Help needed. Simple linear regression simulation in r.

[JUSTICE]

Hi, help needed.
I wish to simulate a simple linear regression using R but I am new user of R and keep struggling with it for some days now.
Below is my task.

Task.

Set up a simulation model according to the following recipe:
i) n=1000
ii) Xi~N(0,1)
iii) Yi = a+bxi+Ɛi i=1,2,3,…,n.
a=0.2
b=0.03
Ɛi~N(0, 0.32)

iv) Calculate the regression slope b and store it.
v) Calculate the t-value, p-value and standard error of regression slope b.
vi) Run this recipe 10000 times in a loop. How many times was P-value <0.05 of power.

My effort so far but am really lost:

# (1)set parameters 
true.intercept<-0.2 
true.slope<-0.03 
true.err.var<-0.09 
n.iter<-1000 
s.sample<-100 
estimates<-matrix(NA,nrow=n.iter,ncol=2) 

x<-10*runif(s.sample) 
for (i in 1:n.iter) { 
  #generate data 
  y<-true.intercept+true.slope*x+rnorm(s.sample,mean=0,sd=true.err.var) 
  #estimate things 
  lm(y~x)->mod 
  #save estimates 
  mod$coef->estimates[i,] 
} 

nms<-c("intercept","slope") 
#compute variance of estimates 
cbind(1,x)->x2 
s2<-solve(t(x2) %*% x2)*true.err.var^2 
estimates

Kindly help me please.
Your assistance is highly appreciated in advance.
Many thanks.

[derksheng]

You should work with lists.

# (1)set parameters
true.intercept<-0.2
true.slope<-0.03
true.err.var<-0.09
n.iter<-1000
s.sample<-100
estimates<-matrix(NA,nrow=n.iter,ncol=2) 

list<-list()
s.sample<-100
x<-10*runif(s.sample) 

for(i in 1:n.iter) {
y<-true.intercept+true.slope*x+rnorm(s.sample,mean=0, sd=true.err.var) 
list[[i]]<-lm(y~x)
}

#Store p-value of coefficient.
sapply(1:length(list), function(X) { summary(list[[X]])$coef[2,4] })

#Store p-value of intercept.
sapply(1:length(list), function(X) { summary(list[[X]])$coef[1,4] })

#Store R squares
sapply(1:length(list), function(X) { summary(list[[X]])$r.squared })

[JUSTICE]

Hi Derksheng,
Thank you very much for the help.
I am very grateful.
Will get in touch should I need further help on that.
Best regards.

[Jake]

Hi derksheng,

Very helpful response. In the future, it is encouraged when posting code to wrap the code in "code tags," which is done like so:

[code]
awesome
code
goes
here
[/code]

This is only for formatting reasons, but it does make it quite a bit easier to read since it preserves the spacing.

[JUSTICE]

Dear Derksheng,
Many thanks for the helpful and quick response yesterday.
I have tried to use your codes to simulate similar problems and it works perfectly.
Could you kindly help me to compute the standard error of the regression slope (b) and the t-value for the same problem.
Best regards.

[derksheng]

Well I'm not 100% sure that this is what the question is after (I don't have time to completely do it), but to get the t-values and the standard errors;

#Retrieve t-values of intercept#
sapply(1:length(list), function(X) { summary(list[[X]])$coef[1,3] })

#Retrieve sample standard-errors of intercept#
sapply(1:length(list), function(X) { summary(list[[X]])$coef[1,2] })

#Retrieve t-values of slope coeff#
sapply(1:length(list), function(X) { summary(list[[X]])$coef[2,3] })

#Retrieve estimated SEs of slope coeff#
sapply(1:length(list), function(X) { summary(list[[X]])$coef[2,2] })

Be careful in your language. This isn't the population standard error. We can't observe the population standard error of the estimator. Also, it's the sample standard error of the ESTIMATOR, not of the ESTIMATE. The estimate is a constant, the estimator is a random variable.

If you want to learn of the meaning of the numbers inside my $coef[number1,number2], just go summary(list[[1]])$coef and observe the dimensions of the matrix and the column names.

[JUSTICE]

Hi Derksheng,
I am very grateful for the response and the notes. They were very helpful.
Best regards.