Building a Multiple Linear Regression Model for Satisfaction Scores

The Interrogation Room: Assembling the Full Lineup

Our initial investigation focused on a simple relationship:
$\text{Satisfaction}={\beta }_0+{\beta }_1\times (\text{Key\_Factor\_Score})+ϵ$

Now, we introduce the other variables from the leaked file:

• Communication_Frequency (rated 1-10)
• Shared_Activities (hours per week)
• Years_Together (number of years)

A multiple linear regression model allows us to model Satisfaction as a function of all these predictors simultaneously. Instead of fitting a line in two dimensions, we are now fitting a plane (or hyperplane) in a multi-dimensional space.

The model takes the form:
$Y={\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+{\beta }_3{X}_3+{\beta }_4{X}_4+ϵ$
Where Y is Satisfaction and $X_1$ through $X_4$ are our four predictors. Each $\beta$ coefficient now represents the effect of its corresponding predictor on Satisfaction, while holding all other predictors constant. This "controlling for" aspect is the most powerful tool in our detective kit.

The following video from StatQuest provides an excellent visual intuition for this leap from a simple line to a multi-dimensional plane.

The New Testimony: Analyzing the Full Model

With all four predictors in the model, we run the analysis again. The statistical software provides a new report. This is the moment of truth. Does the original suspect's story hold up under cross-examination?

Here is the output from our new, comprehensive model:

SUMMARY OUTPUT

Regression Statistics
-------------------
R Square           0.921
Adjusted R Square  0.886
Observations       10

ANOVA
--------------------------------------------------
             df      SS          MS         F      
Regression   4    2490.15      622.54     14.71
Residual     5     212.85       42.57
Total        9    2703.00

Coefficients
---------------------------------------------------------------------------------
                         Estimate   Std. Error   t Stat    P-value   Std. Coef.
Intercept                 5.120       12.850      0.40     0.706        -
Key_Factor_Score          0.198        0.355      0.56     0.599       0.10
Communication_Frequency   3.551        1.102      3.22     0.023       0.45
Shared_Activities         2.890        0.950      3.04     0.029       0.41
Years_Together           -0.885        0.750     -1.18     0.289       -0.15

At first glance, this table is dense with clues. To interpret it, we need a systematic approach. The article below provides a fantastic guide to every number in a regression summary.

Let's apply this knowledge to our case.

The Plot Twist

Compare the new results for Key_Factor_Score with the old ones from our simple model.

• Simple Model: Estimate = 1.896, P-value < 0.0001
• Multiple Model: Estimate = 0.198, P-value = 0.599

This is the smoking gun. The moment we put Key_Factor_Score in the room with the other predictors, its influence evaporated. The coefficient shrank dramatically, and its p-value skyrocketed far above our significance threshold (e.g., 0.05). It is no longer a statistically significant predictor.

Meanwhile, two other suspects have been clearly implicated:

• Communication_Frequency: P-value = 0.023 (Significant)
• Shared_Activities: P-value = 0.029 (Significant)

Years_Together remains not significant, suggesting it's likely an innocent bystander in this particular case.

Why Did the Story Change?

The Key_Factor_Score was never the true driver. It was simply highly correlated with the real drivers, Communication_Frequency and Shared_Activities. In statistical terms, this is an issue of multicollinearity. Because the Key_Factor_Score moved in tandem with the other two variables, our simple model mistakenly attributed their explanatory power to it. Once the true drivers were present to speak for themselves, the Key_Factor_Score had no unique information left to contribute.

Think of it this way: a getaway car's driver might seem important until you realize they were just following directions from the masterminds in the back seat.

Identifying the Primary Drivers

Now that we have our true suspects, Communication_Frequency and Shared_Activities, how do we know which is more important? We cannot simply compare their Estimate values (3.551 vs. 2.890) because the variables are on different scales (a 1-10 rating vs. hours per week).

This is where we use the Standardized Coefficient (also called "beta weight"). This value represents the change in the dependent variable (in standard deviations) for a one-standard-deviation change in the predictor. Because these coefficients are unitless, we can compare them directly.

• Communication_Frequency: Std. Coef. = 0.45
• Shared_Activities: Std. Coef. = 0.41

The standardized coefficients are very close, but Communication_Frequency has a slightly stronger impact on Satisfaction than Shared_Activities. These two are the primary drivers we were looking for.

This process of using multiple regression to find the most important predictors is a powerful technique known as Key Driver Analysis.

Finally, to understand the mechanics of how adding new variables alters the model, the following video gives a clear, step-by-step demonstration.

Summary of Today's Investigation

• Objective: We built a comprehensive multiple linear regression model to test if Elysian's "Key Factor" was the true driver of satisfaction or merely a decoy.
• Method: We included three new predictors (Communication_Frequency, Shared_Activities, Years_Together) alongside the original Key_Factor_Score to model Satisfaction.
• Evidence: The analysis revealed a dramatic shift.
  1. The Confession: The Key_Factor_Score became statistically insignificant (p=0.599) in the multiple regression model.
  2. The Real Culprits: Communication_Frequency and Shared_Activities emerged as the true, statistically significant drivers of satisfaction.
  3. Relative Importance: Using standardized coefficients, we determined Communication_Frequency (Std. Coef. = 0.45) is the single most important predictor, followed closely by Shared_Activities (Std. Coef. = 0.41).
• Conclusion: Elysian's marketing is a deception. Their "Key Factor" is a statistical phantom, a composite variable that was simply reflecting the effects of more fundamental relationship dynamics. The firm likely identified that communication and shared time were key, created a proprietary score based on them, and then marketed that score as their "secret sauce."

Preview of the Next Lesson

We've cracked the case of the fixed survey by identifying the true drivers of satisfaction. We have a compelling model and a strong theory of what happened. But a good detective always double-checks the evidence. A regression model is only reliable if its underlying assumptions are met. The suspiciously high fit of our model, even after accounting for the new predictors, still warrants scrutiny.

In our next session, we will conduct a forensic analysis of our model. We will examine the residuals—the errors our model makes—to search for hidden patterns, outliers, or other signs of tampering that could challenge our conclusion. This is the final step to ensure our case is airtight.