education, a dummy variable, coded as less than bachelor’s degree versus bachelor’s
degree or more.
Results of this regression analysis are shown in Table 8.1.
Table 8.1: Results of regression analysis
Predictors Regression coefficient SE t
Intercept −1.76 0.97−1.81
Pre-service 2.36 1.41 2.06
Education
In-service 1.28 0.43 2.99
Training (Internal)The investigators (Raudenbush et al., 1993) reported that, ‘The model was remarkably
parsimonious, including just two predictors: preservice training, b=2.36, t= 2.06, and
internal supervision [in-service training], b=1.28, t=2.99’ (p. 292). p-values were not
reported in the table or the text but with a sample size of 2111 students and 103
classrooms, any absolute t values greater than 1.96 would indicate statistical significance
at the 5 per cent level. As a quick estimate with large sample sizes, if the regression
coefficient is more than twice the size of its standard error then it is statistically
significant at the 5 per cent level. In this example in-service training is significant and
pre-service education might be although the reported t-value of 2.06 is not consistent with
the ratio of 2.36/1.41 (1.67), which is how t is evaluated (see worked example).
The authors concluded that there is strong evidence that teacher supervision (internal
in-service training) improves teachers’ instructional quality (b 1 =1.28, SEb 1 =0.43). The
authors went on to show that there was no support for the proposition that INSET by
inspectors or other staff outside of the school would be helpful in improving instructional
quality. The subjects in this study were teachers and pupils from small rural primary
schools in Thailand.
In the particular analysis described here the null hypothesis tested is ‘no linear
relationship between instructional quality and internal in-service training’, that is H 0 : β 1
(internal in-service training)=0. The alternative hypothesis is that internal in-service
training makes a significant contribution to the prediction of teachers instructional
competence. Assumptions for linear regression were checked and some variables
including number of sessions of internal in-service training were transformed to a
logarithmic metric because they had positively skewed distributions (as would be
expected with a count variable). Linear relations between log-transformed variables and a
response variable imply a diminishing effect of the predictor as it increases. In this
example, it would mean that when internal in-service training was low, the effect on
instructional quality would be significant and positive but with an increase in exposure to
internal in-service training the beneficial effects would be reduced.
In another study which examined the influence of state-level centralization on
innovation in post-secondary education the authors Hearn and Griswold (1994) provide a
good description of model building and variable selection for a multivariate regression
and describe how checks were made for model fit. Their regression model was developed
from a general theoretical model but was also influenced by empirical analyses which
revealed significant relationships between candidate independent variables. They
Inferences involving continuous data 259