Introduction to Biostatistics and Bioinformatics Regression and Correlation

Learning Objectives Regression – estimation of the relationship between variables Linear regression Assessing the assumptions Non-linear regression

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

Relationships Relationship No Relationship

Relationships Linear RelationshipsNon-Linear Relationship

Relationships Linear, StrongLinear, Weak

Linear Regression Linear, StrongLinear, WeakNon-Linear

Linear Regression - Residuals Linear, StrongLinear, WeakNon-Linear Residuals

Linear Regression Model Linear component Intercept Slope Random Error Dependent Variable Independent Variable Random Error component

Linear Regression Assumptions The relationship between the variables is linear.

Linear Regression Assumptions The relationship between the variables is linear. Errors are independent, normally distributed with mean zero and constant variance.

Linear Regression Assumptions LinearNon-Linear Residuals

Linear Regression Assumptions Constant VarianceVariable Variance Residuals

Linear Regression Model Linear component Intercept Slope Random Error Dependent Variable Independent Variable Random Error component

Linear Regression – Estimating the Line Estimated Intercept Estimated Slope Estimated Value Independent Variable

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.

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.

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.

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.

Linear Regression in Python import scipy.stats as stats slope,intercept,r_value,p_value,std_err = stats.linregress(x,y)

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')

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

Linear Regression Example Outlier

Regression – Non-linear data Solution 1: Transformation Solution 2: Non-linear Regression

Correlation Coefficient A measure of the correlation between the two variables Quantifies the association strength Pearson correlation coefficient:

Correlation Coefficient

Source: Wikipedia

Coefficient of Variation Variance Sample Mean Coefficient of Variation (CV)

Correlation Coefficient and CV Uniform distribution

Correlation Coefficient and CV Uniform distributionNormal distributionLognormal distribution

Correlation Coefficient - Outliers Outlier

Correlation Coefficient – Non-linear Solutions: Transformation Rank correlation (Spearman, r=0.93)

Correlation Coefficient and p-value Hypothesis: Is there a correlation? r rr p pp

Application: Analytical Measurements Theoretical Concentration Measured Concentration

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.

Limit of Detection and Linearity Theoretical Concentration Measured Concentration

Precision and Accuracy Theoretical Concentration Measured Concentration

Summary - Regression Source:

Summary - Correlation

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