Transcript 02. Organization of statistical investigation

Organization of
statistical investigation
Medical Statistics
Commonly the word statistics means
the arranging of data into charts,
tables, and graphs along with the
computations of various descriptive
numbers about the data. This is a
part of statistics, called descriptive
statistics, but it is not the most
important part.
Half the subjects
receive one treatment
and the other half
another treatment
(usually placebo)
Statistical Analysis in a
Simple Experiment
Define population
of interest
Use statistical techniques
to make inferences about
the distribution of the
variables in the general
population and about the
effect of the treatment
Randomly
select sample of
subjects to
study
Measure
baseline
variables in
each group
The most important part
The most important part is
concerned with reasoning in an
environment where one doesn’t know,
or can’t know, all of the facts needed to
reach conclusions with complete
certainty. One deals with judgments and
decisions in situations of incomplete
information. In this introduction we will
give an overview of statistics along with
an outline of the various topics in this
course.
The stages of statistic
investigation
1st stage –
composition of
the program and
plan of
investigation
5th stage –
putting into
practice
2nd stage –
collection of
material
3ed stage –
working up of
material
4th stage – analysis
of material,
conclusions,
proposals
Survival Analysis
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Kaplan-Meier analysis measures the ratio of
surviving subjects (or those without an event)
divided by the total number of subjects at risk for
the event. Every time a subject has an event, the
ratio is recalculated. These ratios are then used
to generate a curve to graphically depict the
probability of survival.
Cox proportional hazards analysis is similar to
the logistic regression method described above
with the added advantage that it accounts for
time to a binary event in the outcome variable.
Thus, one can account for variation in follow-up
time among subjects.
Kaplan-Meier Survival Curves
Why Use Statistics?
Cardiovascular Mortality in Males
1,2
1
0,8
SMR 0,6
0,4
0,2
0
'35-'44 '45-'54 '55-'64 '65-'74 '75-'84
Bangor
Roseto
Percentage of Specimens Testing
Positive for RSV (respiratory syncytial virus)
Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun
South 2
2
5
7
20
30
15
20
15
8
4
3
North- 2
east
West 2
3
5
3
12
28
22
28
22
20
10
9
2
3
3
5
8
25
27
25
22
15
12
2
2
3
2
4
12
12
12
10
19
15
8
Midwest
Descriptive Statistics
Percentage of Specimens Testing Postive for
RSV 1998-99
South
Northeast
West
Midwest
D
ec
Ja
n
Fe
b
M
ar
A
pr
M
ay
Ju
n
Ju
l
Ju
l
A
ug
Se
p
O
ct
N
ov
35
30
25
20
15
10
5
0
Distribution of Course Grades
14
12
10
Number of
Students
8
6
4
2
0
A
A- B+ B
B- C+ C
Grade
C- D+ D
D-
F
The Normal Distribution
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Mean = median =
mode
Skew is zero
68% of values fall
between 1 SD
95% of values fall
between 2 SDs
Mean, Median, Mode
.
1

2
SAMPLING AND ESTIMATION
One of the questions asked was “Do you
try hard to avoid too much fat in your
diet?” They reported that 57% of the
people responded YES to this question,
which was a 2% increase from a similar
survey conducted in 1983. The article
stated that the margin of error of the
study was plus or minus 3%.
Measures of Association
Measures Of Diagnostic Test
Accuracy
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Sensitivity is defined as the ability of the test to identify
correctly those who have the disease.
Specificity is defined as the ability of the test to identify
correctly those who do not have the disease.
Predictive values are important for assessing how
useful a test will be in the clinical setting at the individual
patient level. The positive predictive value is the
probability of disease in a patient with a positive test.
Conversely, the negative predictive value is the
probability that the patient does not have disease if he
has a negative test result.
Likelihood ratio indicates how much a given diagnostic
test result will raise or lower the odds of having a disease
relative to the prior probability of disease.
Measures Of Diagnostic Test
Accuracy
Expressions Used When
Making Inferences About Data
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Confidence Intervals
- The results of any study sample are an estimate of the true value
in the entire population. The true value may actually be greater or
less than what is observed.
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Type I error (alpha) is the probability of incorrectly
concluding there is a statistically significant difference in
the population when none exists.
Type II error (beta) is the probability of incorrectly
concluding that there is no statistically significant
difference in a population when one exists.
Power is a measure of the ability of a study to detect a
true difference.
SAMPLING AND ESTIMATION
This is an example of an inference
made from incomplete information.
The group under study in this survey
is the collection of adult Americans,
which consists of more than 200
million people. This is called the
population.
Group properties of statistical
totality:
Distribution of
characteristic
(criterion – relative
sizes)
Average level of
index (criterions –
Mo-mean, Memedian,
arithmetical mean)
Variety of
characteristic
(criterions –
lim- limit, am
– amplitude, σ
– average
deviation)
Mutual connection
between
characteristics
(criterion – rxy coefficient of
connection
Representation
(criterions – mM –
mistake of average
sizes, m% - mistake
of relative sizes)
SAMPLING AND ESTIMATION
If every individual of this group were to
be queried, the survey would be
called a census. Yet of the millions
in the population, the Harris survey
examined only 1;256 people. Such a
subset of the population is called a
sample.
SAMPLING AND ESTIMATION
We shall see that, if done carefully,
1;256 people are sufficient to make
reasonable estimates of the opinion of
all adult Americans. Samuel Johnson
was aware that there is useful
information in a sample. He said that
you don’t have to eat the whole ox to
know that the meat is tough.
SAMPLING AND ESTIMATION
The people or things in a population
are called units. If the units are
people, they are sometimes called
subjects. A characteristic of a unit
(such as a person’s weight, eye
color, or the response to a Harris Poll
question) is called a variable.
SAMPLING AND ESTIMATION
If a variable has only two possible values (such
as a response to a YES or NO question, or a
person’s sex) it is called a dichotomous
variable. If a variable assigns one of several
categories to each individual (such as person’s
blood type or hair color) it is called a
categorical variable. And if a variable assigns
a number to each individual (such as a person’s
age, family size, or weight), it is called a
quantitative variable.
SAMPLING AND ESTIMATION
A number derived from a sample is
called a statistic,
whereas a number derived from
the population is called a
parameter.
SAMPLING AND ESTIMATION
Parameters are is usually denoted by Greek
letters, such as π, for population
percentage of a dichotomous variable, or
μ, for population mean of a quantitative
variable. For the Harris study the sample
percentage p = 57% is a statistic. It is not
the (unknown) population percentage π,
which is the percentage that we would
obtain if it were possible to ask the same
question of the entire population.
SAMPLING AND ESTIMATION
SAMPLING AND ESTIMATION
Inferences we make about a population based
on facts derived from a sample are uncertain.
The statistic p is not the same as the
parameter π. In fact, if the study had been
repeated, even if it had been done at about
the same time and in the same way, it most
likely would have produced a different value
of p, whereas π would still be the same. The
Harris study acknowledges this variability by
mentioning a margin of error of ± 3%.
SIMULATION
Consider a box containing chips or cards,
each of which is numbered either 0 or 1.
We want to take a sample from this box in
order to estimate the percentage of the
cards that are numbered with a 1. The
population in this case is the box of cards,
which we will call the population box. The
percentage of cards in the box that are
numbered with a 1 is the parameter π.
SIMULATION
In the Harris study the parameter π is
unknown. Here, however, in order to see
how samples behave, we will make our
model with a known percentage of cards
numbered with a 1, say π = 60%. At the
same time we will estimate π, pretending
that we don’t know its value, by examining
25 cards in the box.
SIMULATION
We take a simple random sample with replacement
of 25 cards from the box as follows. Mix the box of
cards; choose one at random; record it; replace it;
and then repeat the procedure until we have
recorded the numbers on 25 cards. Although
survey samples are not generally drawn with
replacement, our simulation simplifies the analysis
because the box remains unchanged between
draws; so, after examining each card, the chance
of drawing a card numbered 1 on the following
draw is the same as it was for the previous draw, in
this case a 60% chance.
SIMULATION
Let’s say that after drawing the 25 cards this way,
we obtain the following results, recorded in 5
rows of 5 numbers:
ERROR ANALYSIS
An experiment is a procedure which
results in a measurement or
observation. The Harris poll is an
experiment which resulted in the
measurement (statistic) of 57%. An
experiment whose outcome depends
upon chance is called a random
experiment.
ERROR ANALYSIS
On repetition of such an experiment one
will typically obtain a different
measurement or observation. So, if the
Harris poll were to be repeated, the
new statistic would very likely differ
slightly from 57%. Each repetition is
called an execution or trial of the
experiment.
ERROR ANALYSIS
Suppose we made three more series of draws,
and the results were + 16%, + 0%, and +
12%. The random sampling errors of the four
simulations would then average out to:
ERROR ANALYSIS
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Note that the cancellation of the positive and
negative random errors results in a small average.
Actually with more trials, the average of the
random sampling errors tends to zero.
ERROR ANALYSIS
So in order to measure a “typical size” of a random
sampling error, we have to ignore the signs. We
could just take the mean of the absolute values
(MA) of the random sampling errors. For the four
random sampling errors above, the MA turns out to
be
ERROR ANALYSIS
The MA is difficult to deal with theoretically because
the absolute value function is not differentiable at
0. So in statistics, and error analysis in general, the
root mean square (RMS) of the random sampling
errors is generally used. For the four random
sampling errors above, the RMS is
ERROR ANALYSIS
The RMS is a more conservative
measure of the typical size of the
random sampling errors in the
sense that MA ≤ RMS.
ERROR ANALYSIS
For a given experiment the RMS of all possible
random sampling errors is called the standard
error (SE). For example, whenever we use a
random sample of size n and its percentages p to
estimate the population percentage π, we have