Midterm exam information, STATS 401 W18

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Scheduling

• The midterm exam is in class on Wednesday 2/21, from 1:10 to 2:30.

• Class is canceled on Monday 2/19, and there will be no lab on 2/22 or 2/23.

• Midterm material will be discussed further in class on 2/14 and in lab on 2/15 or 2/16.

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Instructions

The midterm exam will test the skills covered in homeworks 1–6, lab 1–6, and material covered in class up to Wednesday 2/14. You will have 80 minutes allocated. The exam will be closed book. Any electronic devices in your possession must be turned off and remain in a bag on the floor. Technical skills tested will include both numerical and algebraic manipulation of variance and covariance, as well as the skills covered in quiz 1. R coding will be tested by multiple choice questions and reading given code. The only time you may have to write code in the exam is to write a call to ‘pnorm()’ or ‘qnorm()’ to show how to evaluate a normal distribution calculation. There will be questions on explaining R output related to fitting linear models, as well as questions on writing linear models in matrix form and as equations with subscripts.

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Formulas

• You are not allowed to bring any notes into the exam.

• The following formulas will be provided. To use these formulas properly, you need to make appropriate definitions of the necessary quantities. \[ {\mathbf{b}}= \big( {\mathbb{X}}^{{\scriptscriptstyle \mathrm{T}}}{\mathbb{X}} \big)^{-1}\, {\mathbb{X}}^{{\scriptscriptstyle \mathrm{T}}}{\mathbf{y}}, \] \[ {\mathrm{Var}}(\hat{\mathbf{\beta}})= \sigma^2 \big( {\mathbb{X}}^{{\scriptscriptstyle \mathrm{T}}}{\mathbb{X}} \big)^{-1}, \] \[ {\mathrm{Var}}({\mathbb{A}}{\mathbf{Y}})={\mathbb{A}}{\mathrm{Var}}({\mathbf{Y}}){\mathbb{A}}^{{\scriptscriptstyle \mathrm{T}}}\] From ?pnorm:

  pnorm(q, mean = 0, sd = 1)
  qnorm(p, mean = 0, sd = 1)
  q: vector of quantiles.
  p: vector of probabilities.

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Question categories.

All question categories from the quiz will be included except for the basic matrix exercises (M1,M2,M3).

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Summation exercises

S1. A basic exercise.

S2. An example involving the summation representation of matrix multiplication.

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R exercises

R1. Using rep() and matrix().

R2. Manipulating vectors and matrices in R.

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Fitting a linear model by least squares

[This category is similar, but slightly different, from the F1 and F2 questions in the quiz.]

F1. Write the sample version of a linear model in subscript form given the matrix form.

F2. Write the sample version of a linear model in matrix form given the subscript form.

F3. Write the sample version of a linear model in matrix form subscript form given a dataset and verbal description of the model OR writing the sample version of a linear model in matrix form given a dataset and verbal description of the model.

F4. Explain how to obtain the least square value of the coefficients and the fitted values.

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Properties of variance and covariance

V1. A numerical calculation to find the variance of a linear combination using matrix techniques.

V2. An algebraic calculation using basic definitions of variance & covariance, together with the linearity of expectation.

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Normal probability calculations

N1. A normal approximation to estimate a probability using the mean and variance.

N2. A normal approximation to find a region with a given probability using the mean and variance.

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The population version (or probability version) of the linear model

P1. Describe a suitable probability model, in subscript form, to give a population version of a linear model.

P2. Describe a suitable probability model, in matrix form, to give a population version of a linear model.

P3. Explain how R produces standard errors for coefficients in a linear model. Interpret the standard errors using the probability model.

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Example: patient satisfaction in a hospital

The following survey data on a collection of hospital patients measures self-reported satisfaction, age, a measure of case severity, and a measure of anxiety. The hospital managers want to see whether satisfaction can be explained by the other variables, and, if so, which variables are important.

patients <- read.table("patients.txt",header=T)
dim(patients)

## [1] 46  4

head(patients)

##   Satisfaction Age Severity Anxiety
## 1           48  50       51     2.3
## 2           57  36       46     2.3
## 3           66  40       48     2.2
## 4           70  41       44     1.8
## 5           89  28       43     1.8
## 6           36  49       54     2.9

(F1,F2,F3). Write the sample version of a linear model to address this question, in subscript form and matrix form.

(P1,P2). Write a probability model that can be used to assess the chance variation in the coefficients of the sample linear model. What is the source of this chance variation?

(P3) Explain how this probability model is used to obtain standard errors for the coefficient estimates.

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Exercise related to HW6: Working with a generic \((i,j)\) component of a matrix

• A matrix quantity such as \({\mathrm{Var}}({\mathbf{X}})\) is really a collection of quantities for each row \(i\) and column \(j\).

• Sometimes, when working with matrices, it is helpful to write equations for a generic \((i,j)\) component. Something that you work out for a generic \((i,j)\) component is necessarily true for the whole matrix.

• On questions of the form “Show that \(A=B\)” you can in principle start from \(B\) and show how to get \(A\) or vice versa. Usually, people work left-to-right so the question is likely suggesting that it is simpler to start from \(A\) and show how to get to \(B\).

• Here, we are considering questions of the form “Show that \({\mathbb{A}}={\mathbb{B}}\)”. We can do this by showing that \([{\mathbb{A}}]_{ij}=[{\mathbb{B}}]_{ij}\) for an arbitrary \((i,j)\).

Example. Let \({\mathbf{X}}=(X_1,\dots,X_n)\) and \({\mathbf{Y}}=(Y_1,\dots,Y_n)\) be independent vector random variables with \(n\times n\) variance matrices \({\mathbb{U}}\) and \({\mathbb{V}}\) respectively. Show that \({\mathrm{Var}}({\mathbf{X}}+{\mathbf{Y}})={\mathbb{U}}+{\mathbb{V}}\). This is a version for vector random variables of the formula \({\mathrm{Var}}(X_i+Y_i)={\mathrm{Var}}(X_i)+{\mathrm{Var}}(Y_i)\), which we have already seen for the variance of a sum of two independent random variables. You can use the definitions of variance and covariance. You may also use the basic property of expectation of a product of independent random variables, that \({\mathrm{E}}[X_iY_j]={\mathrm{E}}[X_i]{\mathrm{E}}[Y_j]\).

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License: This material is provided under an MIT license

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