Session 20&21

Class Structure

Last Session

• Testing Differences:

  • Percentages (Two Groups)

    • ​Tyrenol preferences among congestion vs. muscle ache sufferers.

  • Means (Multiple Groups):

    • t-test: Compare means of two groups. e.g., (Male vs. female teens’ sports drink consumption.)

    • ANOVA: Compare means across multiple groups. (e.g., age-based preferences for cars).

  • Paired Samples:

    • Paired t-test: Compare two variables within the same group.

    • Example: Concern for global warming vs. gasoline emissions.

Agenda

• What Are Associative Analyses?

• Types of Relationships

• Correlation Coefficient

• Cross Tabulation and Chi-Square Test

• Special Consideration in Associative Analyses

Introduction to Associative Analyses

• Marketing researchers often want to go beyond descriptive measures, statistical inference, and differences tests.

• Explore relationships among variables in large datasets (hundreds or thousands of survey responses).

• What kinds of people buy Frito-Lay snacks such as Cheetos, Fritos, or Lay’s potato chips?

  • Demographics: Who are the customers?

  • Circumstances: Under what conditions are these products chosen?

• Associative analyses: determine where stable relationships exist between two variables

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.
    • e.g., Older consumers purchase more vitamins.
  • “Labels” refers to the characteristics of description for nominal or ordinal scales.
    • e.g., PayPal customers (Yes/No) tend to also be Amazon Prime customers.

Relationships between Two Variables (cont.)

• Statistical Linkage (our focus): Indicates a consistent pattern or association between variables, but not causation.
  • Most daily exercisers purchase sports drinks → suggests correlation but does not prove that exercising causes sports drink purchases.
• Causal Linkage: Requires certainty that one variable causes the other.
  • Increased advertising spend leads to higher sales → a proven cause-and-effect relationship supported by controlled testing or robust analysis.

Agenda

• What Are Associative Analyses?

• Types of Relationships

• Correlation Coefficient

• Cross Tabulation and Chi-Square Test

• Special Consideration in Associative Analyses

Monotonic Linear Relationship

• A “straight-line association” between two scale variables.

• Formula:
  \[y=a+bx\]

  • \(y\): Dependent variable (predicted/estimated).

  • \(a\): Intercept (value of y when \(x=0\)).

  • \(b\): Slope (rate of change in \(y\) per unit change in \(x\)).

  • \(x\): Independent variable.

  • Predicts \(y\) given any value of \(x\) using known \(a\) and \(b\).

• Linear relationships are commonly used in the analysis of interval or ratio scale variables

• We will learn about it in more detail in the regression analysis.

Monotonic Linear Relationship Example

• Burger King estimates that every customer will spend about $12 per lunch visit

• it is easy to use a linear relationship to estimate how many dollars of revenue will be associated with the number of customers for any given location.

  \[y = 0 + 12\] where x = number of customers.

• 100 customers → 12×100=1,200 revenue.

• 200 customers → 12×200=2,400 revenue.

• Linear relationship provides an average expectation of future revenue.

Nonmonotonic Relationship

• A relationship where the presence or absence of one variable’s label is systematically associated with the presence or absence of another variable’s label.

  • No consistent direction (e.g., increasing or decreasing).

  • The relationship is general and described verbally.

  • Describes patterns of association, not precise relationships or quantities.

• Nonmonotonic relationships are often used for the analysis of norminal scale variables

Nonmonotonic Relationship Example

• Drinks Ordered at McDonald’s

  • Morning customers tend to buy coffee and breakfast foods.
  • Noon customers tend to buy soft drinks and lunch items.

• Not exclusive: Labels are associated on average, but exceptions exist.

Characterizing Relationships between Variables

• Presence:
  Whether a systematic (statistical) relationship exists between two variables.
• Pattern:
  The general nature of the relationship, including its direction.
• Strength of Association:
  Whether the relationship is consistent.

Step-By-Step Procedure for Analyzing the Relationship between Two Variables 

Agenda

• What Are Associative Analyses?

• Types of Relationships

• Correlation Coefficient

• Cross Tabulation and Chi-Square Test

• Special Consideration in Associative Analyses

Correlation Coefficients

• Definition:

  • A measure (r) that quantifies the strength and direction of a linear relationship between two scale variables.

  • Range: −1.0 to +1.0.

• The correlation coefficient communicates both the strength and the direction of the linear relationship between two metric variables.Strength: Determined by the absolute value of
  • Strength: Determined by the absolute value of r.
    • e.g., r=0.8: Strong relationship; r=0.2: Weak relationship.
  • Direction: Indicated by the sign of r.
    • Positive (+): Variables increase together.
    • Negative (−): One variable increases as the other decreases.

Correlation Coefficients (cont.)

• Rules of Thumb about Correlation Coefficient Size

• Regardless of its absolute value, the correlation coefficient must be tested for statistical significance.

Visualizing Covariation Using Scatter Diagrams

• Scatter diagrams visualize covariation between two variables.
  • Vertical axis (y): Dependent variable (e.g., sales).
  • Horizontal axis (x): Independent variable (e.g., number of salespeople).
  • Points: Represent matched pairs of x and y values.
• Systematic covariation forms an ellipse-like pattern.

Visualizing Covariation Using Scatter Diagrams (cont.)

• Patterns can vary depending on the relationship between variables.

The Pearson Product Moment Correlation Coefficient

• Measures the linear relationship between two interval or ratio-scaled variables (scale variables).

  • Based on the closeness of scatter points to a straight line.

• Perfect Correlation: All points fall on a straight line; Correlation coefficient (r) = \(\pm\) 1.00.

• No Correlation: Scatter points form a ball-shaped pattern with no discernable ellipse; r ≈ 0.0.

• Typical Values: Most correlations fall between -1.0 and +1.0 and are interpreted as: Strong, moderate, or weak, if statistically significant.

• Provides insight into the strength and direction of the linear association.

• If you are interested in the math behind it, here is the reference.

  • The calculation of r is VERY tedious, so I don’t even introduce the formula here.

Calculating Correlation Coefficient in R

• Let’s calculate the correlation coefficient using R.
• As always, we will use auto_concept.csv data.
• There are six different lifestyles: Novelist, Innovator, Trendsetter, Forerunner, Mainstreamer, and Classic.
  • Each lifestyle type is measured with a 7-point interval scale
    • 1 = Does not describe me at all to 7 = Describes me perfectly
• Correlation analysis can find out which lifestyle profile is associated with a particular automobile model preference.

  • High Positive Correlations: Indicate consumers prefer a specific model and score high on the associated lifestyle type.

  • Low or Negative Correlations: Suggest consumers do not align with that model or lifestyle type.

Calculating Correlation Coefficient in R (cont.)

• Let’s determine which of the six lifestyle types is associated with the preference for the 5-seat economy gasoline automobile model.

• First step is to select the variables to analyze.

  • Preference for the 4 Seat Economy Gasoline model.
  • All six lifestyle types: Novelist, Innovator, Trendsetter, Forerunner, Mainstreamer, and Classic.

• All of these variables are interval scale variables. -> Use Pearson’s correlation to measure the strength and direction of the linear relationship.

  • Default to a two-tailed test for statistical significance.

Calculating Correlation Coefficient in R (cont.)

• We will rely on psych package for an efficient and clean workflow.

  • Computes both the correlation coefficient and the p-value in a single step.
  • Saves time and reduces redundancy compared to cor() and cor.test().

• Load Data: Import the dataset into R.

• SSelect Variables: Focus on the 4 Seat Economy Gasoline model preference and the six lifestyle types.

  • Rename variables to their original names (e.g., Novelist, Innovator) for clarity.

# Install and load the psych package
# install.packages("psych") #if you didn't install
library(tidyverse)
library(psych)
setwd("~/Documents/GitHub/Marketing-Research-2025-Spring")
# Step 1: Load the dataset
auto_concept <- read_csv("auto_concept.csv")

# Step 2: Select relevant columns
# Include 'economygas4seat' and the six lifestyle variables
selected_data <- auto_concept %>%
  select(economygas4seat, lifestyle1, lifestyle2, lifestyle3, lifestyle4, lifestyle5, lifestyle6) %>%
  rename(
    "Economy Gasoline" = economygas4seat,
    "Novelist" = lifestyle1,
    "Innovator" = lifestyle2,
    "Trendsetter" = lifestyle3,
    "Forerunner" = lifestyle4,
    "Mainstreamer" = lifestyle5,
    "Classic" = lifestyle6
  )

Calculating Correlation Coefficient in R (cont.)

• Use the corr.test() function in the psych package for Pearson correlation.

  • Correlation matrix: Measures strength and direction.
  • P-Value: Indicates statistical significance.

# Step 3: Compute correlations and p-values
# Use rcorr to compute correlation matrix and significance
results <- corr.test(selected_data, method = "pearson")

# Step 4: Extract correlation coefficients and p-values
results
results$stars

Calculating Correlation Coefficient in R (cont.)

• Classic (r=0.63, p<0.01): Strong positive & significant correlation.
  • Highlights that individuals identifying as Classic are the most likely to prefer the economy gasoline model.
• Forerunner (r=0.22, p<0.01): Moderate positive & significant correlation.
  • Suggests Forerunners are more likely to prefer the economy gasoline model
• Novelist (r=0.09, p<0.01): Weak positive & significant correlation
  • Suggests a slight association between preference for the economy gasoline model and the Novelist lifestyle.
• Innovator (r=−0.12, p<0.01): Weak negative & significant correlation
  • Indicates that individuals identifying as Innovators are slightly less likely to prefer the economy gasoline model.

Agenda

• What Are Associative Analyses?

• Types of Relationships

• Correlation Coefficient

• Cross Tabulation and Chi-Square Test

• Special Consideration in Associative Analyses

Cross-Tabulation Analysis

• Cross-tabulation is a statistical method used to examine nonmonotonic relationships between two nominally scaled variables.
• A cross-tabulation table is sometimes referred to as an “r×c” (r-by-c) table because it is comprised of rows and columns.
  • Rows (r): Represent categories of one variable.
  • Columns (c): Represent categories of another variable.
  • Cells: The intersection of rows and columns, containing the frequency of occurrences.

Raw Percentages Table

• In a cross-tabulation table, raw frequencies can be converted into various types of percentages to provide additional insights.
• The raw percentages table is derived by dividing each raw frequency by the grand total (200 in this case) and multiplying by 10 \[\text{Raw Percentage}=\frac{\text{Cell Freqeuncy}}{\text{Grand Total}}\times100\]

Row Percentages Table

• The row percentages table calculates percentages relative to the row totals (e.g., 160 for White Collar, 40 for Blue Collar). \[\text{Row Cell Percentage}=\frac{\text{Cell Freqeuncy}}{\text{Total of Cell Frequencies in that Row}}\times100\]
• e.g., The first column is calculated by \(\frac{152}{160}\times100\) and \(\frac{14}{40}\times100\)

Column Percentages Table

• The row percentages table calculates percentages relative to the row totals (e.g., 160 for White Collar, 40 for Blue Collar). \[\text{Column Cell Percentage}=\frac{\text{Cell Freqeuncy}}{\text{Total of Cell Frequencies in that Column}}\times100\]
• e.g., The first row is calculated by \(\frac{152}{166}\times100\) and \(\frac{8}{34}\times100\)

Chi-Square (χ²) Analysis Overview

• Chi-square (χ²) analysis is a statistical technique used to examine the frequencies of two nominally scaled variables in a cross-tabulation table.

• It assesses nonmonotonic association in a cross-tabulation table based upon differences between observed and expected frequencies

• It determines whether there is a statistically significant relationship between the two variables.

  • Null Hypothesis: Assumes no association between the variables (independence).

    • e.g.,: The distribution of buyers and nonbuyers is independent of occupation.

  • Alternative Hypothesis: Assumes that the variables are associated (dependent).

Observed Frequencies in Chi-Square Analysis

• Chi-square analysis compares observed frequencies (actual counts from the data) to expected frequencies (theoretical counts assuming no association). The degree of difference is expressed in the Chi-square test statistic.

• Observed Frequencies (\(O_{ij}\)):

  • These are the actual cell counts in the cross-tabulation table.
  • Example: 152 White Collar buyers and 8 White Collar nonbuyers.

Expected Frequencies in Chi-Square Analysis

• Expected Frequencies (\(E_{ij}\)): These are theoretical frequencies calculated under the null hypothesis of no association.

• Formula: \(E_{ij} = \frac{\text{Row Total}_{i} \times \text{Column Total}_{j}}{\text{Grand Total}}\)

\[ \text{White-collar buyer} = \frac{160 \times 166}{200} = 132.8 \]

\[ \text{White-collar nonbuyer} = \frac{160 \times 34}{200} = 27.2 \]

\[ \text{Blue-collar buyer} = \frac{40 \times 166}{200} = 33.2 \]

\[ \text{Blue-collar nonbuyer} = \frac{40 \times 34}{200} = 6.8 \]

The Computed χ² Value

• The Chi-square (χ²) value is a single number summarizing how much the observed frequencies deviate from the expected frequencies in a cross-tabulation table.

• It provides a measure of how closely the data align with the null hypothesis of no association. \[ \chi^2 = \sum_{i=1,j=1}^n \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \] Where:

• \(O_{ij}\) : Observed frequency in row i and column j

• \(E_{ij}\): Expected frequency in row i and column j

• \(n\): Number of cells

Degrees of Freedom in Chi-Square Analysis

• Degrees of freedom represent the number of values in a calculation that are free to vary while still meeting the constraints of the data.

• In the context of Chi-square analysis, degrees of freedom are used to determine the critical value from the Chi-square distribution table.

\[ df = (\text{Number of Rows} - 1) \times (\text{Number of Columns} - 1) \]

• 2x2 Table (Michelob Ultra Example): - 2 Rows (White Collar, Blue Collar) and 2 Columns (Buyer, Nonbuyer) \[ df = (2 - 1) \times (2 - 1) = 1 \]

Characteristics of the Chi-Square Distribution

• We will use the chi-square distribution to determine the statistical significance of chi-square value.
• The Chi-square distribution is skewed to the right.

  • The skewness decreases as the degrees of freedom (df) increase.

  • For large df, the Chi-square distribution approximates a normal distribution

• The rejection region is always located in the right-hand tail of the distribution.
• Larger Chi-square values fall in the rejection region, indicating a significant result.

Characteristics of the Chi-Square Distribution

The Computed χ² Value Practice

• Observed Frequencies (O)

• Expected Frequencies (E)

The Computed χ² Value Practice

\[ \chi^2 = \frac{(152 - 132.8)^2}{132.8} + \frac{(8 - 27.2)^2}{27.2} + \frac{(14 - 33.2)^2}{33.2} + \frac{(26 - 6.8)^2}{6.8}=81.70 \] \[ df = (2 - 1) \times (2 - 1) = 1 \]

• Statistical significance is determined by comparing the computed Chi-square value with the critical value from the Chi-square distribution table or by using a p-value.

• Modern statistical software automates the process.

How the Chi-Square Value is Computed

• Compare Observed and Expected Frequencies: Subtract the expected frequency \(E\) from the observed frequency \(O\); \((O - E)\).
• Adjust for Negative Values: Square the difference to eliminate negative values and prevent cancellation effects: \((O - E)^2\)
• Normalize by Expected Frequency: Divide the squared difference by the expected frequency \(E\) to adjust for differences in cell sizes: \(\frac{(O - E)^2}{E}\)
• Summation Across All Cells: Add these normalized values across all cells in the cross-tabulation table: \[ \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \]

Interpreting the Chi-Square Value

• Large Deviations:
  • If observed frequencies deviate significantly from expected frequencies -> the computed Chi-square value increases.
• Small Deviations:
  • If observed frequencies are close to expected frequencies -> the computed Chi-square value remains small.
• The Chi-square value provides a summary measure of the departure of observed frequencies from expected frequencies.

Cross Tabulation in R

• Okay, let’s practice what we have learned in R.

• Recall that we have several nominal variables in our auto_concept.csv data.

  • Marital Status (marital): Unmarried(0) / Married (1) (Nominal Variable)

  • Favorite Newspaper Section: (newspaper): Editorial, Business, Local News, National News, Sports, Entertainment, Do Not Read (Nominal Variable)

• We will use janitor package for the associative analysis

library(tidyverse)
#install.packages("janitor")  # Install if not already installed
library(janitor)
setwd("~/Documents/GitHub/Marketing-Research-2025-Spring")
auto_concept <- read_csv("auto_concept.csv")
# Rename marital and newspaper variables
auto_concept <- auto_concept %>%
  mutate(
    marital = factor(marital,
                     levels = c(0, 1),
                     labels = c("Unmarried", "Married")),
    newspaper = factor(newspaper,
                       levels = c(1, 2, 3, 4, 5, 6, 7),
                       labels = c("Editorial", "Business", "Local News",
                                  "National News", "Sports", "Entertainment", "Do Not Read"))
  )

Cross Tabulation in R (cont.)

• The tabyl() within janitor package provides simple way to calculate the cross table.

# Create a cross-tabulation table
cross_tab <- auto_concept %>%
  tabyl(marital, newspaper)
cross_tab

Cross Tabulation in R (cont.)

• Let’s do the chi-square test.
  • you just put the cross_tab in the chisq.test() function.
• The p-value is essentially zero; it indicates a statistically significant association between marital status and newspaper section preferences.
• However, it does not specify the nature of the relationship.
• To understand this, we must analyze the row or column percentages.

# Perform Chi-square test
chi_test <- chisq.test(cross_tab)

# Display Chi-square test results
chi_test

    Pearson's Chi-squared test

data:  cross_tab
X-squared = 150.24, df = 6, p-value < 2.2e-16

Cross Tabulation in R (cont.)

• We can easily calculate the raw percentages table

• adorn_percentages("all") does the job!

cross_tab_total <- auto_concept %>%
  tabyl(marital, newspaper) %>%
  adorn_percentages("all") %>%
  adorn_pct_formatting(digits = 2)

cross_tab_total

Cross Tabulation in R (cont.)

• Similarly, we can easily calculate the row percentages table

• adorn_percentages("row") does the job!

cross_tab_total <- auto_concept %>%
  tabyl(marital, newspaper) %>%
  adorn_percentages("row") %>%
  adorn_pct_formatting(digits = 2)

cross_tab_total

Cross Tabulation in R (cont.)

• Similarly, we can easily calculate the column percentages table

• adorn_percentages("col") does the job!

cross_tab_total <- auto_concept %>%
  tabyl(marital, newspaper) %>%
  adorn_percentages("col") %>%
  adorn_pct_formatting(digits = 2)

cross_tab_total

Key Observations from Row Percentages Table

• Unmarried Individuals:

  • Sports is the dominant preference, at about 48% of their readership.

  • Do Not Read is the second most frequent choice, at about 23%.

• Married Individuals:

  • Local News is the most common preference, at around 34%.

  • Business comes next, with about 22%; Sports is also significant, at 21%.

Marketing Implications

Agenda

• What Are Associative Analyses?

• Types of Relationships

• Correlation Coefficient

• Cross Tabulation and Chi-Square Test

• Special Consideration in Associative Analyses

Special Considerations in Association Procedures

1. Scaling Assumptions

• Correlation: Requires both variables to have interval-level scaling at a minimum.

• Cross-Tabulation: Used when one or both variables are nominally scaled.

2. Analysis is Limited to Two Variables

• Association analyses focus solely on the relationship between two variables.

• Assumptions: Other variables are assumed constant or “frozen.”

• Limitation: Interactions with other variables are ignored, which can oversimplify complex relationships.

Special Considerations in Association Procedures (cont.)

3. Correlation ≠ Causation

• Neither correlations nor cross-tabulations imply cause-and-effect relationships.

4. Pearson Correlation Only Identifies Linear Relationships

• A correlation coefficient of ~0 does not mean no relationship exists—it simply means no linear relationship exists.

• Nonlinear relationships (e.g., S-shape or J-shape patterns) will not be detected by the Pearson product-moment correlation.

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