Pearson's Correlation - Shippensburg University

Pearson’s Correlation

Correlation – the degree to which two variables areassociated (co-vary). Covariance may be either positive or negative. Its magnitude depends on the units of measurement. Assumes the data are from a bivariate normal population. Does not necessarily imply causation.

The four y variables have the same mean (7.5), standard deviation(4.12), correlation (0.81) and regression line (y 3 0.5x).

Pearson’s correlation coefficient is a measure of theintensity of the linear association between variables. It is possible to have non-linear associations. Need to examine data closely to determine if anyassociation exhibits linearity.LinearNon-linear

Correlation coefficient values range -1 to 1. The closerto 1 the correlation coefficient gets the ‘stronger’ lationyxStrongcorrelationyWeakcorrelationyx

The Pearson’s Correlation Coefficient.𝑟𝑟 𝑥𝑥𝑥𝑥 𝑥𝑥 2 𝑦𝑦 2𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑋𝑋22 𝑥𝑥 𝑋𝑋 𝑛𝑛 𝑌𝑌22 𝑦𝑦 𝑌𝑌 𝑛𝑛22 𝑋𝑋 𝑌𝑌 𝑥𝑥𝑥𝑥 𝑋𝑋𝑋𝑋 𝑛𝑛

The correlation coefficient is a measure of the intensity of theassociation between variables. r is a unit-less number. It can not be used to extrapolate a change in y based on achange in x. If variables are highly correlated, then we may want toinvestigate their association further to determine if there is acausal mechanism operating.

1 versus 2-tailed hypotheses 2-tailed hypotheses concerning r would state that there is asignificant correlation between two variables. e.g. Ho: r 0, Ha: r 0 1-tailed hypotheses concerning r would state that theassociation is either positive or negative. e.g. Ho: r 0, Ha: r 0

Significance Testing for rIf the data are normally distributed we can calculatea t-statistic for the correlation coefficient (r) using theequation:𝑟𝑟𝑡𝑡 ��𝑠𝑠𝑟𝑟 1 (𝑟𝑟)2𝑛𝑛 2df n-2 since there is one df for each column.Here we are testing the null hypothesis that r 0.

Temperature and Dissolved OxygenTemp 7818.3028.5688.724DO 014.46414.14514.76214.48213.166

Temperature and Dissolved OxygenDissolved Oxygen161514135678Temperature (C)9

We will perform a 1-tailed test since our graph suggests that theremay be a significant negative (or inverse) association betweentemperature and dissolved oxygen.Ho: There is not a significant negative correlation betweentemperature and dissolved oxygen.Ha: There is a significant negative correlation between temperatureand dissolved oxygen.

DO emp (X) X*Y

𝑛𝑛 13 𝑋𝑋 93.22𝑑𝑑𝑑𝑑 𝑛𝑛 2, 13 2 11 𝑌𝑌 193.93 (𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜 𝑡𝑡𝑡𝑡𝑡 ��𝑜𝑜𝑜 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑) 𝑋𝑋 2 680.39 𝑌𝑌 2 2899.79 (𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜 𝑡𝑡𝑡𝑡𝑡 ��𝑠 ��𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜) 𝑋𝑋𝑋𝑋 1383.12 (𝑋𝑋 𝑌𝑌 𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠)93.22 𝑥𝑥 680.39 132 𝑥𝑥𝑥𝑥 1383.12 𝑟𝑟 𝑠𝑠𝑟𝑟 7.5111.93 6.8 11.9393.22 193.93 7.5113193.93 𝑦𝑦 2899.79 1322 6.8 0.831 0.832 13 2tCritical 1.79620.311 0.168114.94 1.796 reject Ho𝑡𝑡 0.83 4.94 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠0.168

Since the value of r can be either positive or negative the criticalvalue is a range in this case:-1.796 to 1.796Our t-statistic (-4.94) falls beyond that range so we reject H0.There is a significant inverse (negative) correlation betweentemperature and dissolved oxygen (t-4.94, p 0.005, r -0.83).

Since the range of the correlation coefficient is from -1 to 1, whatdoes an r value of -0.83 tell us?1. The association is inverse meaning as one variable increasesthe other decreases.2. The intensity of this inverse association between temperatureand dissolved oxygen is fairly high.

Be aware that r is influenced by samples size.Therefore if the sample size is large, even smallerr values may be important.

Do by hand on the board:Black LungRateUndergroundMinersKentucky6.0312947West 101Pennsylvania16.416202Colorado1.95923State xy x yr 2 x X22where(X) 22n( Y )yY n222(X )( Y ) xy XY nrt srwheresr SPSS data set s:\GEO\pgmarr\Quantitative Methods\SPSS Data\BlackLung.sav1 (r ) 2n 2

CorrelationsBlackLungUndergroundPearson CorrelationSig. (2-tailed)NPearson CorrelationSig. 1.0171010*. Correlation is s ignificant at the 0.05 level (2-tailed).

Correlation vs RegressionCorrelation measures of association, but no causal relationship isimplied. There are no dependent or independent variables.Regression measures association where a causal relationship isbelieved to exist. A dependent and 1 independent variables are assumed.Both correlation and regression assume that the relationshipunder investigation is linear, but it may be either positive (direct)or negative (inverse).

Boas Native American TribeAnthropometric MeasurementsNote that arm and leg lengths are significantly correlated.However, longer arms do not cause longer legs. The causalmechanism is body proportions, meaning that larger individualstend to have both longer arms and legs.

Remember that causation will result in correlation, but thatcorrelation does not necessarily result in causation.Therefore, correlation is a necessary but not sufficient conditionto make causal inferences regarding our data.Causation can really only be determined through controlled dataanalysis and a firm understanding of the underlying mechanismswhich may result in a causal relationship.

There are situations where correlation is simply by chance, asseen below:Source: ere is no causal link between strangulation by bedsheets andcheese consumption but it has a high r value (0.947).

When performing correlation analysis: Test each variable for normality. Examine you data carefully. Formulate why you think the variable should or should not becorrelated before your analysis. Remember that sample size influences r.e.g. Small r values are important in large samples. Remember that correlation does not equal causation.

The four y variables have the same mean (7.5), standard deviation (4.12), correlation (0.81) and regression line (y 3 0.5. x). Pearson's correlation coefficient is a measure of the. intensity of the . . e.g. Small r values are important in large samples. Remember that correlation does not equal causation. Title: Slide 1