Correlation Flashcards
Correlation….
measures and describes how two variables relate to one another
Three characteristics of correlation
Direction (+/-)
Form (linear)
Strength (0-1)
Pearson correlation (r)
Measures the degree and direction of the linear relationship of the 2 variables
Perfect linear relationship (change in x has a corresponding change in y)
Sums of Products (SP)
Similar to sums of squares
Measures the amount of coverability between two variables
SP definitional formula
SP Computational formula
Pearson Correlation Calculation (Formula)
Ratio comparing coverability of x and y (numerator) with the variability of X and Y separately (denominator)
Partial correlation…
Measures the relationship between two variables while controlling the influence of a third variable by holding it constant
Formula:
Pearson correlation can be expressed as a relationship of z scores (formulas)
Correlation is used for…
Prediction
Validity
Reliability
Theory verification
Correlation does not…
Mean causation
Correlation coefficient is affected by range…
therefore, never generalize correlation beyond the sample range data
An (extreme deviant) outlier has…
A disproportionately large impact on the correlation coefficient (r)
Correlation is not a…
Proportion of variability
Squared correlation (coefficient of determination)…
(r^2) is interpreted as the proportion of shared variability
Pearson correlation is usually computed for sample data, but…
used to test hypothesis’ about the relationship in the population
Population correlation is shown by…
Greek letter rho (ρ)
Non-directional: H0:
ρ = 0 and H1: ρ ≠ 0
Directional: H0: ρ ≤ 0 and H1: ρ > 0
or
Directional: H0: ρ ≥ 0 and H1: ρ < 0
Sample correlation is used to test population ρ
Degree of freedom
n-2
Hypothesis test can be computed using either t or F
Use to t table to find critical value with df=n-2
Reporting Correlation in Literature
Report
- Whether it is statistically significant
Concise test results
- Value of correlation
- Sample Size
- p-value or level
- Type of test (one- or two- tailed)
Example
r=-.76, n=48, p<.01, two tails
What type of data has Pearsons’s correlation been developed for?
Linear relationships
Interval or ratio measurement scales
Other correlations have been developed for…
- Non-linear relationships
- Data with nominal or ordinal measurement scales
Spearman (rs) correlation formula is used with…
Data from an ordinal scale (ranks)
- Both variables are measured on an ordinal scale
- May be used when measurement scale is interval or ration when relationship is consistently directional but not linear
Ranking Tied Scores
Tie scores need for Spearman correlation
Method:
- List ranks smallest to largest
- Assign a rank to each position in the list
- When two (or more) scores are tied, compute the mean of their ranked position, and assign this mean value as the final rank for each score
Special formula for Spearman Correlation
The ranks for scores are simply integers
Use D as the difference between the X rank and the Y rank for each individual to compute the rs statistic
Formula:
Point-Biseral Corrlation…
measures the relationship between two variables
Also r^2
- One continuous variable (e.g., Height)
- One dichotomous variable (e.g., Male=1, female=0)
Phi coefficent…
Measures the relationship between two dichtomous variables
- Both coded 0 and 1
- Regular Persons correlation is used
r=.50
Large correlation
r=.30
Moderate correlation
r=.10
Small correlation
Proportion of variance accounted for (coefficient of determination)
r^2
Confidence Interval on r
It is easy to work out a confidence interval for r too. It works just like the CI on the mean. It is centred on the observed value of r and extends a set number of standard errors in each direction …
Hypothesis Testing (r) steps
- State Hypothesis
- Determine alpha and critical values
- Compute Pearson r
- Compute effect size
- Report the results
