1. Nonparametric Statistics
Lecture 9

2. Small Sample, Non-normal Population
If the sample was large, the Central Limit Theorem would be applicable for testing hypotheses about the mean.
If the population was normal, the sampling distribution of the mean is exactly a normal distribution to start with.
If the sample is small and the population non-normal, what do we do?
Nonparametric statistics is a sub-field of statistics that creates inferences concerning populations that cannot be assumed to follow any particular distribution.

3. One –Sample Example
Suppose that a nurse has been instructed to perform a procedure in a new way . Researchers recorded the change in the number of minutes it took the nurse to perform the procedure.
The data is
0.6, -0.5, 1.1, 2.4, 3.5, 2.0
-0.1, 1.0, 2.1, -0.6, -0.2
We would be hard pressed to say that this data even approximately follows a normal distribution.

4. Assumption of normality for small sample example
There are only 11 observations and we might be uncomfortable claiming that this distribution looks normal. Instead, it looks more uniform.

5. The Sign Test – 5 Steps Assumptions: Random, independent sample
Hypotheses:
Null hypothesis: Median equals zero
Alternative hypothesis: Median does not equal zero
Test statistic:
p=7/11, interested in comparing proportion that are greater than zero with one-half.

6. The Sign Test – 5 Steps, cont.
P-value:
Need exact calculation since CLT doesn’t apply with small samples. 95% CI for p with small samples: (0.308, 0.891)
Conclusion:
Since 0.5 is included in the 95% confidence interval, we can’t say that the median is significantly different than zero at the 0.05 level. (We fail to reject the null hypothesis.)

7. The Signed Rank Test – 5 steps
Assumptions:
The measurement is continuous
Independent, random sample from the population
Distribution is symmetric
Hypotheses:
H0: Median of the distribution is 0
HA: Median of distribution is non-zero
Test Statistic: Minimum of the rank sums
P-value: from the computer!
For this example, p=0.0439
Conclusion: As per usual.

8. Calculation of Signed Rank Test Statistic
Order observations from smallest to largest in absolute value
|Y|(1) ≤ |Y|(2) ≤ … ≤ |Y|(n)
So from example,
|-0.1| < |-0.2| < |-0.5| < |-0.6|
= 0.6 < 1.0 < 1.1 < 2.0 < 2.1
< 2.4 < 3.5
Assign Ranks to these absolute values
1, 2, … , n
In example, 1, 2, … , 11

9. Signed Rank Test Statistic, cont…
Arrange the ranks into two groups: those with actual values that are smaller and those that are larger than zero. Sum the ranks for both the negative and positive valued observations, separately.
Here, for negative values,
sum of ranks = = 10.5
For positive values
sum of ranks = = 55.5
Test Statistic = smallest rank sum

10. P-values for signed rank test
For critical values and p-values, look at tables/computer generated p-values.
This procedure is unavailable in the Student version of SPSS. It is available in SAS and the regular version of SPSS.

11. Comments on Signed Rank Test
More “powerful” than the Sign Test, but requires more assumptions
One-sided tests are possible
Robust to outliers
Some books/programs use the sum of the ranks of the positive values as the test statistic – p-values are always the same
Nonparametric confidence intervals are also available from some software programs.
For tied observations, use average rank for each tied observation.

12. Nonparametric statistics for small, non-normal samples
Paired Data
The same as for univariate data, except perform the test using the differences rather than the raw data.
Two Independent Groups
Mann-Whitney Rank Sum Test (Ch. 24)
Procedure is similar to the Sign Rank test, except that instead of dividing observations according to whether they are positive or negative, we divide observations according to group membership.
Assumptions include (1) independent, random samples, (2) independently selected groups, and (3) the shape and spread of the two distributions are the same

13. Paired Differences Example
Wife
0.4
0.5
1.0
0.2
0.9
1.2
0.1
0.6
Husband
0.7
0.0
Difference
-0.1
0.3
-0.2
Study Hypothesis: Men and women spend different amounts of time reading/watching the news.

14. The Signed Rank Test – 5 steps
Assumptions:
The measurement (difference) is continuous
Independent, random sample from the population
Distribution of difference is symmetric
Hypotheses:
H0: Median of the difference is 0
HA: Median of difference is non-zero
Test Statistic: Minimum of the rank sums
P-value: from the computer!
For this example,
Conclusion: As per usual.

15. Computer Outputs - Paired
Data for wives and husbands are in two separate columns, with matched observations in the same row.
Analyze
Nonparametric tests
2 Related Samples…
Wilcoxon Signed Ranks Test

16. Computer Outputs - Paired
Data for wives and husbands are in two separate columns, with matched observations in the same row.
Analyze
Nonparametric tests
2 Related Samples…
Sign Test

17. Two Independent Groups Example
Wife
0.4
0.5
1.0
0.2
0.9
1.2
0.1
0.6
Husband
0.7
0.0
Study Hypothesis: Men and women spend different amounts of time reading/watching the news.

18. The Mann-Whitney Test – 5 steps
Assumptions:
Independent, random samples
Independently selected groups
The shape and spread of the two distributions are the same
Hypotheses:
H0: Group medians are the same
HA: Group medians are different
Test Statistic: rank sums
P-value: from the table or computer!
For this example,
Conclusion: As per usual.

19. Computer Outputs - Independent
Data for wives & husbands are in the same column; a second column indicates whether each observation is for the wife or husband*.
Analyze
Nonparametric tests
2 Independent Samples…
Mann-Whitney Test
*: Type of this variable must be Numeric in SPSS.

20. Comments on Nonparametric Test for 2 Independent Samples
Robust to outliers
One-sided tests are possible
Nonparametric confidence intervals are also available from some software programs
For tied observations, use average rank for each tied observation.
Possible Names
Mann-Whitney Rank Sum Test
Mann-Whitney Test
Mann-Whitney U Test
Wilcoxon Rank Sum Test

21. Testing for a Relationship between Categorical Variables
Large Sample Size
Chi-square test
Small Sample Size
Chi-square test with Yates’ continuity correction
Fisher’s exact test

22. Urgent Colonoscopy for the Diagnosis and Treatment of Severe Diverticular Hemorrhage New England Journal of Medicine 2000;342:78-82
Severe Bleeding
Medical and Surgical Treatment
Medical and Colonoscopic Treatment
Total No 11 10 21
Yes 6 17 27
Research Hypothesis

23. Fisher’s Exact Test – 5 steps
Assumptions:
Independent, random sample from the population
Two variables are categorical
Hypotheses:
H0: Response and Predictor are Independent
HA: Response and Predictor are Associated
Test Statistic: (p-value)
P-value: from the computer!
For this example, p=0.057
Conclusion: As per usual.

24. Data Entry
Weight the variable: count.
Data
Weight Cases…

25. Computer Outputs - FET Crosstabs
Perform FET (or Chi-square test if sample size is large)
Analyze
Descriptive Statistics
Crosstabs…
Assign “bleeding” for “Row(s)”, “treat” for “Column(s)”
Click “Statistics” to check “Chi- square”

26. The Inexact Use of Fisher’s Exact Test in Six Major Medical Journals
The Inexact Use of Fisher’s Exact Test in Six Major Medical Journals JAMA 1989;261:
Table 1. Specification of Use of Fisher’s Exact Test by Journal
Journal No. of Articles That Specified /
No. of Articles Reviewed
New England Journal of Medicine 8 / 9
Annals of Internal Medicine 2 / 4
British Medical Journal 3 / 6
The Journal of the American 6 / 16
Medical Association
Lancet / 14
American Journal of Medicine 0 / 7

27. Homework To be posted, not graded Solutions will be posted on Monday
Read Chapters 24, 25, 27