Statistics 1 1

Correlations

Definitions:

A correlation is measure of association between two quantitative variables with respect to a single individual

A correlation coefficient is a descriptive statistic that quantifies the degree of the association between two variables

Statistics 1 2

Types of Correlations:

• Positive: high values of one variable are associated with high values of the other variable

• Negative: high values of one variable are associated with low values of the other variable

• Zero: values of one variable are not associated with the values of the other variable

• Perfect: each value of one variable is associated with only a single value of the other variable and plot a straight line

Statistics 1 3

The same set of n = 6 pairs of scores (X and Y values) is shown in a table and in a scatterplot. Notice that the scatterplot allows you to see the relationship between X and Y.

Statistics 1 4

Examples of positive and negative relationships. (a) Beer sales are positively related to temperature. (b) Coffee sales are negatively related to temperature.

Statistics 1 5

(a) shows a strong positive relationship, approximately +0.90; (b) shows a relatively weak negative correlation, approximately –0.40; (c) shows a perfect negative correlation, –1.00; (d) shows no linear trend, 0.00.

Statistics 1 6

Calculating the Correlation

Statistics 1 7

Coefficient of Determination

• After calculating a Pearson Product-moment Correlation Coefficient you can go a step further by calculating the

• Coefficient of Determination which indicates how much of the variability in one variable is proportional to the variability in the other variable.

Statistics 1 8

Definition:

The coefficient of determination is a measure of the proportion of variance that can be accounted for in one variable because of its association with another variable

Calculation:

Square the Pearson Product-moment Correlation Coefficient

Coefficient of Determination = r²

Statistics 1 9

Example:

• Given a correlation of r = +0.5 between IQ and reading speed

• The coefficient of determination (r²) says that 25% of the variation in the reading speed of your subjects is related to the variability in their individual IQ's

• Which also means that 75% of the variation in reading speed of your subjects is related to some other factor(s), i.e. 1 - r²

Statistics 1 10

Final Words

• Correlations DO NOT EVER indicate causation

• The Pearson Product-moment Correlation Coefficient requires the measurement of two quantitative variables on each individual

• The Pearson Product-moment Correlation Coefficient is only applicable to LINEAR relations

Statistics 1 11

Parametric Statistical Tests

• Population parameters are specified – Shape: i.e. normal – Variance: i.e. equal

• Interval scale of measurement

• NonParametric Statistical Tests

• Do not specify the parameters of population

• Most require only an ordinal scale of measurement

Statistics 1 12

Statistical Tests

Data Design One-Sample Two-Sample K-Sample

Nominal

Between Chi-Square One Sample

Test

Chi-Square two sample test

Within

Ordinal

Between

Runs Test

Mann-Whitney U Test

Kruska-Wallis one-way Anova

Within Wilcoxon matched-pairs signed ranks

Friedman two-way Anova

Statistics 1 13

Functional and rational of test

Calculation and interpretation – General – Specific example

Statistics 1 14

FunctionAssume a data set that can be arranged in mutually

exclusive categories Question is whether the number of occurrences in

each category is different than what would be expected by chance if the null hypothesis were true

For example: – Modes of play in children – Opinions about gun control

Statistics 1 15

• Chi square one sample test will allow you to determine whether your observations are different than would be expected by chance

• Are there more aggressive children in this sample than what one would expect from a random sample from the population

• Are there fewer people in this sample in favor of gun control than would be expected from a random sample from the population

Statistics 1 16

Rational and Method

Basically, one sample chi square test compares

Observed frequencies – number of observed occurrences within each

category

Expected frequencies – number of occurrences within each category

expected by chance if null hypothesis is true

Statistics 1 17

The formula to accomplish this comparison:

Where: – O = observed frequencies – E = expected frequencies

Statistics 1 18

The logic of the test is then simple If the differences between O and E are small, chi-

square will be smallIf the differences between O and E are large, chi-

square will be largeAnd if chi-square is large enough

– Your conclusion will be that the observed frequencies are such that your sample does not come from the population from which the null hypothesis was derived

– i.e., you reject the null hypothesis

Statistics 1 19

Evaluate chi-square using the chi-square distribution for your chosen alpha level

• Using degrees of freedom equal to (k-1)

• Reject, at your alpha level, if observed chi-square is greater than tabled value

Statistics 1 20

Question: Does "post position" (1-8) make and difference in the outcome of horse racing?Data: Observe 144 races and record starting post position of winner

Statistics 1 21

Statistics 1 22

Summary

• Calculate chi square

• Determine df, i.e. k-1

• Evaluate null hypothesis with regard to chi square table

• Interpretation:

Statistics 1 23

The End