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Chapter 14 - Correlation and Regression - Coggle Diagram
Chapter 14 - Correlation and Regression
14-1 Introduction
The Characteristics of a Relationship
correlation
- is a numerical value that describes and measures 3 characteristics of the relationship between X and Y
Direction of the relationship: sign of a correlation (+ or -) describes the direction of the relationship
positive correlation - 2 variables tend to change in the same direction; as X increases, Y increases; as X decreases, the Y decreases
negative correlation - 2 variables tend to go in opposite directions; as X increases, Y decreases
Form of the Relationship - tend to have a linear form
The Strength or Consistency of the Relationship - every time X increases by one point, Y also changes by a consistent and predictable amount
perfect correlation
- correlation of 1.00 and indicates a perfectly consistent relationship
envelope
- line sketched around the data points, to help you see the overall trend in the data
14-2 The Pearson Correlation
The Sum of Products of Deviations
Calculation of the Pearson Correlation
Correlation and the Pattern of Data Points
adding a constant to (or subtracting a constant from) each X and/or Y does not change the pattern data points and does not change the correlation
multiplying (or dividing) each X or Y by a positive constant does not change the pattern and does not change the value of the correlation
multiplying by a negative constant produced a mirror image of the pattern and changes the sign of the correlation
The Pearson Correlation and z-Scores
14-3 Using and Interpreting the Pearson Correlation
Where and Why Correlations are Used
Prediction - possible to use one known variable to make accurate predictions about the other
Validity
Reliability -
Theory Verification - prediction of theory can be tested by determining the correlation between the variables
Interpreting Correlations
Correlation describes a relationship between two variables
Value of a correlation can be affected greatly by the range of scores represented in the data
one or two extreme data points (outliers) can have a dramatic effect on the value of a correlation
to describe how accurately one variable predicts the other, you must square the correlation
Correlation and Causation
correlation does not establish causation
to establish a cause and effect relationship, conduct a true experiment
Correlation and Restricted Range
correlation does not represent the full range of possible values
restricted range
should not generalize any correlation beyond the range of data represented in the sample
Outliers
- an individual with X and/or Y values that are substantially different (larger or smaller) from the values obtained for the other individuals in the data set
Correlation and the Strength of the Relationship
square correlation provides a better measure of strength in the relationship - measure the proportion of variability in the data explained by relationship between X & Y (
coefficient of determination
)
measures proportion of variability in one variable the can be determined from the relationship with the other variable
14-4 Hypothesis Tests with the Pearson Correlation
Purpose of hypothesis test is to decide between:
there is no correlation in the population
the nonzero sample correlation accurately represents a real, nonzero correlation in the population
Standard error for r:
t statistic -
14-5 Alternatives to the Pearson Correlation
The Spearman Correlation
measures the consistency of direction in the relationship between X and Y; measure degree to which the relationship is one-directional, or monotonic
computed by: (1) ranking the X and Y scores separately (2) computing the Pearson correlation using the ranks
Ranking Tied Scores
when two (or more) scores are tied, compute the mean of their ranked positions, and assign this mean value as the final rank for each score
Special Formula for the Spearman Correlation
The Point-Biserial Correlation and Measuring Effect Size with r2
: used to measure the strength of the relationship when one of the two variables is dichotomous (coded using values of 0 and 1)
The Phi-Coefficient
correlation between the variables, when X and Y are dichotomous
computed by: (1) convert each dichotomous variable to numerical values by assigning a 0 to one category and a 1 to the other category for each othe variables (2) use the regular formula with the converted scores
14-6 Introduction to Linear Equations and Regression
Linear Equations
b = slope; determines how much the Y variable changes when X is increased by one point
y-intercept - value of a ; determines value of Y when X = 0
Regression
statistical technique for finding the best-fitting straight line for a set of data
regression lin
e - resulting straight line from use of regression technique
goal is to find the best-fitting straight line for a set of data
least-squared-error solution: best fitting line that has the smallest total squared error
total squared error =
Regression equation for y
can be used for prediction
The Standard Error of Estimate
computed using the residual variability; provides a measure of the standard distance (or error) between the predicted y values on the line and the actual data points.
Analysis of Regression: The Significance of the Regression Equation
can use an F-ration to evaluate the significance of the regression equation;
determines whether the equation predicts a significant portion of the variance for the Y scores
