Multiple regression
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Multiple regression. Overview. Simple linear regression SPSS output Linearity assumption Multiple regression … in action; 7 steps checking assumptions (and repairing) Presenting multiple regression in a paper. Simple linear regression. Class attendance and language learning

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Presentation Transcript

Overview
Overview

  • Simple linear regression

    SPSS output

    Linearity assumption

  • Multiple regression

    … in action; 7 steps

    checking assumptions (and repairing)

    Presenting multiple regression in a paper


Simple linear regression
Simple linear regression

Class attendance and language learning

Bob: 10 classes; 100 words

Carol: 15 classes; 150 words

Dave: 12 classes; 120 words

Ann: 17 classes; 170 words

Here’s some data. We expect that the more classes someone attends, the more words they learn.


The straight line is the model for the data. The definition of the line (y = mx + c) summarises the data.


Spss output for simple regression 1 3
SPSS output for simple regression (1/3) of the line (y = mx + c) summarises the data.

Model Summaryb

Model R R Square Adjusted R Square Std. Error of the Estimate

1 .792a .627 .502 25.73131

a. Predictors: (Constant), classes

b. Dependent Variable: vocabulary


Spss output for simple regression 2 3
SPSS output for simple regression (2/3) of the line (y = mx + c) summarises the data.


Spss output for simple regression 3 3
SPSS output for simple regression (3/3) of the line (y = mx + c) summarises the data.

Coefficientsa

Model Unstandardized Coefficients Standrdzd Coefficients t Sig. B Std. Error Beta

1 (Constant) -19.178 64.837 -.296 .787

classes 10.685 4.762 .792 2.244 .111

a. Dependent Variable: vocabulary


Linearity assumption
Linearity assumption of the line (y = mx + c) summarises the data.

  • Always check that the relationship between each predictor variable and the outcome is linear


Multiple regression1
Multiple regression of the line (y = mx + c) summarises the data.

More than one predictor

e.g. predict vocabulary from

classes + homework + L1vocabulary


Multiple regression in action
Multiple regression in action of the line (y = mx + c) summarises the data.

  • Bivariate correlations & scatterplots – check for outliers

  • Analyse / Regression

  • Overall fit (R2) and its significance (F)

  • Coefficients for each predictor (‘m’s)

  • Regression equation

  • Check mulitcollinearity (Tolerance)

  • Check residuals are normally distributed


Bivariate outlier
Bivariate outlier of the line (y = mx + c) summarises the data.


Multivariate outlier
Multivariate outlier of the line (y = mx + c) summarises the data.

Test

Mahalanobis distance

(In SPSS, click ‘Save’ button in Regression dialog)

to test sig., treat as a chi-square value

with df = number of predictors


Multicollinearity
Multicollinearity of the line (y = mx + c) summarises the data.

Tolerance should not be too close to zero

T = 1 – R2

where R2 is for prediction of this predictor by the others

If it fails, you need to reduce the number of predictors (you don’t need the extra ones anyway)


Failed normality assumption
Failed normality assumption of the line (y = mx + c) summarises the data.

If residuals do not (roughly) follow a normal distribution

… it is often because one or more predictors is not normally distributed

 May be able to transform predictor


Categorical predictor
Categorical predictor of the line (y = mx + c) summarises the data.

Typically predictors are continuous variables

Categorical predictors

e.g. Sex (male, female)

can do: code as 0, 1

Compare simple regression with t-test

(vocabulary = constant + Sex)


Presenting multiple regression
Presenting multiple regression of the line (y = mx + c) summarises the data.

Table is a good idea:

Include correlations (bivariate)

R2 adjusted

Report F (df, df), and its p, for the overall model

Report N

Coefficient, t, and p (sig.) for each predictor

Mention that assumptions of linearity, normality, and absence of multicollinearity were checked, and satisfied


Further reading
Further reading of the line (y = mx + c) summarises the data.

Tabachnik & Fidell (2001, 2007) Using Multivariate Statistics. Ch5 Multiple regression


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