Two-Way (Independent) ANOVA

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Two-Way (Independent) ANOVA. Two-Way ANOVA. “Two-Way” means groups are defined by 2 independent variables. These IVs are typically called factors . An experiment in which any combination of values for the 2 factors can occur is called a completely crossed factorial design.

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## Two-Way (Independent) ANOVA

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### Two-Way (Independent) ANOVA

Two-Way ANOVA
• “Two-Way” means groups are defined by 2 independent variables.
• These IVs are typically called factors.
• An experiment in which any combination of values for the 2 factors can occur is called a completely crossed factorial design.
• If all cells have the same n, the design is said to be balanced.
• Still have only 1 dependent variable

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500 ms

500 ms

200 ms

Until Response

Example: Visual Grating Detection in Noise

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2 x 3 Design

0.5

Grating

Frequency

(c/deg)

1.7

50.0%

4.3%

14.8%

Noise Contrast

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Balanced Design

Factor B

Factor A

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Descriptive Statistics

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Interactions
• If there is no interaction between the factors (spatial frequency, noise contrast), the dependent variable (SNR) for each condition (cell) can be predicted from the independent effects of factors A and B:
• Cell mean = Grand mean + Row effect + Column effect

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Interactions
• If there are no interactions, curves should be parallel (effect of noise contrast is independent of spatial frequency).

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Types of Effects

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Interactions
• In the general case,

Cell mean = Grand mean + Row effect + Column effect + Interaction effect

• Score deviations from cell means are considered error (unpredictable).
• Thus:

Score = Grand mean + Row effect + Column effect + Interaction effect + Error

• OR

Score - Grand mean = Row effect + Column effect + Interaction effect + Error

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Sum of Squares Analysis

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### Multiple Subscript and Summation Notation

Single Subscript Notation

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Double Subscript Notation

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Double Subscript Notation
• The first subscript refers to the row that the particular value is in, the second subscript refers to the column.

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Double Subscript Notation
• Test your understanding by identifying in the table below.

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Double Subscript Notation
• We will follow the notation of Howell:

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Multi-Subscript Notation
• In two-way ANOVA, 3 indices are needed:

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Multi-Subscript Notation
• Statistics are calculated by summing over scores within cells, and thus the third subscript (k) is dropped:

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Multi-Subscript Notation

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Pooled Statistics
• Multi-factor ANOVA requires the calculation of statistics that pool, or ‘collapse’ data over one or more factors.
• We indicate the factors over which the data are being pooled by substituting a ‘bullet’ • for the corresponding index.

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Pooled Statistics

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### Six Step Procedure

Group

Noise Contrast (Michelson units)

Total

.043

.148

.500

Mean

Mean

Mean

Mean

Spatial Frequency

.500

Signal to Noise

7.8

6.4

6.5

6.9

(cpd)

at Threshold (%)

1.700

9.5

8.9

9.8

9.4

Group Total

8.7

7.6

8.2

8.2

Noise Contrast (Michelson units)

.043

.148

.500

Std Deviation

Std Deviation

Std Deviation

Spatial

.500

Signal to Noise

0.62

0.29

0.31

Frequency

at Threshold

(cpd)

1.700

Signal to Noise

0.36

0.54

0.80

at Threshold

Example

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Step 1. State the Hypothesis
• Null hypothesis has 3 parts, e.g.,
• Mean SNR at threshold same for both spatial frequencies
• Mean SNR at threshold same for all noise levels
• No interactions

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Step 2. Select Statistical Test and Significance Level
• Normally use same a-level for testing all 3 F ratios.

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Step 3. Select Samples and Collect Data
• Strive for a balanced design
• Ideally, randomly sample
• More probably, random assignment

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Degrees of Freedom Tree

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Step 5. Calculate the Test Statistics

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Step 5. Calculate the Test Statistics

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Step 6. Make the Statistical Decisions
• Note that 3 independent statistical decisions are being made.
• Thus the probability of one or more Type I errors is greater than the α value used for each test.
• It is not common to correct for this.
• You should be aware of this issue as both a producer and consumer of scientific results!

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Main effects

Interaction

SPSS Output

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SPSS Output

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Assumptions of Two-Way Independent ANOVA
• Same as for One-Way
• If balanced, don’t have to worry about homogeneity of variance.

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Advantages of 2-Way ANOVA with 2 Experimental Factors
• One factor may not be of interest (e.g., gender), but may affect the dependent variable.
• Explicitly partitioning the data according to this ‘nuisance’ variable can increase the power of tests on the independent variable of interest.

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Simple Effects
• When significant main effects are discovered, it is common to also test for simple effects.

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Simple Effects
• A main effect is an effect of one factor measured by collapsing (pooling) over all other factors.
• A simple effect is an effect of one factor measured by fixing all other factors.
• Although we found significant main effects, given the significant interaction, these main effects do not necessarily imply similarly significant simple effects.

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Simple Effects
• Thus, particularly when a significant interaction is observed, a factorial ANOVA is often followed up by a series of one-way ANOVAS to test simple effects.
• For our example, there are a total of 5 possible simple effects to test.

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Simple Effects
• To conduct follow-up one-way ANOVA tests of simple effects in SPSS:
• Select Split File … from the Data menu
• Click on Organize Output by Groups
• Transfer the factor to be held constant to the space labeled “Groups Based On.”
• Now proceed with one-way ANOVAS as usual.

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a

Test of Homogeneity of Variances

Signal to Noise at Threshold

a

ANOVA

Levene

Statistic

df1

df2

Sig.

Signal to Noise at Threshold

5.120

2

27

.013

Sum of

a.

Spatial Frequency (cpd) = .500

Squares

df

Mean Square

F

Sig.

Between Groups

.001

2

.001

32.990

.000

Within Groups

.001

27

.000

Total

.002

29

a.

Spatial Frequency (cpd) = .500

b

Robust Tests of Equality of Means

Signal to Noise at Threshold

a

Statistic

df1

df2

Sig.

Welch

21.413

2

16.975

.000

Brown-Forsythe

32.990

2

17.382

.000

a.

Asymptotically F distributed.

b.

Spatial Frequency (cpd) = .500

Simple Effects

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a

Test of Homogeneity of Variances

Signal to Noise at Threshold

Levene

Statistic

df1

df2

Sig.

2.037

2

27

.150

a.

Spatial Frequency (cpd) = 1.700

a

ANOVA

Signal to Noise at Threshold

Sum of

Squares

df

Mean Square

F

Sig.

Between Groups

.000

2

.000

5.899

.007

Within Groups

.001

27

.000

Total

.001

29

a.

Spatial Frequency (cpd) = 1.700

b

Robust Tests of Equality of Means

Signal to Noise at Threshold

a

Statistic

df1

df2

Sig.

Welch

5.527

2

16.511

.015

Brown-Forsythe

5.899

2

19.883

.010

a.

Asymptotically F distributed.

b.

Spatial Frequency (cpd) = 1.700

Simple Effects

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Simple Effects
• Again note that multiple independent statistical decisions are being made.
• Conditioning the test for simple effects on a significant main effect provides protection if only 2 simple effects are being tested.
• Otherwise, the probability of one or more Type I errors is greater than the α value used for each test.
• It is not common to correct for this.
• You should be aware of this issue as both a producer and consumer of scientific results!

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### End of Lecture

April 8, 2009

Planned or Posthoc Pairwise Comparisons
• If significant main (and possibly simple) effects are found, it is common to follow up with one or more pairwise tests.
• It is most common to test differences between marginal means within a factor (i.e., pooling over the other factor).
• In this example, there are only 3 meaningful posthoc tests on marginal means. Why?

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Test of Homogeneity of Variances

Signal to Noise at Threshold

Levene

Multiple Comparisons

Statistic

df1

df2

Sig.

Dependent Variable: Signal to Noise at Threshold

12.229

2

57

.000

LSD

Mean

95% Confidence Interval

(I) Noise Contrast

(J) Noise Contrast

Difference

(Michelson units)

(Michelson units)

(I-J)

Std. Error

Sig.

Lower Bound

Upper Bound

.043

.148

.010120

*

.004492

.028

.00112

.01912

.500

.004800

.004492

.290

-.00420

.01380

.148

.043

-.010120

*

.004492

.028

-.01912

-.00112

.500

-.005320

.004492

.241

-.01432

.00368

.500

.043

-.004800

.004492

.290

-.01380

.00420

.148

.005320

.004492

.241

-.00368

.01432

*.

The mean difference is significant at the .05 level.

Pairwise Comparisons on Marginal Means
• Since there are 3 levels of noise, we can consider using Fisher’s LSD.
• However, since variances do not appear homogeneous, we should not use an LSD based on pooling the variance over all 3 conditions.

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Multiple Comparisons

Dependent Variable: Signal to Noise at Threshold

Games-Howell

Mean

95% Confidence Interval

(I) Noise Contrast

(J) Noise Contrast

Difference

(Michelson units)

(Michelson units)

(I-J)

Std. Error

Sig.

Lower Bound

Upper Bound

.043

.148

.010120

*

.003787

.030

.00085

.01939

.500

.004800

.004573

.552

-.00648

.01608

.148

.043

-.010120

*

.003787

.030

-.01939

-.00085

.500

-.005320

.005028

.546

-.01762

.00698

.500

.043

-.004800

.004573

.552

-.01608

.00648

.148

.005320

.005028

.546

-.00698

.01762

*.

The mean difference is significant at the .05 level.

Pairwise Comparisons on Marginal Means
• Alternative when variances appear heterogeneous:
• Compute Fisher’s LSD by hand, calculating standard error separately for each test (not difficult)
• One of the unequal variance post-hoc tests offered by SPSS

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Planned or Posthoc Pairwise Comparisons
• It is also possible to test differences between cell means. Note that in this design, there are 15 possible pairwise cell comparisons.
• It doesn’t make that much sense to compare 2 cells that are not in the same row or column (i.e. that differ in both factors).
• It is more likely that you would follow a significant simple effect test with a set of pairwise comparisons within a factor while holding the other factor constant. There are 9 such comparisons possible here.
• For example, within a spatial frequency condition, what noise conditions differ significantly?
• This defines a total of 6 pairwise comparisons (2 families of 3 comparisons each).

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a

Multiple Comparisons

Dependent Variable: Signal to Noise at Threshold

Games-Howell

Mean

95% Confidence Interval

(I) Noise Contrast

(J) Noise Contrast

Difference

(Michelson units)

(Michelson units)

(I-J)

Std. Error

Sig.

Lower Bound

Upper Bound

.043

.148

.005890

*

.002067

.030

.00055

.01123

.500

-.003160

.002790

.513

-.01056

.00424

.148

.043

-.005890

*

.002067

.030

-.01123

-.00055

.500

-.009050

*

.003066

.024

-.01697

-.00113

.500

.043

.003160

.002790

.513

-.00424

.01056

.148

.009050

*

.003066

.024

.00113

.01697

*.

The mean difference is significant at the .05 level.

a.

Spatial Frequency (cpd) = 1.700

Planned or Posthoc Pairwise Comparisons
• Alternative when variances appear heterogeneous:
• Compute Fisher’s LSD by hand, calculating standard error separately for each test (not difficult)
• One of the unequal variance post-hoc tests offered by SPSS (assumes all-pairs)

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Interaction Comparisons
• If significant interactions are found in a design that is 2x3 or larger, it may be of interest to test the significance of smaller (e.g., 2x2) interactions.
• These can be tested by ignoring specific subsets of the data for each test (e.g., by using the SPSS Select Cases function).

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Unbalanced Designs for Two-Way ANOVA
• Dealing with unbalanced designs is easy for One-Way ANOVA.
• Dealing with unbalanced designs is trickier for Two-Way.

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Simple Solution
• Let n = harmonic mean of sample sizes.
• Calculate marginal means as an unweighted mean of cell means (not the pooled mean).

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Better Solution
• Regression approach to ANOVA (will not cover)

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