Lesson 6 Study Guide: Relationships Between Categorical Variables

Introduction to Analyzing Categorical Variables

• Lesson 6 focuses on the relationship between categorical variables, specifically how to analyze them visually and numerically to determine if a relationship exists.

• The primary tools discussed include organizing data into tables, calculating probabilities (risk and odds), and understanding the cautions involved in interpreting these statistics.

• This lesson moves away from measurement variables to explore how data categories interact.

Analyzing Categorical Data: The Two-Way Table

• Definition of a Two-Way Table: A summary table where each cell contains the count (frequency) of cases that fall into both a specific row category and a specific column category.

• Example: Binge-Watching Survey:

  • Organization: YouGov.

  • Data Source: A 2022 survey of $1,000$ U.S. adults regarding TV viewing habits.

  • Research Question: Is there a relationship between binge-watching frequency and geographical census regions in the U.S.?

  • Variables: There are exactly two variables shown in this analysis:

  1. Binge-watching frequency (e.g., Always, Never).

  2. Geographical region (Northeast, Midwest, South, West).

  • Sample Size: While $1,000$ adults were surveyed, only $932$ provided responses to this specific question, making $932$ the grand total for the table.

Reading and Testing Knowledge of Two-Way Tables

• Summarized vs. Raw Data: Two-way tables contain summarized data (totals/counts), not raw individual data points.

• Navigating Rows and Columns:

  • To find the total who reported "Never" binge-watching, look at the end of that specific row (Total = $128$).

  • To compare specific regions for the "Always" category, look across the row: there were more "Always" responses from the South than from the West.

  • Total respondents from the Northeast reached $167$ (found at the bottom of the column).

  • The "Always" category totaled $68$ out of $932$, making the statement that it was the "most selected" false.

Calculating Proportions and Conditional Proportions

• Proportion of Total Sample:

  • Proportion who always binge-watch: $\frac{68}{932} \approx 0.073$ or approximately $7.3\%$.

• Combined Proportions:

  • Proportion from the South ($364$) or the West ($209$): $\frac{364 + 209}{932} = \frac{573}{932} \approx 0.615$ or $61.5\%$.

• Conditional Proportions:

  • These limit the denominator to a specific subgroup (a "double conditional").

  • Proportion of Northeast residents who always binge-watch:

  • Number who always watch in the Northeast = $17$.

  • Total in the Northeast = $167$.

  • Calculation: $\frac{17}{167} \approx 0.10$ or $10\%$.

Visualizing Categorical Relationships

• Side-by-Side Bar Charts: Used to compare categories across different groups (e.g., region vs. binge frequency).

• Segmented Bar Charts: A single bar represents a whole group (like a region), and different colors within that bar represent the frequency levels of binge-watching, stacking them to show total proportions.

Quantitative Methods: Risk and Odds

• In research studies, particularly randomized comparative experiments, groups are often divided into treatment and placebo to measure differences.

• Risk:

  • Formula: $\text{Risk} = \frac{\text{Number with the trait}}{\text{Total number in the group}}$.

  • Example: In a set of 5 items where 2 are positive, the risk is $\frac{2}{5} = 0.4$.

• Odds:

  • Formula: $\text{Odds} = \frac{\text{Number with the trait}}{\text{Number without the trait}}$.

  • Example: In a set of 5 items where 2 are positive and 3 are negative, the odds are $2$ to $3$, or $\frac{2}{3}$.

  • Note: Odds can result in improper fractions (values greater than 1), unlike risk.

Relative Risk (RR) and Odds Ratios (OR)

• Relative Risk (RR):

  • Formula: $\text{Relative Risk} = \frac{\text{Risk in Group 1}}{\text{Risk in Group 2}}$.

  • Interpretation:

  • $RR = 1$: The risk is identical for both groups.

  • $RR = 3$: The first group is 3 times more likely to have the outcome than the second.

  • Increased Risk: If $RR = 1.25$, the increased risk is $25\%$ (calculated as $RR - 1$).

• Odds Ratio (OR):

  • Formula: $\text{Odds Ratio} = \frac{\text{Odds in Group 1}}{\text{Odds in Group 2}}$.

  • Interpretation:

  • $OR = 1$: The odds are the same for both groups.

  • $OR = 3$: The odds for the first group are 3 times the odds of the second group.

Case Study: Rosiglitazone for Type 2 Diabetes

• Study Goal: Evaluating the success of glycemic control in youth with Type 2 diabetes using three treatments ($N = 699$ participants randomly assigned).

• Explanatory Variables (3 Groups):

  1. Metformin: Known drug alone ($n = 232$ total).

  2. ROSI: Metformin plus Rosiglitazone ($n = 233$ total; $143$ successes, $90$ failures).

  3. Lifestyle: Metformin plus weight loss/exercise program ($n = 234$ total; $125$ successes, $109$ failures).

• Response Variable: Glycemic control (Success vs. Failure).

• Calculations for the ROSI Group:

  • Risk of Success: $\frac{143}{233}$.

  • Odds of Success: $\frac{143}{90}$.

• Relative Risk Calculation (ROSI vs. Lifestyle):

  • $\text{RR} = \frac{143/233}{125/234} \approx 1.15$.

  • Interpretation: Patients using rosiglitazone had a $1.15$ times greater risk of success, or a $15\%$ increased risk of success compared to the lifestyle group.

• Odds Ratio Calculation (ROSI vs. Metformin Alone):

  • Odds for ROSI = $\frac{143}{90} \approx 1.59$.

  • Odds for Metformin = $\frac{112}{120} \approx 0.93$.

  • $\text{OR} = \frac{1.59}{0.93} \approx 1.71$.

  • Interpretation: The odds of success for ROSI patients are approximately $1.7$ times the odds for those on metformin alone.

Cautions and Interpretations of Risk

• Baseline Risk: It is crucial to know the starting risk level. A reported "5 times higher risk" could mean an increase from $2$ per $100,000$ to $10$ per $100,000$. While significant, the absolute risk remains very low.

• Protective Effect: A relative risk less than 1 (e.g., RR < 1.0) indicates that the factor being studied provides a protective effect, lowering the risk below the baseline.

• Specificity: Risk is rarely universal. It is usually conditional on factors like age, sex, health history, and behavior.

  • Example: Men over 50 have $5$ times the risk of sudden cardiac events during marathons compared to women under 40. This risk does not apply equally to a 22-year-old woman.

• Relative Risk vs. Baseline Risk:

  • Relative risk answers: "Compared to what?"

  • Baseline risk answers: "How big is the problem in the first place?"

Identifying Statistical Statements

• Relative Risk: "Participants taking ROSI were $1.5$ times as likely to maintain control."

• Odds: "17 reported always binge-watching, while 159 did not."

• Odds Ratio: "The odds of reporting 'always' among Northeast respondents were $2.1$ times the odds among respondents from the Midwest."

• Risk: "Across all $932$ respondents, $7.3\%$ reported always binge-watching."

Simpson’s Paradox

• Definition: An observed association between two variables that changes or reverses direction when a third confounding variable is introduced that interacts strongly with both.

• Speed Limit Example:

  • In a scatterplot of speed vs. accident rates, the overall trend might appear negative (as speed limits increase, injuries go down).

  • However, when looking within individual speed limit groups (e.g., $30, 50, 80, 100\,\text{km/h}$), the trend in each group is positive (higher average speed within that limit leads to more injuries).

• Restaurant Recommendation Example:

  • Carlos' restaurant may have a higher recommendation percentage among males and a higher percentage among females individually.

  • Yet, when the data is merged, Sophia's restaurant might have a higher overall percentage due to differences in sample sizes within the gender subgroups (e.g., $200$ vs. $36$).

• Prevention: To avoid Simpson's Paradox, researchers must identify potential confounding variables during the study design phase and analyze data within subgroups rather than just in aggregate.