Example of ANOVA

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Example of ANOVA
[For Independent Groups Design]

Research Question

Will subjects instructed to complete null matches while viewing the Müller-Lyer illusion make different settings than subjects ask to set the variable component to be the same physical length as the fixed component of the illusion?

Experimental Design

Two groups of subjects complete several tasks with a computerized version of the Müller-Lyer illusion using the method of adjustment . The data for one group of subjects is recorded when "null matching" instructions are used (n = 9). Data for the other group of subjects is recorded when "set to same physical length" instructions are used (n = 14).

20 trials are collected from each subject. The average "equality" setting is calculated.

Statistical Hypothesis

Are the mean settings of the two groups sampled from the same population or different populations? Is µ₁ = µ₂?

Hypothesis Testing Model

Ho: µ₁ = µ₂

Ha: µ₁ <> µ₂

The Test Statistic

The test statistic is the F-statistic (aka "variance ratio")

The Sampling Distribution

If the Ho is true then the F value from the experiment will be drawn from the sampling distribution of F with n₁= 1 and n₂ = 21 degrees of freedom.

Type-I Error rate

We'll use a 5% Type-I error rate.

Decision Rule

The Critical Value of the test statistic is _.95F_1,21 = 4.32
Reject Ho if F > 4.32

Data

Descriptive Statistics

Physical Equality	Null Matching
125.15	113.00
96.00	110.30
105.90	98.30
90.35	99.75
109.10	106.90
120.30	119.90
113.10	104.10
112.60	92.75
120.15	104.95
122.65
118.55
127.30
152.90
127.80

	Physical Equality	Null Matching
n	14	9
Mean	117.28	105.55
est sigma	15.200	8.203
est variance	231.035	67.289
SS	3003.454	538.315

ANOVA Source Table

	Source of Variation	SS	df	MS	F
Numerator of F	Between Groups	753.127337	1	753.127337	4.465473381
Denominator of F	Within Groups	3541.76875	21	168.6556548
	Total	4294.896087	22

Statistical Decision

Since the F-value we got from the experiment (4.465) is greater than the critical value (4.38), we reject Ho. There is evidence the independent variable effected the mean setting.

Interpretation

There is evidence that subjects make different mean settings when the "matching" instructions were altered. Since the mean setting for instructions to set the variable stimulus to "physical equality" are closer to the length of the standard stimulus (130 pixels), we can conclude that subjects who know the manner in which the illusion affects settings can adjust the setting more toward physical equality.

Calculations for ANOVA (optional)

SS_Total: Calculate the est sigma for all 23 scores in the data set. Square the value and multiply by 22 to get the Total SS. [est. sigma = 13.9722; n = 23; SS_Total = (13.9722)² X (23 - 1 ) = 4294.896]

SS_{Within Groups}: Calculate the SS for each group. Add the separate SS values together.
[SS_W/I = 3003.454 + 538.315 = 3541.769]

SS_{Between Groups} = SS_Total - SS_{Within Groups}

_{MS = SS ÷ df}[Remember: mean square (aka "Variance") is SS divided by df ].

F = MS_Bet ÷ MS_W/I

Example of ANOVA [For Independent Groups Design]

Example of ANOVA
[For Independent Groups Design]