1. 2011.5.22 1 Medical Statistics Medical Statistics Tao Yuchun Tao Yuchun Practice 2

2. 2011.5.22 2 Review 1. 1. centrosymmetric 2. 2. μ — position parameter I. Normal Distribution character The center is μ. Determine location of the peak (center) for a normal distribution.

3. 2011.5.22 3 3. 3. σ — shape parameter Determine shape of a normal curve. A normal distribution with μ and σ denoted with. 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% 

4. 2011.5.22 4 Standard Normal Distribution A normal distribution with μ=0 and σ=1 denoted with. II. Reference 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

5. 2011.5.22 5 Statistical methods (1) For normal distributed data -- normal Two sides (1-α)range: One side (1-α)range: or Two sides: One side: 

6. 2011.5.22 6 (2) For skew distributed data -- percentiles Two sides (1-0.05)range: One side (1-0.05)range: or Two sides or one side according to professional knowledge.

7. 2011.5.22 7 III. Estimation 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

8. 2011.5.22 8 3. 3. Standard error S is estimation of σ, is estimation of.  4. 4. t distribution Standard error of mean

9. 2011.5.22 9 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.

10. 2011.5.22 10 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%.

11. 2011.5.22 11 (1-α) confidence interval of population meanThe formula of (1-α) confidence interval of population mean for two sides is:  came from the Table of t distribution. Reference Range Distinguish Reference Range from Confidence Interval Standard Deviation Distinguish Standard Deviation from Standard Error Standard Error You can see this table  

12. 2011.5.22 12 IV. Hypothesis testing IV. Hypothesis testing 1. 1. The idea and steps of Hypothesis testing (1) (1) The idea Null hypothesis Alternative hypothesis Null hypothesis and Alternative hypothesis P-value α () P-value and α (level of a test) A small-probability event A small-probability event One-sided test Two-sided test One-sided test or Two-sided test Test statistic Distribution Test statistic and its Distribution

13. 2011.5.22 13 A.Set hypotheses and the level of test B.Select an appropriate test and calculate the test statistic the test statistic C.Determine P-value, and make decision (2) (2) The steps If P ≤α ， then reject H 0 at significance level If P ≤α ， then reject H 0 at significance level α=0.05. α=0.05. If P ＞ α ， then accept H 0 at significance level If P ＞ α ， then accept H 0 at significance level α=0.05. α=0.05. 

14. 2011.5.22 14 2. 2. t tests (part) (1) (1) Comparing to a given population mean ( One-sample t test) H 0 : μ ＝ μ 0 H 1 : μ ≠ μ 0 α= 0.05 When t ≥ t α,ν, then P ≤ α, reject H 0 ; When t ＜ t α,ν, then P ＞ α, accept H 0. t α,ν came from the Table of t distribution.

15. 2011.5.22 15 Excel’s statistical method You can use Excel’s statistical function TINV(Probability,Deg_freedom) TINV(Probability,Deg_freedom) to get t α,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 updated [ stat1(English).xls updated ]

16. 2011.5.22 16 Practice in class Exercise 1 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?

17. 2011.5.22 17 Exercise 2 Exercise 2: the frequency table of latent period (day) from 110 certain infectious disease patients. Please estimate the 95% Reference Range.

18. 2011.5.22 18 Answer Answer See the Excel file ( practice2key.xls ) Exercise 3 Exercise 3: the RBC of 144 healthy male adults, got sample mean and SD: mean= 55.32 × 10 12 / L, SD = 0.44 × 10 12 / L. Please estimate the 95% Reference Range and 95% CI.

19. 2011.5.22 19 Homework 1. 1. The content (mg/L) of CaCo 3 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 ?

20. 2011.5.22 20 C 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?

21. 2011.5.22 21