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One-Way Dependent Groups ANOVA

Research Question

Is the Immediate Memory Span for Digits and Letters different?

Experimental Design

Immediate memory span letters and digits was measured for 56 persons. This data was collected in January 1998.

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 = 55 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_1,55 = 4.016. Note: Your table doesn't have this entry. The rule is to use the nearest tabled value with fewer DF. In this instance that would be F with 1 and 40 DF or 4.08.
Reject Ho if F > 4.016

Data

Descriptive Statistics

Subject	Letter Span	Digit Span
1	xx	xx
2	xx	xx
3	xx	xx
...	...	..
56	xx	xx

Letter	Digit	Diff
56	56	56	Count
6.28	7.34	-1.06	Mean
0.9905.	1.1643.	1.0228.	est Sigma
0.1324.	0.1556.	0.1367.	SEM
	0.4760.		Slope
	2.7829.		y-intercept
	0.5596.		corr

ANOVA Source Table

Concerning the Dependent Groups ANOVA Table:

There are two F-values given. One of the F-values tells me wether or not the Subjects had scores different from one another; the other F-value tells me whether or not the independent variable was effective.
I expect the individual subjects to have scores different from one another, so this information is "trivial". The line is calculated because the "Subjects" variablilty is removed from the "Error" variability, That makes the MSerror smaller and the F-value for Treatments (aka "Columns") larger--at the cost of reducing the MSerror degrees of Freedom (thereby changing the critical value of the test statistic.

ANOVA Summary Table
Source		SS	df	MS	F
Subjects In Excel this row is labeled "Rows" Numerator of F		99.7477679	55	1.81359578	3.46760301	This F "tells" whether or not some subjects had memory spans different from other subjects. Irrelvant or trivial in this instance.
Treatment In Excel this row is labeled "Columns" Numerator of F		31.609375	1	31.609375	60.4372624	This F "tells" whether or not the independent variable had an effect on the value of the dependent variable.
Error Denominator of F		28.765625	55	0.5230
	Total	160.122768	111	MSs do not Total

Statistical Decision

Since the F-value we got from the experiment (60.437) is greater than the critical value (4.08), we reject Ho. There is evidence the independent variable affected the mean setting.

Interpretation

There is evidence the letter and digit spans are different. Since the mean digit span is larger than the mean letter span, we can say that digit span is larger than digit span.

Calculations for ANOVA (optional)

The Dependent Groups ANOVA Source table is calculated using Excel. I do not know of an intuitive calculation scheme for the Dependent Groups ANOVA.