Carlos Lopez
Biostat 310
1/23/17
Homework 3
1. The fraction 12 out of 120 is a measure of prevalence.
2. A. Incidence rate Numerator: 2+4+8 = 14, Denominator: (2 x 1) + (4 x 2) + (8 x 3) + (5
x 4) + (106 x 10) = 1114. 14/1114 = 0.01257
B. Cumulative incid
Carlos Lopez
Biostats 310
2/2/17
Homework 4
1. A. d) All of the above.
2. A. Relative risk = (20/49) / (31/50) = 0.658
B. Odds Ratio = (0.41/0.59) / (0.62/0.38) = 0.43
3. I. A. Proportion (miscarriage given heavy drinking) = 14/18. Not valid.
B. Proportio
Carlos Lopez
Biostat 310
1/16/17
Homework 2
1. A. The plot shown is a scatterplot.
B. ii) Systolic blood pressure tends to be higher at older stages; the correlation is about
0.4.
2. A. Probability of having NYAH I: (200/500) x 100 = 40%.
B. No, the proba
Carlos Lopez
Biostat 310
1/9/17
Homework 1
1. The increase in cancer deaths despite the cancer treatments becoming more effective
since 1970 could be due to a couple reasons: first, this could be due to the fact that the
population itself has increased so
Carlos Lopez
Biostats 310
2/14/17
Homework 5
1. Best description is meta-analysis study.
2. A. Best description is randomized clinical trial because the participants were randomly
assigned to be in the two different groups of either receiving 10 days of a
Confidence Intervals
1
Terminology Reminders
Population: any collection of entities that have at least one characteristic in
common
Parameter: the numbers that describe characteristics of scores in the
population (mean, variance, s.d., etc.)
Sample: a
Error Propagation
November 30, 2016
1 / 63
Motivation
Motivation
Let us measure the area of a circle: A = r2
In measuring the area we note that there is uncertainty in our
measurement of the radius, r. For example r = 2.5 0.2
Given this uncertainty what i
Sampling
Sampling: Experiment
Show that the distribution of averages from a non-normal distribution tends to
follow a normal distribution.
Obtained 20 or so uniform random numbers, sum them and compute the average.
Compute 100,000 of these averages.
Plot
Inference
Inference
Confidence Intervals: Estimating a population parameters
Tests of significance: To assess the evidence provided by data about some
claim on the population.
Test of Significance: A formal procedure for comparing observed data with a
c
Permutations and Combinations
Permutation: The number of ways in which a subset of
objects can be selected from a given set of objects, where
order is important.
Given the set of three letters, cfw_A, B, C, how many possibilities
are there for selecting a
Probability
Distributions
Random Variables
A random variable, X associates a unique numerical value with every
outcome of an experiment. The value of the random variable will vary
from trial to trial as the experiment is repeated.
There are two types of
Introduction to Probability
Biostatistics, 499/599
Sample Space
The sample Space, S
The sample space, S, for a random phenomena is the
set of all possible outcomes.
Examples
Examples
1. Tossing a coin outcomes S =cfw_Head, Tail
2. Rolling a die outcomes
S
Nonlinear Fitting
Linearizing nonlinear Functions
Not recommended unless you have information on the errors in your y data and
you weight the fit according to those errors. Note these errors will change due
to the transformation!
Example
x
y
1
2
3
4
5
0.5
Continuous Distributions
1
Continuous Distributions
Foundations for much of
statistical inference
Normal Distribution
Log Normal Distribution
Gamma Distribution
Chi Square Distribution
F Distribution
t Distribution
Weibull Distribution
Extreme Value Distr
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com be obtculeol S m
fig W menP-a 11mm
'T m
We the vectors
u ye", h
0
6).: (z,
I _
a L ).ntl~thL\ (Pm-try efwvwbm't
.7 M
hwclb em , 3 Oct/(e661 or). cfw_3642
Th2): L7 @
W I? \s the standanl matrix oLsSom'afed
w cfw_fl/x T,
whefLer a map [A cfw_S Unto.
Suppose that A, B and C are square matrices, and that CA = B. Then C is
given by
(a) A1 B
(b) BA1
(c) AB1
(d) B1 A
()
1/2
Suppose that A, B, and C are n n invertible matrices, and that
C1 (A + X)B1 = In . Then the matrix X is given by
(a) BC A
(b) B1 C1 A
When two linear transformations are performed one after another, the
combined effect will always be a linear transformation.
(a) True
(b) False
()
1/5
Let S : Rp Rn and T : Rn Rm be linear transformations. Then another
valid linear transformation is
(a) T
Consider the triangular matrix
a11
A= 0
0
a12
a22
0
a13
a23 ,
a33
so that
a11
0
A I =
0
a12
a22
0
a13
a23 .
a33
Then the eigenvalues of A are given by = a11 , = a22 , and = a33 .
(a) True
(b) False
()
1/3
Suppose that A is an n n matrix that has n dis
[Participation points only] Let R22 denote the space of all possible 2 2
matrices. The zero element is the zero matrix
0 0
.
0 0
Then the set of all diagonal matrices
a 0
,
0 b
a R, b R,
is a subspace of R22 .
(a) True
(b) False
()
1/3
[Participation poin
[Participation points only] Suppose that A is a 4 6 matrix and that
dim Nul A = 2. Then Col A = R4 .
(a) True
(b) False
()
1/3
Suppose that A is a 10 9 matrix and has 6 pivot columns. Then
(a) Col A = R6 .
(b) dim Col A = 6
(c) any basis of Col A consists
[Participation points only] Let R22 denote the space of all possible 2 2
matrices. The zero element is the zero matrix
0 0
.
0 0
Then the set of all singular (non-invertible) matrices is a subspace of R22 .
(a) True
(b) False
()
1/5
Let U be the matrix th
An n n matrix is invertible if its columns are linearly independent.
(a) True
(b) False
()
1/3
An n n matrix is invertible if its rows are linearly independent.
(a) True
(b) False
()
2/3
The following matrix is invertible:
1
A= 2
3
3 2
5 7
6 3
(a) True
(
A is a 4 4 matrix with three distinct eigenvalues. One eigenspace is
one-dimensional, and one of the other eigenspaces is two-dimensional. Is it
possible that A is not diagonalizable?
(a) Yes
(b) No
()
1/2
The eigenspace corresponding to eigenvalue of the
When the sizes of the matrices A, B, and C are such that the product ABC is
defined, then (ABC)T is equal to
(a) AT CT BT
(b) BT AT CT
(c) CT BT AT
(d) CT AT BT
(e) all of the above are equal
()
1/1
Biostat 310
Winter 2017
Homework 1 Solutions
Total Points: 10 points
There are a total of 8 graded points (questions indicated by *) and then an additional 2 points in total
for the additional questions without an asterisk, graded according to: 0= no addi
LECTURE 3B:
NONLINEAR REGRESSION,
CONTINUED
BIOSTATISTICS/STATISTICS 570
Andrew Spieker (Based on slides previously prepared by Barbara
McKnight)
Autumn 2014
OUTLINE
Example: Michaelis-Menten Kinetics Example
The Newton-Raphson Algorithm
The Gauss-Newt
INTRODUCTION
BIOSTATISTICS/STATISTICS 570
ADVANCED REGRESSION METHODS
FOR
INDEPENDENT DATA
Barbara McKnight
September 24, 2014
OUTLINE
Examples
Course Topics
Course Goals
Course Procedures
Course Materials
Course Requirements
Notation
Review of Li
LECTURE 6a:
BAYESIAN INFERENCE part I
BIOSTATISTICS/STATISTICS 570
Barbara McKnight
Autumn 2014
Based on Notes Prepared by Professor Jon Wakefield
OUTLINE
Bayesian Inference
Choosing Priors
Frequentist Properties
Evaluating Posterior Densities and Oth
LECTURE 4:
OPTIMAL LINEAR
ESTIMATING EQUATIONS
BIOSTATISTICS/STATISTICS 570
Barbara McKnight
Autumn 2014
OUTLINE
Notation
Optimal Linear EEs
Examples
Biostatistics/Statistics 570
Autumn 2014
B. McKnight
236
NOTATION
Up until now we have been solving es