2011.5.221

Medical StatisticsMedical Statistics

Tao YuchunTao Yuchun

Practice 2Practice 2

2011.5.222

Review Review

1.1. centrosymmetric

2.2. μ — position parameter

I.I. Normal Distribution Normal Distribution

• character

•The center is μ .

•Determine location of the peak (center) for a

normal distribution.

2011.5.223

3.3. σ — shape parameter

•Determine shape of a normal curve.

•A normal distribution with μ and σ denoted

with . ),( 2N

4.4. The area under the curve

•The area under the curve is probability.•The area of certain range under the curve is:

•μ±1.96σ 95%•μ±2.58σ 99%

2011.5.224

• Standard Normal Distribution

)1,0(~ NX

Z

• A normal distribution with μ=0 and σ=1

denoted with . )1,0(N

II.II. Reference RangeReference Range

• definition • steps of establishment

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

2011.5.225

• Statistical methods

(1)(1) For normal distributed data -- normalFor normal distributed data -- normal

Two sides (1-α)range:

One side (1-α)range:

SZXXSZX 2/2/

SZXX SZXX or

58.2,96.1 2/01.02/05.0 ZZ•Two sides:

•One side: 33.2,65.1 01.005.0 ZZ

2011.5.226

(2)(2) For skew distributed data -- percentilesFor skew distributed data -- percentiles

Two sides (1-0.05)range:

One side (1-0.05)range:5.975.2 PXP

5PX 95PX or

•Two sides or one side according to professional knowledge.

2011.5.227

III. III. Estimation of population parameterEstimation of population parameter

• Sampling error is related to the variation of

the population.

• Sampling error is also related to sample size.

1.1. The sampling error and standard

error of mean

•Sampling errorSampling error

2.2. The distribution of sample mean

•Central Limit Theorem

2011.5.228

),(~ 2XNX

3.3. Standard error

n

SSX

S is estimation of σ, is estimation of .XS X

4. 4. t distribution

nS

X

S

Xt

X /

Standard error of mean

2011.5.229

• character

(1)(1) centrosymmetric

(2)(2) ν — shape parameter

•The center is 0 .

•Determine shape of a t curve.•When ν is increasing, t curve is close to

standard normal curve.

(3)(3) the area under t curve — t Table

α is probability, ν is degree of freedom, ν = n-1. ,t

2011.5.2210

5.5. Confidence Interval of Population Mean

Statistical

inference

Estimation

parameter

Hypothesis testing

point estimation

interval estimation

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%.

2011.5.2211

•The formula of (1-(1-αα) confidence interval of ) confidence interval of population meanpopulation mean for two sides is:

XStX ,2/

came from the Table of t distribution. ,2/t

Distinguish Reference RangeReference Range from Confidence IntervalConfidence Interval

Distinguish Standard Deviation Standard Deviation from

Standard ErrorStandard Error You can see this table

2011.5.2212

IV. Hypothesis testingIV. Hypothesis testing

1.1. The idea and steps of Hypothesis testing

(1)(1) The idea

• Null hypothesisNull hypothesis and Alternative hypothesisAlternative hypothesis

• PP-value -value and α α ((level of a test))

• A small-probability eventA small-probability event

• One-sided test One-sided test or Two-sided testTwo-sided test

• Test statistic Test statistic and its DistributionDistribution

2011.5.2213

A.A. Set hypotheses and the level of testSet hypotheses and the level of test

B.B. Select an appropriate test and calculate Select an appropriate test and calculate

the test statisticthe test statistic

C.C. Determine Determine PP-value, and make decision -value, and make decision

(2)(2) The steps

• IfIf P P ≤ ≤αα ，， then reject then reject HH0 0 at significance levelat significance level

αα=0.05.=0.05.

• IfIf P P ＞＞ αα ，， then accept then accept HH0 0 at significance levelat significance level

αα=0.05.=0.05.

2011.5.2214

2. 2. t tests (part)

(1)(1) Comparing to a given population mean

( One-sample t test)H0: μ ＝ μ0

H1: μ≠ μ0

α= 0.05

)(~0 t

nS

X

S

Xt

X

• When t ≥ tα,ν , then P ≤ α , reject H0 ;

• When t ＜ tα,ν , then P ＞ α , accept H0 .

• tα,ν came from the Table of t distribution.

2011.5.2215

Excel’s statistical method Excel’s statistical method • You can use Excel’s statistical function

TINV(Probability,Deg_freedom) 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 updatedupdated]

2011.5.2216

Practice in class Practice in class Exercise 1Exercise 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?

2011.5.2217

Exercise 2Exercise 2: the frequency table of latent period (day)

from 110 certain infectious disease patients. Please

estimate the 95% Reference Range.

(1) (2)

2~ 26 26 23.644~ 48 74 67.276~ 25 99 90.008~ 6 105 95.45

10~ 3 108 98.1812~14 2 110 100.00

total 110 － －

(4)=(3)/n

tab2 the frequency table of latent period (day) from some infectious disease patients

latentperiod

CumulativeFrequency(∑ f )frequency(f )

CumulativeFrequency(%)

(3)

2011.5.2218

Answer Answer

•See the Excel file (practice2key.xls)

Exercise 3Exercise 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.

2011.5.2219

HomeworkHomework 1.1. 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 ?

2011.5.2220

CC

2.2. 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?

2011.5.2221