Correlation and Regression Doctor of Education (EdD) Analysing, Interpreting and Using Educational Research (Research Methodology)

Examples of correlation coefficients © 2005 Robert Coe, University of Durham

r = 0.3 © 2005 Robert Coe, University of Durham

r = 0.5 © 2005 Robert Coe, University of Durham

r = 0.7 © 2005 Robert Coe, University of Durham

r = 0.9 © 2005 Robert Coe, University of Durham

Grammar School selection A test selects the top 25% at age 11: 11% failed who should not have 18% rightly passed 11% passed who should not have 60% rightly failed © 2005 Robert Coe, University of Durham Based on a correlation of 0.7

Variance accounted for Academic achievement r = 0.7 (r2 = 0.49) Cognitive measure of prior attainment / aptitude © 2005 Robert Coe, University of Durham

Measure of socioeconomic status Academic achievement r = 0.3 (r2 = 0.09) Low SES Measure of socioeconomic status High SES © 2005 Robert Coe, University of Durham

Aggregated or Individual? “Ecological Fallacy” Academic achievement Correlations for: Individual students = 0.3 School means = 0.9 Socioeconomic status School 1 + School 2 + School 3 + School 4 + © 2005 Robert Coe, University of Durham

But add one extreme point ... Beware small samples: r = 0.03 But add one extreme point ... r = 0.33 (n = 30) © 2005 Robert Coe, University of Durham

Restricted range: © 2005 Robert Coe, University of Durham

© 2005 Robert Coe, University of Durham

Regression

One school’s maths GCSE grades: How good are they? A* A B C D E F G U 2 17 25 14 18 11 8 3 45% % FSM? =15% Socioeconomic status? School % 5A*-C? =56% Subject difficulty? Students’ability? Prior attainment? © 2005 Robert Coe, University of Durham

Value Added as we know it: Average performance for people with that test score A* A B C D E F G U RESIDUAL Average residual = 0.26 © 2005 Robert Coe, University of Durham YELLIS test score

Cognitive © 2005 Robert Coe, University of Durham

Social © 2005 Robert Coe, University of Durham

© 2005 Robert Coe, University of Durham

Output from SPSS: © 2005 Robert Coe, University of Durham

Issues in regression Check residuals are Normally distributed Check for collinearity in explanatory variables Use adjusted R2 Which explanatory variables to include? © 2005 Robert Coe, University of Durham

Doctor of Education (EdD) Regression to the mean Doctor of Education (EdD) Analysing, Interpreting and Using Educational Research (Research Methodology)

Measures with less than perfect reliability A test with test-retest correlation r=0.7 is repeated after an interval. What would you expect for the TEST 2 scores of A person who achieved a very high score on TEST 1 A person who achieved a very low score on TEST 1 How will the overall distribution of scores on the two tests compare? © 2005 Robert Coe, University of Durham

© 2005 Robert Coe, University of Durham

Two subgroups with different means Students with high SES tend to get higher test scores. Two students have the same TEST 1 scores, but one is high SES, the other low SES. What would you expect their TEST 2 scores to be? © 2005 Robert Coe, University of Durham

© 2005 Robert Coe, University of Durham

Is social class more important than early ability? Feinstein, L (2003) ‘Inequality in the early cognitive development of British children in the 1970 cohort’. Economica, 70, 277, 73-97. © 2005 Robert Coe, University of Durham Feinstein (2003)

Or is it just regression to the mean? © 2005 Robert Coe, University of Durham