﻿ Two-Way Analysis of Variance

# Two-Way Analysis of Variance

### Statistics > ANOVA > Two-Way

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.

### Dialog box items

Response:
Enter the column containing the response variable.

Factor A:
Enter one of the factor level columns.

Factor B:
Enter the other factor level column

Report:
The display of outputs of VisualStat.

### Data

The response variable must be numeric and in one column. Factors can be numeric, or text.

### Charts

Displays a histogram, a histogram with a normal curve, and a boxplot.

Histogram:
Choose to display a histogram for each variable

Histogram with Normal Curve:
Choose to display a histogram with a normal curve for each variable

Boxplot of data:
Choose to display a boxplot for each variable

Options:
Choose the options you want.

oExclude missing values: Check to excludes rows that have missing values.

oInverted Bar: Check to reverse the axes.

### Options

Display means for Factor A:
Check to compute marginal means and confidence intervals for each level of the row factor

Display means for Factor B:
Check to compute marginal means and confidence intervals for each level of the column factor

Alternative Hypothesis:
Enter Two-sided, Upper One-sided, or Lower One-sided. If you choose an One-sided hypothesis test, an upper or lower confidence bound will be constructed, respectively, rather than a confidence interval.

Confidence Level:
Enter the level of confidence desired. Enter any number between 0 and 100. Entering 90 will result in a 90% confidence interval. The default is 95%.

### Example

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.

1.Open the DataBook anova.vstz

2.Select the sheet Cover

3.Choose the tab Statistics, the group Anova 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

Report window 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.