Download Medical Statistics

Medical Statistics
Practice 2
Tao Yuchun
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Review
I. Normal Distribution
• character
1. centrosymmetric
•The center is μ .
2. μ — position parameter
•Determine location of the peak (center) for a
normal distribution.
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3. σ — shape parameter
•Determine shape of a normal curve.
•A normal distribution with μ and σ denoted
with N (  ,  2 ) .
4. The area under the curve
•The area under the curve is probability.
•The area of certain range under the curve is:
•μ±1.96σ

•μ±2.58σ
95%
99%
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• Standard Normal Distribution
Z
X 

~ N (0,1)
• A normal distribution with μ=0 and σ=1
denoted with N (0,1) .
II. Reference Range
• definition
In health-related fields, a reference range (or reference values or interval) is a set of
values of some measurement that a physician or other health professional can use to
interpret a set of results for a particular patient. It is determined by collecting data
from vast numbers of laboratory tests. ----From Wikipedia, the free encyclopedia
• steps of establishment
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• Statistical methods
(1) For normal distributed data -- normal
Two sides (1-α)range:
X  Z / 2 S  X  X  Z / 2 S
One side (1-α)range:
X  X  Z S
or
X  X  Z S
•Two sides:
Z0.05/ 2  1.96, Z0.01/ 2  2.58
•One side:
Z 0.05  1.65, Z 0.01  2.33
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(2) For skew distributed data -- percentiles
Two sides (1-0.05)range:
P2.5  X  P97.5
One side (1-0.05)range:
X  P5
or
X  P95
•Two sides or one side according to professional
knowledge.
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III. Estimation of population parameter
1. The sampling error and standard
error of mean
•Sampling error
• Sampling error is related to the variation of
the population.
• Sampling error is also related to sample size.
2. The distribution of sample mean
•Central Limit Theorem
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X ~ N (  ,  X2 )
3. Standard error
S
 SX 
n
Standard error of mean
S is estimation of σ, S Xis estimation of  X .
4. t distribution
X  X 
t

SX
S/ n
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• character
(1) centrosymmetric
•The center is 0 .
(2) ν — shape parameter
•Determine shape of a t curve.
 •When ν is increasing, t curve is close to
standard normal curve.
(3) the area under t curve — t Table
t ,
α is probability, ν is degree of freedom, ν = n-1.
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5. Confidence Interval of Population Mean
Estimation
Statistical
inference
parameter
point estimation
interval estimation
Hypothesis testing
 Point estimation of population mean
-- sample mean
 Interval estimation of population mean
-- (1-α) confidence interval
 Confidence level: 1-α, such as 95% or 99%.
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•The formula of (1-α) confidence interval of
population mean for two sides is:

X  t / 2, S X
t / 2, came from the Table of t distribution.
Distinguish Standard Deviation from
Standard Error
You can see this table

Distinguish Reference Range from
Confidence Interval
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IV. Hypothesis testing
1. The idea and steps of Hypothesis testing
(1) The idea
• Null hypothesis and Alternative hypothesis
• P-value and α (level of a test)
• A small-probability event
• One-sided test or Two-sided test
• Test statistic and its Distribution
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 (2) The steps
A. Set hypotheses and the level of test
B. Select an appropriate test and calculate
the test statistic
C. Determine P-value, and make decision
• If P ≤α，then reject H0 at significance level
α=0.05.
• If P ＞α，then accept H0 at significance level
α=0.05.
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2. t tests (part)
(1) Comparing to a given population mean
( One-sample t test)
H0: μ＝μ0
H1: μ≠μ0
α= 0.05
X   X  0
t

~ t ( )
SX
S n
• tα,ν came from the Table of t distribution.
• When t ≥ tα,ν , then P ≤ α , reject H0 ;
• When t ＜ tα,ν , then P ＞ α , accept H0 .
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Excel’s statistical method
• You can use Excel’s statistical function
TINV(Probability,Deg_freedom) to get tα,ν,
here Probability is α，Deg_freedom is ν
(degree of freedom). It is for two sides.
• You can use the t-test method of the macro
of statistical analysis tools.
•See the example
[stat1(English).xls updated]
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Practice in class
Exercise 1: the blood-glucose(mmol/L) values from
12 randomly selected patients.
5.31, 6.12, 6.53, 6.53, 6.65, 6.66, 6.71, 6.93, 7.05,
7.15, 7.21, 7.35
Please to infer the population mean of the patients is
whether greater than or equal the standard value 6.1?
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Exercise 2: the frequency table of latent period (day)
from 110 certain infectious disease patients. Please
estimate the 95% Reference Range.
tab2 the frequency table of latent period (day) from some infectious disease patients
latent
period
(1)
2~
4~
6~
8~
10~
12~14
total
frequency(f )
(2)
Cumulative
Frequency(∑f )
(3)
Cumulative
Frequency(%)
(4)=(3)/n
26
48
25
6
3
2
26
74
99
105
108
110
23.64
67.27
90.00
95.45
98.18
100.00
110
－
－
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Exercise 3: the RBC of 144 healthy male adults,
got sample mean and SD: mean= 55.32×1012/L, SD
= 0.44×1012/L. Please estimate the 95% Reference
Range and 95% CI.
Answer
•See the Excel file
(practice2key.xls)
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Homework
1. The content (mg/L) of CaCo3 within a
material was independently measured 15 times,
resulting in:
20.99, 20.41, 20.62, 20.75, 20.10, 20.00, 20.80,
20.91, 22.60, 22.30, 20.99, 20.41, 20.50, 23.00,
22.60.
Please check whether the true value was
20.7mg/L ?
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2. The temperature（℃ ）for 102 female students
from certain college resulting in:
sample mean= 37.06 ℃, SD= 0.198 ℃
Please estimate the 95% Reference Range and 99%
CI?
C
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