Generalized Linear Models with Restricted Cubic Splines

In medical research new medicine and drug control plays vital role according to age and their body immune system. A medicine have different association with 70 year old and 45 years old person with diabetes and blood pressure patients. For example, if a model suggests that an increase in fasting plasma glucose (FPG) leads to an increase in HbA1c, this increase is the same if FPG increase from 120 to 165 mg/dL or if FPG change from 180 to 220 mg/dL. Assumption of linearity would be true often than the assumption of dichotomising data. Many scenarios this assumption would not be true. In longitudinal randomized clinical trials collection of efficacy or safety data often occur off-schedule beyond protocol allowed visit windows, considering this data at scheduled visits introduce potential bias due to clinical observations being carried forward or backward to the closest planned visits. In this situation it might be advantageous to treat time as continuous (actual time from baseline) rather than category variable in analysis of repeated measures mixed modelling. In these scenarios alternative modelling strategy would assume and explore non-linear continuous associations. There are many regression spline models are available to explore, but here we focus use of natural cubic splines. Loic Desquilbet and Fralcoismariotti [1] in this paper non-linear dose response association are widely criticized and restricted cubic spline functions are powerful tools to characterize a dosce response association between a Continues exposer and an outcome.  SAS software is used for fitting Linear, logistic and cox model as wall as linear and logistic generalized estimating equations and statistical tests for overall non-linear associations.   Hansnyquist [2] Penalty function approach as iterate procedure for obtaining the estimates of generalize linear model under linear restrictions on parameters by using likelihood ratio test, Wald test and Lagrange’s Multiple tests are considered as alternatives for testing hypothesis about linear restrictions on the parameters. Mary C. Meyar Et.al [3] explains about non-parametrical modeling using regression splines with Bayesian Approach to generalized partial linear regression model where Knots may be modeled as fixed or free the R code to implement the methods is described. Arisperoglou Et.al [4], in their paper the discussed about popular splines like cubic splines, B-splines, Penalized splines, natural cubic and cardinal splines and these splines where fitted and tested using R. Laralusa and CRT Ahlin[5], in their paper periodic restricted cubical splines (RCS ) and cubical splines are fitted using R packages for different knots for 500 units of data and obtained simulation  results, model accuracy tested using AKAIKE information criteria. Ulrich Halekoh Et.al [6], explains about generalized estimates fact for R and its applications by using cluster binary data was listed. Chin-schang Li [7], explains about a penalized log-likelihood ratio test statistic is constructed for a null hypothesis of the non-parametric components of a semi parametric generalized linear model component estimates using cubic B splines. The knots for this splines is fixed and its limiting its null distribution is the distribution of a linear combination of independent chi-square random variables each with 1 degree of freedom. A real life data is used for practical use of problem.

MEHODOLOGY:

Spline Regression Modelling

In general cubic splines are parabola curves fitted to data

C(Y|X₁) = β₀ + β₁X₁ + β₂ X₁²

Piece wise regression in basic form is linear function. Data is divided into parts, the intersection part is called knots. Generally, piece wise regression is as follows

f(x) = β₀ + β₁x + β₂(x − a)₊ + β₃ (x − b)₊ + β₄ (x − c)₊

where (u)₊ = u, u > 0,

                  = 0, u ≤ 0.

According to knots is increasing the function f(x) is as follows

f(x) = β₀ + β₁x, x ≤ a

        = β₀ + β₁x + β₂(x − a),  a< x≤ b

       = β₀ + β₁x + β₂(x− a) + β₃(x− b), b< x≤ c

       = β₀ + β₁x + β₂(x − a) +β₃(x− b) + β₄(x − c), c < x

A linear spline function with knots at a=1, b=3, c=5

Cubic Splines (or) cubic piece wise regression:

In this situation cubic polynomial regression is fitted for each and every part of data. If data is divided into three knots is as follows

f(X) = β₀ + β₁X + β₂X² + β₃X³ + β₄(X − a)³₊ + β₅(X − b)³₊ + β₆(X − c)³₊

       = Xβ

with constructed variables: X₁ = X, X₂ = X₂, X₃ = X₃, X₄ = (X − a)³₊, X₅ = (X − b)³₊, X₆ = (X − c)³₊

Restricted Cubic Splines:

The restricted spline function with k knots t₁,...,t_k is given by

f(X) = β₀ + β₁X₁ + β₂X₂ + ... + β_k−1X_k−1,

where X₁ = X and for s = 1,...,m− 2, X_j+1 = (X – t_s)³₊ − (X – t_m−1)³₊  (t_m – t_s) / (t_m – t_m−1) + (X – t_m)³₊ (t_m−1 – t_s )/(t_m – t_m−1)

It can be shown that X_j is linear in X for X≥t_m.  For numerical behaviour and to put all basis functions for X on the same scale, the above terms divide by τ = (t_k − t₁)²

Once β₀,...,β_k−1 are estimated, the restricted cubic spline can be restated in the form

f(X) = β₀ + β₁X + β₂(X − t₁)³₊ + β₃(X − t₂)³₊ + ... + βm₊₁(X – t_m)³₊

by dividing β₂,...,β_k−1 by τ and computing

β_k = [β₂(t₁ – t_m) + β₃(t₂ – t_m)+ ... + β_m−1(t_m−2 – t_m)]/(t_m – t_m−1)

β_k+1 = [β₂(t₁ − tm₋₁)+ β₃(t₂ – t_m−1)+ ... + β_m−1(t_m−2 – t_m−1)]/(t_m−1 – t_m)

A test of linearity in X can be obtained by testing H₀: β₂ = β₃ = ... = β_m−1 = 0

Default values for knots