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Multiple regression

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Multiple regression

- Simple linear regression
SPSS output

Linearity assumption

- Multiple regression
… in action; 7 steps

checking assumptions (and repairing)

Presenting multiple regression in a paper

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.

Model Summaryb

ModelRR Square Adjusted R SquareStd. Error of the Estimate

1.792a .627.50225.73131

a. Predictors: (Constant), classes

b. Dependent Variable: vocabulary

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

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

More than one predictor

e.g. predict vocabulary from

classes + homework + L1vocabulary

- 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

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

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)

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

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)

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

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