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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
  • 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)

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 3 3
SPSS output for simple regression (3/3)


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
  • 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
multivariate outlier
Multivariate outlier


Mahalanobis distance

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

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

with df = number of predictors


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