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**Correlation and Regression**

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

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**Examples of correlation coefficients**

© 2005 Robert Coe, University of Durham

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r = 0.3 © 2005 Robert Coe, University of Durham

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r = 0.5 © 2005 Robert Coe, University of Durham

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r = 0.7 © 2005 Robert Coe, University of Durham

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r = 0.9 © 2005 Robert Coe, University of Durham

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**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

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**Variance accounted for**

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

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**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

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**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

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**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

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Restricted range: © 2005 Robert Coe, University of Durham

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**© 2005 Robert Coe, University of Durham**

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Regression

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**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

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**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

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Cognitive © 2005 Robert Coe, University of Durham

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Social © 2005 Robert Coe, University of Durham

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**© 2005 Robert Coe, University of Durham**

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

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**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

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**Doctor of Education (EdD)**

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

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**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

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**© 2005 Robert Coe, University of Durham**

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**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

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**© 2005 Robert Coe, University of Durham**

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**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, © 2005 Robert Coe, University of Durham Feinstein (2003)

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**Or is it just regression to the mean?**

© 2005 Robert Coe, University of Durham

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Factor P 16 8(8-5ab) 4(d² + 4) 3rs(2r – s) 15cd(1 + 2cd) 8(4a² + 3b²)

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