[PDF] Multiple Linear Regression (2nd Edition) Mark Tranmer Jen Murphy PDF

Calculation of Multiple Regression with Three. Independent Variables Using a Programable Pocket. Calculator. Paul Evenson. Follow this and additional works at

The Simple Linear Regression Model

A Linear Probabilistic Model. Page 9. 9. The only randomness on the right-hand side of the linear model equation is in ε. (x is fixed!) What is μY

Multiple Regression

We'd never try to find a regression by hand and even Assuming that the conditions for multiple regression are met

Simple Linear Regression

The fitted (or predicted) values are obtained by substituting x1. …

Testing Mediation with Regression Analysis

coefficient from a multiple regression. The indirect effect is the in R run the separate regression models described above calculate the indirect effect ...

Predicting Hand Surface Area from a Two-Dimensional Hand Tracing

Nov 3 2017 By applying a linear regression formula between the 2D and 3D areas

Latent Topic Analysis of the Post Property for Sales to Predict a

prediction model of second-hand condominium using multiple linear regression and artificial • Calculate the accuracy of the multiple regression model based ...

Long Hand Regression in R

Linear Regression via Least Squares. So in the following we will be Instead you compute the equivalent of 1 over the matrix

The Omission or Addition of an Independent Variate in Multiple

variates the equations to determine the linear regression coefficients bl numbers in the right-hand column is then calculated. The sum of. Page 5. 1938 ...

The Statistical Software Toolkit: BERDC Seminar Series 1

Multiple linear regression. Yes but hard linear-regression-by-hand/. [17] https://owlcation.com/stem/How-to-Create-a-Simple-Linear-Regression-Equation.

Calculation of Multiple Regression with Three Independent

Evenson Paul

Multiple Regression

We'd never try to find a regression by hand and Second

Interactions in Multiple Linear Regression

On the other hand if men

Week 7: Multiple Regression

i.e. analogous to the simple linear regression case! Disclaimer: the final equation is exactly true for all non-intercept coefficients if you remove the.

The Simple Linear Regression Model

The only randomness on the right-hand side of the linear model equation is in ?. (x is fixed!) What is ?Y

Testing Mediation with Regression Analysis

Psy 523/623 Structural Equation Modeling Spring 2020 Step 4 Conduct a multiple regression analysis with X and M predicting.

Multiple Regression - Introduction

The formula for the SEE is the same as in the bivariate case; however K = 2 in this example

Linear Regression using Stata

Technically linear regression estimates how much Y changes when X changes one unit. In Stata use the command regress

Multiple Regression - Prediction & Confidence Intervals Demystified

With multivariate regression the confidence and prediction interval must account for the simultaneous wiggle of multiple X variables. – To calculate the

Review of Multiple Regression

Jan 3 2022 Before doing other calculations

Multiple Linear Regression by Hand (Step-by-Step) - Statology

18 nov 2020 · This tutorial explains how to perform multiple linear regression by hand including a step-by-step example

[PDF] Calculation of Multiple Regression with Three Independent

This paper describes a multiple re- gression program for an equation with one dependent and three independent variables which was written for a Hewlett-

[PDF] Multiple Regression - UC Berkeley Statistics

We'd never try to find a regression by hand and even calculators aren't really up to the task This is a job for a statistics program on a computer

[PDF] Multiple Linear Regression

All standard statistical software packages compute and show the standard deviations of the regression coefficients Inference concerning a single ?i is based on

[PDF] Multiple Regression 1

The answer depends upon the use to which we wish to put the estimated regression equation The issue is whether the equation is to be used simply for predicting

[PDF] Multiple Regression - Introduction

The formula for the SEE is the same as in the bivariate case; however K = 2 in this example since there are two independent variables MSE = 1 - K - N SSE

[PDF] Multiple Linear Regression (2nd Edition) Mark Tranmer Jen Murphy

The simple linear regression line plot in Figure 5 shows an R2 value of 0 359 at the top right hand side of the plot This means that the variable social

[PDF] Chapter 3 Multiple Linear Regression Model - IIT Kanpur

independent variables called a multiple linear regression model This model generalizes the simple linear regression in two ways

[PDF] Multiple Linear Regression - San Jose State University

Multiple Linear Regression Remark This is the same formula for ˆ ? = (ˆ?0 ˆ ?1) in simple linear regression To demonstrate it consider the toy data

18 nov. 2020 · This tutorial explains how to perform multiple linear regression by hand, including a step-by-step example.

How do you manually calculate multiple regression?
With these variables, the usual multiple regression equation, Y = a + b₁X₁ + b₂X₂, becomes the quadratic polynomial Y = a + b₁X + b₂X². This is still considered a linear relationship because the individual terms are added together.
What is the formula for calculating multiple regression?
Therefore, the formula for calculation is Y = a + bX + E, where Y is the dependent variable, X is the independent variable, a is the intercept, b is the slope, and E is the residual. Regression is a statistical tool to predict the dependent variable with the help of one or more independent variables.
How do you manually calculate regression?
The five steps to follow in a multiple regression analysis are model building, model adequacy, model assumptions – residual tests and diagnostic plots, potential modeling problems and solution, and model validation.

Multiple Linear Regression

(2nd Edition)

Mark Tranmer

Jen Murphy

Mark Elliot

Maria Pampaka

January 2020

License and attribution

This document is open access and made available

under a CC-BY licence; see: https://creativecommons.org/licenses/. You are free to use or remodel the content in any way as long as you credit this document in any such use. When citing please use the following (or equivalent): Tranmer, M., Murphy, J., Elliot, M., and Pampaka, M. (2020) Multiple Linear Regression (2nd Edition); Cathie

Marsh Institute Working Paper 2020-01.

multiple-linear-regression.pdf 3

Contents ..................................................................................................................................... 3

1 The basics - understanding linear regression .................................................................... 6

1.1 Simple Linear Regression - estimating a Bivariate model .............................................. 6

1.2 Hypothesis testing .......................................................................................................... 8

1.3 Residuals ......................................................................................................................... 9

1.4 Multiple Linear Regression - a multivariate model ...................................................... 10

2 Basic analysis using SPSS ................................................................................................... 12

2.1 Variables in the analysis ................................................................................................ 12

2.2 Exploratory data analysis .............................................................................................. 13

2.2.1 Descriptive statistics .......................................................................................... 13

2.2.2 Producing univariate box plots .......................................................................... 15

2.2.3 Bivariate correlations ......................................................................................... 17

2.2.4 Producing scatterplots (in spss) ......................................................................... 18

2.3 Simple Linear Regression .............................................................................................. 22

2.3.1 Regression outputs ............................................................................................ 24

2.3.2 Standardised coefficients ................................................................................... 27

2.3.3 Statistical significance ........................................................................................ 27

2.4 Multiple linear regression analysis ............................................................................... 28

2.4.2 Regression outputs ............................................................................................ 29

2.4.3 Interpreting the results ...................................................................................... 30

3 The assumptions of Linear Regression .............................................................................. 31

3.1 Assumption 1: Variable Types ....................................................................................... 32

3.2 Assumption 2: Linearity ............................................................................................... 32

3.2.1 Checking for non-linear relationships ................................................................ 33

3.2.2 Modelling a non-linear relationship, using linear regression ............................ 33

3.3 Assumption 3: Normal distribution of residuals .......................................................... 34

3.3.1 P-P plots ............................................................................................................. 34

3.3.2 Histograms of residuals...................................................................................... 35

3.4 Assumption 4: Homoscedasticity ................................................................................. 36

3.4.1 Checking for homoscedasticity of the residuals ................................................ 36

3.4.2 What to do if the residuals are not homoscedastic and why does it matter .... 37

3.5 Assumption 5: Multicolinearity .................................................................................... 38

3.5.1 Testing for colinearity - correlations .................................................................. 39

3.5.2 Testing for collinearity - variance inflation factor ............................................. 40

3.5.3 Collinearity - what to do .................................................................................... 40

3.6 Checking the assumptions of linear regression with SPSS ........................................... 40

3.6.1 Requesting plots ................................................................................................ 40

3.6.2 Calculating Variance Inflation Factors ............................................................... 41

3.7 Saving regression values ............................................................................................... 42

3.8 Extreme values .............................................................................................................. 43

3.8.1 Cook's Distance .................................................................................................. 44

4 Moving to a more complex model .................................................................................... 45

4.1 Nominal variables ......................................................................................................... 45

4.2 Interaction effects ......................................................................................................... 47

4.2.1 Scenario A: Same slope, same intercept ........................................................... 47

4.2.2 Scenario B: Different intercept, same slope ..................................................... 48

4.2.3 Scenario C: Different intercept, different slopes .............................................. 48

4.2.4 Scenario D: Different slope, same intercept ..................................................... 49

4.3 Transforming a variable ................................................................................................ 50

4.4 More model selection methods - beyond the default ................................................. 50

4.4.1 Backwards Elimination ....................................................................................... 51

4.4.2 Stepwise ............................................................................................................. 51

4.5 SPSS skills for more advanced modelling ...................................................................... 51

4.5.1 Recoding into a dummy variable ....................................................................... 51

4.5.2 Computing a new variable ................................................................................. 53

5 Further reading ................................................................................................................. 54

6 Appendix A: Correlation, covariance and parameter estimation .................................... 56

7 Glossary ............................................................................................................................. 57

1 THE BASICS - UNDERSTANDING LINEAR REGRESSION

Linear regression is a modelling technique for analysing data to make predictions. In simple linear regression, a bivariate model is built to predict a response variable (ݕ) from an explanatory variable (ݔ)1. In multiple linear regression the model is extended to include This primer presents the necessary theory and gives a practical outline of the technique for bivariate and multivariate linear regression models. We discuss model building, assumptions for regression modelling and interpreting the results to gain meaningful understanding from data. Complex algebra is avoided as far as is possible and we have provided a reading list for more in-depth learning and reference.

1.1 SIMPLE LINEAR REGRESSION - ESTIMATING A BIVARIATE MODEL

A simple linear regression estimates the relationship between a response variable ݕ, and a single explanatory variable ݔ, given a set of data that includes observations for both of these variables for a particular sample. For example, we might be interested to know if exam performance at age 16 - the response variable - can be predicted from exam results at age 11 - the explanatory variable. Table 1 Sample of exam results at ages 11 and 16 (n = 17)

Results at age 16

(Variable name: Exam16)

Results at age 11

(Variable name: Exam11) 45 55
67 77
55 66
39 50
72 55
47 56
49 56
81 90

1 The terms response and explanatory variables are the general terms to describe predictive relationships. You

will also see the terms dependent and independent used. Formally, this latter pair only applies to experimental

designs but are sometimes used more generally. Some statistical software (e.g. SPSS) uses dependent/independent by default. 7 33 40
65 70
57 62
33 45
43 55
55 65
55 66
67 77
56 66
Table 1 contains exam results at ages 11 and 16 for a sample of 17 students. Before we use linear regression to predict a student's result at 16 from the age 11 score, we can plot the data (Figure 1). Figure 1 Scatterplot of exam score at age 16, against score at age 11 We are interested in the relationship between age 11 and age 16 scores - or how they are correlated. In this case, the correlation coefficient is 0.87 - demonstrating that the two variables are indeed highly positively correlated. To fit a straight line to the points on this scatterplot, we use linear regression - the equation of this line, is what we use to make predictions. The equation for the line in regression modelling takes the form: We refer to this as our model. For the mathematical theory underlying the estimation and calculation of correlation coefficients, see Appendix A. 30
40
50
60
70
80
90

30405060708090100

Exam16

Exam11

8 ɴ0 is the intercept also called the constant- this is where the line crosses the ݕ axis of the graph. For this example, this would be the predicted age 16 score, for someone who has scored nil in their age 11 exam.

ɴ1 is the slope of the line - this is how much the value of ݕ increases, for a one-unit increase

in ݔ, or for each additional mark gained in the age 11 exam, how much the student scores in the age 16 exam.

݁௜ is the error term for the ݅௧௛student. The error is the amount by which the predicted

value is different to the actual value. In linear regression we assume that if we calculate the error terms for every person in the sample, and take the mean, the mean value will be zero. The error term is also referred to as the residual (see 1.3 for more detail on residuals).

1.2 HYPOTHESIS TESTING

Our hypothesis is that the age 16 score can be predicted from the age 11 score that is to say that there is an association between the two. We can write this out as null and alternative hypotheses: The null hypothesis is that there is no association - it doesn't matter what the age 11 score would be zero. If there is a relationship, then the slope is not zero - our alternative hypothesis. The relationship between x and y is then estimated by carrying out a simple linear regression analysis. SPSS estimates the equation of the line of best fit by minimising the sum of the squares of the differences between the actual values, and the values predicted by the equation (the residuals) for each observation. This method is often referred to as the ordinary least squares approach; there are other methods for estimating parameters but the technical details of this are beyond this primer.

For this example:

ɴ0 = -3.984

ɴ1 = 0.939

This gives us a regression equation of:

where xi is the value of EXAM11 for the ith student. The ^ symbol over the ݕ௜ is used to show that this is a predicted value. So, if a student has an EXAM11 score of 55 we can predict the EXAM16 score as follows: If we draw this line on the scatter plot, as shown in Figure 2, it is referred to as the line of best fit of y on x, because we are trying to predict y using the information provided by x.

1.3 RESIDUALS

The predicted EXAM16 score of the student with an EXAM11 score of 55 is 47.7;; however, if we refer to the original data, we can see that the first student in the table scored 55 at age

11, but their actual score at age 16 was 45. The difference between the actual or observed

value, and the predicted value is called the error or residual. Remember that ݕො means predicted, and ݕ means actual or observed. The residual for the first student is therefore 45 - 47.7 = -2.7. The residual is the distance of each data point away from the regression line. In Figure 2 the prediction equation is plotted on the scatter plot of exam scores. We can see that very few if any of the actual values fall on the prediction line. 10 Figure 2 Plotting the regression line for age 11 and age 16 exam scores If we calculate the predicted value using the regression equation for every student in the sample, we can then calculate all the residuals. For a model which meets the assumptions for linear regression, the mean of these residuals is zero. More about assumptions and testing data to make sure they are suitable for modelling using linear regression later! Our model has allowed us to predict the values of EXAM16, however it is important to distinguish between correlation and causation. The EXAM11 score value, has not caused the EXAM16 score value, they are simply correlated - there may be other variables through which the relationship is mediated: base intellect, educational environment, parental support, student effort and so on and these could be causing the score, rather than the explanatory variable itself. To illustrate this further, statistically speaking, we would have just as good a model if we used EXAM16 to predict the values of EXAM11. Clearly one would not eǆpect a student's EyAM scores at age 16 to be causing in any sense their eǆam scores at age 11! So a good model does not mean a causal relationship. Our analysis has investigated how an explanatory variable is associated with a response variable of interest, but the equation itself is not grounds for causal inference.

1.4 MULTIPLE LINEAR REGRESSION - A MULTIVARIATE MODEL

Multiple linear regression extends simple linear regression to include more than one the response variable is directly related to a linear combination of the explanatory variables. 30
40
50
60
70
80
90

30405060708090100

Exam16

Exam11

11 The equation for multiple linear regression has the same form as that for simple linear regression but has more terms: As for the simple case, Ⱦ଴ is the constant - which will be the predicted value of y when all explanatory variables are 0. In a model with ݌ explanatory variables, each explanatory variable has its own ɴͺcoefficient. Again, the analysis does not allow us to make causal inferences, but it does allow us to investigate how a set of explanatory variables is associated with a response variable ofquotesdbs_dbs6.pdfusesText_11

[PDF] [PDF] Multiple Linear Regression (2nd Edition) Mark Tranmer Jen Murphy

How do you manually calculate multiple regression?

What is the formula for calculating multiple regression?

How do you manually calculate regression?

Multiple Linear Regression

Mark Tranmer

Jen Murphy

Mark Elliot

Maria Pampaka

January 2020

License and attribution

This document is open access and made available

Marsh Institute Working Paper 2020-01.

CONTENTS

1 The basics - understanding linear regression .................................................................... 6

1.1 Simple Linear Regression - estimating a Bivariate model .............................................. 6

1.2 Hypothesis testing .......................................................................................................... 8

1.3 Residuals ......................................................................................................................... 9

1.4 Multiple Linear Regression - a multivariate model ...................................................... 10

2 Basic analysis using SPSS ................................................................................................... 12

2.1 Variables in the analysis ................................................................................................ 12

2.2 Exploratory data analysis .............................................................................................. 13

2.2.1 Descriptive statistics .......................................................................................... 13

2.2.2 Producing univariate box plots .......................................................................... 15

2.2.3 Bivariate correlations ......................................................................................... 17

2.2.4 Producing scatterplots (in spss) ......................................................................... 18

2.3 Simple Linear Regression .............................................................................................. 22

2.3.1 Regression outputs ............................................................................................ 24

2.3.2 Standardised coefficients ................................................................................... 27

2.3.3 Statistical significance ........................................................................................ 27

2.4 Multiple linear regression analysis ............................................................................... 28

2.4.2 Regression outputs ............................................................................................ 29

2.4.3 Interpreting the results ...................................................................................... 30

3 The assumptions of Linear Regression .............................................................................. 31

3.1 Assumption 1: Variable Types ....................................................................................... 32

3.2 Assumption 2: Linearity ............................................................................................... 32

3.2.1 Checking for non-linear relationships ................................................................ 33

3.2.2 Modelling a non-linear relationship, using linear regression ............................ 33

3.3 Assumption 3: Normal distribution of residuals .......................................................... 34

3.3.1 P-P plots ............................................................................................................. 34

3.3.2 Histograms of residuals...................................................................................... 35

3.4 Assumption 4: Homoscedasticity ................................................................................. 36

3.4.1 Checking for homoscedasticity of the residuals ................................................ 36

3.4.2 What to do if the residuals are not homoscedastic and why does it matter .... 37

3.5 Assumption 5: Multicolinearity .................................................................................... 38

3.5.1 Testing for colinearity - correlations .................................................................. 39

3.5.2 Testing for collinearity - variance inflation factor ............................................. 40

3.5.3 Collinearity - what to do .................................................................................... 40

3.6 Checking the assumptions of linear regression with SPSS ........................................... 40

3.6.1 Requesting plots ................................................................................................ 40

3.6.2 Calculating Variance Inflation Factors ............................................................... 41

3.7 Saving regression values ............................................................................................... 42

3.8 Extreme values .............................................................................................................. 43

3.8.1 Cook's Distance .................................................................................................. 44

4 Moving to a more complex model .................................................................................... 45

4.1 Nominal variables ......................................................................................................... 45

4.2 Interaction effects ......................................................................................................... 47

4.2.1 Scenario A: Same slope, same intercept ........................................................... 47

4.2.2 Scenario B: Different intercept, same slope ..................................................... 48

4.2.3 Scenario C: Different intercept, different slopes .............................................. 48

4.2.4 Scenario D: Different slope, same intercept ..................................................... 49

4.3 Transforming a variable ................................................................................................ 50

4.4 More model selection methods - beyond the default ................................................. 50

4.4.1 Backwards Elimination ....................................................................................... 51

4.4.2 Stepwise ............................................................................................................. 51

4.5 SPSS skills for more advanced modelling ...................................................................... 51

4.5.1 Recoding into a dummy variable ....................................................................... 51

4.5.2 Computing a new variable ................................................................................. 53

5 Further reading ................................................................................................................. 54

6 Appendix A: Correlation, covariance and parameter estimation .................................... 56

7 Glossary ............................................................................................................................. 57

1 THE BASICS - UNDERSTANDING LINEAR REGRESSION

1.1 SIMPLE LINEAR REGRESSION - ESTIMATING A BIVARIATE MODEL

Results at age 16

Results at age 11

1 The terms response and explanatory variables are the general terms to describe predictive relationships. You

30405060708090100

Exam16

Exam11

1.2 HYPOTHESIS TESTING