XLSTAT: Quantile Regression

 

Background

Quantile Regression is part of the broader GLM (Generalized Linear Models) family, like ANCOVA. Its key feature is estimating conditional quantiles rather than the conditional mean, allowing for a more nuanced analysis of the response variable.

QR coefficients, like in OLS, represent rates of change—interpreted as the change in a specific quantile of the dependent variable per unit change in a predictor. QR also accommodates both qualitative and quantitative variables, similar to ANCOVA.

 

Dataset for Example

The data have been obtained in Lewis T. and Taylor L.R. (1967). Introduction to Experimental Ecology, New York: Academic Press, Inc.. They concern 237 children, described by their Gender, Age in months, Height in inches (1 inch = 2.54 cm), and Weight in pounds (1 pound = 0.45 kg).

Using the Quantile Regression, we want to find out how the weight of the children varies with their gender (a qualitative variable that takes value f or m), their height and their age, and to verify if a linear model makes sense.

 

Setting Up

• Open XLSTAT.

• Click on the Modeling data menu and select the Quantile Regression feature.

• Once you've clicked on the button, the Quantile Regression dialog box appears.

In the General tab:

• Select Weight as the Dependent variable.

• Select Height and Age as the Quantitative explanatory variables.

• Select Gender as the qualitative variable.

• Leave the option Variable labels activated, as we selected the column title for the variables.

• The other options have been left at their default value.

• Click on the OK button to launch the computations.

• The results are displayed on a new Excel sheet.

 

This study focuses on quantiles where Quantile Regression coefficients notably differ from those in ANCOVA.

Since no specific quantiles of interest are predefined, a two-step analysis is used. First, the Quantile Process is applied as an exploratory tool to identify relevant quantiles for deeper analysis.

Before proceeding, we briefly recall the main ANCOVA results for this dataset:

 

Interpreting the Results of the Quantile Process Computation

In this preliminary step, only general results are provided.

Around the median, Quantile Regression results are similar in magnitude to those from ANCOVA (which targets the mean):

Focusing on Age and Height, their impact appears stronger for heavier children (α > 0.9), while Gender has a greater influence on the weight of the slimmest ones (α < 0.1), as suggested by the Quantile Regression results.

The quantiles of interest clearly lie in the [0.9, 1] and [0, 0.1] intervals.

These patterns are easily confirmed by the illustrative charts shown at the end of the analysis.

• Then, in a second step, the quantile Selection: 0.95 and 0.05 is done via the Excel sheet.

• Click on the OK button to launch the computations.

• The results are displayed on a new Excel sheet.

 

Interpreting the Results of the Quantile Selection Computation

This first table displays the goodness of fit coefficients of the model for a specific quantile. The R² (coefficient of determination) indicates the % of variability of the dependent variable which is explained by the explanatory variables. The closer to 1 the R² is, the better the fit.

In this case, 91.6% of the variability in Weight is explained by Height, Age, and Gender. The remaining variability likely stems from unmeasured factors, such as genetic or nutritional influences, or possibly from untransformed existing variables.

Reviewing the model significance table is essential. It helps assess whether the explanatory variables add meaningful information (i.e., testing the null hypothesis H0). This determines if the chosen quantile adequately represents the population or if the variables contribute valuable insight. The table also supports prediction and model comparison across populations.

Three tests are used: Maximum Likelihood, Lagrange Multipliers, and Wald. Since the p-value associated with the Chi² statistic is below 0.0001, there's less than a 0.01% risk in rejecting the null hypothesis. We can confidently conclude that the three variables provide significant information.

The next table presents a sample of predictions and standardized residuals, allowing for a closer examination of individual model fits.

The chart below shows the predicted values versus the observed values:

Conclusion

In conclusion, Height, Age, and Gender explain over 90% of the variability in Weight. The Quantile Regression model captures a significant portion of this information effectively.

 

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