# How to quantify the agreement between the same parameter from two different data sets

I am looking at Arctic ice thickness from two different Earth-orbiting satellites A and B. I'm interested in quantifying how well these two datasets agree, but I'm struggling over what parameters to use.

I have plotted the data below.

(a) shows the histograms of thickness from A (blue) and B (red).

(b) shows the difference (B minus A) at each location.

(c) shows a scatter plot of thickness from satellite A vs. thickness from satellite B. I have plotted the first principal component in dashed blue, which shows the primary trend in the data. The equation of this line is y= 0.991 x + 0.024 From this comparison I want to extract two quantities:

1) The 'average' difference in thickness between A and B

Looks by eye to be ~ 2 cms

2) The 'goodness' of the correlation.

As a starting point, maybe I need advice on how to fit a PDF to a histogram that is not Gaussian, when I do not know what kind of distribution it is. Other studies tell me that ice thickness is log-normal, but I have values below zero which contradicts this idea (I think)

Any help on this appreciated. It's easy to eye-ball that these two datasets agree 'quite well' but are off by 'a few cms', but I need numbers!

Many thanks

• Could you explain what the negative thicknesses plotted in $(a)$ might mean? Do these satellites have the same resolutions? What are their positional accuracies? – whuber Jan 14 '19 at 18:46
• The thickness I'm showing in these is the thickness of ice sticking above the water (called the ice freeboard). So where it's negative it means that the sea surface (which we interpolate) lies above the ice surface. We expect negative values because we expect a distribution about the mean. I hope that makes sense. The satellites have the same resolution yes, and where I say same location in the above I mean 'in the same grid cell', i.e. each point represents the average of all points (satellite footprints) that fall in a grid cell. – izzyrizzy Jan 14 '19 at 22:45
• (1) You cannot possibly model these values with lognormal distributions. The real thickness may appear lognormal, but the freeboard cannot. (2) Note that two different satellite images of the same location, even when taken by the same satellite, never have the same cells--it's just not possible to guarantee the sensor pixels will be perfectly aligned. If they are registered perfectly, that means the raw data were post-processed by resampling to a fixed grid. This affects how you should compare the images. – whuber Jan 14 '19 at 23:02
• You could get some ideas by searching this site for agreement-statistics. 2) maybe show us a map of the differences? – kjetil b halvorsen Jan 15 '19 at 13:09

## 1 Answer

The simplest way to formally test if the means of measurements are different is by making a t-test on the measurement differences. I understand that your data represents measurements from the same locations but from two different satellites. This would constitute paired observations. I would do a paired t-test. It is simply doing a t-test on the mean of the difference in measurement between the two satellites. Check out the wiki article on the subject.

An important point is that you do not need your underlying variables (or the difference of those variables) to follow a Gaussian distribution for the t-test on to be valid. The t-test builds on the Central Limit Theorem which (loosely said) states that the sum of ANY independent and identically distributed random variable will approximately be Gaussian. Since the mean is just a re-scaled sum, you're good!

• One problem is that the CLT is unlikely to apply, due to potentially strong correlations among the observations. – whuber Jan 14 '19 at 19:56
• That's true, wasn't thinking. Then I would use the paired t-test. Editing my answer. Thanks. – Duffau Jan 16 '19 at 10:22