Nonparametric Statistics Lecture 9

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.

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.

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.

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.

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.)

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.

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

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 = 1+2+3+4.5 = 10.5 For positive values sum of ranks = 4.5+6+7+8+9+10+11 = 55.5 Test Statistic = smallest rank sum

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.

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.

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

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.

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.

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

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

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.

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.

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.

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

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

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

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.

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

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”

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;261:3430-3433 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 4 / 14 American Journal of Medicine 0 / 7

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