1. Two Samples (K=2) Hypothesis Tests
PhD Özgür Tosun

3. Evidence Based Medicine
“Medical progress has become associated with better technologies, not with better doctors who understand these technologies. Medical students have to memorize tons of facts about common and rare diseases. What they rarely learn is statistical thinking and critical evaluation of scientific articles in their own field. ”
Excerpt From: Gerd Gigerenzer. “Risk Savvy.” iBooks.

4. A Well Known Joke
“Two students, one from biology, one from medicine, are asked to learn the telephone book by heart (memorise).
The biology student asks, why?
The medical student asks, by when?”
Excerpt From: Gerd Gigerenzer. “Risk Savvy.” iBooks.

5. Understand the “EVIDENCE”
Evidence Based Medicine
Know how to get the best and latest “EVIDENCE”
Understand the numbers
Understand the “Materials and Methods”
Understand the “Limitations”
Establish the link between “EVIDENCE” and “YOUR PATIENT”

6. On average, the class is doing well
Misunderstanding the Data
On average, the class is doing well
Half of the students think that
2+2=3 the other half thinks that 2+2=5

8. Paradox – Misuse of the Evidence
Mammography screening reduces breast- cancer mortality by 1 in every women.
Seemingly paradoxical fact that cigarette smoking reduces breast-cancer mortality by the same rate !!!
What would be the reason???
Shall we advice women to have breast- cancer screening and start smoking???

9. The Answer Is:
Smoking kills earlier so that some women don’t live long enough to get breast cancer.

10. Back to Hypothesis Tests Parametric or Nonparametric
Selecting the most appropriate hypothesis test starts with the decision of parametric or nonparametric
Given that the parametric assumptions are met, these tests are more powerful compared to their nonparametric alternatives
One must first check the assumptions before deciding which test to use

11. Parametric Assumptions
Check if every group has at least 10 objects
Check if the number of objects are close to each other for every study group
Be sure that the variable being tested between the groups is continuous
Do normality test for each group and be sure that each of the are normally distributed
Do the homogeneity of variances test and see if the groups have homogenous variances

12. Independent vs Dependent (Unpaired vs Paired )
Two samples are said to be paired when each data point in the first sample is matched and is related to a unique data point in the second sample
The paired samples may represent at least two sets of measurements on the same people. In this case each person is serving as his or her own control
Samples (groups) are independent when the objects in each of them are different individuals and the measurements in one group do not affect the measurements of other groups

13. Examples

16. Few Steps to Systemize Hypothesis Tests Selection
In following few slides, a simple but effective strategy to select appropriate statistical hypothesis testing method is given
Remember ! These steps include only basic and most common methods and selection must be carefully done without violating any critical assumptions about the data

17. DATA
CONTINUOUS
CATEGORICAL

18. CONTINUOUS
ONE SAMPLE
TWO SAMPLES
>2 SAMPLES

19. CONTINUOUS
ONE SAMPLE
TWO SAMPLES
>2 SAMPLES
Independent
Paired

20. CONTINUOUS ONE SAMPLE TWO SAMPLES >2 SAMPLES Parametric Independent
Paired
Student’s
t Test
Paired Samples
One Way
Analysis
Of Variance
Repeated
Measures
ANOVA
Mann Whitney
U Test
Wilcoxon Signed
Rank Test
Kruskal Wallis
Analysis of
Variance
Friedman
Test
One Sample
Sign Test
Parametric
Nonparametric

21. CATEGORICAL
ONE SAMPLE
TWO SAMPLES
>2 SAMPLES

22. CATEGORICAL
ONE SAMPLE
TWO SAMPLES
>2 SAMPLES
Independent
Paired

23. CATEGORICAL N x M Chi ONE SAMPLE TWO SAMPLES >2 SAMPLES Parametric
Independent
Paired
2 x 2 Chi
Square Test
Mc Nemar
Test
N x M Chi
One Sample
Chi Square
Fisher’s
Exact Test
One sample difference
of proportions test
Parametric
Nonparametric

24. Student’s T test

25. Student’s t Test
Parametric test to compare means of two independent study groups
Also called «Independent Samples t Test»
Test statistic is «Student’s t»
Compare hemoglobin levels of pregnant women versus control group
Compare BMI of the children of high versus low socioeconomical class families
Nonparametric alternative is Mann Whitney U test

26. An Example
In a heart study, the systolic blood pressure was measured for two groups:
24 men aged 20 year old (Young Group)
30 men aged 40 year old (Old Group)
Do these data show sufficient evidence to conclude that the older men have a higher systolic blood pressure, at the 0.05 level of significance?

27. The variable concerning systolic blood pressure is continuous
Since
The variable concerning systolic blood pressure is continuous
The sample size of each group is greater than 10
Systolic blood pressure values in each group is normally distributed
There are two groups and they are independent
Independent samples t-test is used

28. 20- year-old
40- year-old
Subject
Sbp 1 95 13
132
150 16
148 2
122 14
100
152 17
116 3
130 15
120
154 18
128 4
125
160 19
136 5
115
164 20
110 6
138
176 21
126 7
105
108 22 8
118 23 9 24
140 10
156
142 25 11 26
124 12
106
146 27
114 28 29 30

29. GROUP 40- year-old 20- year-old 24 122,8333 16,7790 30 133,6667
17,3013
GROUP
20- year-old
40- year-old N
Mean
Std. Deviation 30 24
N =
GROUP
40- year-old
20- year-old
Mean  1 SD SBP
160
150
140
130
120
110
100

30. The older men have higher systolic blood pressure24
122,8333
16,7790 30
133,6667
17,3013
GROUP
20- year-old
40- year-old N
Mean
Std. Deviation
t(52,0.05)= AND
p<0.05, Reject H0.
The older men have higher systolic blood pressure

31. SPSS Output with p Value calculation

32. Mann – whitney u test

33. Mann – Whitney U Test
Nonparametric alternative for Student’s t test when parametric assumptions are not met
Basically compares medians not the means
Compare 100 meter running times of two different high school classes
Compare blood pressures of two treatment groups
Test statistic is U

34. An Example
Cryosurgery is a commonly used therapy for treatment of cervical intraepithelial neoplasia (CIN). The procedure is associated with pain and uterine cramping.
Within 10 min of completing the cryosurgical procedure, the intensity of pain and cramping were assessed on a 100-mm visual analog scale (VAS), in which 0 represent no pain or cramping and 100 represent the most severe pain and cramping.
The purpose of study was to compare the perceptions of both pain and cramping in women undergoing the procedure with and without paracervical block

35. 5 women were selected randomly in each groups and their scores are as follows:
Women without a block 14 88 37 27
Women with a paracervical block 50 70 66 75

36. Since
The variable concerning pain/cramping score is continuous
The sample size is less than 10
There are two groups and they are independent
Mann Whitney U test

37. From the table, critical value is 21
5.5 < accept H0
We conclude that the median pain/ cramping scores are same in two groups.

40. SPSS Output with p Value calculation

41. paIred samples t test

42. Paired Samples t Test
Parametric procedure to compare means of two dependent groups
Nonparametric alternative is «Wilcoxon Signed Ranks Test»
Samples must be related/dependent/paired, which generally means the measurements must be taken from the same individuals at two different time points
Compare the pre and post exercise O2 capacities of a football team
Compare the cholesterol level of a group of patients before and after a certain diet
Test statistic is based on t distribution

43. An Example
A study was conducted to see if a new therapeutic procedure is more effective than the standard treatment in improving the digital dexterity of certain handicapped persons
Twenty-four pairs of twins were used in the study, one of the twins was randomly assigned to receive the new treatment, while the other received the standard therapy. At the end of the experimental period each individual was given a digital dexterity test with scores as follows.

44. Since
The variable concerning digital dexterity test scores is continuous
The sample size is greater than 10
digital dexterity test score is normally distributed
There are two groups and they are dependent
Paired sample t-test

45. New
Standard
Difference 49 54 -5 56 42 14 70 63 7 83 77 6 68 51 17 84 82 2 9 67 62 5 79 71 8 88 48 50 -2 52 41 11 73 57 3 78 72 64 44 40 35 81
Total
129
Mean
65,46
60,08
5,38 SD
14,38
14,46
5,65
t(23,0.05)=1.714
Since, reject H0.
We conclude that the new treatment is effective.

47. SPSS Output with p Value calculation

48. wIlcoxon sIgned ranks test

49. Wilcoxon Signed Ranks Test
Nonparametric alternative for paired samples t test
T distribution is used (not t !)
For low sample sizes “Wilcoxon W” test statistic is used
“Z statistic” can be used for larger sample sizes
Heart rate level difference between before and after a certain drug is administered to the same patient group
Compare pre and post performance levels of students depending on a training program
Compares medians rather than means

51. An Example
To test whether the weight-reducing diet is effective 9 persons were selected. These persons stayed on a diet for two months and their weights were measured before and after diet. The following are the weights in kg:

52. Subject
Weights
Before
After 1 85 82 2 91 92 3 68 62 4 76 73 5 81 6 87 83 7
105 8 93 88 9 98 90

53. The variable concerning weight is continuous.
Since
The variable concerning weight is continuous.
The sample size is less than 10
There are two groups and they are dependent
Wilcoxon signed ranks test

54. We conclude that the diet is effective.
Tcal = 6.0
Tcal = > T(n=9,a =0.05) =1.5
reject H0 , p<0.05
We conclude that the diet is effective.

57. SPSS Output with p Value calculation

58. From the Book, “Risk Savvy”
“During a talk to the Austrian Chamber of Physicians, I once explained the problem of doctors who don’t understand evidence. In the discussion afterward, a man in the audience raised his hand. He introduced himself as a professor at the Medical University of Vienna. The problem with innumerate doctors, he said, may be a problem in New York or London, but not in Vienna. He himself teaches biostatistics and takes care that every medical student understands numbers. Positively surprised, I congratulated him. Then a young woman raised her hand. She identified herself as a former student of the professor. “I attended his course on biostatistics, and I can assure you, we didn’t understand a thing.”