Chapter 15

Question 1

correlation

Answer 1

A statistical value that measures and describes the direction and degree of relationship between two variables. The sign (+/−) indicates the direction of the relationship. The numerical value (0.0 to 1.0) indicates the strength or consistency of the relationship. The type (Pearson or Spearman) indicates the form of the relationship. Also known as correlation coefficient.

Question 1

A correlation is a numerical value that describes and measures three characteristics of the relationship between X and Y.

Answer 1

• The Direction of the Relationship The sign of the correlation, positive or negative, describes the direction of the relationship.
• The Form of the Relationship (eg. linear form)
• The Strength or Consistency of the Relationship

Question 2

positive correlation

Answer 2

In a positive correlation, the two variables tend to change in the same direction: as the value of the X variable increases from one individual to another, the Y variable also tends to increase; when the X variable decreases, the Y variable also decreases.

Question 3

negative correlation

Answer 3

In a negative correlation, the two variables tend to go in opposite directions. As the X variable increases, the Y variable decreases. That is, it is an inverse relationship.

Question 4

pearson correlation

Answer 4

The Pearson correlation measures the degree and the direction of the linear relationship between two variables.

The Pearson correlation for a sample is identified by the letter r

r = degree to which X and Y vary together/degree to which X and Y vary separately = covariability of X and Y / variability of X and Y seperately

Question 5

sum of products

Answer 5

A measure of the degree of covariability between two variables; the degree to which they vary together.

Question 6

definitional formula for the sum of products

Answer 6

SP = sigma (X-Mx)(Y-My)

• Find the X deviation and the Y deviation for each individual.
• Find the product of the deviations for each individual.
• Add the products.

Question 7

computational formula for the sum of products

Answer 7

SP = sigma XY - sigmaX sigma Y / n

Question 8

the formula for the Pearson correlation becomes

Answer 8

r = SP/square root of SSx * SSy

Question 9

sample r

Answer 9

r = sigmazxzy / (n-1)

Question 10

population r

Answer 10

p = sigma zxzy / N

Question 11

Where and why Correlation Are Used

Answer 11

• Prediction If two variables are known to be related in some systematic way, it is possible to use one of the variables to make accurate predictions about the other.
• validity
• reliability
• theory verification

Question 12

When you encounter correlations, there are four additional considerations that you should bear in mind.

Answer 12

• Correlation simply describes a relationship between two variables. It does not explain why the two variables are related.
• The value of a correlation can be affected greatly by the range of scores represented in the data.
• One or two extreme data points, often called outliers, can have a dramatic effect on the value of a correlation.
• Thus, a correlation of r = 0.5 means that one variable partially predicts the other, but the predictable portion is only r^2 = 0.5^2 = 0.25 (or 25%) of the total variability.

Question 13

coefficient of determination

Answer 13

The value r^2 is called the coefficient of determination because it measures the proportion of variability in one variable that can be determined from the relationship with the other variable. A correlation of r = 0.80 (or −0.80), for example, means that R^2 = 0.64 (or 64%) of the variability in the Y scores can be predicted from the relationship with X.

Question 14

regression towards the mean

Answer 14

When there is a less-than-perfect correlation between two variables, extreme scores (high or low) for one variable tend to be paired with the less extreme scores (more toward the mean) on the second variable. This fact is called regression toward the mean.

Question 15

partial correlation between X and Y, holding Z constant, is determined by the formula

Answer 15

rxyz = rxy - (rxz*ryz) / square root of (1-r^2 xz) (1-r^2 yz)

Question 16

partial correlation

Answer 16

A partial correlation measures the relationship between two variables while controlling the influence of a third variable by holding it constant.

Question 17

When you obtain a nonzero correlation for a sample, the purpose of the hypothesis test is to decide between the following two interpretations.

Answer 17

1. There is no correlation in the population and the sample value is the result of sampling error.
2. The nonzero sample correlation accurately represents a real, nonzero correlation in the population. This is the alternative stated in H1.

Question 18

t statistic equation

Answer 18

t = sample statistic - population parameter / standard error

Question 19

standard error for r

Answer 19

r = sr = square root of 1-r^2 / n-2

t = r-p / square root of 1-r^2 / n-2

Question 20

degrees of freedom for the t statistic

Answer 20

df = n-2

Question 21

spearman correlation

Answer 21

A correlation calculated for ordinal data. Also used to measure the consistency of direction for a relationship.

Question 22

To summarize, the Spearman correlation measures the relationship between two variables when both are measured on ordinal scales (ranks). There are two general situations in which the Spearman correlation is used.

Answer 22

• Spearman is used when the original data are ordinal; that is, when the X and Y values are ranks. In this case, you simply apply the Pearson correlation formula to the set of ranks.
• Spearman is used when a researcher wants to measure the consistency of a relationship between X and Y, independent of the specific form of the relationship.

Question 23

Whenever two scores have exactly the same value, their ranks should also be the same. This is accomplished by the following procedure.

Answer 23

1. List the scores in order from smallest to largest. Include tied values in the list.
2. Assign a rank (first, second, etc.) to each position in the ordered list.
3. When two (or more) scores are tied, compute the mean of their ranked positions, and assign this mean value as the final rank for each score.

Question 24

SS for this series of integers can be computed by

Answer 24

n (n^2-1)/12

Question 25

you can put the ranks directly into a simplified formula:

Answer 25

rs = 1 - 6 sigma D^2 / n (n^2-1)

Question 26

point-biserial combination

Answer 26

A correlation between two variables where one of the variables is dichotomous.

• The point-biserial correlation is used to measure the relationship between two variables in situations in which one variable consists of regular, numerical scores, but the second variable has only two values.

Question 27

dichotomous variable

Answer 27

A variable with only two values is called a dichotomous variable or a binomial variable. Some examples of dichotomous variables are:

• male vs female

Question 28

In some respects, the point-biserial correlation and the independent-measures hypothesis test are evaluating the same thing.

Answer 28

• The correlation is measuring the strength of the relationship between the two variables.
• The t test evaluates the significance of the relationship

Question 29

the values for t and r^2 are directly related. In fact, either can be calculated from the other by the equations

Answer 29

r^2 = t^2/t^2 + df

and

t^2 = r^2/(1-r^2)/df

Question 30

phi-coefficient

Answer 30

A correlation between two variables both of which are dichotomous

When both variables (X and Y) measured for each individual are dichotomous, the correlation between the two variables is called the phi-coefficient.

To compute phi , you follow a two-step procedure.

1. 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.
2. Use the regular Pearson formula with the converted scores.