1. A theory of transmission bias
Assume that people report correctly what they know.
The comparison of the data from clergy and others shows that whatever the errors are, they are consistent.
Shortly before his untimely death. Peter Killworth proposed the following: (1) Instead of assuming that people are inaccurate, assume that they report correctly what they know. After all, whatever the errors are in the model’s predictions, we see from slide 16 – which we repeat in the next slide – that those errors are consistent (see next slide). 1

2. Whatever the errors are in the model’s predictions, we see from slide 16 that those errors are consistent. 2

3. Most Americans know a Christopher
It’s likely that you know at least one Christopher
That is, the probability of knowing NO Christophers is close to zero.
Twins are likely to be underreported.
But what’s the truth? How can we draw the curve on that jagged diagram so that the true values are represented?
From the graph in slide 43, we see that Americans are very likely to know at least one person named Christopher. We also see that twins are probably underreported. The population of twins is very large (about 1 in 125 births), but about 30% of Americans reported in our surveys that they did not know anyone who has a twin. The problem is, we don’t know what the truth is. We’d like to be able to re-draw the graph in slide 43 so that the true values were represented, not just what people report. 3

4. Suppose people report accurately
In other words, given the structure of that diagram, we decided to trust our informants and assume that they are reporting correctly what they know.
It’s just that what they know is incorrect.
That jaggedy curve doesn’t tell us where the curve would be if people responded honestly to correct information instead of honestly to incorrect information.
To do this, instead of assuming inaccurate informants, suppose we assume that people are accurate in their reporting. It’s just that what they know is incorrect. 4

5. This means adjusting the x-axis rather than the y-axis
Suppose that widows don’t tell half the people they know about their being a widow.
The 0.13 on the x-axis would remain the same but the number that people would be responding to would be .013/2.
To make the x-axis the effective size of that population, we would slide it to the left while the y-axis would remain the same.
Widows are 0.13 of the population in the U.S. … Suppose that widows only tell half the people they know that they are widows. Then, some people who report that they don’t know any widows would be incorrect, but would still be reporting correctly what they know. To adjust for this, we would slide the x-axis in the graph to the left while keeping the y-axis the same. 5

6. Of course, we have no idea what the transmission error might be – that’s what we tried in vain to get with weightings.
We only know that if the numbers remain the same on the y-axis and we make up the effective sizes on the x-axis, the jaggedy line would go.
How big an adjustment should we make to the x-axis? We don’t know – that’s what we could not find out with any of the weightings. 6

7. Killworth did this analytically by satisfying certain mathematical properties.
We know the probability of knowing none and also of knowing just one person in a subpopulation. These have to be related mathematically, which leads to a well-defined set of values for the effective subpopulation.
We can then compute the predicted distribution of c. This next diagram shows that we may be on the right track.