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^{1}School of Graduate Studies and Research, Meharry Medical College, Nashville, TN 37208, USA

^{2}Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

^{3}Department of Mathematics, New Jersey City University, Jersey City, NJ 07305, USA

^{4}Department of Pathology and Comprehensive Cancer Center, University of Alabama at Birmingham, Birmingham, AL 35294, USA

^{5}Department of Medicine Division of Preventive Medicine and Comprehensive Cancer Center, University of Alabama at Birmingham, Birmingham, AL 35294, USA

- *Corresponding Author:
- Karan P Singh

Department of Medicine Division of Preventive Medicine

and Comprehensive Cancer Center

University of Alabama at Birmingham, Birmingham, AL 35294, USA

**Tel:**205-934-6887

**Fax:**205-934-4262

**E-mail:**[email protected]

**Received date:** July 01, 2014; **Accepted date:** July 27, 2014; **Published date:** July 30, 2014

**Citation:** Tabatabai MA, Li H, Eby WM, Kengwoung-Keumo JJ, Manne U, et al. (2014) Robust Logistic and Probit Methods for Binary and Multinomial Regression. J Biomet Biostat 5:202. doi:10.4172/2155-6180.1000202

**Copyright:** © 2014 Tabatabai MA, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are are credited.

**Visit for more related articles at** Journal of Biometrics & Biostatistics

In this paper we introduce new robust estimators for the logistic and probit regressions for binary, multinomial, nominal and ordinal data and apply these models to estimate the parameters when outliers or influential observations are present. Maximum likelihood estimates don’t behave well when outliers or influential observations are present. One remedy is to remove influential observations from the data and then apply the maximum likelihood technique on the deleted data. Another approach is to employ a robust technique that can handle outliers and influential observations without removing any observations from the data sets. The robustness of the method is tested using real and simulated data sets.

Binary and multinomial regressions are commonly used by medical scientists and researchers for analysis of binary or polytomous outcomes. These methods are routinely used as diagnostic tools in all areas of medicine including oncology and cardiology. Zhou et al. [1] used logistic regression to relate the gene expression with class labels. They also used logistic regression for their microarray-based analysis of cancer classification and prediction. Sator et al. [2] applied a logistic regression model to identify enriched biological groups in gene expression microarray studies. Majid et al. [3] performed logistic regression analysis to predict endoscopic lesions in iron deficiency anemia when there are no gastrointestinal symptoms.

Morris et al. [4] applied multinomial regression technique to analyze the sub-phenotypes by allowing for heterogeneity of genetic effects. Richman et al. [5] investigated the association between European ancestry and renal disease when compared with African Americans, East Asians, and Hispanics. They concluded that European ancestry is protective against the development of renal disease in systematic lupus erythematosus. Their data had some outliers but they were excluded in their final analysis. Timmerman et al. [6] used the logistic regression to distinguish between benign and malignant adnexal mass before surgery. Merritt et al. [7] used the binary and multinomial logistic regressions to investigate the role of dairy food intake and risk of ovarian cancer. The validity of estimation and testing procedures used in the analysis of binary data are heavily dependent on whether or not the model assumptions are satisfied. The maximum likelihood method of estimating binary regression parameters using logistic, probit and many other methods is extremely sensitive to outliers and influential observations.

There is a large literature on the robustness issue of the binary regression. Most of the existing methods attempt to achieve robustness by down weighting observations which are far from the majority of the data, that is, outliers. The reader is referred to papers published by Pregibon [8], Carroll and Pederson [9], and Bianco and Yohai [10]. Bianco and Martinez [11] modified the original score functions of the logistic regression to obtain bounded sensitivity, which is a concept introduced by Morgenthaler [12] using the L1-norm instead of the L2-norm in the likelihood, resulting in a weighted score function of the original score function. Cantoni and Ronchetti [13] focused on robustness of inference rather than the model. Pregibon [8] suggested resistant fitting methods which taper the standard likelihood to reduce the influence of extreme observations. Kordzakhia et al. [14] introduced a robust logistic regression by minimizing the mean-squared deviance for the worst case contamination. Bergesio and Yohai [15] introduced projection estimators for generalized linear model. These estimators have the same asymptotic normal distribution as the M-estimators. Hobza et al. [16] introduced a median estimator to estimate the parameters of the logistic regression.

Robust binary and multinomial regression estimators for analysis of biomedical data are proposed. This robust method has a bounded influence and high breakdown point and efficiency under normal distribution and is able to estimate the parameters of logistic and probit regression models. The proposed model is computationally simple and can easily be used by researchers.

Consider the model y_{i}=π(x_{i};β)+ε_{i} where ε_{1},ε_{2},…, ε_{n} are independent random variables with E(ε_{i})=0 and Var(ε_{i})=π(x_{i};β)(1−π(x_{i};β)), and y_{1},y_{2},…,y_{n} are n independent Bernoulli random variables with Ey_{i}=π(x_{i};β) and Var(y_{i})=π(x_{i};β)(1−π(x_{i};β)) such that the conditional success probability is given by P(y_{i}=1| x_{i})=π(x_{i};β) and x_{i}=(x_{i0},x_{i01},… ,x_{ip})t; 1≤i≤n is a p+1 dimensional vector of predictor variables with β=(β_{0},β_{1},…,β_{p})^{t} as the parameters vector.

There are various estimation methods for the estimation of the parameter vector β. The most commonly used method is the logistic regression which is used to analyze the effects of explanatory variables on the binary response y. In the logistic regression the link function π_{L}(x_{i};β) is assumed to have the following functional form

The logistic transformation of π_{L}(x_{i};β) is called the logit function and is given by

The probit function is a link function of the form

Where Erf function is defined as

which has the probit transformation function

The tabaistic model introduced by Tabatabai and Argyros [17] has the link function:

Where arcsinh (.) represents the inverse hyperbolic sine function.

The tabaistic transformation function is called the tabit function and is defined as

Where Csch (.) denotes the hyperbolic cosecant function and the complementary log-log model link function has the form

with the complementary log-log transformation function (cllogit) defined as

**Figure 1** shows the graph of π_{L}, π_{CLL}, π_{P} and π_{T} where of π_{L}, π_{CCL}, π_{P} and π_{T} take values between zero and one. The solid curve is the graph of πL function, the dotted curve is the graph of πP function, the dot-dashed curve is the graph of π_{CLL} function, and the dashed curve is the graph of π_{T} function. **Figure 2** shows the graph of logit (π_{L}). cllogit (π_{CLL}), probit(π_{P}), and tabit(π_{T}). The solid curve, the dotted curve, the dotdashed curve and the dashed curve are the graph of logit (π_{L}) function, the graph of probit(π_{P}) function, the graph of cllogit (π_{CLL}) function, and the graph of tabit(π_{T}) function, respectively. The principle of maximum likelihood is ordinarily used to estimate the model parameters by maximizing the log-likelihood function of the form

In other words, the estimate of β is

Although the maximum likelihood estimator is asymptotically efficient, it is not recommended as a method of choice when outliers are present. The alternative techniques are robust statistical methods.

Tabatabai et al. [18] defined the one parameter family of differentiable functions ρ_{ω}(x) of the form ρ_{ω}(x)=1−Sech(ωx), where the positive real number ω is called the tuning constant.

The bounded function ρ_{ω}: R→ R is a differentiable function satisfying the following properties:

Under the normality assumption for the error term ε_{i}, the asymptotic efficiency (Aeff) is defined as

(1)

Where ψ_{ω} is the derivative of ρ_{ω} and is equal to

Where Sech and Tanh represent the hyperbolic secant and hyperbolic tangent, respectively.

with

The tuning constant ω can be calculated by solving the following equation (1) for ω. The numerical values for ω at the efficiency levels 0.80, 0.85, 0.90, and 0.95 are approximately 0.721, 0.628, 0.525 and 0.405, respectively. Although the choice for tuning constant ω is left for the investigator to decide, we do recommend an efficiency of approximately 90 percent which corresponds to ω=1/2. We now consider the hat matrix of the form

H = X(X^{t}X)^{-1}X^{t},

where X is the design matrix defined as

If the model has intercept, then the column vector X_{0} has the form

For j=1,2,…,k, define

and for i=1,2,…,n, define

and for ω>0 define the function G_{ω}(u) as

(2)

The estimator

(3)

Where

and h_{ii} is the ith diagonal element in the hat matrix. For the logistic model, we have

So that the above integral (2) can be written as

Where H(k_{1},k_{2},k_{3},t) is the Gauss hypergeometric function 2F1 with parameters k_{1},k_{2} and k_{3}. If ω=1, then we have

and if ω=1/2, then we have

Define the Hessian matrix H_{b} for binary data as

Then an estimate of the variance-covariance matrix for vector is with an estimated variance σ^{2} given by

To perform hypothesis testing, we let be the parameter space and be a subset of {β_{0},β_{1},…,β_{p}}. Define

To test the following hypothesis

against the alternative one can use the Wald type test statistic which is defined as

The null distribution of the statistic is asymptotically a chisquare distribution with q degrees of freedom.

In this section we generalize the robust binary method to multinomial regression where the response y includes k categories. When k=2, this model reduces to the binary regression. Now, consider the response matrix Y given by

where

This means that y_{ij}=1whenever the ith response is in category j.

For i =1,2,…,n, and x_{i}=(x_{i1},x_{i2},…,x_{ip})^{t}. The parameter vector

β_{j}=(β_{j0},β_{j1},…,β_{jp})^{t} and the vector β_{0} is a zero vector in a p+1 dimensional space with β=(β_{0},β_{1},…,β_{k-1})^{t}. The multinomial likelihood function of parameter vector β is defined by

and the multinomial log- likelihood function is

For the generalized logit when the first category is the designated reference category and the intercepts are β_{01},β_{02},…,β_{0k-1}, we get

and for j = 2,…,k,

and for j = 1,…,k−1 the logit function η_{j}(x_{j};β) is the log-odds of membership in category j versus the reference category 1 and is equal to

The principle of maximum likelihood can be used to estimate model parameters. The maximum likelihood estimate of the parameters vector β is

(4)

where

The robust estimate of the model parameters is given by

(5)

Define the Hessian matrix H_{p} as

Then, for the multinomial response, an estimate of the variancecovariance matrix for vector is with an estimated variance σ^{2} given by

For the cumulative logit model for ordinal k-category response, the cumulative probability for the ith response belongs to the response category less than or equal j is

and for j=1,…,k−1 the ordinal logit O_{j}(x_{j};β) is the log-odds of falling into or below category j against falling above it and is given by

where . The ordered logit model is sometimes called the proportional odds model.

Let . Then we have

and for the ordinal probit we have

The robust estimate of ordinal multinomial parameter vector is giveb

(6)

where

**Vasoconstriction example**

Vasoconstriction and vasodilation are two important physiological mechanisms used to control the circulation of blood throughout the body. These mechanisms directly affect both the blood pressure and the distribution of the blood in the body. Vasodilation refers to the expansion of blood vessels through relaxation of smooth muscles in the vessel walls. This allows increased flow of blood through these vessels and also decreases the blood pressure. Contraction of the same muscles tightens the blood vessels, which decreases blood flow and increases pressure. Thus, vasodilation and vasoconstriction work in opposition to adjust both blood flow and blood pressure. The usual controls for vasoconstriction and vasodilation are done by smooth muscles and autonomic nervous system, triggered by the medulla. These responses can also be affected by drugs promoting either constriction or dilation. Furthermore, there is a means of control by circulating hormones in the bloodstream, as well as control by intrinsic mechanisms to vasculature, called the myogenic response. The antagonistic operation of vasoconstriction and vasodilation is used by the body for numerous purposes. Primary among these is regulation of the supply of oxygen and nutrients to the cells of the body, to meet their needs. Furthermore, this regulation of blood flow is also needed for thermoregulation within the body. At times of increased metabolic needs or needs for oxygen in certain organs or systems in the body, the blood flow to these regions will also be modulated. Finally, vasoconstriction is also important in restricting blood flow to regions of the body in cases of traumatic injury.

The data set was analyzed originally by Finney [19]. It consists of 39 observations where the binary response variable y=1 or y=0 represent the presence or absence of vasoconstriction of the skin respectively. This experimental data set considers the effect of inhalation of air in a single deep breath on the presence or absence of vasoconstriction in the digits. Presence or absence of vasoconstriction is considered as a categorical variable, and the study considers the effect of two variables, the volume of inhaled air and the rate of inhalation. (Data has hidden outliers so that the robust logistic regression will be useful in its analysis.) In the remainder of this work, we denote by ML and BY, the Maximum Likelihood and Bianco-Yohai methods, respectively. By examining **Table 1** we conclude that the new robust estimator has produced the closest parameter estimates to the maximum likelihood estimates when the outliers were removed. In addition, the is the lowest for the new robust method. The is defined by Kordzakhia et al. [14] as

Methods | Coefficients b0 b1 b2 |
Standard Error b0 b1 b2 |
---|---|---|

ML ML (Influential observations removed ) |
-2.887 5.191 4.578 (=48.3118) -24.590 39.539 31.928 (=32.0159) |
1.324 1.869 1.843 13.974 23.153 17.687 |

BY c=1.25 |
-5.3214 8.44547.4801 (=43.1853) |
9.7647 14.1903 12.3199 |

New Robust c=0.5 |
-24.1191 38.4639 30.9608 (=32.2671) |
15.3232 24.9410 19.2365 |

**Table 1:** Parameter estimates for Vaso Data Using ML, BY and New Robust
Method.

**Plasma example**

The erythrocyte sedimentation rate (ESR) is very important. It is a common hematology test which is simple and inexpensive but can be used to detect infection or acute phase response, which can alert physicians to a wide variety of conditions. The test is very versatile and can assist physicians in detecting conditions from rheumatoid arthritis to systemic lupus erythematosus to multiple myeloma; however it is non-specific and is usually combined with other tests. In practice, ESR is used widely to test for a range of conditions, including inflammation, trauma, and malignant disease. Studies have also suggested the utility of the ESR among the elderly as a general indicator of level of sickness or disease. Recently ESR has also attracted attention for a potential role as a predictor for the development of cardio-vascular disease and heart failure.

The ESR simply measures the rate at which red blood cells precipitate during a period of one hour. Anticoagulated blood is placed in an upright tube, and the rate at which the erythrocytes settle is measured in mm per hour. Although the test is a direct measurement of rate of sedimentation, the balance between factors stimulating sedimentation and factors resisting sedimentation allows for a number of clinically relevant factors to influence this rate. Fibrinogen is the most important factor promoting sedimentation, and the high level of fibrinogen in the blood during the inflammatory process makes this test sensitive to inflammation. High levels of fibrinogen in the blood decrease the repulsive forces experienced between the negatively charged erythrocytes and favor the formation of rouleaux. These stacks of erythrocytes that stick together will settle faster and lead to an increased ESR. Other acute phase reactants, or other large molecules, especially when positively charged, can have a similar effect, although fibrinogen has been observed to have the largest effect.

A recent focus on the inflammatory nature of artherosclerosis has been accompanied by a recent study of increased levels of ESR and elevated risk of coronary heart disease. Erikssen et al. [20] observed that elevated ESR is a strong predictor of mortality from heart failure, suggesting it may serve as a marker for aggressive forms of coronary heart disease. Andresdottir et al. [21] observed an increased risk of coronary heart disease among the top quintile or ESR rates, with a hazard ratio of 1.57 for men and 1.9 for women. The 2005 paper of Inglesson et al. [22] also observes a significant association between elevated ESR and heart failure, suggesting both that inflammation is involved in the processes leading to heart failure and that the ESR may be used in evaluating this process. In addition to the well-established uses of ESR, Saadeh [23] mentions some potential new applications of this test such as bacterial otitis media, acute hematogenous osteomyelitis, AIDS, pelvic inflammatory disease, prostate cancer, and early prediction of stroke severity.

Although the ESR usually detects acute phase response from fibrinogen in blood in conditions such as those mentioned above, in certain cases there are factors which decrease the rate of sedimentation. One important factor that can slow the rate of sedimentation is irregularity in the erythrocytes, either in shape or unusually small size. As a consequence, ESR can detect certain blood diseases (including sickle cell anemia and spherocytosis) which lead to a lower than normal rate of sedimentation, as observed in Bridgen [24]. Other conditions that may also lower ESR include the extreme levels of white blood cells as observed in chronic lymphocytic leukemia. Furthermore the surplus of erythrocytes found in patients with polycythemia makes rouleau formation difficult and decreases the ESR.

In clinical applications the erythrocyte sedimentation rate may in many cases be treated as a categorical variable, with a normal ESR for values less than some given α and an elevated ESR for values greater than α. When representing such a set of data where ESR depends on one or more variables the logistic regression may be used. For instance in the data set from Collett [25], the ESR is considered as a function of two variables, the level of fibrinogen and the level of γ-globulin. The data for 32 individuals represents the levels of fibrinogen and γ-globulin in the blood and whether the ESR level is healthy (< 20 mm/hr) or unhealthy (≥ 20 mm/hr), and the logistic regression is used to describe how both fibrinogen and γ-globulin affect the ESR variable. Since this data set contains (hidden/influential) outliers, both the probit method of regression and the logit method do not give accurate results. However we observed that our new methods for robust logistic regression do represent the data accurately. The logit, when all 32 observations are included in the study, is given by

When one removes the influential observations 15, and 23, the logit model becomes

The level of γ-globulin was not a statistically significant variable to be included in the model. Thus only the level of fibrinogen f is used in the variable selection.

Again, examining **Table 2** reveals that the new robust estimator has produced the closest parameter estimates to the maximum likelihood estimates when the outliers were removed as well as the lowest value for the

Methods | Coefficients b0 b1 |
Standard Error b0 b1 |
---|---|---|

ML ML (Influential observations removed ) |
-6.8451 1.8271 (=38.9405) -59.62 17.46 (=29.1415) |
2.7703 0.9009
45.5113.50 |

BY c=1.25 |
-8.3774 2.2870 (=37.0421) |
5.4383 1.6632 |

New method c=0.5 |
-60.5094 17.7654 (=29.2719) |
49.7325 14.7409 |

**Table 2:** Parameter Estimates for Plasma Data Using ML, BY and New Robust
Method.

**Mental health example**

The following example involves the ordinal multinomial regression. The data comes from a mental health study for a random sample of adult residents of Alachua County, Florida. This data was appeared in Agresti [26]. The mental impairment is divided into four categories (well, mild symptom formation, moderate symptom formation, and impaired). The explanatory variables are life events index X_{1} and socioeconomic status X_{2}, where X_{2} is binary and takes high and low levels. There is no outlier in this data. We just want to show how the method works even when outliers are not present. The logit is given by

**Table 3** gives the parameter values using maximum likelihood method as well as the new robust ordinal multinomial method. The R program for this example is provided in the Appendix.

Parameter | Estimated | Standard Error | Robust estimate | Standard Error |
---|---|---|---|---|

Intercept 1 | -0.2819 | 0.6423 | -.2374 | 0.7265 |

Intercept 2 | 1.2128 | 0.6607 | 1.1923 | 0.7507 |

Intercept 3 | 2.2094 | 0.7210 | 2.2981 | 0.8364 |

Life | -0.3189 | 0.1210 | -.3181 | 0.1618 |

Socioeconomic status | 1.1112 | 0.6109 | 1.1423 | 0.7789 |

**Table 3:** Parameter estimates for Mental Health data using robust ordinal method.

To evaluate the performance of the new robust method for logistic regression we conduct a Monte Carlo simulation. In the first round of our simulation, we use one explanatory variable and in the next round, we increase the number of explanatory variables to two. We first generate an independent random sample of size 100 from the standard normal distribution with mean 0 and standard deviation equal to 0.5. We call the variable x. Then we generate a sample of error terms ε_{i} of size 100 from the logistic distribution with mean zero and standard deviation equal to 1. The dependent variable y is generated using the formula

The model parameters are 1 (intercept) and 3 (coefficient for x). We then select a random sample (5%) from the generated sample x and contaminate the selected sample by multiplying each x value by a factor of 10. Then we repeat the above procedures 1000 times. Finally, we estimate both the bias and mean squared errors using the following equations

where m is the number of iterations in the simulation. The mean squared error is estimated by

For the two explanatory variables, we generate two independent normal random samples of size 100 from a normal distribution with mean 0 and standard deviation 0.5 and call them x_{1} and x_{2} respectively. Then we select 5% of this random sample (3% from x_{1} and 2% from x_{2}) and multiply the selected samples by 10. Then we generate a sample of error terms εi of size 100 from the logistic distribution with mean zero and standard deviation equal to 1. The dependent variable y is generated using the formula

We calculate the parameter estimates and continue the iteration 1000 times. In addition, we calculate the bias and mean squared errors. **Tables 4 and 5** show the results of simulations using ML, BY and the new robust method with one and two explanatory variables, respectively. For binary logistic regression the simulation results indicate that our new robust method is as good as the BY method. The BY method only covers binary logistic regression whereas our method not only covers binary but also covers multinomial regression for both nominal and ordinal responses.

Method | Bias | MSE | Bias (5% x) | MSE (5%) |
---|---|---|---|---|

ML | ||||

b_{0} |
0.0430 | 0.0876 | 0.08065 | 0.07938 |

b_{1} |
0.1784 | 0..6310 | 1.00777 | 2.16368 |

BY | ||||

b_{0} |
0.0463 | 0.0923 | 0.0027 | 0.0842 |

b_{1} |
0.1857 | 0.6662 | 0.2336 | 0.8885 |

New Robust | ||||

b_{0} |
0.0342 | 0.0897 | 0.0125 | 0.0779 |

b_{1} |
0.1125 | 0.5321 | 0.2642 | 0.9771 |

**Table 4:** Simulation Results for Logistic Regression (b_{0}=1, b_{1}=3, N=100, m=1000).

Method | Bias | MSE | Bias (5%) | MSE (5%) |
---|---|---|---|---|

ML | ||||

b_{0} |
0.04283 | 0.08152 | 0.00312 | 0.07526 |

b_{1} |
0.01071 | 0.25287 | 0.19770 | 0.23694 |

b_{2} |
0.10448 | 0.38209 | 0.36612 | 0.75513 |

BY | ||||

b_{0} |
0.0485 | 0.0856 | 0.0232 | 0.0788 |

b_{1} |
0.0128 | 0.2650 | 0.1711 | 0.2456 |

b_{2} |
0.1143 | 0.4028 | 0.0909 | 0.5125 |

New Robust | ||||

b_{0} |
0.0410 | 0.0786 | 0.0270 | 0.0755 |

b_{1} |
0.0255 | 0.2799 | 0.1045 | 0.2676 |

b_{2} |
0.1034 | 0.4037 | 0.0553 | 0.5032 |

**Table 5:** Simulation Results for Logistic Regression (b_{0}=1, b_{1}=0.5, b_{2}=2, N=100,
m=1000).

In this work we have proposed a new robust method to analyze binary and multinomial regression models. We believe that these new robust methods for binary and multinomial regressions have potential to play a key role in modeling categorical data in medical, biological and engineering sciences. We have shown the lack of robustness of the maximum likelihood technique when outliers are present. In both real examples and simulated ones and when the outliers are present, the new robust method performed well. In conclusion the motivation was to introduce a new robust loss function of residuals which can attain high breakdown value. The method has high efficiency and high breakdown points with bounded influence function.

Research reported in this paper was partially supported by the Center grant of the National Cancer Institute of the National Institutes of Health to the University of Alabama at Birmingham Comprehensive Cancer Center (P30 CA013148), the Cervical SPORE grant (P50CA098252), the Morehouse/Tuskegee University/ UAB Comprehensive Cancer Center Partnership grant (2U54-CA118948), and the Mid-South Transdisciplinary Collaborative Center for Health Disparities Research (U54MD008176). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

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