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Regression & Correlation Analysis of Biological Data Ryan McEwan and Julia Chapman Department of Biology University of Dayton ryan.mcewan@udayton.edu

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Simple linear regression is a standard technique in the Analysis of Biological Data: The main idea is assessing the relationship between two variables, assuming that the relationship is direction and linear…and assuming that one variable is a driver of the relationship. The Response variable (plotted on X) is assumed to respond in a linear relationship to changes in the Predictor variable (plotted on Y). The reverse is not assumed in this analysis (that X drives Y). Think heart rate and exercise. Other examples?

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But if you have a cloud of points…where do you put the line?

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Best fit lines & “Least Squares” regression The idea is to drive the line through the cloud in the area that minimizes the distance between the points and the line.

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Regression residuals You can generate a table of residuals.. a new data set! How much does each point deviate from the regression line?

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Detrending… a scientific siren song

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Regression lines can have varying slopes from a single Y intercept.

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Regression lines can have identical slopes, but different Y intercepts.

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We will be running a test of this sort in R. The thing I want to you to understand is that the statistical test…. The P-value generated… relates to the null hypothesis of NO SLOPE. That the line is indeed flat. That would mean the response variable is NOT changing in relation to the predictor.

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…ruut row…

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IMPORTANT! The P-value from a regression, tells you whether the line is statistically flat….it does not tell you how much variation is captured!

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It may be more useful to calculate a confidence interval

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You might wish to have replicate values

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Your relationship might not be linear! Polynomial Regression

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Regression Diagnostics! A stepwise process of adding factors to the regression. Testing P value, r 2, etc. If you are going to take this on, you need to grind! Read, analyze, read some more

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Correlation is a related form of analysis, but is different in one fundamental way…a correlation is testing for a relationship between two factors, but NOT ASSUMING one causes the other. Thus, no predictor and response

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You would use a correlation analysis if you are not making assumptions about one factor driving another. Pearson correlation for normally distributed data Spearman (rank) correlation for non normally distributed data.

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Logistic regression: To be used if your data are categorical……

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Caution 1: Correlation is not causation!

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Extrapolation is dangerous!!

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