Week 4 Correlations

Why use correlation?

To test for systematic relationships between variables, variables which cannot be experimentally manipulated

Correlation

A relationship between two continuous variables

Why might it not be possible to experimentally manipulate a variable?

Ethical and logistical reasons

Correlation coefficient

single numerical value that describes a relationship

Two qualities of correlation

strengths and direction

Strength

0 -1; the closer to 1 the stronger the relationship

Negative correlation

-1 = as variable 1 goes up, variable 2 goes down

No correlation

0

Positive correlation

+1 as variable 1 goes up, variable 2 goes up

Boxplots

horizontal x axis typically represents the independent variable, vertical y axis typically represents the dependent variable

Pearson Product Moment Correlation (Pearson’s r)

Statistical test of significance that measures linear relationship between two variables (covariance)

Covariance

How much the variables change together, tells us the direction of the relationship

Positive covariance

As the value of one variable increases, so does the other

Negative covariance

As the value of one variable increases, the other decreases

Calculating variance of 1 variable (how much x varies around the mean)

1. Calculate the mean (x̅)

2. Subtract the mean from each score

3. Square the results

4. Add up the squared scores

5. Divide by the number of scores -1

Variance of two variables, x and y

For each pair of scores

1. Subtract the mean of the variable X from each score in X

2. Subtract the mean of variable Y from each score Y

3. Multiply each of the pairs difference scores together

4. Sum the results

5. Divide by the number of scores-1 (N-1)

why divide by N-1 (degrees of freedom)

Small samples tend to underestimate the underlying population covariance

For large samples the -1 has negligible effect on the estimate of the population covariance

Limitations of covariance

The covariance value is dependent on the measurement units of the variables, and doesn’t tell us the strength of the association

Standard deviation

A universal unit of measurement into which any other scale of measurement can be converted

Calculating standard deviation

The square root of its variance

Variance =

1. Calculate the mean (x̅)

2. Subtract the mean from each score

3. Square the results

4. Add up the squared scores

5. Divide by the number of scores -1

Pearson r correlation coefficient

Why r can take a maximum value of 1

the total shared variation represented by cov(x,y) can be as large as the total variation in the variables (SD), but no bigger. If the shared variation is smaller than the total variation, magnitude of r will be less than 1

When are correlations statistically significant

If p <.05 you can reject the null hypothesis,  If you predicted the direction of correlation, then p value can be divided by 2 (one tailed test)

Reporting correlations

If significant, place more emphasis upon interpreting the direction and size (strength) of r

Limitations of Pearson’s r

only measures linear relationships, can find correlation not causation

Checking if data is normally distributed in SPSS

use Shapiro Wilk (<50) or Kolmogorov Smirnov (50+) in SPSS for each variable to check if data is skewed

When to us Spearman’s row

Data is not normally distributed, there are outliers, data is not continuous

Spearman’s rho

Converts data into ranks

Disadvantages of Spearman’s rho

Converting the data to ranks before analysis entails a loss of information, as you are probably reducing ratio-level data to ordinal data

Calculating Spearman’s rho

rank the data, do a Pearson’s r on the ranked data

Correlating 3 or more variables

Do a series of calculation to see how all variables are connected to each other, Correlation for each pair of variables calculated (separate bivariate correlations)

Parametric analysis

assume that the data follows a specific probability distribution (such as a normal or Gaussian distribution) and is defined by a fixed, finite number of parameters, such as the mean and standard deviation

Non-parametric analysis

make minimal or no strict assumptions about the underlying population distribution (often called "distribution-free"). Instead of modelling specific distribution parameters, they often analyse the rank-order or median of the data