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University of Durham D Dr Robert Coe University of Durham School of Education Tel: (+44 / 0) Fax: (+44 / 0) Correlation and Regression Doctor of Education (EdD) Analysing, Interpreting and Using Educational Research (Research Methodology)

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© 2005 Robert Coe, University of Durham 2 r = 0.0 Examples of correlation coefficients

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© 2005 Robert Coe, University of Durham 3 r = 0.3

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© 2005 Robert Coe, University of Durham 4 r = 0.5

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© 2005 Robert Coe, University of Durham 5 r = 0.7

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© 2005 Robert Coe, University of Durham 6 r = 0.9

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© 2005 Robert Coe, University of Durham 7 Grammar School selection A test selects the top 25% at age 11: 11% passed who should not have 18% rightly passed 60% rightly failed 11% failed who should not have Based on a correlation of 0.7

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© 2005 Robert Coe, University of Durham 8 Variance accounted for Cognitive measure of prior attainment / aptitude Academic achievement r = 0.7 (r 2 = 0.49)

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© 2005 Robert Coe, University of Durham 9 Measure of socioeconomic status r = 0.3 (r 2 = 0.09) Academic achievement High SES Low SES

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© 2005 Robert Coe, University of Durham 10 Aggregated or Individual? “Ecological Fallacy” Socioeconomic status School 1 + School 2 + School 3 + School 4 + Academic achievement Correlations for: Individual students = 0.3 School means = 0.9

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© 2005 Robert Coe, University of Durham 11 r = 0.03 (n = 30) Beware small samples: But add one extreme point... r = 0.33

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© 2005 Robert Coe, University of Durham 12 Restricted range:

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© 2005 Robert Coe, University of Durham 13

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University of Durham D Dr Robert Coe University of Durham School of Education Tel: (+44 / 0) Fax: (+44 / 0) Regression

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

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© 2005 Robert Coe, University of Durham 16 A* A B C D E F G U YELLIS test score Average performance for people with that test score Average residual = 0.26 Value Added as we know it: RESIDUAL

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© 2005 Robert Coe, University of Durham 17 Cognitive

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© 2005 Robert Coe, University of Durham 18 Social

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© 2005 Robert Coe, University of Durham 19

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© 2005 Robert Coe, University of Durham 20 Output from SPSS:

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© 2005 Robert Coe, University of Durham 21 Issues in regression Check residuals are Normally distributed Check for collinearity in explanatory variables Use adjusted R 2 Which explanatory variables to include?

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University of Durham D Dr Robert Coe University of Durham School of Education Tel: (+44 / 0) Fax: (+44 / 0) Regression to the mean Doctor of Education (EdD) Analysing, Interpreting and Using Educational Research (Research Methodology)

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© 2005 Robert Coe, University of Durham 23 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?

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© 2005 Robert Coe, University of Durham 24

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© 2005 Robert Coe, University of Durham 25 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?

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© 2005 Robert Coe, University of Durham 26

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© 2005 Robert Coe, University of Durham 27 Is social class more important than early ability? Feinstein (2003)

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© 2005 Robert Coe, University of Durham 28 Or is it just regression to the mean?

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