One-Way ANOVA Model #1

Using MINITAB

Data Entry

Enter the responses in C1, named 'y', by treatment groups from A to D. Enter the treatment groups in C2, named 'brand' by using CalcMake Patterned DataText Values as follows

(rust01.png here)
Then the dialog box:
(rust02.png here)

One-Way ANOVA with pre-plot, residual plots, and pairwise comparisons

Setup
Do StatANOVAOne-Way as follows
(rust03.png here)
Session Output
One-way ANOVA: y versus brand 

Source  DF        SS       MS       F      P
brand    3  15953.47  5317.82  866.12  0.000
Error   36    221.03     6.14               
Total   39  16174.50

S = 2.478   R-Sq = 98.63%   R-Sq(adj) = 98.52%


                          Individual 95% CIs For Mean Based on
                          Pooled StDev
Level   N    Mean  StDev  ----+---------+---------+---------+-----
A      10  43.140  3.000    (*)
B      10  89.440  2.218                                   (*)
C      10  67.950  2.169                    (*)
D      10  40.470  2.436  (*)
                          ----+---------+---------+---------+-----
                             45        60        75        90

Pooled StDev = 2.478


Tukey 90% Simultaneous Confidence Intervals
All Pairwise Comparisons among Levels of brand

Individual confidence level = 97.70%


brand = A subtracted from:

brand   Lower  Center   Upper  -------+---------+---------+---------+--
B      43.667  46.300  48.933                                  *)
C      22.177  24.810  27.443                          (*)
D      -5.303  -2.670  -0.037                 (*)
                               -------+---------+---------+---------+--
                                    -30         0        30        60


brand = B subtracted from:

brand    Lower   Center    Upper  -------+---------+---------+---------+--
C      -24.123  -21.490  -18.857           (*)
D      -51.603  -48.970  -46.337  (*)
                                  -------+---------+---------+---------+--
                                       -30         0        30        60


brand = C subtracted from:

brand    Lower   Center    Upper  -------+---------+---------+---------+--
D      -30.113  -27.480  -24.847         (*)
                                  -------+---------+---------+---------+--
                                       -30         0        30        60


Fisher 95% Individual Confidence Intervals
All Pairwise Comparisons among Levels of brand

Simultaneous confidence level = 80.32%


brand = A subtracted from:

brand   Lower  Center   Upper  -------+---------+---------+---------+--
B      44.053  46.300  48.547                                  *)
C      22.563  24.810  27.057                           *)
D      -4.917  -2.670  -0.423                 (*)
                               -------+---------+---------+---------+--
                                    -30         0        30        60


brand = B subtracted from:

brand    Lower   Center    Upper  -------+---------+---------+---------+--
C      -23.737  -21.490  -19.243           (*)
D      -51.217  -48.970  -46.723  (*
                                  -------+---------+---------+---------+--
                                       -30         0        30        60


brand = C subtracted from:

brand    Lower   Center    Upper  -------+---------+---------+---------+--
D      -29.727  -27.480  -25.233         (*)
                                  -------+---------+---------+---------+--
                                       -30         0        30        60
Individual Value Plot of y vs brand

(rust04.png here)
Note that MINITAB jitters (i.e., adds small random noise around) data points. Treatment means appear to be different with constant variance.

Normplot of Residuals for y

(rust05.png here)
No serious departure from nromality detected.

Residuals vs Fits for y

(rust06.png here)
No apparent violation of and homoscedasticity and additivity.

Reporting The Outcomes

Underlining Using Tukey Method

  D     A     C     B
40.47 43.14 67.95 89.44
Note that all comparisons are significant (with the comparison between D and A being marginally so) and consequently no underline was drawn. The family error rate was set at 10% resulting in individual confidence level of 97.70% with an error rate of only 2.3%.

Underlining Using Fisher LSD Method

  D     A     C     B
40.47 43.14 67.95 89.44
Again, all comparisons are significant. Notice that, with individual error rate set at 5%, the overall error was 19.68% (since overall confidence level was 80.32%).

Test for Equal Variances: y versus brand

Setup

Do StatANOVATest for Equal Variances as follows

(rust07.png here)

Session Output
95% Bonferroni confidence intervals for standard deviations

brand   N    Lower    StDev    Upper
    A  10  1.87765  3.00007  6.63472
    B  10  1.38830  2.21821  4.90560
    C  10  1.35725  2.16859  4.79587
    D  10  1.52481  2.43632  5.38797


Bartlett's Test (normal distribution)
Test statistic = 1.20, p-value = 0.753


Levene's Test (any continuous distribution)
Test statistic = 0.23, p-value = 0.878
Test for Equal Variances: y versus brand

(rust08.png here)
There is no apparent violation of homoscedasticity.

Balanced ANOVA with expected mean squares

Setup
Do StatANOVABalanced ANOVA as follows
(rust09.png here)
Session Output
ANOVA: y versus brand 

Factor  Type   Levels  Values
brand   fixed       4  A, B, C, D


Analysis of Variance for y

Source  DF       SS      MS       F      P
brand    3  15953.5  5317.8  866.12  0.000
Error   36    221.0     6.1               
Total   39  16174.5


S = 2.47787   R-Sq = 98.63%   R-Sq(adj) = 98.52%


                                         
                             Expected
                             Mean Square
                             for Each
                             Term (using
            Variance  Error  unrestricted
   Source  component   term  model)      
1  brand                  2  (2) + Q[1]
2  Error       6.140         (2)


Means

brand   N       y
A      10  43.140
B      10  89.440
C      10  67.950
D      10  40.470

Orthogonal Contrasts

Setup

To create contrast, say 'amed vs generic', do DataCodeText to Numeric as follows

(rust10.png here)
The other two contrasts are defined likewise
(rust11.png here)

Checking Orthogonality

From class notes, we have claimed that these contrasts are orthogonal. This can be verified in MINITAB by copying these contrasts into a matrix, say M1; obtaining its transpose, say M2; then create a product matrix (of M2 by M1), say M3. The resulting matrix M3 should be a diagonal matrix.
To copy columns to matrix, do DataCopyColumns to Matrix as follows

(rust12.png here)
To transpose a matrix, do CalcMatricesTranspose as follows
(rust13.png here)
To perform matrices multiplication, do CalcMatricesArithmetic as follows
(rust14.png here)
To display matrix M3, do DataDisplay Data as follows
(rust15.png here)

Session Output
Data Display 

 Matrix M3

10   0   0
 0  20   0
 0   0  20

Estimates

To obtain estimates of the contrasts, one can perform computation in MINITAB by submitting commands (doing EditCommand Line Editor) as follows

(rust16.png here)
Then display the constants K1 to K3

Data Display 

psi1    36.8900
psi2    2.67000
psi3    21.4900

Estimates and components using regression analysis

As a matter of fact, it's easier just to run regression analysis of the response against these three contrast columns (doing StatRegressionRegression) as follows

(rust17.png here)

Session Output
Regression Analysis: y versus named vs generic, A vs D, B vs C 

The regression equation is
y = 60.3 + 36.9 named vs generic + 1.34 A vs D + 10.7 B vs C


Predictor            Coef  SE Coef       T      P
Constant          60.2500   0.3918  153.78  0.000
named vs generic  36.8900   0.7836   47.08  0.000
A vs D             1.3350   0.5541    2.41  0.021
B vs C            10.7450   0.5541   19.39  0.000


S = 2.47787   R-Sq = 98.6%   R-Sq(adj) = 98.5%


Analysis of Variance

Source          DF       SS      MS       F      P
Regression       3  15953.5  5317.8  866.12  0.000
Residual Error  36    221.0     6.1               
Total           39  16174.5


Source            DF   Seq SS
named vs generic   1  13608.7  C1
A vs D             1     35.6  C2
B vs C             1   2309.1  C3

Note that to get the F statistics of the contrasts, simply square their respective t statistics.