MTB > # Analysis of Hunter L Values and Consumer Panel Scores for Nine Lots
MTB > # of Canned Tuna
MTB > # First I'll input the data.
MTB > set c1
DATA> 44.4 45.9 41.9 53.3 44.7 44.1 50.7 45.2 60.1
DATA> end
MTB > set c2
DATA> 2.6 3.1 2.5 5.0 3.6 4.0 5.2 2.8 3.8
DATA> end
MTB > name c1 'Hunter L' c2 'c. score'
MTB > print c1 c2
ROW Hunter L c. score
1 44.4 2.6
2 45.9 3.1
3 41.9 2.5
4 53.3 5.0
5 44.7 3.6
6 44.1 4.0
7 50.7 5.2
8 45.2 2.8
9 60.1 3.8
MTB > # I'll produce a scatter plot that will give us some idea about the nature
MTB > # of the relationship between Hunter L value and consumer score.
MTB > plot c2 c1
-
- *
5.0+ *
-
c. score-
-
-
4.0+ *
- *
- *
-
- *
3.0+
- *
- * *
-
-
----+---------+---------+---------+---------+---------+--Hunter L
42.0 45.5 49.0 52.5 56.0 59.5
MTB > # There seems to be a positive association, but not a really strong one.
MTB > # Now I'll do a test for association using Kendall's tau. Because of the small
MTB > # sample size, one can easily determine that t = 4/9 (about 0.444) and go to
MTB > # Table L in the Appendix of G&C and obtain that the p-value for a two-tailed
MTB > # test is about 0.12. (StatXact gives that the exact p-value is about 0.119.)
MTB > # Just for fun I'll see how a normal approximation works. First I'll use
MTB > # the z statistic given near the middle of p. 11-6 of the class notes.
MTB > let k1 = 3*sqrt(9*8/(2*23))*(4/9)
MTB > cdf k1 k2;
SUBC> norm 0 1.
MTB > let k2 = 2*(1 - k2)
MTB > name k1 'z' k2 'p-value'
MTB > print k1 k2
z 1.66812
p-value 0.0952928
MTB > # Now I'll apply the continuity correction indicated right below the
MTB > # z formula on p. 11-6 of the class notes.
MTB > let k1 = 3*sqrt(9*8/(2*23))*(4/9 - 1/36)
MTB > cdf k1 k2
MTB > let k2 = 2*(1 - k2)
MTB > print k1 k2
z 1.56386
p-value 0.117851
MTB > # This approximate p-value of about 0.118 is pretty close to the exact p-value
MTB > # of about 0.119, even though the sample size is rather small. StatXact gives
MTB > # an approximate p-value of about 0.0425, which is only about one third of the
MTB > # correct value. (StatXact's approximate value is based on a normal approximation
MTB > # w/o a cont. correction. But they use a different formula for the variance. It
MTB > # isn't clear to me why they do this, since obviously using the formula from G&C
MTB > # seems to work a lot better, particularly when a continuity correction is applied.)
MTB > # The estimate of the variance given a bit below the top of p. 11-7 of the class
MTB > # notes leads to an estimated standard error of about 0.160375. Using the confidence
MTB > # interval formula given at the bottom of p. 11-7 produces (0.130, 0.759) as an
MTB > # approximate 95% c.i. for tau. This differs from the interval supplied by StatXact
MTB > # which results from using their ASE1 value in place of the estimated standard error
MTB > # given by G&C (and given on p. 11-7 of the class notes).
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*** StatXact information ***
To do the test based on Kendall's tau, use
Nonparametrics > Ordinal Response > Kendall's tau and Somer's D...
Assuming Var1 and Var2 contain the two columns of data, click them into the boxes for
Varible 1 and Variable 2. Then select Exact under Compute, and click OK.
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MTB > # Although Minitab doesn't do the test based on Spearman's rho, it can be
MTB > # used as shown below to obtain the value of the statistic.
MTB > rank c1 c11
MTB > rank c2 c12
MTB > corr c11 c12
Correlation of C11 and C12 = 0.600
MTB > # From Table M of G&C it can be determined that the p-value for a two-tailed test is
MTB > # about 0.096. Using StatXact, an exact p-value of about 0.097 is obtained. (The
MTB > # slight discrepancy is due to the book rounding the one-tailed value of about 0.0484
MTB > # to 0.048. If the table in the book had used more accuracy and gave 0.0484, then
MTB > # doubling that value would give us about 0.097 for a two-tailed test p-value.)
MTB > # I'll try the normal and t approximations given on p. 11-13 of the class notes.
MTB > # First the normal approx. given near the middle of p. 11-13 of notes.
MTB > let k1 = sqrt(8)*0.6
MTB > cdf k1 k2;
SUBC> norm 0 1.
MTB > let k2 = 2*(1 - k2)
MTB > # Now the t approx. given at the bottom of p. 11-12.
MTB > let k3 = sqrt(7)*0.6/sqrt( 1 - 0.6**2 )
MTB > cdf k3 k4;
SUBC> t 7.
MTB > let k4 = 2*(1 - k4)
MTB > name k3 't' k4 't p-v'
MTB > print k1-k4
z 1.69706
p-value 0.0896859
t 1.98431
t p-v 0.0875434
MTB > # The t approx. p-value of about 0.088 is in agreement with
MTB > # StatXact's asymptotic p-value. (StatXact uses the same t
MTB > # approximation. The slight difference in values is due to
MTB > # one (or both) of the software packages being a bit inaccurate
MTB > # in obtaining t distribution probabilities.)
-------------------------------------------------------------------------------------
*** StatXact information ***
To do the test based on Spearman's coefficient, use
Nonparametrics > Ordinal Response > Spearman's Correlation...
Assuming Var1 and Var2 contain the two columns of data, click them into the boxes for
Varible 1 and Variable 2. Then select Exact under Compute, and click OK.
-------------------------------------------------------------------------------------
MTB > # We can also do a test using Pearson's sample correlation coefficient.
MTB > # We can get the value of the estimated correlation as shown below.
MTB > corr c1 c2
Correlation of Hunter L and c. score = 0.571
MTB > # A quick way to get a p-value based on the normal theory t statistic is
MTB > # to do a regression and look at the p-value for the slope parameter.
MTB > # (With a simple regression, the t test about the slope is equivalent to
MTB > # the t test based on Pearson's sample correlation coefficient.)
MTB > regress c2 1 c1
The regression equation is
c. score = - 1.02 + 0.0972 Hunter L
Predictor Coef Stdev t-ratio p
Constant -1.024 2.540 -0.40 0.699
Hunter L 0.09718 0.05278 1.84 0.108
MTB > # The p-value is about 0.108. StatXact can be used to get an exact p-value
MTB > # using the exact permutation-based null distribution instead of a null
MTB > # distribution based on an assumption of normality. StatXact's exact p-value
MTB > # is about 0.10.
-------------------------------------------------------------------------------------
*** StatXact information ***
To do the test based on Pearson's coefficient, use
Nonparametrics > Ordinal Response > Pearson's Correlation...
Assuming Var1 and Var2 contain the two columns of data, click them into the boxes for
Varible 1 and Variable 2. Then select Exact under Compute, and click OK.
-------------------------------------------------------------------------------------
MTB > save 'tuna'
Saving worksheet in file: tuna.MTW