So, height is just one determinant and is a contributing factor, but not the only determinant of BMI. For example, body mass index (BMI) is determined by multiple factors ("exposures"), such as age, height, sex, calorie consumption, exercise, genetic factors, etc. How can a correlation be weak, but still statistically significant? Consider that most outcomes have multiple determinants. There is quite a bit of scatter, but there are many observations, and there is a clear linear trend. It suggests a weak (r=0.36), but statistically significant (p<0.0001) positive association between age and systolic blood pressure. The scatter plot below illustrates the relationship between systolic blood pressure and age in a large number of subjects. The four images below give an idea of how some correlation coefficients might look on a scatter plot. Also, keep in mind that even weak correlations can be statistically significant, as you will learn shortly. The table below provides some guidelines for how to describe the strength of correlation coefficients, but these are just guidelines for description. 0.2917043 Describing Correlation Coefficients For example, we could use the following command to compute the correlation coefficient for AGE and TOTCHOL in a subset of the Framingham Heart Study as follows: Instead, we will use R to calculate correlation coefficients. You don't have to memorize or use these equations for hand calculations. Where Cov(X,Y) is the covariance, i.e., how far each observed (X,Y) pair is from the mean of X and the mean of Y, simultaneously, and and s x 2 and s y 2 are the sample variances for X and Y. Nevertheless, the equations give a sense of how "r" is computed. We will use R to do these calculations for us. However, you do not need to remember these equations. The equations below show the calculations sed to compute "r". The scatter plot suggests that measurement of IQ do not change with increasing age, i.e., there is no evidence that IQ is associated with age.Ĭalculation of the Correlation Coefficient Possible values of the correlation coefficient range from -1 to +1, with -1 indicating a perfectly linear negative, i.e., inverse, correlation (sloping downward) and +1 indicating a perfectly linear positive correlation (sloping upward).Ī correlation coefficient close to 0 suggests little, if any, correlation. 2024.Īll rights reserved.The sample correlation coefficient (r) is a measure of the closeness of association of the points in a scatter plot to a linear regression line based on those points, as in the example above for accumulated saving over time. Outliers can badly affect the product-moment correlation coefficient, whereas other correlation coefficients are more robust to them. An individual observation on each of the variables may be perfectly reasonable on its own but appear as an outlier when plotted on a scatter plot. If the association is nonlinear, it is often worth trying to transform the data to make the relationship linear as there are more statistics for analyzing linear relationships and their interpretation is easier thanĪn observation that appears detached from the bulk of observations may be an outlier requiring further investigation. The wider and more round it is, the more the variables are uncorrelated. The narrower the ellipse, the greater the correlation between the variables. If the association is a linear relationship, a bivariate normal density ellipse summarizes the correlation between variables. The type of relationship determines the statistical measures and tests of association that are appropriate. Other relationships may be nonlinear or non-monotonic. When a constantly increasing or decreasing nonlinear function describes the relationship, the association is monotonic. When a straight line describes the relationship between the variables, the association is linear. If there is no pattern, the association is zero. If one variable tends to increase as the other decreases, the association is negative. If the variables tend to increase and decrease together, the association is positive.
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