Statistical Description (1)

1. Statistical Description (1) Xiaojin Yu Introductory biostatistics http://www.hstathome.com/tjziyuan/Introductory%20Biostatistics%20Le%20C.T.%20%20(Wiley,%202003)(T)(551s).pdf Introductory biostatistics for the health science http://faculty.ksu.edu.sa/hisham/Documents/eBooks/Introductory_Biostatistics_for_the_Health.pdf

2. Review • What is Medical statistics about? • key terms in Statistics

3. 1.2 key words in Statistics Population(individual) & sample Variation & random variable Random Variable & data Statistic & parameter Sampling error Probability

4. Framework of statistical analysis sample representative, sampling error population individual, variation Randomly sampling Statistics known parameter unknown Statistical inference based on probabililty Statistical Description

5. Statistical Description CONTENTS • For quantitative(numerical) data • Frequency distribution • Measures of central tendency • Measures of dispersion • For qualitative(categorical) data

6. Raw Data (quantitative) Example: 120 values of height (cm) for 12-year-old boys in 1997: 142.3 156.6 142.7 145.7 138.2 141.6 142.5 130.5 134.5 148.8 134.4 148.8 137.9 151.3 140.8 149.8 145.2 141.8 146.8 135.1 150.3 133.1 142.7 143.9 151.1 144.0 145.4 146.2 143.3 156.3 141.9 140.7 141.2 141.5 148.8 140.1 150.6 139.5 146.4 143.8 143.5 139.2 144.7 139.3 141.9 147.8 140.5 138.9 134.7 147.3 138.1 140.2 137.4 145.1 145.8 147.9 150.8 144.5 137.1 147.1 142.9 134.9 143.6 142.3 125.9 132.7 152.9 147.9 141.8 141.4 140.9 141.4 160.9 154.2 137.9 139.9 149.7 147.5 136.9 148.1 134.7 138.5 138.9 137.7 138.5 139.6 143.5 142.9 129.4 142.5 141.2 148.9 154.0 147.7 152.3 146.6 132.1 145.9 146.7 144.0 135.5 144.4 143.4 137.4 143.6 150.0 143.3 146.5 149.0 142.1 140.2 145.4 142.4 148.9 146.7 139.2 139.6 142.4 138.7 139.9

7. Data Summary For continuous variable data • Numerical methods • Description of tendency of central • Description of dispersion • Tabular and graphical methods

8. Tabular & Graphical Methods • Frequency table • Histogram

9. FREQUENCY TABLE

10. SOLUTION TO EXAMPLE • 1.number of intervals k=10 • 2 calculate the width R=Xmax-Xmin= 160.9- 125.9=35 w=R/k W=35/10=3.5 • 3.form the intervals • 4.counting frequency • A recommended step is to present the proportion or relative frequency.

11. Class intervals

12. Tally and Counting 12

13. Final Frequency Table • A recommended step is to present the proportion or relative frequency. 13

14. Basic Steps to Form Frequency Table • step1: determining the number of intervals 5-15 • step2: calculating the width of intervals • Step3: forming intervals- certain range of values • Step4: count the number of observation with certain interval • the final table consists of the intervals and the frequencies.

15. 40 30 20 10 0 124 164 132 140 148 156 Frequency Figure 2.1 Distribution of heights of 120 boys from China,1997

16. Present data graphically presenting data • visually • intuitively • easy to read and understand • self-explanatory • stand alone from text Statistical table and graph are intended to communicate information, so it should be easy to read and understand. The shape of the distribution is the characteristic of the variable.

17. Application • One lead to a research question • concerns unimodal and symmetry of the distribution

18. Shape of frequency distribution Distribution • Unimodal/bimodal • Symmetry /skew 18

19. Homogeneous /heterogeneous The definition of population or the classification is approapriate. Unimodal/bimodal

20. SYMMETRY & SKEWNESS • Symmetric means the distribution has the same shape on both side of the peak location. • Skewness means the lack of symmetry in a probability distribution. (The Cambridge Dictionary of Statistics in the Medical Sciences.) • An asymmetric distribution is called skew. (Armitage: Statistical Methods in Medical Research.)

21. Figure 2.2 Symmetric And Asymmetric Distribution negative skewness positive skewness

22. Positive & Negative Skewness • A distribution is said to have positive skewness when it has a long thin tail at the right, and to have negative skewnesswhen it has a long thin tail to the left. • A distribution which the upper tail is longer than the low, would be called positively skew.

23. 70 60 50 40 30 20 10 0 1 3 5 7 9 11 13 15 17 19 21 Hg (umol/kg) Fig. The distribution of Hg (hydrargyrum) of 237 adults hair Frequency

24. 400 300 200 100 0 Frequency 0 10 20 30 40 50 60 70 80 90 100 QOL Fig. The distribution of scores of QOL (quality of life ) of 892 senior citizen

25. 40 30 20 10 0 1 5 10 15 20 25 30 35 40 45 Survival time (month) Fig. The distribution of survival times for 102 malignant melanoma patients(恶性黑素瘤) Frequency

26. Frequency 2500 2000 1500 1000 500 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 Age at death (year) Fig. The distribution of ages at death of males in 1990~1992

27. Numerical methods • Central tendency • Tendency of dispersion • arithmetic mean, Median, geometric mean • range, interquartile range, standard deviation, variance, coefficient of variation,

28. Mean • Concept and notation • Calculation • Application

29. CONCEPT OF MEAN • Arithmetic mean, mean • Population mean μ • The Sample mean will be denoted by x (‘‘x-bar’’).

30. CALCULATION OF MEAN • given a data set of size n {x1,x2,…，xn}, • The mean is computed by summing all the x’s and divided the sum by n. symbolically

31. GROUPED DATA • The mean can be approximated using the formula • Where f denotes the frequency ,m the interval midpoint ,and the summation is across the intervals.

32. Midpoint • The midpoint for an interval is obtained by calculating the average of the interval lower true boundary and the upper true boundary. • The midpoint for the first interval is • The midpoint for the second interval is

33. Example 1

34. Average: Limitation in describing data It has been said that a fellow with one leg frozen in ice and the other leg in boiling water is comfortable ON AVERAGE !

35. Geometric Mean-notation • The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers. • G /GM

36. Geometric Mean-calculation • As the definition, the expression is • Example like, the G for 2, 4, 8(n=3) should be like:

37. Geometric mean: 37

38. Geometric Mean-calculation • Example1_geo given a data set consisting of survival times to relapse in weeks of 21 acute leukemia patients that received some drug. • 1,1,2,2,3,4,4,5,5,8,8,8,8,11,11,12,12,15,17,22,23(n=21) • The mean is 8.67 weeks 38

39. Ex. Serum HI antibody dilution from 107 testees after measles vaccination hemagglutination inhibition(HI)

40. application • positive skew data_if log transformation creates symmetric, unimodal • Geometric series.

41. Median • Concept of median • Calculation • Application-disadvantage $ advantage

42. Concept of Median • If the data are arranged in increasing or decreasing order, the median is the middle value, which divided the set into equal halves. M sample median

43. Calculation-how do we get it? Example1 n=11 M=56 a. When n is odd,

44. Calculation-how do we get it? • Example2 n=12 M=(56+58)/2=57 b. When n is even

45. Application-Advantage It is robust to the extreme value. Mean=58.42 mean=149.9 Median=57

46. Application-when is it used? Fig.A skew distribution.

47. Data described by Median • Skew data • Normal distribution data • Ordinal data!!

48. For normal distribution

49. Figure 3 the average of height of basketball players.

50. Disadvantage of median • the precise magnitude of most of the observations are not taken. • if two groups of observations are pooled, the median of the combined group cannot be expressed in terms of the medians of the two component.