# Goodness of fit of abrupt change model

My model has two parameters (pre 1920 rate, post 1920 rate).
Your model has four parameters (starting rate, first decrease, second
decrease, year that rate of decrease changed). The more parameters
that your model has, the easier it is to fit the data.

Frank Crary said:

However, no one is restricting the number of free parameters in your
model, except yourself: You are (or were) using this data to support
your assertions that: The homicide rate in New South Wales dropped
suddenly after the introduction of gun control laws in 1920, and that
there was no pre-existing trend toward lower rates.
The only thing restricting the number of free parameters in your model
is these assertions. If models based on this do not fit the data well, that
would imply that these assertions are not accurate. If you can post a
more accurate model, which is still consistent with your theory, please do.

A model with as many parameters as data points will fit the data
perfectly. Are these the best models? No, of course not. We should
prefer a model with as small a number of parameters as possible. The
only reason to consider a three parameter model is if we can't find a
two parameter model that fits the data adequately.

To test goodness of fit, we need to find the chi-square value

chisquare = sum( (o[i]-p[i]/sd[i])^2 ) where i=1,2,...,n

o[i] is the observed value at i, p[i] is the value predicted by the
model, and sd[i] is the standard deviation of o[i].

Estimating sd[i] is the tricky part. I assume that homicides are
Poisson distributed. This means that the variance is the same as the
expected number of homicides. We still need to know the expected
number of homicides. Using the model we are testing to tell us this
would be naughty, so I just took the average over the period
1910-1930. The resulting standard deviations are at the end of this
posting.

For my model, over the period 1910-1930, the resulting chi-square
statistic is 24.6, with 19 degrees of freedom, which has a probability
of 0.17.

I conclude that my model gives a good fit to the data, and there is no
reason to consider models with more parameters.

Standard deviations for NSW homicide rate

```1910  0.34
1911  0.34
1912  0.33
1913  0.33
1914  0.32
1915  0.32
1916  0.32
1917  0.32
1918  0.32
1919  0.31
1920  0.31
1921  0.30
1922  0.30
1923  0.30
1924  0.29
1925  0.29
1926  0.29
1927  0.28
1928  0.28
1929  0.28
1930  0.28
```
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