a_matrix(scan("kyphosis.dat"),83,5,byrow=T)
y_a[,2]
age_a[,3]
start_a[,4]
number_a[,5]

#Fit a quadratic logistic model of y on age

logitquad_glm(y~poly(age,2),family=binomial)
summary(logitquad)

Call: glm(formula = y ~ poly(age, 2), family = binomial)
Deviance Residuals:
        Min        1Q     Median         3Q      Max 
 -0.9912322 -0.828576 -0.4745756 -0.2718635 2.233231

Coefficients:
                    Value Std. Error    t value 
  (Intercept)  -1.5659259  0.3402633 -4.6021010
poly(age, 2)1   0.5940142  3.4628418  0.1715395
poly(age, 2)2 -10.2320391  4.3235541 -2.3665806

(Dispersion Parameter for Binomial family taken to be 1 )

    Null Deviance: 86.80381 on 82 degrees of freedom

Residual Deviance: 78.21608 on 80 degrees of freedom

Number of Fisher Scoring Iterations: 4 

Correlation of Coefficients:
              (Intercept) poly(age, 2)1 
poly(age, 2)1 0.0201553                
poly(age, 2)2 0.5608006   0.2495414    

#plot the fitted value (linear predictor) against age
quadfit_logitquad$fitted.values
plot(age[sort.list(age)],quadfit[sort.list(age)],type="l",lty=1,xlab="age", 
ylab="fitted value"))

#Fit a logistic nonparametric model of y on age using loess
logitnon_gam(y~lo(age,span=0.7,degree=1),family=binomial)
nonfit_logitnon$fitted.values

cage_cbind(age,age)
cage_cage[sort.list(age),]
fit_cbind(quadfit,nonfit)
fit_fit[sort.list(age),]
matplot(cage,fit,type="l", lty=c(1,2),xlab="age",
ylab="fitted value")
legend(150,0.35,lty=c(1,2),legend=c("quad fit","loess fit"))
title("Logistic Quadratic and Loess Fits of Kynosis on Age")




logitcubic_glm(y~poly(age,3),family=binomial)
summary(logitcubic)

Call: glm(formula = y ~ poly(age, 3), family = binomial)
Deviance Residuals:
        Min         1Q     Median         3Q      Max 
 -0.9974528 -0.8438042 -0.4748754 -0.2226955 2.203141

Coefficients:
                   Value Std. Error    t value 
  (Intercept)  -1.625395  0.3858689 -4.2122994
poly(age, 3)1  -1.295963  6.0719092 -0.2134358
poly(age, 3)2 -12.610944  7.7803757 -1.6208657
poly(age, 3)3  -2.482434  6.3054700 -0.3936953

(Dispersion Parameter for Binomial family taken to be 1 )

    Null Deviance: 86.80381 on 82 degrees of freedom

Residual Deviance: 78.0554 on 79 degrees of freedom

Number of Fisher Scoring Iterations: 5 

Correlation of Coefficients:
              (Intercept) poly(age, 3)1 poly(age, 3)2 
poly(age, 3)1 0.4068619                              
poly(age, 3)2 0.6625294   0.7728313                  
poly(age, 3)3 0.4441241   0.8058684     0.8134943