未命名 2
SIMULATION STUDY
Problem 1
People are notoriously bad at generating random numbers in their heads. In this
problem, we will compare random binary sequences generated experimentally
versus ones that are made up.
A.
Write down a sequence of 100 binary digits (each digit is a ‘0’ or a ‘1’) that you
make up off the top of your head without any experimental or computational help.
B.
Find a way to generate a sequence of 100 binary digits that represent an
independent sample of size 100 from a Bernoulli distribution with p=0.5.
C.
Let be the digit in either of the sequences you generated.
Compute the sample mean, , for both sequences.
What is the expected value for independent Bernoulli random variables
with ?
Compute the sample covariance between adjacent
samples, , for both sequences.
What is the theoretical covariance for independent Bernoulli random variables?
How do the sample statistics from each sequence compare to the theoretical
values?
D.
A run is a sequence of adjacent ‘0’s or ‘1’s.
If a sequence of random Bernoulli variables is drawn independently, after drawing
a first ‘0’ or ‘1’, what is the probability that the first run will be of length 1?
What is the probability that the first run will be of length 2?
What is the probability that the first run will be of length ?
Plot the empirical PMF and empirical CDF of the lengths of runs from both
sequences, and compare these to the theoretical distributions.
E. Discuss how well your made up binary sequence resembles an actual sequence
of independent Bernoulli[0.5] random variables. If you wanted to determine
whether a binary sequence was generated from an independent Bernoulli process
or made up by a person, which statistics would you check?
F. Write code to generate 0s and 1s in the following way.
1. For the first sample, randomly select a 0 or 1 with probability 1/2 for each.
2. If the last sample was a 1, the next sample will be a 1 with probability and
a 0 with probability . If the last sample was a 0, the next sample will be
a 1 with probability and a 0 with probability .
3. Repeat until the sample is length 100.
G. Come up with a statistic to estimate in the above problem by computing the
number of times the following sequential pairs appear in the sequence: 00, 01, 10,
11. For example in the sequence XXXXXXXXXXyou would make this table.
Use this statistics to estimate for your sequence from part A. Do you think the
above model is a good model for your sequence? Is there something different
about your sequence?
DATA ANALYSES
Problem 2 The dataset contains (made up) pilot data for a test of a new
cholesterol drug. 20 high-cholesterol patients were assigned to a drug group and
20 patients were assigned to a placebo group. Their blood cholesterol levels were
measured before and after a 1-month regimen of the drug or placebo. The dataset
contains the patient blood triglyceride levels in mg/dL before and after the
egimen. The group variable contains a ‘0’ for the placebo group and a ‘1’ for the
drug group.
A.
Load the data into a statistics software package (such as R or MATLAB).
Visualize the data before and after the intervention.
Use some descriptive statistics to describe and compare the data between the
efore and after periods.
Describe the structure of the data in words based on your statistics and
visualizations.
B.
Compute the change in triglyceride level for each patient, both as a change in the
aw value (in mg/dL) and as a percentage change from the period before the
egimen.
Visualize and use descriptive statistics to characterize features of the levels of
change for both the drug and placebo groups.
Also compute the fraction of patients who saw a reduction of triglyceride level from
each of the drug and placebo groups.
Is there evidence for an improved effect of the drug relative to a placebo?
0 1
0 2 2
1 1 1
C.
When you show your results to the clinicians studying the drug, they hypothesize
that some of the measurements might be abnormally high because patients did
not fast before their blood was drawn.
Abnormal data points are often called outliers.
Is their any evidence in the data of abnormally high measurements? If so, remove
these outliers and repeat the analyses from part B.
How does this change your results?
D.
What would you recommend to the researchers regarding this dataset? What are
the potential drawbacks of removing the outlier points in part C?