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Relationships Between Two Variables • Nonmonotonic: two variables are associated, but only in a very general sense; don’t know “direction” of relationship, but we do know that the presen

Determining and Interpreting Associations

Among Variables

Associative Analyses

• Associative analyses: determine

where stable relationships exist

between two variables

• Examples

– What methods of doing business are

associated with level of customer satisfaction? – What demographic variables are associated

with repeat buying of Brand A?

– Is type of sales training associated with sales performance of sales representatives?

– Are purchase intention scores of a new product associated with actual sales of the product?

Relationships Between Two

Variables

• Relationship: a consistent, systematic

linkage between the levels or labels for

two variables

• “Levels” refers to the characteristics of

description for interval or ratio scales…the level of temperature, etc.

• “Labels” refers to the characteristics of

description for nominal or ordinal scales, buyers v non-buyers, etc.

• As we shall see, this concept is important

Relationships Between Two

Variables

• Nonmonotonic: two variables are

associated, but only in a very general sense; don’t know “direction” of

relationship, but we do know that the presence (or absence) of one variable

is associated with the presence (or

absence) of another

• At the presence of breakfast, we shall

have the presence of orders for coffee.

• At the presence of lunch, we shall have the absence of orders for coffee.

Nonmonotonic Relationship

– Increasing

– Decreasing

• Shoe store managers know that there is

an association between the age of a child and shoe size The older a child, the

larger the shoe size The direction is

increasing, though we only know general direction, not actual size.

Monotonic Increasing

Relationship

Relationships Between Two

Variables

• Linear: “straight-line” association

between two variables

• Here knowledge of one variable will yield knowledge of another variable

• “100 customers produce $500 in

revenue at Jack-in-the-Box” (p 525)

Relationships Between Two

Variables

• Curvilinear: some smooth curve

pattern describes the association

• Example: Research shows that job

satisfaction is high when one first

starts to work for a company but goes down after a few years and then back

up after workers have been with the same company for many years This would be a U-shaped relationship

Characterizing Relationships

Between Variables

1 Presence: whether any systematic

relationship exists between two

variables of interest

2 Direction: whether the relationship

is positive or negative

3 Strength of association: how strong

the relationship is: strong?

moderate? weak?

• Assess relationships in the order

shown above

• Cross-tabulation table: four types of

numbers in each cell

– Frequency

– Raw percentage

– Column percentage

Cross-Tabulations

Ch 18 14

Cross-Tabulations

• When we have two nominal-scaled

variables and we want to know if

they are associated, we use

cross-tabulations to examine the

relationship and the Chi-Square test

to test for presence of a systematic

relationship

• In this situation: two variables, both with nominal scales, we are testing for a nonmonotonic relationship

Chi-Square Analysis

• Chi-square (X2) analysis: is the

examination of frequencies for two

nominal-scaled variables in a tabulation table to determine whether the variables have a significant

cross-relationship

• The null hypothesis is that the two

variables are not related

• Observed and expected frequencies:

• Example: Let’s suppose we want to know if there is a relationship

between studying and test

performance and both of these

variables are measured using

nominal scales…

– The column percentages table or

– The raw percentages table

Did You St udy f or t he Test ? * How Did You Perf orm on t he

Test ? Crosst abulat ion

Did You Study for the Test?

• Do you “see” a relationship? Do you “see” the

“presence” of studying with the “presence” of passing? Do you “see” the “absence” of

passing with the presence of not studying?

• Congratulations! You have just “seen” a

Did You St udy f or t he Test ? * How Did You Perf orm on t he

Test ? Crosst abulat ion

Did You Study

for the Test?

• But while we can “see” this

association, how do we know there

is the presence of a systematic

association? In other words, is this

association statistically significant?

Would it likely appear again and

again if we sampled other students?

• We use the Chi-Square test to tell us

if nonmonotonic relationships are

really present

• Using SPSS, commands are

ANALYZE, DESCRIPTIVE

STATISTICS, CROSSTABS and

within the CROSSTABS dialog box, STATISTICS, CHI-SQUARE

Chi-Square Analysis

• Chi-square analysis: assesses

nonmonotonic associations in tabulation tables and is based upon differences between observed and

cross-expected frequencies

• Observed frequencies: counts for

each cell found in the sample

• Expected frequencies: calculated on the null of “no association” between the two variables under examination

Chi-Square Analysis

• Computed Chi-Square values:

Chi-Square Analysis

• The chi-square distribution’s shape

changes depending on the number of degrees of freedom

• The computed chi-square value is

compared to a table value to

determine

statistical

significance

Ch 18 26

Chi-Square Analysis

• How do I interpret a Chi-square result?

– The chi-square analysis yields the

probability that the researcher would find evidence in support of the null hypothesis

if he or she repeated the study many, many times with independent samples.

– If the P value is < or = to 0.05, this means there is little support for the null

hypothesis (no association) Therefore,

we have a significant association…we have the PRESENCE of a systematic relationship between the two variables.

Chi-Square Analysis

• Read the P value (Asympt Sig) across from Pearson Chi-Square Since the P value is <0.05, we have a

Chi- Square Test s

39.382b 1 000 35.865 1 000 34.970 1 000

.000 000 100

Pearson Chi- Square

Chi-Square Analysis

• How do I interpret a Chi-square result?

– A significant chi-square result means the researcher should look at the

cross-tabulation row and column percentages to “see” the association pattern

– SPSS will calculate row, column, (or both) percentages for you See the CELLS box at the bottom of the

CROSSTABS dialog box

Chi-Square Analysis

• Look at the ROW %’s: 92% of those

who studied passed; almost 70% of

those who didn’t study failed “See” the

Did You St udy f or t he Test ? * How Did You Perf orm on t he Test ? Crosst abulat ion

Did You Study

for the Test?

Presence, Direction and

Strength

• Presence? Yes, our Chi-Square was

significant This means that the pattern

we observe between studying/not

studying and passing/failing is a

systematic relationship if we ran our

study many, many times

• Direction? Nonmonotonic relationships

do not have direction…only presence and absence

Presence, Direction and

Strength

• Strength? Since the Chi-Square only tells us presence, you must judge the strength by looking at the pattern

Don’t you think there is a “strong”

relationship between study/not

studying and passing/failing?

When can you use Crosstabs and

Correlation Coefficients and

Covariation

• The correlation coefficient: is an

index number, constrained to fall

between the range of −1.0 and +1.0

• The correlation coefficient

communicates both the strength and the direction of the linear relationship between two metric variables

Ch 18 36

Correlation Coefficients and

Covariation

• The amount of linear relationship

between two variables is

communicated by the absolute size

of the correlation coefficient

• The direction of the association is

communicated by the sign (+, -) of

the correlation coefficient

• Covariation: is defined as the amount

of change in one variable

systematically associated with a

change in another variable

Measuring the Association Between Interval- or Ratio-Scaled Variables

• In this case, we are trying to assess

presence, direction and strength of a

monotonic relationship

• We are aided in doing this by using:

• Using SPSS, commands are

ANALYZE, CORRELATE,

BIVARIATE

Pearson Product Moment Correlation

Correlation Coefficients and

Covariation

• Covariation can be examined with

use of a scatter diagram

Pearson Product Moment

Correlation Coefficient (r)

• Presence? Determine if there is a

significant association The P value should be examined FIRST! If it is

significant, there is a significant

association If not, there is no

association

• Direction? Look at the coefficient Is

it positive or negative?

Pearson Product Moment

Correlation Coefficient (r)

• Strength? The correlation coefficient (r) is a number ranging from -1.0 to +1.0 the closer to 1.00 (+ or -), the stronger the association There are

“rules of thumb”…

Rules of Thumb Determining

Strength of Association

• A correlation coefficient’s size indicates the

strength of association between two

variables.

• The sign (+ or -) indicates the direction of

Pearson Product Moment

Pearson Product Moment

– Correlations will not detect

non-linear relationships between

• When there is NO association, the P value for the Pearson r will be >0.05.

• When there IS association, the P value for the Pearson r will be < or =0.05.

• Examples: negative association between sales force rewards and turnover; positive association between length of sales force training and sales.

Ch 18 46

Example

• What items are associated with

preference for a waterfront view

among restaurant patrons?

– Are preferences for unusual entrées,

simple décor, and unusual desserts associated with preference for

waterfront view while dining?

– Since all of these variables are scaled we can run a Pearson

interval-Correlation to determine the association between each variable with the

preference for waterfront view.

• Using SPSS, commands are

ANALYZE, CORRELATE,

BIVARIATE

Ch 18 48

• The output shows presence, direction and strength of the association

• Do you see any managerial

significance to these associations?

Concluding Remarks on Associative Analyses

• Researchers will always test the null hypothesis of NO relationship or no correlation

• When the null hypothesis is rejected, then the researcher may have a

managerially important relationship to share with the manager