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Design and Data Analysis in Psychology I Salvador Chacón Moscoso Susana Sanduvete Chaves School of Psychology Dpt. Experimental Psychology 1
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Lesson 11 Relationship between two quantitative variables 2
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INTRODUCTION When assumptions are accepted (parametric tests): Simple linear regression (it is going to be studied next academic year in the subject Design and Data Analysis in Psychology II). Pearson correlation. When assumptions are not accepted (non- parametric tests): Spearman correlation. 3
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PEARSON CORRELATION: DEFINITION r XY Coefficient useful to measure covariation between variables: in which way changes in a variable are associated to the changes in other variable. Quantitative variables (interval or ratio scale). Linear relationship EXCLUSIVELY. Values: -1 ≤ r XY ≤ +1. Interpretation: +1: perfect positive correlation (direct association). -1: perfect negative correlation (inverse association). 0: no correlation. 4
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5 Perfect positive correlation: r xy = +1 (difficult to find in psychology)
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6 Positive correlation: 0 < r xy < +1
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7 Perfect negative correlation: r xy = -1 (difficult to find in psychology)
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8 Negative correlation: -1 < r xy < 0
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9 No correlation
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Formulas 10 Raw scores Deviation scores Standard scores
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Example X: 2 4 6 8 10 12 14 16 18 20 Y:1 6 8 10 12 10 12 13 10 22 1. Calculate r xy in raw scores. 2. Calculate r xy in deviation scores. 3. Calculate r xy in standard scores. 11
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Example: scatter plot 12
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Example : calculation of r xy in raw scores XYXYX2X2 Y2Y2 21241 46241636 68483664 8108064100 1012120100144 1210120144100 1412168196144 1613208256169 1810180324100 2022440400484 110104139015401342 13
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Example : calculation of r xy in raw scores 14
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Example : calculation of r xy in deviation scores XYxyxyx2x2 y2y2 21-9-9.484.68188.36 46-7-4.430.84919.36 68-5-2.412255.76 810-3-0.41.290.16 10121.6-1.612.56 12101-0.4 10.16 141231.64.892.56 161352.613256.76 18107-0.4-2.8490.16 2022911.6104.481134.56 11010400246330260.4 15
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Example : calculation of r xy in deviation scores 16
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Example : calculation of r xy in standard scores XYZxZyZxZy 21-1.567-1.8422.886 46-1.218-0.8621.051 68-0.870-0.4700.409 810-0.522-0.0780.041 1012-0.1740.314-0.055 12100.174-0.078-0.014 14120.5220.3140.164 16130.8700.5100.443 18101.218-0.078-0.096 20221.5672.2733.561 110104008.391 17
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Example : calculation of r xy in standard scores 18
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Significance Does the correlation coefficient show a real relationship between X and Y, or is that relationship due to hazard? Null hypothesis H 0 : r xy = 0. The correlation coefficient is drawn from a population whose correlation is zero (ρ XY = 0). Alternative hypothesis H 1 :. The correlation coefficient is not drawn from a population whose correlation is different to zero (ρ XY ). 19
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Significance Formula: Interpretation: Null hypothesis is rejected. The correlation is not drawn from a population whose score ρ xy = 0. Significant relationship between variables exists. Null hypothesis is accepted. The correlation is drawn from a population whose score ρ xy = 0. Significant relationship between variables does not exist. Exercise: conclude about the significance of the example. 20
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Significance: example Conclusions: we reject the null hypothesis with a maximum risk to fail of 0.05. The correlation is not drawn from a population whose score ρ xy = 0. Relationship between variables exists. 21
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Other questions to be considered Correlation does not imply causality. Statistical significance depends on sample size (higher N, likelier to obtain significance). Other possible interpretation is given by the coefficient of determination, or proportion of variability in Y that is ‘explained’ by X. The proportion of Y variability that left unexplained by X is called coefficient of non-determination: Exercise: calculate the coefficient of determination and the coefficient of non-determination and interpret the results. 22
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Coefficient of determination: example 70.4% of variability in Y is explained by X. 29.6% of variability in Y is not explained. 23
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Which is the final conclusion? Significant effect Non-significant effect High effect size ( ≥ 0.67) The effect probably exists The non- significance can be due to low statistical power Low effect size ( ≤ 0.18) The statistical significance can be due to an excessive high statistical power The effect probably does not exist 24
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Which is the final conclusion? Significant effect Non-significant effect High effect size ( ≥ 0.67) The effect probably exists The non- significance can be due to low statistical power Low effect size ( ≤ 0.18) The statistical significance can be due to an excessive high statistical power The effect probably does not exist 25
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