Biostatistics ppt.pptx

1. University of Gondar College of medicine and health science Department of Epidemiology and Biostatistics Basic Biostatistics Wullo S. (BSc, MPH, Assistant professor) Tuesday, February 7, 2023 1

2. Chapter One 1.1 Introduction to Biostatistics  Objectives of the chapter  After completing this chapter, the student will be able to: – Define Statistics and Biostatistics – Identify the Branch of biostatistics – Enumerate the importance and limitations of biostatistics – Define and Identify the different types of data and understand why we need to classify variables 2 2/7/2023

3. Definition and classification of Biostatistics  Statistics is the science of  collecting  organizing  Presenting  analysing and drawing conclusion (inferences) from data for the purpose of making decision.  Biostatistics: The application of statistical methods to the fields of biological and health sciences. 3 2/7/2023

4. Classification of Biostatistics Descriptive biostatistics  A statistical method that is concerned with the collection, organization, summarization, and analysis of data from a sample of population. Inferential biostatistics  A statistical method that is concerned with the drawing conclusions/inference about a particular population by selecting and measuring a random sample from the population. 4 2/7/2023

5. Cont… collection organizing sum m arizing presentingof data DescriptiveStatistics m akinginferences hypothesistesting determ iningrelationship m akingtheprediction Inferential Statistics Biostatistics 5 2/7/2023

6. Descriptive Biostatistics • Some statistical summaries which are especially common in descriptive analyses are:  Measures of central tendency  Measures of dispersion  Measures of association  Cross-tabulation, contingency table  Histogram  Quantile, Q-Q plot  Scatter plot  Box plot 2/7/2023 6

7. Inferential Biostatistics 7 2/7/2023

8. 1.2 Stages in statistical investigation There are five stages or steps in any statistical investigation. 1. Collection of data  The process of obtaining measurements or counts. 2. Organization of data Includes editing, classifying, and tabulating the data collected. 3. Presentation of data: overall view of what the data actually looks like. facilitate further statistical analysis. Can be done in the form of tables and graphs or diagrams. 8 2/7/2023

9. Cont… 4. Analysis of data  To dig out useful information for decision making  It involves extracting relevant information from the data (like mean, median, mode, range, variance…), 5. Interpretation of data  Concerned with drawing conclusions from the data collected and analyzed; and giving meaning to analysis results.  A difficult task and requires a high degree of skill and experience. 9 2/7/2023

10. 1.3 Definition of Some Basic terms Population: is the complete set of possible measurements for which inferences are to be made. Census: a complete enumeration of the population. But in most real problems it cannot be realized, hence we take sample. Sample: A sample from a population is the set of measurements that are actually collected in the course of an investigation. Parameter: Characteristic or measure obtained from a population. Statistic: A statistic (rather than the filed of Statistics) refers to a numerical quantity computed from sample data (e.g. the mean, the median, the maximum). 10 2/7/2023

11. Parameter and statistic 11 2/7/2023

12. Cont... Sampling: The process or method of sample selection from the population. Sample size: The number of elements or observation to be included in the sample. variable is a characteristic or attribute that can assume different values in different persons, places, or things. Some examples of variables include:  Diastolic blood pressure,  heart rate, heights,  The weights Data: Refers to a collection of facts, values, observations, or measurements that the variables can assume. 12 2/7/2023

13. Uses of statistics: The main function of statistics is to enlarge our knowledge of complex phenomena. The following are some uses of statistics:  Estimating unknown population characteristics.  Testing and formulating of hypothesis.  Studying the relationship between two or more variable.  Forecasting future events.  Measuring the magnitude of variations in data.  Furnishes a technique of comparison. 13 2/7/2023

14. Limitations of statistics As a science statistics has its own limitations. The following are some of the limitations:  Deals with only quantitative information.  Deals with only aggregate of facts and not with individual data items.  Statistical data are only approximately and not mathematical correct.  Statistics can be easily misused and therefore should be used be experts. 14 2/7/2023

15. 1.5 Types of Variables and Measurement Scales A variable is a characteristic or attribute that can assume different values in different persons, places, or things. Examples :  age,  diastolic blood pressure,  heart rate,  the height of adult males,  the weights of preschool children,  gender of Biostatistics students,  marital status of instructors at University of Gondar,  ethnic group of patients 15 2/7/2023

16. A. Depending on the characteristic of the measurement, variable can be:  Qualitative(Categorical) variable  A variable or characteristic which cannot be measured in quantitative form but can only be identified by name or categories,  for instance place of birth, ethnic group, type of drug, stages of breast cancer (I, II, III, or IV), degree of pain (minimal, moderate, sever or unbearable).  The categories should be clear cut, not overlapping, and cover all the possibilities. For example, sex (male or female), vital status (alive or dead), disease stage (depends on disease), ever smoked (yes or no). 16 2/7/2023

17. Quantitative(Numerical) variable:  is one that can be measured and expressed numerically. Example: survival time, systolic blood pressure, number of children in a family, height, age, body mass index.  they can be of two types Discrete Variables  Have a set of possible values that is either finite or countabl infinite.  The values of a discrete variable are usually whole numbers.  Numerical discrete data occur when the observations are integers that correspond with a count of some sort. 17 2/7/2023

18. Some common examples are:  Number of pregnancies,  The number of bacteria colonies on a plate,  The number of cells within a prescribed area upon microscopic examination,  The number of heart beats within a specified time interval,  A mother’s history of numbers of births ( parity) and pregnancies  The number of episode of illness a patient experiences during some time period, etc. 18 2/7/2023

19. Continuous Variables  A continuous variable has a set of possible values including all values in an interval of the real line.  No gaps between possible values.  Each observation theoretically falls somewhere along a continuum. Example: body mass index, height, blood pressure, serum cholesterol level, weight, age etc. 19 2/7/2023

20. Con…  Observations are not restricted to take on certain numerical values: Often measurements (e.g., height, weight, age).  Continuous data are used to report a measurement of the individual that can take on any value within an acceptable range. 20 2/7/2023

21. Level of measurement which classifies data into mutually exclusive, all inclusive categories in which no order or ranking can be imposed on the data.  Assign subjects to groups or categories  No order or distance relationship  No arithmetic origin  Only count numbers in categories  Only present percentages of categories  Chi-square most often used test of statistical significance Nominal Scale 2/7/2023 21

22. Sex Social status Marital status Days of the week (months) Geographic location Seasons Ethnic group Types of restaurants Brand choice Religion Job type : executive, technical, clerical Other Examples Coded as “0” Coded as “1” Nominal Scale 2/7/2023 22

23. Classifies data according to some order or rank With ordinal data, it is fair to say that one response is greater or less than another. E.g. if people were asked to rate the hotness of 3 chili peppers, a scale of "hot", "hotter" and "hottest" could be used. Values of "1" for "hot", "2" for "hotter" and "3" for "hottest" could be assigned. Ordinal Scale Level of measurement which classifies data into categories that can be ranked. Differences between the ranks do not exist. 2/7/2023 23

24. Ordinal Scales • Arithmetic operations are not applicable but relational operations are applicable. • Ordering is the sole property of ordinal scale. Examples: Letter grades (A, B, C, D, F). Rating scales (Excellent, Very good, Good, Fair, poor). Military status. 2/7/2023 24

25. Interval Scales • Level of measurement which classifies data that can be ranked and differences are meaningful. However, there is no meaningful zero, so ratios are meaningless. • All arithmetic operations except division are applicable. • Relational operations are also possible. Examples: IQ Temperature in oF. 2/7/2023 25

26. assumes that the measurements are made in equal units. i.e. gaps between whole numbers on the scale are equal. e.g. Fahrenheit and Celsius temperature scales an interval scale does not have a true zero. e.g. A temperature of "zero" does not mean that there is no temperature...it is just an arbitrary zero point. permissible statistics: count/frequencies, mode, median, mean, standard deviation Interval Scale Numerically equal distances on the scale represent equal values in the characteristic being measured. An interval scale contains all the information of an ordinal scale, but it also allows you to compare the differences between objects. 2/7/2023 26

27. Ratio Scales • Level of measurement which classifies data that can be ranked, differences are meaningful, and there is a true zero. True ratios exist between the different units of measure. • All arithmetic and relational operations are applicable. Examples: Weight Height Number of students Age 2/7/2023 27

28. Primary Scales of Measurement 4 81 9 Nominal Numbers assigned to runners Ordinal Rank order of winners Third Place Second Place First Place Interval Performance rating on a 0 to 10 Scale 8.2 9.1 9.6 Ratio Time to finish in 20 seconds 15.2 14.1 13.4 2/7/2023 28

29. STATISTICS SCALE DESCRIPTIVE INFERENTIAL Nominal Percentages, Mode Chi-square, Binomial test Ordinal Percentile, Median Rank-order, Correlation, ANOVA Interval Range, Mean, SD Correlations, t-tests, ANOVA Regression, Factor Analysis Ratio Geometric Mean, Coefficient of Variation (CV) Harmonic Mean 2/7/2023 29

30. Excercise Categorize the following variables into nominal, ordinal, interval or ratio  Gender  Grade(A, B, C, D and F )  Rating scale(poor, good, excelent)  Eye colour  Political affilation  Religious affilation  Ranking of tennis players  Majour field  Nationality 30 Height Weight Time Age IQ Temprature Salary 2/7/2023

31. Chapter 2 Organization and Presentation of data • Having collected and edited the data, the next important step is to organize it. • The process of arranging data in to classes or categories according to similarities is called classification • Classification is a preliminary and it prepares the ground for proper presentation of data. • The presentation of data is broadly classified in to the following two categories: • Tabular presentation • Diagrammatic and Graphic presentation. Tuesday, February 7, 2023 Wullo S. 31

32. Tabular presentation of data • Frequency distribution: is the organization of raw data in table form using classes and frequencies. • There are three basic types of frequency distributions  Categorical frequency distribution Ungrouped frequency distribution Grouped frequency distribution Tuesday, February 7, 2023 Wullo S. 32

33. Categorical frequency Distribution: Used for data that can be place in specific categories such as nominal, or ordinal. Example1: a social worker collected the following data on marital status for 25 persons.(M=married, S=single, W=widowed, D=divorced) M S D W D S S M M M W D S M M W D D S S S W W D D Class (1) Frequency (3) Percent (4) M 6 24 S 7 28 D 7 28 W 5 20 Tuesday, February 7, 2023 Wullo S. 33

34. Example 2 • Consider for example, the variable birth weight with levels ‘Very low ’, ‘Low’, ‘Normal’ and ‘Big’. The frequency distribution for newborns is obtained simply by counting by the number of newborns in each birth weight category. Table 2. Distribution of birth weight of newborns between 1976-1996 at TAH. BWT Freq. Rel.Freq(%) Cum. FreCum.rel.freq.(%) Very low 43 0.4 43 0.4 Low 793 8.0 836 8.4 Normal 8870 88.9 9706 97.3 Big 268 2.7 9974 100 Total 9974 100 Tuesday, February 7, 2023 Wullo S. 34

35. 2. Ungrouped frequency Distribution • Is a table of all the potential raw score values that could possible occur in the data along with the number of times each actually occurred. • Example: The following data represent the mark of 20 students. 80 ,76, 90 ,85 ,80 ,70 ,60 ,62 ,70 ,85, 65 ,60 ,63 ,74 ,75 ,76 ,70 ,70 ,80, 85 Construct ungrouped frequency distribution • Mark Frequency 60 2 62 1 63 1 65 1 70 4 74 1 75 2 76 1 80 3 85 3 90 1 Tuesday, February 7, 2023 Wullo S. 35

36. 3.Grouped frequency distribution • When the range of the data is large, the data must be grouped in to classes that are more than one unit in width. Definitions of same terms: • Grouped Frequency Distribution: a frequency distribution when several numbers are grouped in one class. • Class limits: Separates one class in a grouped frequency distribution from another. The limits could actually appear in the data and have gaps between the upper limits of one class and lower limit of the next. • Units of measurement (U): the distance between two possible consecutive measures. It is usually taken as 1, 0.1, 0.01, 0.001, -- ---. Tuesday, February 7, 2023 Wullo S. 36

37. Cont… • Class boundaries: Separates one class in a grouped frequency distribution from another. The boundaries have one more decimal places than the row data and therefore do not appear in the data. There is no gap between the upper boundary of one class and lower boundary of the next class. The lower class boundary is found by subtracting U/2 from the corresponding lower class limit and the upper class boundary is found by adding U/2 to the corresponding upper class limit. • Class width: the difference between the upper and lower class boundaries of any class. It is also the difference between the lower limits of any two consecutive classes or the difference between any two consecutive class marks. Tuesday, February 7, 2023 Wullo S. 37

38. Cont… • Class mark (Mid points): it is the average of the lower and upper class limits or the average of upper and lower class boundary. • Cumulative frequency: is the number of observations less than/more than or equal to a specific value. • Cumulative frequency above: it is the total frequency of all values greater than or equal to the lower class boundary of a given class. • Cumulative frequency blow: it is the total frequency of all values less than or equal to the upper class boundary of a given class. Tuesday, February 7, 2023 Wullo S. 38

39. Cont… • Cumulative Frequency Distribution (CFD): it is the tabular arrangement of class interval together with their corresponding cumulative frequencies. It can be more than or less than type, depending on the type of cumulative frequency used. • Relative frequency (rf): it is the frequency divided by the total frequency. • Relative cumulative frequency (rcf): it is the cumulative frequency divided by the total frequency. Tuesday, February 7, 2023 Wullo S. 39

40. Steps for constructing Grouped frequency Distribution 1. Find the largest and smallest values 2. Compute the Range(R) = Maximum - Minimum 3. Select the number of classes desired, usually between 5 and 20 or use Sturges rule where k is number of classes desired and n is total number of observation. 4. Find the class width by dividing the range by the number of classes and rounding up, not off. Tuesday, February 7, 2023 Wullo S. 40

41. Cont… 5. Pick a suitable starting point less than or equal to the minimum value. The starting point is called the lower limit of the first class. Continue to add the class width to this lower limit to get the rest of the lower limits. 6. To find the upper limit of the first class, subtract U from the lower limit of the second class. Then continue to add the class width to this upper limit to find the rest of the upper limits. 7. Find the boundaries by subtracting U/2 units from the lower limits and adding U/2 units from the upper limits. The boundaries are also half-way between the upper limit of one class and the lower limit of the next class. !may not be necessary to find the boundaries. Tuesday, February 7, 2023 Wullo S. 41

42. Cont… 8. Tally the data. 9. Find the frequencies. 10. Find the cumulative frequencies. Depending on what you're trying to accomplish, it may not be necessary to find the cumulative frequencies. 11. If necessary, find the relative frequencies and/or relative cumulative frequencies Tuesday, February 7, 2023 Wullo S. 42

43. Example*: • Construct a frequency distribution for the following data. 11, 29 , 6 ,33 , 14 ,31, 22 , 27 , 19 ,20 ,18 ,17 ,22 ,38 ,23 ,21 ,26 ,34 ,39 ,27 Solutions: Step 1: Find the highest and the lowest value H=39, L=6 Step 2: Find the range; R=H-L=39-6=33 Step 3: Select the number of classes desired using Sturges formula; Tuesday, February 7, 2023 Wullo S. 43

44. • Step 4: Find the class width; w=R/k=33/6=5.5=6 (rounding up) • Step 5: Select the starting point, let it be the minimum observation. 6, 12, 18, 24, 30, 36 are the lower class limits. Step 6: Find the upper class limit; e.g. the first upper class=12-U=12-1=11 11, 17, 23, 29, 35, 41 are the upper class limits. So combining step 5 and step 6, one can construct the following classes. Tuesday, February 7, 2023 Wullo S. 44

45. • Class limits 6 – 11 12 – 17 18 – 23 24 – 29 30 – 35 36 – 41 • Step 7: Find the class boundaries; • E.g. for class 1 Lower class boundary=6-U/2=5.5 Upper class boundary =11+U/2=11.5 Tuesday, February 7, 2023 Wullo S. 45

46. • Then continue adding w on both boundaries to obtain the rest boundaries. By doing so one can obtain the following classes. • Class boundary 5.5 – 11.5 11.5 – 17.5 17.5 – 23.5 23.5 – 29.5 29.5 – 35.5 35.5 – 41.5 Step 8: tally the data. Step 9: Write the numeric values for the tallies in the frequency column Tuesday, February 7, 2023 Wullo S. 46

47. Cont… • . Step 10: Find cumulative frequency. • Step 11: Find relative frequency or/and relative cumulative frequency. • The complete frequency distribution follows: Tuesday, February 7, 2023 Wullo S. 47

48. Diagrammatic and Graphic presentation of data. • These are techniques for presenting data in visual displays using geometric and pictures for a categorical / qualitative types of data. Importance: • They have greater attraction. • They facilitate comparison. • They are easily understandable. Diagrams are appropriate for presenting discrete data. Tuesday, February 7, 2023 Wullo S. 48

49. Cont… • The three most commonly used diagrammatic presentation for discrete as well as qualitative data are: • Pie charts • Pictogram • Bar charts Pie chart A pie chart is a circle that is divided in to sections or wedges according to the percentage of frequencies in each category of the distribution. The angle of the sector is obtained using: angle of the sector =RF*360 Tuesday, February 7, 2023 Wullo S. 49

50. Cont… • Example: Draw a suitable diagram to represent the following population in a town. Men Women Girls Boys 2500 2000 4000 1500 Solutions: Step 1: Find the percentage. Step 2: Find the number of degrees for each class. Step 3: Using a protractor and compass, graph each section and write its name corresponding percentage. Tuesday, February 7, 2023 Wullo S. 50

51. Cont… Tuesday, February 7, 2023 Wullo S. 51 1 2 3 4

52. Bar Charts  The frequency distribution of a categorical variable is often presented graphically as a bar chart or pie chart.  Bar charts: display the frequency distribution for nominal or ordinal data.  In a bar chart the various categories into which the observation fall are represented along horizontal axis and a vertical bar is drawn above each category such that the height of the bar represents either the frequency or the relative frequency of observation within the class.  The vertical axis should always start from 0 but the horizontal can start from any where. 52 Wullo S. Tuesday, February 7, 2023

53. Cont… • There are different types of bar charts. The most common being : • Simple bar chart • Component or sub divided bar chart. • Multiple bar charts. Simple bar chart: Are used to display data on one variable. They are thick lines (narrow rectangles) having the same breadth. The magnitude of a quantity is represented by the height /length of the bar. Tuesday, February 7, 2023 Wullo S. 53

54. Cont… • Example: The following data represent sale by product, in 1957 a given company for three products A, B, C. Product In 1957 Sales($) A 12 B 24 C 24 Tuesday, February 7, 2023 Wullo S. 54

55. Cont… • Component Bar chart -When there is a desire to show how a total (or aggregate) is divided in to its component parts, we use component bar chart. • Example: Example: The following data represent sale by product, 1957- 1959 of a given company for three products A, B, C. Product In 1957 Sales($) In 1958 Sales($) In 1959 A 12 14 18 B 24 28 18 C 24 30 36 Tuesday, February 7, 2023 Wullo S. 55

56. Cont… • Multiple Bar charts - These are used to display data on more than one variable. - - They are used for comparing different variables at the same time. Example: Draw a component bar chart to represent the sales by product from 1957 to 1959. Tuesday, February 7, 2023 Wullo S. 56

57. Graphical Presentation of data The histogram, frequency polygon and cumulative frequency graph or ogive are most commonly applied graphical representation for continuous data. Procedures for constructing statistical graphs: • Draw and label the X and Y axes. • Choose a suitable scale for the frequencies or cumulative frequencies and label it on the Y axes. • Represent the class boundaries for the histogram or ogive or the mid points for the frequency polygon on the X axes. • Plot the points. • Draw the bars or lines to connect the points. Tuesday, February 7, 2023 Wullo S. 57

58. Stem-and-Leaf Represents data by separating each value into two parts: the stem (the left most digit) and the leaf (such as the rightmost digit). • Are most effective with relatively small data sets • Are not suitable for reports and other communications, • Help researchers to understand the nature of their data Example: 43, 28, 34, 61, 77, 82, 22, 47, 49, 51, 29, 36, 66, 72, 41 Tuesday, February 7, 2023 Wullo S. 58

59. Histogram • A graph which displays the data by using vertical bars of various heights to represent frequencies. • Class boundaries are placed along the horizontal axes. • Class marks and class limits are some times used as quantity on the X axes. Table 3: Distribution of the age of women at the time of marriage Age group No. of women 15-19 11 20-24 36 25-29 28 30-34 13 35-39 7 40-44 3 45-49 2 Tuesday, February 7, 2023 Wullo S. 59

60. Cont… • A histogram of the age of women at the time of marriage Tuesday, February 7, 2023 Wullo S. 60 Age of women at the time of marriage 0 5 10 15 20 25 30 35 40 14.5-19.5 19.5-24.5 24.5-29.5 29.5-34.5 34.5-39.5 39.5-44.5 44.5-49.5 Age group No of women

61. Frequency Polygon • Frequency Polygon : - A line graph. The frequency is placed along the vertical axis and classes mid points are placed along the horizontal axis. • It is customer to the next higher and lower class interval with corresponding frequency of zero, this is to make it a complete polygon. Example: Draw a frequency polygon for the above data Tuesday, February 7, 2023 Wullo S. 61

62. Ogive (cumulative frequency polygon) - A graph showing the cumulative frequency (less than or more than type) plotted against upper or lower class boundaries respectively. - That is class boundaries are plotted along the horizontal axis and the corresponding cumulative frequencies are plotted along the vertical axis. - The points are joined by a free hand curve. - Example: Draw an ogive curve(less than type) for the above data. Tuesday, February 7, 2023 Wullo S. 62

63. Chapter 3 Numerical summary measures A. Measures of location • It is often useful to summarize, in a single number or statistic, the general location of the data or the point at which the data tend to cluster. • Such statistics are called measures of location or measures of central tendency. • We describe them mean, median and mode. 63 Wullo S. Tuesday, February 7, 2023

64. Cont… Arithmetic mean • The arithmetic mean, usually abbreviated to ‘mean’ is the sum of the observations divided by the number of observations. • We use the following data set of 10 numbers to illustrate the computations:

65. Arithmetic Mean 19 21 20 20 34 22 24 27 27 27 • Then, mean = (19 + 21 + … +27) = 24.1 10 • General formula a) Ungrouped mean 65 Wullo S. Tuesday, February 7, 2023 Estimation of the mean from a grouped frequency distribution In calculating the mean from grouped data, we assume that all values falling into a particular class interval are located at the mid-point of the interval. It is calculated as follow:

66. Cont… Tuesday, February 7, 2023 Wullo S. 66 where, k = the number of class intervals xi = the mid-point of the ith class interval fi = the frequency of the ith class interval

67. Properties of the arithmetic mean • The mean can be used as a summary measure for both discrete and continuous data, in general however, it is not appropriate for either nominal or ordinal data. • For given set of data there is one and only one arithmetic mean. • The arithmetic mean is easily understood and easy to compute. • Algebraic sum of the deviations of the given values from their arithmetic mean is always zero. • The arithmetic mean is greatly affected by the extreme values. • In grouped data if any class interval is open, arithmetic mean can not be calculated. 67 Wullo S. Tuesday, February 7, 2023

68. Reading Assignment A. Weighted mean B. Correct and wrong mean C. Combined mean D. Geometric mean E. Harmonic Tuesday, February 7, 2023 Wullo S. 68

69. Median • With the observations arranged in increasing or decreasing order, the median is defined as the middle observation. • If the number of observations is odd, the median is defined as the [(n+1)/2]th observation. • If the number of observations is even the median is the average of the two middle {(n/2)th +[(n/2)+1]th }/2 values i.e • If the number of observations is even, so that there is no middle observation, the median is defined as the average of the two middle observations. • Example : 19 20 20 21 22 24 27 27 27 34 • Then, the median = (22 + 24)/2 = 23 69 Wullo S. Tuesday, February 7, 2023

70. Median for Grouped data In calculating the median from grouped data, we assume that the values within a class-interval are evenly distributed through the interval. – The first step is to locate the class interval in which it is located. – Find n/2 and see a class interval with a minimum cumulative frequency which contains n/2. • Whereas: • LCB= lower class boundary of the median class • Fc= cumulative frequency just before the median class • fc=frequency of the median class • W =class width and n=number of observations. 70 Wullo S. Tuesday, February 7, 2023

71. Properties of median • The median can be used as a summary measure for discrete and continuous data, in general however, it is not appropriate for nominal data. • There is only one median for a given set of data • The median is easy to calculate • Median is a positional average and hence it is not drastically affected by extreme values • Median can be calculated even in the case of open end intervals 71 Wullo S. Tuesday, February 7, 2023

72. Mode • Any observation of a variable at which the distribution reaches a peak is called a mode. • Most distributions encountered in practice have one peak and are described as uni-modal. • E.g. Consider the example of ten numbers 19 21 20 20 34 22 24 27 27 27 • In the above data set, the mode is 27, because the value 27 occurs three times (the most frequent). • The mode of grouped data, usually refers to the modal class, where the modal class is the class interval with the highest frequency. • If a single value for the mode of grouped data must be specified, it is taken as the mid point of the modal class interval 72 Wullo S. Tuesday, February 7, 2023

73. Properties of mode • The mode can be used as a summary measure for nominal, ordinal, discrete and continuous data, in general however, it is more appropriate for nominal and ordinal data. • It is not affected by extreme values • It can be calculated for distributions with open end classes • Often its value is not unique • The main drawback of mode is that often it does not exist 73 Wullo S. Tuesday, February 7, 2023

74. proportion • As we have seen from the previous section, a variable can be either categorical or numerical • Proportion is one of summery measures for categorical variable and also numerical variable if we are counting the number of cases under the specific category. • If we denote “x” as the number of success in an experiment and “n” is the number of trials then the proportion of success from n number of trials is given by x/n. 74 Wullo S. Tuesday, February 7, 2023

75. Percentiles, Quartiles and Inter-quartile Range • The quartiles are sets of values which divide the distribution into four parts such that there are an equal number of observations in each part. – Q1 = [(n+1)/4]th – Q2 = [2(n+1)/4]th – Q3 = [3(n+1)/4]th • The inter-quartile range is the difference between the third and the first quartiles. – Q3 - Q1 Example1: We use the data set of 11 numbers: 19 21 20 20 34 22 24 27 27 27 28 – The first quartile is 20 and the third quartile is 27 – The inter quartile range = 27 – 20 = 7. 75 Wullo S. Tuesday, February 7, 2023

76. Percentiles, Quartiles and Inter-quartile Range • Percentiles divide the data into 100 parts of observations in each part. • It follows that the 25th percentile is the first quartile, the 50th percentile is the median and the 75th percentile is the third quartile. 76 Wullo S. Tuesday, February 7, 2023

77. B. Measures of Dispersion (Variation) • The scatter or spread of items of a distribution is known as dispersion or variation. • In other words the degree to which numerical data tend to spread about an average value is called dispersion or variation of the data. • The most commonly used measures of dispersions are: 1) Range and relative range 2) Quartile deviation and coefficient of Quartile deviation 3) Mean deviation and coefficient of Mean deviation 4) variance 5) Standard deviation and coefficient of variation. Tuesday, February 7, 2023 Wullo S. 77

78. Range • The range is the largest score minus the smallest score. • It is a quick and dirty measure of variability • Because the range is greatly affected by extreme scores and its only depends on two observations Relative Range (RR) • It is also some times called coefficient of range and given: Tuesday, February 7, 2023 Wullo S. 78

79. Cont.. • Example: 1. If the range and relative range of a series are 4 and 0.25 respectively. Then what is the value of Smallest observation and Largest observation The Quartile Deviation (Semi-inter quartile range) The inter quartile range is the difference between the third and the first quartiles of a set of items and semi-inter quartile range is half of the inter quartile range. Tuesday, February 7, 2023 Wullo S. 79

80. Variance • variance is the "average squared deviation from the mean". • A good measure of dispersion should make use of all the data. • For the case of frequency distribution it is expressed as: the variance is limited as a descriptive statistic because it is not in the same units as in the observations. Tuesday, February 7, 2023 Wullo S. 80

81. Cont…. Tuesday, February 7, 2023 Wullo S. 81 For the case of frequency distribution it is expressed as:

82. Coefficient of Variation (C.V) Is defined as the ratio of standard deviation to the mean usually expressed as percents. The distribution having less C.V is said to be less variable or more consistent. Tuesday, February 7, 2023 Wullo S. 82

83. Which measures to use? • When the distribution of the data is symmetric and unimodal (i.e. the data are approximately normally distributed), it is usual to summarize the data using means and standard deviations. • However when the data are skewed, it is preferable to use the median and quartiles as summary statistics. • Median and quartiles are not easily influenced by extreme values in a skewed distribution unlike means and standard deviations. • Remark: – The mean and median of symmetric distribution coincide. – When the distribution is skewed to the right, its mean is larger than its median. – When the distribution is skewed to the left, its mean is smaller than its median. [See Figures 7(a-c)]. 83 Wullo S. Tuesday, February 7, 2023

84. Median Mode Mean Fig. 2(a). Symmetric Distribution Mean = Median = Mode Mode Median Mean Fig. 2(b). Distribution skewed to the right Mean > Median > Mode Mean Median Mode Fig. 2(c). Distribution skewed to the left Mean < Median < Mode 84 Wullo S. Tuesday, February 7, 2023

85. Chapter 4 Introduction to probability Objectives • After completing this chapter, you should be able to – Determine sample spaces and ﬁnd the probability of an events – Understand the different properties of probability – Explain the various types of probability distributions – emphasis will be given to the two widely used probability distributions Tuesday, February 7, 2023 Wullo S. 85

86. Introduction • Many medical decisions are made on a statistical basis since individuals differ in their reactions to medications or surgery in an unpredictable way. • In that case the treatment applied is based on getting the best outcome for as many patients as possible – The life experienced consists of a series of events – “Probability” is a very useful concept and are used in everyday communication. 86 Wullo S. Tuesday, February 7, 2023

87. Introduction con'td…  An understanding of probability is fundamental for  quantifying the uncertainty in the decision-making process  drawing conclusions about a population of patients based on known information about a sample of patients drawn from that population. Probability can be deﬁned as the chance of an event occurring. Many people are familiar with probability from observing or playing games of chance, such as card games, slot machines, or lotteries. Probability theory is used in the various ﬁelds of area like insurance, investments, and weather forecasting and other areas. 87 Wullo S. Tuesday, February 7, 2023

88. Basic concepts • The following definitions and terms are used in studying the theory of probability. – Random experiment: a chance process that leads to well-deﬁned results called outcomes, that is the result cannot be predicted. E.g. Tossing of coins, throwing of dice are some examples of random experiments. – Trial: Performing a random experiment is called a trial. – Outcomes: The results of a random experiment are called its outcomes. When two coins are tossed the possible outcomes are HH, HT, TH, TT. 88 Wullo S. Tuesday, February 7, 2023

89. Basic concepts con'td…. – Sample space: Each conceivable outcome of an experiment is called a sample point. The totality of all sample points is called a sample space and i s denoted by S. – Event: An outcome or a combination of outcomes of a random experiment is called an event. It is a subset of the sample space of a random experiment. – Equally-likely Approach: If an experiment must result in n equally likely outcomes, then each possible outcome must have probability 1/n of occurring. – Mutually exclusive events: when the occurrence of any one event excludes the occurrence of the other event. Mutually exclusive events cannot occur simultaneously. 89 Wullo S. Tuesday, February 7, 2023

90. Basic concepts cont’d….. • Some sample spaces for various probability experiments are. • Probability attempts to quantify an uncertain situation and relative tries to make it more concrete the occurrence of events. • Probability is used to quantify the likelihood, or chance, that an outcome of a random experiment will occur. • Probability is a number between 0 and 1 that expresses how likely the event is occur. 90 Wullo S. Tuesday, February 7, 2023

91. Basic concepts cont’d….. • Example: Find the sample space for the gender of the children if a family has three children. Use B for boy and G for girl. – Solution: There are two genders, male and female, and each child could be either gender. Hence, there are eight possibilities, as shown here. S= {BBB, BBG, BGB, GBB, GGG, GGB, GBG, BGG} • Note: the way to ﬁnd all possible outcomes of a probability experiment (the sample spaces) – by observation and reasoning; – use a tree diagram (a device consisting of line segments emanating from a starting point and also from the outcome point.) 91 Wullo S. Tuesday, February 7, 2023

92. Tree diagram of the above example 92 Wullo S. Tuesday, February 7, 2023

93. Types of probability 1. Classical Probability If the number of outcomes belonging to an event E is NE, and the total number of outcomes is N, then the probability of event E is defined as P(E)=NE/N Example: A couple wants to have exactly 3 children. Assume that each child is either a Boy or a Girl and that there are no duplicate births. Find the probability that two of them will be boys? List all possible orderings for the three children. Solution: S {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG} Then the event E={BBG, BGB, GBB} P(E)=3/8 2. Relative Frequency Probabilities This approach to probability is well-suited to a wide range of scientific disciplines. It is based on the idea that the underlying probability of an event can be measured by repeated trials Tuesday, February 7, 2023 Wullo S. 93

94. • In this view, probability is treated as a quantifiable level of belief ranging from 0 (complete disbelief) to 1 (complete belief) • For instance, an experienced physician may say “this patient has a 50% chance of recovery.” • An appreciation of the various types of probability are not mutually exclusive. And fortunately, all obey the same • mathematical laws, and their methods of calculation are similar. • All probabilities are a type of relative frequency—the number of times something can occur divided by the total number of possibilities or occurrences. 3. Subjective probability Tuesday, February 7, 2023 Wullo S. 94

95. Rules of Probability • Any probability assigned must be a nonnegative number. • The probability of the sample space (the collection of all possible outcomes) is equal to 1. • The probability of A or B involves addition. • P(A or B) = P(A) + P(B) if the two are mutually exclusive. • The probability of A and B involves multiplication • P(A and B) = P(A) P(B) if the two are independent • P( Not A) = 1- P(A) • P(At least one) = 1- P(none) • P(none) = P(each event not happening)^number of events Tuesday, February 7, 2023 Wullo S. 95

96. Addition Rule • General rule: if two events, A and B, are not mutually exclusive, then the probability that event A or event B occurs is: – P(A or B) = P(A) + P(B) – P(A and B) – P(AuB) = P(A) + P(B) – P(AnB) • Special case: when two events, A and B, are mutually exclusive, then the probability that event A or event B occurs is: – P(A or B) = P(A) + P(B) – P(AuB) = P(A) + P(B) • since P(A and B) = 0 for mutually exclusive events Wullo S. 96 Tuesday, February 7, 2023

97. Conditional Probabilities • Sometimes the chance a particular event happens depends on the outcome of some other event. This applies obviously with many events that are spread out in time. • The probability that an event occurs subject to the condition that another event has already occurred is called conditional probability • If A and B are events with Pr(A) > 0, the conditional probability of B given A is • Example: Drug test • Given A is independent from B, what is the relationship between Pr(A|B) and Pr(A)? Pr( ) Pr( | ) Pr( ) AB B A A  97 Women Men Success 200 1800 Failure 1800 200 A = {Patient is a Women} B = {Drug fails} Pr(B|A) = 1800/2000 Pr(A|B) = Wullo S. Tuesday, February 7, 2023

98. Conditional probability: • The conditional probability of an event A given an event B is present is: – P(A | B) =P(AnB)/P(B) • where P(B)≠0 • Joint probability • The joint probability of an event A and an event B is – P(AnB) = P(A and B) • When events A and B are mutually exclusive, then – P(A and B) = 0 Wullo S. 98 Tuesday, February 7, 2023

99. Multiplication rule • General rule: the multiplication rule specifies the joint probability as: • P(AnB)=P(B)P(A/B) • Special case: When events A and B are independent, then: – P(A|B) = P(A) – P(AnB)=P(A)P(B) Wullo S. 99 Tuesday, February 7, 2023

100. Example 1. Find the Probability of at least one male birth in ten consecutive births? Solution • P(At least one male) = 1- P(all females) • P(all females) = P(each single birth is a female)10 = (0.5)10 = 9.77 x 10-4 • So P(At least one male) = 1 – 9.77 x 10-4 = 0.999023. Tuesday, February 7, 2023 Wullo S. 100

101. Exercises 1. 50% of the students in a school weigh more than 140 pounds, but 70% weigh less than 170. If a student is randomly selected, what is the probability the student will weigh between 140 and 170? • Solution • P(w<140)=0.5, P(w>170)=0.3, and P(w<140)+P(140<w<170)+ P(w>170)= 1 • Then • 0.5+P(140<w<170)+ 0.3= 1 P(140<w<170)=1-0.8=0.2 Tuesday, February 7, 2023 Wullo S. 101

102. Probability Distribution • A probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence. • Probability distribution for discrete variable • Probability distribution for continuous variable Wullo S. 102 Tuesday, February 7, 2023

103. Probability Distribution for discrete Variable 1. Binomial distribution • A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties: • The experiment consists of n repeated fixed number of trials. • Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. • The probability of success, denoted by P, is the same on every trial. • The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials. • The probability distribution of this experiment is known as binomial probability distribution. • The binomial distribution describes the distribution of "success" in a series of trials, that is, out of N tries, what is the probability that X of them succeed. 103 Wullo S. Tuesday, February 7, 2023

104. Binomial formula • If the probability of success on an individual trial is P, then the binomial probability is defined by: • Where • K=the number of success • P=probability of success • n=the number of experiments • 1-p=probability of failure 104 Wullo S. Tuesday, February 7, 2023

105. 2. Poisson Distribution • The Poisson Distribution is a discrete distribution which takes on the values X = 0, 1, 2, 3, and so on. • It is often used as a model for the number of events in a specific time period. • It is determined by one parameter, lambda. The Poisson random variable satisfies the following conditions: • The number of successes in two disjoint time intervals is independent. • The probability of a success during a small time interval is proportional to the entire length of the time interval. • Some of the examples in which Poisson distribution is appropriate are:  birth defects and genetic mutations  rare diseases (like Leukemia, but not AIDS because it is infectious and so not independent)  car accidents  traffic flow and ideal gap distance, and so on Tuesday, February 7, 2023 Wullo S. 105

106. Poisson formula • The probability distribution of a Poisson random variable X representing the number of successes occurring in a given time interval or a specified region of space is given by the formula: Where • X=Number of successes per unit time • e=The base of the natural log • λ= The expected number of successes per unit time • If λ is the average number of successes occurring in a given time interval or region in the Poisson distribution, then the mean and the variance of the Poisson distribution are both equal to λ. Tuesday, February 7, 2023 Wullo S. 106

107. Probability Distribution for continuous Variables • If a random variable is a continuous variable, its probability distribution is called a continuous probability distribution. • A continuous probability distribution differs from a discrete probability distribution in several ways by: • Under different circumstances, the outcome of a random variable may not be limited to categories or counts. • Because a continuous random variable X can take on an uncountable infinite number of values, the probability associated with any particular one value is almost equal to zero. • As a result, a continuous probability distribution cannot be expressed in tabular form. 107 Wullo S. Tuesday, February 7, 2023

108. Normal distribution • The normal distribution refers to a family of continuous probability distributions described by the normal equation and described as follows: where • X is a normal random variable, • μ is the mean • σ is the standard deviation • pi is approximately 3.14159, and e is approximately 2.71828. • The random variable X in the normal equation is called the normal random variable. 108 Wullo S. Tuesday, February 7, 2023

109. Characteristics of Normal Distribution • It links frequency distribution to probability distribution • Has a Bell Shape Curve and is Symmetric • It is Symmetric around the mean: Two halves of the curve are the same (mirror images) • Hence Mean = Median • The total area under the curve is 1 (or 100%) • Normal Distribution has the same shape as Standard Normal Distribution. 109 Wullo S. Tuesday, February 7, 2023

110. Normal Curve • The graph of the normal distribution depends on two factors: the mean and the standard deviation. • The mean of the distribution determines the location of the center of the graph, and the standard deviation determines the height and width of the graph. • When the standard deviation is large, the curve is short and wide; when the standard deviation is small, the curve is tall and narrow. • All normal distributions look like a symmetric, bell-shaped curve. 110 Wullo S. Tuesday, February 7, 2023

111. Standard Normal Distribution • It makes life a lot easier for us if we standardize our normal curve, with a mean of zero and a standard deviation of 1 unit. • We can transform all the observations of any normal random variable X with mean μ and variance σ to a new set of observations of another normal random variable Z with mean 0 and variance 1 using the following transformation: 111 Wullo S. Tuesday, February 7, 2023

112. • About 95% of the area under the curve falls within 2 standard deviations of the mean. • About 99.7% of the area under the curve falls within 3 standard deviations of the mean. • A graph of this standardized (mean 0 and variance 1) normal curve is given in Graph: 112 Wullo S. Tuesday, February 7, 2023

113. Table of normal • Example 1: Suppose we want to compute the area under the normal curve to the right of 1.45 • This area can be computed by finding the probability under the normal curve. The probability can be read at the normal curve by combining the value of 1.4 under the first column and 0.05 under the first row. • The green shaded area in the diagram represents the area that is within 1.45 standard deviations from the mean. The area of this shaded portion is 0.4265 (or 42.65% of the total area under the 113 Wullo S. Tuesday, February 7, 2023

114. z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359 0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753 0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141 0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517 0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879 0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224 0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549 0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852 0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133 0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3304 0.3365 0.3389 1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621 1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830 1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015 1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177 1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319 1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441 1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545 1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633 1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706 1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767 2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817 114 Wullo S. Tuesday, February 7, 2023

115. Example 2 • Assuming the normal heart rate (H.R) in normal healthy individuals is normally distributed with Mean = 70 and Standard Deviation =10 beats/min Then: 1) What area under the curve is above 80 beats/min? Now we know, Z =X-M/SD, Z=? X=80, M= 70, SD=10 . So we have to find the value of Z. For this we need to draw the figure…..and find the area which corresponds to Z. 115 Wullo S. Tuesday, February 7, 2023

116. • Since M=70, then the area under the curve which is above 80 beats per minute corresponds to above + 1 standard deviation. • • The total shaded area corresponding to above 1+ standard deviation in percentage is 15.9% or Z= 15.9/100 =0.159. • Or we can find the value of z by substituting the values in the formula Z= X-M/ standard deviation. • Therefore, Z= 70-80/10 -10/10= -1.00 is the same as +1.00. The value of z from the table for 1.00 is 0.159. How do we interpret this? • • This means that 15.9% of normal healthy individuals have a heart rate above one standard deviation (greater than 80 beats per minute). 116 Wullo S. Tuesday, February 7, 2023

117. 13.6% 2.2% 0.15 -3 -2 -1 μ 1 2 3 Diagram of Exercise 0.159 33.35% 117 Wullo S. Tuesday, February 7, 2023

118. Example … 2) What area of the curve is above 90 beats/min? 3) What area of the curve is between 50-90 beats/min? 4)What area of the curve is above 100 beats/min? 5) What area of the curve is below 40 beats per min or above 100 beats per min? 118 Wullo S. Tuesday, February 7, 2023

119. solution 2) 2.3% or 0.023 3) 95.4% or 0.954 4) 0.15 % or 0.015 5) 0.3 % or 0.015 (for each tail) 119 Wullo S. Tuesday, February 7, 2023

120. Example 3 • Suppose scores on an IQ test are normally distributed. If the test has a mean of 100 and a standard deviation of 10, what is the probability that a person who takes the test will score between 90 and 110? • Solution: 120 Wullo S. Tuesday, February 7, 2023

121. Application/Uses of Normal Distribution • It’s application goes beyond describing distributions • It is used by researchers and modelers. • The major use of normal distribution is the role it plays in statistical inference. • The z score along with the t –score, chi-square and F- statistics is important in hypothesis testing. • It helps managers/management make decisions. 121 Wullo S. Tuesday, February 7, 2023

122. Exercises • Find the probability under the normal curve of the following: • The area greater than 1.25 • The area lower than 0.87 • The area greater than -2.36 • The area lower than -1.96 • The area between 0.87 and 1.25 • The area between -1.96 and 1.25 • The value of z which cuts the lower 20% • The value of z which cuts the upper 10% • The values of z which cut the middle 80% 122 Wullo S. Tuesday, February 7, 2023

123. Chapter s Sampling methods and sample size determination • Sampling is a procedure by which some members of the given population are selected as representative of the entire population  Theoretical population  Target population  Study population  sampling frame  Sampling unit  Study unit

124. Hierarchy of sampling Wullo S. 124 Tuesday, February 7, 2023

125. Error in sampling • No sample is the exact mirror image of the population • Potential Source of Error in research are two. 1. Sampling error is any type of bias that is attributable to mistakes in either drawing a sample or determining the sample size  Sampling error (chance ) Can not be avoided or totally eliminated  The causes are – One is chance: That is the error that occurs just because of bad luck – Design error – Un representativeness of the sample Wullo S. 125 Tuesday, February 7, 2023

126. Error in sampling con’d.. 2. Non-sampling error is any error which will be committed during data collection, coding, entry, and so on  Observational error  Respondent error  Lack of preciseness of definition  Error in editing and tabulation of the data Wullo S. 126 Tuesday, February 7, 2023

127. Wullo S. 127 Tuesday, February 7, 2023

128. Advantage of sampling Feasibility it may be the only feasible method of collecting data Reduced cost sampling reduces demands on resource such as finance, personal and material Greater accuracy sampling may lead to better accuracy of collecting data/ detailed information Greater speed data can be collected and summarized more quickly Wullo S. 128 Tuesday, February 7, 2023

129. Disadvantage of sampling • There is always sampling error • Sampling may create a feeling of discrimination within the population • It may be inadvisable where every unit in the population is legally required to have a record • Demands more rigid control in undertaking sample operation • The presence of bias creates difference between the parameter and the statistic • Minority and smallness in number of sub-groups often render study to be suspected. • Sample results are good approximations at best. Wullo S. 129 Tuesday, February 7, 2023

130. Types of sampling I. Probability sampling • Is any method of sampling that utilizes some form of random selection. • probability sampling is a procedure for sampling from a population in which – The selection of a sample unit is based on chance – Every element of the population has a known and non-zero probability of being selected – Random sampling helps produce representative samples by eliminating voluntary response bias and guarding against under coverage bias  Every individual of the target population has equal chance to be included in the sample. Wullo S. 130 Tuesday, February 7, 2023

131. 1. Simple random sample (SRS) • Objective: To select n units out of N • If the population is homogenous • If frame is available • If the study area is not very wide – Note: Homogeneity refers to the similarity of the population with regard to the outcome variable . • Procedure: Use a table of random numbers: takes on values 0,1,2,…….,9 with equal probability  a computer random number generator  mechanical device to select the sample. RAND() function from Excel sheet if frame is available Lottery method Wullo S. 131 Tuesday, February 7, 2023

133. 2. Stratified random sampling • Stratified Random Sampling involves dividing your population into homogeneous subgroups and then taking a simple random sample in each subgroup. • Objective: Divide the population into non-overlapping groups (i.e., strata) N1, N2, N3, ... Ni, such that N1 + N2 + N3 + ... + Ni = N. Then do a simple random sample depending on the type of allocation  Proportional allocation:  Equal allocation Example:  An agency has clients from three ethnic groups and the agency wants to asses clients view of quality of service for the last year. i i N N n n *  Wullo S. 133 Tuesday, February 7, 2023

134. Stratified random sampling Wullo S. 134 Tuesday, February 7, 2023

135. 3. Systematic random sampling  Here are the steps you need to follow in order to achieve a systematic random sample: number the units in the population from 1 to N decide on the n (sample size) that you want or need k = N/n = the interval size randomly select an integer between 1 to k then take every kth unit • Assumptions – Homogeneous population – Frame is not available – If the study area is not very wide Wullo S. 135 Tuesday, February 7, 2023

136. Systematic random sampling • Example Wullo S. 136 Tuesday, February 7, 2023

139. 4. Cluster (area) random sampling • The problem with random sampling methods when we have to sample a population that's disbursed across a wide geographic region is that you will have to cover a lot of ground geographically in order to get to each of the units you sampled • In cluster sampling, we follow these steps:  divide population into clusters (usually along geographic boundaries)  randomly sample clusters  measure all units within sampled clusters Wullo S. 139 Tuesday, February 7, 2023

140. Example • In the figure we see a map of the counties in New York State. Let's say that we have to do a survey of town governments that will require us going to the towns personally. If we do a simple random sample state-wide we'll have to cover the entire state geographically Wullo S. 140 Tuesday, February 7, 2023

141. 5. Multi-stage sampling • The four methods we've covered so far -- simple, stratified, systematic and cluster -- are the simplest random sampling strategies • When we combine sampling methods, we call this multi-stage sampling • Consider the problem of sampling students in grade schools. We might begin with a national sample of school districts stratified by educational level. Within selected districts, we might do a simple random sample of schools. Within schools, we might do a simple random sample of classes or grades. And, within classes, we might even do a simple random sample of students. In this case, we have three or four stages in the sampling process and we use both stratified and simple random sampling. Wullo S. 141 Tuesday, February 7, 2023

143. II. Non probability sampling • Non probability sampling does not involve random selection • Does that mean that non probability samples aren‘t representative of the population? • It does mean that non probability samples cannot depend upon the rationale of probability theory • Most sampling methods are purposive in nature because we usually approach the sampling problem with a specific plan in mind. Wullo S. 143 Tuesday, February 7, 2023

144. Convenience sampling Example  Man in the street (attitude of foreigners about Ethiopia)  College students for psychological study Wullo S. 144 Tuesday, February 7, 2023

145. Purposive sampling • In purposive sampling, we sample with a purpose in mind 1. Modal Instance Sampling  In statistics, the mode is the most frequently occurring value in a distribution.  we are sampling the most frequent case, or the "typical" case  We could say that the modal voter is a person who is of average age, educational level, and income in the population Wullo S. 145 Tuesday, February 7, 2023

146. Purposive sampling… 2. Expert Sampling  Expert sampling involves the assembling of a sample of persons with known or demonstrable experience and expertise in some area. 3. Quota Sampling  In quota sampling, you select people non randomly according to some fixed quota  There are two types of quota sampling: proportional and non proportional 4. Heterogeneity Sampling  We sample for heterogeneity when we want to include all opinions or views, and we aren't concerned about representing these views proportionately.  Another term for this is sampling for diversity. Wullo S. 146 Tuesday, February 7, 2023

147. Purposive sampling… 5. Snowball sampling  In snowball sampling, you begin by identifying someone who meets the criteria for inclusion in your study  You then ask them to recommend others who they may know who also meet the criteria  Snowball sampling is especially useful when you are trying to reach populations that are inaccessible or hard to find  For instance, if you are studying the homeless, you are not likely to be able to find good lists of homeless people within a specific geographical area. However, if you go to that area and identify one or two, you may find that they know very well who the other homeless people in their vicinity are and how you can find them Wullo S. 147 Tuesday, February 7, 2023

148. Summary  Selecting a sampling method depends on • Population to be studied  Size and geographic distributions  Heterogeneity with respect to the variable studied • Resource available • Level of precision required • Importance of having a precise estimate of sampling error Wullo S. 148 Tuesday, February 7, 2023

149. Sample size determination • Determining the sample size for a study is a crucial component of study design. • The goal is to include sufficient numbers of subjects so that statistically significant results can be detected. • Among the questions that a researcher should ask when planning a survey or study is that "How large a sample do I need?" • The answer will depend on the aims, nature and scope of the study and on the expected result. • All of which should be carefully considered at the planning stage Wullo S. 149 Tuesday, February 7, 2023

150. Sample size determination …. • In general, sample size depends on – Objective of the study – Design of the study – Plan for statistical analysis – Accuracy of the measurement to be made – Degree of precision required for generalization – Degree of confidence • We can use three approaches to determine sample size – Rules of thumb for determining the sample size – Statistical formula • Confidence interval approach • Hypothesis testing approach Wullo S. 150 Tuesday, February 7, 2023

151. Rules of thumb • For smaller samples(N < 100), there is little point in sampling. Survey the entire population. • If the population size is around 500 (give or take 100), 50% should be sampled. • If the population size is around 1500, 20% should be sampled. • Beyond a certain point (N = 5000), the population size is almost irrelevant and a sample size of 400 may be adequate. • Statistician maximalist: at least 500 • To make generalizations about entire population, need a total sample size of 200-400 (depending on total population and confidence level desired) Wullo S. 151 Tuesday, February 7, 2023

152. confidence interval • Hence the absolute precision denoted by d is given as • Where s.e is the standard error of the estimator of the parameter of interest. • The margin of error (d) measures the precision of the estimate – Small value of w indicates high precision – It lies in the interval (0%; 5%] – For p close to 50%, w is assumed to be close to 5% – For smaller value of p, w is assumed to be lower than 5% e s z proportion mean . ) ( 2   e s z d . 2   Wullo S. 152 Tuesday, February 7, 2023

153. Estimating a single population mean Where the standard deviation δ can be estimated by; From previous study, if there is From pilot study From educate guess Wullo S. 153 Tuesday, February 7, 2023

154. Single population proportion • Let p denotes proportion of success, then Where the standard deviation p can be estimated by; From previous study, if there is From pilot study P=50% Wullo S. 154 Tuesday, February 7, 2023

155. Point to be considered Wullo S. 155 Tuesday, February 7, 2023

157. Example Wullo S. 157 Tuesday, February 7, 2023

158. Chapter six Inferential statistics • After complete this session you will be able to do – Parameter estimations – Point estimate – Confidence interval – Hypothesis testing – Z-test – T-test – Testing associations – Chi-Square test

159. Introduction 159 Wullo S. Tuesday, February 7, 2023

160. Introduction con'td…….. • Before beginning statistical analyses – it is essential to examine the distribution of the variable for skewness (tails), – kurtosis (peaked or ﬂat distribution), spread (range of the values) and – outliers (data values separated from the rest of the data). • Information about each of these characteristics determines to choose the statistical analyses and can be accurately explained and interpreted. 160 Wullo S. Tuesday, February 7, 2023

161. Introduction con’td ……. • Statistical tests can be either parametric or non- parametric • The path way for the analysis of continuous variables is shown below 161 Wullo S. Tuesday, February 7, 2023

162. Sampling Distribution • The frequency distribution of all these samples forms the sampling distribution of the sample statistic 162 Wullo S. Tuesday, February 7, 2023

163. Sampling distribution .......  Three characteristics about sampling distribution of a statistic its mean its variance its shape  If we repeatedly take sample of the same size n from a population the means of the samples form a sampling distribution of means of size n is equal to population mean.  In practice we do not take repeated samples from a population i.e. we do not encounter sampling distribution empirically, but it is necessary to know their properties in order to draw statistical inferences. 163 Wullo S. Tuesday, February 7, 2023

164. The Central Limit Theorem • Regardless of the shape of the frequency distribution of a characteristic in the parent population, • the means of a large number of samples (independent observations) from the population will follow a normal distribution (with the mean of means approaches the population mean μ, and standard deviation of σ/√n ). • Inferentialstatisticaltechniqueshavevariousassumptionst hatmustbemetbeforevalid conclusions can be obtained • Samples must be randomly selected. • sample size must be greater (n>=30) • the population must be normally or approximately normally distributed if the sample size is less than 30. 164 Wullo S. Tuesday, February 7, 2023

165. Sampling Distribution...... • E.g. Sampling Distribution of the mean • Suppose we choose a random sample of size n, the sampling distribution of the sample mean x posses the following properties. – The sample mean x will be an estimate of the population mean μ. – The standard deviation of x is σ/√n (called the standard error of the mean). – Provided n is large enough the shape of the sampling distribution of x is normal. 165 Wullo S. Tuesday, February 7, 2023

166. Sampling Distribution .......... • Proportion  Suppose we choose a random sample of size n, the sampling distribution of the sample means p posses the following properties.  The sample proportion p will be an estimate of the population mean p. ________  The standard deviation of p is = √p(1-p) /n called the standard error of the proportion).  Provided n is large enough the shape of the sampling distribution of p is normal. 166 Wullo S. Tuesday, February 7, 2023

167. Parameter Estimations • In parameter estimation, we generally assume that the underlying (unknown) distribution of the variable of interest is adequately described by one or more (unknown) parameters, referred as population parameters. • As it is usually not possible to make measurements on every individual in a population, parameters cannot usually be determined exactly. • Instead we estimate parameters by calculating the corresponding characteristics from a random sample estimates . • the process of estimating the value of a parameter from information obtained from a sample. • Point estimation • Interval estimation 167 Wullo S. Tuesday, February 7, 2023

168. Point estimation • A point estimate is a speciﬁc numerical value estimate of a parameter. • Sample measures (i.e., statistics) are used to estimate population measures (i.e., parameters). These statistics are called estimators. • Point estimate for population mean µ is • Point estimate for population proportion is given by • Where x is the total number of success (events) 168 n x = x n 1 = i i  n = p x  Wullo S. Tuesday, February 7, 2023

169. Three Properties of a Good Estimator • The estimator should be an unbiased estimator. That is, the expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated. • The estimator should be consistent. For a consistent estimator, as sample size increases, the value of the estimator approaches the value of the parameter estimated. • The estimator should be a relatively efficient estimator. That is, of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance. 169 Wullo S. Tuesday, February 7, 2023

170. Some BLUE estimators 170 Wullo S. Tuesday, February 7, 2023

171. Confidence Interval estimate • However the value of the sample statistic will vary from sample to sample therefore, to simply obtain an estimate of the single value of the parameter is not generally acceptable. – We need also a measure of how precise our estimate is likely to be – We need to take into account the sample to sample variation of the statistic • A confidence interval defines an interval within which the true population parameter is like to fall (interval estimate). 171 Wullo S. Tuesday, February 7, 2023

172. Confidence intervals… • Confidence interval therefore takes into account the sample to sample variation of the statistic and gives the measure of precision. • The general formula used to calculate a Confidence interval is Estimate ± K × Standard Error, k is called reliability coefficient • Confidence intervals express the inherent uncertainty in any medical study by expressing upper and lower bounds for anticipated true underlying population parameter. • The conﬁdence level is the probability that the interval estimate will contain the parameter, assuming that a large number of samples are selected and that the estimation process on the same parameter is repeated • Most commonly the 95% confidence intervals are calculated, however 90% and 99% confidence intervals are sometimes used. 172 Wullo S. Tuesday, February 7, 2023

173. Confidence interval …… ] / ) 1 ( . , / ) 1 ( . [ ] . , . [ 2 2 2 2 n p p z p n p p z p n z x n z x                 173 A (1-α) 100% confidence interval for unknown population mean and population proportion is given as follows; Wullo S. Tuesday, February 7, 2023

174. Interval estimation 174 Wullo S. Tuesday, February 7, 2023

175. 175 Wullo S. Tuesday, February 7, 2023

178. Confidence intervals… • The 95% confidence interval is calculated in such a way that, under the conditions assumed for underlying distribution, the interval will contain true population parameter 95% of the time. • Loosely speaking, you might interpret a 95% confidence interval as one which you are 95% confident contains the true parameter. • 90% CI is narrower than 95% CI since we are only 90% certain that the interval includes the population parameter. • On the other hand 99% CI will be wider than 95% CI; the extra width meaning that we can be more certain that the interval will contain the population parameter. But to obtain a higher confidence from the same sample, we must be willing to accept a larger margin of error (a wider interval). 178 Wullo S. Tuesday, February 7, 2023

179. Confidence intervals… • For a given confidence level (i.e. 90%, 95%, 99%) the width of the confidence interval depends on the standard error of the estimate which in turn depends on the – 1. Sample size:-The larger the sample size, the narrower the confidence interval (this is to mean the sample statistic will approach the population parameter) and the more precise our estimate. Lack of precision means that in repeated sampling the values of the sample statistic are spread out or scattered. The result of sampling is not repeatable. 179 Wullo S. Tuesday, February 7, 2023

180. Confidence intervals… - To increase precision (of an SRS), use a larger sample. You can make the precision as high as you want by taking a large enough sample. The margin of error decreases as√n increases. • 2. Standard deviation:-The more the variation among the individual values, the wider the confidence interval and the less precise the estimate. As sample size increases SD decreases. • Z is the value from SND • 90% CI, z=1.64 • 95% CI, z=1.96 • 99% CI, z=2.58 180 Wullo S. Tuesday, February 7, 2023

181. Confidence interval …… • If the population standard deviation is unknown and the sample size is small (<30), the formula for the confidence interval for sample mean is: x ± t (s/√n) – x is the sample mean – s is the sample standard deviation – n is the sample size – t is the value from the t-distribution with (n-1) degrees of freedom 181 Wullo S. Tuesday, February 7, 2023

182. Mean Example  A SRS of 16 apparently healthy subjects yielded the following values of urine excreted (milligram per day); 0.007, 0.03, 0.025, 0.008, 0.03, 0.038, 0.007, 0.005, 0.032, 0.04, 0.009, 0.014, 0.011, 0.022, 0.009, 0.008 Compute point estimate of the population mean Construct 90%, 95%, 98% confidence interval for the mean (0.01844-1.65x0.0123/4, 0.01844+1.65x0.0123/4)=(0.0134, 0.0235) (0.01844-1.96x0.0123/4, 0.01844+1.96x0.0123/4)=(0.0124, 0.0245) (0.01844-2.33x0.0123/4, 0.01844+2.33x0.0123/4)=(0.0113, 0.0256)182 01844 . 0 16 295 . 0 n x = x then , values observed n are x ..., , x , x If n 1 = i i n 2 1    Wullo S. Tuesday, February 7, 2023

183. Proportion example • In a survey of 300 automobile drivers in one city, 123 reported that they wear seat belts regularly. Estimate the seat belt rate of the city and 95% confidence interval for true population proportion. • Answer : p= 123/300 =0.41=41% n=300, Estimate of the seat belt of the city at 95% CI = p ± z ×(√p(1-p) /n) =(0.35,0.47) 183 Wullo S. Tuesday, February 7, 2023

184. Summary • Students sometimes have difficulty deciding whether to use Za/2 or t a/2 values when ﬁnding conﬁdence intervals 184 Wullo S. Tuesday, February 7, 2023

185. HYPOTHESIS TESTING Introduction – Researchers are interested in answering many types of questions. For example, A physician might want to know whether a new medication will lower a person’s blood pressure. – These types of questions can be addressed through statistical hypothesis testing, which is a decision-making process for evaluating claims about a population. 185 Wullo S. Tuesday, February 7, 2023

186. Hypothesis Testing • The formal process of hypothesis testing provides us with a means of answering research questions. • Hypothesis is a testable statement that describes the nature of the proposed relationship between two or more variables of interest. • In hypothesis testing, the researcher must deﬁned the population under study, state the particular hypotheses that will be investigated, give the signiﬁcance level, select a sample from the population, collect the data, perform the calculations required for the statistical test, and reach a conclusion. 186 Wullo S. Tuesday, February 7, 2023

187. Idea of hypothesis testing 187 Wullo S. Tuesday, February 7, 2023

188. type of Hypotheses • Null hypothesis (represented by HO) is the statement about the value of the population parameter. That is the null hypothesis postulates that ‘there is no difference between factor and outcome’ or ‘there is no an intervention effect’. • Alternative hypothesis (represented by HA) states the ‘opposing’ view that ‘there is a difference between factor and outcome’ or ‘there is an intervention effect’. 188 Wullo S. Tuesday, February 7, 2023

189. Methods of hypothesis testing • Hypotheses concerning about parameters which may or may not be true • The three methods used to test hypotheses are • The traditional method • The P-value method (New approach) • The conﬁdence interval method 189 Wullo S. Tuesday, February 7, 2023

190. Steps in hypothesis testing 1 Identify the null hypothesis H0 and the alternate hypothesis HA. 190 3 Select the test statistic and determine its value from the sample data. This value is called the observed value of the test statistic. Remember that t statistic is usually appropriate for a small number of samples; for larger number of samples, a z statistic can work well if data are normally distributed. 4 Compare the observed value of the statistic to the critical value obtained for the chosen a. 5 Make a decision. 6 Conclusion 2 Choose a. The value should be small, usually less than 10%. It is important to consider the consequences of both types of errors. Wullo S. Tuesday, February 7, 2023

191. Test Statistics  Because of random variation, even an unbiased sample may not accurately represent the population as a whole.  As a result, it is possible that any observed differences or associations may have occurred by chance. • A test statistics is a value we can compare with known distribution of what we expect when the null hypothesis is true. • The general formula of the test statistics is: Observed _ Hypothesized Test statistics = value value . Standard error • The known distributions are Normal distribution, student’s distribution , Chi-square distribution …. 191 Wullo S. Tuesday, February 7, 2023

192. Critical value • The critical value separates the critical region from the noncritical region for a given level of significance 192 Wullo S. Tuesday, February 7, 2023

193. Decision making • Accept or Reject the null hypothesis • There are 2 types of errors • Type I error is more serious error and it is the level of significant • power is the probability of rejecting false null hypothesis and it is given by 1-β 193 Type of decision H0 true H0 false Reject H0 Type I error (a) Correct decision (1- β) Accept H0 Correct decision (1- a) Type II error (β) Wullo S. Tuesday, February 7, 2023

197. H0: m  m0 H1: m < m0 0 0 0 H0: m  m0 H1: m > m0 H0: m  m0 H1: m  m0   /2 Critical Value(s) Rejection Regions One tailed test Two tailed test Types of testes Wullo S. 197 Tuesday, February 7, 2023

198. Two tailed test 198             o tab cal o tab cal tabulated cal A H reject not do z z if H reject z z if Decision test tailed two for z z n x z H H | | | | : ) ( : ) ( : 1 2 0 0 0 1 0 0 0   m   m m   m m Wullo S. Tuesday, February 7, 2023

199. Steps in hypothesis testing….. 199 If the test statistic falls in the critical region: Reject H0 in favour of HA. If the test statistic does not fall in the critical region: Conclude that there is not enough evidence to reject H0. Wullo S. Tuesday, February 7, 2023

200. One tailed tests 200                        o tab cal o tab cal A o tab cal o tab cal tabulated cal A H reject not do z z if H reject z z if Decision H H H reject not do z z if H reject z z if Decision test tailed one for z z n x z H H : ) ( : ) ( : 3 : , ) ( : ) ( : 2 0 0 1 0 0 0 0 0 0 1 0 0 0   m m   m m  m   m m   m m  Wullo S. Tuesday, February 7, 2023

201. The P- Value • In most applications, the outcome of performing a hypothesis test is to produce a p-value. • P-value is the probability that the observed difference is due to chance. • A large p-value implies that the probability of the value observed, occurring just by chance is low, when the null hypothesis is true. • That is, a small p-value suggests that there might be sufficient evidence for rejecting the null hypothesis. • The p value is defined as the probability of observing the computed significance test value or a larger one, if the H0 hypothesis is true. For example, P[ Z >=Zcal/H0 true]. 201 Wullo S. Tuesday, February 7, 2023

202. P-value…… • A p-value is the probability of getting the observed difference, or one more extreme, in the sample purely by chance from a population where the true difference is zero. • If the p-value is greater than 0.05 then, by convention, we conclude that the observed difference could have occurred by chance and there is no statistically significant evidence (at the 5% level) for a difference between the groups in the population. 202 Wullo S. Tuesday, February 7, 2023

203. How to calculate P-value o Use statistical software like SPSS, SAS…….. o Hand calculations —obtained the test statistics (Z Calculated or t- calculated) —find the probability of test statistics from standard normal table —subtract the probability from 0.5 —the result is P-value Note if the test two tailed multiply 2 the result. Wullo S. 203 Tuesday, February 7, 2023

204. P-value and confidence interval • Confidence intervals and p-values are based upon the same theory and mathematics and will lead to the same conclusion about whether a population difference exists. • Confidence intervals are referable because they give information about the size of any difference in the population, and they also (very usefully) indicate the amount of uncertainty remaining about the size of the difference. • When the null hypothesis is rejected in a hypothesis-testing situation, the conﬁdence interval for the mean using the same level of signiﬁcance will not contain the hypothesized mean. 204 Wullo S. Tuesday, February 7, 2023

205. The P- Value ….. • But for what values of p-value should we reject the null hypothesis? – By convention, a p-value of 0.05 or smaller is considered sufficient evidence for rejecting the null hypothesis. – By using p-value of 0.05, we are allowing a 5% chance of wrongly rejecting the null hypothesis when it is in fact true. • When the p-value is less than to 0.05, we often say that the result is statistically significant. 205 Wullo S. Tuesday, February 7, 2023

206. SUMMARY 206 Wullo S. Tuesday, February 7, 2023

207. Hypothesis testing for single population mean • EXAMPLE 1: A researcher claims that the mean of the IQ for 16 students is 110 and the expected value for all population is 100 with standard deviation of 10. Test the hypothesis . • Solution 1. Ho:µ=100 VS HA:µ≠100 2. Assume α=0.05 3. Test statistics: z=(110-100)4/10=4 4. z-critical at 0,025 is equal to 1.96. 5. Decision: reject the null hypothesis since 4 ≥ 1.96 6. Conclusion: the mean of the IQ for all population is different from 100 at 5% level of significance. 207 Wullo S. Tuesday, February 7, 2023

208. Hypothesis testing for single proportions • Example: In the study of childhood abuse in psychiatry patients, brown found that 166 in a sample of 947 patients reported histories of physical or sexual abuse. a) constructs 95% confidence interval b) test the hypothesis that the true population proportion is 30%? • Solution (a) • The 95% CI for P is given by 208 ] 2 . 0 ; 151 . 0 [ 0124 . 0 96 . 1 175 . 0 947 825 . 0 175 . 0 96 . 1 175 . 0 ) 1 ( 2            n p p z p  Wullo S. Tuesday, February 7, 2023

209. Example…… • To the hypothesis we need to follow the steps Step 1: State the hypothesis Ho: P=Po=0.3 Ha: P≠Po ≠0.3 Step 2: Fix the level of significant (α=0.05) Step 3: Compute the calculated and tabulated value of the test statistic 209 96 . 1 39 . 8 0149 . 0 125 . 0 947 ) 7 . 0 ( 3 . 0 3 . 0 175 . 0 ) 1 (            tab cal z n p p Po p z Wullo S. Tuesday, February 7, 2023

210. Example…… • Step 4: Comparison of the calculated and tabulated values of the test statistic • Since the tabulated value is smaller than the calculated value of the test the we reject the null hypothesis. • Step 6: Conclusion • Hence we concluded that the proportion of childhood abuse in psychiatry patients is different from 0.3 • If the sample size is small (if np<5 and n(1-p)<5) then use student’s t- statistic for the tabulated value of the test statistic. 210 Wullo S. Tuesday, February 7, 2023

211. Two sample mean and proportion • Still now we have seen estimate for only single mean and single proportion. However it is possible to compute point and interval estimation for the difference of two sample means. • let x1, x2, …, xn1 are samples from the first population and y1, y2, …, yn2 be sample from the second population. • Sample mean for the first population be • Sample mean for the second population • Then the point estimate for the difference of means (µ1-µ2) is given by 211 ) ( Y X  Y X Wullo S. Tuesday, February 7, 2023

212. Two sample estimation • Confidence interval estimation • A (1-α)100% confidence interval for the difference of means is given by • If are unknown, then can be estimated by 212 2 2 2 1 2 1 2 ) ( n n z y x       2 1,   and 2 1, s and s Wullo S. Tuesday, February 7, 2023

213. Hypothesis testing for two sample means • The steps to test the hypothesis for difference of means is the same with the single mean Step 1: state the hypothesis Ho: µ1-µ2 =0 VS HA: µ1-µ2 ≠0, HA: µ1-µ2 <0, HA: µ1-µ2 >0 Step 2: Significance level (α) Step 3: Test statistic 213 2 2 2 1 2 1 2 1 ) ( ) ( n n y x zcal   m m      Wullo S. Tuesday, February 7, 2023

214. Hypothesis …                         o cal cal o cal cal A o cal cal o tab cal A o tab cal o tab cal A tabulated tabulated H reject not do z z if H reject z z if H For H reject not do z z if H reject z z if H For H reject not do z z if H reject z z if H For test tailed one for z z test tailed two for z z 0 : 0 : | | | | 0 : 2 1 2 1 2 1 2 m m m m m m   214 Wullo S. Tuesday, February 7, 2023

215. Small sample size and population variance is not given • The test statistic will be student’s t-statistic with degree of freedom equals to n1+n2 -2 • Hence the tabulated value of t is read from the table. • The decision remains the same 215 Wullo S. Tuesday, February 7, 2023

216. Example • A researchers wish to know if the data they have collected provide sufficient evidence to indicate a difference in mean serum uric acid levels between normal individual and individual with down’s syndrome. The data consists of serum uric acid readings on 12 individuals with down’s syndrome and 15 normal individuals. The means are 4.5mg/100ml and 3.4 mg/100ml with standard deviation of 2.9 and 3.5 mg/100ml respectively. 216 0 : 0 : 2 1 2 1     m m m m A O H H Wullo S. Tuesday, February 7, 2023

217. SOLUTION 217 96 . 1 33 . 5 23 . 1 6 . 1 5178 . 1 6 . 1 15 5 . 3 12 9 . 2 0 ) 4 . 3 3 . 4 ( ) ( ) ( 025 . 0 2 2 2 2 2 2 1 2 1 2 1               z z n n y x zcal    m m Wullo S. Tuesday, February 7, 2023

218. Estimation and hypothesis testing for two population proportion • Let n1 and n2 be the sample size from the two population. If x and y are the out come of interest then the point estimate for each population is given by p1=x/n1 and p2=y/n2 respectively. • The point estimates π1-π2 =p1-p2 • The interval estimate for the difference of proportions is given by • If the sample size is large and n1p1>5, n1 (1-p1)>5, n2p2>5, then 218              2 2 2 1 1 1 2 2 1 ) 1 ( ) 1 ( n p p n p p z p p  Wullo S. Tuesday, February 7, 2023

219. Hypothesis testing for two proportions • To test the hypothesis Ho: π1-π2 =0 VS HA: π1-π2 ≠0 The test statistic is given by 219 2 2 2 1 1 1 2 1 2 1 ) 1 ( ) 1 ( ) ( ) ( n p p n p p p p zcal          Wullo S. Tuesday, February 7, 2023

220. Small sample size • If the sample size is small and n1p1>5, n2p2<5, then use student’s t-statistic at n1+ n2-2 degrees of freedom with the given level of significant. 220 Wullo S. Tuesday, February 7, 2023

221. Chi-square test • In recent years, the use of specialized statistical methods for categorical data has increased dramatically, particularly for applications in the biomedical and social sciences. • Categorical scales occur frequently in the health sciences, for measuring responses. • E.g. • patient survives an operation (yes, no), • severity of an injury (none, mild, moderate, severe), and • stage of a disease (initial, advanced). • Studies often collect data on categorical variables that can be summarized as a series of counts and commonly arranged in a tabular format known as a contingency table 221 Wullo S. Tuesday, February 7, 2023

222. Chi-square Test Statistic cont’d… • As with the z and t distributions, there is a different chi-square distribution for each possible value of degrees of freedom. Chi-square distributions with a small number of degrees of freedom are highly skewed; however, this skewness is attenuated as the number of degrees of freedom increases. The chi-squared distribution is concentrated over nonnegative values. It has mean equal to its degrees of freedom (df), and its standard deviation equals √(2df ). As df increases, the distribution concentrates around larger values and is more spread out. The distribution is skewed to the right, but it becomes more bell- shaped (normal) as df increases. 222 Wullo S. Tuesday, February 7, 2023

223. The degrees of freedom for tests of hypothesis that involve an rxc contingency table is equal to (r-1)x(c-1); 223 Wullo S. Tuesday, February 7, 2023

224. Test of Association • The chi-squared (2) test statistics is widely used in the analysis of contingency tables. • It compares the actual observed frequency in each group with the expected frequency (the later is based on theory, experience or comparison groups). • The chi-squared test (Pearson’s χ2) allows us to test for association between categorical (nominal!) variables. • The null hypothesis for this test is there is no association between the variables. Consequently a significant p-value implies association. 224 Wullo S. Tuesday, February 7, 2023

225. Test Statistic: 2-test with d.f. = (r-1)x(c-1) 225      j i ij ij ij E E O , 2 2  n C R i E j i th ij     total grand al column tot j total raw th Oij=observed frequency, Eij=expected frequency of the cell at the juncture of I th raw & j th column Wullo S. Tuesday, February 7, 2023

226. Procedures of Hypothesis Testing 1. State the hypothesis 2. Fix level of significance 3. Find the critical value (x2 (df, α)) 4. Compute the test statistics 5. Decision rules; reject null hypothesis if test statistics > table value. 226 Wullo S. Tuesday, February 7, 2023

227. Test of associations for 2x2 tables • If we call the frequencies in the four cells of 2x2 table a, b, c and d then the table is given by 227 Disease status Row total D ND Exposur e Status E a b a+b NE c d c+d Column total a+c b+d N Wullo S. Tuesday, February 7, 2023

228. Test of association • If the contingency table is 2x2 • Is the table is rxc then 228   ) )( )( )( ( 2 2 d c b a d b c a bc ad n             j i ij ij ij E E O , 2 2  Wullo S. Tuesday, February 7, 2023

229. Assumptions of the 2 - test The chi-squared test assumes that • Data must be categorical • The data be a frequency data – the numbers in each cell are ‘not too small’. No expected frequency should be less than 1, and – no more than 20% of the expected frequencies should be less than 5. • If this does not hold row or column variables categories can sometimes be combined (re-categorized) to make the expected frequencies larger or use Yates continuity correction. 229 Wullo S. Tuesday, February 7, 2023

230. Yates correction • It is a requirement that a chi-squared test be applied to discrete data. Counting numbers are appropriate, continuous measurements are not. Assuming continuity in the underlying distribution distorts the p value and may make false positives more likely. • Frank Yates proposed a correction to the chi-squared formula. Adding a small negative term to the argument. This tends to increase the p-value, and makes the test more conservative, making false positives less likely. However, the test may now be *too* conservative. • Additionally, chi squared test should not be used when the observed values in a cell are <5. It is, at times not inappropriate to pad an empty cell with a small value, though, as one can only assume the result would be more significant with no value there. 230 Wullo S. Tuesday, February 7, 2023

231. Examples Consider hypothetical example on smoking and symptoms of asthma. The study involved 150 individuals and the result is given in the following table: 231 Wullo S. Tuesday, February 7, 2023

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