﻿ VisualStat® Examples - Two-Way Analysis of Variance

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## Two-Way Analysis of Variance

Performs a balanced two-way analysis of variance.

Two-way analysis of variance is a direct extension of one-way analysis of variance. In this case, data are grouped according to two factors rather than a single factor.

#### Data

An evaluation of a new coating applied to 3 different materials was conducted at 2 different laboratories. Each laboratory tested 3 samples from each of the treated materials. The results are given in a dataset.

#### Analysis

1. Open the DataBook anova.vstzz
open this data file via the Help / Open Examples menu; it is in the Sample Data
2. Select the sheet Cover
3. Choose the menu Statistics and the command Two-Way
4. In Response, select Coating. In Factor A, select Materials. In Factor B, select LABS
5. Click Options page
6. Check Display means for Factor A and check Display means for Factor B
7. Click OK

#### Output

```Two-Way Analysis of Variance

-=-=-=-= Coating / Materials / LABS =-=-=-=-

ANOVA TABLE
df  SumOfSquares  Mean Square          F          P
Materials     2        2.1811       1.0906    21.8111     0.0001
LABS          1        5.0139       5.0139   100.2778     0.0000
Interaction   2        0.1344       0.0672     1.3444     0.2973
Error        12           0.6       0.0500
Total        17        7.9294

S         2.236068E-001
R-Sq            92.43 %
R-Sq Adj        78.56 %

Materials   N    Mean   StDev  SE Mean   Effect  95% LCL  95% UCL  Minimum  Maximum
1           6  3.4500  0.7423   0.3030   0.4444   2.6710   4.2290   2.6000   4.3000
2           6  2.6000  0.5514   0.2251  -0.4056   2.0214   3.1786   1.9000   3.3000
3           6  2.9667  0.5428   0.2216  -0.0389   2.3970   3.5363   2.3000   3.6000
<total>    18  3.0056  0.6830   0.1610   0.0000   2.6659   3.3452   1.9000   4.3000

LABS      N    Mean   StDev  SE Mean   Effect  95% LCL  95% UCL  Minimum  Maximum
1         9  3.5333  0.4924   0.1641   0.5278   3.1548   3.9119   2.8000   4.3000
2         9  2.4778  0.3492   0.1164  -0.5278   2.2094   2.7462   1.9000   3.1000
<total>  18  3.0056  0.6830   0.1610   0.0000   2.6659   3.3452   1.9000   4.3000

```

#### Interpreting the results

For the coating data, there is no significant evidence for a materials*labs interaction effect if your acceptable a value is less than 0.2973 (the p-value for the interaction F-test). There is significant evidence for materials main effects, as the F-test p-value is 0.0001.