The Outlier Detection for Ordinal Data Using Scalling Technique of Regression Coefficients

The aims of this study is to detect the outliers by using coefficients of Ordinal Logistic Regression (OLR) for the case of k category responses where the score from 1 (the best) to 8 (the worst). We detect them by using the sum of moduli of the ordinal regression coefficients calculated by jackknife technique. This technique is improved by scalling the regression coefficients to their means. R language has been used on a set of ordinal data from reference distribution. Furthermore, we compare this approach by using studentised residual plots of jackknife technique for ANOVA (Analysis of Variance) and OLR. This study shows that the jackknifing technique along with the proper scaling may lead us to reveal outliers in ordinal regression reasonably well.


Introduction
On the category scale, the panellists are asked to give a score, for an example 1, 2, …, 8 on his opinion about the taste of a certain food. This scale is a discrete number and arranged as an ordinal scale which usually ranges from 1 (the best) to 8 (the worst), or from 'extremely tender' to 'extremely tough' for tenderness of meat, for example. Joo et al. [8] reviewed the definition of fresh meat quality, meat quality traits and variations of meat quality. They also presented the recent knowledge underlying the relationship between fresh meat quality traits and muscle fiber characteristics. In the research, they proposed several potential factors including breed, genotype, sex, hormone, growth performance, diet, muscle location, exercise and ambient temperature that can be used to manipulate muscle fiber characteristics and subsequently meat quality in animals.
This problem is a very popular topic on sensory evaluation. A comprehensive review of sensory and consumer science has been conducted by Naes et al. [10]. They also used a 8-category scale for scoring quality attributes. Some researchers omit a central, neutral category to produce a more even distribution of responses. A line or an unstructured scale is a judge response on a solid line from 'little' to 'much' or from 'none' to extremely'.
One of the common used response models for ordinal data analysis is the proportional odds model. This model is an extension of logistic model for binary data where their responses are more than two categories [3]. He discussed loglinear models for ordinal variables which have some advantages compared to the model treating all variable as nominal. Whenever the relationship between response and explanatory variables needs to be distinguished then we use the 'logit'model. The model is based on the success probability on response variables, hence, the procedure has to ensure that fitted probabilities are in the range (0,1). Logit model can be found using logit transformation based on logistic function. In term of the power test, Avery and Adnan [4] have simulated several cases and investigated the predictor effects on response variable. When comparing this approach with the ANOVA procedure we can say that OLR was more powerful then ANOVA procedure except for very smal sample sizes (n=6) and very large sample sizes (n=600).
Furthermore, Adnan and Avery [1] have discussed various problems with analysing ordinal data using ANOVA and OLR and proposed several possible solutions such that these approaches could be used. We suggest the use of sample averages to avoid the violation of most of the ANOVA assumptions. By applying the proportional odds model, it was found that the estimated effects from these two methods were quite similar [9].
The most commonly used method to assess the fit of a model is by constructing a goodness-of-fit test. This test determines the adequacy of the fitted model in describing the relationship between response and predictor variables. We can say that a large value of the statistic means that tehre is a big discrepancy between observations and fitted value. Consequently, it gives an indication that the model has not fitted very well. In logistic regression, goodness-of-fit test using the residual deviance is very common but care needs to be taken since the number of parameters is increasing at the same rate as the sample size. If the deviance follows a χ 2 -distribution then we expect that the ratio of the mean and the degrees of freedom will be approximately one. Unfortunately, the simulation shows that the deviance does not follow a χ 2 -distribution [1]. Hence, jackknifing technique for ordinal logistic regression is an alternative goodness-of-fit test. By applying this method for detecting some outliers along with the proper scaling may lead us to reveal outliers in ordinal regression reasonably well.

Methods
The term `outlier' refers to values of the response variables which are unusually large or small [3,7]. Since all of the explanatory variables in our case are factors, any abnormal values are likely to be outliers. The method that we propose here is a resampling method, the so called jackknife technique, i.e. repeated analysis leaving out one observation at a time [6]. This technique is usually used for estimating the bias and standard error of an estimate.
One of the most popular models for logistic regression is a proportional odds model [9] which can be stated in response Y with j category (j=1,2,…k) and predictor X as ( The equation can also be written as a linear logistic regression: Furthermore, we use this model to identify if any outliers on ordinal data set by using goodness-offit test. Now consider the deviance: approximately distributed as χ 2 distribution with a particular degreee of freedom. The deviance is also useful for testing the parameters of two models by examining the difference in the two deviances. The overall discrepancy between observed and predicted values can be calculated using a generalised likelihood ratio statistic. This idea comes from the fact that a generalised linear model is defined and fitted using ideas based on likelihood. Now consider n binomial observations of the forms of proportion . i=1,2,…n. Let the fitted value of is ̂ ̂ . The counterpart of standardised residual for logistic model can be stated as [5]: and is the ith diagonal element of nxn matrix . For the ordinal data with (k+1) categories we have k residuals for each multinomial observations. Since these residuals are correlated it is not clear how to analyze them using standardized residuals mentioned above. Unusual observations in taste panel data can occur in various ways, e.g. misreading of scores, erratic and inconsistent scoring by a panelist. However, since we combine the scores over the panellists for a particular sample, then the important effect is when a sample's score are all high or all low relative to the average for that treatment combination.
Since the treatment effects in ANOVA and OLR are compared to the first level as a reference coefficient, i.e. the first coefficient is zero, then it would be interesting to check if changing the reference coefficient has any effect on the analysis. The sum of the modulus of the three coefficients is plotted again the number of the observation removed.
Since the attribute consists of four levels, we can find four sum of moduli when applying the jackknife technique, depending on which is the reference level, but two of these will be the same as each other. This can be illustrated by imagining the four treatment effects, e.g. their coefficients, which are denoted by b 1 , b 2 , b 3 , and b 4 . Without loss of generality, we can assume b 1 < b 2 <b 3 <b 4 . We can now scale numbers to put different classes equal to zero giving four possible scaling as listed in Table 1. Table 1: Table of four possible scaling  Number  1  2  3 The sum of the moduli for each column are as follows: We can see that the second scaling (2) and the third scaling (3) have the same sum of moduli, i.e. when we scale to either of the middle numbers. In general, if the numbers of factor levels is even then the number of different plot is the number of level minus 1.
We now propose an alternative procedure by scaling the coefficients to an average. For example let the estimated coefficients S ki , k=1,2,3,4; i=1,2,3,4 by choosing the first level to be zero, be: 0, b 2 , b 3 , and b 4 . The possible scaling using average, putting each leavel to zero, is listed in Table 2.
This is because the estimated effect of differences between levels is always the same. If we calculate the average in each case, we obtain for each possibilities: For k=2 (choose the second level to be zero), k=3, and k=4, we have the same scaled values as for k=1. Since all of the cases give the same scaled values then it is sufficient to calculate the scaled sum of moduli from the first coefficient only.

Main Results
Let us know consider the meat data set that consist of 948 observations [1]. This dataset has four predictors with different numbers of levels: factor1 (fac1) and favtor3 (fac3) consist of 3 levels while factor2 (fac2) and factor4 (fac4) have 2 levels. The response scores are classified into 8 categories start from 1 (very tenderness) to 8 (very untenderness). We use the reference distribution for this stage by scoring the tenderness of meat. The estimated distributions are listed on Table 3. Scores are generated using uniform random numbers conditioned on the estimated probility of a reference distribution. We use the inverse probability integral transform method for generating observations for these discrete cases. We then define k x X  if and only if 1 ,..., . Based on the probability distribution in Table   3 data were simulated for various numbers of observations per factor. Basically, these observations, i.e. the ordinal responses, were attained using uniform random number generation conditioned on estimated probabilities. Furthermore, we analysed the outliers, if exist, on modelling fac1 and fac2 only. Following the procedure as the previous discussion, the sum of moduli of the difference between each coefficient ki S and their average k S will be the same for each case. Studentised residual plots of jackknife technique for ANOVA and OLR for (fac1) are shown in Figure 1.

Conclusions
The residual deviance probably does not have a χ 2 -distribution . The main reason for using jackknife technique is because residuals are difficult to define in OLR and we would expect that removing abnormal points to have the largest effect on estimated tretament coefficients. For ANOVA and OLR along with the proper scaling for regression coefficients may lead us to reveal outliers in ordinal regression reasonably well. The choice of the scalling play an important result to find the best method to use.