6. CORRELATION SCATTER PLOTS. PEARSON S CORRELATION COEFFICIENT: Definition

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1 6. CORRELATION Scatter plots Pearson s correlation coefficient (r ). Definition Hypothesis test & CI Spearman s rank correlation coefficient rho (ρ) Correlation & causation Misuse of correlation Two techniques are commonly used to examine the association between 2 quantitative variables: Example: Blood concentration of GSH & % ideal body weight of children with cystic fibrosis 1) Correlation do 2 variables covary (are they interdependent)? - what is the direction of the relationship between the 2 variables and or and - what is the strength of the relationship? 2) Regression Prediction: knowing the value of one variable, can the value of the other be predicted? The 2 techniques have a close mathematical relationship but are used to offer different questions. The 1 st step when examining the possible relation between 2 quantitative variables is the creation of a scatter plot or scattergram. Example 1. Measurements of «% ideal body weight», FEV and GSH in a sample of children with cystic fibrosis. SCATTER PLOTS *lower levels of GSH are related to increased oxidative stress in the lungs. L Lands et al (1999) «Lymphocyte Glutathione Levels in Children With Cystic 4 Fibrosis» Chest SPSS 15.0 : Graphs Legacy Dialogs Scatter/Dot By looking at a scatter plot, one can tell: 1) Whether or not the relationship appears to be linear 2) If there are outliers that may affect the estimate of the correlation PEARSON S CORRELATION COEFFICIENT: Definition 5 1

2 Pearson s correlation coefficient r ranges from 1 to 1. The correlation coefficient shows the extent to which two quantitative variables are linearly related. Perfect negative correlation. r = -1 Perfect positive correlation. r = 1 2 variables are negatively correlated when one decreases as the other increases (r<0). 2 variables are positively correlated when the variables increase together (r>0). Complete lack of a linear correlation. r = 0 If r=0 then there is no linear correlation. 7 8 If we wish to label the strength of the association, for absolute values of r, is regarded as very weak, as weak, as moderate, as strong and as very strong correlation, but these are rather arbitrary limits, and the context of the results should be considered. (Swiscow, «Statistics at Square 1») Scattergrams of 100 observations where both variables have a normal distribution a) r = 0, b) r = 0.3, c) r = 0.6 d) r = Reproduced from Rice (page 128). NOTE The value of r depends on the variability of the data ( restriction in range ). With a reduced range, r is expected to be lower. 10 Pearson s correlation coefficient is calculated using the following formula: mean%βw = where x i and y i are the values for subject i (i=1,2,..,n). Mean GSH = 1.3 Can be calculated for any pair of continuous variables (but only useful when the variables appear to have a linear relationship). + - The denominator keeps r between 1 and

3 Example 1. (cont) Example 1 (cont). Scatter plot of GSH with % ideal body weight. r=0.49 Numerator {The authors calculated Pearson s r but we will see that it is not appropriate here} Denominator 176,1= (5,2*5943,2) 13 L Lands et al (1999) «Lymphocyte Glutathione Levels inchildren With Cystic 14 Fibrosis» Chest Non-linear correlation. If the variables are correlated but not linearly DON T calculate r! An outlier: r = r = 0! (y=x 2 ) Removing one observation The correlation coefficient should always be calculated in r = combination with a graphical representation of the correlation PEARSON S CORRELATION COEFFICIENT: Hypothesis test & CI General procedure in hypothesis testing 1. Form the null (Η 0 ) and alternative hypotheses. 2. Check the assumptions of the test 3. Define the significance level (α) 4. Calculate the test statistic that corresponds to H0. 5. Refer the test statistic to a known distribution that it would follow if the null hypothesis were true. 6. Calculate the probability of a value of the test statistic arising which is as or more extreme than that observed, if the null hypothesis were true. Hypothesis test for r H 0 : the population correlation coefficient is 0 Assumptions : 1. At least one variable has a normal distribution 2. There are no outliers that influence r. SPSS : Analyse Correlate Bivariate 18 3

4 Test statistic: t = Compared to a t distribution with n 2 d.f. r 1 r 2 n 2 = r n 2 1 r 2 Example 1 (cont). Measurements of «% ideal body weight», FEV and GSH in a sample of children with cystic fibrosis. Example 2. Scatter plot of the incidence rates of ulcerative colitis and Crohn s disease in 52 regions in Mannitoba, Canada JF Blanchard et al (2001) Am J Epidemiol r = 0.49, p< r = -0.45, p<0.05 There is strong evidence of a moderate positive correlation between UC and CD incidence rates in Canada (r=0.49, p<0.001) Statistical significance does not necessarily imply a close relationship. A statistically significant result (eg p=0.003), does not tell us much about the strength of the correlation. Even if the correlation is weak (e.g. r = 0.1), when the sample is large enough (eg 1000), the result could be statistically significant. A CI can be found but it is relatively complicated to do so. CI s are not usually presented. ASSUMPTION: Both variables are normally distributed EXAMPLE 3. Scatter plot of ear length against age in 400 Japanese men.. Fig 1--Scatter plot of ear length divided by height against age SPEARMAN S RANK CORRELATION COEFFICIENT Asai, Y. et al. BMJ 1996;312:582c Copyright 1996 BMJ Publishing Group Ltd. 23 4

5 EXAMPLE 1 (cont). The authors chose to present Pearson s r (r=0.49, p<0.05). What can be done if the assumptions for the validity of the hypothesis test do not hold? Scatter plot of GSH with % ideal body weight. If these 2 points are removed, r=0.15 (p=0.55, n=18). 1) Transformation of one of the variables (or both). 2) Calculation of Spearman s non-parametric correlation coefficient, ρ. {Why not always use this? Because the parametric method theoretically has greater power} These 2 outliers have a strong influence on the estimate of r. It does not help that the sample size is small. 25 SPSS : Analyse Correlate Bivariate (tick the Spearman box). 26 EXAMPLE 1. (cont) The procedure for calculating the correlation coefficient is the same, but the ranks of the observations are used instead of the actual values. 27 r=0.3 (n=20, p>0.2) 663.5= (664*663) 28 Correlation is not causation CORRELATION & CAUSATION e.g. 1. Consumption of «fast food» and the frequency of divorces in Crete, a positive correlation! 2. No. of churches and the number of reported burglaries in 50 defined regions in Greece 30 5

6 Correlation is not causation MISUSE OF CORRELATION Scattergram of the correlation between fibre intake and 25- year mortality from colorectal cancer in countries participating in the Seven Countries Study {reproduced from Figure 2 in Jansen et al, IJC, 1999}. [Ecological study] 31 Correlation should not be used Misuse of correlation 1) For the comparison of 2 methods of measurement. The correlation coefficient is a measure of association. Here, a measure of agreement is needed. The correlation coefficient α) ignores possible systematic bias. e.g. if we add 10 to all the values, r is unchanged. We have perfect correlation (r=1 ή -1) if all observations lie on a straight line BUT the 2 methods will give the same results only if the points lie on y=x β) r depends on the dispersion of the points. If the sample is spilt into 2 according to whether reuoc <0 or not, we find : Correlation should not be used: 2) When the sample contains subgroups whose characteristics are known to differ. When the range is wide, r is greater than when there is a small range

7 EXAMPLE 4. Association of serum leptin concentration with body mass index and waist circumference in non-diabetic and diabetic men and women Copyright 1996 BMJ Publishing Group Ltd. Zimmet, P. et al. BMJ 1996;313: