Flashcards
Regression - Chapter Seven
Cards in this deck (15)
- 1We know that Var(B_hat) = ? So if the ith diagonal element of [X^TX]^-1 is denoted v(i+1,i+1), then Var(Bihat) = ? So it is estimated by [Var(Bihat)]hat = ? The standard error is denoted the ... of this so s.e.(Bihat) = ?Var(B_hat) = [(X^TX)]^-1*sigma^2
Var(Bihat) = [v(i+1,i+1)]*sigma^2
[Var(Bihat)]hat = [v(i+1,i+1)]*s^2
s.e.(B1hat) = sqrt[v(i+1,i+1)]*s - 2A 100(1-alpha)% confidence interval for Bi is given by ?Bihat + or - t(n-p-1,alpha/2)*s.e.(Bihat) = Bihat + or- t(n-p-1,alpha.2)*s(sqrt[v(i+1,i+1)]
- 3where t(n-p-1,alpha/2) is obtained from ... To test the null hypothesis H0:Bi = bi versus H1:Bi =/ bi we calculate T = ? and compare with t tables on ... degrees of freedom We then calculate P = ? LAST POINT DON'T UNDERSTAND?!... the tables or in R
T = modulus(Bihat -bi)/s*sqrt[v(i+1,i+1)] ... n-p-1 degrees of freedom
P = Pr(modulus[t(n-p-1)]>/modulus[T]) - 4A large value of modulus(T) and hence a small value of P will lead to ...... a rejection of the null hypothesis and acceptance of the alternative hypothesis that Bi =/ bi
- 5The basic regression line concept is ... which can be written as ... Where the first term is a measure of ... the second term is a measure of ... the third term is the ...Data = Model + Regression ... (yi-ybar) = (yihat - ybar) + (yi -yibar)
... total variation in the response, y
... the variation in the mean response
... residual value - 6What can then be done to the equation so that it equals something else? What do we then get the equation to equal? This new equation can then be written as?Square each term, adding them all over n and then noting that SIGMA(i=1,n)[(yihat - ybar)(yi - yibar)] = 0
SIGMA(i=1,n)[yi-ybar]^2 = SIGMA(i=1,n)[yihat - ybar]^2 + SIGMA(i=1,n)[yi-yihat]^2
TSS = MSS + RSS - 7Where SS is the notation for ... T is the notation for ... M is the notation for ... R is the notation for ... R^2 = ? R^2 is the proportion of ... The range of R^2 is ?... sum of squares
... Total
... Model
... Residual (error)
R^2 = MSS/TSS = 1 - RSS/TSS
... the variation in y explained by the regression on the explanatory variables
0 to 1 - 8The adjusted coefficient of determination is defined as ? This rescales R^2 using the degrees of freedom as ?R^2 adj = 1 -[RSS/(n-p-1)]/[TSS/(n-1)]
R^2 adj = 1 - [RSS/(n-p-1)]/[TSS/(n-1)]
= 1 - [RSS/TSS * (n-1)/(n-p-1)]
= 1 - [TSSRSS/TSS * (n-1)/(n-p-1)]
= 1 - (1-R^2)*[(n-1)/(n-p-1)]
= [n-p-1-(n-1)+(n-1)R^2]/(n-p-1)
= [(n-1)R^2 -p]/(n-p-1) - 9R^2 adj and R^2 are directly ... R^2 adj need not always increase as ... R^2 adj tends to stabilise around ... The simplest model with R^2 adj near the upper limit is usually chosen as ...... related
... variables are added to the model
... some upper limit as variables are added
... the 'best' model - 10Adding in a quadratic is similar to ... Why does care have to be taken however when interpreting the p-values?... including an additional explanatory variable
There is a typically high correlation between the linear and the quadratic term - 11Sometimes what are used as explanatory variables? How is this accomplished? Suppose we have male and females, how would we denote these using indicator variable x1? An indicator value always takes the values ...Qualitative (categorised) variables
Using indicator variables
x1=0 denotes male
x1=1 denotes female
... 0 or 1 - 12A common use of indicator variables is when ...... comparing two simple linear regression equations
- 13Consider the model Y = B0 + B1x1 + B2x2 + epsilon, where x1 is an indicator variable and x2 is a quantitative explanatory variable. Then: a) E[Y|x1=0] = ? b) E[Y|x1=1] = ? Thus what we are really doing is what? B1 measures what ?a) E[Y|x1=0] = B0 + B2x2
b) E[Y|x1=1] = B0 + B1 + B2x2
Fitting two parallel lines with a common slope
B1 measures the difference in the intercepts - 14We can test the null hypothesis H0: B1 = 0 versus H1:B1=/0, and if we cannot reject H0 then ... If we want to consider two lines with different intercepts and different slopes we can fit the model ... Then: a) E[Y|x1=0] = ? b) E[Y|x1=1] = ?... a single common line describes both groups
... Y = B0 + B1x1 + B2x2 +B3x1x2 + epsilon
a) E[Y|x1=0] = B0 + B2x2
b) E[Y|x1=1] = B0 +B1 + B2x2 +B3x2 - 15B3 measures the ... We can test the null hypothesis H0: B3 =0 versus H1: B3 =/0, if we cannot reject H0 then ...... difference in the slopes
... we remove B3 and go back and then test the null hypothesis for H0: B1=0 versus H1: B1 =/0
Information
- Description
- Flashcards on Regression - Chapter Seven, created by Cara Wales on 26/05/2016.
- Created by
- Cara Wales
- Views
- 0
- Rating