2Outline Basics Visualization Covariance Significance testing and interval estimationEffect sizeBiasFactors affecting correlationIssues with correlational studies
3CorrelationResearch question: What is the relationship between two variables?Correlation is a measure of the direction and degree of linear association between 2 variables.Correlation is the standardized covariance between two variables
4Questions to be asked… Is there a linear relationship between x and y? What is the strength of this relationship?Pearson Product Moment Correlation Coefficient rCan we describe this relationship and use this to predict y from x?y=bx+aIs the relationship we have described statistically significant?Not a very interesting one if tested against a null of r = 0
5Other stuff Check scatterplots to see whether a Pearson r makes sense Use both r and R2 to understand the situationIf data is non-metric or non-normal, use “non- parametric” correlationsCorrelation does not prove causationTrue relationship may be in opposite direction, co- causal, or due to other variablesHowever, correlation is the primary statistic used in making an assessment of causality‘Potential’ Causation
6Possible outcomes -1 to +1 As one variable increases/decreases, the other variable increases/decreasesPositive covarianceAs one variable increases/decreases, another decreases/increasesNegative covarianceNo relationship (independence)r = 0Non-linear relationship?
7ScatterplotsAs we discussed previously, scatterplots provide a pictorial examination of the relationship between two quantitative variablesPredictor variable on the X-axis (abscissa); Criterion variable on the Y-axis (ordinate)Each subject is located in the scatterplot by means of a pair of scores (score on the X variable and score on the Y variable)Plot each pair of observations (X, Y)X = predictor variable (independent)Y = criterion variable (dependent)Check for linear relationship‘Line of best fit’y = a + bxCheck for outliers
8Example of a Scatterplot The relationship between scores on a test of quantitative skills taken by students on the first day of a stats course (X- axis) and their combined scores on two semester exams (Y-axis)
9Example of a Scatterplot The two variables are positively relatedAs quantitative skill increases, so does performance on the two midterm examsLinear relationship between the variablesLine of best fit drawn on the graph - the ‘regression line’The ‘strength’ or ‘degree’ of the liner relationship is measured by a correlation coefficient i.e. how tightly the data points cluster around the regression lineWe can use this information to determine whether the linear relationship represents a true relationship in the population or is due entirely to chance factors
10What do we look for in a Scatterplot? Overall pattern: EllipseAny striking deviations (outliers)Form: is it linear? (curved? clustered?)Direction: is it positive…high values of the two variables tend to occur together)Or negativehigh values of one variable tend to occur with low values of the other variable)?Strength: how close the points lie to the line of best fit (if a linear relationship)
19Linear Correlation / Covariance How do we obtain a quantitative measure of the linear association between X and Y?The Pearson Product-Moment Correlation Coefficient, r, comes from the covariance statistic, it reflects the degree to which the two variables vary together
20Covariance The variance shared by two variables When X and Y move in the same direction (i.e. their deviations from the mean are similarly pos or neg)cov (x,y) = pos.When X and Y move in opposite directionscov (x,y) = neg.When no constant relationshipcov (x,y) = 0
21CovarianceCovariance is not easily interpreted on its own and cannot be compared across different scales of measurementSolution: standardize this measurePearson’s r:
22Significance test for correlation All correlations in a practical setting will be non- zeroA significance test can be conducted in an effort to infer to a populationKey Question: “Is the r large enough that it is unlikely to have come from a population in which the two variables are unrelated?”Testing the null hypothesis thatH0: r = 0 vs. alternative hypothesis H1: r ≠ 0r=population product-moment correlation coefficient
23Significance test for correlation However with larger N, small, possibly non-meaningful, correlations can be deemed ‘significant’So the better question is: Is a test against zero useful?Tests of significance for r have typically have limited utility if testing against a zero valueGo by the size1 and judge worth by what is seen in the relevant literaturedf criticalN-2 =.051. It’s not too hard to remember that with ~50 cases you only need ~.20 to ‘flag’ a correlation as statistically significant with the typical tailed approach, and in many situations that’s probably not considered a practical effect. Asterisks aren’t really necessary, are ugly, and make small correlations that are probably unimportant misleadingly displayed as such. Flagging isn’t necessarily a bad approach, it’s just used poorly and often incorrectly (in the case of identification of multiple alpha levels).
24Significance test for correlation Furthermore, using the approaches outlined in Howell, while standard, are really not necessaryUsing the t-distribution as described we would only really be able to test a null hypothesis of zeroIf we want to test against some specific value1, we have to convert r in some odd fashion and test using these new valuesFisher transformationdf = N - 21. Regarding testing against a non-nil r, Howell states “You probably can’t think of many situations in which you would like to do that, and neither can I.” This is definitely one place I would disagree with Howell. I typically can’t think of any reason to test against a nil value except in purely exploratory areas of research.
25Test of the difference between two rs While those new values create an r′ that approximates a normal distribution, why do we have to do it?The reason for this transformation is that since r has limits of +1, the larger the absolute value of r, the more skewed its sampling distribution about the population (rho)
26Sampling distribution of a correlation Via the bootstrap, we can a see for ourselves that the sampling distribution becomes more and more skewed as we deviate from a null value of zero
27The better approachNowadays, we can bootstrap the r or difference between two rs and do hypothesis tests without unnecessary (and most likely problematic) transformations and assumptionsEven for small samples of about 30 it performs as well as the transformation in ideal situations (Efron, 1988)Furthermore, it can be applied to other correlation metrics.
28CorrelationTypically though, for a single sample correlations among the variables should be considered descriptive statistics1, and often the correlation matrix is the data set that forms the basis of an analysisA correlation can also be thought of as an effect size in and of itselfStandardized measure of amount of covariationThe strength and degree of a linear relationship between variablesThe amount some variable moves in standard deviation units with a 1 standard deviation change in another variableR2 is also an effect sizeAmount of variability seen in y that can be explained by the variability seen in xAmount of variance they share2Hence no need for flagging since an inferential analysis is not performed.Interesting example: For IQ it is said that 80% is accounted for by heredity, 20% by environment. 80% and 20% are the variance accounted for in IQ by each, i.e. R2. So heredity is 4 times as important as environment for IQ right? If you look at the correlations (and we can for the most part assume independence between genetics and environment), r(heredity,IQ) = sqrt(.8) = .894, r(environment,IQ) = sqrt(.2) = So in terms of how much we expect IQ to move as a result of a 1 standard deviation unit change in the predictor, the difference is only ~2:1 in favor of heredity (.894 to .447)
29Biased estimate- Adjusted r r turns out to be upwardly biased, and the smaller the sample size, the greater the biasWith large samples the difference will be negligibleWith smaller samples one should report adjusted r or R2
31LinearityNonlinear relationships will have an adverse effect on a measure designed to find a linear relationship
32Heterogeneous subsamples Sub-samples may artificially increase or decrease overall r, or in a corollary to Simpson’s paradox, produce opposite sign relations for the aggregated data compared to the groupsSolution - calculate r separately for sub-samples & overall, look for differences
34Range restrictionLimiting the variability of your data can in turn limit the possibility for covariability between two variables, thus attenuating r.Common example occurs with Likert scalesE.g vsHowever it is also the case that restricting the range can actually increase r if by doing so, highly influential data points would be kept outWilcox 2001
35Effect of Outliers Outliers can artificially increase or decrease r OptionsCompute r with and without outliersConduct robustified R!For example, recode outliers as having more conservative scores (winsorize)Transform variables (last resort)
36Advantages of correlational studies Show the amount (strength) of relationship presentCan be used to make predictions about the variables studiedOften easier to collect correlational data, and interpretation is fairly straightforward.
37Disadvantages of correlational studies Can’t assume that a cause-effect relationship existsLittle or no control (experimental manipulation) of the variables is usually seenRelationships may be accidental or due to a third variable, unmeasured factorCommon causesSpurious correlations and Mediators