Three Cs in measurement models: causal indicators, composite indicators, and covariates

Abstract

In the last 2 decades attention to causal (and formative) indicators has grown. Accompanying this growth has been the belief that one can classify indicators into 2 categories: effect (reflective) indicators and causal (formative) indicators. We argue that the dichotomous view is too simple. Instead, there are effect indicators and 3 types of variables on which a latent variable depends: causal indicators, composite (formative) indicators, and covariates (the "Three Cs"). Causal indicators have conceptual unity, and their effects on latent variables are structural. Covariates are not concept measures, but are variables to control to avoid bias in estimating the relations between measures and latent variables. Composite (formative) indicators form exact linear combinations of variables that need not share a concept. Their coefficients are weights rather than structural effects, and composites are a matter of convenience. The failure to distinguish the Three Cs has led to confusion and questions, such as, Are causal and formative indicators different names for the same indicator type? Should an equation with causal or formative indicators have an error term? Are the coefficients of causal indicators less stable than effect indicators? Distinguishing between causal and composite indicators and covariates goes a long way toward eliminating this confusion. We emphasize the key role that subject matter expertise plays in making these distinctions. We provide new guidelines for working with these variable types, including identification of models, scaling latent variables, parameter estimation, and validity assessment. A running empirical example on self-perceived health illustrates our major points.

Figure 1. Measurement Models for Perceived Health Example

Notes: η₁ = perceived health. y₁ (x₁) = ill enough to go to a doctor in past year, y₂ (x₂) = hospitalized/disabled last year, y₃ (x₃) = hospitalized/disabled in prior 4 years, y₄ = self-rated health, y₅ = health satisfaction, C₁ and C₂ are the composites representing x₁ through x₃.

Figure 1. Measurement Models for Perceived Health Example

Notes: η₁ = perceived health. y₁ (x₁) = ill enough to go to a doctor in past year, y₂ (x₂) = hospitalized/disabled last year, y₃ (x₃) = hospitalized/disabled in prior 4 years, y₄ = self-rated health, y₅ = health satisfaction, C₁ and C₂ are the composites representing x₁ through x₃.

Figure 2

Types of Measurement Models.

Figure 3

Comparison of Composite and Causal Indicators.

Figure 4

Identification with Causal Indicators.