Application of Quantile Regression: Modeling Body Mass Index in Ethiopia

2.1. Study Variable

The response variable in this study is under-five children’s BMI, which is a continuous variable. The explanatory variables used in this study are:- child’s age, sex of child, weight of child at birth, mother’s current age, mother’s BMI, educational attainment of mother, mother’s work status, religion, region, wealth index, place of residence (rural or urban), and current marital status. The socio-economic and demographic factors used in this study were supported by several researchers as most likely to be referred to as intermediate variables for the determinants of children’s nutritional status [18].

The main objective of this study is applying quantile regression to identify factors associated with different quantiles of under-five children’s BMI as a function of age and other relevant factors. It will assist policy makers to know and understand the areas they need to focus on in order to enhance the planning and evaluation of health policies to prevent children’s deaths and to enhance children’s health, diet and growth.

2.2. Statistical Methods

Quantiles are a generalization of percentiles for continuous random variables. Quantiles are cut points dividing the range of a probability distribution into contiguous intervals with equal probabilities or dividing the observations in a sample in the same way. In SAS the QUANTREG procedure models the effect of covariates on the conditional quantiles of a response variable by means of quantile regression. Ordinary least squares regression models the relationship between one or more covariates X and the conditional mean of the response variable given X = x or E[Y|X=x]. Quantile regression, which was introduced by Koenker and Bassett in 1978, extends the regression model to conditional quantiles of the response variable, such as 0.25 quantile or 25^th percentile, 0.5 quantile or 50^th percentile, 0.75 quantile or 75^th percentile and so on other than just the conditional mean of the response variable. Quantile regression is desired if conditional quantiles are of interest. It is also particularly useful when the rate of change in the conditional quantile, expressed by the regression coefficients, depends on the quantile [9].

Quantile regression, which includes median regression as a special case, provides a complete picture of the covariate effect when a set of percentiles is modeled. So it can capture important features of the data that might be missed by models that average over the conditional distribution [9-11]. Quantile regression methods are applied to continuous-response data with no zero values and possibly utilize them in the context of count data [12, 13].

Suppose Y is the response variable, and X is the p-dimensional predictor: Let F_Y(Y/X=x) = P(Y Y/X=x) denote the conditional cumulative distribution function of Y given X=x. Then the τ^th conditional quantile of Y is defined as:

(1)

where the quantile level τ ranges between 0 and 1. In particular, the median is Q_1/2.

The quantile regression model is described by the conditional τ^th quantiles of the response Y for given values of predictors x₁,x₂,...,x_k. It is a natural extension of the traditional mean model in Eq (1):

(2)

where is the unknown parameter vector.

Eq (2) specifies the changes in the conditional quantiles. Since any τ^th quantile can be used, it is possible to model any predetermined position of the distribution and to achieve a more complete understanding of how the response distribution is affected by the predictors. Thus, it allows us to choose positions on the response distribution for their specific inquiries.

For a random sample {y₁,...,y_n} of Y, it is well known that the sample median minimizes the sum of absolute deviations Eq. (3).

(3)

Likewise, the general τ^th sample quantile ξ(τ), which is the analogue of Q(τ), is formulated as the minimizer:

where ρ_τ(Z)=Z(τ-I(Z<0)), 0<τ<1, and where I(·) denotes the indicator function. The loss function ρ_τ assigns a weight of τ to positive residuals y_i-ξ and a weight of 1-τ to negative residuals. Using this loss function, the linear conditional quantile function extends the τ^th sample quantile ξ(τ) to the regression setting in the same way that the linear conditional mean function extends the sample mean.

OLS regression estimates the linear conditional mean function E(Y|X = x) = x'β, by solving: Eq (4)

(4)

The estimated parameter minimizes the sum of squared residuals in the same way that the sample mean minimizes the sum of squares:

Quantile regression also estimates the linear conditional quantile function, (τ|X = x) = x'β(τ), by solving: Eq. (5)

(5)

For any quantile τ (0,1) the quantity (τ) is called the τ^th regression quantile. The case τ = 0.5, which minimizes the sum of absolute residuals, corresponds to median regression, which is also known as L₁ regression. The set of regression quantiles {β(τ):τ (0,1)} is referred to as the quantile process.

Quantile regression minimizes: Eq. (6)

(6)

Where ∑_iτ|e_i| is a sum that gives the asymmetric penalties τ|e_i| for under prediction and (1 - τ)|e_i| for over prediction.

The SAS QUANTREG procedure computed the quantile function Q(τ|X = x) and conducts statistical inference on the estimated parameters (τ).

The τ^th quantile regression estimator (τ) minimizes over β_τ the objective function is: Eq. (7)

(7)

where 0<τ<1,i:y_i≥x'_iβ for under prediction, i:y_i < x'_iβ for under prediction. We have β_τ instead of β, because different choices of τ estimates different values of β.

Since the τ^th conditional quantile of Y given x is given by Q_τ(y_i|x_i) = x'_iβ_τ, its estimate is given by . As one increases τ continuously from 0 to 1, one traces the entire conditional distribution of Y, conditional on x. Note that various quantile regression estimates are correlated. The parameter estimates in quantile regression models have the same interpretation as those of any other linear model as rates of changes. Therefore, in a similar way to the OLS model, the β_i(τ) coefficient of the quantile regression model can be interpreted as the rate of change of the τ^th quantile of the dependent variable distribution per unit change in the value of the i^th regressor [14].

2.3. Advantages of Quantile Regression

Quantile regression is a useful model if the interest is on conditional quantile functions. The main advantage of quantile regression in comparison to the ordinary least squares regression, is that the estimates of quantile regression are more robust against outliers. Nevertheless, the main use of quantile regression is based on different measures. These measures are central tendency and statistical dispersion and these can be useful to obtain a more all-inclusive analysis of the relationship between variables [10]. Because quantile regression does not assume a particular distribution for the response, nor does it assume a constant variance for the response, unlike ordinary least squares regression, quantile regression offers considerable model robustness. The BMI considered for this study is the continuous outcome. It also allows us to study the impact of predictors on different quantiles of the response distribution, and thus provides a complete picture of the relationship between the dependent and explanatory variables. Quantile regression is also flexible because it does not involve a link function and distributional assumption (such as the normal or poisson distribution) that relates the variance and the mean of the response variable.

Details about quantile regression can be found in various literatures [10, 12, 13, 15-17, 19].

3.1. Quantile Regression Analysis

Table 3 shows the estimates and significant effect of the parameters across quantile levels. It was found that current age of mother, mother’s BMI, region (Addis Ababa, SNNPR and Somali), and child size at birth (average and large) were found to have significant effect on under-five children’s BMI at 0.05 quantile. At 0.5 quantile current age of children, current age of mother, mother’s BMI, region (Addis Ababa, Afar, Dire Dawa, Gambela, SNNPR, Somali), place of residence, wealth index (poor and middle) and weight of child at birth (average and large) were found to have significant effect on under-five children’s BMI. Similarly, at 0.85 quantile current age of child, mother’s current age, mother’s BMI, region (Addis Ababa, Dire Dawa, Oromia, SNNPR and Somali) and weight of child at birth (average and large) were found significantly affecting under-five children’s BMI. The findings using quantile regression across quantile levels (0.25 quantile, 0.75 quantile and 0.95 quantile) were also indicated Table 3.

At 0.25 quantile, intercept = 13.27, which is the predicted value of the 0.25 quantile under-five children BMI when all the explanatory variables are zero. _{1(0.25 quantile)} = -0.19 indicates the rate of change of the 0.25 quantile (Q₁) of the dependent variable distribution per unit change in the value of the first regressor (current age of child), keeping all the other explanatory variables constant. In other words, the Q₁ regression coefficient indicates that 25% of the under-five children’s BMI will decrease by -0.19 for every one unit change in current age of a child, setting all the other explanatory variables constant. Q₁ is a value that has 25% of the observations smaller or equal to it.

At 0.5 quantile, intercept = 14.61, which is the predicted value of the 0.5 quantile under-five children’s BMI when all the explanatory variables are zero. _{3(0.25 quantile)} = 0.07 indicates the rate of change of the 0.5 quantile (Q₂) of the dependent variable distribution per unit change in the value of the third regressor (mother’s BMI), keeping all the other explanatory variables constant. In other words, the Q₂ regression coefficient indicates that 50% of the under-five children’s BMI will increase by 0.07 for every one-unit change in mother’s BMI, setting all the other explanatory variables constant. Q₂ is a value that has 50% of the observations smaller or equal to it.

At 0.75 quantile, intercept = 15.93, which is the predicted value of the 0.75 quantile of under-five children’s BMI when all the explanatory variables are zero. _{2(0.75 quantile)} = -0.01 indicates the rate of change of the 0.75 quantile (Q₃) of the dependent variable distribution per unit change in the value of the second regressor (current age of mother), keeping all the other explanatory variables constant. In other words, the Q₃ regression coefficient indicates that 75% of the under-five children BMI will decrease by -0.01 for every one unit change in current age of mother, setting all the other explanatory variables constant. Q₃ is a value that has 75% of the observations smaller or equal to it; in other words, 25% of the observations are greater than it.

The rate of change of the coefficients across quantile levels for other significant predictors can be interpreted in the same way as above.

3.2. Graphical Assessment of the Explanatory Variables

Figs. (1 to 5) present a concise summary of the quantile regression results of the study variables. Each plot depicts one coefficient in the quantile regression model, the shaded area depicting a 95% pointwise confidence band. In the first panel of the Figure, the intercept of the model may be interpreted as the estimated conditional quantile function of the under-five children BMI across quantile levels. It has a positive effect in the upper quantiles rather than the lower quantiles; the graph indicates a positively upward sloped line across the quantiles. The second plot shows that the effect of current age of child on under-five children’s BMI has a negative effect, especially in the upper rather than the lower quantiles. The third plot shows the effect of mother’s current age in the model. The plot shows a negative effect across quantile levels. The fourth plot shows that the effect of mother’s BMI in the model has a positive effect in the upper quantiles; the graph indicates a positively upward sloped line across the quantiles.

The effect of female child compared to male in the model has a negative effect across the quantiles. Type of place of residence has a positive effect in the middle quantiles. Quantile plot related to region, wealth index, working status, wealth index, religion and weight of child at birth were also presented in Fig. (1-5). The values on the Y-axis in each graph indicate the estimated value of the variables across quantiles levels.