Correlation Beyond Linearity: Exploring Pre-Post Program Satisfaction

The Linear Hypothesis: Pearson's Correlation

Our first line of inquiry is to test for a simple, linear relationship. Does a higher Before score correspond to a higher After score in a straight-line fashion? The standard tool for this is the Pearson correlation coefficient ($r$). It measures the strength and direction of a linear association between two continuous variables.

Let's revisit the evidence from the 12 case files:

By applying the Pearson formula, which you'll recall is based on the covariance of the two variables scaled by their standard deviations, we get:

This value suggests a moderate, positive linear relationship. As Before scores increase, After scores also tend to increase. However, a value of 0.58 is far from a perfect +1, indicating the data points do not fall neatly on a single line. This could be due to random noise, or it could be a clue that a linear model isn't the best fit for the data.

Uncovering a Deeper Pattern: Spearman's Rank Correlation

What if the relationship isn't linear? For example, the program might provide a huge boost to couples starting with very low satisfaction, but offer diminishing returns for those already doing moderately well. This would create a relationship that is consistently increasing (monotonic) but curved, not straight.

To investigate this, we turn to Spearman's rank correlation coefficient ($\rho$ or $r_s$). This non-parametric measure is a powerful tool because it assesses the relationship between the ranks of the data, not their raw values. It effectively asks: "Does a higher rank on the 'Before' scale tend to correspond with a higher rank on the 'After' scale?" This makes it robust to outliers and, crucially for our case, sensitive to monotonic relationships that are not linear.

The first step is to convert our raw scores into ranks. For each variable, the lowest score gets rank 1, the next lowest gets rank 2, and so on. When scores are tied, they all receive the average of the ranks they would have occupied.

After performing this ranking process on both the Before and After scores and then calculating the correlation on these ranks (which is mathematically equivalent to applying the Pearson formula to the ranked data), we find:

This is a significantly stronger correlation than the Pearson coefficient we calculated. This discrepancy is a major lead in our investigation.

Comparing the Evidence: Linear vs. Monotonic

We have two pieces of evidence: a Pearson $r$ of 0.58 and a Spearman $\rho$ of 0.71. When the Spearman correlation is notably stronger than the Pearson correlation, it's a classic sign that the underlying relationship between the variables is monotonic but not linear.

The video below from the Learn2Stats channel provides an excellent visual walkthrough of how these two coefficients behave with different data patterns.

The images below illustrate this principle perfectly. In the top panel of each image, where the data follows a clear linear trend, the Pearson and Spearman coefficients are nearly identical. However, in the bottom panel, where the data follows a non-linear (but still increasing) curve, the Spearman coefficient is much higher than the Pearson coefficient. This is because Pearson is "penalized" by the deviation from a straight line, while Spearman only cares that as X increases, Y also consistently increases.

Our findings ($r\approx 0.58$ and $\rho \approx 0.71$) mirror the non-linear scenarios in these exhibits. This suggests the effect of Dr. Eve's program is not a simple straight-line improvement.

To deepen our understanding of the mechanics, let's turn to some more detailed intelligence.

The fact that satisfaction scores are ordinal strengthens the case for trusting the Spearman coefficient more. It doesn't assume that the interval between a score of 2 and 3 is the same as the interval between 7 and 8—an assumption that is implicit when using Pearson's $r$.

The TileStats video below provides a step-by-step calculation of Spearman's $\rho$ and an excellent summary of when to use each method.

Conclusion

Our investigation into the relationship between Before and After scores has revealed a subtle but important pattern. While a simple linear correlation exists ($r\approx 0.58$), a much stronger rank-based correlation is also present ($\rho \approx 0.71$). This discrepancy is our key finding.

Key Takeaways:

• Pearson's correlation ($r$) measures the strength of a linear relationship. For our data, it was moderate.
• Spearman's correlation ($\rho$) measures the strength of a monotonic relationship (one that consistently moves in the same direction). For our data, it was strong.
• When $\rho >r$, it provides strong evidence that the relationship between the variables is monotonic but non-linear.
• This suggests Dr. Eve's program does not provide a uniform, linear benefit. The effect of the program on satisfaction likely changes depending on a couple's starting point, a clue we would have missed if we had only calculated Pearson's $r$.

Preview of the Next Lesson

We've established that the program's effect is significant and likely non-linear. But our analysis so far has been limited to just two variables: Before and After satisfaction. In the real world, client outcomes are more complex. Dr. Eve's files contain more data that she collected from her clients: their communication frequency, number of shared activities, and years together. Could these variables help us predict the final satisfaction score more accurately? Next, we will venture into the world of predictive modeling by building a multiple linear regression model to see if we can find the hidden drivers of client success—or uncover statistical evidence of something amiss.