Download Introduction to Biostatistics

H. James Norton, PhD
[email protected]
Website:
www.jimnortonphd.com
Introduction to
Biostatistics
A young fellow from
had committed a grievous crime.
He had murdered his
In Texas there is no excuse for murdering a
horse. If the jury finds you guilty there is only
one punishment. Hanging.
On the day of his hanging the warden talked to the
prisoner and said, “I am one of the few wardens that
follows the law of 1889. The law states that I was to
allow you your last meal of anything you wanted.
And I did that, didn’t I.”
The prisoner said, “Yes, steak and eggs.”
The warden said “And the law requires that I gather a
random sample of 100 Texans.” The warden dragged
the prisoner to the auditorium and sure enough there
were 100 Texans sitting there. The warden said,
“ The law requires that I now give you an hour to
speak to these people. You can discuss whatever
you want, for instance your guilt or innocence, or the
Texas prison system.”
The prisoner said, “I have nothing to say.”
Whereupon a professor in the audience stood
up and said, “ I don’t think the people of Texas
know enough about statistics. If you are not
going to use the hour, would you mind if I use
it to educate these people about statistics?”
The prisoner said, “ No, go ahead.”
But the prisoner turned to the warden
and said,
“BUT HANG ME FIRST!”
3 Types of Measurement Scales
• Interval Scale – the size of the difference between 2 values on
the scale has a meaning. For example, the difference between
1° F and 2° is identical to the difference between 100° and 101°.
• What are other examples of interval measurements a physician
would collect or make on a patient?
• Examples - age, blood pressure, heart rate, weight.
• Nominal (categorical) Scale – place the person or thing in a
category that is not ordered.
• Example – eye color puts people in a category but there is no
natural ordering.
• What are other examples of nominal measurements that would
be collected about a patient?
• Examples – gender, race, country of birth.
• Ordinal scale – Observations ore ordered but
differences may not have a meaning or the
difference between scores are not equal.
• Example #1 - Cancer Stages. The differences
between Stage 0, Stage I, Stage II, Stage III,
and Stage IV are not the same.
• Example #2 - the scale with choices strongly
disagrees, disagrees, neutral, agrees, and
strongly agrees.
• What is the name of this scale?
• The Likert Scale.
• Example #3 - What is the name of the scale
that is used to evaluate the health of a baby
sometimes taken a few minutes after birth?
APGAR SCORING
SIGN
0
1
2
A
COLOR
Blue pale
Body pink
Extremities blue
Completely pink
P
HEART RATE
Absent
Slow below 100/min
Over 100/min
G
EMEX RESPONSE
1. Response to
catheter in nostril
tested after
oropharynx is clear)
2. Tangential foot slap
No response
Grimace
Cough or sneeze
No response
Grimace
Cry and withdrawal
of foot
A
MUSCLE TONE
Limp
Some flexion of
extremities
Well Flexed
R
RESPIRATORY
EFFORT
Absent
Weak cry,
hypoventilation
Good strong cry
Hierarchy for scientific evidence in
medical studies
•
•
•
•
Case series
Retrospective case-control studies
Prospective observational studies
Randomized clinical trials
“Adenocarcinoma of the vagina:
Association of ?? with tumor appearance
in young women”
• Retrospective study of 8 young women
from Boston, ages 15-22, with
adnocarcinoma of the vagina
• Each case matched to 4 control women
born in the same hospital during the
same week of birth
• Ask students if they were epidemiologists,
“What questions would you ask the cases
and controls?”
• Family history of disease, sexual history,
drug use, co-morbid conditions, treatment
with chemo-therapy or radiation.
• What caused the disease?
• Mom’s use of diethylstilbestrol (DES)
during pregnancy (7/8 cases, 0/32
controls)
Doll & Hill’s two famous studies
• Case-control study of smoking and
lung cancer (1950)
• Prospective study of 34,000 British
physicians (1954). Detailed
questionnaires on smoking habits
• Student workbook published by CDC at
www.cdc.gov/eis/casestudies/xsmoke.s
tudent.731-703.pdf
Requirements for Gold Standard Clinical Trial
• Randomized
• Placebo controlled (if ethical)
• Else standard of care
• Double Blind
• Sufficient Power (adequate sample size)
From: Statistics Concepts and Controversies by David Moore
Contrast Nurses’ Heath Study (NHS)
& Women’s Health Initiative (WHI)
• NHS – observational cohort of 127,000
nurses ages 30 to 55
• WHI – randomized clinical trial of 50,000
women ages 50 to 79
• NHS – women taking estrogen after
menopause had reduced risk of heart
disease.
• WHI – stopped study prematurely. Women
taking Prempro had increased risk of heart
disease.
Descriptive Statistics
Suppose our sample Xi ‘s are:
2, 7, 1, 11, 2, 5, 2,10
Compute the mode, median, mean, range, variance, standard
deviation, and standard error of the mean.
Mode – value that occurs most often. Mode = 2 .
Range = (minimum value, maximum value). Range = (1,11).
Sample Mean = ∑ Xi / N where N = sample size.
Mean = (2+7+1+11+2+5+2+10) / 8 = 40/8 = 5.
Median is the middle most value of the data set after the data are
ordered from low to high.
If there are an odd number of data points such as 7, 5, -1, 10, 11
then order -1, 5, 7, 10, 11 and Median = 7.
With and even number of values, order the data and then take the
average of the two middle points.
1, 2, 2, 2, 2, 5, 7, 10, 11 Take the average of 2, 5. Median = is 3.5
Sample variance = S2 = ∑ (Xi – mean)2 / (N-1)
Xi
Xi - mean
(Xi – mean)2
2
2 – 5 = (-3)
(-3)2 = 9
7
7–5=2
22 = 4
1
1 – 5 = (-4)
(-4) 2 = 16
11
11 – 5 = 6
62 = 36
2
2 – 5 = (-3)
(-3) 2 = 9
5
5–5=0
02 = 0
2
2 – 5 = (-3)
(-3) 2 = 9
10
10 – 5 = 5
52 = 25
Total = 108
S2 = 108/7 = 15.43
standard deviation = s = √ s2
s
= √ 15.43 = 3.93
SEM = s / √ N
SEM = 3.93 / √ 8 = 1.39
Suppose data are 5, 5, 5, 5, 5, 5, 5, 5
What is the mean?
mean = (5+5+5+5+5+5+5+5)/8 = 5
What is the variance?
0
When will the variance of a set of number = 0 ?
Only if all the numbers are the same.
Suppose data are 2, 7, 1, 11, 2, 5, 2,10 and we add an outlier
or a 9th number to the data set that is = 1000.
What changes the most, the mode, median, or mean?
Mode stays the same = 2.
1, 2, 2, 2, 2, 5, 7, 10, 11, 1000 (median now 5)
Mean now 1040/9 = 115.56
From: Statistics Concepts and Controversies by David Moore
REALITY
Conclusion of Test
Null Hypothesis
True
Null Hypothesis
False
Do not
reject Ho
NO ERROR
TYPE II ERROR
Reject Ho
TYPE I ERROR
NO ERROR
Ho is the null hypothesis
α = Probability of a Type I error (most often chosen = .05)
β = Probability of a Type II error
Power = 1 - β
Examples of H0 & H1
Comparing 2 groups where outcome is on interval scale:
H0: μ1 = μ2 (the means are equal)
H1: μ1 ≠ μ2 (the means are ≠ )
Statistical test employed is Student’s t-test.
(Data for both groups must be normally distributed.)
Outcome variable is systolic blood pressure
after 6 months of treatment.
Patients randomized to diuretic or new drug.
Comparing 2 groups where outcome is dichotomous:
H0: P1 = P2 (the proportions are equal)
H1: P1 ≠ P2 (the proportions are ≠ )
(or equivalently)
H0: The two variables are independent
H1: The two variables are not independent
Statistical test employed is the chi-square test.
Outcome variable is whether or not infection is
cured after 2 weeks of treatment.
Children are randomized to once-daily or
twice-daily antibiotic.
Comparing 2 groups where outcome is ordinal scale.
H0: The two distributions are identical
H1: The two distributions are not identical
Statistical test employed is the Wilcoxon rank sum test.
Outcome variable is stage of colon cancer
upon diagnosis.
Do whites and African-Americans present with similar
stages of colon cancer at diagnosis?
Sampling
• Simple random sample – of n elements is a sample selected
from a population in such a manner that each combination of n
elements has the same chance or probability of being selected
as every other combination.
Examples – flipping coin, random, lottery drawings, random
number tables, computer generated “random numbers”.
• Systematic sample - Choosing every nth person in a list of N
people. Example – Suppose from a random number table the
number 3 is chosen. A sample of 100 people are chosen from a
list of 1000 people by taking the 3rd, 13th,23rd,33rd, …persons
from the list. Is this a simple random sample?
• Stratified sample – Divide the population into groups and then
take a random sample from within each of the groups. For
instance divide the population into males and females and then
randomly assigning them to a treatment or placebo.
• Haphazard sample or sample of convenience.
Hypothesis Testing
H0: Lucy is not a liar
H1: Lucy is a liar
What’s the p-value?
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