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Two-way ANOVA
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Up: Analysis of Variance Next: Time Series Analysis Previous: One-way ANOVA with Repeated Measures Contents
Two-Way ANOVA
An ANOVA design with two factors is called a two-way analysis of variance. The two-way analysis of variance is
implemented by the TwoWayAnovaModel class.
Constructing One-Way ANOVA Models
The TwoWayAnovaModel class has three constructors.
The first constructor takes three parameters: a NumericalVariable that specifies the dependent variable, and two
CategoricalVariable objects that specify the independent
variables. All three variables must have the same number of observations.
As an example, we construct an ANOVA model to measure the effect of package color and shape on sales. Our data
comes from 12 stores. The two categorical variables are the shape color and the shape shape. The dependent variable
is the total sales of the product in the store.
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string[] shapeData = {"square", "circle", "rectangle"...};
CategoricalVariable shape = new CategoricalVariable("shape", shapeData);
string[] colorData = {"red", "green", "blue"...};
CategoricalVariable color = new CategoricalVariable("color", colorData);
double[] salesData = {1256, 3611...};
NumericalVariable sales = new NumericalVariable("sales", salesData);
TwoWayAnovaModel anova1 = new TwoWayAnovaModel(sales, color, shape); |
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Dim shapeData As String() = {"square", "circle", "rectangle"...}
Dim shape As CategoricalVariable = _
New CategoricalVariable("shape", shapeData)
Dim colorData As String() = {"red", "green", "blue"...}
Dim color As CategoricalVariable = _
New CategoricalVariable("color", colorData)
Dim salesData As Double() = {"red", "green", "blue"...}
Dim salesAs NumericalVariable = _
New NumericalVariable("sales", salesData)
Dim anova1 As TwoWayAnovaModel = New TwoWayAnovaModel(sales, color, shape) |
The second constructor takes four parameters. The first parameter is a VariableCollection that contains the variables you wish to use in the
analysis. The second parameter is the name of the dependent variable in the collection. The third and fourth
parameters are the names of the independent variable in the collection. Using the variables we created in the
previous example, we get:
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VariableCollection variables = new VariableCollection();
variables.Add(sales);
variables.Add(color);
variables.Add(shape);
TwoWayAnovaModel anova2 = new TwoWayAnovaModel(variables, "sales", "color", "shape"); |
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Dim variables As VariableCollection = New VariableCollection()
variables.Add(sales)
variables.Add(color)
variables.Add(shape)
Dim anova2 As TwoWayAnovaModel = _
New TwoWayAnovaModel(variables, "sales", "color", "shape") |
The third constructor also takes four parameters. The first parameter is a DataTable that contains
the data for the analysis. The second parameter is the name of the column that contains the dependent variable data.
The third and fourth parameters are the names of the columns that contain the independent variable data.
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DataTable table = new DataTable();
// Fill the DataTable from some data source...
TwoWayAnovaModel anova3 = new TwoWayAnovaModel(table, "sales", "color", "shape"); |
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Dim table As DataTable = New DataTable()
' Fill the DataTable from some data source...
Dim anova3 As TwoWayAnovaModel = New TwoWayAnovaModel(table, "sales", "color", "shape") |
All three examples produce the same ANOVA model.
Performing the analysis
The
Compute method performs the actual analysis.
The results of the analysis can be obtained through the model's AnovaTable property. The ANOVA table for a two-way design has five rows.
There are three rows that describe the contribution of the model to the varation. There is one row for each of the
factors and one row for the interaction between the two factors. These can be retrieved through the GetModelRow method. As always, there is one row for the
residuals, and one for the complete model. Index 0 gives the row for the first factor, index 1 gives the row for the
second factor, and index 3 gives the row for the interaction.
The ModelRow property is not part of the ANOVA
table. It shows the contribution of the complete model to the variation.
The AnovaModelRow objects obtained in this way show the
results of the test for significance of the variation due to the factor compared to the variation not explained by
the model. The FStatistic property gives the value of
the F statistic for this ratio, while the PValue gives
the actual significance of the F statistic.
The Within Groups row shows the variation of the data around the group means. It corresponds to the error
or residual of the variation in the data after the model has been taken into account. The row is available through
the ANOVA table's ErrorRow property.
The Total row contains the summary data for the entire data set. It can be retrieved through the
TotalRow property of the ANOVA table.
The example below illustrates these properties:
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anova.Compute();
Console.WriteLine("F statistic: {0}", anova.AnovaTable.ModelRow.FStatistic);
Console.WriteLine("P-value : {0}", anova.AnovaTable.ModelRow.PValue);
Console.WriteLine("Sum of sq. total: {0}",
anova.AnovaTable.TotalRow.SumOfSquares);
Console.WriteLine("Sum of sq. error: {0}",
anova.AnovaTable.ErrorRow.SumOfSquares);
Console.WriteLine("Sum of sq. color: {0}",
anova.AnovaTable.GetModelRow(0).SumOfSquares);
Console.WriteLine("Sum of sq. shape: {0}",
anova.AnovaTable.GetModelRow(1).SumOfSquares);
Console.WriteLine("Sum of sq. interaction: {0}",
anova.AnovaTable.GetModelRow(2).SumOfSquares);
Console.WriteLine(anova.AnovaTable.ToString()); |
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anova.Compute();
Console.WriteLine("F statistic: {0}", anova.AnovaTable.ModelRow.FStatistic)
Console.WriteLine("P-value : {0}", anova.AnovaTable.ModelRow.PValue)
Console.WriteLine("Sum of sq. total: {0}", _
anova.AnovaTable.TotalRow.SumOfSquares)
Console.WriteLine("Sum of sq. error: {0}", _
anova.AnovaTable.ErrorRow.SumOfSquares)
Console.WriteLine("Sum of sq. color: {0}", _
anova.AnovaTable.GetModelRow(0).SumOfSquares)
Console.WriteLine("Sum of sq. shape: {0}", _
anova.AnovaTable.GetModelRow(1).SumOfSquares)
Console.WriteLine("Sum of sq. interaction: {0}", _
anova.AnovaTable.GetModelRow(2).SumOfSquares)
Console.WriteLine(anova.AnovaTable.ToString()) |
For the example using the packaging, we find that the F statistic for the color is 2.5049, corresponding to a
p-value of 0.1619. For the shape, we find an F statistic of 7.7670 with a p-value of 0.0317, and for the interaction,
we find an F statistic of 0.1359 with a p-value of 0.8755. The conclusion is that the color of the packaging does not
contribute significantly to the sales of the product, but the impact of the shape is significant at the usual 0.05
level.
The group means can be accessed through the model's Cells property. In the example below, we first
obtain the CategoricalScale object corresponding to the color factor. We then iterate through the levels
of the scale, and print the group means for the square boxes:
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CategoricalScale colorFactor = anova.GetFactor(0);
foreach(object level in colorFactor.GetLevels())
Console.WriteLine("Mean for square boxes group '{0}': {1:F4}",
level, anova.Cells[level, "Square"].Mean); |
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Dim colorFactor As CategoricalScale = anova.GetFactor(0)
For Each level As Object In colorFactor.GetLevels()
Console.WriteLine("Mean for square boxes group '{0}': {1:F4}", _
level, anova.Cells(level, "Square").Mean)
Next |
Using the special index Cell.All, we can obtain the summary statistics for all rectangular packages
of all colors:
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Console.WriteLine("Variance of square packages: {0:F4}",
anova.Cells[Cell.All, "square"].Variance); |
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Console.WriteLine("Variance of square packages: {0:F4}", _
anova.Cells[Cell.All, "square"].Variance) |
The grand mean is obtained by setting both indices to Cell.All:
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Console.WriteLine("Grand mean: {0:F4}",
anova.Cells[Cell.All, Cell.All].Mean); |
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Console.WriteLine("Grand mean: {0:F4}", _
anova.Cells[Cell.All, Cell.All].Mean) |
Up: Analysis of Variance Next: Time Series Analysis Previous: One-way ANOVA with Repeated Measures Contents
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