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Applied Biostatistics
Correlation and regression
Martin Bland
Professor of Health Statistics University of York http://www-users.york.ac.uk/~mb55/msc/
Correlation
Example: Muscle strength and height in 42 alcoholics
A scatter diagram:
How close is the relationship? Correlation: measures closeness to a linear relationship.
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Quadriceps strength (N) 155 160 165 170 175 180 Height (cm)
Correlation coefficient
Subtract means from observations and multiply.
Sum of products about the means. Like the sum of squares about the means used for measuring variability.
Mean strength
Mean height
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Muscle strength (newtons) 155 160 165 170 175 180 Height (cm)
Correlation coefficient
Subtract means from observations and multiply.
Products in top right and bottom left quadrants positive.
Mean strength
Mean height +ve
+ve 100
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400
500
Muscle strength (newtons) 155 160 165 170 175 180 Height (cm)
Correlation coefficient
Subtract means from observations and multiply.
Products in top right and bottom left quadrants positive. Products in top left and bottom right quadrants negative.
Mean strength
Mean height +ve
+ve
-ve
-ve 100
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300
400
500
Muscle strength (newtons) 155 160 165 170 175 180 Height (cm)
Correlation coefficient
Subtract means from observations and multiply.
Sum of products positive. Correlation positive.
Mean strength
Mean height +ve
+ve
-ve
-ve 100
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Muscle strength (newtons) 155 160 165 170 175 180 Height (cm)
Correlation coefficient
Divide sum of products by square roots of sums of squares. Correlation coefficient, denoted by r. Maximum value = 1.00. Minimum value = โ1.00.
r = 0.42. Positive correlation of fairly low strength
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Muscle strength (newtons) 155 160 165 170 175 180 Height (cm)
Correlation coefficient
Divide sum of products by square roots of sums of squares. Correlation coefficient, denoted by r. Maximum value = 1.00. Minimum value = โ1.00.
r = โ0.42. Negative correlation of fairly low strength.
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Muscle strength (newtons) 20 30 40 50 60 70 Age (years)
Correlation coefficient
Positive when large values of one variable are associated with large values of the other.
Variable Y
Variable X
r = 0.
Correlation coefficient
Positive when large values of one variable are associated with large values of the other.
Variable Y
Variable X
r = 0.
Correlation coefficient
Negative when large values of one variable are associated with small values of the other.
Variable Y
Variable X
r = -0.
Correlation coefficient
Negative when large values of one variable are associated with small values of the other.
Variable Y
Variable X
r = -0.
Correlation coefficient
r = 0.00 when there is no linear relationship.
Variable Y
Variable X
r = 0.
Correlation coefficient
It is possible for r to be equal to 0.00 when there is a relationship which is not linear.
Variable Y
Variable X
r = 0.
Correlation coefficient
We can test the null hypothesis that the correlation coefficient in the population is zero. Simple t test, tabulated. Assume: independent observations, one of the variables is from a Normal distribution. Large deviations from assumption P very unreliable.
r = 0.42, P = 0.006. Easy to do, simple tables. Computer programmes almost (^100) always print this.
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Quadriceps strength (N) 155 160 165 170 175 180 Height (cm)
Correlation coefficient
We can find a confidence interval for the correlation coefficient in the population. Fisherโs z transformation. Assume: independent observations, both of the variables are from a Normal distribution. Large deviations from assumption CI very unreliable.
r = 0.42, approximate 95% confidence interval: 0.13 to 0. Tricky, approximate. Computer programmes rarely print this.
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Quadriceps strength (N) 155 160 165 170 175 180 Height (cm)
Regression
Example: Muscle strength and height in 42 alcoholics
What is the relationship? Regression: predict strength from observed height.
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Quadriceps strength (N) 155 160 165 170 175 180 Height (cm)
Simple Linear Regression
Example: Muscle strength and height in 42 alcoholics What is the relationship? Regression: predict strength from observed height. What is the mean strength for people with any given observed height? Strength is the outcome , dependent , y , or left hand side variable. Height is the predictor , explanatory , independent , x , or right hand side variable. Linear relationship: strength = intercept + slope ร height Equation of a straight line.
Simple Linear Regression
Which straight line should we choose?
Minimise the sum of the squares of these differences. Principle of least squares, least squares line or equation.
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Quadriceps strength (N) 155 160 165 170 175 180 Height (cm)
Simple Linear Regression
Strength = โ908 + 7.20 ร height
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Quadriceps strength (N) 155 160 165 170 175 180 Height (cm)
Simple Linear Regression
Strength = โ908 + 7.20 ร height
The intercept and slope are call coefficients. The slope of the line is sometimes called the regression coefficient.
These coefficients have units, unlike the correlation coefficient. They can take any value.
Strength in newtons = โ908 newtons + 7.20 newtons per cm ร height in cm
We can find confidence intervals and P values for the coefficients subject to assumptions.
Simple Linear Regression
We can find confidence intervals and P values for the coefficients subject to assumptions. Deviations from line should have a Normal distribution with uniform variance.
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Quadriceps strength (N) 155 160 165 170 175 180 Height (cm)
Simple Linear Regression
We can find confidence intervals and P values for the coefficients subject to assumptions. Slope = 7.20, 95% CI = 2.15 to 12.25 newtons/cm, P=0.006 against zero. Intercept = โ908, 95% CI = โ45 to โ1771 newtons.
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Quadriceps strength (N) 155 160 165 170 175 180 Height (cm)
Simple Linear Regression
Assumptions: independent observations, deviations from line should have a Normal distribution with uniform variance. Calculate the deviations or residuals, observed minus predicted. Check Normal distribution: Check uniform variance:
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Frequency
Strength residual^ -200^0
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Residual strength
-200Inverse Normal 0 200
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Strength residual
155 160 165 170 175 180 height
