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Correlation and Regression, Alternatives to the Pearson Correlation -…
Correlation and Regression
Correlation: a statistical technique that is used to measure and describe the relationship between two variables.
Scores are identified as X and Y; and can be listed in a table or presented on a scatter plot.
If on a scatter plot, X variable is on the horizontal axis and Y on the vertical axis.
characteristics of the relationship between X and Y:
1-Direction of relationship: the sign of the correlation is either positive or negative.
2-Form of the relationship: the relationship tends to be linear, there are other forms of relationships that exist and there are special correlations used to measure them.
3-Strength or consistency of the relationship: the perfect correlation is always identified by a correlation of 1.00 and indicates a perfectly consistently relationship.There is a perfectly negative correlation of -1.0, there is a no linear trend of 0.00, and a relatively weak negative correlation with a value of about -.40.
Pearson Correlation: measures the degree and he direction the linear relationship between two variables. The Pearson correlation for a sample is identified by the letter r.
Sum of products of deviation (SP): used to measure the amount of coveriability between two variable.
Definitional formula: SP = L(X - Mx) (Y - My)
Computational formula: SP = ΣXY - ΣXΣY/n
Multiplying a negative constant is the only one to produce a mirror image of the pattern and changes the sign of the correlation.
X and Y value transforming into a zero score if a sample:
X and Y value transforming into a zero score if a population:
Where and why correlations are used:
1-Prediction: if two variables are known to be related in some systemic way, it is possible to use one of the variables to make accurate predictions about the other.
2-validity: correlation is one of the most common techniques for demonstrating validity.
3-reliability: a way to evaluate reliability is to use correlation to determine the relationship between two sets of measurements. When reliability is high, the correlation between two measurements should be strong and positive.
4-theory verification: the prediction of a theory could be tested by determining the correlation between two variables.
Correlation Interpretation:
1-correlation describes the relationship between two variables, it does not interpret cause and effect between two variables.
2-the values of a correlation can be affected by the range of scores represented in the data.
3-outliers have a dramatic effect on the values of a correlation
4- the numerical value does not explain the relationship between two variables. In order to interpret, r value needs to be squared to provide the total variability.
r square:
0.01 indicates a small correlation
0.09 indicates a medium correlation
0.25 indicates a large correlation.
Hypothesis (population):
Hypothesis (directional or one-tailed test):
Hypothesis test:
degrees of freedom: df=n-2
Regression: the statistical technique for finding the best-fitting straight line for a set of data.
Regression line: the resulting straight line.
Alternatives to the Pearson Correlation
Spearman correlation measures the relationship between two variables when both are measured on ordinal scales (ranks).
1-used when original data for X and Y are ranked
2-used when researcher wants to measure the degree to which the relationship between X and Y is consistently one-directional and independent of the specific form of the relationship.
-represented as rs
Point-Biserial Correlation: used to measure th relationship between two variables in situations in which one variable consists of regular, numerical scores, but the second variable only has two values
Phi-coefficient: when both variables x and y measured for each individual are dichotomous, the correlation between the two variables.
