Item Parceling

1. Fit the model at the item level in Mplus, with every item as an indicator of the construct it belongs to, and add this line to the input file, exactly as it is written here: OUTPUT: STDYX MODINDICES;
2. Run that input in Mplus. The program writes a plain text file with the same name as the input and the extension .out, in the same folder as the input.
3. Drag the .out file onto the box below, or click that box, or move the focus to it and press Enter, to choose the file from your computer. It is read inside your browser and semstudio.tools does not transmit the file contents; processing occurs in the browser.
4. Choose which items are to be parceled. Parcels are built for one construct at a time, so an output with several factors opens on the first of them and the menu holds the others. The reading that takes every factor at once puts the items of all of them into the same parcels and treats each factor as a facet of one construct: it belongs to a model whose factors are facets of a single construct, not to one whose factors are distinct constructs. The item names, their standardized loadings and their facet all come from that choice.
5. Choose how the items are to be allocated, how many parcels are wanted, and whether a parcel is the mean or the sum of its items. These are decisions about the scheme, not values from the file.
6. Press Build the parcels from this file, then read the allocation table, the Mplus input written under it, and the paragraph that describes what was done.

The page requires STDYX loadings for item ordering. Purposeful parceling additionally requires modification indices, and the facet-based strategies require a facet assignment for every selected item. When a required section is absent, the page reports the corresponding Mplus output option.

A parcel is a variable this page creates, so its name is not in the output; the construct keeps the name the output gives the factor.

The paragraph below states what was done, in the terms the strategy is described in, and is meant to be edited rather than copied unchanged: the name of the scale and the sample are yours to supply.

Everything in this section is an account of what the cited authors argue. None of it is a rule, and none of it settles a particular application.

The question the decision turns on

Little, Rioux, Odejimi and Stickley (2022) distinguish uses by analytic aim. When item behavior is itself the object of study, they advise retaining item-level analysis. When the aim is relations among latent constructs, they treat parceling as a possible modeling strategy, provided the rationale, construction and reporting are explicit. The page reports this position rather than treating parceling as a default.

The items in a parcel have to be unidimensional

The strategies for a unidimensional item set — random assignment, purposeful parceling, the balancing technique — are described by Little, Rioux, Odejimi and Stickley (2022) for items that measure one construct. Random assignment in particular is said to rest on the strong assumption that the items are interchangeable in the population, and to be recommended for unidimensional indicators, or for a large set of items with high communality, when sampling variability is low; the sample size they attach to that condition is above about 200. When items are not interchangeable, the allocation is no longer arbitrary and the choice of allocation begins to matter.

The arguments against parceling are reviewed at length in Little, Cunningham, Shahar and Widaman (2002) and Matsunaga (2008), to which Little, Rioux, Odejimi and Stickley (2022) refer. The consequence of parceling items that are not unidimensional is the main argument against parceling that these authors report, and they attribute it to Bandalos (2002): with multidimensional or multifaceted constructs, parcels can bias the estimated factor loadings in the measurement model and confuse the interpretation of the structural relations. Their own position is that this holds when parcels are created randomly or improperly, and that parcels built by a strategy suited to the dimensionality of the items — the two multidimensional strategies below — can make the structure of a multidimensional construct easier rather than harder to interpret, a position they attribute to Graham, Tatterson and Widaman (2000).

What to do with a multidimensional item set

Little, Rioux, Odejimi and Stickley (2022) describe two schemes, and the choice between them is a choice about what the latent construct is to mean. Both have older names, which they list: the first has been called homogeneous parceling, the isolated uniqueness strategy (Hall, Snell & Foust, 1999) and factored homogeneous item dimensions (Comrey, 1961); the second heterogeneous parceling and the distributed uniqueness strategy (Hall, Snell & Foust, 1999), and its present name comes from Kishton and Widaman (1994). In facet-representative parceling, the items of one facet go into one parcel, so the uniqueness of the facet is isolated inside that parcel and the common factor captures only what the facets share; this is an implicit way of estimating what would otherwise be a higher-order model, and they recommend it when the interest is in the higher-order construct alone. In domain-representative parceling, each facet is distributed across parcels. Little et al. (2022), citing Kishton and Widaman (1994), recommend this strategy when facet-specific variance is substantively relevant to relations with other constructs. They also report that in the comparison of the two schemes on an Internal–External Locus of Control measure by Kishton and Widaman (1994) the domain-representative scheme performed better.

The effect on fit and on parameter estimates

Parceling can improve global fit even when item-level misspecification remains. Little et al. (2022) illustrate this and note that additional parcel indicators also add degrees of freedom that can improve fit mechanically. A better-fitting parcel model therefore does not by itself establish a better measurement model.

The second effect is the argument that parcels camouflage misspecification, which they attribute to Bandalos (2002) and to Marsh, Lüdtke, Nagengast, Morin and von Davier (2013): cross-loadings and residual correlations are harder to detect with parcels, so estimates may be biased, and the better fit may be read as the absence of misspecification. Against this they cite the simulation of Rhemtulla (2016), in which parcel-level models produced similar or less biased estimates more often than item-level models, and in which parcel-level models had lower power to detect a within-factor misspecification or a single cross-loading but higher power to detect multiple cross-loadings and misspecifications in the structural model.

Results can vary across parcel allocations. Little et al. (2022) therefore recommend reporting the allocation rule and, where relevant, examining sensitivity across alternative allocations (Sterba, 2019; Sterba & MacCallum, 2010). For random parceling, the generation rule and the criterion used to summarize or select allocations should be stated.

What they recommend reporting

Little et al. (2022) recommend reporting the preliminary item-level analyses, the rationale for parceling and for the number of parcels, and the item composition of each parcel. They also recommend examining the item-level model before parcel construction and checking local fit after parceling. Supplementary materials can be used when the allocation details do not fit in the main report.

Two further pieces of guidance from the same source bear on what this page writes. They recommend averaging the items rather than summing them, so that the parcel keeps the metric of the response scale; with effects-coded scaling the mean of the construct is then the mean of the items. And they recommend against proration, that is against averaging the items that are present when some are missing, on the ground that proration is a within-person mean imputation whose assumptions are often untenable and which is associated with bias even under data missing completely at random, referring to Lee, Bartholow, McCarthy, Pedersen and Sher (2015), Mazza, Enders and Ruehlman (2015) and Schafer and Graham (2002). Their suggestion is that the parcel be computed only when all its items are present, and that the incomplete items be entered as auxiliary variables when full information maximum likelihood is used.

These are the recommendations of Little, Rioux, Odejimi and Stickley (2022), reported as such.

• Three parcels per construct is the configuration they call optimum. With three indicators the construct is just identified once a scaling constraint is chosen, so there are no within-construct degrees of freedom and all the misfit of the model is between constructs.
• Four or more parcels leave the construct overidentified. They note that the leftover degrees of freedom typically give an arbitrary improvement in fit, and that overidentified constructs have been known to be awkward under model constraints because the orientation among the loadings can shift.
• Two parcels can be identified in longitudinal or multiple-group models when invariance constraints provide the required information. In a single-group, single-occasion model, a two-indicator factor needs an additional identification constraint, such as equal loadings; Little, Lindenberger and Nesselroade (1999) report simulation results for this case.
• The number of items in a parcel does not have to be the same for every parcel, and not every item has to be assigned to a parcel. An item that sits at the center of the construct — their example is “I feel lonely” in a loneliness scale — can be left as an indicator on its own.
• They give no minimum number of items per parcel. Their examples run from two items upward, and their remark on designing a measure with parceling in mind is that multiples of three are a good starting point for the number of items, with at least three indicators for each subscale.

Each strategy below is the procedure described in the section of Little, Rioux, Odejimi and Stickley (2022) named with it, together with the rules this page uses to settle the cases the description leaves open. Those rules are marked as such.

Reading the output file

Everything about the items is read from the .out file of the item-level model, and nothing about them can be entered on this page. The STDYX Standardization section supplies the item names and their standardized loadings, SUMMARY OF ANALYSIS supplies the name of the data file that the written input then names in turn, the input echoed under INPUT INSTRUCTIONS supplies the NAMES and MISSING statements that the written input repeats, and MODEL MODIFICATION INDICES supplies the pairs of items the purposeful strategy keeps together. Windows-1252, UTF-8 and UTF-16 encodings are decoded, Fortran D exponents are accepted, an estimate printed with asterisks in place of its standard error is still read, and the 999.000 sentinel, which Mplus prints for a quantity it could not compute, is discarded rather than read as a loading of 999. Only the first group of a multiple-group output is read, and when the invariance shortcut has written several models into one file the first of them is read.

Choosing one factor gives its items and their loadings on it. Choosing every factor treats each factor name as a facet label for the multidimensional strategies. If an item loads on more than one factor, the page assigns the facet with the largest absolute loading and reports the item, because its facet membership is not unique. The construct in the written input is the factor the items came from; with every factor taken as a facet the output holds no name for the construct the facets share, so the input calls it G.

The balancing technique

Balancing. Items are ordered by standardized loading and assigned in a serpentine sequence across parcels, pairing relatively high and low loadings. Ties follow output order. Items without a loading are placed last and reported; negative loadings are ordered numerically and trigger a coding check. These deterministic rules make the same input reproduce the same allocation.

Random assignment

Random assignment. Items are shuffled with Fisher–Yates and split into parcel sizes that differ by at most one item. A seedable counter-based generator makes allocations reproducible. With multiple draws, the page reports the allocation at the median loading-balance dispersion rather than selecting the most favorable draw. This dispersion concerns parcel loading balance; assessing variability in fitted parameters requires fitting each allocation separately.

Purposeful parceling

Purposeful parceling. The implementation uses two patterns described by Little et al. (2022): opposite-sign cross-loadings on the same other factor and elevated residual covariances. Shared cross-loadings in the same direction are displayed but not used automatically because retaining or removing that cross-loading is a model-specification decision. Pairs are selected from the modification indices at the user-set threshold. For opposite-sign cross-loadings, the smaller expected parameter change must be at least half the larger in absolute value. Directly or transitively linked items form a group and stay in one parcel; remaining items are assigned by the balancing rule. If the requested number of parcels cannot be populated under these grouping constraints, the page reports the maximum feasible number.

The operation this strategy performs is also the objection raised against it. Placing two items with a high residual covariance in the same parcel removes that covariance from the parcel-level model, and Coffman and MacCallum (2005) and Little, Cunningham, Shahar and Widaman (2002) hold that aggregating items whose residuals are correlated hides the misspecification and inflates the fit of the parcel-level model. Little, Rioux, Odejimi and Stickley (2022) present the same operation as the way to keep local dependence from distorting the measurement model, on the condition that the items are unidimensional and that the decision is reported. Both readings describe the same consequence: the parcel-level model is no longer able to show what was folded into a parcel. The page therefore prints, beside the allocation, every pair it kept together with the modification index it was read from, so that the pairs can be reported and the fit of the parcel-level model interpreted with them in view.

Facet-representative parceling

Section 3.3.1. Every parcel is built from the items of one facet. When the number of parcels equals the number of facets there is one parcel per facet. When more parcels are asked for, the extra ones are given to the facets with the most items per parcel, and the items of a facet split over two or more parcels are divided by the balancing rule; this follows the source's remark that a facet with more than three scores can itself be parceled. Fewer parcels than facets is refused, because two facets sharing a parcel would defeat the purpose of the scheme, which is to isolate the uniqueness of each facet inside its own parcel. A facet with a single item yields a parcel that is that item, which the source allows and the page reports.

Domain-representative parceling

Domain-representative allocation. Items from each facet are distributed across parcels in decreasing loading order. Within each round, assignment favors a parcel that has not yet received that facet, then the parcel with fewer items and, if needed, the lower loading sum. When a facet contains at least as many items as there are parcels, every parcel receives that facet; otherwise the missing facet–parcel combinations are reported.

What became of the a priori scheme

A priori allocation. Section 3.3.3 describes parcel membership fixed by the measurement design. Because that information is not recoverable from an Mplus output, this page does not generate it. An a priori allocation should be written directly in DEFINE and checked against the observed covariance structure, as recommended by the source.

Balance, and the comparison shown beside it

For each parcel the page reports the number of items in it, the mean of their loadings and the sum of their loadings. Writing P_p for the set of items allocated to parcel p and n_p for how many there are, the two quantities are

Here λ̄_p is the mean loading of parcel p, n_p the number of items in it, the sum over i runs over those items, and λ_i is the standardized loading of item i read from the output.

Here S_p is the sum of the loadings of parcel p, the sum over i running over the same items.

The mean says how strongly the parcel marks the construct; the sum is what an equal split of total strength would equalize, and the balancing technique reduces the spread of it, that is

Here range(S) is the spread reported under the allocation table, and the maximum and the minimum are taken over the parcels of the scheme.

A sequential split is shown only as a comparison. In 2,500 generated loading configurations, the balancing algorithm produced a smaller loading-sum spread than the sequential split in 98.2% and the same spread in 1.8%; this simulation is a software check, not a literature-based cutoff. The page flags a range of parcel mean loadings above .10 as a display convention and reports parcel-size differences greater than one item because size alone can affect parcel reliability and loading magnitude.

The syntax, and what is checked in it

Parcel averages are written explicitly as the sum of all assigned items divided by their count, so a parcel is missing when any contributing item is missing; this follows the source’s recommendation against proration. Variable names are validated against Mplus naming rules, and parcel-name collisions are resolved visibly. NAMES and MISSING are copied from the echoed input when available; otherwise the generated input marks the lines that require completion. The generated syntax is then parsed again to verify variable declarations, USEVARIABLES, parcel definitions, BY statements, parameter labels, line length, and name length.

What the page does not do

The page constructs parcel allocations and Mplus syntax; it does not fit the item-level or parcel-level model, test unidimensionality, or compute new modification indices. It does not read raw data, so it cannot create parcel scores or evaluate missingness. Standardized loadings and facet assignments are taken from the fitted item-level output. One construct is handled at a time, and facet-based strategies assume the supplied facet structure. Whether parceling is appropriate for the construct remains a substantive modeling decision.

• Bandalos, D. L. (2002). The effects of item parceling on goodness-of-fit and parameter estimate bias in structural equation modeling. Structural Equation Modeling, 9(1), 78–102. https://doi.org/10.1207/S15328007SEM0901_5
• Bandalos, D. L., & Finney, S. J. (2001). Item parceling issues in structural equation modeling. In G. A. Marcoulides & R. E. Schumacker (Eds.), New developments and techniques in structural equation modeling (pp. 269–296). Erlbaum.
• Coffman, D. L., & MacCallum, R. C. (2005). Using parcels to convert path analysis models into latent variable models. Multivariate Behavioral Research, 40(2), 235–259. https://doi.org/10.1207/s15327906mbr4002_4
• Comrey, A. L. (1961). Factored homogeneous item dimensions in personality research. Educational and Psychological Measurement, 21(2), 417–431. https://doi.org/10.1177/001316446102100215
• Graham, J. W., Tatterson, J. W., & Widaman, K. F. (2000). Creating parcels for multi-dimensional constructs in structural equation modeling [Conference presentation]. Annual meeting of the Society of Multivariate Experimental Psychology, Saratoga Springs, NY.
• Hall, R. J., Snell, A. F., & Foust, M. S. (1999). Item parceling strategies in SEM: Investigating the subtle effects of unmodeled secondary constructs. Organizational Research Methods, 2(3), 233–256. https://doi.org/10.1177/109442819923002
• Kishton, J. M., & Widaman, K. F. (1994). Unidimensional versus domain representative parceling of questionnaire items: An empirical example. Educational and Psychological Measurement, 54(3), 757–765. https://doi.org/10.1177/0013164494054003022
• Landis, R. S., Beal, D. J., & Tesluk, P. E. (2000). A comparison of approaches to forming composite measures in structural equation models. Organizational Research Methods, 3(2), 186–207. https://doi.org/10.1177/109442810032003
• Lee, M. R., Bartholow, B. D., McCarthy, D. M., Pedersen, S. L., & Sher, K. J. (2015). Two alternative approaches to conventional person-mean imputation scoring of the Self-Rating of the Effects of Alcohol Scale (SRE). Psychology of Addictive Behaviors, 29(1), 231–236. https://doi.org/10.1037/adb0000015
• Little, T. D., Cunningham, W. A., Shahar, G., & Widaman, K. F. (2002). To parcel or not to parcel: Exploring the question, weighing the merits. Structural Equation Modeling, 9(2), 151–173. https://doi.org/10.1207/S15328007SEM0902_1
• Little, T. D., Lindenberger, U., & Nesselroade, J. R. (1999). On selecting indicators for multivariate measurement and modeling with latent variables: When “good” indicators are bad and “bad” indicators are good. Psychological Methods, 4(2), 192–211. https://doi.org/10.1037/1082-989X.4.2.192
• Little, T. D., Rhemtulla, M., Gibson, K., & Schoemann, A. M. (2013). Why the items versus parcels controversy needn't be one. Psychological Methods, 18(3), 285–300. https://doi.org/10.1037/a0033266
• Little, T. D., Rioux, C., Odejimi, O. A., & Stickley, Z. L. (2022). Parceling in structural equation modeling: A comprehensive introduction for developmental scientists. Cambridge University Press. https://doi.org/10.1017/9781009211659
• Marsh, H. W., Lüdtke, O., Nagengast, B., Morin, A. J. S., & von Davier, M. (2013). Why item parcels are (almost) never appropriate: Two wrongs do not make a right — camouflaging misspecification with item parcels in CFA models. Psychological Methods, 18(3), 257–284. https://doi.org/10.1037/a0032773
• Matsunaga, M. (2008). Item parceling in structural equation modeling: A primer. Communication Methods and Measures, 2(4), 260–293. https://doi.org/10.1080/19312450802458935
• Mazza, G. L., Enders, C. K., & Ruehlman, L. S. (2015). Addressing item-level missing data: A comparison of proration and full information maximum likelihood estimation. Multivariate Behavioral Research, 50(5), 504–519. https://doi.org/10.1080/00273171.2015.1068157
• Rhemtulla, M. (2016). Population performance of SEM parceling strategies under measurement and structural model misspecification. Psychological Methods, 21(3), 348–368. https://doi.org/10.1037/met0000072
• Schafer, J. L., & Graham, J. W. (2002). Missing data: Our view of the state of the art. Psychological Methods, 7(2), 147–177. https://doi.org/10.1037/1082-989X.7.2.147
• Sterba, S. K. (2019). Problems with rationales for parceling that fail to consider parcel-allocation variability. Multivariate Behavioral Research, 54(2), 264–287. https://doi.org/10.1080/00273171.2018.1522497
• Sterba, S. K., & MacCallum, R. C. (2010). Variability in parameter estimates and model fit across repeated allocations of items to parcels. Multivariate Behavioral Research, 45(2), 322–358. https://doi.org/10.1080/00273171003680302
• Sterba, S. K., & Rights, J. D. (2017). Effects of parceling on model selection: Parcel-allocation variability in model ranking. Psychological Methods, 22(1), 47–68. https://doi.org/10.1037/met0000067

Sterba and Rights (2017) appeared online in 2016; the year, volume, issue and pages given are those of the printed issue, as in the reference list of Little, Rioux, Odejimi and Stickley (2022).