Power and sample size estimation for the clustered wilcoxon test

The Wilcoxon rank sum test is widely used for two-group comparisons of nonnormal data. An assumption of this test is independence of sampling units both within and between groups, which will be violated in the clustered data setting such as in ophthalmological clinical trials, where the unit of randomization is the subject, but the unit of analysis is the individual eye. For this purpose, we have proposed the clustered Wilcoxon test to account for clustering among multiple subunits within the same cluster (Rosner, Glynn, and Lee, 2003, Biometrics 59, 1089-1098; 2006, Biometrics 62, 1251-1259). However, power estimation is needed to plan studies that use this analytic approach. We have recently published methods for estimating power and sample size for the ordinary Wilcoxon rank sum test (Rosner and Glynn, 2009, Biometrics 65, 188-197). In this article we present extensions of this approach to estimate power for the clustered Wilcoxon test. Simulation studies show a good agreement between estimated and empirical power. These methods are illustrated with examples from randomized trials in ophthalmology. Enhanced power is achieved with use of the subunit as the unit of analysis instead of the cluster using the ordinary Wilcoxon rank sum test.     

Incorporation of clustering effects for the Wilcoxon rank sum test: a large-sample approach

The Wilcoxon rank sum test is frequently used in statistical practice for the comparison of measures of location when the underlying distributions are far from normal or not known in advance. An assumption of the ordinary rank sum test is that individual sampling units are independent. In many ophthalmologic clinical trials, the Early Treatment for Diabetic Retinopathy Scale (ETDRS) is a principal endpoint used for measuring the level of diabetic retinopathy. This is an ordinal scale, and it is natural to consider the Wilcoxon rank sum test for the comparison of the level of diabetic retinopathy between treatment groups. However, under this design, unlike the usual Wilcoxon rank sum test, the subject is the unit of randomization, but the eye is the unit of analysis. Furthermore, a person will tend to have different, but correlated, ETDRS scores for fellow eyes. Thus, we propose a correction to the variance of the Wilcoxon rank sum statistic that accounts for clustering effects and that can be used for both balanced (same number of subunits per cluster) or unbalanced (different number of subunits per cluster) data, both in the presence or absence of ties, with p-value adjusted accordingly. In this article, we present large-sample theory and simulation results for this test procedure and apply it to diabetic retinopathy data from type I diabetics in the Sorbinil Retinopathy Trial.     

Exact Calculation of Permutational Distributions for Two Dependent Samples I

A simple Fortran subroutine is given for the calculation of permutational distributions. Important special cases are Fisher's randomization test, the Wilcoxon signed ranks test and the sign test. The algorithm works in polynomial time. Thus it can be used even for micro-computers within justifiable time limits.     

Extension of the rank sum test for clustered data: two-group comparisons with group membership defined at the subunit level

The Wilcoxon rank sum test is widely used for two-group comparisons for nonnormal data. An assumption of this test is independence of sampling units both between and within groups. In ophthalmology, data are often collected on two eyes of an individual, which are highly correlated. In ophthalmological clinical trials, randomization is usually performed at the subject level, but the unit of analysis is the eye. If the eye is used as the unit of analysis, then a modification to the usual Wilcoxon rank sum variance formula must be made to account for the within-cluster dependence. For some clustered data designs, where the unit of analysis is the subunit, group membership may be defined at the subunit level. For example, in some randomized ophthalmologic clinical trials, different treatments may be applied to fellow eyes of some patients, while the same treatment may be applied to fellow eyes of other patients. In general, binary eye-specific covariates may be present (scored as exposed or unexposed) and one wishes to compare nonnormally distributed outcomes between exposed and unexposed eyes using the Wilcoxon rank sum test while accounting for the clustering. In this article, we present a corrected variance formula for the Wilcoxon rank sum statistic in the setting of eye (subunit)-specific covariates. We apply it to compare ocular itching scores in ocular allergy patients between eyes treated with active versus placebo eye drops, where some patients receive the same eye drop in both eyes, while other patients receive different eye drops in fellow eyes. We also present comparisons between the clustered Wilcoxon test and each of the signed rank tests and mixed model approaches and show dramatic differences in power in favor of the clustered Wilcoxon test for some designs.     

Topical tocopherol acetate reduces post-UVB, sunburn-associated erythema, edema, and skin sensitivity in hairless mice.

Exposure of the skin of the back of skh-1 hairless mice to UVB (310 nm peak) irradiation at doses of 0.115-0.23 J/cm2 results after 24-48 h in an erythema which can be quantified using an erythema meter, providing a useful model of sunburn. Application of pure d-alpha-tocopherol acetate, a thick oil, to the skin immediately following the exposure to UVB significantly reduces the increase in erythema index, by 40-55%. At the lower dose (0.115 J/cm2), skin thickness (associated with edematous swelling of the sunburned skin) was measured by a novel non-invasive technique not previously reported for this purpose--magnetic resonance imaging (MRI). In two experiments the UVB-induced increase in skin thickness was significantly reduced at 24 hr by 29 and 54%, and at 48 hr by 26 and 61%. After 8 days the untreated irradiated mouse skin still showed a significant increase in thickness (24%) compared to the untreated unirradiated control, while the treated irradiated control was not significantly thicker than the unexposed control. Skin sensitivity was tested using a modification of the technique of esthesiometry, by observing rapid avoidance responses of the mouse to a pressure of 0.96 g/cm2 exerted by applying to the skin the tip of a nylon esthesiometer fiber extended to 60 mm in length. The untreated irradiated mice were more sensitive (p less than 0.07, Wilcoxon test) than the treated irradiated mice, and also significantly different from the untreated unirradiated control mice (p less than 0.04, Wilcoxon test), but the treated irradiated mice were not significantly differently sensitive when compared to the unirradiated controls (p less than 0.32). Taken together these data indicate that the erythema, edema, and skin sensitivity commonly associated with UVB-induced sunburn are significantly reduced by topical application of tocopherol acetate even after the exposure has occurred. This observation suggests that treatment of sunburn may be possible even after the irradiation has stopped, by a derivative of d-alpha-tocopherol which is stable to autooxidation.     

Clearance of a dermal Huffmanela sp. in a sandbar shark (Carcharhinus plumbeus) using levamisole

A wild-caught captive sandbar shark Carcharhinus plumbeus developed a contiguous network of darkly pigmented linear tracks that progressed from the snout to the ventral cervical region. Microscopic examination of a skin scraping revealed nematode eggs of the genus Huffmanela, a group of histozoic nematodes that is known to parasitize requiem sharks and marine and freshwater teleosts. The fresh eggs were darkly pigmented with bipolar plugs, contained a larva, and measured 73.3 to 86.4 by 39.0 to 47.4 microm (n = 10). Formalin-fixed and paraffin-embedded eggs were significantly smaller (Wilcoxon rank sums test, p < 0.005), measuring 70.5 to 78.9 by 33.6 to 41.3 microm (n = 13). These measurements do not correlate with previously reported species of Huffmanela. Serial treatment with levamisole (10 mg kg(-1), intramuscular [i.m.]) cleared the egg tracks within 21 d, with no recurrence or apparent complications.     

Reduction of enantioselectivity in the kinetic disposition and metabolism of verapamil in rats exposed to toluene

Toluene and verapamil are subject to extensive oxidative metabolism mediated by CYP enzymes, and their interaction can be stereoselective. In the present study we investigated the influence of toluene inhalation on the enantioselective kinetic disposition of verapamil and its metabolite, norverapamil, in rats. Male Wistar rats (n = 6 per group) received a single dose of racemic verapamil (10 mg/kg) orally at the fifth day of nose-only toluene or air (control group) inhalation for 6 h/day (25, 50, and 100 ppm). Serial blood samples were collected from the tail up to 6 h after verapamil administration. The plasma concentrations of verapamil and norverapamil enantiomers were analyzed by LC-MS/MS by using a Chiralpak AD column. Toluene inhalation did not influence the kinetic disposition of verapamil or norverapamil enantiomers (p > 0.05, Kruskal-Wallis test) in rats. The pharmacokinetics of verapamil was enantioselective in the control group, with a higher plasma proportion of the S-verapamil (AUC 250.8 versus 120.4 ng x h x mL(-1); p < or = 0.05, Wilcoxon test) and S-norverapamil (AUC 72.3 versus 52.3 ng x h x mL(-1); p < or = 0.05, Wilcoxon test). Nose-only exposure to toluene at 25, 50, or 100 ppm resulted in a lack of enantioselectivity for both verapamil and norverapamil. The study demonstrates the importance of the application of enantioselective methods in studies on the interaction between solvents and chiral drugs.     

The Wilcoxon signed rank test for paired comparisons of clustered data

The Wilcoxon signed rank test is a frequently used nonparametric test for paired data (e.g., consisting of pre- and posttreatment measurements) based on independent units of analysis. This test cannot be used for paired comparisons arising from clustered data (e.g., if paired comparisons are available for each of two eyes of an individual). To incorporate clustering, a generalization of the randomization test formulation for the signed rank test is proposed, where the unit of randomization is at the cluster level (e.g., person), while the individual paired units of analysis are at the subunit within cluster level (e.g., eye within person). An adjusted variance estimate of the signed rank test statistic is then derived, which can be used for either balanced (same number of subunits per cluster) or unbalanced (different number of subunits per cluster) data, with an exchangeable correlation structure, with or without tied values. The resulting test statistic is shown to be asymptotically normal as the number of clusters becomes large, if the cluster size is bounded. Simulation studies are performed based on simulating correlated ranked data from a signed log-normal distribution. These studies indicate appropriate type I error for data sets with > or =20 clusters and a superior power profile compared with either the ordinary signed rank test based on the average cluster difference score or the multivariate signed rank test of Puri and Sen. Finally, the methods are illustrated with two data sets, (i) an ophthalmologic data set involving a comparison of electroretinogram (ERG) data in retinitis pigmentosa (RP) patients before and after undergoing an experimental surgical procedure, and (ii) a nutritional data set based on a randomized prospective study of nutritional supplements in RP patients where vitamin E intake outside of study capsules is compared before and after randomization to monitor compliance with nutritional protocols.     

Efficacy of cranberry juice on Helicobacter pylori infection: a double-blind, randomized placebo-controlled trial

BACKGROUND: Helicobacter pylori infection is a major cause of peptic ulcer disease and gastric cancer. This study postulated that cranberry juice would be effective in the suppression of H. pylori in an endemically infected population at high risk for gastric cancer. MATERIALS AND METHODS: A prospective, randomized, double-blind, placebo-controlled trial was conducted in Linqu County of Shandong Province, China, where 189 adults aged 48.9 +/- 11.2 years (mean +/- SD) with H. pylori infection were randomly divided into two groups: cranberry juice (n = 97) and placebo (n = 92). Participants were assigned to orally receive two 250-ml juice boxes of cranberry juice or matching placebo beverage daily for 90 days. The degree of H. pylori infection was determined using the 13C-urea breath test before randomization at 35 and 90 days of intervention to assess the efficacy of cranberry juice in alleviating infection. RESULTS: A total of 189 subjects with positive 13C-urea breath test results prior to randomization completed the study. At day 35 of intervention, 14 of the 97 (14.43%) from the the cranberry juice treatment group and 5 of the 92 (5.44%) of the placebo recipients had negative 13C-urea breath test results. After 90 days, the study concluded that 14 of the 97 subjects in the cranberry juice treatment group versus 5 of the 92 in the placebo group yielded negative test results. Eleven individuals from the cranberry juice treatment group and only two from the placebo group were negative at 35 and 90 days of experiment. These results are significant (p < .05). CONCLUSIONS: Regular consumption of cranberry juice can suppress H. pylori infection in endemically afflicted populations.     

A novel statistical test for treatment differences in clinical trials using a response-adaptive forward-looking Gittins Index Rule

The most common objective for response-adaptive clinical trials is to seek to ensure that patients within a trial have a high chance of receiving the best treatment available by altering the chance of allocation on the basis of accumulating data. Approaches that yield good patient benefit properties suffer from low power from a frequentist perspective when testing for a treatment difference at the end of the study due to the high imbalance in treatment allocations. In this work we develop an alternative pairwise test for treatment difference on the basis of allocation probabilities of the covariate-adjusted response-adaptive randomization with forward-looking Gittins Index (CARA-FLGI) Rule for binary responses. The performance of the novel test is evaluated in simulations for two-armed studies and then its applications to multiarmed studies are illustrated. The proposed test has markedly improved power over the traditional Fisher exact test when this class of nonmyopic response adaptation is used. We also find that the test's power is close to the power of a Fisher exact test under equal randomization.     

The expression of CD 15 antigen on myeloid leukaemia cells--a new prognostic index for complete remission and for survival

377 untreated acute leukaemia patients were categorized according to FAB and cytochemical criterials and simultaneously phenotyped with the use of 6-21 monoclonal antibodies (MoAb) of VI series (W. Knapp, Vienna). The leukaemia phenotype was compared with the patients outcome after treatment. In adult ANLL patients a positive relationships was proved statistically between the expression of the CD 15 cell differentiation antigen on leukaemic blasts and the CR rate (p less than 0.01, chi 2 test). Also a comparison of the Kaplan-Meier survival curves revealed that the CD 15 positive group of ANLL patients has a better outcome than the CD 15 negative one (p less than 0.01, by Wilcoxon and Log-rank tests). Thus, examination of cell differentiation antigens could be a useful addition to existing risk assignment in acute leukaemia.     

Testing whether an identified treatment is best

We consider the problem of testing whether an identified treatment is better than each of K treatments. Suppose there are univariate test statistics Si that contrast the identified treatment with treatment i for i = 1, 2,...., K. The min test is defined to be the alpha-level procedure that rejects the null hypothesis that the identified treatment is not best when, for all i, Si rejects the one-sided hypothesis, at the alpha-level, that the identified treatment is not better than the ith treatment. In the normal case where Si are t statistics the min test is the likelihood ratio test. For distributions satisfying mild regularity conditions, if attention is restricted to test statistics that are monotone nondecreasing functions of Si, then regardless of their covariance structure the min test is an optimal alpha-level test. Tables of the sample size needed to achieve power .5, .8, .90, and .95 are given for the min test when the Si are Student's t and Wilcoxon.     

Clinical Report on the First Prototype of a Photoacoustic Tomography System with Dual Illumination for Breast Cancer Imaging

Photoacoustic tomography is a recently developed imaging modality that can provide high spatial-resolution images of hemoglobin distribution in tissues such as the breast. Because breast cancer is an angiogenesis-dependent type of malignancy, we evaluated the clinical acceptability of breast tissue images produced using our first prototype photoacoustic mammography (PAM) system in patients with known cancer. Post-excisionally, histological sections of the tumors were stained immunohistochemically (IHC) for CD31 (an endothelial marker) and carbonic anhydrase IX (CAIX) (a marker of hypoxia). Whole-slide scanning and image analyses were used to evaluate the tumor microvessel distribution pattern and to calculate the total vascular perimeter (TVP)/area for each lesion. In this clinical study, 42 lesions were primarily scanned using PAM preoperatively, three of which were reported to be benign and were excluded from statistical analysis. Images were produced for 29 out of 39 cancers (visibility rate = 74.4%) at the median depth of 26.5 (3.25–51.2) mm. Age, menopausal status, body mass index, history of neoadjuvant treatment, clinical stage and histological tumor angiogenesis markers did not seem to affect the visibility. The oxygen saturation level in all of the measured lesions was lower than in the subcutaneous counterpart vessels (Wilcoxon test, p value<0.001), as well as in the counterpart contralateral normal breast region of interest (ROI) (Wilcoxon test, p value = 0.001). Although the oxygen saturation level was not statistically significant between CAIX-positive vs. -negative cases, lesional TVP/area showed a positive correlation with the oxygen saturation level only in the group that had received therapy before PAM. In conclusion, the vascular and oxygenation data obtained by PAM have great potential for identifying functional features of breast tumors.     

A genome-wide association study identifies 5 loci associated with frozen shoulder and implicates diabetes as a causal risk factor

Frozen shoulder is a painful condition that often requires surgery and affects up to 5% of individuals aged 40–60 years. Little is known about the causes of the condition, but diabetes is a strong risk factor. To begin to understand the biological mechanisms involved, we aimed to identify genetic variants associated with frozen shoulder and to use Mendelian randomization to test the causal role of diabetes. We performed a genome-wide association study (GWAS) of frozen shoulder in the UK Biobank using data from 10,104 cases identified from inpatient, surgical and primary care codes. We used data from FinnGen for replication and meta-analysis. We used one-sample and two-sample Mendelian randomization approaches to test for a causal association of diabetes with frozen shoulder. We identified five genome-wide significant loci. The most significant locus (lead SNP rs28971325; OR = 1.20, [95% CI: 1.16–1.24], p = 5x10^-29) contained WNT7B. This variant was also associated with Dupuytren’s disease (OR = 2.31 [2.24, 2.39], p<1x10^-300) as were a further two of the frozen shoulder associated variants. The Mendelian randomization results provided evidence that type 1 diabetes is a causal risk factor for frozen shoulder (OR = 1.03 [1.02–1.05], p = 3x10^-6). There was no evidence that obesity was causally associated with frozen shoulder, suggesting that diabetes influences risk of the condition through glycemic rather than mechanical effects. We have identified genetic loci associated with frozen shoulder. There is a large overlap with Dupuytren’s disease associated loci. Diabetes is a likely causal risk factor. Our results provide evidence of biological mechanisms involved in this common painful condition.     

Coenzyme Q10 in maternal plasma and milk throughout early lactation

In contrast to other lipophilic antioxidants Coenzyme Q10 originates from food intake as well as from endogenous synthesis. The CoQ10 concentration and lipid content of maternal milk and maternal plasma was investigated during early lactation. Breast milk was obtained from 23 women: A: colostrums (24-48 hours postpartum), B: transitional milk (day 7 pp), C: mature milk (day 14 pp). At the same time capillary blood specimens were collected. Milk and plasma were stored at -84 degrees C until CoQ10 was analysed after hexane extraction by HPLC. The lipid content was determined by PAP-analysis of cholesterol. The plasma content of CoQ10 was the highest soon after delivery (A: 1.29, B:1.20, C:1.07 pmol/microl; Wilcoxon p < 0.05 A vs. C and B vs. C). This tendency was still evident after lipid-adjustment (A:209, B:180, C:175 micromol CoQ10/mol cholesterol; Wilcoxon p < 0.01 A vs. B and C). The level of CoQ10 in milk showed a gradual decline during early lactation (A:0.80, B:0.57, C:0.44 pmol/microl; Wilcoxon p < 0.02 A vs. B and C). After lipid-adjustment this tendency became even more evident (A: 137, B:86, C:67 micromol CoQ10/mol cholesterol; Wilcoxon p < 0.002 A vs. B and C, p < 0.05 B vs. C). The content of CoQ10 in plasma and milk showed a correlation with early milk (Spearman p < 0.005) but not with mature milk. Although lipid content is low the colostrums is a rich source for the lipophilic antioxidant CoQ10.     

The review of randomization in the Canadian National Breast Screening Study: What does the verdict mean for clinicians?

What is the practising clinician to make of the review by Drs. John C. Bailar III and Brian MacMahon (see pages 193 to 199 of this issue) of the randomization procedure used in the Canadian National Breast Screening Study? Their conclusion that any flaws in randomization would not have affected the published data is reassuring. Nevertheless, the review has not resolved the controversy surrounding the recommendations for screening mammography for women aged 40-49. Recommendations must be based on strong evidence that the benefits of population-based testing outweigh the harms. The absence of such evidence for women aged 40-49 should not, however, preclude the use of mammography as a diagnostic test for women in their 40s whose clinical signs require follow-up. Mammography could also be considered for women whose family history or other factors suggest an increased risk for breast cancer, provided that the limitations and potential disadvantages of testing are explained.     

Minimize the use of minimization with unequal allocation

Summary Minimization as an alternative to randomization is gaining popularity for small clinical trials. In response to critics’ questions about the proper analysis of such a trial, proponents have argued that a rerandomization approach, akin to a permutation test with conventional randomization, can be used. However, they add that this computationally intensive approach is not necessary because its results are very similar to those of a t ‐test or test of proportions unless the sample size is very small. We show that minimization applied with unequal allocation causes problems that challenge this conventional wisdom.     

Randomization Analysis of Incomplete Data in Some Basic Designs

The first two randomization moments of W = Tt.S.S./(Tt.S.S. + Error S.S.) are derived in case of (a) a completely randomized design with one observation missing using (I) Yates method of fitting constants and (II) the available observations only and in case of (b) a randomized block design in which one treatment is tested twice by mistake in a block and another treatment is not tested at all in that block using (I) all the available observations and (II) the data after deleting the observation due to the treatment tested by mistake in place of another treatment in a block. It is concluded that in each of the two cases for a completely randomized design, the F-test is unbiased whereas in each of the two cases for a randomized block design the F-test is in general not unbiased. However, if one treatment is tested twice by mistake in randomly chosen plots of some block, the F-test is unbiased in both cases for a randomized block design. For a completely randomized design the F-test is found to be a good approximation to the corresponding randomized test if N ≧ 80 in case (I) whereas in case (II) this approximation is of the same accuracy as that of case (I) if N ≧ 90. For a randomized block design it is seen that in case (I), the F-test provides a good approximation to the corresponding randomization test if v≧r≧7 and in case (II) this approximation is of the same accuracy if r≧6 and r(v-1)≧45. The analysis of several uniformity trial data shows that for values of “N” in the neighbourhood of 50 the agreement of the first two moments of “W” under the randomization theory and normal theory are reasonably close in all the four cases considered.