# Residual Variance Ratio Test

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```					                                                                                           Tutorial 9 Solutions (Roy, Nov. 21st, 2008)

To solve this problem, we can use the following two ratio tests: Residual Variance Ratio Test
and Mean Square Regression Ratio Test.

1. Residual Variance Ratio Test
As we have replicates in the study, so we can use the residual variance ratio test to
assess the adequacy of the fitted model. The test statistic is
2
sresiduals MSE
2
 2
sinherent sinherent
The MSE (mean squared error of residuals) is calculated as
n                           n              n

 y  y                   e             e
2
ˆ i     i
2
i
2
i
595.37
MSE      i 1
   i 1
   i 1
           33.076
n p                n p           20  2             18
where there are 20 observations ( n  20 ) and two parameters fitted ( p  2 ).
We will use replicate data to estimate the inherent variance. By observing the data, we
know that there are four sets of replicates, each with 2 data points (as shown in the
following table).
xk                             ykj        j  1, 2             yk                    s k2
22
k 1                         2.1                                                             20                     8
18
18
k 2                         2.9                                                            21.5                  24.5
25
25
k 3                         3                                                              21.5                  24.5
18
27
k 4                         3.4                                                             24                    18
21

where y k and s k2 are the sample average and sample variance within the k th set of

replicates, and are calculated using following formula
nk
1
yk 
nk
y
j 1
kj

1 nk
  ykj  yk 
2
sk 
2

nk  1 j 1

where nk is the number of replicated data in the k th set of replicates (in this problem

nk  2 for k  1, 2,3, 4 ).
1
Tutorial 9 Solutions (Roy, Nov. 21st, 2008)

So we estimate the inherent variance as
m

 (n            k    1) sk 2
sinherent 
2            k 1
m
 18.75
n k 1
k   m

where m is the total number of replicate sets (in this problem, m  4 ).
Now we can calculate the test statistic as
MSE 33.076
2
        1.7641
sinherent   18.75

which will be compared to the fence value from F distribution with n  p and v degrees
of freedom, where
n  p  20  2  18
m
v   nk  m  4
k 1

From the F table, we know that
F ,18,4  F0.05,18,4  5.82                         (Note: interpolation is needed)

where we use 5% significance level (   0.05 ). As the test statistic value ( 1.7641) is less
than the fence value, so with 5% significance level, we cannot say that MSE is relatively
2
large than sinherent , which means no evidence of inadequacy is detected.

2. Mean Square Regression Ratio Test
The test statistic is
MSR
MSE
where the mean square regression is
n

 y  y 
2
ˆ          i
651.83
MSR        i 1
           651.83
p 1                      1
As MSE  33.076 calculated above, so the test statistic value can be calculated as
MSR 651.83
        19.71
MSE 33.076

2
Tutorial 9 Solutions (Roy, Nov. 21st, 2008)

The fence value is from the F distribution with p  1 and n  p degrees of freedom,
where
p 1  2 1  1
n  p  20  2  18
Check the F Table, we have
F , p 1,n  p  F0.05,1,18  4.41

Since the test statistic value is larger than the fence value, with 5% significance level, we
conclude that MSR (variability accounted by the model) is statistically larger than MSE
(variability left over after fitting the model), which means a significant trend has been
modeled.

End of Solution

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