1. 2014.3.13 1 Medical Statistics Medical Statistics Tao Yuchun Tao Yuchun 5 http://cc.jlu.edu.cn/ms.html

2. 2014.3.13 2 Statistical inference Statistical inference 1. Estimation of population parameter

3. 2014.3.13 3 1.1 Sampling error and standard error of mean Sampling Study Sampling error Sampling error: Sample → sample mean (different from population mean) Different samples → Different sample means (different from each other )

4. 2014.3.13 4 (1) Sampling error is related to the variation of the population variation of the population No variation, no error, also no Sampling error! Example Example: The sample means of systolic blood pressure. For adult population (age 25~90) -- vary substantially For young population (age 18~25) -- not vary too much No variation, no statistics, too !

5. 2014.3.13 5 (2) Sampling error is also related to sample size sample size If sample size = population size there is no sampling error! If sample size = 1 Sampling error ≡ variation of population!

6. 2014.3.13 6 How changes for sample mean? See simulative experiment below Sampling from N(4.6602, 0.5746 2 ) by computer (100 times)

7. 2014.3.13 7

8. 8 Sampling from a skew distribution by computer 123456789 (a) 1234578 n=5 (b) 123456789 n=10 (c) 123456789 n=20 (d) 123456789 n=30 (e)

9. 2014.3.13 9 1.2 The distribution of sample mean If the variable ~ a normal distribution sample means ~ a normal distribution If the distribution of variable ~ skew, For small sample distribution of sample mean – skew For large sample sample mean close to a normal distribution --Came from Central Limit Theorem

10. 2014.3.13 10 1.3 standard error Standard deviation of the population: Standard deviation of the sample mean or Standard error of sample mean or Standard error: In any case: Standard error of sample mean = or

11. 2014.3.13 11 For application S is estimation of σ, is estimation of.

12. 2014.3.13 12 1.4 Student’s t distribution The t distribution was discovered William S. Gosset by William S. Gosset in 1908. “Student” is his pen name. 1876 - 1937  For a normal distribution William S. Gosset If Z follows a standard normal distribution ---N(0,1).

13. 2014.3.13 13 When σ is unknown, t follows a t distribution. t curve

14. 2014.3.13 14 The Property of t DistributionThe Property of t Distribution I. centrosymmetric Center is 0. II. ν — shape parameter degree of freedom also called degree of freedom, ν = n-1. determine shape of a t curve. different ν, different t curve. When ν is increasing, t curve is close to standard normal curve; when ν →∞, t curve became standard normal curve. See this animation In statistics, the number of degrees of freedomstatistics is the number of values in the final calculation of a statistic that are free to vary. --Came from Wikipedia

15. 2014.3.13 15 The different t curves ν= ∞ (standard normal curve) ν= 4 ν= 1 f(t)f(t)

16. 2014.3.13 16 III. The area under the t curve The Table for t distribution. t value denotes, α is probability, ν is degree of freedom, ν = n-1. The area under the t curve means: One side ： P(t≤-t α,ν )=α or P(t≥t α,ν )=α Two sides ： P(t≤-t α,ν )+P(t≥t α,ν )=α See next figure

17. 2014.3.13 17 ν The meanings of the area under the t curve for two sides

18. 2014.3.13 18 1.5 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%.

19. 2014.3.13 19 From P(t≤-t α,ν )+P(t≥t α,ν )=α We can get (1- α) confidence intervalIt is the formula of (1- α) confidence interval of population mean of population mean for two sides.

20. 2014.3.13 20 (1- α) confidence interval of population mean95% CI99% CI mean can abbreviate to 95% CI or 99% CI. Whenever we get a mean and standard deviation from a sample, put them into then The two extreme values are called confidence limits.

21. 2014.3.13 21 Example Systolic blood pressures of 20 healthy males were measured. What is 95% confidence interval of the population mean? came from the Table of t distribution

22. 2014.3.13 22 95% CI: What does “confidence interval” mean? (1-α) CI Not include μ

23. 2014.3.13 23 C  You should know  You should know: Once you got a 95% confidence interval of the certain population mean, the μ for this population may be in it, also may not be in it, but the probability probability being in it is 95% ! (Guilin Pagodas http://en.wikipedia.org/wiki/Guilin)

24. 2014.3.13 24 In figure, the red curve is standard normal curve ， the blue curve is t curve ， df is ν (degree of freedom).