Correlation

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Correlation / regression

• Correlation
• Regression
• Multiple Regression
• Curve fitting

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Correlation

• Represents the relationship between two
  measurements
• Examples height and weight, education level and
  income, BMI and skin fold thickness, wealth and
  fertility
• Correlation does not represent one causing the
  other, usually is present if both measurements
  are influenced by a common factor
• The value is from -1 to 1
• 0 no relationship
• 1 perfect relationship
• -1 perfect inverse relationship

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Confidence interval for r

• Correlation is not a linear measurement
• It stretches near 0 and compresses neat 1 or -1
• It has to be
• Transformed into a normally distributed linear
  measurement
• Have Standard Error estimated
• Have CI estimated
• Transformed back to the original format

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Sample size

• Iterative procedure that satisfy two equations

Where Za z value for Type I error zb z value
for Type II error
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Correlation / regression

• Correlation
• Regression
• Multiple Regression
• Curve fitting

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Regression

• Draw a line which best fits the relationship
  between x and y
• The line takes the form y a bx
• Where a is the y value when x0
• Where b is the slope of the line, or how much y
  changes for one unit of change in x
• It assumes that y is dependent on x
• It explains how changes in y values are governed
  by changes in x values
• It allows x to predict y
• Note x a by is not the mirror image of ya
  bx, as how best fit is calculated differs

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Regression - example
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RegressionBest fit ya bx
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RegressionBest fit xa by
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Confidence interval for b
t Students t for sample size and Type I Error
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Confidence interval for predicted y

• SE 2 components and changes with x value
• SE of regression slope b
• SE of departure from residual variation

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Confidence interval for predicted y
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Correlation / regression

• Correlation
• Regression
• Multiple Regression
• Curve fitting

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Multiple Regression

• Outcome, particularly clinical outcome
• Are subjected to multiple influences
• All of which are related to each other
• Multiple regression model is therefore commonly
  needed

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• BMI is influenced by mother and grandparents, but
• People who married tend to have comparable BMI
• Parents BMI tend to be dependent influenced by
  grandparents
• Multiple regression y a b1x1 b2x2 b3x3
  bixi

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Multiple Regression

• Starts with a matrix of Sum/products Sk,k from
  k measurements
• where Si,j is the Sxy between any pair I and j
• Where Si,i is the SSqx of variable i
• This matrix is inverted V S-1
• The Partial Regression Coefficient bi

• The constant a

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Correlation / regression

• Correlation
• Regression
• Multiple Regression
• Curve fitting

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Curve fit

• In cases where the relationship between x and y
  are not linear
• y function(x)
• y Log(x)
• y sine(x)
• Polynomial curve fit
• A special case of multiple regression
• Will fit into any shape where y increases with x
• y a b1x b2x2 b3x3 ..bkxk
• In most biological systems fitting to the power
  of 3 is sufficient

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Polynomial curve fit
Data point
y a bx
y a b1x b2x2 b3x3
y
y a b1x b2x2
x
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CI of polynomial fit

• Complexity of calculating Standard Error
• Summing of each individual coefficients
• Residual
• Solution 2 stage procedure
• Do polynomial curve fit
• Calculate error (distance between each datapoint
  from the regression line)
• Curve fit error

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y 14.45 16.66x 5.83x2 0.45x3
SD 0.29 0.18x
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Curve fitFemur length according to gestational
age
Femur length (cms)
Gestation (days)