Welcome to the second part of this video lecture on disease mapping of spatial analysis. In the following slides, you will be introduced to some of the methods of spatial smoothing, which can be considered when mapping disease rates. When we want to smooth out disease data, the first thing we need to do is to conceptualize what we consider to be our spatial surrounding. How this works, I will show using the following example. Consider the following five municipalities, called A, B, C, D, and E. Each of these municipalities share some borders with their neighboring municipalities, while some do not share borders. This can be conceptualized using the proximity matrix shown in the lower right side.
Here, each row considers a municipality, and each of the column represents its neighbors. So for example, A shares a border with B and D, and does not share a border with A, C, and E. And the proximity matrix is represented by a 1 for B and D and a 0 for A, C, and E. This means that we believe that A might be influenced by what happens in B and D. The strength of this association and how this influence effects A is given in a spatial weights matrix.
The effect of one area on its neighboring areas is defined by the spatial weights. Spatial weights can be calculated in different ways. The first method we commonly use is by taking the inverse of the distance between two neighboring areas, or more commonly, the inverse of the square distance between two neighboring areas. The second method to define spatial weights is by adjacency. In this case, an area can either be influenced or not be influenced by any of its spatial neighbors. Adjacency is defined as the order of adjacency. For example, you have direct neighbors or neighbors of your neighbors which might influence you. The third method to calculate spatial weight is by the length of the shared border.
Here, we see a map of the Northwestern Europe with the Netherlands showing at the upper left side. The Netherlands shares 2/3 of its border with Germany and about 1/3 third of its border with Belgium. Its spatial weights, in this case would be 2/3 of Germany and 1/3 of Belgium. Germany, in contrast, shares its border with a lot of other countries. So the effect which Netherlands has on Germany will be far smaller than the effect of Germany on the Netherlands. The fourth and final methods which we present here is by interaction, or gravity model. In this case, the effect of one area on its neighbors is determined by a property of that specific area. For example, the population size.
A larger population having a larger influence on its neighboring areas than a smaller population size.
Now, we have conceptualized a spatial neighborhood. And if given some examples on how to define spatial weights, we can continue to calculate some smoothing averages. This formula shows you how to calculate a smooth of local attribute values. The value mu, which is the expected rate in a certain area, for example, a disease incidence rate, equals the sum of incidence rates in its surrounding neighboring areas multiplied by their relative weights, divided by the sum of all weights.
How this works is shown in the following example. Consider again the five municipalities, A, B, C, D and E, with values 1, 3, 4, 2 and 6. To calculate the expected value for example for area A, we use to spacial proximity matrix, which is shown in the middle. Here, the spatial weights are based on the length of the shared border. So for example, A shares 70% of its border with B at 30% of its border with D. Now to calculate the expected value for A, we take 70% times the value of B, which is 3, and 30% times the value of D, which is 2. Adding these up comes to the final expected value for area A, 2.7.
Another, and more sophisticated method to calculate smooth rates is by the method of the empirical Bayes. Here, global smoothers can be used to adjust each area estimate towards a global mean. Likewise, local smoothers based on the directly neighboring areas can be used to adjust each area estimate up or down, depending on the data in neighboring areas.