Question

How can one design a future sampling study to have a lower bound? How could more information be used?

Answer 1

LOWER BOUNDS ON SAMPLE SIZE IN STRUCTURAL EQUATION MODELING

Computationally intensive structural equation modeling (SEM) approaches have been in development over much of the 20th century, initiated by the seminal work of Sewall Wright. To this day, sample size requirements remain a vexing question in SEM based studies. Complexities which increase information demands in structural model estimation increase with the number of potential combinations of latent variables; while the information supplied for estimation increases with the number of measured parameters times the number of observations in the sample size – both are non‐linear. This alone would imply that requisite sample size is not a linear function solely of indicator count, even though such heuristics are widely invoked in justifying SEM sample size. This paper develops two lower bounds on sample size in SEM, the first as a function of the ratio of indicator variables to latent variables, and the second as a function of minimum effect, power and significance. The algorithm is applied to a meta‐study of a set of research published in five of the top MIS journals. The study shows a systematic bias towards choosing sample sizes that are significantly too small. Actual sample sizes averaged only 50% of the minimum needed to draw the conclusions the studies claimed. Overall, 80% of the research articles in the meta‐study drew conclusions from insufficient samples. Lacking accurate sample size information, researchers are inclined to economize on sample collection with inadequate samples that hurt the credibility of research conclusions

1. INTRODUCTION

The past two decades have seen a remarkable acceleration of interest in structural equations modeling (SEM) methods in management research, including partial least squares (PLS) and implementations of Jöreskog’s SEM algorithms (LISREL, AMOS, EQS). The breadth of application of SEM methods has been expanding, with SEM increasingly applied to exploratory, confirmatory and predictive analysis with a variety of ad hoc topics and models.    SEM is particularly useful in the social sciences where many if not most key concepts are not directly observable. Because many key concepts in the social sciences are inherently latent, questions of construct validity and methodological soundness take on a particular urgency.

2. PRIOR LITERATURE SEM

evolved in three different streams: (1) systems of equation regression methods developed mainly at the Cowles Commission; (2) iterative maximum likelihood algorithms for path analysis developed mainly at the University of Uppsala; and (3) iterative least squares fit algorithms for path analysis also developed at the University of Uppsala. Figure 1 provides a chronology of the pivotal developments in latent variable statistics in terms of method (pre‐computer, computer intensive and SEM) and objectives (exploratory / prediction or confirmation).

3. SAMPLE SIZE AND THE RATIO OF INDICATORS TO LATENT VARIABLES

Structural equation modeling in MIS has taken a casual attitude towards choice of sample size. Since the early 1990s, MIS researchers have alluded to an ad hoc rule of thumb requiring the choosing of 10 observations per indicator in setting a lower bound for the adequacy of sample sizes. Justifications for this rule of 10 appear in several frequently cited publications (Barclay, et al. 1995; Chin 1998; Chin, and Newsted 1999; Kahai and Cooper 2003) though none of these researchers refers to the original articulation of the rule by Nunnally (1967) who suggested (without providing supporting evidence) that in SEM estimation ‘a good rule is to have at least ten times as many subjects as variables

4. SAMPLE SIZE WITH PAIRED LATENT VARIABLES

This section develops an algorithm for computing the lower bound on sample size required to confirm or reject the existence of a minimum effect in an SEM at given significance and power levels. Where SEM studies are directed towards hypothesis testing for complex models, with some level of significance ߙ and power 1െߚ ,calculating the power requires first specifying the effect size ߜ you want to detect. Funding agencies, ethics boards and research review panels frequently request that a researcher perform a power analysis, the argument is that if a study is inadequately powered, there is no point in completing the research. Additionally, in the framework of SEM the assessment of power is affected by the variable information contained in social science data.

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