Abstract: In this research, the bootstrap methods are used to investigate the effects of sparsity of the data for the binary regression models. The artificial data was created by the bootstrapping vector. We also used the percentile confidence intervals as a tool for inference, because they combine point estimation and hypothesis testing in a single inferential statement of great intuitive appeal. We found that the bootstrap confidence intervals are shorter than classical confidence intervals with the same confidence coefficient. We also found that some parameters that are non-significant when using classical confidence interval become significant with the bootstrapping sampling methods and vice versa. Moreover the bootstrap confidence intervals provided robust results for the sparse data. We also found that the sparsity of data results in the bad behaviour of the tail of the bootstrap sampling distribution, but reduction of confidence coefficient results to obtained robust confidence interval.
INTRODUCTION
Binary regression is widely used in the applied statistics (Agresti,
2002; Kleinbaum and Klein, 2002; Farrell and Rogers-Stewart, 2008). In
the standard setting, subject to regularity conditions, the Maximum Likelihood
Estimator (MLE) of the unknown parameters in the binary regression are
known to be consistent, asymptotically normal and efficient (Lehmann and
Casella, 1998). However, this situation is not satisfied if the data is
sparse (Zadkarami, 2000). This problem is encountered in medical (Allardice,
2001; Ainsworth and Dean, 2006), social (King and Zeng, 2001) and geography
(Begueria, 2006; Griffith, 2006; Vanwalleghem, 2007) studies which involve
the modeling of the outcomes of rare diseases or events because they are
usually dealing with small numbers of deaths or events. Therefore, sparse
binary response is of great interest (Fleiss et al., 2003; Farrell
and Sutradhar, 2006). This phenomenon, sparsity, was pointed out firstly
by Hauck and Donner (1977). Let Yi ε {0,1} be the binary
response variable. The probability of success (event) is p1
= Pr(Yi = 1) = F(βXi) where, F(.) is a cumulative
distribution function. If there are some maximum likelihood estimator
of parameters,
The sparsity of data depends on a range of factors and models which may, or may not, be under the control of the researchers. Sparsity of the data creates some problems in using standard asymptotic methods and it also creates some computational problems in the estimation of the unknown parameters in the models.
The bootstrap is a nonparametric technique that can be used to provide statistical inference about the parameter estimates, especially when the standard asymptotic methods are not satisfied properly (Hansen et al., 1999; Yousef et al., 2005; Modarres et al., 2006; Gerard and Schucany, 2007; Tang et al., 2007). This approach is based on using repeated samples from the data to generate an empirical sampling distribution for a statistic. In response to sparsity of the data, a completely nonparametric bootstrap approach is applied to resample cases. Then we can determine the MLE of the parameters in the model and make inferences about them. Nonparametric simulation requires the generation of artificial data without assuming that the original data have some particular parametric distribution. We use confidence intervals as a tool for inference, because they combine point estimation and hypothesis testing in a single inferential statement of great intuitive appeal.
In almost every case the accuracy of the confidence intervals depends on parametric assumptions that we are considered. However if parametric assumptions are not satisfied properly, the bootstrap methods may be used to obtain a more robust nonparametric estimate of the confidence intervals to assumptions commonly made about data (Moulton and Zeger, 1991; DiCiccio and Efron, 1996). The bootstrap confidence intervals are not only asymptotically more accurate than the standard confidence intervals; they are also more correct (DiCiccio and Efron, 1996). Over the past decade, substantial attention has been paid to the development of techniques using the bootstrap sampling distribution to build confidence intervals around various population parameters (Efron and Tibshirani, 1986; Stine, 1989; Moulton and Zeger, 1991; DiCiccio and Efron, 1996; Tu and Zhou, 2002; Henderson, 2005; Hsieh et al., 2007).
The bootstrap confidence intervals are divided into two groups: parametric and nonparametric, according to the assumptions which are used in their construction (Cojbasic and Tomovic, 2007; Andrews et al., 2006; Karlis and Patilea, 2008). The choice of which bootstrap confidence interval method to use is highly dependent on the particular research situation facing an analyst (Manichaikul et al., 2006). None of these techniques offers the best confidence intervals in every situation, because the criteria for judging the quality of their results vary widely (DiCiccio and Romano, 1988). In this research percentile confidence is used because the data is sparse and the analytic formulae are not satisfied properly. The percentile approach provides a nonparametric method that does not require any assumption, such as normality, in order to build a confidence interval (Mudelsee and Alkio, 2007).
From the point of view of demands on computational resources, bootstrap methods are particularly useful for small data sets because the procedure usually needs at least 1,000 repeated bootstrap samples in order to provide robust results for making confidence intervals (Efron and Tibshirani, 1993).
MATERIALS AND METHODS
Let Yi ε {0,1} be the binary response variable. The probability of success is given by
where, ηi = βXi denotes the linear predictor, Xi denotes the vector of explanatory variables that characteristics of the individual i, β is a interested parameter vector and Φ(.) is the standard normal cumulative distribution. The log likelihood function is
The probit link function is used because this link function produces better results compared to the other link functions in the sparse data (Zadkarami, 2000).
Percentile confidence interval: There are two methods the bootstrapping residual (parametric bootstrap approach) and the bootstrapping vector (nonparametric bootstrap approach) for Generalized Linear Models (GLMs), (Salibian-Barrera, 2005; Pardo-Fernaudez, 2007; Shen and Zhu, 2008). However, the definitions of the residuals for GLMs and consequently nonparametric residual resampling approach for GLMs, are not unique (Davison and Hinkley, 1997). These residuals are: standardized Pearson residual, standardized residual on the linear predictor and standardized deviance residual. Consequently, three different approaches for bootstrapping samples for GLMs can be used, corresponding to these three residuals. These residuals are scaled implicitly or explicitly. However, Davison and Hinkley (1997) reported that none of these methods is perfect.
The disadvantage of the parametric bootstrap approaches for a GLM is that these methods involve the simulation of a new data set from the fitted parametric model. In fact, the generated data sets from a poorly fitting model may not have the statistical properties of the original data, particularly when count data are overdispersed relative to a Poisson or binomial model (Davison and Hinkley, 1997).
In response to these difficulties, a completely nonparametric approach is to resample cases in the manner described as follows. Nonparametric simulation requires the generation of artificial data without assuming that the original data have some particular parametric distribution. Let the vector zi = (Xi, Yi), i = 1, 2, ...,n, be a sample from some multivariate distribution F(.) of (X, Y). In this approach, the regression coefficients are viewed as a statistical function of F(.), but with no assumptions on the random errors of model other than independence. The pairs (Xi, Yi) are bootstrapped to provide the bootstrap data set z* so that
for i1, i2,....in, a random sample of the integers 1 to n. Then, following steps 1, 2, 3 and 4 will allow us to make inferences about parameters in the model.
| • | Selected B replicated bootstrap samples of size n, z1*, z2*, ...zB*. |
| • | The binary response with probit link function was fitted to each
bootstrapping sample z* = (Y*, X*) and the vector of the parameters,
|
| • | The 100(1-2α)% percentile confidence interval for the parameter β is obtained as |
where,
(1) |
| • | The BC (Bias Correction) confidence interval is calculated as |
where, was introduced in (1) and α1 and α2 are functions of zα, the α th percentile of the standard normal distribution. In fact, the percentile interval may be improved by a simple adjustment, the BC (bias-corrected) method (Efron and Tibshirani, 1986; Pituch et al., 2006; Cerin and Leslie 2008; Timmerman and Ter Braak, 2008). In the BC method, α1 and α2 are calculated by
(2) |
| Where: |
When half of the bootstrap distribution of
The bootstrap confidence intervals are also divided into two groups: parametric (normal approximation method and the bootstrap-t method) and nonparametric (the percentile method), according to the assumptions which are used in their construction. The choice of which bootstrap confidence interval method to use is highly dependent on the particular research situation facing an analyst. None of these techniques offers the best confidence intervals in every situation, because the criteria for judging the quality of their results vary widely (DiCiccio and Romano, 1988). However, normal approximation confidence intervals rely on a strong parametric assumption when the bootstrap procedure actually was designed to be a nonparametric technique. Confidence intervals developed in this way are no better than those developed using the traditional parametric approach when this particular assumption is violated (Mooney and Duval, 1993).
The bootstrap-t method raises two problems. First, we need to calculate
| • | This method is free from parametric assumptions and
is simple to execute. We need no complex analytical formulae to estimate
the parameters of |
| • | If a statistic is distributed asymmetrically, it does not in
theory adversely affect the accuracy of the percentile method`s
confidence interval. The bootstrap percentile method allows |
| • | The percentile interval for any (monotone) parameter transformation m(β) is simply the percentile interval for β mapped by m(β), because this method is transformation-respecting (Efron and Tibshirani, 1993). |
However, the percentile method does have some problems. First, it may
perform poorly with small samples, because the tails of the sampling distribution
in these types of confidence interval calculation are important. The second
potential problem with the percentile method is that we must assume that
the bootstrapped sampling distribution is an unbiased estimate of
RESULTS AND DISCUSSION
The National Child Development Survey (NCDS) data set is used to investigate the effect of the sparsity of data on parameter estimates for the binary model. This data set was collected on babies born in one week (3-9 March 1958) in England, Wales and Scotland (Wildschut et al., 1997; Spencer, 2006). We selected a sample of 10,141 individuals for whom we have complete information on forty variables associated with perinatal mortality. The interval between 28 weeks (196 days) of gestation and week 4 after birth is divided into four subintervals, allowing for the possibility of death before delivery (antepartum stillbirth), during delivery (fresh stillbirth), in the first week after birth (early neonatal deaths), or between the first and fourth weeks after birth (late neonatal deaths). We call these stages 1, 2, 3 and 4, respectively. The survivors beyond stage one are further divided into two groups, those with assisted delivery and those with natural delivery to allow for the differential effects of the type of delivery in the two groups (Zadkarami, 2000). The data in stage 3, early neonatal duration, of the assisted delivery cases are selected as an illustration in the investigation of the effect of sparsity of data on the parameter estimates for the binary models. There are 1132 cases of the assisted delivery that include 12 deaths, Y = 1 and 1120 survival cases, Y = 0.
In the nonparametric approach the bootstrapping vector is used to construct the percentile confidence interval for parameters in the model. Although the bootstrapping of cases demands extensive computational resources, this method has the properties of the original data set and the results are not affected by poorly fitting model.
Efron and Tibshirani (1993) suggested at least 1,000 repeated bootstrap samples were needed in order to provide robust results for making confidence intervals. In order to protect the results from the effect of sparseness in the data, 2,000 bootstrapping samples z* = (X*, Y*) were selected from the original data set (X, Y). Moreover, the bootstrapping sample size was subsequently increased to 5,000 and 10,000 samples in order to obtain a better bootstrap confidence interval. However, increasing the number of bootstrap samples from 5,000 to 10,000 improved the results only slightly. The binary response with probit link function was fitted to each bootstrapping sample z* = (X*, Y*). Various starting values were used to assess the effect of starting values on the results. Finally the percentile confidence interval for each parameter was obtained.
As empirical results demonstrated, bootstrap samples with far fewer numbers of (early neonatal) deaths result in poorly estimated parameters. Consequently, the tails of the bootstrap sampling distribution are badly behaved at levels of 95% and higher. However, the behaviour the tails of the distribution at the 90% level is satisfactory enough to allow us to construct the percentile confidence interval. The bad behaviour of the tail of the bootstrap sampling distribution also does not allow the BC methods to improve the endpoints of the percentile confidence interval. Therefore the sparsity of the data results in the bad behaviour of the tail of the bootstrap sampling distribution at levels of 95% and higher. However, the reduction of confidence coefficient results to find the robust confidence interval.
Table 1 displays the 90% percentile confidence interval and classical confidence interval based on the results of the package GLIM (Aitkin et al., 2005). As we can see, the two confidence intervals are completely different and the percentile confidence intervals are shorter than classical confidence intervals with the same confidence coefficient. Shorter size is one of optimality in using confidence intervals (Casella and Berger, 2002).
In the percentile confidence interval, the variable “baby birth order (2nd or later baby)” is associated positively with early neonatal death. Uddian and Hossain (2008) reported similar results. A previous caesarean delivery does not increase the risk of neonatal death (Richter et al., 2007). We also find similar results as reported in Table 1.
We observed that “the week of 1st antenatal visit” also is associated negatively with neonatal death. Uddian and Hossain (2008) reported that the negative association between neonatal mortality and timing of first antenatal check was highly significant such that the neonatal mortality was highest among the babies whose mother had not received antenatal check during pregnancy. Improving antenatal and neonatal care are resulted in the decline in perinatal mortality (Forssas et al., 1998; Cruz-Anguiano et al., 2004). Moreover, every year many women die due to pregnancy and delivery-related complications. Many of these deaths and related morbidity can be avoided through effective appropriate maternity care and preventive, diagnostic and timely therapeutic interventions (Petrou et al., 2003).
| Table 1: | The results of bootstrap and classical confidence intervals |
| APH2: Accidental antepartum haemorrhage | |
Whether the baby is delivered naturally or with assistance reflects both the effect of previous pregnancies and the current pregnancy. If the health of the mother or baby would be endangered by allowing the pregnancy to continue, the labour will be induced by using artificial means. Sanchez-Ramos et al. (2003) reported that women whose labour was induced experienced a lower perinatal mortality rate. We found that the variable whether labour induced (oxytocin+surgical, O.B.E.+oestrogen and oxytocin in labour) are negatively associated with early neonatal deaths for assisted delivery babies.
We also found the variables abnormality during pregnancy (placenta praevia), past complication of pregnancy (other abnormality) are negatively associated with early neonatal deaths for assisted delivery babies. In fact the association between variables “abnormality during pregnancy, past complication of pregnancy with early neonatal deaths is the effect of type of delivery on the surviving baby during the first week after delivery (Zadkarami, 2000). Moreover, a previous caesarean delivery does not increase the risk of neonatal death (Richter et al., 2007). We obtained similar results as reported in Table 1.
However, in the classical confidence interval, the terms birthweight and (birthweight)**2 are the only significant variables. Considering the results that are displayed in Table 1 and discussion about variables in the model, the percentile confidence interval presents better results for the sparse data.
CONCLUSION
Our results confirm that sparsity of the data can affect the results of fitting binary models. We found that some parameters that are non-significant when using classical confidence interval become significant with the bootstrapping sampling methods and vice versa. We also found that the bootstrap confidence intervals are shorter than classical confidence intervals with the same confidence coefficient. Therefore, the bootstrap confidence intervals are more robust and the bootstrap methods can be useful to confirm results in the analysis of sparse data.