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14 Correlation(1)

# 14 Correlation(1) - Biol 214 Statistics for Biology Majors...

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Biol 214 Statistics for Biology Majors Correlation November 29, 2011 1 John J. Kopecky, Ph.D. Fall 2011

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Correlation What have we done so far? 1) Compared “categorical” variable and “continuous” variable using t-tests. For example, height and sex, or concentration of dopamine and disease 2) Compared “categorical” variable and “categorical” variable: contingency tables. For example, disease and risk of dying. What are we going to do now? 1) Compare a “continuous” variable with another continuous variable. 2) There are several different ways of doing this. The first is correlation. 2
Correlation Correlation. A) Simply, we have two continuous variables, neither of which we consider a “predictor” or “independent”. We just want to figure out, for example, if one variable goes up, does the other go up? down? not change? 3

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Correlation 4 Scatter Plot
Correlation 1. Graphical description: 5

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Correlation 2) Problem - just describing the graphs is subjective, so we describe the correlation using what is called a correlation coefficient . 3) The correlation coefficient is designated by the Greek letter “rho”, and is estimated by “r”. In other words: r -> 6
Correlation B)Calculation of “r” 1) Here's the formula: 1 22 11 ( )( ) ( ) ( ) n ii i nn x x y y r x x y y    7

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Correlation Correlation Coefficient: 8
Correlation 2) How does it work? For the numerator: a) first notice that the coordinates for and are somewhere inside our points on the graph b) if x i < , and y i < , that implies that the point is below and to the left of ( , ), and for that value of i, the numerator is positive (you’re multiplying two negative numbers) y y x x x y 9 1 22 11 ( )( ) ( ) ( ) n ii i nn x x y y r x x y y   

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Correlation c) similarly, if x i > , and y i > , the point is above and to the right, and the numerator is positive. - so if all the points pair up below and to the left and above and to the right, r will be positive (you’re adding up a bunch of positive numbers. 10 x y
Correlation d) now what happens if x i < and y i > ? The point is now above and to the left of ( , ), and the numerator is negative (you’re multiplying a (-) and (+) number). e) similarly, if x i > and y i < , the point is below and to the right, and again the numerator is negative - if all the points are above to the left and below to the right of ( , ), r will be negative. Also, - if you have a mix of points, then it depends on where most of the points are, and how far away from ( , ) they are.

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14 Correlation(1) - Biol 214 Statistics for Biology Majors...

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