The EffectLiteR Approach for Analyzing Average and Conditional Effects

ABSTRACT We present a framework for estimating average and conditional effects of a discrete treatment variable on a continuous outcome variable, conditioning on categorical and continuous covariates. Using the new approach, termed the EffectLiteR approach, researchers can consider conditional treatment effects given values of all covariates in the analysis and various aggregates of these conditional treatment effects such as average effects, effects on the treated, or aggregated conditional effects given values of a subset of covariates. Building on structural equation modeling, key advantages of the new approach are (1) It allows for latent covariates and outcome variables; (2) it permits (higher order) interactions between the treatment variable and categorical and (latent) continuous covariates; and (3) covariates can be treated as stochastic or fixed. The approach is illustrated by an example, and open source software EffectLiteR is provided, which makes a detailed analysis of effects conveniently accessible for applied researchers.


Introduction
Evaluating the effectiveness of a treatment or intervention is a central goal of empirical research in many disciplines. Researchers are interested in average effects of a treatment on an outcome as well as in conditional effects given values of categorical or continuous covariates. In this article, we consider experimental and quasi-experimental designs, where we want to estimate average and conditional effects of a discrete treatment variable on a continuous (manifest or latent) outcome variable, taking into account categorical and continuous (manifest and latent) covariates. To illustrate the analysis of average and conditional effects, we repeatedly refer to an example considering average and conditional effects of two psychotherapies.
In the social sciences, a prominent approach to estimate average and conditional effects is based on multiple regression with interactions (Aiken & West, 1996;Cohen, Cohen, West, & Aiken, 2003;Preacher & Hayes, 2004). The average effect is obtained by mean centering the covariates, and conditional effects can be obtained at certain values of continuous covariates. For example, conditional effects are computed at the mean of covariates ± 1 SD or by stratifying the sample according to values of categorical covariates or discretisized continuous covariates. The term moderation analysis is commonly used to refer to an analysis of conditional (total) effects (Baron & Kenny, 1986;Edwards & Lambert, 2007;Fairchild & CONTACT Axel Mayer axel.mayer@ugent.be Ghent University, Faculty of Psychology and Educational Sciences, Department of Data Analysis, Henri Dunantlaan , Ghent, B-, Belgium MacKinnon, 2009;Hayes, 2008;Little, Card, Bovaird, Preacher, & Crandall, 2007), and some computational tools as well as graphical techniques have been proposed (Fox, 2003;Preacher, Curran, & Bauer, 2006). Approaches for testing interactions have been extended to hierarchical linear models (Bauer & Curran, 2005), to multigroup structural equation models (e.g., Rigdon, Schumacker, & Wothke, 1998), to latent growth curve models (Curran, Bauer, & Willoughby, 2004), and to structural equation models with interactions among latent variables (e.g., Jöreskog & Yang, 1996;Klein & Moosbrugger, 2000;Marsh, Wen, & Hau, 2004).
In this article, we present a framework for conducting a detailed analysis of the effects of a discrete treatment variable on a continuous outcome variable, conditioning on categorical and continuous covariates. We term this approach EffectLiteR approach, named after the program EffectLite (Steyer & Partchev, 2008), where the capital letter R indicates "revised" and hints at the software environment R (R Core Team, 2013) that will be used to implement the new approach. The analysis of average and conditional effects is based on a multigroup structural equation model (multigroup SEM) with groups obtained by combinations of treatment groups and values of categorical covariates. In our framework, we present definitions and computations for average and various kinds of conditional effects. Some of the key advantages of the new approach are that it allows for latent covariates and out-come variables, that it includes (higher order) interactions between the treatment variable and categorical and continuous covariates, and that covariates can be treated as stochastic or fixed.
The article is structured as follows: First, we introduce a motivating example with an unbalanced 3 (treatment groups) by 2 (values of a categorical covariate) design, a latent outcome variable, and a latent pretest. Second, we point out the strengths of the EffectLiteR approach and describe how it extends existing approaches for analyzing average and conditional effects. Third, we introduce multigroup structural equation modeling with stochastic or fixed group sizes, which is the basis for our approach. Fourth, we define average and conditional effects in terms of generalized ANCOVA (Steyer & Partchev, 2008). Fifth, we describe the details of the EffectLiteR approach and show how it can be used to estimate average and conditional effects. Sixth, we illustrate a detailed effects analysis using a data set simulated according to the motivating example. Finally, we discuss limitations and practical issues that can arise in using this kind of analysis.

Motivating example
To demonstrate a detailed analysis of effects, we consider a motivating example throughout this article, where the effectiveness of two therapies (a conventional therapy, X = 1 and an innovative therapy, X = 2) is evaluated with respect to mental health in comparison to a wait list control group (X = 0) that receives no treatment during the course of the study. The continuous latent outcome variable mental health (η) is measured by three continuous manifest variables Y 12 , Y 22 , Y 32 , where higher values indicate better mental health. We use a pre-post intervention design in which patients can indicate a preference for either the conventional therapy or the innovative therapy and will then be assigned to one of the three conditions according to their preference and availability of therapists. In this illustration of a (nonrandomized) observational study, we restrict ourselves to two covariates, the continuous latent pretest variable (ξ ) measured by three manifest variables Y 11 , Y 21 , Y 31 , and the dichotomous covariate gender with values male (K = 0) and female (K = 1). In real applications, an applied researcher would need to carefully select covariates that may bias the effect estimates (e.g., Shadish, Cook, & Campbell, 2002).
In this example, there are several kinds of treatment effects of interest to an applied researcher. First, there are the conditional treatment effects given values of the two covariates-that is, the effects of the conventional therapy (X = 1 vs. X = 0) and the effects of the innovative therapy (X = 2 vs. X = 0) given a concrete value of the pretest and a concrete value of gender. 1 An example is the expected effect of conventional therapy on a woman with a high pretest value of mental health. These effects may vary depending on the values of the covariates considered and provide applied researchers with the most specific information. Second, there is the average effect of conventional therapy and the average effect of innovative therapy. These are the corresponding effects in the entire population and will later be defined as (unconditional) expectations of the conditional effects over the joint distribution of the two covariates. Third, there are the treatment effects given a treatment condition. As an example, consider the effect of innovative therapy (X = 2 vs. X = 0) given innovative therapy (X = 2): This is the effect of innovative therapy for those who actually receive innovative therapy. These so called effects on the treated (e.g., Geneletti & Dawid, 2011) are particularly interesting in observational studies where participants choose a treatment (see our discussion section "Effects given a treatment condition" for details). Fourth, we may ask whether the effects of conventional therapy and innovative therapy differ between men and women and look at conditional treatment effects given gender. Like all other effects considered, the conditional treatment effects given gender are aggregates of the conditional treatment effects given gender and the pretest. These aggregates are obtained by averaging over the gender-conditional distribution of the pretest values and will later be defined as conditional expectations. Fifth, we may be interested in some combination of the aforementioned effects and for example look at the effects of conventional therapy for males receiving innovative therapy, or we can be interested in the effects of conventional therapy for females in the control group and so forth. In summary, a detailed analysis of average and conditional effects is much more informative than just looking at the average effect.

Basic idea of the EffectLiteR approach
In this section, we introduce the EffectLiteR approach that allows for a detailed analysis of all these average and conditional treatment effects. The basic idea is to condition on a set of continuous and categorical covariates to obtain the conditional treatment effects-that is, treatment effects given the values of all covariates in the analysis-and then consider various aggregates of these conditional treatment effects such as average effects, effects on the treated, or aggregated conditional effects given values of a subset of covariates. In observational studies, conditioning on covariates for analyzing treatment effects is crucial because failure to control for relevant confounders can result in causally biased treatment effects. In addition, in randomized experimental designs, researchers may benefit from including covariates in the analysis of effects in order to examine differential effectiveness of a focused cause (e.g., a manipulation or an intervention) and/or to increase power. The computation of average and conditional effects in the EffectLiteR approach is based on a multigroup SEM, where the groups (sometimes also referred to as cells) are formed by all possible combinations of treatment groups and levels of categorical covariates. In the structural part of the SEM, the regressions of the (manifest or latent) dependent variable on the (manifest and latent) continuous covariates are estimated in each cell.

Strengths of the EffectLiteR approach
The EffectLiteR approach provides researchers with a structural equation modeling framework for an analysis of effects in experimental and quasi-experimental designs that incorporates desirable aspects of existing approaches and extends these approaches in various ways. From traditional analysis of variance methods, we incorporate the ideas of omnibus tests for average effects and interactions (using Wald tests instead of sums of squares tables for main effects and interactions) and the inclusion of higher order interactions; from the causal inference literature, we incorporate adjustment for continuous and categorical covariates and the definitions of average and conditional effects; from structural equation modeling, we incorporate latent variables and stochastic continuous regressors (predictors). Furthermore, multigroup analysis enables us to allow for heterogeneous residual variances across groups.
In addition to the integration and further development of these aspects, the EffectLiteR approach extends previous approaches in various unique ways: It extends conventional multigroup SEM by allowing for stochastic categorical regressors in addition to the possibility of including stochastic continuous regressors. Stochastic categorical regressors are implemented by adding a part for modeling group sizes to the multigroup SEM and become relevant for estimating average and conditional effects, since erroneously treating stochastic regressors as fixed may lead to biased standard errors of effects in designs in which the distribution of the categorical regressors in the sample is a result of sampling and not fixed by design (Kröhne, 2009;Sampson, 1974). Another unique feature is higher order interactions between the treatment variable, categorical covariates, and latent covariates. Such higher order interactions are implemented by estimating the regression of the outcome variable on the continuous covariates in each cell, which automatically allows regression coefficients to differ across values of the treatment and values of categorical covariates, unless constraints are imposed on the regression coefficients across cells. In models ignoring interactions, the treatment effect is not equal to the average treatment effect (Rogosa, 1980). In addition, the inclusion of such interactions opens up the opportunity for applied researchers to thoroughly examine the differential effectiveness of the treatment.

Software
To the best of our knowledge, there is currently no software that provides such a comprehensive analysis out of the box. As a supplement to this article, we provide an easy to use software including a graphical user interface that allows for a quick and detailed analysis of average and conditional effects. The software we provide is an R package (R Core Team, 2013) called EffectLiteR, which automatically generates lavaan (Rosseel, 2012) code and provides estimates and tests for all the different average and conditional effects presented in this article. It is available from CRAN and can be installed as usual (see Supplemental Appendix A for details). EffectLiteR is a new program based on some of the ideas that are implemented in EffectLite (Steyer & Partchev, 2008).

Multigroup SEM with stochastic group sizes
Before we present the definitions of average and conditional effects and the details of the EffectLiteR approach, we introduce the multigroup structural equation model with stochastic group sizes, which provides the statistical framework for our approach. The multigroup structural equation model with stochastic group sizes extends traditional multigroup structural equation modeling (Jöreskog, 1971) by incorporating an optional part for the group sizes and is a special case of a finite mixture structural equation model (Arminger, Stein, & Wittenberg, 1999;Jedidi, Jagpal, & DeSarbo, 1997) with observed (stochastic) class membership. Let G denote a group variable whose values g = 1, …, h represent all possible combinations of values of stochastic categorical independent variables. The complete model consists of a groupspecific measurement model relating manifest variables in the vector y to latent variables in the vector η, a groupspecific structural model specifying structural relations among latent variables η, and a part for the group sizes: Measurement model for group g y = ν g + g η + ε (1) Structural model for group g η = α g + B g η + ζ Model for group sizes, where ν g is a vector of measurement intercepts; g is a matrix of loadings; α g is a vector of structural intercepts; B g is a matrix of structural coefficients; ε is a vector of measurement error variables with zero mean vector and covariance matrix g ; ζ is a vector of structural residuals with zero mean vector and covariance matrix g ; and κ g is a parameter for the log-transformed expected group frequency of group g; that is, κ g = log(n g ). The probabilities for each group are P(G = g) = exp(κ g )/ࢣ g exp(κ g ). 2 All continuous covariates are treated as stochastic and therefore appear on the y-side of the structural equation model. In addition, if the model contains manifest dependent variables and/or manifest continuous covariates involved in a regression, we automatically "upgrade" these variables to latent variables by specifying a trivial measurement model without measurement error for them. This commonly used strategy allows us to model structural relations between latent variables and manifest variables using the B g matrix. Manual upgrading of manifest variables can also be used to account for measurement error in single-indicator variables, if prior information on reliability is available (e.g., Kline, 2011, p. 276).
The multigroup SEM with stochastic group sizes implies the group-specific covariance structure for continuous manifest variables, and the group-specific mean structures, The parameters of the model can be estimated simultaneously according to a sample of N independent and identically distributed realizations of y using standard  This model as presented here with a saturated Poisson model for group sizes is implemented in lavaan (Rosseel, ) in versions ࣙ.-. We chose a Poisson model with the canonical log link because it leads to a relatively easy-tocompute likelihood and does not require constrained optimization. A similar model using a multinomial model for group sizes is available in Mplus (Muthén & Muthén, -) via the KNOWNCLASS option. The KNOWN-CLASS option was first used to analyze average effects by Kröhne () for cases without categorical covariates and was extended by Dietzfelbinger (), who also provides automatic generation of software code.
maximum-likelihood theory. Let θ = (θ 1 , θ 2 ) denote the vector of all parameters for all groups, where θ 1 contains all parameters except for parameters related to group sizes, and θ 2 contains just these. Assuming a multivariate normal distribution for y, a Poisson distribution for group sizes n g , and independence of group sizes and other model parameters, the total log-likelihood function log L(θ) for a sample is the sum of the log likelihood for the multigroup SEM part log L 1 (θ 1 ) (cf. Browne & Arminger, 1995;Hartley & Hocking, 1971;Jöreskog, 1971) and the log likelihood for the Poisson distributed group sizes log L 2 (θ 2 ) (e.g., Agresti, 2002, p. 7): where H g = S g + (ν g − μ g )(ν g − μ g ) , and for each of the groups, S g is the sample variance-covariance matrix; g is the model-implied variance-covariance matrix;ν g is the sample mean vector; μ g is the model-implied mean vector; n g is the group size, and c is a constant. To estimate θ from a sample, we compute the group sizes and the model-implied variances, covariances, and means according to Equations (4) and (6) for all admissible values of the parameters. The maximum likelihood estimator θ of θ is then the set of values that maximizes the loglikelihood function log L(θ) of the sample. All parameters of the model are estimated simultaneously. The variancecovariance matrix of parameter estimates can be obtained following standard maximum-likelihood theory.
This framework of a multigroup SEM with stochastic group sizes will be used to estimate average and conditional effects and to test hypotheses about these effects. Average and conditional effects and their standard errors will be computed according to parameter estimates and their variance-covariance matrix using the multivariate delta method. For omnibus hypothesis tests, we will use the Wald test since it does not require fitting multiple models (in contrast to the likelihood ratio test), which often leads to estimation problems if the restricted model is grossly wrong.

Generalized ANCOVA and definitions of average and conditional effects
Before we present the details of the EffectLiteR approach, we first introduce generalized ANCOVA to define average and conditional effects of a discrete treatment variable X Conditional effect of treatment X = t vs. X =  given a value ξ of the multivariate covariate ξ and a value k of K Conditional effect of treatment X = t vs. X =  given X = x * ; x * = , …, p (effect given a treatment condition, see the corresponding discussion section for details) Conditional effect of treatment X = t vs. X =  given X = x and K = k (e.g., effect on the "treated males") with values x = 0, 1, …, p on a continuous (latent) outcome variable η, taking into account a single categorical covariate K with values k = 0, 1, …, j, and a vector of (latent) covariates ξ = (1, ξ 1 , ξ 2 , . . . , ξ q ) with z = 0, 1, …, q denoting the (z + 1)th entry in ξ. 3 The regression E(η | X, K, ξ) can always be written: In this representation of the regression, g 0 (K, ξ), . . . , g p (K, ξ) are functions of (K, ξ), and I X = x is an indicator variable for the event that X takes on value x. We call X = 0 the reference or control group and all other treatment groups X = t; t = 1, …, p are compared to it.
The intercept function g 0 (K, ξ) describes the conditional regressive dependency of η on the covariates in the control group (i.e., for X = 0). The values of the effect functions g t (K, ξ) are the conditional treatment effects of treatment X = t versus treatment X = 0 given values of K and ξ. We define average and various conditional effects of treatment X = t versus X = 0 as (conditional) expectations of the effect functions g t (K, ξ). Table 1 gives an overview of the definitions of various kinds of average and conditional effects. For example, the unconditional expectation of the g 1 (K, ξ) effect function, E[g 1 (K, ξ)], is the average effect of treatment X = 1 compared to X = 0, and the (K = 0) −conditional expectation of the g 1 (K, ξ) function, , is the conditional effect of treatment X = 1 compared to X = 0 given the value 0 of the categorical covariate K.  The term generalized ANCOVA refers to an ANCOVA with interactions between treatment and covariates, which is typically implemented using a multigroup structural equation modeling approach. It has been introduced by researchers at the Department of Methodology and Evaluation Research under supervision of Rolf Steyer, and there are several theses about this topic (e.g., Dietzfelbinger, ; Flory, ; Hartenstein, ; Kröhne, ; Nagengast, ). Generalized ANCOVA is also described in the software manual for EffectLite (Steyer & Partchev, ). The consideration of just a single covariate K is not a restriction to the number of categorical covariates, because K may be obtained by unfolding multiple discrete covariates K  , K  , …; that is, the values of K then represent all possible combinations of values of multiple categorical covariates.
In order to estimate treatment effects, we need to assume a parameterization for all g x (K, ξ) functions. In this article, we parameterize the g x functions in such a way, that higher order interactions can be included: where γ xkz are regression coefficients; γ xk = (γ xk0 , γ xk1 , . . . , γ xkq ) is a vector of regression coefficients; and I K=k is an indicator variable for (K = k).

Motivating example
Consider our motivating example: The effect function g 1 (K, ξ ), whose values are the conditional treatment effects of conventional therapy (X = 1 vs. X = 0) on the posttest η given values of the latent pretest variable ξ and gender variable K, is given by and the average effect AE 10 can be obtained by taking the expectation of the effect function g 1 (K, ξ ): Hence, the average effect of conventional therapy AE 10 is identified by a function of regression coefficients and the expectations of the covariates and the interaction term. Similarly, the conditional effect of conventional therapy for males, CE 10;K=0 , can be obtained by which is a function of the two regression coefficients γ 100 and γ 101 and the conditional expectation E(ξ |K = 0). The parameterization and identification of average and conditional effects of innovative therapy (X = 2 vs. X = 0) is analogous.

The EffectLiteR approach
The key for estimating these average and conditional effects will be the computation of the parameters of the functions g t (K, ξ) and the computation of the expectations of the covariates and interaction terms. In the EffectLiteR approach, we will use the multigroup SEM with stochastic group sizes for this purpose. The groups are formed by all possible combinations of values x of the treatment variable X and values k of the unfolded categorical covariate K. From now on we will therefore use (X = x, K = k) in place of g to indicate groups. The specification of the latent variables is done in the groupinvariant measurement model (Equation [1]). The structural model (Equation [2]) is used to specify the regression of the dependent variable η on the continuous covariates in each cell (combination of x and k) and the cellspecific expectations of continuous covariates. Together with the model for the group sizes, the EffectLiteR model is Group-invariant measurement model Group sizes for group (X = x, K = k) where α xkz are regression coefficients of the group-specific regressions of η on ξ; μ xkz = E(ξ z |X = x, K = k); κ xk denote the parameters for the group sizes (see also Equations [1]-[3]); and the residual ζ 0 is independent of the vector of continuous covariates ξ.

Motivating example
In our example, we specify a six-group model that is obtained by all possible combinations of the three values of the treatment variable X (control group, conventional therapy, innovative therapy) and the two values of the categorical covariate K (male, female). We choose a group-invariant τ -equivalent measurement model for the latent pre-and post-measures of mental health; that is, all measurement intercepts are fixed at zero, and all loadings are fixed at one. In the structural model, we estimate the regressions of the latent posttest η on the latent pretest ξ in each of the six groups. In addition, the EffectLiteR model includes the group sizes of the six groups. The complete specification for our example is shown in Supplemental Appendix B, and a path diagram is depicted in Figure 1. From the parameters of the EffectLiteR model, we can compute all parameters needed for the computation of average and conditional effects. First, we show how to compute the regression coefficients γ xkz of the intercept and effect functions according to the regression coefficients α xkz of the multigroup SEM. Second, we show how to compute conditional and unconditional expectations of categorical covariates, continuous covariates, and product terms according to the conditional expectations μ xkz of the multigroup SEM and the group sizes κ xk . Third, we identify different kinds of average and conditional effects according to these derived parameters. Fourth, we introduce overall hypothesis tests that can be used to test null hypotheses concerning average effects and interactions.

Effect functions
First, we compute the parameters γ xkz of the g x (K, ξ) functions (see Equation [8]). As specified in the structural model, the conditional regression in each cell (combination of x and k) can also be written as where α xkz are regression coefficients and α xk = (α xk0 , α xk1 , . . . , α xkq ) is a vector of regression coefficients. Note that we assume linearity in ξ of the (x, k)-conditional regression. As an intermediate step to computing the coefficients γ xkz , we first consider the conditional regressions of η on K and ξ in a treatment group x with x = 0, 1, . . . , p: where β xkz are regression coefficients and β xk = (β xk0 , β xk1 , . . . , β xkq ) is a vector of regression coefficients. β xk can be computed based on α xk from Equation (15): that is, β x0 is equal to α x0 , and β xk for k = 1, . . . , j can be computed as the difference between the corresponding vectors α xk and α x0 (see Dietzfelbinger, 2014 for a step-by-step illustration of this result and our illustrative example for a numerical example). For the elements of the vectors, Equation (17) turns into β x0z = α x0z and β xkz = α xkz − α x0z for k = 1, . . . , j. Returning to the computation of γ xk , recall that we parameterized the functions g x (K, ξ) (cf. Equation [8]): Similarly as shown above, γ xk can be computed based on β xk from Equation (16): which turns into γ 0kz = β 0kz and γ xkz = β xkz − β 0kz for x = 1, . . . , p for the elements of the vectors. Combining Equations (17) and (18) yields the main result of this section-that is, how to compute the coefficients of the intercept and effect functions γ xk according to the parameters α xk of the multigroup SEM.

Motivating example
Using our example of estimating the effects of conventional therapy and innovative therapy, we need to compute the regression coefficients of the effect function for conventional therapy γ 1kz and the regression coefficients of the effect function for innovative therapy γ 2kz . Each of the two effect functions is parameterized using four regression coefficients (cf. Equation [9]). The coefficients are computed according to Equations (17) and (18). The complete computations are shown in Supplemental Appendix B.

Expectations and conditional expectations of covariates
For the identification of average and conditional effects, we also need conditional and unconditional expectations of the covariates. Apart from E(ξ z |X = x, K = k), these (conditional) expectations are not parameters of the model and must therefore also be computed according to other parameters of the EffectLiteR model. The (conditional) expectations of interest for the corresponding effects and their general computation are shown in Table 2 Motivating example For brevity, we mention the necessary (conditional) expectations for our motivating example here while the complete computation is given in Supplemental Appendix B. For computing the average effects of conventional therapy and of innovative therapy in our applied example, we need the unconditional expectation of the indicator variable for females, which is equal to the marginal probability P(K = 1), the unconditional expectation of pretest mental health, E(ξ ), and the expectation of the product term, E(ξ I K=1 ). To compute the gender-specific conditional effects, we need the (K = k)-conditional expectations of Additional conditional expectations needed for X = x-conditional effects Additional conditional expectations needed for K = k-conditional effects

Identification of average and conditional effects
In this section, we show how to identify average and conditional effects according to the γ xkz coefficients (see Equation [18]) and the conditional expectations (see Table 2). The average effect of treatment X = t versus X = 0 can be identified: where E ξ = 1, E(ξ 1 ), E(ξ 2 ), . . ., E(ξ q ) and E ξ I K=k = E(I K=k ), E(ξ 1 I K=k ), E(ξ 2 I K=k ), . . . , E(ξ q I K=k ) . The average effect AE t0 is a function of the γ tk coefficients and the corresponding unconditional expectations of covariates and product terms of covariates. In contrast to traditional approaches (e.g., Aiken & West, 1996), we do not mean center covariates or product terms, but compute their expectation according to model parameters. This way, we can account for uncertainty in the estimates of the expectations and do not run the risk of underestimating standard errors of average effects (Kröhne, 2009). The identification of other kinds of conditional effects is shown in Table 3.

Motivating example
and the effect of innovative therapy on typical females treated with conventional therapy is identified by For a detailed explanation of effects given a treatment condition, see also the corresponding subsection in the discussion.

Main hypotheses
Similar to testing multiple contrasts in ordinary ANOVA, a comprehensive analysis of all sorts of average and conditional effects includes many tests, and the analysis may therefore suffer from cumulative Type 1 error. To prevent this, we suggest testing some overall hypotheses first, before going into detailed analyses of average and conditional effects. These overall hypotheses are similar in spirit to a traditional sums of squares table but very different from a statistical perspective. We suggest using a Wald test procedure to test the following hypotheses.

H1: No average treatment effects
H 0 : E(g t (K, ξ )) = 0∀t = 1, 2, . . . , p H2: No treatment-covariate interaction H 0 : g t (K, ξ) = constant ∀ t = 1, 2, . . . , p H3: No treatment effects H 0 : g t (K, ξ ) = 0∀t = 1, 2, . . . , p The first hypothesis is about average treatment effects. The corresponding null hypothesis is that there are no average treatment effects. For a treatment variable with three values (x = 0, 1, 2), this test has two degrees of freedom, and it is simultaneously tested whether there is an average treatment effect of X = 1 versus X = 0 or there is an average treatment effect of X = 2 versus X = 0. In general, it has degrees of freedom df = p.
The second hypothesis tests whether there are treatment by covariate interactions. In other words, it tests whether the treatment effects depend on values of categorical or continuous covariates. If the treatment effects do not differ between values of covariates, the g t (K, ξ) functions are constants. Consequently, we test simultaneously whether all coefficients in the g t (K, ξ) are zero, except for the first regression coefficient in each g t (K, ξ) function. The second hypothesis has degrees of freedom d f = p · (q + j + q · j). Note that even if there were no average treatment effects, there could still be significant (and even strong) (K = k, ξ = ξ)-conditional treatment effects. If these conditional treatment effects are, for example, positive for high values of ξ, but negative for low values of ξ, they (partly) cancel each other out, which can result in a very small average treatment effect or no average treatment effect at all. This is one reason why it is crucial to consider conditional effects in addition to average effects.
The null hypothesis for our third suggested overall test is that there are no treatment effects, neither average effects nor conditional effects. Consequently, we test simultaneously whether all regression coefficients γ xk in the g t (K, ξ) are zero. It has degrees of freedom d f = p + p · (q + j + q · j).

Motivating example
In our illustration, the hypothesis test of "no average treatment effects" has two degrees of freedom, and the null hypothesis is that the average effect of conventional therapy is zero and that the average effect of innovative therapy is zero, that is, H 0 : The null hypothesis test of "no treatment-covariate interaction" is that neither the effects of conventional therapy nor the effects of innovative therapy depend on values of the latent pretest or gender; that is, H 0 : g 1 (K, ξ ) = γ 100 ∧ g 2 (K, ξ ) = γ 200 , which has six degrees of freedom. This would still allow for average effects of the treatments but not for differential effectiveness of the treatment depending on pretreatment covariates in the model. The hypothesis test of "no treatment effects" combines the first two hypothesis tests. Its null hypothesis is that there are no treatment effects whatsoever; that is, neither conventional therapy nor innovative therapy has any average or conditional effect on the outcome. In formal terms, H 0 : g 1 (K, ξ ) = g 2 (K, ξ ) = 0, which has eight degrees of freedom.

Effect sizes
We suggest to compute effect sizes (ES) by dividing the effect estimates by the standard deviation of η in the control group X = 0, but other ways to standardize the effects are possible as well. All approaches to standardization require computation of the conditional or unconditional variance of the dependent variable according to the model-implied group-specific variances Var(η|K = k, X = x) (cf. Equation [4]). The variance of η in the control group, Var(η|X = 0), can then be computed using the law of total conditional variance: The standard deviation of η given X = 0 is the square root of Var(η|X = 0). Alternatively, one may use the unconditional variance of η for standardizing the effects, which can be computed using the same formula but without conditioning on X = 0.

Simulated data example
To illustrate the EffectLiteR approach, we simulated data (N = 1, 000) according to our motivating example. Recall that the data reflect a hypothetical observational study to assess the effects of conventional and innovative therapy in a pre-post intervention design with a control group. The variables used in the analyses are the three manifest variables measuring the latent posttest mental health, Y 12 , Y 22 , Y 32 ; the three manifest variables measuring the latent pretest mental health, Y 11 , Y 21 , Y 31 ; the treatment variable, X; and the gender variable, K. The R syntax used to generate the data and the true population values are given in Supplemental Appendix C. The estimated coefficients and effects are close to the population values in the sense that the confidence intervals in the sample all include the corresponding true values, which is not surprising because we used a similar model for data generation as for the analysis.

Results
We analyzed the data with the EffectLiteR approach. Table 4 shows the average and conditional treatment effects and the parameters of the conditional effect functions with standard errors and effect sizes.

Main hypotheses
We first tested the three main hypotheses described previously. According to the Wald test, all three null hypotheses are rejected at α = .05 (no average treatment effects: χ 2 = 266, df = 2, p < .001; no treatment covariate interaction: χ 2 = 214, df = 6, p < .001; no treatment effects: χ 2 = 557, df = 8, p < .001). The first hypothesis test suggests that at least the average effect of conventional therapy and/or the average effect of innovative therapy is not zero in the population. The second hypothesis test indicates that at least one of the two effect functions is not a constant. Formally, this indicates that at least one of the six slope coefficients γ 101 , γ 110 , γ 111 , γ 201 , γ 210 , γ 211 is not zero in the population. From a substantive point of view, this means that the true (K = k, ξ = ξ )-conditional effects of conventional therapy and/or innovative therapy depend on the values of gender and/or the pretest and gives an indication for applied researchers to further investigate the differential effectiveness of the two therapies. The last hypothesis test also emphasizes that there are average and/or conditional effects of at least one of the two therapies, which is not surprising in this case since the first two hypothesis tests are significant as well. Drawing on the results of tests of the three main hypotheses, we examine average and conditional effects in more detail.

Average effects AE t0
In this example, the estimated average effect of conventional therapy ( AE 10 = 0.

Conditional effects CE t0;K=k
In Table 4, estimates for the two conditional effects of conventional therapy for males CE 10;K=0 and females CE 10;K=1 , and the two conditional effects of innovative therapy for males CE 20;K=0 and females CE 20;K=1 are shown. These effects are aggregates of the conditional effects CE t0;ξ =ξ,K=k described in the previous paragraph. For both treatments, the conditional effect estimates are higher for females than for males. Testing whether these differences are significant revealed differences between males and females with regard to the effectiveness of both treatments (Wald χ 2 = 6.56, df = 1, p = .0104 for gender differences in the effect of conventional therapy, and Wald χ 2 = 56.41, df = 1, p < .001 for gender differences in the effect of innovative therapy). From these results we conclude that both conventional therapy and innovative therapy are more beneficial for females than for males.

Conditional effects CE t0;X =x
In an observational study, such as our example, patients are not randomly assigned to treatment conditions-they self-select a preferred therapy and are then assigned to a condition according to availability of therapists. This assignment policy may be less than ideal in the sense that patients may not get the most effective treatment for them. We examine this assignment policy by considering conditional treatment effects given values x of X. First, we consider the conditional effects CE 10;X=x -that is, the effects of conventional therapy for those in the control group, for those receiving conventional therapy, and for those receiving innovative therapy (see Table 4). A test indicates no differences (Wald χ 2 = 1.41, df = 2, p = .493). Similarly, we also retain the null hypothesis, that the conditional effects of innovative therapy are equal across the three treatment groups (Wald χ 2 = 1.77, df = 2, p = .413).

Conditional effects CE t0;X =x,K=k
To further examine the differential effectiveness of the two therapies, we tested whether the (X = x, K = k) -conditional effects differ at all and find significant differences for the effects of conventional therapy (Wald χ 2 = 12.89, df = 5, p = .024) and for the effects of innovative therapy (Wald χ 2 = 67.79, df = 5, p < .001) across (X = x, K = k)−conditions (see also Table 4).

Stochastic nature of group sizes and regressors
Traditional methods to examine treatment effects based on the general linear model (GLM), with ANOVA and ANCOVA models being special cases of this model, assume fixed values for both categorical and continuous regressors. This means that we would always obtain exactly the same design matrix across replications of the experiment, which is only a plausible assumption in controlled experimental designs without continuous regressors, where group sizes are chosen and set by the experimenter. The GLM assumption of fixed regressors is not plausible when groups evolve naturally or when continuous regressors are included because those can usually not be set by the experimenter. 4 While ignoring the stochastic nature of regressors does not distort statistical inference for unstandardized regression coefficients (e.g., Casella & Berger, 2002, p. 550), the distinction between stochastic and fixed regressors becomes important when (conditional) expectations of regressors are needed to compute average and conditional effects. In this case, (conditional) expectations of regressors are ingredients of the effect computations, and it therefore can make a difference for the standard errors (but not the point estimates) of average and conditional effects. Usually the standard errors ignoring stochasticity of regressors are too small because uncertainty that stems from estimating their (conditional) expectations is not taken into account (Kröhne, 2009).
In the EffectLiteR approach, we allow stochastic categorical and continuous regressors. For stochastic categorical regressors, we implemented a separate part of the model, the model for group sizes. In addition, we can treat continuous regressors as stochastic by including them in our multigroup SEM, which gives us the opportunity to estimate their group-specific expectations and functions thereof. By default, we are treating all regressors as stochastic, but a model with fixed group sizes and/or fixed continuous regressors is still a special case of our model and can easily be obtained by fixing the corresponding parameter to its value in the sample. In a balanced design for example, the researcher may choose to use fixed group sizes instead of stochastic group sizes. As a side note, the  An exception to this rule is a controlled experiment where dose is varied systematically along a continuum.
classic mean centering approach always assumes fixed values of continuous covariates and ignores stochasticity in the expectation of the regressor. This may lead to underestimated standard errors of average effects, especially in applications with small samples. In larger samples, the uncertainty related to estimating the population mean of a covariate diminishes and the differences between the two methods are less noticeable. To sum up, we only recommend models with fixed regressors in experimental designs where the values of the regressors can be chosen and set by the experimenter.

Higher order interactions
By default, we parameterize the regression and effect functions in the EffectLiteR approach in such a way that higher order interactions are included, which may lead to rather complex models if there are many cells and/or continuous covariates. The inclusion of higher order interactions between the treatment variable and categorical/continuous covariates in our model is advantageous in those empirical applications where these interactions actually exist. Ignoring interactions can lead to incorrect effect estimates and can obscure the differential effectiveness of treatments. However, in applications where some of these interaction terms are not needed, we can set the corresponding regression coefficients to zero in order to make the model more parsimonious. If, for example, we do not want to include three-way interactions, we would constrain certain coefficients γ xkz to be zero-namely, the ones with x > 0, k > 0, and z > 0.

Sample size
If there are many combinations of treatment conditions and values of categorical covariates, large sample sizes may be needed for stable estimation of parameters. The requirements for sample size in the EffectLiteR model depend on many aspects, including the number of groups, unbalancedness of group sizes, effect sizes of average and conditional effects of interest, the size and order of interactions, and the amount of measurement error. Therefore, it is not possible to give general rules of thumb for the required sample size in such complex models.
To get an impression about how large the required sample size needs to be, we recommend to consider the regression model in each cell and count the numbers of parameters that will be estimated for the regression model. Using 10 observations per parameter will give you a vague idea about the sample size. In our motivating example with medium to large average effects, we estimate the intercept and slope of the regression of η on ξ in each cell together with the residual variance. We do not include the parameters for the conditional mean and variance of ξ and the group-invariant parameters of the measurement model for our guess on sample size. This leads us to a sample size of about 30 for each cell, and considering the fact that the smallest cell has probability P(X = 0, K = 0) = 0.1, we conclude that in our motivating example an approximate total sample size of about 300 participants is needed. Constraints on regression coefficients across cells may further reduce the required total sample size.
If sample size is important in planning the study, we recommend running a Monte Carlo simulation to get a much better idea about the sample size needed for the concrete scenario of interest. In Supplemental Appendix D, we provide an R code template to conduct a Monte Carlo simulation for our motivating example. We used 5,000 replications and a sample size of N = 150. This is pushing the limits because N = 150 implies an expected cell count of 15 for the smallest cell, which in fact led to simulated data sets with fewer than 10 observations per cell. The results show no bias for point estimates and standard errors of average and conditional effects and acceptable power to detect the effects. However, because of the small cell sizes, there were convergence issues: Out of the 5000 replications, 1,965 converged normally; 27 did not converge; and in 3,008 runs we had at least one negative variance estimate. Many of these problems could probably be resolved in an application by carefully inspecting the model and adding appropriate constraints.

Missing data and sparse cells
Missing data is a commonly encountered problem in experimental and quasi-experimental studies. The EffectLiteR model builds on structural equation modeling, making it possible to use contemporary methods, such as full information maximum likelihood estimation or multiple imputation to deal with missingness. An overview of modern missing data methodology is given in Molenberghs, Fitzmaurice, Kenward, Tsiatis, and Verbeke (2014).
A limitation, which is a result of the relatively complex EffectLiteR model, is the difficulty to model sparse or empty cells. Sparse cells, empty cells, or structural zero cells pose a particular challenge. There are both frequentist and Bayesian approaches to deal with these problems in other settings; it is currently unclear what is the best solution in the context of the EffectLiteR approach. An option to deal with sparse cells is to add restrictions to the model (e.g., equal residual variances across cells or constraints for interactions as discussed previously) to make it more parsimonious in applications with an unfavorable ratio of cell sample sizes and number of estimated parameters. Adding restrictions, however, does not solve the problem of empty cells. Yet another option would be to reduce the complexity and the number of cells by estimating a model with as many groups as there are treatment conditions and add indicator variables for categorical covariates. This has been the original EffectLite (Steyer & Partchev, 2008) approach and is still a special case of the revised approach we presented in this article. Care must be taken when indicator variables are treated just like continuous covariates because some parameters are deterministic functions of other parameters in the model (e.g., the variance of an indicator variable is a deterministic function of its mean), and this may cause bias in standard errors for average and conditional effects. Dietzfelbinger (2014) shows derivations of necessary constraints in such cases. Bootstrap based standard errors may also provide a less error-prone alternative.

Linearity assumption
It should be noted that the EffectLiteR approach relies on the assumption that the (X = x, K = k)-conditional regressions of η on ξ are linear. While this is already much more general than the usual linearity assumptions made in traditional SEMs for assessing the effects of a treatment on an outcome variable, this linearity assumption of the cell-specific regressions could be violated in some empirical applications. If the (X = x, K = k)-conditional regressions of η on ξ are in fact not linear, the estimates of the average and conditional effects may be distorted. However, we would like to point out that the EffectLiteR approach permits the analysis of nonlinear relationships between the outcome variable and continuous covariates given X = x and K = k if one specifies the cellspecific regressions and expectations accordingly. The linear model is of course appropriate in the illustrative simulated example we used due to the linear data-generating mechanism.
The EffectLiteR approach is to some extent robust against misspecification of the functional form of the conditional regressions because the multigroup model is relatively flexible. A detailed evaluation of its robustness is beyond the scope of this article, but researchers can use the template for the Monte Carlo simulation in Supplemental Appendix D to investigate the issue for a particular scenario by changing the true data-generating mechanism. In case there are strong deviations from conditional linearity, it is straightforward to extend the EffectLiteR approach by including quadratic or cubic terms as covariates-for models with latent variables one would need to use appropriate techniques for nonlinear effects (e.g., Kenny & Judd, 1984;Klein & Moosbrugger, 2000;Marsh et al., 2004).
For more complicated functional forms, semiparametric and double robust approaches provide further options to protect against violations of the conditional linearity assumption. The EffectLiteR model can be written as a finite mixture structural equation model by allowing for latent categorical variables in the model for group sizes. This then permits modeling nonlinear relations among latent variables (see our discussion section on semiparametric methods for details and references).

Large sample statistics
In this article, we used the maximum likelihood estimator to estimate the parameters of the EffectLiteR model. The methods we used for computing average and conditional effects and for testing hypothesis-namely, the delta method and Wald tests-are large-sample methods and rely on asymptotic distribution theory. Their behavior in the context of the EffectLiteR approach for small samples is less clear and needs additional research. Various small-sample corrections have been proposed in the literature for structural equation modeling (Fouladi, 2000;Herzog, Boomsma, & Reinecke, 2007;Nevitt & Hancock, 2004), and these may provide valuable additions to the EffectLiteR approach as presented here.

Discussion
In this article, we presented the EffectLiteR approach, a multigroup structural equation modeling approach with stochastic group sizes for analyzing average and conditional treatment effects in experimental and quasi-experimental designs. We demonstrated how the EffectLiteR approach builds on and extends existing approaches such as (generalized) ANCOVA and multigroup SEM. Our approach combines the following strengths: (1) It permits accounting for measurement error in the continuous covariate(s) and the outcome variable; (2) it naturally includes two way and three way interactions between the treatment variable and categorical and continuous covariates; and (3) both categorial and continuous covariates can be treated as stochastic or fixed. We demonstrated how to estimate average and (various) conditional effects according to the EffectLiteR model, both for the general case and in an illustrative example.

Types of effects
We introduced various types of effects that may be of interest for applied researchers, such as effects given certain combinations of covariate values, aggregated effects, and effects given a treatment condition. In cases without any interaction between the treatment variable and any covariate, all these effects will be identical, and a comprehensive analysis of the differential effectiveness would be not needed. However, we think that in most empirical studies evaluating interventions, the assumption of a uniform treatment effect for all persons is rather unrealistic. Therefore, we recommend to carefully examine conditional effects in addition to average effects.

Effects given a treatment condition
Effects given a treatment condition are an important special case of conditional effects, especially in observational studies. Some of these effects have also been termed effects on the treated in the literature (e.g., Geneletti & Dawid, 2011;Heckman & Robb, 1985;Hotz, Imbens, & Klerman, 2006;Rubin, 1977). We can illustrate the intuitive notion of treatment effects given a treatment condition in two equivalent ways. First, since treatment effects may depend on values of covariates, we can think of a treatment effect given a treatment condition as an effect for a person with covariate values as is typical in a particular treatment group. Second, we can imagine predicting a treatment effect for each person in our study, then take the subset of persons who receive a particular treatment and compute the "average" treatment effect for this subset of persons. In the EffectLiteR approach, we compute such effects given a treatment condition analytically according to the model parameters.
Referring to our motivating example, consider the effect of innovative therapy (X = 2 vs. X = 0) given conventional therapy (X = 1): This is the effect of innovative therapy for those who are assigned to conventional therapy. We can intuitively think of this effect as the effect that is obtained by first computing the conditional effect for X = 2 vs. X = 0 for all individuals and then taking the average for those who are in treatment X = 1. To put it another way, the question an applied researcher might have is "What would be the expected effect of innovative therapy on a typical patient that receives conventional therapy, if he/she would have been assigned to innovative therapy instead?".
The treatment effects given a treatment condition are of interest for two reasons. First, it is important to know how effective a treatment is for those people who actually receive it, and second, comparing the (X = x)-conditional treatment effects allows us to draw conclusions about the appropriateness of the assignment policy. Ideally, we want those people to be in treatment group t who benefit most from treatment t. For example, if it turns out that innovative therapy would be more beneficial for those people who did not receive the innovative therapy than for those who did, one might want to change the assignment policy.
Note that the (X = x)-conditional effects can only differ if there is an interaction between the treatment and covariates and if the distribution of the covariates differs between treatment groups. Consequently, in randomized experiments, the (X = x)-conditional treatment effects are expected to be equal. A treatment-covariate interaction and unequal distribution of covariates between groups will likely yield unequal (X = x)-conditional effects, but this is not a strict mathematical implication and counterexamples, where several terms cancel out can be found.
Another closely related conditional effect that is sometimes of particular interest are complier average causal effects (Angrist, Imbens, & Rubin, 2000;Jo, 2002;Little & Rubin, 2000). These effects can be estimated in EffectLiteR when information on complying status is available or can be estimated.

Causal interpretation of effects
In nonrandomized observational studies, the treatment effects that we dealt with in this article are not necessarily causal effects. The causal interpretation of these effects rests on strong and typically untested causality assumptions. Theories of causality, such as Rubin's causal model (Rubin, 1974), Pearl's graphical approach (Pearl, 2009), and the stochastic theory of causal effects (Steyer, Mayer, & Fiege, 2014), provide definitions of causal effects and describe how and under which conditions we can compute estimates of average and conditional causal effects according to empirically estimable quantities. Some of the causality conditions are testable in the sense that they can be falsified in empirical applications, and we encourage researchers to carefully examine the plausibility of causality conditions whenever possible and let causality conditions guide the selection process of covariates that need to be controlled. Once important covariates are identified, we can use several techniques to control for the selected covariates. The technique we employed in the EffectLiteR approach is regressing the outcome variable on the covariates and the treatment variable. Other techniques include, for example, methods based on propensity scores (Rosenbaum & Rubin, 1983), marginal structural models , and doubly robust approaches (for an overview, see Schafer & Kang, 2008).

Propensity score methods
Methods based on propensity scores are widely used for causal inference. Propensity scores have been introduced by Rosenbaum and Rubin (1983) and represent the individual treatment probabilities. Oftentimes, logistic regression with a large set of covariates is used to estimate the individual treatment probabilities, but estimation using alternative methods such as boosted regression, neural networks, or classification trees have been suggested in the literature (for a review, see Westreich, Lessler, & Funk, 2010). Most of these methods do not allow for latent variables, and using factor scores might currently be the only way to circumvent this limitation (Raykov, 2012).
The estimation of propensity scores is usually an iterative procedure, where multiple underlying models are fitted and balance checks are conducted after each model. Once reasonable estimates for propensity scores are available, there are several ways in which they can be used to adjust for confounding, including weighting, matching, subclassification, and as a covariate in a regression (for a review, see Schafer & Kang, 2008). Propensity score methods and in particular matching and subclassification can also be used to estimate conditional effects, such as effects on the treated or effects given a categorical covariate. A detailed discussion of each of these techniques is beyond the scope of this article.
We do not consider propensity score methods a competing method for the EffectLiteR approach but rather a complimentary method that is useful to summarize information from a large set of covariates. In principle, all the previously mentioned methods can be used in conjunction with EffectLiteR to estimate average and conditional effects. While the EffectLiteR model aims at modeling the relationship between the outcome and treatment/covariates using structural equation models, propensity score methods aim at modeling the relationship between the treatment and covariates. Combining both approaches results in a doubly robust method. According to our personal experiences, a promising approach is to include the logit transformed propensity scores as a continuous covariate in the EffectLiteR model in addition to other relevant covariates such as pretests. For an application of this strategy, see Kirchmann et al. (2011). In the software package EffectLiteR, we make including logit transformed propensity scores conveniently accessible for applied researchers.

Semiparametric methods for causal inference
A key question in applied data analysis is the robustness of the adjustment procedures against possible misspecifications of the functional form of either the model for the outcome variable or the model for the treatment assignment. The consequences of model misspecification are subtle and difficult to investigate. James Robins put forward the idea of semiparametric causal inference and developed together with colleagues a semiparametric approach based on dual modeling of treatment assignment and outcome (Robins, 2000;Robins, Rotnitzky, & Zhao, 1995;van der Laan & Robins, 2003). These models are double robust in the sense that they still yield unbiased estimates for causal effects if either of the two models (but not both) is misspecified (see Kang & Schafer, 2007, and the subsequent discussion of this article). In a recent article, Vermeulen and Vansteelandt (2014) further developed double robust estimators that prevent large bias even when both parametric models are misspecified. In a related line of research, Mark van der Laan and colleagues introduced targeted maximum likelihood learning for semiparametric causal inference (van der Laan & Rose, 2011;van der Laan & Rubin, 2006).
The EffectLiteR approach as presented in this article is a flexible parametric approach that builds on traditional structural equation modeling making it possible to incorporate many methodological advances that have been made in this area, including sophisticated measurement models, quadratic effects, multigroup-multilevel SEM, stochastic regressors, robust maximum likelihood estimation, and tests of model fit (for an illustration of how some of these aspects can be incorporated in the EffectLiteR model, see Mayer, Nagengast, Fletcher, & Steyer, 2014).
While researchers have argued that causal effects are often nonparametrically unidentifiable in models with latent variables (e.g., Díaz & van der Laan, 2013), it is possible to combine a semiparametric approach with the EffectLiteR approach to get the best of both worlds. There are already some steps in this direction: As described previously, we can use propensity score methods in conjunction with the EffectLiteR approach, and we can increase the flexibility of the functional form by adding polynomial terms. A very flexible option is to consider the EffectLiteR model as a finite mixture structural equation model where the groups are formed by manifest and latent categorical variables (Arminger et al., 1999;Jedidi et al., 1997;Muthén, 2001). This then permits modeling nonlinear relations among latent variables (Bauer, 2005). Recent advances in semiparametric structural equation modeling (e.g., Kelava & Brandt, 2014) provide further possibilities to model nonlinear spline relationships between latent variables and can in the future be integrated with EffectLiteR.

Conclusions
We hope that the EffectLiteR approach further advances the use of modern methods to obtain a detailed picture about the effectiveness of a treatment, intervention, or exposition. We wish to provide researchers with clear definitions of effects of interest and offer a useful toolbox that facilitates the computation of various average and conditonal effects. By using a detailed example, we showed the relevance and potential of the EffectLiteR approach and exemplified the necessary steps for such an analysis.

Article information
Conflict of Interest Disclosures: Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.
Ethical Principles: The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.
Funding: This work was not supported.

Role of the Funders/Sponsors
: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.