EPIDEMIOLOGY AND BIOSTATISTICS REVIEW, PART I

Transcription

EPIDEMIOLOGY AND
BIOSTATISTICS
REVIEW, PART I
Tommy Byrd MSII
http://www.usmle.org/pdfs/step-1/2013midMay2014_Step1.pdf
Know the 4 scales of data measurement
• Nominal
• Ordinal
• Interval
• Ratio
Nominal scale data are divided into
qualitative categories or groups
Male
Female
Black
White
Suburban
Rural
Ordinal scale data has an order
•  Class rankings data (1st / 2nd / 3rd…)
•  Answers to these types of questions:
**But it does not describe the size of the interval (eg. it
cannot tell by how many percentage points Tommy is
ranked 1st in his class)
Interval scale data has order and a set
interval
• Celsius (and Fahrenheit)
temperatures
• Anno Domini years (1990,
1991, 1992, etc.)
**But ratios of this kind of data are not meaningful
•  100°C is not twice as hot as 50°C because 0°C does not
indicate a complete absence of heat
Ratio scale data has order, a set interval,
and is based on an absolute zero
•  Kelvin temperatures
•  MOST BIOMEDICAL VARIABLES
•  Weight (grams, pounds)
•  Time (seconds, days à ‘zero’ is the starting point of measurement)
•  Age (years)
•  Blood pressure (mmHg)
•  Pulse (beats per minute)
•  With these types of data ratios are valid:
•  300K is twice as hot as 150K
•  A pulse rate of 120 beats/min is twice as fast as a pulse rate of 60
beats/min
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Many naturally occurring phenomena are
distributed in the bell-shaped normal or
Gaussian distribution
Score
(Blood pressure, cholesterol, etc.)
Skewed distributions are described by the
location of the tail of the curve, not the
location of the hump
a.k.a. “Left skew”
a.k.a. “Right skew”
Know the measures of central tendency
• Mode
• Median
• Mean
Score
Mode is the value that occurs with the
greatest frequency
2 4 5 7 4 2 3 6 8 9 7 5 4 4 2 4 6 7 7 7
Bimodal
distribution!
2
3
4
5
6
7
8
9
Median is the value that divides the
distribution in half
•  Odd # total elements: the median is the middle one
•  Even # total elements: the median is the average of the
two middle ones
**Very useful measure of central tendency for highly skewed distributions
Mean (the average) is the sum of all
values divided by the total # of values
•  Unlike median and mode, it is very sensitive to extreme
scores
•  Therefore NOT good for measuring skewed distributions
•  Repeated samples drawn from the same population will
tend to have very similar means
•  Therefore the mean is the measure of central tendency that BEST
resists the influence of fluctuation between different samples
Match the mean, median, and mode each
with its corresponding hash mark
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Glaser, Anthony N. High-yield Biostatistics, Epidemiology,
& Public Health. N.p.: n.p., n.d. 9. Print.
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Normal distributions with identical
measures of central tendency can have
different variabilities
•  Variability = the extent to which their scores are clustered
together or scattered about
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again.
How do we measure this variability???
Standard deviation (σ) measures how far
away, on average, that values lay away
from the mean of the population
•  Remember the last infectious disease quiz?
•  Let’s assume the mean (average) grade was a 70% with a normal
distribution
•  If the σ was really HIGH, there was probably a bunch of A’s and a bunch
So, since
gun hard,
how
we use standard deviation
of F’s we
in addition
to B’s and
C’scan
and D’s
•  Ifexactly
the σ washow
reallywe
LOW,
people probably to
goteverybody
a high D or low
C
to tell
didmost
in comparison
else?
By MEMORIZING these numbers!
•  Approx. 68% of the distribution falls within ±1 standard deviations
•  Approx. 95% of the distribution falls within ±2 standard deviations
•  Approx. 99.7% of the distribution falls within ±3 standard deviations
So, out of a
class of 100,
about how
many people
got an A?
(assume
extra credit
was possible)
A)  9-11
B)  2-3
C)  14-16
D)  4-6
E)  19-21
Therefore,
assuming the σ
of the test
scores was 10
points, we can
assume the
following:
Grade (%)
The z score is simply how many standard
deviations the element lies above or
below the mean
A table of z scores
compares the z score to
the “Area beyond Z”…
65
z = − 0.5
Grade (%)
85
z = + 1.5
The z score is simply how many standard
deviations the element lies above or
below the mean
A table of z scores
compares the z score to
the “Area beyond Z”…
6.7% got ‘beyond’ an 85%
on our startlingly realistic,
made-up test
~7
people
here
Therefore the z score can be used to
specify probability
We know that 6.7% of the class
has a grade above 85%, so the
probability of one randomly
selected person from this
population having a grade above
85% is 6.7%, or 0.067
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What if we don’t know every single
person’s score on the test?
•  But, through some stealthy looking-over-shoulders while
people check their online test scores, we can get a
sample of random scores
•  How close to the actual class average will our sample be?
One sample
representing
one score
The # of times that
the average of a
sample of 4 scores
is ~80%
n = the size of
each sample
0%
70%
100%
The standard error of the mean (SEM)
is the standard deviation over the square
root of the sample size
SEM = σ/√n
SEM = 10/√1 = 10
Recall that the
standard
deviation (σ) of
this test was 10
percentage
points
SEM = 10/√4 = 5
SEM = 10/√7 = 3.8
SEM = 10/√10 = 3.2
0%
70%
100%
Standard error (SEM) can be used in the
same way as standard deviation
•  But remember that SEM decreases as n é
•  Now we have gathered a sample of 10 random scores
from our classmates, so:
SEM = σ/√n
SEM = 10/√10 = 3.2
**Do you remember how much of the population falls within 2 standard
deviations (or SEMs) of the mean?
95% confidence limits are
approximately equal to the sample mean
plus or minus 2 standard errors
Practically, the 95%
confidence interval is
the range in which the
means 95% of
samples would be
expected to fall
In other words,
there is a 95%
chance that the
average of our
random sample
would be in this
range
µ − 3 SEM µ − 2 SEM µ − 1 SEM
µ + 1 SEM µ + 2 SEM µ + 3 SEM
95% confidence limits are
approximately equal to the sample mean
plus or minus 2 standard errors
•  Remember, the σ on our test was 10%, and the mean was
a 70%. We are randomly sampling 10 scores (n=10)
•  So the standard error (SEM) = σ/√n = 10/√10 = 3.2%
•  We just decided that our sample has a 95% chance of
falling within 2 SEMs of the average
•  So our 95% confidence interval is 70% ± 2(SEM)
= 70% ± 2(3.2%) = 70% ± 6.4%
= 63.6% - 76.4%
A random sample of 10 people’s scores on this test has
a 95% chance of averaging between 63.6% and 76.4%
The width of the confidence interval reflects precision
How would we double the precision of an
estimate?
• Double the sample size?
• We need to quadruple the sample
size!
SEM = σ/√n
If we do not know the σ of our population,
can we still calculate SEM?
•  Pretend we don’t have any fancy ExamSoft statistics from
our test, only our sample of 10 scores
•  We can calculate the standard deviation of the 10 scores
in our sample (S), and substitute it in for σ in the SEM
equation to come up with the estimated standard error of
the mean
Estimated standard error = S / √n
The t score is to the z score as the
estimated standard error is to the σ
Similar to P – values !
For USMLE
purposes,
consider
degrees of
freedom
(df) to
equal n-1
So what do
we do with
all this?
t = the number of estimated standard
errors away from the sample mean
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There are 7 steps in hypothesis testing
•  1) State the null and alternative hypothesis, H0 and HA
•  H0 = no difference
•  HA = there is a difference
•  2) Select the decision criterion α (“level of significance”)
•  3) Establish the critical values of t
•  4) Draw a random sample, find its mean
•  5) Calculate the standard deviation of the sample (S) and
find the estimated standard error of the sample
•  6) Calculate the value of the test statistic t that
corresponds to the mean of the sample (tcalc)
•  7) Compare the calculated value of t with the critical
values of t, then accept or reject the null hypothesis
Step 1: State the null and alternative
hypotheses
•  We want to test Julia Silva’s claim: “Because of Tommy
and Danielle’s amazing biostats presentation, the average
Step 1 score of our class will be 260”
•  Null hypothesis = The mean score is 260
•  Alternative hypothesis = The mean score is not 260
•  We could ask for the score of every student, but we would
rather take a random representative sample so we can
save time
•  Again, our sample size will be 10 randomly selected students
Step 2: Select the decision criteria α
•  Random sampling error (this is normal) will always cause
our sample mean to deviate slightly from the true mean
•  We have to decide what an acceptable level of this chance
deviation is
•  α is conventionally set at 0.05
•  If the probability of obtaining the sample mean is greater than 0.05,
H0 is accepted:
•  The class indeed scored an average of 260
•  If the probability of obtaining the sample mean less than 0.05, H0 is
rejected:
•  The class average is either above or below 260
Step 3: Establish the critical values of t
α = 0.05
Sample size
(n) = 10
students, so
df = 9
So tcrit = ±2.262
Step 4: Draw a random sample and
calculate the mean of the sample
284 234 268 254 246 264 266 265 245 244 Average = 257
Step 5: Calculate standard deviation and
estimated standard error of the sample
•  In our sample, standard deviation (S) = 15
•  (You don’t have to know the equation for standard deviation on the
USMLE)
•  Estimated standard error = S / √n
= 15 / √10
= 4.747
Step 6: Calculate t from the data
•  Remember, similar to a z-value, the t-score represents the
# of estimated standard means that the sample mean lays
away from the hypothesized mean
•  Our average score was 257, which is 3 points away from
our hypothesized average of 260
•  Therefore, our t-value is “the # of estimated standard errors
contained in 3 points”
•  Our estimated standard error from the last slide is 4.747
•  This gives a t-score (tcalc) of:
−3
/ 4.747 = −0.632
Step 7: Compare t-values and be very
concerned that Julia Silva is a psychic
•  Our calculated t-value (same thing as t-score) is −0.632
•  Our critical t-value is ±2.262
•  Clearly our calculated t lies between +2.2 and −2.2,
therefore:
•  H0 is accepted and reported as follows: “The hypothesis that the
mean Step 1 score of the medschool class is 260 was accepted,
t = 0.632, df = 9, p ≤ 0.5”
+2.262
−2.262
★
t=0
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Error types indicate that you accepted the
wrong hypothesis
Type I Error
•  “False-positive” error
•  You accept the alternative
hypothesis when there is no
difference
•  Also known as alpha (α)
error à yes, this is
referring to the α we just
talked about
•  The p-value is the
probability of making a
type I error
Type II Error
•  “False-negative” error
•  You fail to reject the null
hypothesis when there
actually is a difference
•  Also known as β error
•  β is the probability of
making a type II error
A study with greater power has less
type II (β) error
•  The power of a statistical test = 1 − β
•  The power represents the probability of rejecting the null
hypothesis when it is in fact false (vs. accepting it in β
error); we want this to happen!
•  Conventionally, a study is required to have a power of 0.8
(or a β of 0.2) to be acceptable
•  Power increases as α increases à trade off
•  High-yield point: Increasing the sample size is the
most practical and important way of increasing the
power of a statistical test
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Nonexperimental (descriptive or analytic)
study designs – Cohort studies
•  Group without disease are selected and followed for an
extended period
•  Some members may have already been exposed to risk
factor
•  Exception: “Inception Cohorts” follow those recently
diagnosed to track progression
•  Can estimate incidence
•  Not good for rare diseases
•  Historical cohort study = retrospective cohort study
study designs – Case-control studies
•  All are retrospective
•  Compare people who do have the disease (the cases) w/
otherwise similar people who do not have the disease
•  Start w/ outcome then LOOK BACK into the past for
possible independent variables that may have caused the
disease
•  Cheap, good for rare or that take a long time to develop
study designs – Case-series studies
•  Essentially a series of case reports that may link disease
to exposure, but NOT controlled, as in case-control (no
group w/o the disease compared to)
•  Eg. Kaposis’s sarcoma
study designs – Prevalence survey
•  Survey (“snap shot”) of a whole population, also asks
about risk factors individually
•  Prevalence ratio = the prevalence of a disease in people
who have and have not been exposed to a risk factor
•  Likely to overrepresent chronic diseases and
underrepresent acute diseases
study designs – Ecological studies
•  Check non-individual info (eg. study of the rate of
diabetes in countries with different levels of automobile
ownership)
•  May be experimental:
•  Community intervention trials
•  Experimental group consists of an entire community, while the control
group is an otherwise similar community that is not subject to any kind
of intervention
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Bias occurs from systemic (rather than
random) errors when one outcome is
systematically favored over another
e
What is th
difference
between
bias
n
o
i
t
c
e
l
e
s
ling
and samp
bias?
(Magazine
subscribers in
great
depression)
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(Referral bias)
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(Putting all whites in drug
group and blacks in
control group for treating a
racially selective disease)
Race = confounding
variable
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or the image may have been corrupted. Restart your computer, and then open the file again. If the
red x still appears, you may have to delete the image and then insert it again.
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♯
♯
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EPIDEMIOLOGY AND BIOSTATISTICS REVIEW, PART I

Transcription

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