# Autocorrelation in the NSW homicide statistics?

Charles Scripter writes:

BTW, I notice that your web page still seems to purport that your
analysis was correct, even though your friends over in sci.stat.edu
pointed out that it was not correct;

That's an interesting interpretation of the discussion.

Perhaps you'd like to correct this "oversight".

No problem, here's something from one of my friends in sci.stat.edu:

Barry McDonald writes:

THE ARGUMENT ABOUT AUTOCORRELATION IN THE NSW HOMICIDE STATISTICS

One complaint by Scripter about your data was to do with the evident
You see sometimes autocorrelation is not real, but just apparent- arising from
having omitted a variable from the analysis. The Minitab output shows this
is the
case for your data. First I should explain that a Durbin-Watson statistic
shows significant autocorrelation if D< DL where DL is a number from tables,
clear non-significant autocorrelation if D>DU where DU is also from tables,
and inconclusive results if DL<= D <= DU.

Suppose we just fit a single line to the data:

MTB >  Regress 'homocide' 1 'year';
SUBC>  Constant;
SUBC>  DW.

Regression Analysis

The regression equation is
homocide = 47.3 - 0.0236 year

Predictor       Coef       Stdev    t-ratio        p
Constant       47.30       16.08       2.94    0.006
year       -0.023646    0.008381      -2.82    0.008

s = 0.5666      R-sq = 18.1%     R-sq(adj) = 15.8%

Durbin-Watson statistic = 1.23

Notice that if one just fits a straight line in terms of year then the
overall trend is downwards!! but the Durbin-Watson statistic
indicates significant autocorrelation at the 5% level (D< DL=1.41) and
nearly at the 1% level (1.21).

The autocorrelation is cleared up by fitting a more appropriate analysis
(see after graphs)

MTB > GStd.
MTB > Plot 'homocide' 'year';
SUBC> Symbol 'x'.

Character Plot

3.20+
-         x   x xx

homocide-
- x x x
- x x
2.40+ x x
- x
- x x x x
- x x
- x x xx x
1.60+ x x x x
- x x x x x x
- x
- x
- x x x
0.80+
-
--------+---------+---------+---------+---------+--------year
1904.0 1911.0 1918.0 1925.0 1932.0

MTB GPro.
MTB GStd.
MTB Plot 'FITS1' 'year';
SUBC Symbol 'x'.

Character Plot

2.40+   x
-     xx

FITS1 - xxx
- xx
- xx x
2.10+ xx
- xx x
- x xx
- x x
- x xx
1.80+ xx
- x xx
- xx
- xxx
- xx
1.50+ xx
-
--------+---------+---------+---------+---------+--------year
1904.0 1911.0 1918.0 1925.0 1932.0

MTB GPro.

Allowing for a change in intercept level with the law change in 1920:

MTB > Regress 'homocide' 2 'year' 'lawchnge';
SUBC>  Constant;
SUBC>  DW.

Regression Analysis

The regression equation is homocide = - 40.4 + 0.0223 year - 1.18 lawchnge

Predictor Coef Stdev t-ratio p Constant -40.37 26.94 -1.50 0.143 year 0.02233 0.01410 1.58 0.122 lawchnge -1.1769 0.3111 -3.78 0.001

s = 0.4841 R-sq = 41.9% R-sq(adj) = 38.6%

Durbin-Watson statistic = 1.63

There is clear evidence not to reject the hypothesis of zero
autocorrelation if D>DU=1.52. This is indeed the case so the addition of
this extra variable has simultaneously
given us a much more believable analysis (see two lines below: not going
down this time!!) and removed the apparent autocorrelation.
The choice of this cut point (1920) has had a very significant effect
(p-value approx 0.001). Note however that the trend in year is not
significantly different to zero.

MTB > GStd.
MTB > Plot 'FITS2' 'year';
SUBC> Symbol 'x'.

Character Plot

-
-                             xx x
2.40+                       x xx x
-                   xx x

FITS2 - x xx x
- xx x
- x xx x
2.00+
-
-
-
- x xx
1.60+ xxx x
- x xx
- xxx x
- x xx
-
--------+---------+---------+---------+---------+--------year
1904.0 1911.0 1918.0 1925.0 1932.0

If we just assume constant rates of homicides before 1921 and constant
after 1920, (i.e. zero slopes) then we retain a significant drop in
the level of homicides, and the autocorrelation is still not
significant. (though I would be cautious). The conclusion from this
test is exactly the same as you were trying to do by a two-sample
t-test except that in this analysis we have the added feature of
checking whether the autocorrelation is significant.

MTB >Regress 'homocide' 1 'lawchnge';
SUBC>  Constant;
SUBC>  DW.

Regression Analysis

The regression equation is
homocide = 2.28 - 0.753 lawchnge

Predictor       Coef       Stdev    t-ratio        p
Constant      2.2762      0.1078      21.11    0.000
lawchnge     -0.7527      0.1612      -4.67    0.000

s = 0.4941      R-sq = 37.7%     R-sq(adj) = 36.0%

Analysis of Variance

SOURCE       DF          SS          MS         F        p
Regression    1      5.3221      5.3221     21.80    0.000
Error        36      8.7887      0.2441
Total        37     14.1108

Unusual Observations
Obs. lawchnge   homocide        Fit  Stdev.Fit   Residual    St.Resid
4      0.00     1.0000     2.2762     0.1078    -1.2762      -2.65R

R denotes an obs. with a large st. resid.

Durbin-Watson statistic = 1.52

(D= DU=1.52 so there is no proof of autocorrelation )

Of course this analysis does not clear up Scripter's complaint that
(in his eyes) the law change year is irrelevant to homicides and so
an adjacent year could be used to give a similar significant
result. That is a causal matter that I cannot comment on as a
statistician. - except that since a highly significant effect is
apparent in the data, it really behooves him to come up with a
better explanation than yours, especially as to why he postulates
any other year as the change point.

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