cor = c(rep(1, 8), rep(.05, 8), 0) #for cor
names(par_init_cor) = rep(c('l1', 'l2', 'dist', 'cor'), c(4, 4, 8, 1))
```
```{r cfa-cor-est}
fit_cor = optim(
par = par_init_cor,
fn = cfa_cor,
data = y,
method = "L-BFGS-B",
lower = 0,
upper = 1
)
loadings_cor = matrix(
c(fit_cor$par[1:4], rep(0, 4), rep(0, 4), fit_cor$par[5:8]),
ncol = 2
)
disturbances_cor = fit_cor$par[9:16]
```
## Comparison
Gather results for summarizing.
```{r cfa-all}
results = list(
raw = list(
loadings = round(data.frame(loadings_cov, Variances = disturbances_cov), 3),
cov.fact = round(matrix(c(fit_cov$par[c(15, 16, 16, 17)]), ncol = 2) , 3)
),
standardized = list(
loadings = round(
data.frame(
loadings_cor,
Variances = disturbances_cor,
Rsq = (1 - disturbances_cor)
), 3),
cor.fact = round(matrix(c(1, fit_cor$par[c(17, 17)], 1), ncol = 2), 3)
),
# note inclusion of intercepts for total number of par
fit_lav = data.frame(
ll = fit_cov$value,
AIC = 2*fit_cov$value + 2 * (length(par_init_cov) + ncol(y)),
BIC = 2*fit_cov$value + log(nrow(y)) * (length(par_init_cov) + ncol(y))
)
)
results
```
Compare with <span class="pack" style = "">lavaan</span>.
```{r cfa-lavaan}
library(lavaan)
y = data.frame(y)
model = '
F1 =~ X1 + X2 + X3 + X4
F2 =~ X5 + X6 + X7 + X8
'
fit_lav = cfa(
model,
data = y,
mimic = 'Mplus',
estimator = 'ML'
)
fit_lav_std = cfa(
model,
data = y,
mimic = 'Mplus',
estimator = 'ML',
std.lv = TRUE,
std.ov = TRUE
)
# note that lavaan does not count the intercepts among the free params for
# AIC/BIC by default, (can get its result via -2 * as.numeric(lls) + k *
# attr(lls, "df")), but the mimic='Mplus' should have them correspond to optim's
# results
summary(fit_lav,
fit.measures = TRUE,
standardized = TRUE)
```
### Mplus
If you have access to Mplus you can use Mplus Automation to prepare the data. The following code is in Mplus syntax and will produce the above model.
```{r mplus, eval=FALSE}
library(MplusAutomation)
prepareMplusData(data.frame(y), "factsim.dat")
```
```
MODEL:
F1 BY X1-X4;
F2 BY X5-X8;
results:
STDYX;
```
## Source
Original code available at
https://github.com/m-clark/Miscellaneous-R-Code/blob/master/ModelFitting/cfa.R