GRÁFICOS DE CONTROLE BASEADO NOS RESÍDUOS DO MODELO DE REGRESSÃO POISSON* CONTROL CHART BASED ON THE POISSON REGRESSION MODELS RESIDUALS*

The control chart based on the Poisson residuals has been useful for monitoring the number of nonconforming in an industrial process. The Poisson regression model is the most popular used by Generalized Linear Models, which is used to model the event count data. The Poisson regression model has an assumption that the variance is equal the mean, but it does not always happen, in many situations, it has found that the variance is greater than the mean, and this phenomenon is called as overdispersion. The data used in this study are the number of nonconforming at weaving section in the Têxtil Oeste Industry Ltda. It was observed that these data have a great variability and have overdispersion. Thus, it was necessary to apply the Poisson regression models before the use of the control charts techniques.


INTRODUCTION
In SPC (Statistical Process of Control) two forms of variation are identified. Common causes, that are sources of variation that are unavoidable, and special causes of variation that can be corrected or eliminated. A process that is subject only to common causes of variation is said to be in-control. When special causes of variation occur, the process is said to be out of control. The purpose of SPC is to determine when a process is going out of control, so the process may be corrected (MONTGOMERY, 1997).
The traditional methodology of SQC (Statistical Quality Control) is based on a fundamental supposition that the process of the data is independent statistically; however, the data not always are independent. When a process follows an adaptable model, or when the process is a deterministic function, the data will be autocorrelated.
According to McCullagh and Nelder (1989) the Poisson regression models is a kind of specific generalized linear models (GLM), and the maximum likelihood method is used to estimate the parameters of Poisson regression models. When there is a difference in the parameters, that is, the variance value is bigger than the mean value, one considers that there is overdispersion, the variance value is smaller than the mean value, suggest that there is underdispersion. Count data are analyzed through Poisson regression models that very often show overdispersion, so that the assumption of equality between average and variance is not valid.
Evidences of overdispersion or underdispersion suggest that there is not a very good fit for the Poisson regression model. One solution to this phenomenon is a random effects approach of finite mixture. In contrast to random effects models which treat overdispersion as a nuisance factor that complicates statistical inference, finite mixture models may provide additional insights about different sources of heterogeneity in the population (BÖCKENHOLT , 2000).
In this paper we present the methodology of Poisson regression models applied to real data sets in the number of nonconforming at weaving section in the Industry Têxtil Oeste Ltda in September, 2001, and analyze if the data have overdispersion.

POISSON REGRESSION MODEL
In accordance with McCullagh and Nelder (1989), the Poisson regression model is a  Statistical of interest -After convergence (which it can be made through Newton-Raphson algorithm), it is necessary to examine the following statistics:

EXAMPLE
The series used in this study number of nonconforming at weaving section in the Industry Têxtil Oeste Ltda, in September, 2001. The used method in the adjustment involves Poisson regression models. The daily number of nonconforming in the weaving section, 30 values, is shown in figure 1. Notice that the series have a great variability, and the data, apparently don't present the trend over time that was confirmed later. The number of nonconforming mean is 0,9508, and standard deviation is 1,16.
Suppose that there is a tendency on the time, it is needed to have a polynomial of degree 4 to capture this effect. It was made the adjustment of the data through the Poisson regression models, where we found the following results:  Table 3 indicates that the Poisson model is adjusted. In table 4, we find the summary of the parameter model. According to Piegorsch (1998)

graphic diagnoses and other operations about residuals
help to indicate the discrepancy of the potential model.

Application of the control chart
Now the productive process' behavior can be verified. Figures 4 and 5 show the control conditions for the observations in the C chart.    Figure 4 shows the C chart for real data and figure 5 shows the X and R chart for transformed data. Verifications revealed that the system wasn't adjusted and the C control chart was replotted with the found residuals of Poisson regression model to correct the problem. The problem was in the 3 rd, , 4 th and 5 th observations, which after having modeled for the Poisson regression model they were verified inside of the control limits, except the 2 nd observation.
Acording to Wardell, Moskowitz and Plante (1994) it is entirely possible in traditional control charts, the points are out of the limits because of the systematic or the common causes and not because of occurrence of special causes.

CONCLUSION
In a monitored process, the variable must be independents, but if the intervals of time of monitoring are small, the data will be certainly autocorrelated. A chart of these data will frequently have a type in control, denoting a growth or decrease in the tendency. It means that these signs either they are really only false alarms out of the control, or the control limits are very conservative or even the data will have to be transformed because they are data autocorrelated.
The proposed methodology was shown satisfactory statistically; when data autocorrelations are studied a new learning perspective was generated on the productive process through of the information contained in the autocorrecion's structure, of the Poisson regression models, which were unknown for the classic model of monitoring. With this there was a growth of information for the correct decision and it can be detected that there was an improvement in the points of control exit.
As suggestion for further research, we recommend the use of the intervention analysis, in case the data have outliers; the use of non-parametric control charts and the use of other link functions.