Correlation. Chapter 9 (omit 9.5, 9.10, 9.13, 9.14) Aplia (week 6 Correlation; omit question 6)

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1 Chapter 9 (omit 9.5, 9., 9.13, 9.1) Aplia (week Correlation; omit question ) Research report details in course syllabus due October 1

2 Assessing relationship between variables level of neurotransmitter X and amount of selfstimulation behavior in autistic children higher scores on one variable associated with higher scores on the other Graphical representation of relationship between variables scatter diagram or scatter plot

3 Construction of a scatter plot X = neurotransmitter; Y = self-stimulation X Y Y (self-stim.) X (Neurotrans.) 3

4 Interpreting relationships in a scatter plot Y (self-stim.) Y (self-stim.) X (Neurotrans.) X (Neurotrans.) positive linear nonlinear Y (self-stim.) X (Neurotrans.) X (Neurotrans.) negative linear no relationship

5 Measuring the degree of linear relationship between variables is a high score on one variable associated with a high score on the other? what is high and what is low? e.g., X = IQ score, Y = self-confidence (7-pt scale) IQ = 9, self-conf. = 5 Consider each score relative to its mean deviation scores X M X Y M Y 5

6 Implications of high X M X paired with high Y M Y and low X deviations paired with low Y deviations Implications of high X M X paired with low Y M Y and low X deviations paired with high Y deviations Implications of inconsistent pairings mixture of all combinations of high and low Y Y Y X X X

7 Use pattern of paired deviations to quantify strength of linear relationship Cross-product of deviation scores (X M X )(Y M Y ) Average cross-product or covariance as sample size increases, sum of cross-products gets arbitrarily large solution: take the average cov XY = (X M X )(Y M Y ) Division by N 1 due to using estimated value for the mean of the bivariate distribution: (M X, M Y ) estimates (µ X, µ Y ) 7

8 Covariance for neurotransmitter and self-stimulation data Case 1: positive linear relationship X Y X M X Y M Y (X-M X )(Y-M Y ) Y (self-stim.) 1 cov XY = (X M X )(Y M Y ) = = X (Neurotrans.) M X = 9.9, M Y = 7.7

9 Covariance for neurotransmitter and self-stimulation data Case : negative linear relationship X Y X M X Y M Y (X-M X )(Y-M Y ) M X = 9.9, M Y = 7.7 Y (self-stim.) cov XY = (X M X )(Y M Y ) = = X (Neurotrans.) 9

10 Covariance for neurotransmitter and self-stimulation data Case 3: nonlinear relationship X Y X M X Y M Y (X-M X )(Y-M Y ) M X = 9.9, M Y = 7.7 Y (self-stim.) cov XY = (X M X )(Y M Y ) 1 =.7 9 = X (Neurotrans.)

11 Covariance for neurotransmitter and self-stimulation data Case : no relationship X Y X M X Y M Y (X-M X )(Y-M Y ) M X = 9.9, M Y = 7.7 Y (self-stim.) cov XY = (X M X )(Y M Y ) 1 = = X (Neurotrans.) 11

12 Size of covariance depends on size of deviations cov XY = (X M X )(Y M Y ) X Y X M X Y M Y (X-M X )(Y-M Y ) M X = 3, M Y = 3, cov XY =.5 X Y X M X Y M Y (X-M X )(Y-M Y ) M X =, M Y =, cov XY = X Y X 1

13 Size of covariance depends on size of deviations How can we standardize the size of these deviations? conversion of raw scores to z scores produces s = 1 for any distribution divide deviations by the relevant standard deviation cov XY = (X M X )(Y M Y ) (X M X ) s X (Y M Y ) s Y Correlation coefficient (r ) 13

14 Apply the modified covariance formula to the two examples (X M X )(Y M Y ) s X X Y X M X Y M Y [(X-M X )/s X ][(Y-M Y )/s Y ] (-/1.5)(-1/1.5) = (-1/1.5)(-/1.5) = (0/1.5)(0/1.5) = (1/1.5)(1/1.5) = (/1.5)(/1.5) = 1.0 S = 3.0 s Y M X = 3, M Y = 3 s X = 1.5, s Y = 1.5 (X M X ) s X (Y M Y ) s Y = 3.0 =.90 1

15 Apply the modified covariance formula to the two examples (X M X )(Y M Y ) s X X Y X M X Y M Y [(X-M X )/s X ][(Y-M Y )/s Y ] - - (-/3.1)(-/3.1) = (-/3.1)(-/3.1) = (0/3.1)(0/3.1) = 0.00 (/3.1)(/3.1) = 0.0 (/3.1)(/3.1) = 1.0 S = 3.0 s Y M X =, M Y = s X = 3.1, s Y = 3.1 (X M X ) s X (Y M Y ) s Y = 3.0 =.90 15

16 So, from Correlation (X M X ) s X (Y M Y ) s Y we have: r =.90 r = X Y X 1

17 The formula for r can be simplified into two versions = 1 s X = 1 s X 1 s Y 1 s Y (X M X ) s X (Y M Y ) (X M X )(Y M Y ) cov XY = cov XY s X s Y s Y r = = z X z Y z Xz Y r = cov XY s X s Y 17

18 Correlation demonstration using R and data file test.txt data = read.table(file.choose(new = T), header = T) data library(psych) describe(data) hist(data$x) hist(data$y) plot(data) cor(data) 1

19 Most extreme values for r are 1.00 consider a case of a perfect correlation: r = 1.00 X Y z X z Y z X z Y M X = 3 M Y = s X =.1 s Y =.3 r = z X z Y = 3 3 = 1 Y Equivalent to computing variance of z scores 19 X

20 Correlation coefficient Correlation application to data sets involving neurotransmitter and self-stimulation Case 1: positive linear relationship r = cov XY s X s Y = 9.30 (.75)(3.00) =.5 Case : negative linear relationship r = cov XY s X s Y = 9. (.75)(3.00) =.5 Case 3: nonlinear relationship r = cov XY s X s Y =.7 (.75)(3.00) =.05 Case : no relationship r = cov XY s X s Y =.0 (.75)(3.00) =.00 0

21 Correlation and causation Correlation basis for inferring causal relationships: random assignment studies reporting correlation coefficients rarely involve random assignment consider an exception subjects randomly assigned to use cell phone vs. no phone (phone use = 1 or 0) data on a speeded cognitive task (component of driving simulation) each subject has a score representing treatment and a score representing cognitive latency 1

22 Correlation and causation Correlation hypothetical data: X = treatment, Y = cog. latency X Y X M X Y M Y (X-M X )(Y-M Y ) M X = M Y = cov XY = 31/11 = 1 Y (cog. latency) r = cov XY s X s Y = 1 (.5)(5.7) = X (treatment)

23 Restriction of range Correlation correlation between GRE and success in graduate school Grad. School 9 r = GRE Grad. School 9 r = GRE 3

24 Extreme groups Correlation correlation between GRE and success in graduate school Grad. School 9 r = GRE Grad. School 9 r = GRE

Statistics Introductory Correlation

Statistics Introductory Correlation Session 10 oscardavid.barrerarodriguez@sciencespo.fr April 9, 2018 Outline 1 Statistics are not used only to describe central tendency and variability for a single variable.