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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Multiple regression

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Multiple regression

Multiple regression


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.


Multiple regression

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)

Model Summaryb

ModelRR Square Adjusted R SquareStd. Error of the Estimate

1.792a .627.50225.73131

a. Predictors: (Constant), classes

b. Dependent Variable: vocabulary


Spss output for simple regression 2 3

SPSS output for simple regression (2/3)


Spss output for simple regression 3 3

SPSS output for simple regression (3/3)

Coefficientsa

Model Unstandardized Coefficients Standrdzd Coefficients tSig. B Std. Error Beta

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

classes 10.6854.762 .792 2.244 .111

a. Dependent Variable: vocabulary


Linearity assumption

Linearity assumption

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


Multiple regression1

Multiple regression

More than one predictor

e.g. predict vocabulary from

classes + homework + L1vocabulary


Multiple regression in action

Multiple regression in action

  • 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


Multivariate outlier

Multivariate outlier

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

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

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

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

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

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


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