## A Practical Guide to Conditional Dependence in Latent Class Models- Introduction
- Detecting Conditional Dependence
- Modeling Conditional Dependence
- Summary and Conclusions
- Examples
- References
(Back to Latent Class Analysis main page) ## Introduction
A common, serious problem with studies that use Latent Class Analysis
(LCA) is local independence. The basic latent class model assumes
that manifest variables are independent of each other within latent
classes. This assumption is often untrue, and when it is the latent
class model must be modified.
Often some manifest variables (for brevity, "items") are related or
dependent. Examples include items on related symptoms, multiple
indicators, or repetitions of the same item. Such items are termed
"conditionally dependent" or "locally dependent" because they are
associated within latent classes. If this dependence is not accounted
for, model fit indices (such as the G Previous literature has underemphasized this issue. That is changing though, especially as new software makes it easier to consider conditional dependence models. It isn't hard to handle conditional dependence in latent class models (LCMs). Here we consider in simple practical terms how to do so. For more technical discussion see Hagenaars, 1988, or an extended version of this document. First we will see how to detect conditionally dependent items. A simple computer program for this is provided. Second, we will see how to incorporate conditional dependence into LCMs with standard latent class analysis software. Examples and program command files are provided. The main LCA software considered here is LEM (Vermunt, 1997). LEM's flexible command language is well suited to conditional dependence LCMs. Another reason for focusing on LEM is that it can be downloaded for free, making it widely available. The new Latent GOLD program (Vermunt & Magidson, 2000) has special features for handling conditional dependence in LCMs; using the program's Windows, one can easily detect and specify conditional dependence for pairs of items. I will update this page once I've had a chance to use Latent GOLD more. As will be described, conditional dependence can also be modeled with PANMARK (van de Pol, Langeheine & de Jong, 1998) and MLLSA (Clogg, 1977) by recoding variables or with use of special parameter restrictions. This is satisfactory for simpler problems, but can become cumbersome if there are many conditionally dependent items or complex dependencies. For more information on these LCA programs, see the Latent Class Analysis Software page.) ## Types of Conditional DependenceFigure 1 shows types of associations among variables in a latent class model. X and Y represent latent class variables, and A to H represent items. An arrow from X or Y to an item indicates that response probabilities for the item depend on the latent class. A diagram with only these arrows (e.g., part i) would portray a standard LCM with conditional independence.
Diagram parts (ii) and (iii) illustrate two types of conditional dependence. The two-headed arrow connecting items D and E denotes simple conditional dependence of these items; no assumption is made about the cause of the dependence. The situation is similar to "correlated error" among continuous variables in a LISREL-type model. In contrast, items G and H are conditionally dependent because of mutual dependence on a second latent variable, Y. This model would apply, for example, if items G and H were multiple indicators of a latent variable Y. For only two items, the model used to represent their conditional dependence makes little practical difference--the same degree of model fit and the same basic parameter values will be obtained. (Still, one should try to select a model consistent with theory). When there are three or more conditionally dependent items, however, the model chosen does matter. This will be clearer in later discussion. ## Detecting Conditional DependenceAttention here focuses on LCMs with dichotomous items. To some extent, the methods here can be generalized to polytomous-item models.For illustration we consider data on four diagnostic tests for human HIV virus (Table 1) reported by Alvord et al. (1988).
RIA-ag121 tests for a specific antigen, ag121, to the AIDS virus. RIA-p24 and RIA-gp120 test for presence of specific proteins of the virus itself. The ELISA method also tests for directly for the virus' presence. Table 2 summarizes test results for 428 subjects.
The many unobserved patterns suggest that a more deterministic model for the data might be appropriate. However, for the sake of a computational example, we put this reservation aside. ## Standard LCM with conditional independenceAlvord et al. (1988) reasoned there should be two latent classes, corresponding to presence and absence of the HIV virus. Model 1 is a standard two-class LCM with conditional independence. A LEM command file estimate this model is as follows:
The last four lines, optional, give start values for the conditional
response probabilities. Their purpose here is to make Latent Class 1
correspond to the disease-negative cases and Latent Class 2 to the
disease-positive cases.
Model 1 poorly fits; the G Let X1 and X2 denote the two latent classes and let, say, A1 and A2 denote a negative and positive outcome on Test A. One commonly reports results as the conditional probabilities of a positive item response in each latent class, or P(A2|X1), P(A2|X2), P(B2|X1), etc. Table 3 summarizes these values:
Note that although it shows significant lack of fit to the data, Model 1 still gives parameter estimates close to those of Model 2. This is consistent with the finding of Torrance-Rynard and Walter (1998), who noted that, even when technically incorrect, a conditional independence latent class model may supply parameter estimates reasonably close to "true" values. However this is not always so, and it is difficult to predict when this will be the case and when not. ## DiagnosticsA number of authors have discussed diagnostics for detecting conditionally dependent items (Espeland & Handelman, 1988; Garrett & Zeger, 2000; Hagenaars, 1988; Qu et al., 1996; Vermunt & Magidson, 2000). Several methods compare observed and model-predicted crossclassification frequencies for pairs of items; that is, for each (dichotomous) item pair I and J, one compares the observed 2 x 2 crossclassification table for item responses with the corresponding table predicted by the LCM being considered. Higher association of the items in the observed two-way table than in the expected two-way table implies conditional dependence.We use a modified version of Garrett and Zeger's (2000) Log-Odds Ratio Check (LORC). (Their complete method involves advanced statistical methods, including Markov Chain Monte Carlo [MCMC] estimation; the simpler version here should usually produce similar conclusions). The modified method, summarized below, can be easily applied with use of a program that is supplied here.
- Construct the observed and model-predicated two-way
cross-classification frequency tables for the two items.
- Calculate the log-odds ratio (psi) in both the observed and
expected two-way tables.
- Calculate the standard error of psi for the expected data,
using the formula sigma(psi) = sqrt(1/a + 1/b + 1/c + 1/d) where a, b, c
and d denote the four frequencies of the two-way table. The
observed-data psi is then expressed as a z-score relative to the
expected-data psi. That is,
psi(observed) - psi(expected) z = ---------------------------- (1) sigma[psi(expected)] - Examine whether the z-value exceeds a critical value of, say +/- 1.645 or +/- 1.96. If so, that is evidence that the items are conditionally dependent. These z-values, however, are only guides; one should not interpret the p-values too literally. The main issue is their relative magnitude.
Results of this method applied to the Alvord et al. data are shown in Table 4. Tests B and C appear to be conditionally dependent, as indicated by the relatively large z-value associated with them. (For comparison a G ^{2} statistic is also shown for each pair of
items; this is the likelihood-ratio chi-squared statistic comparing the
observed and expected two-way table for the items [Espeland & Handelman,
1988]; again, this is much larger for Tests B and C.)
Table 4 was constructed by the CONDEP program. Program input consisted of the first 6 columns of Table 2, which were cut-and-pasted from the LEM output into a separate file. Click here to download an executable version of the program and documentation (ZIPped file, about 39k). Vermunt and Magidson (2000) recently described a promising new way to diagnose conditional dependence. This estimates the improvement in model fit obtained by allowing conditional dependence of each pair of items. The Latent GOLD program includes this method. ## Modeling Conditional DependenceThere are several ways to relax conditional independence assumptions in LCMs. We will illustrate three methods: - The joint item method
- The multiple indicator method
- The loglinear formulation of LCA
## The joint item methodFor simple kinds of conditional dependence, such as among pairs of items, the method is useful. The principle is to replace two or
more dependent items with a joint item, created by considering all
combinations of levels of the original items. In the present case, we
create a new item BC with four levels, as shown in Table 5.
joint itemOne then recodes the data accordingly, and estimates a standard LCM on the new item set--i.e., here, the three items A, BC and D. The LCM estimated is a standard LCM model with no special provisions: the conditional dependance of B and C is handled by the recoding. The Alvord et al. data so recoded, in indexed-frequency format with unobserved response patterns dropped, is as follows:
With LEM, however, it is not necessary to recode the data; the joint item method can be used in a way that is mostly transparent to the user. To illustrate with the Alvord et al. data, let Model 2 denote a two-latent class LCM with conditional dependence between Tests B and C. The LEM commands to estimate this model via the joint item method is as follows: The main difference between this and the commands for Model 1 is that the model (mod) line substitutes BC|X for B|X and C|X.
For Model 2, LEM obtains a G Parameter estimates for the Model are shown in Table 3. The estimated conditional response probabilities for Tests A and D can be obtained from the CONDITIONAL PROBABILITIES output section, for example, LEM shows:
However, since B and C comprise a joint item, their output is different. In the STATISTICS section, LEM shows: The conditional probabilities we seek are marginal sums this table. For example, LEM supplies these in the LATENT
CLASS OUTPUT section. There we find:
marginal response probabilities
So we can directly see that:
P(B2|X1) = .0357 P(B2|X2) = .5699 P(C2|X1) = .0000 P(C2|X2) = .9112 Neither LEM nor PANMARK supplies the standard errors for the marginal response probabilities (MLLSA supplies no standard errors at all). This is so regardless of which method is used to model conditional dependence. They can be calculated by the delta method, but this may entail a fair amount of work. Latent GOLD supplies these standard errors automatically. Many applied studies using LCA do not report parameter standard errors, so the importance of this is unclear. By analogy, note that one seldom reports standard errors of factor loadings in factor analysis. In any case, this issue should not be seen as a reason to avoid incorporating conditional dependence into LCMs.
In Table 3 one sees that the parameter estimates for Model 2 are very
close to those for Model 1. This similarity of results sometimes occurs
with a conditional dependence LCM and sometimes it does not. However note the
substantial drop in the G One can also construct joint variables that combine responses three or more items. Further, there can be several joint items in a given model. However there cannot be overlap in the original items that define different joint items. For example we could not have one joint item defined by items A and B and another joint item defined by items B and C because both joint items would include item B. ## The multiple indicator methodThis method works by modeling conditionally dependent items as multiple indicators of a common latent variable. For our data, the situation is as shown in Figure 2.
The latent variable Y is an unobserved construct or entity which both B
and C measure. The principle is familiar from LISREL-type modeling; the
difference is that here Y is a discrete latent variable--i.e., a latent
class variable with two or more levels.
X / | \ / | \ / | \ A Y D / \ B C The LEM command file to estimate the model in Figure 2 is as follows: We could instead use the model (mod) line: However specifying X Y|X rather than XY has the effect of structuring output in more helpful way. Because two latent variables are specified (X and Y), LEM will actually estimate a four-class LCM, each class representing a combination levels on X and Y. However with LEM this is mainly transparent and the substantive interpretation of a two-class model with Latent Class 1 = AIDS virus absent and Latent Class 2 = AIDS virus present is not obscured. When the commands above are submitted to LEM, a model is estimated that fits the data to the same degree as the joint item method. That is expected, since the models are identical at a fundamental level. In the CONDITIONAL PROBABILITIES output section we find that the estimates for latent class probabilities and the conditional response probabilities for items A and D are the same as before. We must manipulate output results slightly to get the conditional response probabilities for items B and C. Taking item B, for example, the CONDITIONAL PROBABILITIES section shows the following: From these values we calculate In this way we get the same conditional response probabilities for items B and C as obtained by the joint item method. Goodman (1974a) first described how to implement this method with special restrictions on the latent class model (see also Hagenaars, 1988). With that technique, the multiple-indicator method can be used with programs like PANMARK and MLLSA. The restrictions make equal various conditional response probabilities across pairs of latent classes. To see the parameter restriction file used to estimate this model with PANMARK, click here. ## The loglinear formulation of LCAClogg (1995) distinguished between the and
the classical
formulation of LCA. The classical method is the
standard parameterization of Lazarsfeld and Henry (1968) and Goodman
(1974b), with the basic model parameters being the latent class
prevalences and conditional response probabilities. The loglinear
formulation (Haberman, 1979) reparameterizes the latent class model as a
type of loglinear model. With the loglinear formulation it becomes much
simpler to specify certain types of conditional dependence models.
loglinear formulationA LEM command file to model conditional dependence of Test B and C using the loglinear formulation is as follows: Again, the only difference here is with the mod line. Model effects are specified in the conventional notation for loglinear models (see, for example, Hagenaars, 1990). In particular, the term BCX means that the model includes terms associated with the main effects of tests B and C, the two-way interactions BX, CX and BC, and the three-way interaction between B, C and X. Again, this model is fundamentally identical to those specified in Examples 2 and 3, and the same model fit and parameter estimates are obtained. Estimated latent class prevalences and conditional response probabilities are found directly in the LATENT CLASS OUTPUT section. ## Comparison of methodsWhen there are only pairs of conditionally dependent dichotomous variables, all three methods discussed here give the same results. However when there is conditional interdependence among three or more items the methods give potentially different results. To appreciate this, consider a two-latent class LCM with a joint variable created for dichotomous manifest variables A, B, and C. The joint variable ABC has eight levels and is associated with seven independent estimated response probabilities per latent class. Suppose instead that we wish to model conditional dependence between each pair of items, (A, B), (B, C), and (A, C), as shown in Figure 3. Here there are only six estimated parameters per latent class associated with the response probabilities of these items. (These correspond to the three paths from X to the items and the three paths between the items). With the loglinear formulation we could specify either model by including or not including the three-way interaction between A, B and C. That is, in the model line we could specify the effects associated with these items either by ABCX or by the three terms ABX, ACX and BCX. Alternatively we could view A, B and C as multiple indicators, as illustrated in Figure 4. Then their conditional dependence could be modeled using only four parameters per latent class (that is, assuming only two levels for the latent variable Y). Thus when there are sets of more than two conditionally dependent items the choice of models becomes more of an issue. Theory will of course help guide the choice. In some cases one may wish to compare results using different methods. ## Summary and ConclusionsThe present discussion may be summarized as follows:- Conditional dependence of items is an important issue in LCA. Before
analysis, a researcher should consider whether any items are likely
conditionally dependent.
- Diagnostics to detect conditionally dependent items should be used.
- The joint item, multiple indicator and loglinear methods are all
effective for modeling conditional dependence. All three are easily
implemented using LEM. The joint item and multiple indicator methods
can also be used with PANMARK or MLLSA.
- For pairs of conditionally dependent dichotomous items, all three methods
produce fundamentally the same results. The output for the each method
may differ slightly, however.
- Standard errors of conditional response probabilities for conditionally dependent items cannot be obtained directly from LEM, PANMARK, or MLLSA. They can be obtained directly from Latent GOLD.
## ReferencesClogg CC. Unrestricted and restricted maximum likelihood latent structure analysis: a manual for users. Working paper 1977-09, Pennsylvania State University, Population Issues Research Center. Clogg CC. Latent class models. In Arminger G, Clogg CC, Sobel ME (eds), Handbook of statistical modeling for the social and behavioral sciences (Ch. 6; pp. 311-359). New York: Plenum, 1995. Espeland MA, Handelman SL. Using latent class models to characterize and assess relative error in discrete measurements. Biometrics, 1989, 45, 587-599. Garrett ES, Zeger SL. Latent class model diagnosis. Biometrics, in press. Goodman LA. The analysis of systems of qualitative variables when some of the variables are unobservable. Part I--a modified latent structure approach. American Journal of Sociology, 1974a, 79, 1179-1259. Goodman LA. Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika, 1974b, 61, 215-231. Haberman SJ. Qualitative data analysis: Vol. 2. New developments. New York: Academic Press, 1979. Hagenaars JA. Latent structure models with direct effects between indicators: local dependence models. Sociological Methods and Research, 1988, 16, 379-405. Hagenaars JA. Categorical longitudinal data. Newbury Park, California: Sage, 1990. Heinen T. Latent class and discrete latent trait models: Similarities and differences. Thousand Oaks, California: Sage, 1996. Lazarsfeld PF, Henry NW. Latent structure analysis. Boston: Houghton Mifflin, 1968. Torrance-Rynard VL, Walter SD. Effects of dependent errors in the assessment of diagnostic test performance. Statist Med 1997 Oct 15;16(19):2157-75. Uebersax JS. Statistical modeling of expert ratings on medical treatment appropriateness. Journal of the American Statistical Association, 1993, 88, 421-427. Uebersax JS, Grove WM. A latent trait finite mixture model for the analysis of rating agreement. Biometrics, 1993, 49, 823-835. Van de Pol F, Langeheine R, de Jong W. PANMARK 3 user manual. Netherlands Central Bureau of Statistics, Voorburg, The Netherlands, 1998. Vermunt JK. LEM: A general program for the analysis of categorical data. Tilburg University, Department of Methodology, 1997. Vermunt JK, Magidson J. Latent GOLD User's Guide. Belmont, Mass.: Statistical Innovations, Inc., 2000. Go to Latent Structure Analysis Go to Agreement Statistics
rev: 10 Aug 2000 (c) 2000-2009 John Uebersax PhD email |