Correlation and regression Rob Davies Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster Correlation analysis - 2013 Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 1 / 27 

Outline I 1 Introduction 2 Correlation Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 2 / 27 

Introduction Correlation and regression concern the analysis of relationships between variables Figure : Creative Commons (CC) flickr user Cougar Studio Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 3 / 27 

People vary in performance from test to test, stimulus to stimulus, person to person, study to study Figure : A relationship between word reading ability and nonword reading ability? Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 4 / 27 

Correlation and regression concern the analysis of relationships between variables Each point in the scatterplot represents the scores recorded for a person on two tests The height of a point – its position on the y-axis – represents the person’s score on a test of word reading accuracy The horizontal (left to right) position of a point – its position on the x-axis – represents the same person’s score on a test of nonword reading accuracy Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 5 / 27 

What relationship should we see between participants’ word reading ability and their age, nonword reading ability or reading experience? Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 6 / 27 

When there is nothing there . . . Word reading ability varies (high or low) as print exposure – for each person – increases (from left to right) The trend is flat – whether people are high or low is not associated systematically with variation in ability q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 70 80 90 100 0 10 20 30 40 ART score TOWRE word naming accuracy Word reading skill vs. print exposure Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 7 / 27 

When there is something there? The scatterplot shows how word reading ability values (y-axis) increase as nonword reading ability increases A positive trend – higher scores on one variable are associated with higher scores on the other q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 70 80 90 100 20 40 60 TOWRE nonword naming accuracy TOWRE word naming accuracy Word vs. nonword reading skill Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 8 / 27 

Key concept: validity 1 We hypothesize that some variables are associated 2 Nonword reading reflects phonological coding knowledge which should also feed into word reading ability or vice versa 3 Do not expect to see a relationship if the hypothesized relationship is not valid Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 9 / 27 

Key concepts: variability and relationships 1 Human behaviour varies: in social science, we know some of that variation is random, while some might be due to the variables we test 2 Correlation and regression analyses are techniques to examine the relationship between behaviours we observe and potential explanations 3 Observations = average + important sources of variance + sampling error Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 10 / 27 

Correlation Looking at relationships Figure : CC flickr ephelt Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 11 / 27 

Correlation we use correlation to examine the relationship between two variables the correlation coefficient measures the extent to which the variables vary together coefficients vary between −1 (perfect negative relation) and +1 (perfect positive relation) through 0 (no relation) Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 12 / 27 

Correlation: Is the relationship between two variables positive or negative and how strong is it? Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 13 / 27 

If two variables vary together then they covary Covariance How do we calculate a measure of the extent to which two variables vary together: we begin with covariance Covariance is a number that reflects the degree to which two variables vary together COVXY = (X − ¯ X)(Y − ¯ Y) N − 1 (1) Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 14 / 27 

What does co-variance mean? We start by thinking about how a variable varies on its own variance of a single variable s2 x is the average amount that values vary from the mean ¯ X for that variable is the value of each observation higher or lower than the mean? add up the observed deviations and take into account the size of the sample, dividing by N − 1 s2 x = (X − ¯ X)2 N − 1 (2) Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 15 / 27 

Do two variables vary together? Covariance Covariance – reflects the degree to which two variables vary together if positive relationship: if values in one variable are higher than the mean, values in the other variable also are higher if negative relationship: if values in one variable are higher than the mean, values in the other variable are lower if no relationship: if values in one variable are higher, values in the other variable are higher or lower about equally Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 16 / 27 

Do two variables vary together? Covariance Covariance is a number that reflects the degree to which two variables vary together The top half of the equation gives the sum of how (1) values of X vary (2) values of Y vary The bottom half takes into account the size of the sample N-1 COVXY = (X − ¯ X)(Y − ¯ Y) N − 1 (3) Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 17 / 27 

How do we calculate a measure of the extent to which two variables vary together: better with correlation: r The top half of the equation estimates covariance The bottom half of the equation deals with the need to make the variables generally useful Scales by the SDs of X and Y to ensure we focus on how the variables vary in ways that are comparable r = COVXY sX sY (4) Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 18 / 27 

Pearson product-moment correlation coefficient r Covariance of two variables scaled in the same units: in standard deviations r varies between −1 and +1 r = COVXY sX sY (5) Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 19 / 27 

Pearson product-moment correlation coefficient r r varies between −1 and +1 if r is +1 there is a perfect positive relationship between the two variables: if values in one increase, values in the other variable increase if r is −1 there is a perfect negative relationship between the two variables: if values in one increase, values in the other variable decrease if r is 0 there is no relationship between the two variables: if values in one increase, values in the other variable increase or decrease r = COVXY sX sY (6) Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 20 / 27 

How do we know that a correlation matters? Correlation significance r varies between −1 and +1 we test whether the correlation number (coefficient) is different from 0 tr = r √ N − 2 √ 1 − r2 (7) Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 21 / 27 

How do we know that a correlation matters? Correlation significance usually, we transform r into t – and we look for the probability of finding a t as large as the one we see tr = r √ N − 2 √ 1 − r2 (8) Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 22 / 27 

Correlation and causality in any correlation, a causal relationship between two variables cannot be assumed there may be other measured or unmeasured variables that cause values in one variable to vary with values in another so: could reading ability decrease as adults age? q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 70 80 90 100 20 40 60 Age (years) TOWRE word naming accuracy Word reading vs. age Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 23 / 27 

Correlation assumptions for Pearson’s correlation coefficient to be an accurate measure of correlation both variables should be interval measures for the evaluation of significance to be valid the variables should be normally distributed Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 24 / 27 

Writing about correlation Example: “There was a significant correlation between reaction time and word frequency, r = .87, p = 0.05.” You report the correlation coefficient “r = .87” You report the significance “p = 0.05.” Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 25 / 27 

Summary we use correlation to examine the relationship between two variables the correlation coefficient measures the extent to which the variables vary together coefficients vary between −1 (perfect negative relationship) and +1 (perfect positive relationship) Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 26 / 27 

Rob Davies (Department of Psychology, Oxford Brookes University: from August 2013, University of Lancaster) Correlation and regression Correlation analysis - 2013 27 / 27