WMWodds()
Example 1. Replicate analysis of Newcombe (2006b)
Example 2. Data of Holmes and Williams (1954), used by Agresti (1980)
Example 3. Should "Super HDL" have been touted as effective?
Example 4. Statistical planning: Monte Carlo studies with WMWodds()
Example 5. A "congruent" plot: Using Q-scores to visualize WMWprob.
Example 6. Medians equal, yet WMWodds = 1.44, and 95% CI: [1.23, 1.68]
Example 7. Medians unequal, yet WMWodds = 1.0 & 95% CI: [0.794, 1.260]
Example 1. Replicate analysis of Newcombe (2006b)
Example 2. Data of Holmes and Williams (1954), used by Agresti (1980)
Example 3. Should "Super HDL" have been touted as effective?
Example 4. Statistical planning: Monte Carlo studies with WMWodds()
Example 5. A "congruent" plot: Using Q-scores to visualize WMWprob.
Example 6. Medians equal, yet WMWodds = 1.44, and 95% CI: [1.23, 1.68]
Example 7. Medians unequal, yet WMWodds = 1.0 & 95% CI: [0.794, 1.260]
Example 4. Statistical planning focusing on a confidence interval
A key component of statistical planning involves assess how the proposed statistical methods are likely to perform in the given situation. Too often, we rely on asymptotic results and/or on distributional assumptions for the data that bear no relationship to the study being planned. Indeed, the original cardinal mathematical assumption for the WMW test is that the data are perfectly continuous and thus have no ties. The Newcombe 5 method accommodates ties, and as demonstrated in Examples 4 and 8, behaves "well enough," even in extreme distributional conditions.
Examples 4 and 8 use Monte Carlo experimentation to assess several characteristics of confidence intervals for WMWprob and WMWodds.
- True CI coverage rate. The chance that the CI captures the true value of WMWprob and WMWodds. If the confidence level is 0.95, then the true coverage should be approximately 95%, allowing for Monte Carlo sampling error.
- Average confidence limits. For WMWprob, the ordinary (arithmetic) means of LCL and UCL, the lower and upper confidence limits. For WMWodds, the geometric means.
- CI full width or half width. The average tightness of the confidence interval.
- For CIs of the form [LCL, UCL], the arithmetic mean of UCL - LCL, a full width.
- For CIs of the form [LCL, 1.00), the arithmetic mean of est.WMWprob - LCL, a half width.
- For CIs of the form [0, UCL], the arithmetic mean of UCL - WMWprob, a half width.
For WMWodds:
- For CIs of the form [LCL, UCL], the geometric mean of UCL/LCL, a relative full width.
- For CIs of the form [LCL, Inf), the geometric mean of est.WMWodds/LCL, a relative half width.
- For CIs of the form [0, UCL], the geometric mean of UCL/est.WMWodds, a relative half width.
est.WMWprob, est.WMWodds are the estimates obtained in each trial.
- Traditional power. The chance that the traditional null value, H0.WMWprob = 0.50 or H0.WMWodds = 1.0 falls outside the CI . When considered thoughtfully, such tests may have little or no relevance to the research question.
- Essential power. The chance that a custom null value, H0.WMWprob or H0.WMWodds, falls outside the CI . Essential questions often are better addressed using one-sided CIs, such as computing [LCL, 1.00] for WMWprob or [LCL, Inf] for WMWodds to obtain a more optimal LCL when that is of paramount interest. A CI of the form [0, UCL] begets a more optimal UCL when that is the primary focus.
Example 4 is a set of three examples showing how to perform Monte Carlo sample-size analyses around CI.WMWparms().
- Example 4a, "Planning an efficacy study," deals with the most common question, assessing how much, if any, Y1 and Y2 differ in stochastic magnitude, in this case, how much WMWodds differs from 1.0.
- Example 4b, "Planning a non-inferiority study," investigates a proposed study that is intended to show that WMWodds is at least below 1.25.
- Example 4c, "Planning an equivalence study," investigates a proposed study that is intended to show that WMWodds is within [0.80, 1.25].
WMWodds()
Example 1. Replicate analysis of Newcombe (2006b)
Example 2. Data of Holmes and Williams (1954), used by Agresti (1980)
Example 3. Should "Super HDL" have been touted as effective?
Example 4. Statistical planning: Monte Carlo studies with WMWodds()
Example 5. A "congruent" plot: Using Q-scores to visualize WMWprob.
Example 6. Medians equal, yet WMWodds = 1.44, and 95% CI: [1.23, 1.68]
Example 7. Medians unequal, yet WMWodds = 1.0 & 95% CI: [0.794, 1.260]
Example 1. Replicate analysis of Newcombe (2006b)
Example 2. Data of Holmes and Williams (1954), used by Agresti (1980)
Example 3. Should "Super HDL" have been touted as effective?
Example 4. Statistical planning: Monte Carlo studies with WMWodds()
Example 5. A "congruent" plot: Using Q-scores to visualize WMWprob.
Example 6. Medians equal, yet WMWodds = 1.44, and 95% CI: [1.23, 1.68]
Example 7. Medians unequal, yet WMWodds = 1.0 & 95% CI: [0.794, 1.260]