Model-Based Recursive Partitioning for Subgroup Analyses

2.1 Model and algorithm

We started with a model m((Y,X),ϑ) that describes the conditional distribution of the primary endpoint Y (or certain characteristics of this distribution) as a function of the treatment arm and potentially further covariates (both contained in X) through parameters ϑ as defined in the study protocol. The parameter vector ϑ=(α,β,γ,σ)⊤ typically contains one or more intercept parameters α, one or more treatment effect parameters β, other model parameters γ, e.g. effects of covariates, and potential nuisance parameters σ, e.g. the error variance in a linear model. The estimator is defined as the minimizer of an objective function Ψ, which usually is the negative log-likelihood:

Estimating ϑ is equivalent to solving the score equation

where ψ is the score function, i.e. the gradient of the objective function Ψ with respect to ϑ. The model framework is more general than the log-likelihood framework because Ψ is not necessarily a negative log-likelihood function.

In the presence of patient subgroups that differ in their treatment effect β, an estimate βˆ obtained for all patients i=1,…,N in the study only reflects the mean treatment effect but ignores that the success or failure of a specific treatment might depend on additional characteristics of each individual patient. We describe patient subgroups as a partition {ℬb} (b=1,…,B) of all patients i=1,…,N. The subgroup-specific model parameters are then ϑ(b). These parameters can in general be seen as varying coefficients [8], however they may depend on several patient characteristics and are always step functions with a different level for each subgroup and not only a smoothly varying coefficient for one single predictive variable.

Since we are searching for predictive and prognostic factors, we are only interested in subgroups that differ in the intercept or the treatment effect or both as explained in Section 2.2. With ϑ(b)=(α(b),β(b),γ,σ)⊤ we assume that the effects of covariates and nuisance parameters are constant for all patients. The partition {ℬb} is defined by J partitioning variables Z=(Z1,…,ZJ)∈z; in other words, {ℬb} is a hypercube in the J-dimensional sample space z. These partitioning variables Z are the additional patient characteristics that potentially influence α(b) and β(b). If for example gender were a predictive factor in a given treatment-endpoint relationship, it would be a patient characteristic that is involved in forming the partitions. If the partition {ℬb} is known, the partitioned model parameters ϑ(b) could be estimated by minimising the segmented objective function:

where Z denotes the indicator function and (y,x)i,zi are the realisations of (Y,X) and Z for the i-th patient. This allows us to write the subgroup-specific intercept and treatment parameters as a function of the partitioning variables

Without any a priori knowledge about the partition {ℬb}, we want to estimate the functions α(z) and β(z) by means of model-based recursive partitioning. The main idea underlying this method is the ability to detect parameter instabilities, i.e. non-constant parameters in a parametric or semiparametric model, by looking at the score function. Because we are only interested in detecting non-constant intercepts α(z) and treatment effects β(z), we focus on the partial score functions ψα((Y,X),ϑ)=∂Ψ((Y,X),ϑ)/∂α and ψβ((Y,X), ϑ)=∂Ψ((Y,X), ϑ)/∂β. If the model parameters are in fact constant and do not depend on any of the partitioning variables Z, the partial score functions ψα((Y,X),ϑ) and ψβ((Y,X),ϑ) are independent of Z. Consequently, parameter instability corresponds to a correlation between either of the partial score functions and at least one of the partitioning variables Z1,…,ZJ. In order to formally detect deviations from independence between the partial score functions and the partitioning variables, model-based recursive partitioning utilises independence tests. The null hypotheses

for a given model m((Y,X),ϑˆ) state that the partial score functions with respect to α and β, respectively, are independent of the partitioning variable Zj (j=1,…,J). Hence, these null hypotheses correspond to an appropriate model fit regarding the intercept and treatment parameter. Because the partial score functions under the null hypotheses are at least asymptotically normal in many model families, asymptotic M-fluctuation tests with appropriate correction for multiplicity were introduced for model-based recursive partitioning by Zeileis and coworkers [9, 7]. Alternatively, permutation tests can be applied in situations where asymptotic normality of the partial score is not guaranteed [10] or in cases with small numbers of observations [11–13], which are common in medicine. Also in this case procedures for multiple testing are used to cope with a possibly large number of partitioning variables J.

If we can reject at least one of the 2×J null hypotheses for the global model m((Y,X),ϑˆ) at a pre-specified nominal level, model-based recursive partitioning selects the partitioning variable Zj∗ associated with the highest correlation to any of the partial score functions. This is usually done by means of the smallest p-value. The dependency structure between the partitioning variable Zj∗ and either one of the partial score functions is described by a simple cut-point model. Once we find an optimal cut-point Zj∗<μ using a suitable criterion [13, 7], we split the patients into two subgroups according to Zj∗<μ. For both subgroups, we estimate two separate models with parameters ϑˆ(1) and ϑˆ(2), respectively, obtain the corresponding partial score functions, and test the independence hypotheses. If we find deviations from independence, we in turn estimate a cut-point in the most highly associated partitioning variable, and split again. The procedure of testing independence of partial score functions and partitioning variables is repeated recursively until deviations from independence can no longer be detected.

Since model-based recursive partitioning is a tree method, in the following we use topic-specific vocabulary, such as nodes. The root node contains all patients and is the basis for the initial model, inner nodes represent splits and leaf nodes contain the patients of the different subgroups and specify the partition-specific models. The paths from root to leaf nodes define the subgroups.

2.2 Content interpretation

A clearer picture of the interpretation of subgroup-dependent model parameters and distribution of the partial scores under unstable parameters is best given by means of a partitioned linear model discussed in the following.

Here xA is a contrast that indicates whether a subject was treated with treatment A (active) but not C (control) in a two-armed trial and xstratum is a stratum with x=(xA,xstratum). The conditional distribution of the primary endpoints Y given treatment and stratum is normal

The segmented model we want to fit using model-based recursive partitioning reads

where γ is the effect of the stratum and the variance σ2 is a nuisance parameter. The objective function for a patient with observations (y,x) is the negative log-likelihood, when maximum likelihood estimation is used, or the error sum of squares, when ordinary least squares is used. Yet, both methods lead to the same scores

and thus to the same solution. Note that the partial score function with respect to the intercept is proportional to the least-square residuals and all further scores are proportional to the product of the residuals and the respective variable.

A partitioning variable can be predictive, prognostic, or both, and we have to consider the parameters in the model to understand the nature of a partitioning variable. Figure 1 shows examples for mean primary endpoints and the corresponding intercept α and treatment effect β. If α(z) varies over z, but β(z) is constant, then the components of z are prognostic because the mean primary endpoint varies but not the treatment effect (see first column of Figure 1). If β(z) varies over z and α(z) is constant, then the variables in z are predictive since it means that the mean primary endpoint in one treatment arm stays the same but the treatment effect changes over z (second column). If both parameters vary, then z is predictive (third column) or predictive and prognostic at the same time (last column). In the latter situation, the mean primary endpoint of the second subgroup changes over z and the intercept also changes.

It is also interesting to take a closer look at the partial scores. Figure 2a shows the partial scores with respect to intercept and treatment parameter that result from a linear model Y|X=x  n(α+βxA, σ2) plotted against a partitioning variable z1, which is predictive and prognostic. The data-generating process of this model was suggested by Loh et al. [14] and is defined as

with XA from ℬ(1,0.5) and Z1 from n(0,1). For the example, we used this process to draw a sample of 200 observations.

Figure 1:

Possible mean primary endpoint within subgroups resulting from a predictive, prognostic, or predictive and prognostic variable.

Figure 2:

Partial scores of different kinds of variables. The symbols represent the treatment arms C and A as indicated. (a) Partial scores of a predictive and prognostic variable (eq. (7)). (b) Partial scores of a predictive and prognostic variable (eq. (8)). (c) Partial scores of a prognostic variable (eq. (9)).

The partial scores with respect to the intercept ψα fluctuate randomly around zero over the whole range of z1. The partial scores with respect to the treatment parameter ψβ change. Hence, in this situation, model-based recursive partitioning would detect a deviation from independence between ψβ and z1 and implement a split at approximately z1<0. There is no chance of finding this cut-point by looking at the least-square residuals only, since a deviation of independence between ψα and z1 is hardly visible in the scatterplot in the left panel of Figure 2a. Figure 2b shows the partial scores obtained with a slightly modified data-generating process, where instead of 1(z1>0)⋅xA, one has 1(z1<0)⋅xA:

Here the procedure would split the partial score with respect to the intercept, although z1 is still prognostic and predictive at the same time.

If we focus on the prognostic variable z1 in the model

we see non-random patterns in both scores (see Figure 2c). Since the partial scores with respect to the treatment parameter are set to zero for treatment arm A, we would split on basis of the scores with respect to the intercept, just as a consequence of a higher power.

These three examples show that splitting in the partial score with respect to the intercept does not give any information about whether the partitioning variable is predictive or prognostic. It also does not make sense to choose to split only in the score with respect to the treatment parameter because one might miss important cut-points. In order to be able to say whether a partitioning variable is predictive or prognostic, it is not enough to know which partial scores are responsible for the split. It is necessary to consider the model parameters in the segmented model. If the treatment parameter β varies in the subgroups, then the chosen partitioning variables are predictive or both predictive and prognostic. If β is constant, the variables are only prognostic.

2.3 Relation to established procedures

Traditional approaches for subgroup identification are also based on a model for the primary endpoint, but the segmentation is implemented by means of varying coefficients. More precisely, the model includes interactions between treatment and the patient characteristics z in addition to the main effects

with α=α0,β=α1−α0,γprognosticT=γ0,zT and γpredictiveT=γ1,zT−γ0,zT. The model is known as the “classical approach” for subgroup analyses [15, 16]. Significant interaction terms γpredictive are in this case subject to the choice of relevant partitioning variables. However, patient subgroups can only be identified directly in this model for categorical variables zj since the model has no notion of optimal cut-off points. As the number of potential partitioning variables J might be large, the simultaneous estimation of all parameters in the model might be computationally burdensome and associated with a large variance. Regularisation procedures may be applied for selecting relevant interaction parameters that deviate considerably from zero.

RECPAM [17, 18] goes a step further and fits such models by trees. In every node, a likelihood-ratio test is computed that compares the segmented model

to the constant model

for every possible segment ℬk (k=1,…,K) induced by all possible cut-off points in zj, i.e. an exhaustive search is performed. The procedure is applied to all partitioning variables zj (j=1,…,J). The algorithm then chooses the variable and segmentation that comes along with the highest test statistic. The method is so far limited to linear models and Cox proportional hazards models, and parameter instabilities can only be detected in β but not in α.

A method that is similar in spirit to model-based recursive partitioning, but is limited to normal linear models, is the Gs method [14] based on the GUIDE algorithm [19, 20]. Instead of using partial scores with respect to intercept and treatment effect, Gs uses only the least-square residuals (that is, only the partial score with respect to the intercept). In contrast to model-based recursive partitioning, Gs looks at the dichotomised (at zero) residuals separately in the two treatment arms. The independency between positive/negative residual signs and each partitioning variable is tested using a chi-squared test separately for each treatment. If the partitioning variable is at least ordinal, it is dichotomised by splitting at the mean. The optimal split variable chosen is the one that induces the highest sum of chi-squared statistics. Looking at the left panels of Figures 2a and 2b, one can imagine that in these situations the procedure may successfully find the subgroups. However, in a less clear situation and where the optimal cut-point is not near the mean of z1, the method will have lower power or will not be able to find a split at all.

Another recently proposed tree algorithm is qualitative interaction trees (QUINT [21]). QUINT searches for instabilities in the treatment parameter β only, but the resulting partitions have to have different signs in the parameter. In other words, QUINT aims at finding subgroups in which the treatment effect is the reverse of that of the other subgroups. The current implementation of QUINT [22] is limited to continuous primary endpoints. It would be possible to enforce splits that are qualitatively different in model-based recursive partitioning. This could be achieved by incorporating a criterion that implements a split only if the treatment effects in the two new subgroups have different signs.

SIDES (subgroup identification based on differential effect search) [23] and SIDEScreen [24] aim at identifying subgroups of patients with high benefit from a novum compared to the standard treatment. Although the subgroups are linked to hypercubes in the sample space of Z, they are overlapping and can therefore not be represented as a tree structure. The methods are based on a cross-validated implementation of subgroups that were obtained on independent learning samples.

More general approaches blending recursive partitioning with traditional models (known as hybrid, model, or functional trees in machine learning Gama [25]), include M5 [26], GUIDE [19], CRUISE [27], LOTUS [28] and maximum likelihood trees [29]. (Bayesian approaches can be found in Chipman, George, and McCulloch [30]) and (Bernardo, Bayarri, Berger, Dawid, Heckerman, Smith, and West [31]). Except GUIDE, none of these methods has been studied in the specific context of subgroup analyses so far.

3.1 ALSFRS

The ALSFRS six months after treatment start (ALSFRS6) defined the functional endpoint. The sum-score is positive, and the model needs to adjust for the baseline ALSFRS obtained at treatment start (ALSFRS0). We used a normal GLM with log-link and offset log(ALSFRS0) such that the model

describes the expected relative change in the ALSFRS over the first six months under treatment. The treatment (Riluzole/no Riluzole) is indicated by xR. The model was fitted by maximum likelihood.

The time between disease onset and start of treatment, the forced vital capacity (FVC), and the phosphorus balance are the three partitioning variables selected for the tree given in Figure 3. The FVC value gives the volume of air in liters that can forcibly be blown out after full inspiration to the lung. A normal phosphorus balance is between 1 and 1.5 mmol/L. The tree indicates a negative treatment effect of Riluzole for patients with fewer days between disease onset and start of treatment that have a higher FVC value (node 4). Therefore, the FVC value is predictive in the group of patients with less than 468 days between disease onset and treatment start. Patients with more days between disease onset and treatment start do not seem to have a treatment effect. The fact that the time since onset plays an important role is not surprising since it is a surrogate for the speed of disease progression [38]. Patients with a slow progression were seldom included early in one of the studies. Hence a long time between onset and start of treatment usually stands for a slow progression.

Figure 3:

Results of application of model-based recursive partitioning with a Gaussian GLM with log link and offset on the data from the PRO-ACT database with the ALSFRS score as primary endpoint variable. Inner nodes give the split variable selected and the associated permutation test based p-value for the split. Terminal nodes give the model coefficients including standard confidence intervals.

3.2 ALSFRS items

The model for the ALSFRS sum-score assumes that the effect of Riluzole is the same for the ten items that define the score. In a more fine-grained analysis, we decomposed the score into its ten items (each ranging between zero and four) and modelled each item by means of a proportional odds model. For one of the ten items assessed at six months, e.g. Y6, the model reads

where r=0,…,4 is one of the five possible values of Y. The intercept parameters are now α=(α0,…,α3) and the partial score function ψα is now four dimensional.

As in the previous example, we needed to adjust for the baseline value Y0, i.e. the value of the ALSFRS item read at the beginning of treatment. This adjustment was implemented by computing separate models; one each for the observations with a start value k, which allows a baseline-specific intercept and treatment effect :

Therefore, we had a total of five different treatment parameters and 20 different intercepts for each of the ten different items. Model-based recursive partitioning was used to assess the parameter instability of all 250 parameters simultaneously. Note that some of these parameters could not be estimated owing to too small of sample sizes; these were simply discarded.

The implementation of the non-standard model in the theoretical and computational framework of model-based recursive partitioning was straightforward. For every node, we computed the five separate models for the respective baseline values for each of the ten items and extracted the partial scores. A stratified permutation test using the baseline values as independent blocks was used to assess parameter instability. The same procedure was applied for cut-off selection.

The resulting tree (on top of Table 1) contains splits in time between disease onset and treatment start and in the FVC value. The tree is in good agreement with the tree based on the ALSFRS (Figure 3). The third split variable is the lymphocyte percentage. Normal lymphocyte concentrations range from 16 to 33%. Table 1 shows the coefficient values of the models in the terminal nodes for every item and every starting value of the given item. Empty fields indicate that it was not possible to compute the model. Obviously, there were not enough observations in models with zero as starting value for any items in any nodes. The colours in the table indicate whether the effect of Riluzole was positive (blue), negative (pink) or zero (grey). The colours were assigned on the basis of confidence intervals of the coefficient in the given model. Riluzole had a positive effect on patients in the partition of terminal node 3 who had a starting value of 4 in item 1 (speech), 3 (swallowing) or 9 (climbing stairs) and on patients in the partition of terminal node 7 that had a starting value of 3 in item 5 (cutting food and handling utensils). Patients in node 4 who had a starting value of 3 in item 6 (dressing and hygiene) had a negative effect of Riluzole. Riluzole had no effect on patients in the partition of node 6 which are the patients with more than 584 days between disease onset and treatment start who have a lymphocyte concentration under 21.5%.

3.3 Survival time

We used both a Weibull model and a Cox model to identify subgroups with differing effects of Riluzole on the survival endpoint. The application of the model-based recursive partitioning framework in the Weibull model is straightforward and was introduced by Zeileis et al. [7]. Since the Cox model is a semiparametric model, where the intercept is a function of time, treated as a nuisance parameter omitted in the partial likelihood, there is no direct way of obtaining ψα. Because, conceptually, deviance residuals are always defined as the derivative of the log-likelihood with respect to the intercept, we applied martingale residuals as ψα. Also worth noting is that both models assume proportional hazards. For the segmented model, proportional hazards are only assumed within each partition. This has to be kept in mind when interpreting the treatment effect in different nodes: Parameters with different signs are clearly linked to opposing treatment effects, but when the parameters only differ in size, it is hard to say whether it is because the groups differ in treatment effect or because they differ in the hazard function.

3.3.1 Weibull model

The Weibull model is a transformation model of the form

(16)P(Y≤y|X=x)=Flog(y)−α1−βxRα2,

where F is the cumulative distribution function of the Gompertz distribution. Weibull models are fitted via maximum-likelihood estimation, and therefore the objective function in this case is the negative log-likelihood and the score function has one column per parameter, i.e. intercept α1, slope parameter β and scale parameter α2. In the Weibull model, we take the usual intercept as well as the scale parameter as “intercept”-parameter α=(α1,α2)T because they define the shape of the baseline hazard and hence in some respect take the role of an intercept. Splitting in the intercept or scale parameter score suggests non-proportional hazards.

Figure 4 shows that the patient’s age and again the time between onset and treatment start play a role in the partitioning. Older patients (>55.7 years) for whom the time between onset and treatment was longer than 757 days and very young patients did not seem to benefit at all from the treatment. In the remaining two groups, life expectancy seemed to be prolonged for patients treated with Riluzole.

Figure 4:

Results of application of model-based recursive partitioning with a Weibull model and data from the PRO-ACT database with survival time as primary endpoint variable. Inner nodes give the split variable selected and the associated permutation test based p-value for the split. Terminal nodes give the model coefficients, including standard confidence intervals and the survival curves in the two groups of treatment. Rugs indicate event times.

3.3.2 Cox model

The use of the Cox model in model-based recursive partitioning is a rather special case, since the baseline hazard in the Cox model is treated as an infinite-dimensional nuisance parameter and estimation is performed by minimisation of the negative partial log-likelihood. The Cox proportional hazards model is given by

(17)λ(y|x)=λ0(y)exp(βxR),

where λ is the hazard function and λ0 the baseline hazard function. The partial score function ψα (or better, ψλ0) cannot be easily derived. As surrogate score function, we propose using the martingale residuals as a score for the baseline hazard, which takes the role of an intercept in the Cox model, and the score residuals for the treatment parameters β. The score residuals are an intuitive choice because they are the first derivative of the partial log-likelihood with respect to the parameters. We used martingale residuals to check whether there is a general difference in the endpoint for different patients, which in parametric models is usually shown by the score with respect to the intercept. Instability in the martingale residuals indicates a violation of the proportional hazards assumption. Since the martingale residuals are not normally distributed, the application of permutation tests is more appropriate than the use of M-fluctuation tests.

Age and the time between disease onset and start of Riluzole treatment form the segments in this example. The tree in this example has almost the same splits as the tree in the previous example. Also estimates support the results of the Weibull example. Again, we did not see much difference between treated and untreated very young patients. For all other groups, Riluzole treatment led to a slight tendency for a lower risk of death (Figure 5).

Figure 5:

Results of application of model-based recursive partitioning with a Cox model and data from the PRO-ACT database with the survival time as primary endpoint variable. Inner nodes give the split variable selected and the associated permutation test based p-value for the split. Terminal nodes give the model coefficients including confidence intervals and the survival curves in the two groups of treatment. Rugs indicate event times.