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Introduction to Biostatistics and Bioinformatics Regression and Correlation
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Learning Objectives Regression – estimation of the relationship between variables Linear regression Assessing the assumptions Non-linear regression
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Learning Objectives Regression – estimation of the relationship between variables Linear regression Assessing the assumptions Non-linear regression Correlation Correlation coefficient quantifies the association strength Sensitivity to the distribution
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Relationships Relationship No Relationship
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Relationships Linear RelationshipsNon-Linear Relationship
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Relationships Linear, StrongLinear, Weak
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Linear Regression Linear, StrongLinear, WeakNon-Linear
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Linear Regression - Residuals Linear, StrongLinear, WeakNon-Linear Residuals
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Linear Regression Model Linear component Intercept Slope Random Error Dependent Variable Independent Variable Random Error component
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Linear Regression Assumptions The relationship between the variables is linear.
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Linear Regression Assumptions The relationship between the variables is linear. Errors are independent, normally distributed with mean zero and constant variance.
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Linear Regression Assumptions LinearNon-Linear Residuals
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Linear Regression Assumptions Constant VarianceVariable Variance Residuals
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Linear Regression Model Linear component Intercept Slope Random Error Dependent Variable Independent Variable Random Error component
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Linear Regression – Estimating the Line Estimated Intercept Estimated Slope Estimated Value Independent Variable
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Least Squares Method Find slope and intercept given measurements X i,Y i, i=1..N that minimizes the sum of the squares of the residuals.
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Least Squares Method Find slope and intercept given measurements X i,Y i, i=1..N that minimizes the sum of the squares of the residuals.
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Least Squares Method Find slope and intercept given measurements X i,Y i, i=1..N that minimizes the sum of the squares of the residuals.
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Least Squares Method Find slope and intercept given measurements X i,Y i, i=1..N that minimizes the sum of the squares of the residuals.
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Linear Regression in Python import scipy.stats as stats slope,intercept,r_value,p_value,std_err = stats.linregress(x,y)
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Linear Regression Example Linear, Strong Residuals x=np.linspace(-1,1,points) y=x+0.1*np.random.normal(size=points) slope,intercept,r_value,p_value,std_err = stats.linregress(x,y) y_line=slope*x+intercept fig, (ax1) = plt.subplots(1,figsize=(4,4)) ax1.scatter(x,y,color='#4D0132',lw=0,s=60) ax1.set_xlim([-1.5,1.5]) ax1.set_ylim([-1.5,1.5]) ax1.plot(x,y_line,color='red',lw=2) fig.savefig('linear.png') fig, (ax1) = plt.subplots(1,figsize=(4,4)) ax1.scatter(x,y-y_line, color='#963725',lw=0,s=60) ax1.set_xlim([-1.5,1.5]) ax1.set_ylim([-1.5,1.5]) fig.savefig('linear-residuals.png')
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Linear Regression Example x=np.linspace(-1,1,points) y=x+0.4*np.random.normal(size=points) slope,intercept,r_value,p_value,std_err = stats.linregress(x,y) y_line=slope*x+intercept fig, (ax1) = plt.subplots(1,figsize=(4,4)) ax1.scatter(x,y,color='#4D0132',lw=0,s=60) ax1.set_xlim([-1.5,1.5]) ax1.set_ylim([-1.5,1.5]) ax1.plot(x,y_line,color='red',lw=2) fig.savefig('linear-weak.png') fig, (ax1) = plt.subplots(1,figsize=(4,4)) ax1.scatter(x,y-y_line, color='#963725',lw=0,s=60) ax1.set_xlim([-1.5,1.5]) ax1.set_ylim([-1.5,1.5]) fig.savefig('linear-weak-residuals.png') Linear, Weak Residuals
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Linear Regression Example Outlier
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Regression – Non-linear data Solution 1: Transformation Solution 2: Non-linear Regression
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Correlation Coefficient A measure of the correlation between the two variables Quantifies the association strength Pearson correlation coefficient:
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Correlation Coefficient
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Source: Wikipedia
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Coefficient of Variation Variance Sample Mean Coefficient of Variation (CV)
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Correlation Coefficient and CV Uniform distribution
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Correlation Coefficient and CV Uniform distributionNormal distributionLognormal distribution
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Correlation Coefficient - Outliers Outlier
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Correlation Coefficient – Non-linear Solutions: Transformation Rank correlation (Spearman, r=0.93)
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Correlation Coefficient and p-value Hypothesis: Is there a correlation? r rr p pp
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Application: Analytical Measurements Theoretical Concentration Measured Concentration
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A Few Characteristics of Analytical Measurements Accuracy: Closeness of agreement between a test result and an accepted reference value. Precision: Closeness of agreement between independent test results. Robustness: Test precision given small, deliberate changes in test conditions (preanalytic delays, variations in storage temperature). Lower limit of detection: The lowest amount of analyte that is statistically distinguishable from background or a negative control. Limit of quantification: Lowest and highest concentrations of analyte that can be quantitatively determined with suitable precision and accuracy. Linearity: The ability of the test to return values that are directly proportional to the concentration of the analyte in the sample.
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Limit of Detection and Linearity Theoretical Concentration Measured Concentration
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Precision and Accuracy Theoretical Concentration Measured Concentration
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Summary - Regression Source: http://xkcdsw.com/content/img/2274.png
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Summary - Correlation
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Next Lecture: Experimental Design & Analysis Experimental Design by Christine Ambrosino www.hawaii.edu/fishlab/Nearside.htm
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