Multigroup Models

Beaujean Chapter 4Brown Chapter 7

Terms

• Invariant – Equivalent – Means that the structures, items, etc. are equal– Even better definition:• Under different conditions, do the measurements yield

the same measurement of attributes?

Terms

• Means and covariances model– For these type of models, you must estimate both

the means and covariances– This change lets you test many different pieces of

the model, rather than just the correlations between items

Terms

• Intercepts – think about regular regression:– The Y-intercept is the mean of Y – that is your best

guess for what someone is going to score on that item without any other X information.

– Same idea here, it’s the starting value for a manifest variable.

Terms

• Latent mean– The score of each item * the path for each item all

averaged together. – Think about this value as a weighted mean.

Terms

• Endogenous questions will get intercepts, while exogenous variables will get means.

Identification

• When you add means and intercepts, you are obviously estimating more things

• Ways to make sure it’s identified:– Standardize the latent (sets the latent mean to

zero, estimates all the intercepts)– Scale on a manifest variable (fixes one variable to

one and that intercept to zero)– Effects coding (eww)

Questions

• Do items act the same across groups?• Is the factor structure the same across

groups?• Are the paths equal across groups?• Are the latent means equal across groups?• Does this replicate?

Quick Note

• Most people only consider two groups at a time.– You can do more than two groups but it gets very

complex

So how do we test this?

• We start by working with the least restrictive model– This model is the base CFA you are working with.– You start by putting everyone together regardless

of group (because if the regular CFA is bad, multigroup testing is not appropriate).

So how do we test this?

• We start by working with the least restrictive model– Then you put them all together in one big model,

separated by group but:– You let the estimates be different across groups

So how do we test this?

• Then we move to more restrictive models– Force estimates to be equal across groups– Pick one estimate at a time, so you can see where

the model breaks (or doesn’t)

Steps

• First, test the model as a regular CFA with everyone in the dataset – Do not group them– You must establish that the CFA is good first

Steps

• Second, test each group separately– Subsetting!

Steps

• Second, test each group separately– Pick the one group, get the fit indices– Switch to the other group, get the fit indices

• Fit?– If the fit for one group is extremely different (or

bad), you would stop here. You need them to be roughly equal.

Steps

• Nesting!– The next steps will be to nest the two models

together.– Nesting is like stacking the models together (like

pancakes).

Steps

• For the nested model steps, most people use Brown’s terminology and procedure.

• The Beaujean terms are a mix of different ones (there’s nothing wrong with them, but I’ve mostly seen Brown’s published).

Steps

• All the possible paths:– The whole model (the picture)– Loadings (regression weights)– Intercepts (y-intercept for each item)– Error variances (variance)– Factor variances (variances for the latents)– Factor covariances (correlation)– Factor means (latent means)

Steps

• Y = a + bx + e– A = intercept– B = Loading– E = residual

Steps

• Equal form / configural invariance– In this model, you put the two groups together

into the same model.– You do not force any of the paths to be the same,

but you are forcing the model picture to be the same.

– You are testing if both groups show the same factor structure (configuration).

Steps

• Metric Invariance– In this model, you are forcing all the factor

loadings (regression weights) to be exactly the same

– This step will tell you if the groups have the same weights for each question – or if some questions have different signs or strengths

Steps

• Scalar Invariance– In this model, you are forcing the intercepts of the

items to be the same.– This step will tell you if items have the same

starting point – remember that the y-intercept is the mean of the item.

– If a MG model is going to indicate non-invariance – this step is usually the one that breaks.

Steps

• Strict Factorial Invariance– In this model, you are forcing the error variances

for each item to be the same. – This step will tell you if the variance (the spread)

of the item is the same for each item. If you get differences, that indicates one group has a larger range of answers than another. (means they are more heterogeneous).

Steps

• Population Heterogeneity – Equal factor variances • Testing if latents have the same set of variance – means

that the overall score has the same spread

– Equal factor covariances• Testing if the correlations between factors is the same

for each group

– Equal latent means• Testing if the overall latent means are equal for each

group

How to tell?

• How can I tell if steps are invariant?– You will expect fit to get worse as you go because

you are being more and more restrictive.– You can use a change in chi-square test (not

suggested too much because of chi-square issues we’ve discussed before).

– Most people use the change in CFI test.

What next?

• What do I do if steps are NOT invariant?• Partial invariance – when strict invariance

cannot be met, you can test for partial invariance

Partial Invariance

• Partial invariance occurs when most of the items are invariant but a couple.– You have to meet the invariance criteria, so you

trying to bring your bad step “up” to the invariant level

– You want to do as few of items as possible (see table in handout).

Partial Invariance

• So, how do I test partial invariance?– You will change ONE item at a time.

Lavaan

• We’ve talked before about setting paths equal to each other by calling them the same parameter name– Cheese =~ a*feta + a*swiss– So feta and swiss will be estimated at the same

value.

Lavaan

• A way to set all things equal– group.equal = “keyword”– Put in the sem() or cfa() function

Lavaan

• Options for group.equal– loadings (for manifest)– intercepts (for manifest)– means (for latents)– residuals (for manifest)– residual.covariances (for manifest)– lv.variances (for latents)– lv.covariances (for latents)– regressions (for manifest)

Example

• RS14 Scale!– Men = 1– Women = 2– Black = 1– White = 2

Overall Model

• First, let’s program the overall model to make sure it fits ok.

• NEW: meanstructure = TRUE– Gives you the intercepts and means in the model.

Overall Model

• Fit for the model– CFI = .927– RMSEA = .086– SRMR = .041– X2(77) = 368.984

Separate Group Models

• Test men and then women separately before nesting them together.

Separate Group ModelsMen Women

CFI .920 .895

RMSEA .102 .094

SRMR .041 .052

X2(77) 271.457 258.610

The women model fit is a little bit worse, but we won’t know if it’s significant (or where) until we combine them.

Configural Invariance

• Now, combine the models together and determine if they have the same factor structure. – Use the group function in the CFA code.– group = “variable name”

Configural InvarianceConfigural Metric Scalar Strict

CFI .909

RMSEA .098

SRMR .047

X2 (154) = 530.067

Metric Invariance

• For metric invariance, we are going to set loadings across groups equal.– Use the code group.equal=c(“loadings”) in the

sem() or cfa() function. • Check out what that does in the output

Metric InvarianceConfigural Metric Scalar Strict

CFI .909 .909

RMSEA .098 .094

SRMR .047 .057

X2 (154) = 530.067 (167) = 545.518

DIFFERENT? n/a NOPECFI = 0.000

Scalar Invariance

• Intercepts are set to equal by adding “intercepts” to the group.equal part.– We are slowly making the model more restrictive,

so you want to add constraints, not just change them.

Scalar Invariance

Scalar InvarianceConfigural Metric Scalar Strict

CFI .909 .909 .902

RMSEA .098 .094 .094

SRMR .047 .057 .060

X2 (154) = 530.067 (167) = 545.518 (180) = 585.327

DIFFERENT? n/a NOPECFI = 0.000

NOPECFI = .007

Strict Invariance

• Residuals for the manifest variables are set to equal by adding in residuals to the group.equal part.

Strict InvarianceConfigural Metric Scalar Strict

CFI .909 .909 .902 .888

RMSEA .098 .094 .094 .097

SRMR .047 .057 .060 .062

X2 (154) = 530.067 (167) = 545.518 (180) = 585.327 (194) = 660.398

DIFFERENT? n/a NOPECFI = 0.000

NOPECFI = .007

YESCFI = .014

So what now?

• Partial invariance– We have to figure out where the problem is for

our residuals that says they aren’t equal.– We’ll use the modindices function to find the

biggest issue but only for the items in the step you were having problems with.

Partial Invariance

• Figure out which level you need:– ##metric =~ for loadings– ##scalar ~1 for intercepts– ##strict ~~ for variances

• Subset out that information:– variances = partialmod[partialmod$op == "~~", ]– variances[order(variances$mi,

decreasing=TRUE), ]

Partial Invariance

• Remember that not everything is useful!– So, in this case, adding correlated variances is not

helpful.– We want to look for the times that variances are

not equal to variances– RS9~~RS9

Partial Invariance

• Since that constraint is the problem, we are going to let it go free!– group.partial = c("RS9 ~~ RS9")– You can add more here if you want, but do one of

them at a time to determine when to stop.

Partial Invariance

• Women

• Men

Think about partial invariance as mini t-tests.

Strict InvarianceScalar Strict RS9 Free

CFI .902 .888 .891

RMSEA .094 .097 .096

SRMR .060 .062 .062

X2 (180) = 585.327

(194) = 660.398

(193) = 646.559

DIFFERENT? NOPECFI = .007

YESCFI = .014

YESCFI = .011

NOTE: compare to scalar not strict.

Partial Invariance

• Women

• Men

Think about partial invariance as mini t-tests.

Strict InvarianceScalar Strict RS9 Free RS13 Free

CFI .902 .888 .891 .894

RMSEA .094 .097 .096 .095

SRMR .060 .062 .062 .062

X2 (180) = 585.327

(194) = 660.398

(193) = 646.559

(192) = 633.111

DIFFERENT? NOPECFI = .007

YESCFI = .014

YESCFI = .011

NOCFI = .008

NOTE: compare to scalar not strict.

Latent Means

• You can if latent means are different across groups using the same procedures with group.equal = c(“means”)– However, let’s say you want to use the weighted

means as a predictor in a later analysis – Or we can calculate a t.test to determine if they

are different and get effect sizes.

Latent Means

• Think about this for a second:– We normally use EFA/CFA to show that each

question has a nice loading and the questions “go together”

– And then we totally ignore the fact that the loadings are different and just create total scores or average scores.

– Why lose that information?

Latent Means

• You will calculate the latent score for each person by multiplying their individual item score times the loading– Then you can average them or total them

depending on how the scale is traditionally scored

Latent Means

• You will take the estimates for the loadings for each factor– So, if you have two factors you will need the

weights for each one but separately, aka don’t just average them all together.

– Make sure you are multiplying the right things!

Latent Means

• First, save the loadings:– loadings = parameterestimates(partial.fit)– Now view the loadings – it’s very similar to saving

the modinces, which means we can subset them.

Latent Means

• Subset out the loadings by group and type.• We want to use =~ for regression weights. • The select part gets you only the estimates.– menload = subset(loadings, group == "1" & op ==

"=~", select = "est")– womenload = subset(loadings, group == "2" & op

== "=~", select = "est")

Latent Means

• Now, we have the loadings and want to multiply them by the original scores.

• Be sure you’ve separated the data into groups!

Latent Means

• Apply function:– Arguments (dataset, 1 for rows / 2 for columns,

function)– Make sure the columns in the same order!– menmultiply = apply(men[, 4:ncol(men)], 1,

function(x) { x * menload })– womenmultiply = apply(women[, 4:ncol(women)],

1, function(x) { x * womenload })

Latent Means

• Apply is great! But it saves as a list. Lame.• So change that over to a data frame.– menmultiply = as.data.frame(menmultiply)– menmultiply = t(menmultiply) – ##flip it so it's people per row rather than

question by row

Latent Means

• To get the mean for each person:– menlatent = rowMeans(menmultiply)– womenlatent = rowMeans(womenmultiply)

• If you want to get the sum:– menlatent = rowSums(menmultiply)– womenlatent = rowSums(womenmultiply)

Latent Means

• Let’s do a t.test to see if they are different– t.test(group 1, group 2, …)• The rest of the arguments are not required but wanted

to show them to you if you wanted to change them to one tailed.

– t.test(menlatent, womenlatent, alternative = "two.sided", paired=FALSE, na.action=TRUE)

Latent Means

• Effect sizes!• Install the effsize package– library(effsize)– cohen.d(menlatent, womenlatent)– cohen.d(group 1, group 2, paired=FALSE)