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Modelling data from Multiple Sclerosis longitudinal studies is a challenging topic since the phenotype of interest is typically ordinal; time intervals between two consecutive measurements are nonconstant and they can vary among individuals. Due to these unobservable sources of heterogeneity statistical models for analysis of Multiple Sclerosis severity evolve as a difficult feature. A few proposals have been provided in the biostatistical literature (Heijtan (1991); Albert, (1994)) to address the issue of investigating Multiple Sclerosis course. In this paper Bayesian P-Splines (Brezger and Lang, (2006); Fahrmeir and Lang (2001)) are indicated as an appropriate tool since they account for nonlinear smooth effects of covariates on the change in Multiple Sclerosis disability. By means of Bayesian P-Spline model we investigate both the randomness affecting Multiple Sclerosis data as well as the ordinal nature of the response variable.

Multiple Sclerosis (MS) is a progressive neurological disorder classified among complex diseases. Investigating MS causes and potential triggers is a difficult task since the clinical manifestations and course vary considerably. Therefore longitudinal studies, both clinical trial as well as natural history studies, become crucial to assess the disease evolution over time. How to measure MS phenotype has been a major problem [

This paper proposes a new statistical perspective to model both the longitudinal nature of the data as well as the individual heterogeneity. The novelty is due both to the statistical framework chosen to model these data and to the random variable used to describe MS evolution (see Section

In Section

Multiple Sclerosis (MS) is a chronic progressive disease that affects the brain and spinal cord (central nervous system). This disease is classified among the multifactorial genetic diseases (or complex diseases); the causes and potential triggers of MS are thought to be based both on genetic predisposition and on biological and environmental patients characteristics. The variability of the MS symptoms and the potentially long duration of the latent period of the disease from onset make MS extremely difficult to measure. As mentioned above, the disease markers used in MS literature to measure disease activity are typically related either to impairments of functional status or to dissemination of lesions. This latter, which is not the object of our analysis, are becoming crucial to measure early disease activity. In this class of measures are included magnetic resonance imaging (MRI), cerebrospinal fluid (CSF), and visual lesions. In this paper we consider as outcome variable the degree of functional disability usually measured by the so-called

The Kurtzke Expanded Disability Status Scale (EDSS) is a method of measuring disability in multiple sclerosis. This scale quantifies disability in eight Functional Systems (FSs) and allows neurologists to assign a Functional System Score (FSS) in each of these. (the Functional Systems are: pyramidal, cerebellar, brainstem, sensory, bowel and bladder, visual, cerebral, and other.) EDSS steps 1.0 to 4.5 refer to people with MS who are fully ambulatory. EDSS steps 5.0 to 9.5 are defined by the impairment to ambulation. The value 10 represents death due to MS. The EDSS has many shortcomings such as its nonlinearity and its discontinuity. Common ways to overcome data related problems mentioned above is to put MS data in survival analysis frameworks, modelling the time to a certain EDSS level (4.0 or 6.0), or time to worsening, defined by an increase of 1 point in EDSS. Dynamic approaches have been developed [

To investigate MS evolution we introduce a new variable “EDSS change” that we model over time. This is the ratio between two subsequent EDSS measurements. In addition, since higher EDSS values (such as

In the paper we model the “

In the statistical literature (Pinheiro and Bates (2000), [

A random slope model is needed to allow the intensity of evolution to vary among subjects since the coefficient of one or more explanatory variable varies randomly across higher-level units. Thus, in a longitudinal setting, the evolution profiles for each subject have specific intercepts and slopes (see Figure

The random slopes model.

The GAMMs are here adopted to investigate through mixed effects modelling the MS data structure within a nonparametric Bayesian framework. This is done by modelling the dependence between the response variable

Expression (

In this paper we deal with a particular class of smooth functions out of the big set of Splines, the P-Splines [

The Fisher-Scoring Algorithm is used to conduct the maximization on the Penalized Likelihood with respect to the unknown regression coefficients. The smoothness of the function is regulated by the smoothing parameters

In the next subsections we use a Bayesian version of P-Splines in the case of a Gaussian as well as ordinal responses [

A Bayesian version for P-Splines has the advantage of allowing for simultaneous estimation of smooth functions and smoothing parameters. It can easily be extended to complex formulations like mixed effect models. This is a flexible way to use P-Spline since no constant smoothing parameters are assumed and they are locally adaptive. This can be very useful in MS context, where the smooth function may change curvature. Inference is fully Bayesian using MCMC simulation technique to draw sample from the posterior. In the Bayesian approach the unknown P-splines parameters

Note that priors in (

This concept is intuitively illustrated in Figure

Prior distribution for

In this case the joint distribution of the prior is given by

To relate the full Bayesian setting to these results a set of coherent convergence assumptions is needed on

Suppose that repeated measurements have been taken on

In a Bayesian context, in addition to the above discussed variance component for the random walk regulating the smoothness of the P-Splines, prior distributions are assigned to all the parameters in expression (

We remark that in this framework two assumptions are required: (i) conditional independence of

The threshold model is based on the idea that the observable variable

The additive mixed effect model with an ordinal response does not differ from (

Drawing out a truncated normal distribution evolves as numerically difficult and almost not solvable together with random effects. Thus, reparametrization strategies are used to overcome the numerical problems [

In this section we show how Bayesian P-Splines with mixed effects constitute an efficient method to model the heterogeneity in MS clinical data and to state the role of covariates in determining the severity of the disease. The covariates included are chosen among the most important prognostic factors in MS (Table

P-Splines random intercept model:

with a Gaussian response,

with an ordinal response.

P-Splines random slope model:

with a Gaussian response,

with an ordinal response.

Results from the latter modelling are here omitted since they do not really add additional information for interpreting the covariates role.

The influence of the covariates on

Let the response variable be normally distributed. The prior distribution functions for the parameters are those chosen according to the previous section.

The model can be specified by the formula

The prior distributions were chosen in the usual way, that is, diffuse priors for the fixed effects

Autocorrelations of fixed effects and parameters for time (a,b) and mixing behavior of the estimate for gender and one time parameter (c,d).

We notice in Table

Estimates of variance components.

source of variation | Mean | Std. dev. | 10% qu. | 50% qu. | 90% qu. |
---|---|---|---|---|---|

Within patients | 0.593373 | 0.009933 | 0.580887 | 0.593407 | 0.606655 |

Between patients | 0.536405 | 0.000929 | 0.498304 | 0.535726 | 0.573733 |

Table

Estimates of constant effects.

Variable | Mean | Std. var. | 10% qu. | 50% qu. | 90% qu. |
---|---|---|---|---|---|

gender | |||||

0.323480 | 0.104238 | 0.185648 | 0.321651 | 0.460414 | |

0.339837 | 0.098472 | 0.210783 | 0.339599 | 0.469657 |

Posterior means are plotted in Figures

Overall, it has to be noted that not all included effects influence the response variable significantly. This is also affected by the modelling approches. The plot of the population residuals (Figure

Histograms and normal-quantile plots for population residuals (a,b) and individual residuals (c,d).

Plot of observed against fitted values (dashed line: linear regression line of the scatter plot; full line: the diagonal).

P-Spline posteriors for random intercept models with Gaussian response. Posterior means and confindence interval are plotted.

P-Spline for time (in weeks)

P-Spline for age

P-Spline for EDSS

P-Spline for duration

Let now include in the P-Splines analysis the ordinal nature of the variable

In Section

As mentioned in the Introduction, to focus on the severity of the disease change values of the variable “

In accordance with the Gaussian response P-Splines model in (

Thresholds for ordinal response variable are described in Table

Thresholds | Changeord | |
---|---|---|

Big decrease | ||

Small decrease | ||

Stable | ||

Small increase | ||

Big increase |

The ordinal mixed effect model results are obtained by a combination of Bayesian and classical estimation procedures. First, Bayesian estimates for the fixed effects are derived. These estimates constitute the basis for the marginal likelihood estimation of the random effect, as implemented by the software MIXOR (

Sampling plots of threshold parameters.

Bayesian estimates of the fixed effects provide information to reduce the number of parameters and to construct an appropriate ordinal regression model with smooth functions chosen as polynomial. Let now present the final estimation results obtained by this mixing two-step procedure.

By this mixing two-step procedure, the estimated threshold parameters are given in Table

Estimates of threshold parameters.

Threshold | Estimator | Std. error | z-value | |
---|---|---|---|---|

0 | — | — | — | |

1.46894 | 5.66944 | 44.35629 | <.0001 | |

4.59204 | 0.03922 | 117.09799 | <.0001 | |

5.66944 | 0.04102 | 138.22608 | <.0001 |

The fixed effect estimates are reported in Table

Estimates of constant effects.

Estimate | Std. error | z-value | ||
---|---|---|---|---|

gender | ||||

0.54872 | 0.17777 | 3.08677 | .00202 | |

0.62386 | 0.12768 | 4.88632 | .00000 |

In Figure

Estimation results for Gaussian model with no random effect (left) and with random effect (right).

Bayesian P-Spline for time

Regression Spline for time

Bayesian P-Spline for age

Regression Spline for age

Bayesian P-Spline for edss

Regression Spline for edss

Bayesian P-Spline for duration

Regression Spline for duration

Preserving the ordinal nature of the EDSS weighted change did not provide evidence of a change in the interpretation of the estimated parameters when compared to the Gaussian model. Results (Figure

As in the Gaussian model, the model fit is analyzed by comparing the fitted values against the observed values. Table

Crosstab of observed and fitted response.

Fitted category | ||||||||
---|---|---|---|---|---|---|---|---|

Total | ||||||||

Observed category | 61 | 47 | 0 | 0 | 118 | |||

16 | 556 | 4 | 0 | 696 | ||||

85 | 128 | 278 | 6 | 5181 | ||||

2 | 15 | 704 | 17 | 1399 | ||||

0 | 6 | 154 | 453 | 874 | ||||

Total | 113 | 330 | 6145 | 1396 | 284 | 8268 |

The fitted values plotted against observed values revealed a systematic bias in both random intercept models. The analysis of residuals also suggested that additional random components should be included in the analysis. We next investigate a random slopes model as last step of our modelling.

Heterogeneity in individual MS progression is observed as regarding both the magnitude and the speed. The disability may rise fast in some patients in the beginning and then stabilize; whereas for other patients it rises steadily but slows thereafter. The random intercept models proposed before may be debatable for fitting repeated measures of weighted change in EDSS, since they underestimate the change for patients, whose disability greatly decreased or increased within the time frame of a clinical study. This could cause the bias in the fitted values seen previously within the random intercept models. Of course, the introduction of a random slope alone is not sufficient, as it assumes a constant slope over the whole range of the time frame. By adding a quadratic random effect, the curvature in the progression of disability can be reflected. The splines for the time effect in the previous model (Figures

In the random slopes model the MS patients are considered to differ from the average trend of the populations as regarding both the initial disability level (random intercept) and the intensity of the MS clinical progression (random slopes). The proposed model is therefore given by

With the introduction of a random slope component we notice that the variance components explained by the random intercept and slope shown in Table

Estimates of variance components.

Source of variation | Mean | Std. dev. | 10% qu. | 50% qu. | 90% qu. |
---|---|---|---|---|---|

Scale | 0.360485 | 0.006067 | 0.352562 | 0.360414 | 0.368427 |

Intercept | 0.024117 | 0.009096 | 0.012335 | 0.024135 | 0.036359 |

Linear slope | 0.000449 | 0.000404 | 0.000448 | 0.000495 | |

Quadr. slope |

Histogram of random slope estimates.

Histogram of quadratic random slope estimates.

To ensure comparability of the random intercept and random slopes model, calculation in BayesX was performed with the same number of iterations, that is, a burn-in of 20 000 and a step width of 500. The convergence and mixing behavior were comparable to the ones obtained in the random intercepts model (Figure

Let next present the results of this modelling. In Table

The estimation of the fixed effect is provided in Table

Estimates of constant effects.

Variable | Mean | Std. var. | 10% qu. | 50% qu. | 90% qu. |
---|---|---|---|---|---|

Intercept | |||||

Time | |||||

Gender |

Mean of random slope estimates, stratisfied for courses.

Course | Mean of random linear slope | Mean of random quadratic slope |
---|---|---|

pp or pr | ||

sp ( | ||

rr (reference category) |

Description of the covariates included in the analysis of SLCMSR data set (

Changew | Weigthed change in EDSS from first observation |

t | Time in weeks from first observation |

Edss | EDSS at first observation |

Age | Age at disease onset |

Dur | Duration in months from onset to first observation |

Gender | |

Course | |

Reference category is relapsing remitting |

Categorization of EDSS change.

weighted change | ordinal change | label | number of observations |
---|---|---|---|

big decrease | 126 | ||

small decrease | 721 | ||

stable | 5438 | ||

small increase | 1440 | ||

big increase | 893 |

P-Splines curves plotted for the metric variables “age at onset,” “baseline EDSS,” and “duration” are here omitted since it did not change substantially the information obtained in the previous analyses, except for the credible intervals that resulted noticeably narrower than before.

Finally, by looking at the plot of fitted against observed values (Figure

Plot of observed against fitted residuals in a random slope model.

Another question was whether it is justified to use the change in EDSS as a metric outcome variable. However it is still debatable whether a 1 point change, although weighted, does really have the same meaning over the whole EDSS range. Results on ordinal modelling are here omitted. Indeed, combining levels of the outcome variable to 5 ordered categories not only accounts for the ordinal structure in the response, but also ensures comparability of the responses. However, P-spline results are very similar and do not justify the use of such computationally demanding and time consuming procedure. Reducing the number of ordered categories to three could be an alternative. In this case, the levels of the response are reparametrized to obtain stable estimates and then, analysis can be carried out in BayesX.

In the biostatistic literature a few attempts of statistical modelling for investigating MS course are provided Heijtan (1991). In this paper the focus of the interest lies on modelling the unobserved heterogeneity in MS longitudinal data for the better understanding of the impact of prognostic factors on MS severity. A nonparametric approach is suggested to avoid restrictive assumptions about the analytical form of the relation between prognostic factors and outcome of interest. Furthermore, the introduction of random as well as fixed effects of the covariates addressed the issue of including both observed and unobserved heterogeneity. Hence, generalized additive mixed models (GAMMs) have been presented as a natural statistical tool to investigate nonparametrically, by means of Splines, the role of MS prognostic factors.

We have been mainly addressing two fundamental features.

Most of the statistical modelling in MS consider EDSS as a metric variable, regardless of the ordinal nature of this measure. Does this assumption affect the estimation of the effect of the prognostic factors?

Unobserved sources of heterogeneity affect individual MS development. Does this source of heterogeneity create difference among patients as regarding how they enter the study or also how large and fast the progression is?

An answer to the first question is provided by comparing results from Bayesian P-Splines models performed with a Gaussian as well as an ordinal response. A first conclusion is that the numerically demanding and time consuming estimation of an ordinal mixed effect model is not justified by a real gain in the results. Actually, the interpretation of the role of prognostic factors did not change dramatically. Thus, a Gaussian model is suggested. Overall, the “baseline EDSS” appears to have a strong influence on the weighted change in EDSS. Patients enrolled in a study with an EDSS lower than

To address the second question the variance components in the random slope model and in the random intercept model are compared. The introduction of a random slope leads to a better estimate of the “within-patients” variability which is much higher than the “between-patients” variability; whereas this was not the case in the random intercept model. This implies that accounting for the variability in the progression of the individual disability allows for a much better classification of the patients. Furthermore, a comparison between the estimates of the fixed effect of time in random slopes model (Table

In conclusion Bayesian approach to random slope models, based on MCMC algorithms, emerged as extremely flexible in the context of MS data as presented here. In particular BayesX is implemented to allow for smooth P-Splines for metric covariates. This is a noteworthy advantage with respect to common techniques based on simple linear models or on other strict assumptions on the functional form since nonlinear effects like the influence of the EDSS at first observation could not have been detected.

A fully Bayesian method for P-splines has been used according to Brezger (2000). This emerges as extremely flexible in situations where large data sets are used and a moderate number of smoothing parameters are to be estimated. Thus, Bayesian P-splines are recommended as a suitable and flexible tool in addressing complex data like MS frameworks, where the form of the smoothing function

See Tables

This manuscript was based on a subsample of data sets and knowledge provided by the SLCMSR; however, it reflects only the opinions or views of the authors. The authors have sole responsibility for their work. they would like to thank the referees for the extremely accurate revision of this paper and their contribution.