Using Substantive Diagnostics to Evaluate the Validity of Micro-level Latent Class Indicators of Measurement Error Clyde Tucker and Brian Meekins U.S. Bureau of Labor Statistics and Paul Biemer Research Triangle Institute

Background Developed by Lazarsfeld (1950)—unobserved or “latent” variable drawn from relationships between two or more “manifest” variables Lazarsfeld and Henry (1968) and Goodman (1974) extended mathematics of theory Software for latent class analysis (LCA) developed (MLLSA, lEM, M-PLUS) LCA used to study measurement or response error (VandePol and deLeeuw 1986; Tucker 1992; Van de Pol and Langeheine 1997; Bassi et al. 2000; Biemer and Bushery 2000; Tucker, et al. 2002, 2003, 2004, 2005, 2006, and 2008)

Creation of Manifest Variables Try to create at least three Try to avoid direct relationships with outcome variable (expenditures, in this case) Use LCA to “triangulate” them to produce a latent variable with more information than any one of them alone

Statistical Logic In mathematical terms, when manifest variables A and B are not independent, the following relationship will not hold: where i indexes the classes of A, j indexes the classes of B, π ij AB is the probability an individual is in cell ij, π i A is the probability an individual is in class i, and π j B is the probability an individual is in class j.

Statistical Logic For the above expression to be true, A and B must be independent. The purpose of the latent variable X is to achieve that independence. Thus, the following latent class model is desired: where t indexes the classes of X, π ijt ABX is the probability of being in cell ijt of the unobserved ABX table, π t X is the probability that an individual is in one of the mutually exclusive and exhaustive classes of X, π it AX and π jt BX are the conditional probabilities that an individual is in a particular class of A and B, respectively, given that a person is in a certain class of X. Equation (2) indicates that, within a class of X, A and B are independent.

Purpose of Paper Concept of LCA relatively straightforward— create a variable to account for common variance among observed variables Issues:  What is the new variable?  What do its classes mean?  Does it really tell us anything useful? Statistical diagnostics don’t help us here. We need substantive ones. Paper explores some of this type of diagnostics

Data Sources CED  2 week diaries  All expenditures  Small items and grocery expenditures  Used for CPI cost weights CEQ  5 quarters (first for bounding) PV  All consumer expenditures  2 hours  Larger consumer items  Used for CPI cost weights

Three Examples 1985 CED Operational Test (micro level)  3 treatments—specific, nonspecific, control  800 households in each  Latent response error measure of underreporting of grocery expenditures using manifest performance indicators CEQ ( ) (micro level)  Only analyzed the 2 nd wave  43,000 completed 2 nd wave interviews  Latent response error measure of underreporting for 7 expenditure categories for purchasers using manifest performance indicators CEQ ( ) (micro level)  Analyzed all four waves  14,877 remained in sample throughout  Latent response error measure of underreporting for almost 30 expenditure categories for all households (purchasers and nonpurchasers) using manifest performance indicators and indicators of pattern of wave nonresponse

Critical Assumption Response errors in CE only come from underreporting of expenditures and not overreporting  Tedious  Time-consuming  Recall problems  Lack of knowledge

Methodological Issues Weighted vs. unweighted Variances for complex sample design vs. SRS Local vs. global maxima Sparse cells (too many manifest variables) Restricted vs. unrestricted models Boundary problems (no overreporting)

1985 Diary Test Manifest variables  Difference in first and second week grocery expenditures  Difference in usual and average weekly grocery expenditures  Amount of expenditure information collected by recall  Respondent’s attitudes and behavior with respect to diarykeeping Latent variable  3 classes (low, moderate, high response error)

CEQ Micro-level Manifest Indicators for First Study Interview level indicators considered: 1. Number of contacts 2. Ratio of respondents/household members 3. Missing income data 4. Type and frequency of records used 5. Length of interview 6. Ratio of expenditures in last month to quarter 7. Combination of type of record and interview length

Indicator Coding #contacts (1=0-2; 2=3-5; 3=6+) Resp/hh size (1= <.5; 2=.5+) Income missing (1=present; 2=missing) Records use (1=never; 2=single type or sometimes; 3=multiple types and always) Interview length (1= <45; 2=45-90; 3= 90+) Month3 expn/all (1= <.25; 2=.25-.5; 3= +.5) Combined records and length (1= poor; 2= fair; 3=good)

Latent Variables Three-class latent variables (poor, fair, good reporting) for  Kid’s Clothing  Women’s Clothing  Men’s Clothing  Furniture  Electricity  Minor Vehicle Repairs  Kitchen Accessories

Second CEQ Micro-level Study Based on results of first CEQ study, analysis of purchasers and nonpurchasers together Used Interviews 2-5 data. Not limited to within-interview indicators Developed model using all Interview 2 respondents Latent variable is still intended to represent quality of reporting

New Manifest Indicators Overall Panel level indicators considered 1. Number of completed interviews (1-4) 2. Attrition combined with # of complete interviews 3. Average number of commodity categories for which CU had expenditure 4. Number of interviews the ratio of third month expenditure to quarter was between Panel averages of interview level indicators from first CEQ study

Model Selection Ran both ordered (fixed or restricted ordinal constraints) latent class models and unordered.  Order was determined based on theoretical relationship between values of indicators and level of underreporting. Ran all combinations of indicators in groups of 3 & 4, using 3 or 4 category LC variable for each commodity category & overall Multiple iterations to avoid local maxima Best model candidates were selected based on fit From those candidates, models selected based on relationship of indicators to latent construct

Application of Model For the final models for each commodity:  Each combination of indicators was assigned to a latent class based on probability of being in that class given the value of the indicators  Ran demographic analysis to identify characteristics of members of each latent class  Expenditure means were found for each latent class  Examined the pattern of mean expenditure and the contribution of the latent variable in predicting these expenditures

Second CEQ Micro-level Study– Expenditure Categories Cable/satellite TV Men’s apparel Women’s apparel Men’s clothing only Women’s clothing only Men’s accessories Women’s accessories Men’s shoes Women’s shoes Kid’s apparel Kid’s clothing only Kid’s Accessories Kid’s shoes Dental care Drugs and medical supplies Electricity Gas (household) Eye care Sports equipment Televisions, video, & sound equip. Vehicle service, major Vehicle service, minor Vehicle service, oil changes only Vehicle expenses, other Pets and pet supplies Sports equipment Trash collection Televisions, video, & sound equip. Vehicle service, major Vehicle service, minor Vehicle service, oil changes only Vehicle expenses, other Pets and pet supplies Kitchen accessories Other household items

Conclusions When doing LCA for measuring response error, one cannot rely on statistical diagnostics alone. Substantive diagnostics are needed to judge the meaningfulness of the results. Sometimes the models work and sometimes they don’t. Unfortunately, this is likely to depend on the characteristic you’re analyzing. We need better manifest variables to explain more variance. We have been unable to develop meaningful latent variables with more than three or four categories, and, in some cases, we could only identify two. LCA software really does work best with large sample sizes. Besides only defining a few latent classes, we certainly will not progress beyond the most rudimentary ordinal rankings any time soon. LCA problems are likely to be multiplied many times for response error measures for non-factual items such as attitudes or opinions.

Contact Information Clyde Tucker Senior Survey Methodologist OSMR