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One-way ANOVA
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One Way ANOVA
The simplest ANOVA design considers one factor only. This is called one-way ANOVA. The one-way analysis of
variance is implemented by the OneWayAnovaModel class.
Constructing One-Way ANOVA Models
The OneWayAnovaModel class has three constructors.
The first constructor takes two parameters: a NumericalVariable that specifies the dependent variable, and a
CategoricalVariable that specifies the independent
variable. The two variables must have the same number of observations.
As an example, we construct an ANOVA model for measurements of the resistance of silicon wafers using 5 different
instruments. The factor is the instrument. The numerical value is the resistance.
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int[] instrumentData =
{1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5};
CategoricalVariable factor = new CategoricalVariable("Instrument", instrumentData);
double[] resistanceData = {
196.3052, 196.1240, 196.1890, 196.2569, 196.3403,
196.3042, 196.3825, 196.1669, 196.3257, 196.0422,
196.1303, 196.2005, 196.2889, 196.0343, 196.1811,
196.2795, 196.1748, 196.1494, 196.1485, 195.9885,
196.2119, 196.1051, 196.1850, 196.0052, 196.2090};
NumericalVariable dependent = new NumericalVariable("Resistance", resistanceData);
OneWayAnovaModel anova1 = new OneWayAnovaModel(dependent, factor); |
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Dim instrumentData As Integer() = _
{1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5}
Dim factor As CategoricalVariable = _
New CategoricalVariable("Instrument", instrumentData)
Dim resistanceData As Double{} = { _
196.3052, 196.1240, 196.1890, 196.2569, 196.3403, _
196.3042, 196.3825, 196.1669, 196.3257, 196.0422, _
196.1303, 196.2005, 196.2889, 196.0343, 196.1811, _
196.2795, 196.1748, 196.1494, 196.1485, 195.9885, _
196.2119, 196.1051, 196.1850, 196.0052, 196.2090}
Dim dependent As NumericalVariable = _
New NumericalVariable("Resistance", resistanceData)
Dim anova1 As OneWayAnovaModel = New OneWayAnovaModel(dependent, factor) |
The second constructor takes three 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 parameter is the
name 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(dependent);
variables.Add(factor);
OneWayAnovaModel anova2 = new OneWayAnovaModel(variables, "Resistance", class="str">"Instrument"); |
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Dim variables As VariableCollection = New VariableCollection()
variables.Add(dependent)
variables.Add(factor)
Dim anova2 As OneWayAnovaModel = New OneWayAnovaModel(variables, "Resistance", class="str">"Instrument") |
The third constructor also takes three 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 parameter is the name of the column that contains the independent variable data.
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DataTable table = new DataTable();
// Fill the DataTable from some data source...
OneWayAnovaModel anova3 = new OneWayAnovaModel(table, "Resistance", class="str">"Instrument"); |
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Dim table As DataTable = New DataTable()
' Fill the DataTable from some data source...
Dim anova3 As OneWayAnovaModel = New OneWayAnovaModel(table, "Resistance", class="str">"Instrument") |
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 one-way design has three rows.
These are commonly labeled Between Groups, Within Groups, and Total.
The crucial data is provided by the Between Groups row, which shows the contribution to the total
variation of the factor under consideration. It is the first row in the ANOVA table, and therefore has index 0. It
can be retrieved through the GetModelRow method. Since
the entire model of a one-way analysis of variance consists of the contribution of one factor, this property is also
available as the CompleteModelRow property.
The AnovaModelRow obtained in this way shows the results of
the test for significance of the variation due to the factor compared to the variation not explained by the factor.
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. model: {0}",
anova.AnovaTable.ModelRow.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. model: {0}", _
anova.AnovaTable.ModelRow.SumOfSquares)
Console.WriteLine(anova.AnovaTable.ToString()) |
For the example using the silicon wafers, we find that the F statistic is 1.1804, corresponding to a p-value of
???.
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 factor. We then iterate through the levels of
the scale, and print the group means:
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CategoricalScale instrumentFactor = anova.GetFactor(0);
foreach(object level in instrumentFactor.GetLevels())
Console.WriteLine("Mean for group '{0}': {1:F4}",
level, anova.Cells[level].Mean); |
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CategoricalScale instrumentFactor = anova.GetFactor(0)
For Each level As Object In instrumentFactor.GetLevels()
Console.WriteLine("Mean for group '{0}': {1:F4}", _
level, anova.Cells[level].Mean)
Next |
The grand mean can be obtained through the cell with index Cell.All:
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Console.WriteLine("Grand mean: {0:F4}", anova.Cells[Cell.All].Mean); |
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Console.WriteLine("Grand mean: {0:F4}", anova.Cells[Cell.All].Mean) |
Up: Analysis of Variance Next: One-way ANOVA with Repeated Measures Previous: ANOVA Models Contents
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