# in regression analysis, the response variable is the _____.

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## Correlation and Linear Regression

**Correlation and Linear Regression**

Author: Lisa Sullivan, PhD

Professor of Biostatistics

Boston University School of Public Health

## Introduction

In this section we discuss correlation analysis which is a technique used to quantify the associations between two continuous variables. For example, we might want to quantify the association between body mass index and systolic blood pressure, or between hours of exercise per week and percent body fat. Regression analysis is a related technique to assess the relationship between an outcome variable and one or more risk factors or confounding variables (confounding is discussed later). The outcome variable is also called the **response** or **dependent variable,** and the risk factors and confounders are called the **predictors**, or **explanatory** or **independent variables**. In regression analysis, the dependent variable is denoted "Y" and the independent variables are denoted by "X".

[ **NOTE:** The term "predictor" can be misleading if it is interpreted as the ability to predict even beyond the limits of the data. Also, the term "explanatory variable" might give an impression of a causal effect in a situation in which inferences should be limited to identifying associations. The terms "independent" and "dependent" variable are less subject to these interpretations as they do not strongly imply cause and effect.

## Learning Objectives

After completing this module, the student will be able to:

Define and provide examples of dependent and independent variables in a study of a public health problem

Compute and interpret a correlation coefficient

Compute and interpret coefficients in a linear regression analysis

## Correlation Analysis

In correlation analysis, we estimate a **sample correlation coefficient**, more specifically the **Pearson Product Moment correlation coefficient**. The sample correlation coefficient, denoted r,

ranges between -1 and +1 and quantifies the direction and strength of the linear association between the two variables. The correlation between two variables can be positive (i.e., higher levels of one variable are associated with higher levels of the other) or negative (i.e., higher levels of one variable are associated with lower levels of the other).

The sign of the correlation coefficient indicates the direction of the association. The magnitude of the correlation coefficient indicates the strength of the association.

For example, a correlation of r = 0.9 suggests a strong, positive association between two variables, whereas a correlation of r = -0.2 suggest a weak, negative association. A correlation close to zero suggests no linear association between two continuous variables.

It is important to note that there may be a non-linear association between two continuous variables, but computation of a correlation coefficient does not detect this. Therefore, it is always important to evaluate the data carefully before computing a correlation coefficient. Graphical displays are particularly useful to explore associations between variables.

The figure below shows four hypothetical scenarios in which one continuous variable is plotted along the X-axis and the other along the Y-axis.

Scenario 1 depicts a strong positive association (r=0.9), similar to what we might see for the correlation between infant birth weight and birth length.

Scenario 2 depicts a weaker association (r=0,2) that we might expect to see between age and body mass index (which tends to increase with age).

Scenario 3 might depict the lack of association (r approximately = 0) between the extent of media exposure in adolescence and age at which adolescents initiate sexual activity.

Scenario 4 might depict the strong negative association (r= -0.9) generally observed between the number of hours of aerobic exercise per week and percent body fat.

## Example - Correlation of Gestational Age and Birth Weight

A small study is conducted involving 17 infants to investigate the association between gestational age at birth, measured in weeks, and birth weight, measured in grams.

Infant ID #

Gestational Age (weeks)

Birth Weight (grams)

1 34.7 1895 2 36.0 2030 3 29.3 1440 4 40.1 2835 5 35.7 3090 6 42.4 3827 7 40.3 3260 8 37.3 2690 9 40.9 3285 10 38.3 2920 11 38.5 3430 12 41.4 3657 13 39.7 3685 14 39.7 3345 15 41.1 3260 16 38.0 2680 17 38.7 2005

We wish to estimate the association between gestational age and infant birth weight. In this example, birth weight is the dependent variable and gestational age is the independent variable. Thus y=birth weight and x=gestational age. The data are displayed in a scatter diagram in the figure below.

## Answer the following questions with either (True/False) 1. In the regression analysis, the dependent variable is used to predict the value of the independent variable(s) .... 2. In a multiple regres

Answer to: Answer the following questions with either (True/False) 1. In the regression analysis, the dependent variable is used to predict the...

Regression analysis

## Answer the following questions with either (True/False) 1. In the regression analysis, the...

Answer the following questions with either (True/False) 1. In the regression analysis, the... Question:

Answer the following questions with either (True/False)

1. In the regression analysis, the dependent variable is used to predict the value of the independent variable(s) _____.

2. In a multiple regression analysis, it is possible for the value of the coefficient of determination to be greater than one _____.

3. The coefficient of determination can range only -1.0 to +1.0 _____

4. R-square (R2) represents the correlation coefficient rather than the coefficient of determination _____.

5. A linear regression model that has more than one dependent variables is called multiple regression model _____.

## Regression Analysis

The term literally meaning . It is a mathematical measure of the average relationship between two or more variables in terms of the original units of the data. In regression analysis there are two types of variables, one is the independent variable, used to create a model for the dependent variable, the variable that is being modeled.

## Answer and Explanation:

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**1 - False**

**Reason:**The independent variable is used to predict the dependent variable.

**2 - False**

**Reason:**The coefficient of determination always...

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### Learn more about this topic:

Regression Analysis: Definition & Examples

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Chapter 21 / Lesson 4

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Regression analysis is used in graph analysis to help make informed predictions on a bunch of data. With examples, explore the definition of regression analysis and the importance of finding the best equation and using outliers when gathering data.

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## Explanatory Variable & Response Variable: Simple Definition and Uses

An explanatory variable is another term for an independent variable. The two terms are often used interchangeably. However, there is a subtle difference.

Statistics How To

## Statistics for the rest of us!

## Explanatory Variable & Response Variable: Simple Definition and Uses

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## What is an Explanatory Variable?

Watch the video or read on below:

Explanatory Variables Explained

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Can’t see the video? Click here.

An explanatory variable is a type of independent variable. The two terms are often used interchangeably. But there is a subtle difference between the two. When a variable is independent, it is not affected at all by any other variables. **When a variable isn’t independent for certain, it’s an explanatory variable.**

Let’s say you had two variables to explain weight gain: fast food and soda. Although you might think that eating fast food intake and drinking soda are independent of each other, they aren’t really. That’s because fast food places encourage you to buy a soda with your meal. And if you stop somewhere to buy a soda, there’s often a lot of fast food options like nachos or hot dogs. Although these variables aren’t completely independent of each other, they do have an effect on weight gain. They are called explanatory variables because they may offer some explanation for the weight gain.

The line between independent variables and explanatory variables is usually so **unimportant **that no one ever bothers. That’s unless you’re doing some advanced research involving lots of variables that can interact with each other. It can be very important in clinical research. For most cases, especially in statistics, the two terms are basically the same.

## Explanatory Variables vs. Response Variables

The **response variable** is the focus of a question in a study or experiment. An explanatory variable is one that explains changes in that variable. It can be anything that might affect the response variable.

Let’s say you’re trying to figure out if chemo or anti-estrogen treatment is better procedure for breast cancer patients. The question is: which procedure prolongs life more? And so survival time is the response variable. The type of therapy given is the explanatory variable; it may or may not affect the response variable. In this example, we have only one explanatory variable: type of treatment. In real life you would have several more explanatory variables, including: age, health, weight and other lifestyle factors.

A scatterplot can help you see trends between paired data. If you have both a response variable and an explanatory variable, the explanatory variable is always plotted on the x-axis (the horizontal axis). The response variable is always plotted on the y-axis (the vertical axis).

If you look at the above image, you should be able to tell that wrist size isn’t a very good explanatory variable to predict body fat (the response variable). The red line in the image is the “line of best fit.” Although it runs through the middle of the spread of dots, most of the dots aren’t anywhere near it. This means that the explanatory variable really isn’t explaining anything.

On the other hand, how large a person’s thighs are is a better predictor of body fat. Even this isn’t perfect. Many very fit people have large thighs! See how closer the dots are to the red line of best fit.

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## References

Levine, D. (2014). Even You Can Learn Statistics and Analytics: An Easy to Understand Guide to Statistics and Analytics 3rd Edition. Pearson FT Press

J Wilson at UGA COE. Assignment 2012.

**CITE THIS AS:**

**Stephanie Glen**. "Explanatory Variable & Response Variable: Simple Definition and Uses" From

**StatisticsHowTo.com**: Elementary Statistics for the rest of us! https://www.statisticshowto.com/probability-and-statistics/types-of-variables/explanatory-variable/

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