Modelling monotonic effects of ordinal predictors in Bayesian regression models

May combine two claims.

In simulation studies, the model is well calibrated and, in the case of monotonicity, has similar or even better predictive performance than other approaches designed to handle ordinal predictors.

“In simulation studies, we show that the model is well calibrated and, in case of monotonicity, has similar or even better predictive performance than other approaches designed to handle ordinal predictors.”

May combine two claims. The quote only partly states this claim.

Increasing the maximum treedepth can resolve the convergence problem in the N=1000 and D=50 interaction model but increases computation time noticeably.

“Increasing the maximum treedepth can resolve this problem but increases the computation time noticeably.”

May combine two claims.

In the models estimated for the purpose of this paper, sampling efficiency and convergence were good and rarely worse than when using a purely linear approach.

“Indeed, in the models estimated for the purpose of this paper, sampling efficiency and convergence were good and rarely worse that when using a purely linear approach.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

The variable which should be modeled as monotonic may either be integer valued or an ordered factor; in the latter case, the ordered factor is transformed to an integer variable with the lowest factor level identified with zero.

“The variable which should be modeled as monotonic may either be integer valued or an ordered factor.In the latter case, the ordered factor will be transformed to an integer variable with the lowest factor level being identified with zero as described above.”

May combine two claims.

Monotonic effects made similar or better predictions than other penalized ordinal approaches such as ordinal splines and much better predictions than unpenalized approaches such as standard isotonic regression or approaches treating the predictors as categorical.

“More precisely, monotonic effects made similar or better predictions than other penalized ordinal approaches such as ordinal splines and much better predictions than unpenalized approaches such as standard isotonic regression or approaches treating the predictors as categorical.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For models including interactions with monotonic variables, brms uses different simplex parameters for different terms of the same monotonic variable.

“Please note that for models including interactions with monotonic variables, brms will use different simplex parameters for different terms of the same monotonic variable (e.g., for the main effect of x and the interaction of x and z).”

May combine two claims.

The summary estimates of the regression parameters confirm that simo[1] and simo[3] have the largest estimates.

“This impression is confirmed by the summary estimates of the regression parameters (see Table 1) as simo[1] and simo[3] have the largest estimates.”

May combine two claims.

For the default a=1 and D+1=5 categories, SD(ζ_i)=0.19; for a=5, SD(ζ_i)=0.09.

“For the default of a = 1 and a total of D + 1 = 5 categories, we get a rather large standard deviation of SD(ζ i ) = 0.19.If we set, for example, a = 5, we get SD(ζ i ) = 0.09 and thus much higher certainty in changes of equal size.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Monotonic predictors can be used like any other predictor variable and, with respect to the formula syntax, behave like a numeric predictor.

“Monotonic predictors can be used like any other predictor variable and, with respect to the formula syntax, behave like a numeric predictor.”

May combine two claims.

The effective number of parameters is somewhat smaller for the monotonic model than for the categorical model; about the same difference as in the corresponding ELPD difference.

“Looking closer at the results, we see that the effective number of parameters is somewhat smaller for the monotonic model than those of the categorical model; about the same difference as we see in the corresponding ELPD difference.”

May combine two claims.

The monotonic model fits better than the categorical model, followed by the linear model, although the differences between the three models are not substantial.

“As shown in Table 2, the monotonic models fits better than the categorical model followed by the linear model although the differences between the three models is not that substantial.”

Probably not a substantial claim: may not pass S3 (load-bearing).

brms has a multilevel formula syntax allowing to model hierarchically nested data structures such as multiple observations per person in a longitudinal study.

“An especially well developed feature of brms is its multilevel formula syntax allowing to model, for instance, hierarchically nested data structures such as multiple observations per person in a longitudinal study.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Monotonic effects integrate into the framework of generalized linear regression and can be used within multilevel models.

“Moreover, monotonic effects nicely integrate into the framework of generalized linear regression and can even be used within multilevel models.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In brms, monotonic effects are fully integrated into the formula syntax, which builds on and extends standard R formula syntax as well as the multilevel formula syntax initially created for lme4.

“In brms, monotonic effects are fully integrated into the formula syntax, which builds on and extends standard R formula syntax as well as the multilevel formula syntax initially created for the lme4 package (Bates, Mächler, Bolker, & Walker, 2015).”

• cites Bates, Mächler, Bolker, & Walker, 2015Not checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

Monotonic effects are fully supported in the brms R package, which fits Bayesian regression models using Stan and provides an intuitive user interface based on widely known R formula syntax.

“Monotonic effects are fully supported in the brms R package, which fits Bayesian regression models using Stan and provides an intuitive user interface based on widely known R formula syntax.”

May combine two claims.

In the counterexample with varying simplex parameters, the effect of x is monotonic for group G, but non-monotonic for group H.

“Consider the situation shown in Figure 13, where quite clearly, the effect of x is monotonic for group G, but non-monotonic for group H.”

May combine two claims.

Under true linearity, LIN and CS models performed better than all other models, closely followed by MO, CS and LS.

“From Figure9we see that, under true linearity, LIN and CS models performed 492 better than all other models, closely followed by MO, CS and LS.”

May combine two claims.

All penalized/regularized approaches generally performed well in the simulations and differences between them were comparably small.

“It has to be noted that all penalized/regularized approaches generally performed well in our simulations and differences between them were comparably small (which we believe is a good sign for the validity of these methods in general).”

Probably not a substantial claim: may not pass S3 (load-bearing).

In the counterexample, b 0 = 0.

“Clearly, b 0 = 0.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In the counterexample, for group H, b 1 = 0.

“As the curve of group H starts at the origin, we have b 1 = 0.”

May combine two claims.

Working in a fully Bayesian framework increases the complexity of models in which monotonic effects can be incorporated and allows specifying prior distributions on the corresponding parameters.

“In contrast to existing approaches, we work in a fully Bayesian framework for model specification and estimation, which increases the complexity of models in which monotonic effects can be incorporated and also allows to specify prior distributions on the corresponding parameters.”

May combine two claims.

The Dirichlet prior with constant α=1 puts equal probability on all valid simplexes and centers ζ around a linear trend with large uncertainty.

“The Dirichlet prior with a constant α = 1 puts equal probability on all valid simplex and can thus be understood as the multivariate generalization of the uniform prior on simplexes.Since we have w i = 1/D, this prior centers ζ around a linear trend with large uncertainty and thus appears to be a good default prior in the absence of any problem specific information.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In the parameterization, b represents the size and sign of the effect similar to an ordinary regression coefficient.

“In the above parameterization, b represents the size and the sign of the effect similar to an ordinary regression coefficient.”

May combine two claims.

For the Dirichlet prior, higher values of α_i relative to the sum over α imply higher a priori values of ζ_i, and a higher sum over α implies higher certainty in each proportion w_i.

“As the a-priori expectation of ζ i is given by w i = E(ζ i ) = α i /α 0 , with α 0 = D i=1 α i , higher values of α i in comparison to the sum over α imply higher a priori values of ζ i .Moreover, the higher the sum over α, the higher the certainty in each of the proportions w i .”

• cites Balakrishnan, 2014Not checked yet

  Not checked in the free run.

May combine two claims.

The data must provide enough evidence for a non-zero effect to overcome a zero-centered prior; a stronger prior in favor of zero requires more evidence.

“That is, the data has to provide enough evidence for a non-zero effect in order to overcome the prior.The stronger the prior in favor of a zero-effect, the more evidence we need from the data in order to be convinced of a non-zero effect.”

Classified as probably not a substantial claim. May combine two claims.

The proposed parameterization is designed to fit naturally into generalized linear modeling frameworks and can be combined with other predictor terms, interactions, and multilevel structures.

“The proposed parameterization is designed to fit naturally into generalized linear modeling frameworks and their extentions.As such, it can be seamlessly combined with other types of predictor terms to model parameters of arbitrary response distributions, and may even be used within interactions or multilevel structures.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

The formulation generalizes to interaction terms as well as multilevel structures.

“This formulation generalizes to interaction terms as well as multilevel structures.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Any family of prior distributions typically applied to regression coefficients can be applied to b.

“Any family of prior distributions typically applied to regression coefficients can be applied on b, as well.”

May combine two claims.

Priors on monotonic effect parameters can incorporate subject matter knowledge and regularize predictions to make them robust against overfitting even without specific knowledge.

“The latter may not only be used to incorporate additional subject matter knowledge into the model that would otherwise remain unused, but also to regularize the model's predictions and make it robust against overfitting even in the absense of such specific knowledge.”

May combine two claims.

If the monotonic effect is used in a linear model, b can be interpreted as the expected average difference between two adjacent categories of x, while ζ_i describes the expected difference between categories i and i-1 as a proportion of the overall difference between lowest and highest categories.

“If the monotonic effect is used in a linear model, b can be interpreted as the expected average difference between two adjacent categories of x, while ζ i describes the expected difference between the categories i and i -1 in the form of a proportion of the overall difference between lowest and highest categories.”

• AssumptionRests on an assumption: Ordinal variables contain no information about the distance between adjacent categories.

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Using Stan, the authors developed a Bayesian estimation method for monotonic effects that allows incorporating prior information and checking the assumption of monotonicity.

“Using Stan, we developed a Bayesian estimation method for monotonic effects, which allows to incorporate prior information and to check the assumption of monotonicity.”

Classified as probably not a substantial claim.

The parameterization uses a scale parameter b representing the direction and size of the effect and a simplex parameter ζ modeling the normalized differences between categories.

“The parameterization of such monotonic effects is realized in terms of a scale parameter b representing the direction and size of the effect and a simplex parameter ζ modeling the normalized differences between categories.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Monotonic effects may be applied not only to ordinal predictors but also to other discrete variables for which a monotonic relationship is plausible.

“Monotonic effects may not only be applied to ordinal predictors, but also to other discrete variables for which a monotonic relationship is plausible.”

May combine two claims.

Ordinal predictors are often incorrectly treated as either nominal or metric, leading to under- or overestimation of contained information and potentially worse inference and predictions than methods specifically designed for ordinal predictors.

“They are often incorrectly treated as either nominal or metric, thus under-or overestimating the contained information.Such practices may lead to worse inference and predictions compared to methods which are specifically designed for this purpose.”

• cites Liddell & Kruschke, 2017Not checked yet

  Not checked in the free run.

May combine two claims.

The parameterization ensures that predictions increase or decrease monotonically, while changes between adjacent categories may vary across categories.

“This ensures that predictions increase or decrease monotonically, while changes between adjacent categories may vary across categories.”

• AssumptionRests on an assumption: Ordinal variables contain no information about the distance between adjacent categories.

Probably not a substantial claim: may not pass S3 (load-bearing).

brms provides an interface to fit Bayesian generalized (non-)linear (multilevel) regression models using Stan.

“The brms package (Bürkner, 2017;Bürkner, 2018) provides an interface to fit Bayesian generalized (non-)linear (multilevel) regression models using Stan (Carpenter et al., 2017;Stan Development Team, 2019), which is a C++ package for performing full Bayesian inference”

• cites Bürkner, 2017Not checked yet

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• cites Carpenter et al., 2017Not checked yet

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• cites Stan Development Team, 2019Not checked yet

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May combine two claims.

Both impairments in walking (d450) and moving around (d455) were strong predictors of phcs in the analysis of Gertheiss et al. (2011).

“Both of these variables were strong predictors of phcs in the analysis of Gertheiss et al. (2011).”

• cites Gertheiss et al., 2011Not checked yet

  Not checked in the free run.

May combine two claims.

The proposed parameterization not only ensures monotonicity but also naturally regularizes the model and its predictions even without the usage of strong prior information.

“The proposed parameterization not only ensures monotonicity but also naturally regularizes the model and its predictions even without the usage of strong prior information.”

May combine two claims.

Monotonic effects avoid the problems of under- or overstating information from ordinal predictors while still allowing for an intuitive interpretation of the estimated parameters.

“Monotonic effects avoid these problems but still allow for an intuitive interpretation of the estimated parameters.”

May combine two claims.

Elements of a simplex tend to be negatively correlated, sometimes rather strongly, thus making MCMC sampling more difficult.

“One potential problem in the Bayesian estimation of monotonic effects is that elements of a simplex tend to be negatively correlated, sometimes rather strongly, thus making MCMC sampling more difficult (Hoffman & Gelman, 2014).”

• cites Hoffman & Gelman, 2014Not checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S4 (based).

Using different simplex parameters for different terms of the same monotonic variable results in much greater modeling flexibility.

“This results is much greater modeling flexibility as explained in the former section.”

May combine two claims.

The spikes in the histograms for the N=1000 and D=50 interaction model indicate strong autocorrelation in the chains and thus convergence problems of the model.

“This indicates strong autocorrelation in the chains and thus convergence problems of the model (Talts et al., 2018).”

• cites Talts et al., 2018Not checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S4 (based).

By making an informed decision about the parameterization of interactions with monotonic effects, different kinds of monotonicity can be modeled depending on the research question and a-priori information available.

“By making an informed decision about the parameterization of interactions with monotonic effects, different kinds of monotonicity can be modeled depending on the research question and a-priori information available.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Monotonic effects share important aspects with existing methods for modeling ordinal predictors.

“Thus, monotonic effects share important aspects with existing methods for modeling ordinal predictors.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For a very high number of predictor categories, it may be easier to fit (monotonic) splines or similar models which require fewer parameters.

“For that many predictor categories, it may be easier to fit (monotonic) splines or similar models which require fewer parameters.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The present results provide no guarantee that monotonic effects will be well calibrated in cases not studied in the present paper.

“Of course, monotonic effects can be applied in a lot of other modelling settings and so the present results provide no guarantee that they will be well calibrated in cases not studied in the present paper.”

Probably not a substantial claim: may not pass S3 (load-bearing).

A similar idea is used in I-splines, whose basis functions represent integrals over the non-negative basis functions of another spline.

“A similar idea is used in I-splines (i.e., intergral splines) whose basis functions represent integrals over the non-negative basis functions of another spline (Ramsay, 1988).”

• cites Ramsay, 1988Not checked yet

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Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

For very few time points, using a monotonic effect may be a more reliable solution than modeling time as continuous, without strong assumptions outside of monotonicity.

“Of course, time may also be modeled as continuous, but for very few time points, using a monotonic effect may be a more reliable solution without strong assumptions outside of monotonicity.”

• AssumptionThe authors state this as an assumption: Ordinal predictors are treated as manifest variables, not as categorizations of a latent continuous variable.

Probably not a substantial claim: may not pass S3 (load-bearing).

For simple cases such as regression models with only a single monotonic effect and normally distributed errors, maximum likelihood estimators can be developed.

“For simple cases such as regression models with only a single monotonic effect and normally distributed errors, maximum likelihood estimators can be developed as well (Barlow et al., 1972;Robertson et al., 1988).”

• cites Barlow et al., 1972Not checked yet

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• cites Robertson et al., 1988Not checked yet

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May combine two claims.

If categorical coding is used for the interaction of two ordinal predictors, the number of regression coefficients increases quadratically with the number of predictor categories, which complicates interpretation.

“If we used any type of categorical coding (e.g., dummy coding) for the interaction of two ordinal predictors, the number of regression coefficients would increase quadratically with the number of predictor categories, which complicates interpretation.”

May combine two claims.

Under the stated assumptions, the linear predictor η(x) is monotonic in x with the sign of the effect determined by the sign of b.

“If we define b = K i=1 b i D i we see that η(x) is monotonic in x with the sign of the effect determined by the sign of b.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

For continuous data, the sum in the definition of monotonic effects becomes an integral and ζ becomes a non-negative function to be integrated over.

“In this case, the sum in the definition of monotonic effects becomes an integral and ζ a non-negative function to be integrated over.”

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).

The general idea of monotonic effects should generalize to continuous data.

“Lastly, we want to note that the general idea of monotonic effects should also generalize to continuous data.”

May combine two claims.

For all values x between 0 and D-1, the linear predictor η(x) is monotonically increasing if b ≥ 0 and monotonically decreasing if b ≤ 0.

“Since D > 0 and ζ x+1 > 0, the linear predictor η(x) is monotonically increasing if b ≥ 0 and monotonically decreasing if b ≤ 0.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Monotonic effects may be applied to other kinds of discrete variables besides strictly ordinal predictors.

“Although our primary focus was the use of monotonic effects for modeling strictly ordinal predictors, we want to point out that they may be applied to other kinds of discrete variables, as well.”

• AssumptionThe authors state this as an assumption: Ordinal predictors are treated as manifest variables, not as categorizations of a latent continuous variable.