Patterns of associations of clinical features in neurofibromatosis 1 (NF1)

Abstract

Neurofibromatosis 1 (NF1) is a common, fully penetrant autosomal dominant disease. The clinical course is generally progressive but highly variable, and the pathogenesis is poorly understood. We studied statistical associations among 13 of the most common or important clinical features in data from four separate sets of NF1 patients: a "developmental sample" of 1,413 probands from the NNFF International Database, an independent "validation sample" of 1,384 probands from the same database, 511 affected relatives of these probands, and 441 patients from a population-based registry in northwest England. We developed logistic regressive models for each of the 13 features using the developmental sample and attempted to validate these models in the other three samples. Age and gender were included as covariates in all models. Models were successfully developed and validated for ten of the 13 features analysed. The results are consistent with grouping nine of the clinical features into three sets: (1) café-au-lait spots, intertriginous freckling and Lisch nodules; (2) cutaneous, subcutaneous and plexiform neurofibromas; (3) macrocephaly, optic glioma and other neoplasms. In addition, three-way interactions among café-au-lait spots, intertriginous freckling and subcutaneous neurofibromas indicate that the first two groups are not independent. Our studies show that some individuals with NF1 are more likely than others to develop certain clinical features of the disease. Some NF1 features appear to share pathogenic mechanisms that are not common to all features.

Appendix

The first step in developing the model for each of the NF1 clinical features was to treat the selected feature as the response variable (Y) as a function of age in a logistic regression equation:

$$\log \left( {{{p\left( {1\left| x \right.} \right)} \over {1 - p\left( {1\left| x \right.} \right)}}} \right) = \alpha + \beta _1 AGE,$$

where p(1| x) is Pr(Y=1| covariates x). Maximum likelihood techniques were used to generate parameter estimates (SAS Institute 1996). Linearity in the logit was examined in each model, and age was transformed when necessary to meet the requirement of linearity in the logit.

$$\log \left( {{{p\left( {1\left| x \right.} \right)} \over {1 - p\left( {1\left| x \right.} \right)}}} \right) = \alpha + \beta _1 AGETRF$$

where

$$AGETRF = \exp \left( { - c \times AGE} \right).$$

At AGE = 0, the value of this function is α + β₁. For negative values of β₁, the value of AGETRF approaches α as AGE gets larger. This function approximates the frequency-by-age curves of the NF1 features considered in this study (DeBella et al. 2000). It was necessary to use this transformation of AGE to maintain linearity of the logit for most outcome variables in this study.

At the second stage, a series of bivariate analyses was performed using the equation

$$\log \left( {{{p\left( {1\left| x \right.} \right)} \over {1 - p\left( {1\left| x \right.} \right)}}} \right) = \alpha + \beta _1 AGETRF + \beta _2 x,$$

where each of the 13 features was set in turn as the response variable (Y), and AGETRF and one of the 12 remaining features ( x) were used as explanatory variables to screen for potential main effects. Variables with parameters (β values) with P <0.2 were included as explanatory variables (x _i values) in multivariate analyses at the third stage. AGETRF and gender were included as covariates in all of these third stage multivariate models:

$$\log \left( {{{p\left( {1\left| x \right.} \right)} \over {1 - p\left( {1\left| x \right.} \right)}}} \right) = \alpha + \beta _1 AGETRF + \beta _2 MALE + \beta _3 x_3 + \beta _4 x_4 + \beta _5 x_5 ...$$

Following maximum likelihood estimation of the parameters in the multivariate model, the importance of each explanatory variable was reassessed at the fourth stage. Explanatory variables with parameters greater than zero with P <0.2 were used to refit the model, and interaction terms (δ values) among the explanatory variables were considered by forward selection. For example,

$$\log \left( {{{p\left( {1\left| x \right.} \right)} \over {1 - p\left( {1\left| x \right.} \right)}}} \right) = \alpha + \beta _1 AGETRF + \beta _2 MALE + \beta _3 x_3 + \beta _4 x_4 + \delta _1 x_3 x_4 $$

Interpretation

The strength of interaction between the response variable (Y) and an explanatory variable ( x ₁ ) in a univariate model is measured by β ₁ . Subjects with variable x ₁ coded as "present" are exp( β ₁ ) times more likely also to have feature Y than are subjects with feature x ₁ absent. The strength of interaction between Y and explanatory variables ( x ₁ and x ₂ ) in a bivariate model is measured by β ₁ , β ₂ , and δ ₁ . Subjects with variables x ₁ and x ₂ present are exp( β ₁ + β ₂ + δ ₁ x₁x2) times more likely also to have the response variable.