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- MIMIC models (with formative and reflective manifest variables connected to a latent construct)
- Confirmatory factor analysis with both continuous and ordinal data

Contributed by tim.bates@ed.ac.uk, any errors or questions, please post on the forums.

This is an example from chapter 13 of Schumacker and Lomax’s Beginner’s guide to SEM

The fitted model is as follows:

mimic model

Not shown on this diagram is the fact that the variance of the formative variables (income, occupation, education) are fixed to 1 (the data input are a correlation matrix) (hat-tip to Andreas Brandmaier for this!)

require(sem) require(OpenMx) # Often you will see data presented as a lower diagonal. # the readMoments() function in the sem package is a nice helper to read this from the screen: data = sem::readMoments(file = "", diag = TRUE) 1 .304 1 .305 .344 1 .100 .156 .158 1 .284 .192 .324 .360 1 .176 .136 .226 .210 .265 1 # terminates with an empty line: see ?readMoments for more help # now letsfill in the upper triangle with a flipped version of the lower data[upper.tri(data, diag=F)] = t(data)[upper.tri(data, diag=F)] # Set up manifest variables manifests = c("income", "occup", "educ", "church", "member", "friends") # Use these to create names for our dataframe dimnames(data) = list(manifests, manifests) # And latents latents = "social" # 1 latent, with three formative inputs, and three reflective outputs (each with residuals) # Just to be helpful to myself, I've made lists of the formative sources, and the reflective receiver variables in this MIMIC model receivers = manifests[4:6] sources = manifests[1:3] MIMIC <- mxModel("MIMIC", type="RAM", manifestVars = manifests, latentVars = latents, # Factor loadings mxPath(from = sources , to = "social" , values), mxPath(from = "social", to = receivers, values), # Correlated formative sources for F1, each with variance = 1 mxPath(from = sources, connect = "unique.bivariate", arrows = 2), mxPath(from = sources, arrows = 2, values = 1, free=F ), # Residual variance on receivers mxPath(from = receivers, arrows = 2), mxData(data, type = "cov", numObs = 530) ) MIMIC <- mxRun(MIMIC); summary(MIMIC)

free parameters: name matrix row col Estimate Std.Error lbound ubound 1 A social income 0.13244947 5.51420454 2 A social occup 0.05701774 2.37389629 3 A social educ 0.19511219 8.12304135 4 A church social 0.60971866 25.38489597 5 A member social 1.28671307 53.56883563 6 A friends social 0.86519046 36.02017596 7 S income occup 0.30399991 0.03764547 8 S income educ 0.30499977 0.03761111 9 S occup educ 0.34399974 0.03618180 10 S church church 0.96770471 0.05954356 11 S member member 0.85617184 0.05267749 12 S friends friends 0.93497178 0.05749125 observed statistics: 21 estimated parameters: 12 degrees of freedom: 9 -2 log likelihood: 2893.752 saturated -2 log likelihood: 2804.686 number of observations: 530 chi-square: 89.06589 p: 2.506216e-15 Information Criteria: df Penalty Parameters Penalty Sample-Size Adjusted AIC 71.06589 113.0659 NA BIC 32.61000 164.3404 126.249 CFI: 0.7740255 TLI: 0.6233758 RMSEA: 0.1295581

Contributed by tim.bates@ed.ac.uk, any errors or questions, please post on the forums.

Data: Built-in example data frame

Variables: One continuous variable, and two binary ordinal variables

The fitted model is as follows:

mimic model -->

require(OpenMx) require(psych) # Read in an example dataset supplied with the OpenMx package data(myFADataRaw) # Grab three variables df <- myFADataRaw[, c("x1","z1", "z3")] # rename them to be more memorable manifests = c("cont1", "ord1","ord2") names(df) <-manifests # For OpenMx to work with ordinal data, they must be ordered factors! df$ord1 <- mxFactor(df$ord1, levels=0:1) df$ord2 <- mxFactor(df$ord2, levels=0:2) # Let's see what we're working with now str(df) 'data.frame': 500 obs. of 3 variables: $ cont1: num 1 1.63 1.95 2.95 3.66 ... $ ord1 : Ord.factor w/ 2 levels "0"<"1": 2 2 2 2 2 2 2 2 2 1 ... $ ord2 : Ord.factor w/ 3 levels "0"<"1"<"2": 1 2 2 1 2 2 1 2 1 1 ... hist(df$cont1) # nice and normal plot(~df$ord1) # binary - 4x as many 1s as 0s plot(~df$ord2) # three-values: low endorsement of the highest value # Just to help keep things simple in the model, let's list up the continuous # and the ordinal manifests separately ordinalVars = manifests[2:3] continuousVars = manifests[1] m1 <- mxModel("joint CFA", type="RAM", manifestVars = manifests, latentVars = "F1", # Factor loadings mxPath(from="F1", to = manifests, arrows=1, free=T, values=1, labels=c("l1","l2","l3")), # Manifest continuous mean & residual variance (both free) mxPath(from="one", to = continuousVars, arrows=1, free=T, values=0, labels="meancont1"), mxPath(from=continuousVars , arrows=2, free=T, values=1, labels="e1" ), # Ordinal manifests mean and variance fixed at 0 & 1 # The thresholds will be computed as being based off this underlying normal distribution) mxPath(from="one", to = ordinalVars, arrows=1, free=F, values=0, labels=c("meanord1","meanord2")), mxPath(from=ordinalVars , arrows=2, free=F, values=1, labels=c("e2","e3")), # fix latent variable variance @1 & mean @0 mxPath(from="F1" , arrows=2, free=F, values=1, labels ="varF1"), mxPath(from="one", to="F1", arrows=1, free=F, values=0, labels="meanF1"), # Make a matrix as wide as the number of ordinal variables (2), and as many rows as the max # of thresholds (3) - 1 # In this case that is 2 * 2 # leave it free where there is a real threshold to be set, and fill the rest of each column fixed @NA # set the column names to the name of the ordinal variables: that is how OpenMx maps into this threshold matrix mxMatrix("Full", nrow=2, ncol=2, byrow=T, name="thresh", dimnames = list(c(), ordinalVars), free = c(T, T , F, T), values = c(0, -1, NA, 1) ), # Add some data mxData(df, type="raw"), # tell the mxRAMObjective that we have not only A,S, F and M matrices, but also some ordinal vars with thresholds mxRAMObjective(A="A", S="S", F="F", M="M", thresholds="thresh") ) # run! m1 = mxRun(m1); summary(m1) free parameters: name matrix row col Estimate Std.Error lbound ubound 1 l1 A cont1 F1 -0.1953311 0.04115263 2 l2 A ord1 F1 -1.0135426 NaN 3 l3 A ord2 F1 -4.3049347 NaN 4 e1 S cont1 cont1 0.9571266 0.06018277 5 meancont1 M 1 cont1 2.9876221 0.04461568 6 <NA> thresh 1 ord1 -1.3038918 NaN 7 <NA> thresh 1 ord2 -2.1373824 NaN 8 <NA> thresh 2 ord2 5.7570834 NaN observed statistics: 1500 estimated parameters: 8 degrees of freedom: 1492 -2 log likelihood: 2678.115 saturated -2 log likelihood: NA number of observations: 500 chi-square: NA p: NA Information Criteria: df Penalty Parameters Penalty Sample-Size Adjusted AIC -305.8846 2694.115 NA BIC -6594.0798 2727.832 2702.44 CFI: NA TLI: NA RMSEA: NA

Attachment | Size |
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MIMIC.png | 29.38 KB |

MIMIC.R | 4.03 KB |

joint_ordinal_CFA.R | 3.75 KB |

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