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Outline. Hypothesis testingExamplesInterpreting resultsResources. Hypothesis testing. Steps:Select a one-sided or two-sided test.Establish the level of significance (e.g., ? = .05).Select an appropriate test statistic.Compute test statistic with actual data.Calculate degrees of freedom (df)
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1. Topics in Biostatistics Part 2 Sarah J. Ratcliffe, Ph.D.
Center for Clinical Epidemiology and Biostatistics
University of Penn School of Medicine
2. Outline Hypothesis testing
Examples
Interpreting results
Resources
3. Hypothesis testing Steps:
Select a one-sided or two-sided test.
Establish the level of significance (e.g., ? = .05).
Select an appropriate test statistic.
Compute test statistic with actual data.
Calculate degrees of freedom (df) for the test statistic.
4. Hypothesis testing Steps cont’d:
Obtain a tabled value for the statistical test.
Compare the test statistic to the tabled value.
Calculate a p-value.
Make decision to accept or reject null hypothesis.
5. Hypothesis testing Steps:
Select a one-sided or two-sided test.
Establish the level of significance (e.g., ? = .05).
Select an appropriate test statistic.
Compute test statistic with actual data.
Calculate degrees of freedom (df) for the test statistic.
6. Hypothesis testing: One-sided versus Two-sided Determined by the alternative hypothesis.
Unidirectional = one-sided
Example:
Infected macaques given vaccine or placebo. Higher
viral-replication in vaccine group has no benefit of
interest.
H0: vaccine has no beneficial effect on viral-replication levels at 6 weeks after infection.
Ha: vaccine lowers viral-replication levels by 6 weeks after infection.
7. Hypothesis testing: One-sided versus Two-sided Bi-directional = two-sided
Example:
Infected macaques given vaccine or placebo.
Interested in whether vaccine has any effect on viral-
replication levels, regardless of direction of effect.
H0: vaccine has no beneficial effect on viral-replication levels at 6 weeks after infection.
Ha: vaccine effects the viral-replication levels.
8. Hypothesis testing Steps:
Select a one-sided or two-sided test.
Establish the level of significance (e.g., ? = .05).
Select an appropriate test statistic.
Compute test statistic with actual data.
Calculate degrees of freedom (df) for the test statistic.
9. Hypothesis testing: Level of Significance How many different hypotheses are being examining?
How many comparisons are needed to answer this hypothesis?
Are any interim analyses planned?
e.g. test data, depending on results collect more data and re-test.
=> How many tests will be ran in total?
10. Hypothesis testing: Level of Significance ?total = desired total Type-I error (false positives) for all comparisons.
One test
?1 = ?total
Multiple tests / comparisons
If ?i = ?total, then ??i > ?total
Need to use a smaller ? for each test.
11. Hypothesis testing: Level of Significance Conservative approach:
?i = ?total / number comparisons
Can give different ?’s to each comparison.
Formal methods include: Bonferroni, Tukey-Cramer, Scheffe’s method, Duncan-Walker.
O’Brien-Fleming boundary or a Lan and Demets analog can be used to determine ?i for interim analyses.
Benjamini Y, and Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. JRSSB, 57:125-133.
12. Hypothesis testing Steps:
Select a one-tailed or two-tailed test.
Establish the level of significance (e.g., ? = .05).
Select an appropriate test statistic.
Compute test statistic with actual data.
Calculate degrees of freedom (df) for the test statistic.
13. Hypothesis testing: Selecting an Appropriate test How many samples are being compared?
One sample
Two samples
Multi-samples
Are these samples independent?
Unrelated subjects in each sample.
Subjects in each sample related / same.
14. Hypothesis testing: Selecting an Appropriate test Are your variables continuous or categorical?
If continuous, is the data normally distributed?
Normality can be determined using a P-P (or Q-Q) plot.
Plot should be approximately a straight line for normality.
If not normal, can it be transformed to normality?
Blindly assuming normality can lead to wrong conclusions!!!
15. Hypothesis testing: Selecting an Appropriate test
16. Hypothesis testing: Selecting an Appropriate test
17. Hypothesis testing: Selecting an Appropriate test
18. Hypothesis testing: Geometric versus Arithmetic mean Geometric mean of n positive numerical values is the nth root of the product of the n values.
Geometric will always be less than arithmetic.
Geometric better when some values are very large in magnitude and others are small.
If geometric is used, log-transform the data before analyzing.
Arithmetic mean of log-transformed data is the log of the geometric mean of the data
E.g. t-test on log-transformed data = test for location of the geometric mean
Langley R., Practical Statistics Simply Explained, 1970, Dover Press
20. Hypothesis testing: Selecting an Appropriate test Other tests are available for more complex situations. For example,
Repeated measures ANOVA: >2 measurements taken on each subject; usually interested in time effect.
GEEs / Mixed-effects models: >2 measurements taken on each subject; adjust for other covariates.
21. Hypothesis testing Steps:
Select a one-tailed or two-tailed test.
Establish the level of significance (e.g., ? = .05).
Select an appropriate test statistic.
Run the test.
22. Example 1 Expression of chemokine receptors on CD14+/CD14- populations of blood monocytes.
Percent of cells positive by FACS.
24. Example 1 cont’d Continuous data, 2 samples
=> t-test, if normal OR
=> Wilcoxon rank sum or signed-rank sum test, if non-normal
Are samples independent or paired?
If independent, can test for equality of variances using a Levene’s test
25. Example 1 cont’d T-tests in excel
=TTEST(L6:L15,M6:M15,2,2)
27. Example 1 cont’d Possible results for different assumptions:
28. Example 1 cont’d Which result is correct?
Data are paired
The differences for each subject are normally distributed.
=> Paired t-test
p = .0095
There is a difference in the percentage of positive CD14+ and CD14- cells.
29. A graph of the 95% CIs for the means would give the impression there is no difference …
30. When it’s really the differences we are testing.
31. Example 1 cont’d Note: paired tests don’t always give lower p-values.
A 1-sided test on the CCR5 values would give p-values of:
p = 0.06 independent samples
p = 0.11 paired samples
WHY?
32. Example 1 cont’d The differences have a larger spread than the individual variables.
33. Example 2 Does the level of CCR5 expression on PBLs (basal or upregulated using lentiviral vector) determine the % of entry that occurs via CCR5?
Two viruses
89.6
DH12
34. Example 2 cont’d
35. Example 2 cont’d How do we know if the slope of the line is significantly different from 0?
Can perform a t-test on the slope estimate. For simple linear regression, this is the same as a t-test for correlation (= square root of R2).
36. Example 2 cont’d
37. Interpreting Results P-values
Is there a statistically significant result?
If not, was the sample size large enough to detect a biologically meaningful difference?
38. Online Resources Power / sample size calculators
http://calculators.stat.ucla.edu/powercalc/
http://www.stat.uiowa.edu/~rlenth/Power/
Free statistical software
http://members.aol.com/johnp71/javasta2.html#Freebies
39. BECC – Consulting Center www.cceb.upenn.edu/main/center/becc.html
Hourly fee service
Design and analysis strategies for research proposals;
Selecting and implementing appropriate statistical methods for specific applications to research data;
Statistical and graphical analysis of data;
Statistical review of manuscripts.
