Chapter 15: Correlation

Correlations: Measuring and Describing Relationships A correlation is a statistical method used to measure and describe the relationship between two variables. A relationship exists when changes in one variable tend to be accompanied by consistent and predictable changes in the other variable.

Correlations: Measuring and Describing Relationships

Correlations: Measuring and Describing Relationships (cont'd.) A correlation typically evaluates three aspects of the relationship: –The direction: positive or negative –The form: linear or nonlinear –The strength: degree of correlation

Correlations: Measuring and Describing Relationships (cont'd.) The direction of the relationship is measured by the sign of the correlation (+ or -). A positive correlation means that the two variables tend to change in the same direction; as one increases, the other also tends to increase. A negative correlation means that the two variables tend to change in opposite directions; as one increases, the other tends to decrease.

Correlations: Measuring and Describing Relationships (cont'd.)

The most common form of relationship is a straight line or linear relationship which is measured by the Pearson correlation.

Correlations: Measuring and Describing Relationships (cont'd.) The strength or consistency of the relationship is measured by the numerical value of the correlation. A value of 1.00 indicates a perfect relationship and a value of zero indicates no relationship. r  [-1, 1]

Correlation Analysis – Measuring the Relationship Between Two Variables

Correlations: Measuring and Describing Relationships (cont'd.) r = - 1 r = 0.9 r = 0 r = - 0.4

p positive or negative relationship? a. model year and price for a used Honda? + b. IQ and grade point average? + c. daily high temperature and energy consumption in winter? - 2. r=-0.8 and r= clustered close to a line, slopes up, r= circular pattern, r=0

Correlations: Measuring and Describing Relationships (cont'd.) To compute a correlation you need two scores, X and Y, for each individual in the sample. The Pearson correlation requires that the scores be numerical values from an interval or ratio scale of measurement. Other correlational methods exist for other scales of measurement.

The Pearson Correlation The Pearson correlation measures the direction and degree (strength) of the linear relationship between two variables. –To compute the Pearson correlation, you first measure the variability of X and Y scores separately by computing SS for the scores of each variable ( SS X and SS Y ). –Then, the covariability (tendency for X and Y to vary together) is measured by the sum of products (SP). –The Pearson correlation:

The Pearson Correlation (cont'd.) Thus, the Pearson correlation is comparing the amount of covariability (variation from the relationship between X and Y) to the amount X and Y vary separately. The magnitude of the Pearson correlation ranges from 0 (indicating no linear relationship between X and Y) to 1.00 (indicating a perfect straight-line relationship between X and Y). The correlation can be either positive or negative depending on the direction of the relationship.

The Pearson Correlation (cont'd.)

Computing the Correlation Coefficient: X M Y X>M X, Y>M Y X>M X, Y<M Y X<M X, Y<M Y +, + −, + −, − +, −

correlation coefficient: r r: Pearson correlation (13-46)

correlation coefficient: r

Pearson correlation: r  [-1, +1] If Y = a + bX

Example 15.2 (p ) n=4, M X =3, M Y =5  SP = Σ(X-M X )(Y-M Y ) = 6 or SP = ΣXY – (ΣX ΣY)/n = 66 – (12*20)/4 = 6

Example 15.3 (p ) n=5, SP=28, M X =6, M Y =4, SS X =64, SS Y =16  r = (because SP > 0) See Fig 15.5 and Table 15.1

r and z-score (p..518) another formula for r (sample) another formula for ρ (population)

Alternatives to the Pearson Correlation The Spearman correlation is used in two general situations: –It measures the relationship between two ordinal variables; that is, X and Y both consist of ranks. –It measures the consistency of direction of the relationship between two variables. In this case, the two variables must be converted to ranks before the Spearman correlation is computed.

Alternatives to the Pearson Correlation (cont’d.) The calculation of the Spearman correlation requires: 1.Two variables are observed for each individual. 2.The observations for each variable are rank ordered. Note that the X values and the Y values are ranked separately. 3.After the variables have been ranked, the Spearman correlation is computed by either: a)Using the Pearson formula with the ranked data. b)Using the special Spearman formula (assuming there are few, if any, tied ranks).

26 16-* Spearman’s Rank-Order Correlation Spearman’s coefficient of rank correlation reports the association between two sets of ranked observations. The features are: It can range from –1.00 up to It is similar to Pearson’s coefficient of correlation, but is based on ranked data. As with all nonparametric statistics, Spearman’s correlation coefficient does not require any assumptions about the distribution of the populations. It is computed using the formula:

Alternatives to the Pearson Correlation (cont’d.) The Pearson correlation formula can also be used to measure the relationship between two variables when one or both of the variables is dichotomous. A dichotomous variable is one for which there are exactly two categories: for example, men/women or succeed/fail.

Alternatives to the Pearson Correlation (cont’d.) The point-biserial correlation is used in situations where one variable is dichotomous and the other consists of regular numerical scores (interval or ratio scale).

Alternatives to the Pearson Correlation (cont’d.) The calculation of the point-biserial correlation proceeds as follows: –Assign numerical values to the two categories of the dichotomous variable(s). Traditionally, one category is assigned a value of 0 and the other is assigned a value of 1. –Use the regular Pearson correlation formula to calculate the correlation.

Alternatives to the Pearson Correlation (cont’d.) The point-biserial correlation is closely related to the independent-measures t test introduced in Chapter 10. When the data consists of one dichotomous variable and one numerical variable, the dichotomous variable can also be used to separate the individuals into two groups. Then, it is possible to compute a sample mean for the numerical scores in each group.

Alternatives to the Pearson Correlation (cont’d.) In this case, the independent-measures t test can be used to evaluate the mean difference between groups. If the effect size for the mean difference is measured by computing r 2 (the percentage of variance explained), the value of r 2 will be equal to the value obtained by squaring the point- biserial correlation.

Alternatives to the Pearson Correlation (cont’d.)

The phi-coefficient is used when both variables are dichotomous. The calculation proceeds as follows: –Convert each of the dichotomous variables to numerical values by assigning a 0 to one category and a 1 to the other category for each of the variables. –Use the regular Pearson formula with the converted scores.

Alternatives to the Pearson Correlation (cont’d.)