3.1.2. Heckit Model

Heckman (1979) proposed a two-step estimation method to resolve the problem of sample selection bias caused by using observable sample data. The two-step estimation method first uses the probit method to estimate the coefficients of all observed values and calculates the inverse Mills ratio (IMR). It has subsequently used the ordinary least squares method to estimate nonzero observed values, to include the IMR as an explanatory variable, and to estimate the coefficients of the model. The Heckit model mainly comprises a selection equation and an outcome equation:

Selection equation:

$$d\_i^\* = z\_i \alpha + \mu\_i \,\,\mu\_i \sim \mathcal{N}(0, 1) \tag{9}$$

$$d\_i = 1 \text{ if } d\_i^\* > 0 \tag{10}$$

$$d\_{\bar{1}} = 0\_{\prime} \,\, else\,\,\tag{5a}$$

In Equation (9), *d* ∗ *i* is the latent variable, *zi* is the explanatory variable influencing participation and consumption, and *α* is the corresponding coefficient. Equation (9) reflects the relationship between *d* ∗ *i* , the latent variable of the selection mechanism, and *di* , the dichotomous dummy variable actually observed (Huang and Wang 2016).

Outcome equation:

$$y\_i^\* = \mathfrak{x}\_i \mathfrak{E} + v\_i \quad v\_i \sim \mathcal{N}\left(0, \sigma^2\right) \tag{11}$$

$$y\_i = y\_i^\* \text{ if } \ d\_i = 1 \tag{12}$$

In Equation (11), *y* ∗ *i* is the latent consumption expenditure variable, *yi* is the observed consumption expenditure variable, *xi* is the variable influencing consumption expenditure, and *β* is the corresponding coefficient. The Heckit model assumes that the error terms (*µi* and *νi* ) of the selection equation and the outcome equation are correlated, with the degree of correlation being expressed by *ρ*. The normal distribution of the error terms of the two equations is represented in Equation (5).

Apart from the two-step estimation method, the Heckit model can also adopt the maximum likelihood method to estimate the parameters, and its log-likelihood function is as follows (Aristei et al. 2008; Wodjao 2007):

$$\ln L = \sum\_{0} \ln \left[ 1 - \Phi(z\_i a) \right] + \sum\_{+} \ln \left[ \Phi \left( \frac{z\_i a + \frac{\rho}{\sigma} (y\_i - x\_i \beta)}{\sqrt{1 - \rho^2}} \right) \frac{1}{\sigma} \phi \left( \frac{y\_i - x\_i \beta}{\sigma} \right) \right] \tag{13}$$


Contents

#### **Preface XI**

Piotr Henryk Skarżyński and Maciej Ludwikowski

#### **X** Contents

### **Section 3 Treatment Strategies for Hearing Loss 135**

Chapter 8 **Cochlear Implants: An Excursus into the Technologies and Clinical Applications 137** Mohammad Hossein Khosravi, Ali Kouhi, Sasan Dabiri, Pedram Borghei and Masoumeh Saeedi

Preface

throughout each chapter.

The objective of this volume is to diffuse the information related to hearing loss, which is among the most prevalent chronic disabilities worldwide. Nowadays, it is clear that the identification and rehabilitation of hearing impairment, when possible, have to be adequate‐ ly and promptly managed because hearing loss can seriously interfere with psychosocial de‐

This volume was made possible through the substantial contributions of 10 groups of authors. The format of this volume is purely educational, targeting graduate courses in audiology, speech pathology and hearing science, and neurosciences and basic graduate course in oto‐ laryngology. The authors have been asked to present a generous background of their topic in the introduction section of their chapter and an extensive list of technical terms used

1. Section I: "Hearing Loss and Its Etiology." In this section, the problems associated with various types of hearing impairment are presented, particularly focusing on

2. Section II: "Early Diagnosis and Prevention of Hearing Loss." This section focuses on the description of a newborn-hearing screening as an important tool to prevent problems resulting from hearing loss and on the assessment and prevention of oto‐ toxicity (the various side effects of drugs related to cochlear-induced damage). 3. Section III: "Treatment Strategies for Hearing Loss." Hearing aids, cochlear im‐ plants, and implantable devices represent the currently used treatments for reha‐ bilitation of patients affected by hearing impairment, and the specific indication

It was our intention to use multimedia material wherever possible, but for the majority of contributors, this was not feasible. Further developments (such as multimedia materials and updates) will probably appear on IntechOpen website (https://www.intechopen.com/).

We would personally like to thank all the contributing authors, the IntechOpen personnel, and specifically Ms. Renata Sliva and Ms. Dajana Pemac for their continuous and generous

**Stavros Hatzopoulos PhD and Andrea Ciorba MD PhD**

Clinic of Audiology & ENT University of Ferrara, Italy

velopment, family dynamics, and social interactions.

The volume is divided into three main sections as follows:

the genetic background of hearing loss.

of each solution is presented and discussed.

assistance during the preparation of this volume.