3rd year Flashcards
(29 cards)
In a generalised linear model, how is the mean of the response variable related to the linear
predictor?
In a GLM, the mean response µ is related to the linear predictor η according to
g(µ) = η,
where g(·) is the link function.
How is the variance related to the linear predictor?
The variance of the response is related to the linear predictor η through the mean µ as
Var{y} = V (µ)a(φ) = V (g^−1(η))a(φ),
where V (µ) is the variance function.Explain what the canonical link is
The canonical link is the same function of µ as θ is (a function of µ).
The Newton-Raphson algorithm becomes identical to which
algorithm named after a famous statistician when the canonical link is used?
The Newton-Raphson algorithm becomes identical to the Fisher scoring algorithm when the canonical
link is used.
Explain the role of the link function g(µ) in a generalised linear model
The role of the link function g is to transform the mean response µ so that g(µ) = η, the linear
predictor.
Write down the general form of a distribution from the exponential family
f(y) = exp
[(yθ − b(θ))/a(φ)] + c(y, φ)
Explain how a generalised linear model is more general than a linear model with normal errors
A generalised linear model is more general in two ways.
(i) The distribution of the response variable can be any member of the exponential family.
(ii) The mean response µ need not be a linear function of the explanatory variables, as long as g(µ)
is, for some link function g
Write down the log linear regression model with Poisson responses yi ∼ Pois(λi) on a single explanatory
variable xi with intercept, i = 1, . . . , n, stating any further assumptions on y1, . . . , yn
Poisson response log linear model with intercept:
yi = µi + εi ∼ Pois(µi) independent,
log(µi) = β0 + β1xi
, i = 1, . . . , n.
Explain what the updating equation β(k+1) = (X′W(k)X)^(−1)X ′W^(k)ξ^(k) , k = 0, 1, 2, . . . does
The updating equation calculates β^(k+1) as weighted least squares estimates of regression coefficients, using working responses ξ^(k) with predictor values in X and weights in W(k) on the diagonal.
Explain the connection between Poisson regression and logistic regression with binomial responses.
In large scale studies of rare events a Poisson regression with log link and offset gives nearly the same
results as logistic regression
Write down the linear regression model as a GLM for yi ∼ N(µi, σ2) on a single explanatory variable x with values xi, i = 1, 2, . . . , n and state any further assumption on y1, y2, . . . , yn. An
intercept or constant term should be included in the model.
As a GLM, the linear regression model with normal errors is given by
yi = µi + εi ∼ N(µi, σ2) independent, µi = β0 + β1xi, i = 1, . . . , n.
Give three reasons why a Poisson model can be used to analyse contingency table data.
(i) It has been shown in class that yij ∼ Pois(λij )|Pij yij = n ∼ Multinomial with πij = λij/Pij λij. (ii) The MLE problems are equivalent. (iii) An additive Poisson model with log link is equivalent to independence between row and column classifications.
The F-distribution with n1 and n2 degrees of freedom is defined as
…
In the full (saturated) model
all the θi
’s are free to vary, so that the fitted value
for yi equals yi
, i = 1,…,n
The scaled deviance of a GLM
is twice the difference in maximum loglikelihood
value when comparing it with the full (saturated) model.
The variance function is
V (µ) = b′′(θ) written as a function of µ
What is the likelihood equation in matrix form?
The likelihood equation in matrix form is
X′W (ξ − Xβ) = 0
where X = (xi j) is the n × p design/data matrix, W is a diagonal matrix with
wi =1/[V(µi) . g′(µi)^2], i = 1,…,n
on the diagonal line, and ξ is a column vector with elements
ξi = x′iβ + g′(µi)(yi −µi), i = 1,…,n.
What are the projection errors y − yˆ
called
residuals
How is the deviance related to the residual sum of squares? How do you estimate
σ^2 using the deviance?
The deviance equals the residual sum of squares: D = SSE.
σ^2 is estimated by σˆ2 = SSE/(n − p) = D/(n − p).
For f (y;θ,ϕ) in the exponential family, the Fisher information in y about θ is
i(θ) =b′′(θ)/a(ϕ)
Properties of ξi
E[ξi] = x′_iβ
Var{ξi} = g′(µi)^2 . V(µi)a(ϕ) = w^(−1)_i . a(ϕ)The scaled deviance is
SSE/σ^2
Coefficient of determination
R^2 = SSR/SST
The logit link is preferred to the probit link Φ−1(π) because
- it provides a canonical link within the framework of a GLM;
- it makes it easy to compute the parameter estimates,
- it has interpretation in terms of odds ratio.
