HW 14 Stat 705  Fall 2015

Assigned Saturday 11/14/15 DUE Monday 11/23/15


(A) Start by running the following lines to generate a small 
illustrative simulated dataset exactly the same as one I generated.

> set.seed(3)
  x = runif(100)
  y = 3 - 7*x + 10*abs(x-.5)^1.5*rnorm(100)
  Dset = cbind.data.frame(y, x)


VERIFY that weighted least squares models  y ~x  fitted to this 
dataset in two different ways, 

(1) using lm with weights = b(x),  and 

(2) using lm (without an intercept!) with  ystar = y*sqrt(b(x))  
and model-matrix columns all also multiplied componentwise by sqrt(b(x))
    
give IDENTICALLY the same results in estimated coefficients and 
significance levels and (after you transform back from ystar to 
y under method (2)) also in "fitted" (ie predictors) and residuals.   
Here your choice of weights b(x) should be based on the model 
assumption in such a way that  Y*b(X)  conditionally given X is 
normally distributed with constant variance (and mean depending on X).


(B) Generate a dataset the same as mine using the statements

> set.seed(3577)
  x = rgamma(250, 0.7)
  z = sample(1:4, 250, replace=T, prob=c(0.2,0.3,0.3,0.2))
  nv = rep(c(5,10),125)
  y = rbinom(250, nv, plogis(0.3 + 1.2*x - 0.7*z))
  AuxFr = cbind.data.frame(y,x,z,nv)

-- Fit a probit model for  y  (or y/nv) in terms of predictor 
variables  x  and  z.  
-- Obtain the NULL deviance and the Residual Deviance from the 
model. (The residual deviance is simply called  Deviance  in 
the model outputs unless you run anova(.)  on the model.
-- Show how to obtain these deviances by calculating the Probit 
Model logLikelihoods, either directly or by using other R functions 
that calculate log-likelihoods, or both.