Confidence on standardized parameters

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This Page will discuss getting CIs on standardized parameters. Often, raw parameters are valuable: They give us a read-out of effects in the natural units of the variables in the model, e.g. change in miles per gallon per kg of extra vehicle weight, instead of change in SDs of miles per gallon per SD of weight, which is less meaningful.

However, especially in social science, readers are used to standardized effects. In addition, readers might want an idea of the relative size of an effect, and standardized parameters convey this.

To standardize models in umx, you typically add std=TRUE to the summary method. Getting SEs around these standardized values can be more involved. There are four ways umx can give CIs on standardized values.

  1. Using SEs from umxSummary (for RAM models).
  2. Running the model on standardized data.
  3. Using mxSE to compute a standardized effect, along with mxCIs on these.
  4. Adding mxAlgebras to compute standardized effects, along with mxCIs added for these algebras. This approach is most accurate, can be time consuming, is not guaranteed (models can have problems moving parameters to non-significant levels without them hitting boundaries or illegal values.)

1. Using SEs

Method 1 is easiest, and this is what umxSummary does for you for RAM models.

Given a RAM model, umxSummary can report either raw or standardized parameter estimates, and also the SE or standardized SE.

m1 = umxRAM("tim", data = mtcars,
	umxPath(c("wt", "disp"), to = "mpg"),
	umxPath("wt", with = "disp"),
	umxPath(v.m. = c("wt", "disp", "mpg"))

umxSummary(m1, std=T)
name Std.Estimate Std.SE CI
disp_to_mpg -0.36 0.18 -0.36 [-0.71, -0.02]
wt_to_mpg -0.54 0.17 -0.54 [-0.89, -0.2]
mpg_with_mpg 0.22 0.07 0.22 [0.08, 0.35]
disp_with_disp 1.00 0.00 1 [1, 1]
disp_with_wt 0.89 0.04 0.89 [0.81, 0.96]
wt_with_wt 1.00 0.00 1 [1, 1]
one_to_mpg 5.89 0.68 5.89 [4.56, 7.22]
one_to_disp 1.89 0.30 1.89 [1.31, 2.47]
one_to_wt 3.34 0.45 3.34 [2.45, 4.23]

χ²(0) = 0, p = 1.000; CFI = 1; TLI = 1; RMSEA = 0

The calculate 95% confidence intervals around the standardized parameter values are based on std estimate - (1.96 × std.SE) and std estimate + (1.96 × std.SE).

PS: If you haven’t tried umxAPA have a look now: it can take many objects, and turn them into APA-style report format. For instance, data, lm results, and also effects/SE pairings.

2. Using scaled data

A second method is to scale() your data (the easiest way is with umxScale, which handles skipping over binary and factor variables properly), and run the model on this z-scored data. All estimates are automatically in standardized terms. You can also add mxCI calls to the model to get profile-based estimates of confidence rather than extrapolate from the SEs.

2. Adding algebras which compute the scaled value and calling mxCI

If you are an advanced user, you might add mxAlgebra calls which compute the standardized parameters (umxACE does this, for instance).

To get CIs around these algebras, you can either call mxSE(), giving it the model and the algebra you wish to estimate CIs for, or add mxCI calls to the model requesting CIs for the cells you want from these algebras. The underlying parameters of the model are then varied when the model is run seeking the values which drive the model-fit to the edge of the requested confidence limit. This is done for each of the CIs you request.

nb: For large, complex, raw-data or ordinal models, profile CIs can be very time-consuming.

data(myFADataRaw, package="OpenMx")
manifests = paste0("x",1:6)
a1 = umxRAM("m1", 	data = myFADataRaw, type = "cov",
	umxPath(from = "g", to = manifests),
	umxPath(var  = manifests),
	umxPath(var  = "g", fixedAt = 1)

Now, we can add a set of standardization algebras

a1 = umxConfint(a1, parm = "all", run=TRUE)
parameter lbound estimate ubound
g_to_x1 0.731 0.803 0.880
g_to_x2 0.729 0.803 0.881
g_to_x3 0.697 0.770 0.848
g_to_x4 0.754 0.825 0.902
g_to_x5 0.737 0.810 0.888
g_to_x6 0.748 0.820 0.898
x1_with_x1 0.302 0.350 0.407
x2_with_x2 0.327 0.379 0.439
x3_with_x3 0.337 0.389 0.450
x4_with_x4 0.274 0.320 0.374
x5_with_x5 0.319 0.370 0.430
x6_with_x6 0.297 0.346 0.403