Statistical Analysis for Education and Psychology Researchers

regression line would be zero. In a test of significance of a predictor (explanatory)
variable the null hypothesis would be, H 0 : β 1 =0. The alternative hypothesis, that is X and
Y are linearly related, is H 1 : β 1 ≠ 0, and X makes a significant contribution to the
prediction of Y. To test this null hypothesis we evaluate the ratio of b 1 /standard error of
b 1 , and compare this with the sampling distribution of the t-statistic with n−2 degrees of
freedom. To use the sampling distribution of the regression test statistic, b 1 , certain
assumptions about the random error term in the regression model must be met, these are
discussed under test assumptions. It has already been stated in previous chapters that
whenever possible confidence intervals should be used in conjunction with tests of
significance. A confidence interval for the population regression slope is estimated from
sample data using the following formula:
b−[t 1 −α/2 SE(b 1 )] to b+[t 1 −α/2 SE(b 1 )]
Confidence
interval for
the regression
slope—8.2

with df=n−2.
If a 95 per cent confidence interval was required then t 1 −α/2 would equal t0.025. The
confidence interval for the intercept of the regression line is similar to formula 8.2 except
that SE(b 1 ) is changed to SE(b 0 ), the standard error of the intercept.

Multiple Regression

A simple linear regression model can be easily extended to accommodate two or more
explanatory variables. Practical applications of regression analysis often require two or
more predictor variables. The general notation for a multiple regression model is
y=β 0 +β 1 x 1 +β 2 x 2 +...+βkxk+ε

The intercept, β 0 , sometimes called the constant term, is the value of the response
variable y when all the explanatory variables are zero. The regression statistics, b 1 , b 2 ...bk
as in simple linear regression estimate the unknown parameters β 1 , β 2 ...βk.

Steps in Regression Analysis

There are seven steps in a typical regression analysis; the first two can be regarded as part
of initial data analysis:

1 Check the reasons for fitting a regression model—is it a description of a linear
relationship, estimation of parameters and significance of explanatory variables or the
prediction of an individual or mean response value?
2 Examine the means and standard deviations of the response and explanatory variable(s)
and explore the main features of the data using scatterplots of response variable
against each explanatory variable (and plots of pairs of explanatory variables in
multiple regression) to see whether there appears to be any relationship between the
variables and to check for linearity (see test assumptions in section 8.2).

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