Factors Affecting Plasma Beta Carotene

[Date]
FACTOR AFFECTING PLASMA BETA CAROTENE

Table of Contents
Introduction.................................................................................................................................................2
LASSO (Least Absolute shrinkage and selection operator)..........................................................................2
LAR (Least Angle Regression).......................................................................................................................3
RESULTS.......................................................................................................................................................5
Population characteristics.......................................................................................................................5
Descriptive statistics................................................................................................................................7
Inferential statistics.................................................................................................................................9
Correlation analysis.............................................................................................................................9
Regression analysis............................................................................................................................13
Discussion..................................................................................................................................................14
Conclusion.................................................................................................................................................17
References.................................................................................................................................................18

Introduction
This paper aims to investigate factors that affect plasma beta-carotene. The factors are
investigated using the regression techniques of LASSO and LAR. LASSO is an extension of
linear regression, but using shrinkage. On the other hand, LAR is a technique of fitting a
regression model for a linear combination of a subset of potential covariates. Correlation analysis
was used to investigate association between variables.
LASSO (Least Absolute shrinkage and selection operator)
Lasso is a shrinkage and variable selection method for linear models. It is a is a regression
analysis technique that performs both variable determination and regularization so as to improve
the forecast exactness and interpretability of the factual model it produces. It is an extension of
linear regression using shrinkage.
Stepwise method was used for selecting the best model. This is a method of fitting regression
models in which the choice of predictive variables is carried out by an automatic procedure. In
each step, a variable is considered for addition to or subtraction from the set of explanatory
variables based on some predetermined criterion.
Considering a sample of N cases, the objective of LASSO is to solve the following function:
Min ( B 0 , B ) { 1
N ∑ ( yi−B 0−xiB )2
} subject to ∑ Bj ≤ t ,
where B 0 refers¿ the intercept , B refers ¿ the slope intercept ,
yi refer ¿ the dependent variable data points∧xi refer ¿ theindependent variables .
The bound t is a tring parameter .
The procedure for computing LASSO is given as;
 Start with all coefficients Bj equal to zero.

 Find the predictor xi most correlated with y, and add it into the model. Take residuals r=
y-yhat.
 Continue, at each stage adding to the model the predictor most correlated with r until all
predictors are in the model.
The method shall select variables to be included in the model one after another.
Model 1: Betaplasma = fiber
Model 2: Betaplasma = fiber + calories
Model 3: Betaplasma = fiber + calories + alcohol
Model 4: Betaplasma = fiber + calories + alcohol + retdiet
Model 5: Betaplasma = fiber + calories + alcohol + retdiet + vituse
Model 6: Betaplasma = fiber + calories + alcohol + retdiet + vituse + sex.
Model 6 is the best LASSO model as it includes only the variables that are statistically
significant.
LAR (Least Angle Regression)
LAR is a technique of fitting a regression model for a linear combination of a subset of potential
covariates. The calculation is like forward stepwise regression, however as opposed to including
factors at each progression, the evaluated parameters are increased toward a path equiangular to
every one's relationships with the residual.
To select the best method, we use forward regression. This is a technique which involves starting
with no variables in the model, testing the addition of each variable using a chosen model fit
criterion.
The procedure for LAR is;
 Start with all coefficients Bj equal to zero.
 Find the predictor xi most correlated with y

 Increase the coefficient Bj in the direction of the sign of its correlation with y. Take
residuals r=y-yhat along the way. Stop when some other predictor xj has as much
correlation with r as xi has.
 Increase (Bj, Bk) in their joint least squares direction, until some other predictor xk has as
much correlation with the residual r. Continue until all predictors are in the model
Similar to the LASSO, 6 models were obtained with the following model being the best;
Betaplasma = fiber + calories + alcohol + retdiet + vituse + sex.
Hypotheses
Given the models selected from BIC, AIC and LASSO, we shall investigate the following
hypotheses;
i. H0: Fibre consumption does not affect plasma beta-carotene.
H1: Fibre consumption affects plasma beta-carotene.
ii. H0: Calories consumption does not affect plasma beta-carotene.
H1: Calories consumption affect plasma beta-carotene.
iii. H0: Alcohol consumption does not affect plasma beta-carotene.
H1: Alcohol consumption affects plasma beta-carotene.
iv. H0: Retdiet does not affect plasma beta-carotene.
H1: Retdiet affects plasma beta-carotene.
v. H0: Vituse does not affect plasma beta-carotene.
H1: Vituse affects plasma beta-carotene.
vi. H0: Sex does not affect plasma beta-carotene.
H1: Sex affects plasma beta-carotene.
vii. H0: Cholesterol consumption does not affect plasma beta-carotene.
H1: Cholesterol consumption affects plasma beta-carotene.

RESULTS
Based on the three information criterion (BIC), AIC and LASSO, we shall investigate the factors
that affect the concentration of beta-carotene plasma. The following variables shall be examined
on their effect on concentrations of beta-carotene plasma; age, sex, smoke, quetelet, vituse,
calories, Fatt, fiber, alcohol, colesetrol, betadiet, retdiet and retplasma.
Population characteristics
Age
The minimum age was found to be 19 years while the maximum age was found to be 83 years.
The mean age was found to be 50.01 and the standard deviation 14.463.
Descriptive Statistics
N Minimum Maximum Mean Std. Deviation
age 315 19 83 50.01 14.463
Valid N (listwise) 315