A method enabling computation of linear rates of change of spatial averages on visual field patterns that have varying test locations over time

This paper introduces and validates a new method for accurately computing linear rates of change in spatial averages across visual field series with varying test locations over time, demonstrating that it achieves slope estimation errors comparable to those of standard fixed-pattern analyses.

The Gist

The Big Problem: Measuring a Moving Target

Imagine you are trying to measure the average temperature of a large garden to see if it is getting hotter or colder over time.

In the medical world, doctors use a test called a "visual field test" to check how well a person sees across their entire vision, similar to checking different spots in that garden. Usually, they check the same 50 or 60 specific spots every time, like measuring the temperature at the same 50 trees. If the trees get hotter, the average goes up, and the doctor knows the garden is warming.

But what if the list of trees you check keeps changing?
Imagine that in January, you check 50 trees. In February, you decide to check those same 50 trees plus 10 new ones. In March, you check the original 50 plus 10 more new ones.

If you just take a simple average of all the trees you checked that month, your average temperature will look like it's dropping, even if the garden isn't changing at all. Why? Because the new trees you added might be in a shady, cool spot that you didn't check before. By adding them to the math, you are "diluting" the average with new, cooler data.

This is exactly the problem the authors (Andrew Turpin and Allison McKendrick) are solving. In eye care, doctors sometimes need to add new test spots to a patient's vision map to get a better look at a specific defect. The old math tricks for calculating "how fast is vision getting worse?" break down when the list of test spots changes.

The Solution: A Smarter Way to Do the Math

The authors propose a new method to calculate the rate of change that ignores the "noise" caused by adding new spots. They call this method sMD + sMD'.

Here is how it works, using a "Garden Party" analogy:

1. The Old Way (Simple Average): You ask everyone at the party (all the test spots) how much they are enjoying the music. If you add 10 new people who just arrived and haven't heard the music yet, their silence lowers the average enjoyment score, even if the original guests are having a great time.
2. The New Way (sMD + sMD'): The authors suggest a two-step check:
   • Step 1: Calculate the average enjoyment of everyone currently at the party (including the new people).
   • Step 2: Calculate the average enjoyment of only the people who were there last week.
   • The Trick: To figure out if the music is getting better or worse, you compare the "everyone" score from this week against the "old-timers" score from last week.

By doing this, you ignore the fact that new people just arrived. You only measure the change in the people who have been there the whole time. This prevents the math from being tricked by the addition of new test spots.

The "Spatial" Secret: Weighting the Map

The paper also mentions that not all spots on the vision map are equal. Some spots cover a bigger area of your vision than others.

• The Analogy: Imagine your vision is a map of a country. Some test spots are like tiny villages; others are like massive cities. If you just count the "happiness" of every village and city equally, your average is skewed because the cities (which cover more land) are under-represented.
• The Fix: The authors use a "spatial weighting" system. They give more importance to the test spots that cover larger areas of the eye, just like you would give more weight to the temperature of a massive city than a tiny village when calculating the country's average temperature.

Did It Work? (The Simulation)

The authors didn't just guess; they ran a computer simulation to test their idea.

• The Setup: They created 50 fake eyes. Some were perfectly stable (not changing), and some were slowly getting worse at random spots. They simulated a scenario where the test pattern kept changing, adding 3 to 10 new spots at every visit.
• The Comparison: They compared three methods:
  1. Simple Average: Just averaging all numbers (The "Bad" way).
  2. Weighted Average: Counting spots by size, but still using the old math (The "Okay" way).
  3. The New Method (sMD + sMD'): Weighting by size and ignoring new spots in the change calculation (The "Good" way).
• The Result: The new method was almost perfect. It calculated the rate of change with almost zero error, matching the results you would get if you had stuck to a fixed, unchanging test pattern the whole time. The other methods were way off, often making stable eyes look like they were getting worse just because new spots were added.

The Bottom Line

The paper claims that it is now possible to accurately measure how fast a patient's vision is changing, even if the doctor changes the pattern of test spots from visit to visit.

By using a clever math trick that:

1. Weighs spots based on how much vision they cover, and
2. Ignores the "shock" of new spots when calculating the change from last time,

Doctors can get a true picture of disease progression (like glaucoma) without being fooled by the fact that they are testing more areas of the eye over time. The authors state this works for both adding new spots and removing them, making it a flexible tool for future, more personalized eye tests.

Technical Summary

Technical Summary: A Method for Computing Linear Rates of Change on Variable Visual Field Patterns

Problem Statement
Clinical visual field (VF) testing typically employs summary metrics, such as Mean Deviation (MD), to monitor disease progression (e.g., glaucoma) over time. These metrics are usually calculated as a simple arithmetic average of sensitivity deviations across a fixed grid of test locations (e.g., the standard 24-2 pattern). While effective for fixed grids, this approach fails when the test pattern changes between visits—a scenario increasingly common in adaptive testing or when switching between patterns (e.g., 24-2 to 24-2C).

The core issue is that a simple average (MD) or even a spatially weighted average (sMD) of newly added test locations introduces artificial changes in the global metric. When a location is first measured, its value is implicitly assumed to be the population mean for all prior visits. If that location is subsequently measured and found to be abnormal, the calculation of the average changes not because the eye has progressed, but because the "sample" has expanded to include a previously unmeasured area. This creates a bias in the linear regression slope used to determine the rate of change (dB/year), potentially misdiagnosing a stable eye as progressing or vice versa.

Methodology
The authors propose a method to compute linear rates of change on spatial averages that remain valid even when test locations vary over time. The approach involves two main components:

1. Spatial Weighting (sMD): Instead of a simple arithmetic mean, the method calculates a spatially weighted average (sMD). Each test location is weighted by the spatial area it represents (e.g., based on the number of adjacent 2x2 degree squares it covers). This ensures that locations in sparse regions do not disproportionately influence the average compared to densely sampled regions.

2. Exclusion of New Locations in Slope Calculation (sMD'): To prevent the artificial change caused by the initial measurement of a new location, the authors introduce a second metric, sMD'. This metric is the spatially weighted average calculated only on locations that were also tested in the previous visit.

   • For a given visit $i$, the change in the field is computed using the difference between the sMD' of visit $i$ (which includes only previously tested points) and the sMD of visit $i-1$.
   • This effectively isolates the change in sensitivity for locations that have been measured at both time points, ignoring the "shock" of adding a new data point to the global average.

3. Regression Implementation: The authors utilize Abel's Lemma (summation by parts) to rewrite the standard linear regression slope formula. This allows the slope to be computed as a weighted sum of the differences between consecutive time points ($y_i - y_{i-1}$), where $y_i$ is derived from sMD' and $y_{i-1}$ from sMD. This mathematical reformulation facilitates the exclusion of newly added locations from the slope calculation while still utilizing all available data for the final global average reporting.

Key Contributions

• Novel Algorithm: The paper introduces a specific algorithm (sMD + sMD') to calculate linear rates of change that accounts for evolving test patterns, addressing a gap in current perimetry analysis where fixed grids are assumed.
• Simulation Framework: The authors developed a simulation environment generating 50 series of visual fields with random progression patterns and dynamically evolving test grids (adding 3–10 random locations per visit).
• Validation: The method was rigorously tested against "ground truth" slopes in simulations involving stable fields, randomly progressing fields, and "extreme" cases (e.g., progression restricted to specific sub-regions).

Results

• Stable Fields: In simulations of stable eyes where the true rate of change is 0 dB/year, standard linear regression on MD or sMD yielded significant false-positive slopes (e.g., -0.48 dB/visit for MD). The proposed sMD+sMD' method correctly yielded a slope of 0.00 dB/visit with no statistical significance for change.
• Progressing Fields: For fields with random progression, the error in slope estimation using the new method was comparable to that of a fixed 24-2 pattern.
  • The difference in error between the fixed 24-2 MD method and the variable-grid sMD+sMD' method was less than 0.003 dB/year (paired t-test, p < 0.001).
  • Standard sMD (without the exclusion of new points) and simple MD showed significantly larger errors in slope estimation due to the inclusion of newly tested locations.
• Extreme Cases: The method performed robustly in "extreme" scenarios where progression was localized to specific areas (e.g., only the 24-2 locations or only non-24-2 locations), accurately reflecting the underlying true rate of change.

Significance and Claims
The paper claims that it is possible to calculate linear rates of change on spatial averages that accurately reflect the true underlying rate of change, even when the test pattern evolves over time. The authors assert that their method eliminates the bias introduced by the addition or removal of test locations, which is a critical limitation in current practices involving adaptive perimetry or pattern switching.

The significance lies in enabling the use of more sophisticated, patient-specific, or adaptive visual field testing strategies without sacrificing the ability to accurately quantify global disease progression. The authors note that while their simulations used random progression and random grid additions, the method is theoretically applicable to any spatial scale (including cluster averages) and works equally well when points are removed or when there is a mix of adding and removing locations. They conclude that the new method produces rates of change similar to those derived from fixed patterns, validating its utility for clinical monitoring in non-standard testing scenarios.