## NCA ceiling techniques

What is a ceiling?

A ceiling is the border between an empty space without observations and a full space with observations in the multi-dimensional space. This can be mathematically expressed as Y ≤ f(X), where Y is the outcome, X are the conditions, and f(X) is the ceiling function.

What is a floor?

A floor is the border between an empty space without observations and a full space with observations in the multi-dimensional space and that is mathematically expressed as Y ≥ f(X), where Y is the outcome, X are the conditions, and f(X) is the floor function.

What is a ceiling line?

A ceiling line is a ceiling in the two-dimensional space, where the ceiling function f(X) is a line.

What is a ceiling surface?

A ceiling surface is a ceiling in the three-dimensional space, where the ceiling function f(X) is a surface.

What is a ceiling technique?

A ceiling technique is a mathematical or statistical approach to approximate the ceiling (see table). In the NCA software package for R, the CE-FDH ceiling technique is the default ceiling technique for dichotomous and discrete (with few levels) necessary conditions. The CR-FDH ceiling technique is the default ceiling technique for discrete (with many levels) and continuous necessary conditions.

 Ceiling Technique Name in NCA software Name LH lh Low-High COLS cols Corrected Ordinary Least Squares QR qr Quantile Regression SFA sfa Stochastic Frontier Analysis CE-VRS ce_vrs Ceiling Envelopment with Varying Return to Scale CR-VRS cr_vrs Ceiling Regression with Varying Return to Scale CE-FDH ce_fdh Ceiling Envelopment with Free Disposal Hull CR-FDH cr_fdh Ceiling Regression with Free Disposal Hull

What is the LH ceiling technique?

The Low-High (LH) ceiling technique is a ceiling approximation that connects the observation with the maximum Y for the lowest X with the observation of the maximum Y for the highest X.

What is the COLS ceiling technique?

The Corrected Ordinary Least Squares (COLS) ceiling technique is a ceiling approximation that moves the Ordinary Least Squares (OLS) regression line through the middle of the data upwards toward the most upper observations such that all observations are below this line.

What is the QR ceiling technique?

The Quantile Regression (QR) ceiling technique is a ceiling approximation that splits the observations in an upper part and a lower part. For high levels of quantiles (e.g., 90) the majority of the points is below the ceiling.

What is the SFA ceiling technique?

The Stochastic Frontier Analysis (SFA) ceiling technique is a ceiling approximation that makes probability assumptions about observations below the ceiling.

What is the CE-VRS ceiling technique?

The Ceiling Envelopment - Varying Return to Scale (CE-VRS ) ceiling technique is a ceiling approximation obtained from Data Envelopment Analysis (DEA) that assumes that the ceiling is convex, resulting in a piecewise linear convex ceiling function.

What is the CR-VRS ceiling technique?

The Ceiling Regression - Varying Return to Scale (CE-VRS ) ceiling technique is a ceiling approximation that smooths the piecewise linear function obtained by the CE-VRS technique by using OLS regression through the corners of the piecewise linear function.

What is the CE-FDH ceiling technique?

The Ceiling Envelopment – Free Disposal Hull (CE-FDH ) ceiling technique is a ceiling approximation obtained from the Free Disposal Hull (FDH) data envelopment technique that assumes that the ceiling is non-decreasing, resulting in a non-decreasing step function. In the NCA software package for R, the CE-FDH ceiling technique is the default ceiling technique for dichotomous and discrete (with few values) necessary conditions.

What is the CR-FDH ceiling technique?

The Ceiling Regression - Free Disposal Hull (CR-FDH) ceiling technique is a ceiling approximation that smooths the step function obtained by the CE-FDH technique by using OLS regression through the upper-left corners of the step function. In the NCA software package for R, the CR-FDH ceiling technique is the default ceiling technique for discrete (with many values) and continuous necessary conditions.