Purpose – Partial least squares (PLS) path modeling has become a pivotal empirical research method in international marketing. Owing to group comparisons' important role in research on international marketing, we provide researchers with recommendations on how to conduct multigroup analyses in PLS path modeling.

Methodology/approach – We review available multigroup analysis methods in PLS path modeling and introduce a novel confidence set approach. A characterization of each method's strengths and limitations and a comparison of their outcomes by means of an empirical example extend the existing knowledge of multigroup analysis methods. Moreover, we provide an omnibus test of group differences (OTG), which allows testing the differences across more than two groups.

Findings – The empirical comparison results suggest that Keil et al.'s (2000) parametric approach can generally be considered more liberal in terms of rendering a certain difference significant. Conversely, the novel confidence set approach and Henseler's (2007) approach are more conservative.

Originality/value of paper – This study is the first to deliver an in-depth analysis and a comparison of the available procedures with which to statistically assess differences between group-specific parameters in PLS path modeling. Moreover, we offer two important methodological extensions of existing research (i.e., the confidence set approach and OTG). This contribution is particularly valuable for international marketing researchers, as it offers recommendations regarding empirical applications and paves the way for future research studies aimed at comparing the approaches' properties on the basis of simulated data.

Multigroup Analysis in Partial Least Squares (PLS) Path Modeling: Alternative Methods and Empirical Results

In this article, I introduce the itsa command, which performs interrupted time-series analysis for single- and multiple-group comparisons. In an interrupted time-series analysis, an outcome variabl...

https://journals.sagepub.com/doi/pdf/10.1177/1536867X1501500208

Conducting Interrupted Time-series Analysis for Single- and Multiple-group Comparisons:

This article reports the results of a study that located, digitized, and coded all 809 single-case designs appearing in 113 studies in the year 2008 in 21 journals in a variety of fields in psychology and education. Coded variables included the specific kind of design, number of cases per study, number of outcomes, data points and phases per case, and autocorrelations for each case. Although studies of the effects of interventions are a minority in these journals, within that category, single-case designs are used more frequently than randomized or nonrandomized experiments. The modal study uses a multiple-baseline design with 20 data points for each of three or four cases, where the aim of the intervention is to increase the frequency of a desired behavior; but these characteristics vary widely over studies. The average autocorrelation is near to but significantly different from zero; but autocorrelations are significantly heterogeneous. The results have implications for the contributions of single-case designs to evidence-based practice and suggest a number of future research directions.

https://link.springer.com/content/pdf/10.3758%2Fs13428-011-0111-y.pdf

Characteristics of single-case designs used to assess intervention effects in 2008

It has been recognized that the two-phase version of the interrupted time-series design can be frequently modeled using a four-parameter design matrix. There are differences across writers, however, in the details of the recommended design matrices to be used in the estimation of the four parameters of the model. Various writers imply that different methods of specifying the four-parameter design matrix all lead to the same conclusions; they do not. The tests and estimates for level change are dramatically different under the various seemingly equivalent design specifications. Examples of egregious errors of interpretation are presented and recommendations regarding the correct specification of the design matrix are made. The recommendations hold whether the model is estimated using ordinary least squares (for the case of approximately independent errors) or some more complex time-series approach (for the case of autocorrelated errors).

Design Specification Issues in Time-Series Intervention Models

Multiple-baseline studies are prevalent in behavioral research, but questions remain about how to best analyze the resulting data. Monte Carlo methods were used to examine the utility of multilevel models for multiplebaseline data under conditions that varied in the number of participants, number of repeated observations per participant, variance in baseline levels, variance in treatment effects, and amount of autocorrelation in the Level 1 errors. Interval estimates of the average treatment effect were examined for two specifications of the Level 1 error structure (σ2I and first-order autoregressive) and for five different methods of estimating the degrees of freedom (containment, residual, between—within, Satterthwaite, and Kenward—Roger). When the Satterthwaite or Kenward—Roger method was used and an autoregressive Level 1 error structure was specified, the interval estimates of the average treatment effect were relatively accurate. Conversely, the interval estimates of the treatment effect variance were inaccurate, and the corresponding point estimates were biased.

https://link.springer.com/content/pdf/10.3758/BRM.41.2.372.pdf

Making treatment effect inferences from multiple-baseline data: the utility of multilevel modeling approaches.

A new method for the analysis of linear models that have autoregressive errors is proposed. The approach is not only relevant in the behavioral sciences for analyzing small-sample time-series intervention models, but it is also appropriate for a wide class of small-sample linear model problems in which there is interest in inferential statements regarding all regression parameters and autoregressive parameters in the model. The methodology includes a double application of bootstrap procedures. The 1st application is used to obtain bias-adjusted estimates of the autoregressive parameters. The 2nd application is used to estimate the standard errors of the parameter estimates. Theoretical and Monte Carlo results are presented to demonstrate asymptotic and small-sample properties of the method; examples that illustrate advantages of the new approach over established time-series methods are described.

A double bootstrap method to analyze linear models with autoregressive error terms.

Several issues regarding the effects of autocorrelated errors on Type I error in ordinary least-squares models are clarified. Although autocorrelated errors have a large effect on both omnibus F tests and tests on individual intervention effect coefficients in many applications, there are exceptions that have not been pointed out previously. It is demonstrated that under certain conditions, distortion in Type I error is far less than is predicted by asymptotic theory. It is shown that these exceptions occur because the effect of autocorrelated errors is dependent on (a) the type of parameters (e.g., level change and/or slope change) required in the model, (b) the number of variables in the design matrix, and (c) the sample size. Because existing time-series methods perform poorly with small samples, this may be a useful finding in some situations; however, a better general solution is to use a recently developed small-sample method.

Scott D. McKnight

Papers

A double bootstrap method to analyze linear models with autoregressive error terms.

Autocorrelation Effects on Least-Squares Intervention Analysis of Short Time Series