How to Tell Which Way Correlation in Regression Continuous Dichotomous
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Correlation and Regression
Correlation andregression analyses are used to determine how much of the variation in one variable is explained by variation in the other (correlation), and how to predict one variable (the response) from the other (predictor) using an equation (regression). These are two of the most widely used statical analyses in biology. The basic assumptions of these analyses are that the response and predictor variables are normally distributed, and that the relation between them is linear (falls along a straight line). When the correlation coefficient (r is large (close to –1.0 or +1.0) then we say the correlation is strong and thus that the prediction of the response from the predictor is quite accurate. When the slope of the regression line is 1.0 then each increment in the predictor results in an equal increment in the response. Usually correlation and regression analyses are run simultaneously, and it would be unusual to report the regression results without the correlation results. Correlation results often appear alone, however, because we may not be interested in actually making a prediction or knowing the functional relation (slope) between the two variables.
To perform these analyses online you can enter your raw data:
- here for lots of detailed output and even some graphs
- here for a simple regression calculator that gives minimal output (with no correlation) and a simple graph (enter data with one sample per line, x then y separated by a space)
- here and here for some really nice output and detailed explanations for both regression and correlation. The first site allows you to enter data one at a time online and the second allows you to upload or paste your own data file into the calculator
A worked example
small mammals
Dr Chris Moyes has measured the body mass and COX enzyme activity in a sample of individuals from 12 species of small mammals. He calculated the mean body mass and enzyme activity for each species, and log-transformed both variables to make the distributions closer to normal. He then ran a correlation and a regression analysis and got the following results:
- r = –0.72 (P = 0.009)
- a = –0.01 (P = 0.85)
- b = –0.16 (P = 0.009)
wherer is the correlation coefficient and can vary between –1 and +1,a is the intercept of the regression line, andb is the slope of that line. The P-values tell us that r and b are significantly different from zero (note that both always have the same P), and that the intercept is not significantly different from zero, where P≤0.05 is considered to be statistically significant.
You can see from the graph (and the slope) that COX activity decreases with larger body sizes. By squaring the r-value we get an estimate of the amount of variation in the response variable (logCOX) explained by variation in the predictor variable (logMASS). Here r-squared is 0.52, meaning that 52% of the variation in COX activity is explained by body mass.
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Source: https://biology.queensu.ca/academics/undergraduate/resources-for-courses/analyzing-data/correlation-and-regression/
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