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MARE 250Dr. Jason Turner
Correlation & Linear Regression
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Means Tests Vs. AssociationsMeans tests – t-test, ANOVA – test for differences between/among means (Responses among/between factors)
Associations – tests for relationships between/among variables (responses)
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Linear RegressionLinear regression investigates and models the linear relationship between a response (Y) and predictor(s) (X)Both the response and predictors are continuous variables (“Responses”)
Linear regression analysis is used to: - determine how the response variable changes as
a particular predictor variable changes
- predict the value of the response variable for any value of the predictor variable
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Regression vs. CorrelationLinear regression investigates and models the linear relationship between a response (Y) and predictor(s) (X)Both the response and predictors are continuous variables (“Responses”)
Correlation coefficient (Pearson) – measures the extent of a linear relationship between two continuous variables (“Responses”)
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When Regression vs. Correlation?Linear regression - used to predict relationships, extrapolate data, quantify change in one versus other is weighted direction
Correlation coefficient (Pearson) – used to determine whether there is a relationship or not
IF Regression – then it matters which variable is the Response (Y) and which is the predictor (X)
Y – (Dependent variable) X – (Independent)X causes change in Y (Y outcome dependent upon X)Y Does Not cause change in X (X –Independent)
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Linear RegressionRegression provides a line that "best" fits the data (from response & predictor)
The least-squares criterion (method used to draw this "best line“) requires that the best-fitting regression line is the one with the smallest sum of the squared error terms (the distance of the points from the line).
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Linear Regression
The R2 and adjusted R2 values represent the proportion of variation in the response data explained by the predictors
Adjusted R2 is a modified R2 that has been adjusted for the number of terms in the model. If you include unnecessary terms, R2 can be artificially high
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y
Is This Them? Are These They?
y = b0 + b1xy = dependent variable
b0 + b1 = are constants
b0 = y intercept
b1 = slope
x = independent variable
Urchin density = b0 + b1(salinity)
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Effects of OutliersOutliers may be influential observations
A data point whose removal causes the regression equation (line) to change considerably
Consider removal much like an outlier
If no explanation – up to researcher
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Warning on Regression
Regression is based upon assumption that data points are scattered about a straight line
What can we do to determine if a Regression is warranted?
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Correlation Coefficient (r)(Pearson) – measures the extent of a linear relationship between two continuous variables (responses)
Pearson correlation of cexa Ant and cexa post = 0.811P-Value = 0.000
IF p < 0.05 THEN the linear correlation between the two variables is significantly different than 0
IF p > 0.05 THEN you cannot assume a linear relationship between the two variables
Correlation Coefficient
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Correlation Coefficient
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“R2 D2 it is you, it is you”
Coefficient of Determination (R2) - Expression of the proportion of the total variability in the response (s) attributable to the dependence of all of the factors
R2 – used for assessing the “goodness of fit” of a regression model
Should use Adjusted R2 as it is a more conservative measure
R2 values range from 0 to 100%. An R2 of 100% means that all of the variability in the data can be explained by the model
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Coefficient Relationships
The coefficient of determination (r2) is the square of the linear correlation coefficient (r)
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Next Week
Regression Analysis: _ Urchins versus % Rock
The regression equation is_ Urchins = - 0.557 + 0.0361 % Rock
Predictor Coef SE Coef T PConstant -0.5569 0.3820 -1.46 0.146% Rock 0.036116 0.0062 5.80 0.000
S = 3.27363 R-Sq = 11.0% R-Sq(adj) = 10.6%