Section 3.4. Robust correlation and regression.

We work with the aluminum data in ALMPIN.mtw
MTB > Retrieve  'C:\MTBSEW\INDUST~1\MTW\ALMPIN.MTW'.

We find the robust correlations as it is done in example 3.8 of the textbook. 
We only use the 4th and 7-th boards of the data. We take x-variable=tet-dev 
(c4) and y-variable y-dev (c3)

First, we select those boards.
MTB > Retrieve  'C:\MTBSEW\INDUST~1\MTW\PLACE.MTW'.
MTB > copy c3 c13;
SUBC> use c1=4,7.
MTB > copy c4 c14;
SUBC> use c1=4,7.

If we do the boxplots of c13 and c14, we find that these data has outliers. 
So, it make sense to use robust methods. We are goind to find the robust 
correlation of c13 and c14, using the method in pages 63-66 of the book.
Next, we find the trimmed sample mean and trimmed sample standard deviation
MTB > sort c13 c23
MTB > sort c14 c24
MTB > delete 1,2 31,32 c22-c24
MTB > desc c24
                N     MEAN   MEDIAN   TRMEAN    STDEV   SEMEAN
C24            28  0.01612  0.00063  0.01453  0.02366  0.00447

              MIN      MAX       Q1       Q3
C24      -0.00332  0.07677 -0.00144  0.03400
MTB > desc c23
                N     MEAN   MEDIAN   TRMEAN    STDEV   SEMEAN
C23            28 -0.00209 -0.00210 -0.00210  0.00050  0.00010

              MIN      MAX       Q1       Q3
C23      -0.00281 -0.00113 -0.00257 -0.00176
MTB > let c31=((c14-mean(c24))/stde(c24))+((c13-mean(c23))/stde(c23))
MTB > let c32=((c14-mean(c24))/stde(c24))-((c13-mean(c23))/stde(c23))
If we print c14 c13 c12 c31 c32 we get the Table 3.12 in page 85 of the book: 
MTB > print c14 c13 c31 c32

Next, we find the robust variances of the samples Z_1 and Z_2 
and find the robust correlation:
MTB > sort c31 c35
MTB > sort c32 c36
MTB > delete 1,2 31,32 c35 c36
MTB > let k1=stdev(c35)**2
MTB > let k2=stdev(c36)**2
MTB > let k3=(k1-k2)/(k1+k2)
MTB > print k1 k2 k3

K1       2.04002
K2       1.70318
K3       0.0899896

MTB > corre c13 c14

Correlation of C13 and C14 = 0.366
We get that the robust correlation is 0.0899896, 
which is much smaller than the Pearson correlation 0.366.

We can also find the Spearman rank-order correlation:
MTB > rank c13 c23
MTB > rank c14 c24
MTB > corre c24 c23

Correlation of C24 and C23 = 0.148

****************************************************
Next, we find the robust regression line for the solar cell (socell.dat). 
We want to find the regression line of c2 by c1.
Since there are 16 observations, we use the first 5 observations to find 
the median for the first group of observations and the 5 last observations 
to find the other medians:

MTB > sort c1 c8
MTB > sort c2 c9;
SUBC> by c1.
MTB > Copy C8 c10;
SUBC> Use 1:5.
MTB > Copy C9 c11;
SUBC> Use 1:5.
MTB > Copy C8 c14;
SUBC> Use 12:16.
MTB > Copy C9 c15;
SUBC> Use 12:16.
MTB > let k1=medi(c8)
MTB > let k2=medi(c9)
MTB > let k3=medi(c10)
MTB > let k4=medi(c11)
MTB > let k5=medi(c14)
MTB > let k6=medi(c15)
MTB > let k7=(k6-k4)/(k5-k3)
MTB > let k8=k2-(k7*k1)
MTB > print k8 k7

K8       0.595205
K7       0.904110
We get that the resistant regression line is
Time_2 = 0.595205+ 0.904110Time_1

Meanwhile the usual regression line is given by this equation:
MTB > regr c2 1 c1

The regression equation is
Time_2 = 0.536 + 0.929 Time_1

Comments to: Miguel A. Arcones