Abdul Rahman Othman

The New and Improved Two-Sample t Test

This article considers the problem of comparing two independent groups in terms of some measure of location. It is well known that with Students two-independent-sample t test, the actual level of significance can be well above or below the nominal level, confidence intervals can have inaccurate probability coverage, and power can be low relative to other methods. A solution to deal with heterogeneity is Welchs (1938) test. Welchs test deals with heteroscedasticity but can have poor power under arbitrarily small departures from normality. Yuen (1974) generalized Welchs test to trimmed means; her method provides improved control over the probability of a Type I error, but problems remain. Transformations for skewness improve matters, but the probability of a Type I error remains unsatisfactory in some situations. We find that a transformation for skewness combined with a bootstrap method improves Type I error control and probability coverage even if sample sizes are small.

Comparing measures of the 'typical' score across treatment groups.

Researchers can adopt one of many different measures of central tendency to examine the effect of a treatment variable across groups. These include least squares means, trimmed means, M-estimators and medians. In addition, some methods begin with a preliminary test to determine the shapes of distributions before adopting a particular estimator of the typical score. We compared a number of recently developed adaptive robust methods with respect to their ability to control Type I error and their sensitivity to detect differences between the groups when data were non-normal and heterogeneous, and the design was unbalanced. In particular, two new approaches to comparing the typical score across treatment groups, due to Babu, Padmanabhan, and Puri, were compared to two new methods presented by Wilcox and by Keselman, Wilcox, Othman, and Fradette. The procedures examined generally resulted in good Type I error control and therefore, on the basis of this critetion, it would be difficult to recommend one method over the other. However, the power results clearly favour one of the methods presented by Wilcox and Keselman; indeed, in the vast majority of the cases investigated, this most favoured approach had substantially larger power values than the other procedures, particularly when there were more than two treatment groups.

Unravelling the temporal association between lameness and body condition score in dairy cattle using a multistate modelling approach

Recent studies have reported associations between lameness and body condition score (BCS) in dairy cattle, however the impact of change in the dynamics of BCS on both lameness occurrence and recovery is currently unknown. The aim of this longitudinal study was to investigate the effect of change in BCS on the transitions from the non-lame to lame, and lame to non-lame states. A total of 731 cows with 6889 observations from 4 UK herds were included in the study. Mobility score (MS) and body condition score (BCS) were recorded every 13-15 days from July 2010 until December 2011. A multilevel multistate discrete time event history model was built to investigate the transition of lameness over time. There were 1042 non-lame episodes and 593 lame episodes of which 50% (519/1042) of the non-lame episodes transitioned to the lame state and 81% (483/593) of the lame episodes ended with a transition to the non-lame state. Cows with a lower BCS at calving (BCS Group 1 (1.00-1.75) and Group 2 (2.00-2.25)) had a higher probability of transition from non-lame to lame and a lower probability of transition from lame to non-lame compared to cows with BCS 2.50-2.75, i.e. they were more likely to become lame and if lame, they were less likely to recover. Similarly, cows who suffered a greater decrease in BCS (compared to their BCS at calving) had a higher probability of becoming lame and a lower probability of recovering in the next 15 days. An increase in BCS from calving was associated with the converse effect, i.e. a lower probability of cows moving from the non-lame to the lame state and higher probability of transition from lame to non-lame. Days in lactation, quarters of calving and parity were associated with both lame and non-lame transitions and there was evidence of heterogeneity among cows in lameness occurrence and recovery. This study suggests loss of BCS and increase of BCS could influence the risk of becoming lame and the chance of recovery from lameness. Regular monitoring and maintenance of BCS on farms could be a key tool for reducing lameness. Further work is urgently needed in this area to allow a better understanding of the underlying mechanisms behind these relationships.

Testing the Equality of Location Parameters for Skewed Distributions Using S1 with High Breakdown Robust Scale Estimators

A simulation study had been carried out to compare the Type I error and power of S 1, a statistic recommended by Babu et al. (1999) for testing the equality of location parameters for skewed distributions. Othman et al. (in press) showed that this statistic is robust to the underlying populations and is also powerful. In our work, we modified this statistic by replacing the standard errors of the sample medians with four alternative robust scale estimators; the median absolute deviation (MAD) and three of the scale estimators proposed by Rousseeuw and Croux (1993); Q n , S n , and T n These estimators were chosen based on their high breakdown value and bounded influence function, and in addition, they are simple and easy to compute. Even though MAD is more appropriate for symmetric distributions (Rousseeuw and Croux, 1993), due to its popularity and for the purpose of comparison, we decided to include it in our study. The comparison of these methods was based on their Type I error and the power for J = 4 groups in an unbalanced design having heterogeneous variances. Data from the Chi-square distribution with 3 degrees of freedom were considered. Since the null distribution of S 1 is intractable, and its asymptotic null distribution may not be of much use for practical sample sizes, bootstrap methods were used to give a better approximation. The S 1 statistic combined with each of the scale estimators was shown to have good control of Type I errors.

Area of hock hair loss in dairy cows: Risk factors and correlation with a categorical scale

Data from 3691 dairy cows from 76 farms were used to investigate the risk factors associated with the area of hair loss over the lateral aspect of the hock and the correlation between the area of hair loss (as calculated using a hock map) and hock lesion scores determined using a pre-existing categorical scale. Six factors were associated with a greater area of hair loss, including cows with locomotion score 3, a cleanliness score (10/28 to 18/28), high daily milk yield (25.1-58.1 kg), poor body condition score (1-1.5), duration of winter housing (≥41 days) and some combinations of cubicle base and bedding materials. Compared with cows housed in cubicles with a concrete base and whole straw or rape straw bedding, cows housed in cubicles with concrete bases with sand or chopped straw bedding had smaller areas of hair loss and cows housed on a mattress base with whole straw or rape straw bedding had larger areas of hair loss. Area of hair loss, as measured on hock maps, was not significantly different between cows with score 1 (median 23.6 cm(2)) and score 2 (median 20.3 cm(2)) on the categorical scale for hock lesions. This suggests that the categorical scale was not reflecting the extent of hair loss and that hock maps are a good alternative for studying the dynamics of hock lesions over time.

The Phylogenetic Tree of RNA Polymerase Constructed Using MOM Method

Phylogenetic analysis is a study of evolutionary relationship between organisms or species, and the evolutionary is displayed as a phylogenetic tree. In this paper, the phylogenetic tree was constructed by using distance-based method; UPGMA and the proposed method, MOM estimator. The UPGMA calculates the average of possible pairwise distances to get a new distance in the clustering process. If outliers exist in the possible pairwise distances, new mean distances are calculated, and the result is not robust. To overcome this problem, we implement a checking process to detect the outliers using MADn criteria on (based on median), and the new distances using the modified one-step M-estimator (MOM). In order to evaluate the branch of the tree constructed, the bootstrap method is used and the p-value (bootstrap value) for both methods is compared.

Modified distance in average linkage based on M-estimator and MADn criteria in hierarchical cluster analysis

The process of grouping a set of objects into classes of similar objects is called clustering. It divides a large group of observations into smaller groups so that the observations within each group are relatively similar and the observations in different groups are relatively dissimilar. In this study, an agglomerative method in hierarchical cluster analysis is chosen and clusters were constructed by using an average linkage technique. An average linkage technique requires distance between clusters, which is calculated based on the average distance between all pairs of points, one group with another group. In calculating the average distance, the distance will not be robust when there is an outlier. Therefore, the average distance in average linkage needs to be modified in order to overcome the problem of outlier. Therefore, the criteria of outlier detection based on MADn criteria is used and the average distance is recalculated without the outlier. Next, the distance in average linkage is calculated based on a modified one step M-estimator (MOM). The groups of cluster are presented in dendrogram graph. To evaluate the goodness of a modified distance in the average linkage clustering, the bootstrap analysis is conducted on the dendrogram graph and the bootstrap value (BP) are assessed for each branch in dendrogram that formed the group, to ensure the reliability of the branches constructed. This study found that the average linkage technique with modified distance is significantly superior than the usual average linkage technique, if there is an outlier. Both of these techniques are said to be similar if there is no outlier.

Adaptive and automatic trimming in testing the equality of two group case

In testing the equality of two independent groups, t-test plays a very important role for the purpose. This test is reliable when the data is normally distributed. Based on central limit theorem, the normality assumption is fulfilled with large data set, but getting large data set is not always feasible. Most of the time, the researchers have to make do with small sample sizes which are hardly normally distributed. There are many causes of non normality, and one of it is the presence of outliers. One way to handle outliers is by using robust estimator with trimming approach. In this study, robust estimators using different trimming approaches namely adaptive and automatic trimming were proposed as the center measures in Alexander-Govern (AG) test. The results of the Type I error rate was then compared with the original AG test and the classical t-test. The AG test with the adaptive and automatic trimming showed robustness across distributions. The two trimming approaches are comparable to each other in most conditions. As expected the original AG test and classical t-test cannot maintain their robustness especially under skewed distribution.

P-method post hoc test for adaptive trimmed mean, HQ

Adaptive trimmed mean, HQ, which is one of the latest additions in robust estimators, had been proven to be good in controlling Type I error in omnibus test. However, post hoc (pairwise multiple comparison) procedure for HQ was yet to be developed then. Thus, we have taken the initiative to develop post hoc procedure for HQ. Percentile bootstrap method or P-Method was proposed as it was proven to be effective in controlling Type I error rate even when the sample size was small. This paper deliberates on the effectiveness of P-Method on HQ, denoted as P-HQ. The strength and weakness of the proposed method were put to test on various conditions created by manipulating several variables such as shape of distributions, number of groups, sample sizes, degree of variance heterogeneity and pairing of sample sizes and group variances. For such, a simulation study on 2000 datasets was conducted using SAS/IML Version 9.2. The performance of the method on various conditions was based on its ability in controlling Ty...

A power investigation of Alexander Govern test with adaptive trimmed mean as a central tendency measure

This study investigated on power of the test for Alexander Govern method using adaptive trimmed mean as a central measurement (AH). The power of the test is controlled by three parametric specification namely significance level (α), sample size (n) and effect size (EF). Previous studies on Type I error rates found that this method is robust and perform very well even under extreme conditions. To check on the strength and weakness of the method with regards to power of test, variables such as the shape of the distribution of data, variance ratio, sample sizes and the pair between sample sizes and unequal variance were manipulated to create various conditions. Results from this study show that the power of the AH test can be considered high in all of the normal distribution. High power test is also consistent under nonnormal data for the case of unbalanced sample sizes and equal variance of positive pairing.