PSGY1014 Lecture 6: Correlations 2024-2025 PDF

PSGY1014 Lecture 6: Correlations Dr Wong Hoo Keat ([email protected])  Overview What are correlations? –Uses & Limitations Covariance Pearson’s r Spearman’s Rho Using SPSS –Getting r and Rho –Significance Variance accounted for PSG...

PSGY1014 Lecture 6: Correlations Dr Wong Hoo Keat ([email protected])  Overview What are correlations? –Uses & Limitations Covariance Pearson’s r Spearman’s Rho Using SPSS –Getting r and Rho –Significance Variance accounted for PSGY1014 2 Looking for associations T-test (and other test statistics) explore differences between conditions Differences between male and females in spatial reasoning Differences between grades before and after a learning programme … Sometimes we want to explore the association between variables Is there an association between caffeine and alertness? Is there an association between smoking and cancer? PSGY1014 3 Correlation does not tell us about causation Often people imply that A causes B because they co-vary That logic doesn’t hold. We know nothing about cause Maybe B depends A instead? Maybe A and B depend on some other joint variable That doesn’t mean correlations aren’t useful: Often correlations highlight possible causal relations Certainly null effects (i.e. absence of correlation) may allow us to discount certain theories But usually we need an experiment to find out why the correlation exists PSGY1014 4 Examples Easy: The number of fire engines at a burning house correlates with the amount of damage done Shoe size correlates with reading skills Trickier: Imagine the headline “Bottled water linked to baby health” PSGY1014 5 Looking for associations We have various measures of association: Covariance (parametric, calculated on the way to Pearson’s r) Pearson’s Coefficient of Correlation (parametric) Variance explained (variance accounted for) Spearman’s Coefficient of Rank Correlation (non-parametric) PSGY1014 6 Covariance (supplementary) Covariance is the mean of the product of the deviations: X Y 10 40 10 – 20 40 – 50 100 20 50 20 – 20 50 – 50 0 30 60 30 – 20 60 – 50 100 Sum 200 Cov 200/3 = 66.7 PSGY1014 7 Covariance Covariance is the mean of the product of the deviations: X Y 1 4 1–2 4–5 1 2 5 2–2 5–5 0 3 6 3–2 6–5 1 Sum 2 Cov 2/3 = 0.66 PSGY1014 8 Covariance Positive value: as variable “a” increases, variable “b” increases Negative value: as variable “a” increases, variable “b” decreases Problem: it is a not standardized measure Covariance values cannot be compared each other PSGY1014 9 Correlations Best fit line Line that best represent our data Correlation considers how closely the data points fall to this line Correlations have two characteristics Magnitude: Between 0 and 1 A value of ‘0’ means no relationship A value of ‘1’ means a perfect relationship Direction: Negative Correlation vs. Positive Correlation For example, -.67 or.56 PSGY1014 10 Correlations There are two key tests of correlation: Pearson’s Correlation Coefficient This can be used to explore the linear relationship between two continuous variables Spearman’s Rho Correlation Coefficient This is similar to Pearson’s but uses ranked scores (ordinal data) rather than continuous data PSGY1014 11 Pearson’s Correlation Coefficient It is based on covariance But, by dividing by the standard deviations of the variables, it is independent of the overall variability cov( x, y) X Y 66.7/(sY*sX) r= 10 40 66.7/(8.16*8.16) sxsy 20 30 50 60 1.000 PSGY1014 12 Pearson’s Correlation Coefficient It is based on covariance But, by dividing by the standard deviations of the variables, it is independent of the overall variability cov( x, y) X Y 66.7/(sY*sX) r= 1 4 66.7/(8.16*8.16) sxsy 2 3 5 6 1.000 PSGY1014 13 Properties of Pearson’s r It is only dependent on how well-related the variables are, not how much they vary It takes values between -1 and +1 -1 means when X goes up, Y goes down +1 means when X goes up, Y goes up too 0 signifies no (linear) relationship between X and Y PSGY1014 14 Spearman’s Rho Correlation Coefficient Non-parametric test It does not assume that data are evenly spaced or normally distributed (so use this test for ordinal data) It works by first ranking the data and then working out Pearson’s r on the ranks PSGY1014 15  But first thing first: Scatterplots To get a pictorial view Each data point is Excel is a great tool to of the relationship marked on a graph make scatterplots between two variables you can create a scatter plot PSGY1014 16  Examples of perfect correlation 30 30 30 20 20 20 Var2 Var2 Var3 10 10 10 r = +1 0 0 0 0 10 20 0 10 20 0 10 20 Var1 Var3 Var1 30 30 30 20 20 20 Var2 Var3 Var2 r = -1 10 10 10 0 0 0 0 10 20 0 10 20 0 10 20 Var1 Var3 Var1 17  Examples of smaller r r=0.153, df=16, p>0.05 30 20 Var3 30 30 10 20 Var4 20 0 Var2 10 0 10 20 10 Var1 0 0 0 10 20 0 10 20 Var1 Var1 r can’t be calculated r=0.199, df=16, p>0.05 (it is zero divided by zero) PSGY1014 18 Correlation in SPSS Correlation between number of errors in two different tasks Must have at least two columns of data Analyze < correlate < bivariate PSGY1014 19 Correlation in SPSS Correlation between number of errors in two different tasks Must have at least two columns of data Analyze < correlate < bivariate PSGY1014 20 Correlation in SPSS Each row is a variable and the way it correlates with each other variable (columns) The correlation of a variable with itself is always +1.0 PSGY1014 21 Reporting a correlation coefficient A positive correlation between both variables was found , r(8) =.97, p <.001 PSGY1014 22 Variance accounted for Another way to think of correlations is in terms of variance accounted for This is expressed as a % or a fraction It’s simply calculated by squaring Pearson’s r If the correlation value is squared, you have the amount of variance explained by your data If this value is close to 1, it means that your variables explain almost all the variation in your data PSGY1014 23 Partial correlations Sometimes the correlation between two variables (A & B) might be explained by another that we have measured (C) For example, we want to know whether excessive eating affects your performance adversely in exams: – Correlate how many times you eat too much (A) with your exam score – But what about the fact that the excessive eating stopped you from studying? (C) A partial correlation can tell you the part of the correlation that is not related to variable C PSGY1014 24 Partial correlations The variable C is referred to as being ‘partialized out’ or ‘held constant’ A simple (bivariate) correlation is called a zero-order correlation A first-order (partial) correlation is one that ‘partials out’ a single variable constant A second-order (partial) correlation holds two variables constant, etc. PSGY1014 25 Partial correlations Analyze < correlate < partial… PSGY1014 26 To recap… Correlations help us to understand how closely two things vary together They don’t allow us to infer causality Relationships can be described in terms of covariance, correlation and variance accounted for Using a correlation coefficient does not mean you had a correlational design Significant is not the same as important (as with all tests). You can have a significant but very weak correlation. PSGY1014 27