**2.2 Data description**

The variables were obtained directly from the questionnaire applied to the dataset of the sample that responded to the Complete Questionnaire and can be found on the website www.ibge.gov.br in the 2010 Census, sample, and microdata with more details about its description in Oliveira [1].

#### **2.3 Ordinal logistic regression**

A good number of the variables used in the social sciences and humanities are ordinal. Often, the dependent variable takes discrete values, or sortable categories, but the distance between them is neither known nor constant. For example, in epidemiological studies, the level of severity of visual, hearing, or physical is set out in the 2010 Demographic Census Sample Questionnaire, which can be classified as ¨*can not at all,*¨ ¨*he succeeds, but with great difficulty,*¨ ¨*can, but with a little difficulty*,¨ and, finally, "*no problem*" to hear, see, or get around. In the case of intellectual disability, it is divided into ¨*has*¨ or ¨*has not.*¨

Among possible adjustment models for ordinal logistic regression, the following ones stand out: proportional probability model, more suitable for interpretation when the response variable is continuous and has been categorized; continuous ratio model, suitable in situations where there is specific interest in a particular category of the response variable; partial proportional probability model that allows to moderate covariates with the assumption of proportional probabilities, and for other variables in which this assumption is not satisfied, specific parameters that vary for the different categories compared and an extension of the proportional probability model are included in the model; and finally, stereotype model, proposed by [5–7]) used in situations where the response variable is ordinal, which is not a discrete version of some continuous variable that was considered in this research.

For this work, we have response variables: visual, hearing, physical, and intellectual disabilities, which are ordinal variables. In view of this, we adopted the stereotyped model in this work.

#### *2.3.1 Stereotype model specification*

Imagine that the dependent variable consists of *J* categories (*m* = *1, … , J*) and consider *K* predictors (*J = 1, … ,K*). The stereotype ordinal model is defined at an early stage with the multinomial regression model to which the condition is added *<sup>β</sup>m J*<sup>j</sup> � *φmβ*^, where *J* is the reference category, that is, we have that the multinomial regression model is given by:

$$\text{Prob}\left(\mathbf{y} = \mathbf{m} | \mathbf{x}\right) = \frac{\exp\left(\boldsymbol{\beta}\_{m|\mathcal{J}}^{\prime}\mathbf{x}\right)}{\sum\_{j=1}^{J} \exp\left(\boldsymbol{\beta}\_{m|\mathcal{J}}^{\prime}\mathbf{x}\right)}, \text{with } m = 1, \ldots, J \tag{1}$$

Replacing *<sup>β</sup>m J*<sup>j</sup> <sup>¼</sup> *<sup>φ</sup>*m*β*<sup>~</sup> in Eq. (1) results in the stereotype model that can be written mathematically.

$$\begin{aligned} \text{Prob}\left(|\mathbf{y} = \mathbf{m}|\mathbf{x}\right) &= \frac{\exp\left(\rho\_m \tilde{\rho}^\prime \mathbf{x}\right)}{\sum\_{j=1}^l \exp\left(\rho\_1 \tilde{\rho}^\prime \mathbf{x}\right)} = \frac{\exp\left(\rho\_m \tilde{\rho}\_0 + \rho\_m \tilde{\rho}\_1 \mathbf{x}\_1 + \dots + \rho\_m \tilde{\rho}\_k \mathbf{x}\_k\right)}{\sum\_{j=1}^l \exp\left(\rho\_m \tilde{\rho}\_0 + \rho\_m \tilde{\rho}\_1 \mathbf{x}\_1 + \dots + \rho\_m \tilde{\rho}\_k \mathbf{x}\_k\right)}, \\ \text{with } m = 1, \dots, l \end{aligned} \tag{2}$$

For some parameters of Eq. (2) that are not identifiable, we consider as constraints *<sup>φ</sup>m*~*β*<sup>0</sup> � *<sup>θ</sup>m*ð Þ *<sup>m</sup>* <sup>¼</sup> 1, … , *<sup>J</sup>* , where *<sup>ϕ</sup><sup>J</sup> = 0*; and *<sup>φ</sup>m*~*β<sup>j</sup>* � �*θmβ<sup>j</sup>* ð Þ *m* ¼ 1, … , *J* e *j* ¼ 1, … , *k* , where *φ<sup>J</sup>* ¼ 0 e *φ* ¼ 1*:* Thus, from Eq. (2), the stereotype model can be written as follows:

$$\text{Prob}\left(\mathbf{y} = \mathbf{m} | \mathbf{x}\right) = \frac{\exp\left(\theta\_m - \rho\_m \beta' \mathbf{x}\right)}{\sum\_{j=1}^{J} \exp\left(\rho\_1 \beta' \mathbf{x}\right)},\tag{3}$$

with *m* = 1, … , *J* where *θ<sup>J</sup>* = 0, *φ<sup>J</sup>* ¼ 0 where *φ* ¼ 1.

#### *2.3.2 Interpretation of estimated coefficients*

Applying logarithm in function (3) to any two categories, we get:

$$\log \left[ \frac{p(Y = q/\varkappa)}{p(Y = r/\varkappa)} \right] = \left(\theta\_q - \theta\_r\right) - \left(\rho\_q - \rho\_r\right)\beta'\infty. \tag{4}$$

Applying the exponential function to the exponential function to Eq. (4), it follows

$$\Omega\_{q/r} = \frac{p(Y = q/\varkappa)}{p(Y = r/\varkappa)} = \exp\left\{ (\theta\_q - \theta\_r) - \left(\wp\_q - \wp\_r\right) \beta' \varkappa \right\}.\tag{5}$$

Eq. (5) allows us to evaluate the odds ratio before and after we add a unit to the variable *xj*, that is,

*Logistic Regression: Risk Question for Disabled People DOI: http://dx.doi.org/10.5772/intechopen.106212*

$$\frac{\Omega\_{q/r}(\mathbf{x}, \mathbf{x}\_k + \mathbf{1})}{\Omega\_{q/r}(\mathbf{x}, \mathbf{x}\_k)} = \exp\left\{ \left(\boldsymbol{\rho}\_r - \boldsymbol{\rho}\_q\right) \boldsymbol{\beta}\_x \right\}.\tag{6}$$

The value obtained in expression (6) can be interpreted as adding a unit to the variable *xk*, the odds ratio of category *r* varies exp *φ<sup>r</sup>* � *φ<sup>q</sup>* � �*β<sup>k</sup>* n o, keeping all other variables constant.

## *2.3.3 Estimation of estimated coefficients*

The parameters of the stereotype model are estimated by the maximum likelihood method, in which the estimators are obtained by the system of equations given in (7) as follows:

$$\mathbf{p}\_{i} = \begin{cases} \text{Prob}(\mathbf{y}\_{i} = \mathbf{1} | \mathbf{x}\_{i}, \boldsymbol{\rho}, \boldsymbol{\theta}) & \text{if } \boldsymbol{y}\_{i} = \mathbf{1} \\ & \vdots \\ \text{Prob}(\mathbf{y}\_{i} = \mathbf{m} | \mathbf{x}\_{i}, \boldsymbol{\rho}, \boldsymbol{\theta}) & \text{if } \boldsymbol{y}\_{i} = \boldsymbol{m} \\ & \vdots \\ \text{Prob}(\mathbf{y}\_{i} = \mathbf{J} | \mathbf{x}\_{i}, \boldsymbol{\rho}, \boldsymbol{\theta}) & \text{if } \boldsymbol{y}\_{i} = \mathbf{J} \end{cases} \tag{7}$$

where *pi* is the probability of observing any value of *<sup>y</sup>*, and the Prob yi <sup>¼</sup> 1 xi <sup>j</sup> , *<sup>φ</sup>*, *<sup>θ</sup>* � � was defined in expression (3). Assuming that the sample is independent and identically distributed, the likelihood function is given by the following expression (8):

$$\mathcal{L}(\beta,\varphi,\theta|\mathbf{y},\mathbf{x}) = \prod\_{i=1}^{N} p\_i = \prod\_{m=1}^{J} \prod\_{\mathbf{y}=m} \text{Prob}\left(\mathbf{y} = \mathbf{m}|\mathbf{x},\boldsymbol{\varphi},\boldsymbol{\theta}\right) \tag{8}$$

on what Q *y*¼*j* indicates the multiplications over all cases where *y = m (m = 1, … ,J)*.

Applying logarithm to the likelihood function obtained in (8), we obtain the logarithm of the likelihood function given in (9) as follows:

$$\log\left(\mathcal{L}(\boldsymbol{\beta},\boldsymbol{\varphi},\boldsymbol{\theta}|\boldsymbol{y},\boldsymbol{\chi})\right) = \sum\_{m=1}^{J} \sum\_{\mathbf{y}=\mathbf{m}} \log\left[\text{Prob}\left(\mathbf{y}=\mathbf{m}|\boldsymbol{\chi},\boldsymbol{\varphi},\boldsymbol{\theta}\right)\right].\tag{9}$$

The parameters *ϕ*'s and *θ*'s of Eq. (9) are estimated by the Newton–Raphson method.

The odds ratio formed will have an upward trend, as the weights can be produced by sorting. Thus, the effect of covariates on the first odds ratio is smaller than the effect on the second, and so on.

These weights can be done a priori, being estimated by a pilot study or by a set of properly chosen values.

In the case of this work, the number of disabilities that a person may have can vary from 0 to 4, and there may be five response options.

In order to assess the goodness of fit for ordinal models, it can be done using tests such as Pearson's or deviation. These tests involve creating a contingency table in which the rows consist of all possible configurations of the model's covariates and the columns are the ordinal response categories [8]. The expected counts (*Elj*) from this

table are expressed by *Elj* <sup>¼</sup> <sup>P</sup>*NL <sup>l</sup>*¼1*p*^*ij*, on where *NL* is the total number of individuals classified in the row *l* and *p*^*ij* represents the probability of an individual in line l having the answer *j* calculated from the adopted model.

Pearson's test to assess the adequacy of fit compares these expected counts with those observed by the formula:

$$\chi^2 = \sum\_{l=1}^{L} \sum\_{j=1}^{k} \frac{\left(O\_{lj} - E\_{lj}\right)^2}{E\_{lj}} \tag{10}$$

The deviance stat also compares observed (*Olj*) and expected counts, but using the formula:

$$D^2 = 2\sum\_{l=1}^{L} \sum\_{j=1}^{k} \mathcal{O}\_{lj} \log \frac{O\_{lj}}{E\_{lj}} \tag{11}$$

The tests to assess the goodness of fit of the model are given by approximations of statistics (10) and (11) for chi-square distribution with (*L* – 1)(*k* – 1)*p* degrees of freedom, where *L* and *k* are as defined earlier and p is the number of model covariates. Significant differences lead to the conclusion that the model does not fit the data studied.

As an alternative, we will use the Wald test which is given by:

$$\mathcal{W} = \left(\hat{p} - \hat{p}\_0\right)' \hat{V}\_p^{-1} \left(\hat{p} - \hat{p}\_0\right) \tag{12}$$

on where *V*^ *<sup>p</sup>* is the consistent estimator of the variance-covariance matrix of the estimator *p*^ of the proportion vector *p*^. An estimator *V*^ *<sup>p</sup>* can be obtained by linearization method.

## *2.3.4 Significance test for the model*

The Wald test for the parameters considered individually can be obtained by comparing the estimate of maximum likelihood of a given coefficient *β*^*<sup>j</sup>* � � with the estimate of its standard error (based on the asymptotic distribution of the maximum likelihood estimators). Thus, the null hypothesis and the alternative hypothesis of the test are respectively:

$$H\_0: \hat{\boldsymbol{\beta}}\_{\mathbf{j}} = \boldsymbol{\beta}\_{\mathbf{j}}^\* \quad \text{vs} \quad H\_1: \hat{\boldsymbol{\beta}}\_{\mathbf{j}} \neq \boldsymbol{\beta}\_{\mathbf{j}}^\* \qquad \quad (\mathbf{j} = \mathbf{2}, \dots, \mathbf{k}), \tag{13}$$

the respective statistic under the null hypothesis:

$$T = \frac{\hat{\boldsymbol{\beta}}\_{j} - \boldsymbol{\beta}\_{j}^{\*}}{\sqrt{\text{var}\left(\hat{\boldsymbol{\beta}}\_{j}\right)}} \sim \mathbf{N}(\mathbf{0}, \mathbf{1}) \tag{14}$$

By rejecting H0, for a significance α, we conclude that the estimated parameter is statistically different from *β* <sup>∗</sup> *<sup>j</sup>* . Generally, use *β* <sup>∗</sup> *<sup>j</sup>* ¼ 0 which, under these conditions,

we conclude that the parameter is relevant to explain the behavior of the dependent variable.

### *2.3.5 Selection of variables*

Selecting variables means choosing a subset that retains the most important predictor variables in such a way that we seek to avoid problems such as multicollinearity and that this subset fits as well as the complete model and contains the most important predictor variables.

Among different procedures that can be used to select variables, we highlight forward stepwise and backward stepwise. Forward stepwise starts with the constant *β<sup>0</sup>* and sequentially adds the predictor *Xi* most correlated with Y to the model so that it improves the fit according to the evaluation of the *F* statistic and the introduction of variables when it fails to produce an F statistic greater than the 90th or 95th percentile of the distribution, *F1, N – <sup>k</sup> – <sup>2</sup>*, where *N* is the sample size and *k* is the number of variables.

On the other hand, the backward stepwise selection strategy starts with the model with all independent variables, and sequentially, excludes variables using the F statistic to choose the predictors to be eliminated. The predictor that has the smallest F statistic is eliminated, and the process stops when each predictor eliminated from the model has an F value greater than the 90th or 95th percentile of the distribution, *F1, N – <sup>k</sup> – <sup>2</sup>*. For this work, forward backward and the Wald statistic were chosen.

In ordinal logistic regression, the TRV (likelihood ratio test) ensures the significance of the fit. Thus, at each stage of the process, the most important variable, in statistical terms, is one that produces the greatest change in the logarithm of the likelihood in relation to the model without the variable [9].

After estimating the parameters, the next step is to verify if the covariates used for modeling are statistically significant for the modeled event, for example, condition of an individual becoming a disability person.

To test the significance of the coefficient of a covariate, it is sufficient to compare the observed values of the response variable with the predicted values obtained by the models with and without the variable of interest [10].

The comparison between observed and predicted values is made using the likelihood ratio test, which is widely applicable by the maximum likelihood estimation. For test *H*<sup>0</sup> : *θ* ∈ Θ<sup>0</sup> versus *Ha* : *θ* ∈ Θ*<sup>c</sup>* 0, we calculated the statistics [11]:

$$\lambda(\mathbf{x}) = \frac{\sup\_{\theta\_0} L(\theta/\mathbf{x})}{\sup\_{\theta} L(\theta/\mathbf{x})}. \text{For } n \to \infty,\\ -2\ln \lambda(\mathbf{x}) \to \chi^2\_v. \tag{15}$$

where *ν* is obtained through the difference between the number of parameters existing in the tested model and the number of parameters existing in the saturated model [12].

To verify the quality of the adjusted model, it is sufficient to compare the observed and predicted values for the response variable (in this case, one of the different deficiencies already mentioned).

When choosing a particular model, it means that we must include as many independent variables as possible to improve the forecast; simultaneously, we want to include a smaller number of variables for reasons of cost and simplicity [10].

According to Draper and Smith [13], to select the best model is to reconcile two objectives (incorporating a certain number of variables that can improve the

predictability of the model, at the same time, discarding variables that are not significant as a way of simplifying the model to reduce costs). This selection involves a dose of subjectivity, and the result may be different if the procedure is used for selection changes.

### *2.3.6 Model selection*

Selecting a model means, after the formulation and adjustment of different plausible models, to select the model that ¨best¨ fits the data of a certain experiment according to a certain criterion adopted [14].

In statistics, there is a vast literature relevant to the selection of models [15–17]. An alternative for model selection is the use of methods based on the likelihood function that provides several statistical measures that help in the comparison between different models. The most common of these measures are as follows: Akaike Information Criterion (AIC) proposed by Paulino et al. [18] and Sakamoto et al. [19] with penalty given discounting the value of twice the difference between the number of parameters between the two models; Bayesian Information Criterion (BIC) discussed by Paulino et al. [18] and having as a penalty the value of double the number of parameters between the two models multiplied by the Naperian logarithm of the sample size; and, finally, Deviation Information Criterion (DIC) also discussed by Paulino et al. [18] and the penalty is given by the sum of the difference value between the number of parameters between the two models.

In this text, for each of the AIC, BIC, and DIC criteria, the model with the lowest value for each one of them is chosen.

### **2.4 Epidemiology**

According to the International Epidemiology Association (IEA), epidemiology is defined as the study of the different factors involved in the spread and propagation of diseases, frequency, their mode of distribution, their evolution, and the placement of the necessary means for their prevention in human communities.

According to Suser [20], epidemiology is essentially a population science, which is based on the social sciences for the understanding of social structure and dynamics, on mathematics for statistical, probability, inference, and estimation notions, and, on the biological sciences, the knowledge of the environment organic substrate where the observed manifestations will find individual expression.

A single and precise definition of epidemiology as a scientific field ends up not being possible due to the increasing complexity and scope of its current practice:

Science that studies the health-disease process in society, analyzing population distribution and determining factors of risk, diseases, injuries, and events associated with health, proposing specific measures for the prevention, control, or eradication of diseases, damages, or health and protection problems, promotion or recovery of individual and collective health, producing information and knowledge to support decision-making in the planning, administration, and evaluation of health systems, programs, services, and actions [21].

Epidemiology is a basic discipline of public health aimed at understanding the health-disease process within populations, an aspect that differentiates it from clinical practice, which aims to study this same process, but in individual terms and that studies the different factors that intervene in the spread and propagation of diseases,

their frequency, their mode of distribution, their evolution, and the placement of the necessary means for their prevention.

In scientific terms, epidemiology is based on causal reasoning; as a public health discipline, focusing on the development of a sequence of actions aimed at protecting and promoting the health of the community.

Epidemiology is also an important tool for policy development in the health sector. Its application in this case must be taken into account the available knowledge, adapting it to local realities.

Among the possibilities of applications of epidemiology, we highlight: the analysis of the health situation; identify profiles and risk factors; carry out epidemiological assessment of services; study and understand the causality of health problems; describe the clinical spectrum of diseases and their natural history; assess the performance of health services in responding to the problems and needs of populations; test the efficacy, effectiveness, and impact of intervention strategies, as well as the quality, access, and availability of health services to control, prevent, and treat health problems in the community; identify risk factors for a disease and groups of individuals who are at greater risk of being affected by a particular disease; define modes of transmission; identify and explain patterns of geographic distribution of diseases; establish methods and strategies to control health problems; establish preventive measures; assist in the planning and development of health services; and, finally, establish criteria for health surveillance.

In the discussion about disabilities, epidemiological views on social points of view, accessibility, assistive technology, among others, were used in these researches, and, physicians, from the perspective of prevention, treatment, and control.
