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Correlation and Regression00:53

Correlation and Regression

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In statistics, correlation describes the degree of association between two variables. In the subfield of linear regression, correlation is mathematically expressed by the correlation coefficient, which describes the strength and direction of the relationship between two variables. The coefficient is symbolically represented by 'r' and ranges from -1 to +1. A positive value indicates a positive correlation where the two variables move in the same direction. A negative value suggests a...
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Multiple Regression01:25

Multiple Regression

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Multiple regression assesses a linear relationship between one response or dependent variable and two or more independent variables. It has many practical applications.
Farmers can use multiple regression to determine the crop yield based on more than one factor, such as water availability, fertilizer, soil properties, etc. Here, the crop yield is the response or dependent variable as it depends on the other independent variables. The analysis requires the construction of a scatter plot...
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Regression Analysis01:11

Regression Analysis

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Regression analysis is a statistical tool that describes a mathematical relationship between a dependent variable and one or more independent variables.
In regression analysis, a regression equation is determined based on the line of best fit– a line that best fits the data points plotted in a graph. This line is also called the regression line. The algebraic equation for the regression line is called the regression equation. It is represented as:
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Regression Toward the Mean01:52

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Regression toward the mean (“RTM”) is a phenomenon in which extremely high or low values—for example, and individual’s blood pressure at a particular moment—appear closer to a group’s average upon remeasuring. Although this statistical peculiarity is the result of random error and chance, it has been problematic across various medical, scientific, financial and psychological applications. In particular, RTM, if not taken into account, can interfere when...
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Calibration Curves: Linear Least Squares01:20

Calibration Curves: Linear Least Squares

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A calibration curve is a plot of the instrument's response against a series of known concentrations of a substance. This curve is used to set the instrument response levels, using the substance and its concentrations as standards. Alternatively, or additionally, an equation is fitted to the calibration curve plot and subsequently used to calculate the unknown concentrations of other samples reliably.
For data that follow a straight line, the standard method for fitting is the linear...
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Calculating and Interpreting the Linear Correlation Coefficient01:11

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The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable, x, and the dependent variable, y. Hence, it is also known as the Pearson product-moment correlation coefficient. It can be calculated using the following equation:
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Related Experiment Video

Updated: Jan 8, 2026

Lexical Decision Task for Studying Written Word Recognition in Adults with and without Dementia or Mild Cognitive Impairment
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Lexical Decision Task for Studying Written Word Recognition in Adults with and without Dementia or Mild Cognitive Impairment

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CONTRASTIVE LINEAR REGRESSION.

Boyang Zhang1, Sarah Nyquist2, Andrew Jones3

  • 1Department of Genetics, Stanford University.

The Annals of Applied Statistics
|December 12, 2025
PubMed
Summary
This summary is machine-generated.

We introduce contrastive regression, a new method for analyzing case-control data with response variables. This approach identifies key biological predictors linked to outcomes like autism severity and tumor stages.

Keywords:
Contrastive modelscase-control studieslinear regression

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Area of Science:

  • Biostatistics
  • Genomics
  • Computational Biology

Background:

  • Case-control studies are common in biomedical research.
  • Existing dimension reduction methods identify variations between cases and controls.
  • A gap exists in analyzing case-control data with response variables.

Purpose of the Study:

  • To develop contrastive regression for case-control data with response variables.
  • To capture shared variation between cases and controls.
  • To explain case-specific responses using remaining predictor variance.

Main Methods:

  • Developed a contrastive regression model.
  • Applied the model to single-cell RNA sequencing data (chronic rhinosinusitis).
  • Applied the model to single-nucleus RNA sequencing data (autism severity).

Main Results:

  • The contrastive linear regression model effectively ranks features.
  • Identified biologically informative predictors associated with response variables.
  • These predictors were not identifiable with other methods.

Conclusions:

  • Contrastive regression is a powerful tool for analyzing complex case-control data.
  • The method enhances understanding of disease mechanisms and patient stratification.
  • It offers novel insights in datasets like autism severity and cellular differentiation.