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Semi-parametric regression as a tool for assessing moderation: an analysis of the Fast Track intervention

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Abstract

When assessing the impact of a family or youth intervention, program evaluators often consider whether and how baseline characteristics moderate the effect of treatment. Parametric and non-parametric approaches are both possible, but each has limitations. In this article, we argue for and illustrate the use of semi-parametric methods. A hybrid of parametric and non-parametric approaches, semi-parametric modeling techniques combine the strengths of each approach. This article demonstrates how this approach can be used to evaluate whether an intervention effect was larger for those with most serious problems at baseline. Using data from the Fast Track project—a comprehensive intervention designed to prevent serious conduct problems among children at high risk—we investigated the moderating effect of two key baseline characteristics: parent-reported and teacher-reported problem behavior. Our analyses do not demonstrate a significant moderation of the intervention effect.

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Notes

  1. The literature on semi-parametric and non-parametric methods is large, diffuse and growing. We found these three sources particularly helpful (Yatchew 2003; Wood 2006; Fox 2000). Our presentation of this methodology draws heavily on those sources. We also refer the reader to Li and Racine (2007) for a detailed treatment.

  2. The mean square error of an estimate of an univariate function diminishes to zero with \( \sqrt N \). (The estimator is said to be “root-N consistent.”) Because of the curse of dimensionality, a function of two variables is \( \sqrt[3]{T} \) consistent—the estimator converges to the “truth” much more slowly. If one can separate the function into two pieces, each a function of one variable, the overall estimator is still \( \sqrt N \)-consistent (Yatchew 1998).

  3. Estimation is somewhat more complicated than indicated by Eq. 3. In particular, differencing can involve subtracting a weighted average the k preceding observations. Doing so smoothes out additional variation in the relationship between the covariate and the outcome: doing so improves the efficiency of the resulting estimator. In our empirical example below, we employ fifth-order differencing using the optimal weights derived in Yatchew (1997). For additional details, see that article.

  4. Researchers typically transform the polynomial into an alternative format or basis—that is, a set of functions that provide an equivalent fit to the piecewise cubic polynomial. Those functions have been refined over time in order to improve the computational properties of estimation and include models such as the B-Splines (Wood 2006). Other models set the second derivative at the knots to zero. These are labeled “natural splines”. Other models restrict the model to be linear in the tails (Durrleman and Simon 1989).

  5. The intuition is the same as in linear regression—adding a square term or an interaction can never cause the R-square to decrease.

    In an extreme case, if λ were set to zero and the knots were set at the actual data points, the fitted model would involve simple interpolation “connect the dots”. If λ were set to infinity, the resulting line would be the least squares line.

  6. Alternative methods exist for selecting the right level of smoothing. For details (Wood 2006; Wood and Wood 2007).

  7. The denominator of the GCV includes the term, \( [n - \gamma tr(A)] \). When γ equals one, this is the residual degrees of freedom. The research literature suggests that there is some tendency for the GAM to overfit even when the GCV is used to determine λ. For that reason, researchers often set γ equal to 1.4. We have followed that convention in our empirical illustration below.

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Acknowledgments

This work was supported by National Institute of Mental Health (NIMH) grants R18 MH48043, R18 MH50951, R18 MH50952, and R18 MH50953. The Center for Substance Abuse Prevention and the National Institute on Drug Abuse also has provided support for Fast Track through a memorandum of agreement with the NIMH. This work was also supported in part by Department of Education grant S184U30002 and NIMH grants K05MH00797 and K05MH01027. The economic analysis of the Fast Track project is supported through R01MH62988. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute On Drug Abuse or the National Institutes of Health.

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Authors and Affiliations

  1. Department of Maternal and Child Health, School of Public Health, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

    E. Michael Foster

  2. Department of Epidemiology, School of Public Health, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

    Stephanie Watkins

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  1. E. Michael Foster
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Correspondence to E. Michael Foster.

Appendix 1: R code for the model

Appendix 1: R code for the model

  • # outcome—indicates the outcome of interest

  • # parent—parent-reported behavior problems at baseline

  • # teacher—teacher-reported behavior problems at baseline

  • # trt—treatment status

  1. (#1)

    Estimate the GAM using the variables that have been differenced and residualized

    1. #

      Variables that have been differenced have the prefix DT or DP, indicating whether

    2. #

      the variable has been differenced using the parent (DP) or teacher (DT) variables.

    3. #

      All variables have a suffix of R, indicating that they have been residualized using the

    4. #

      other covariates (such as site).

    $$ {\tt{Model1 <- gam}}\, {\tt{(DT}} {\underline{\tt{outcome}}} {\tt{R \sim s}} ({\tt{DT}}{\underline{\tt{parent}}}{\tt{R}}), \quad {\tt{bs = tp,}} \quad {\tt{gamma = 1.4)}} $$
  2. (#2)

    Estimate a standard linear model for comparison purposes

    $$ {\tt{Model2 <- lm}} ({\tt{DT}} {\underline{{\tt{outcome}}}} {\tt{R\sim DT}} {\underline{\tt{parent}}} {\tt{R}}) $$
  3. (#3)

    Estimate the test statistic using an F statistic

    $$ {\tt{anova}}{\tt{(Model2}},\quad {\tt{Model1}},\quad {\tt{test}} = {\tt{\hbox{``}}}{\tt{F}}{\tt{\hbox{''})}}$$
    1. #

      We conducted similar steps using the teacher report as the outcome.

  4. (#4)

    Estimate a model with each of the predictors included flexibly

    $$ {\tt{Model3<-gam}}{\tt{(}}{\underline{{\tt{outcome}}}}{\tt{R}}\sim {\tt{s}}{\tt{(}}{\underline{{\tt{parent)}}}}{\tt{R}}+{\tt{s}}{\tt{(}}{\underline{{\tt{teacher}}}}{\tt{R)}},\quad{\tt{gamma}}=1.4,\quad{\tt{bs}}={\tt{tp)}}$$
    1. #

      note that differenced variables are no longer used; each of the two baseline variables

    2. #

      are now included as predictors

  5. (#5)

    Estimate a model that allows for flexible interaction between the parent and teacher

    1. #

      reports

      $$ {\tt{Model4<-gam}}{\tt{(}}{\underline{{\tt{outcome}}}}{\tt{R}}\sim {\tt{s}}{\tt{(}}{\underline{{\tt{parent}}}}{\tt{R}},\quad{\underline{{\tt{teacher}}}}{\tt{R}}{\tt{)}},\quad{\tt{gamma}}=1.4,\quad {\tt{bs}}={\tt{tp)}}$$
  6. (#6)

    Estimate the test statistic for the additivity assumption

    $$ {\tt{anova}}{\tt{(Model3}},\quad {\tt{Model4}},\quad {\tt{test}} = {\tt{\hbox{``}}}{\tt{F}}{\tt{\hbox{''})}}$$
  7. (#7)

    Estimate a model that estimates the smoothed model separately for the

    1. #

      treatment and comparison cases

      $$ {\tt{Model5<-gam}}{\tt{(outcomeR}}\sim {\tt{s}}{\tt{(}}{\underline{{\tt{parent}}}}{\tt{R}},{\underline{{\tt{teacher}}}}{\tt{R)}}+{\tt{s}}{\tt{(}}{\underline{{\tt{parent}}}}{\tt{R}},{\underline{{\tt{teacher}}}}{\tt{R}},{\tt{by}}={\tt{trt}}),\quad{\tt{gamma}}=1.4,{\tt{bs}}={\tt{tp)}}$$
  8. (#8)

    Estimate the test statistic for the assumption that the same model fits both groups

    $$ {\tt{anova}}{\tt{(Model4}},\quad {\tt{Model5}},\quad {\tt{test}} = {\tt{\hbox{``}}}{\tt{F}}{\tt{\hbox{''})}}$$

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Michael Foster, E., Watkins, S. Semi-parametric regression as a tool for assessing moderation: an analysis of the Fast Track intervention. Health Serv Outcomes Res Method 10, 67–85 (2010). https://doi.org/10.1007/s10742-010-0064-0

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