ANOVA mixed split plot -formula and practice example

What is ANOVA mixed split plot design

ANOVA mixed split plot designs are complex designs that employ both independent and repeated measures. The best way to explain this is to present the layout of these experimental designs.

Subjects	Drug 1	Drug 2
DEPRESSIVE 1 2 3 4 5 6	50 55 56 50 56 54	68 63 65 67 69 68
SCHIZOPHRENIC 7 8 9 10 11 12	99 100 110 90 105 115	122 125 130 135 140 131

Observe that there are two independent groups, depressive, and schizophrenic. Also observe that each subject of the depressive and schizophrenic groups is repeatedly tested, once with Drug 1, and later with Drug 2. This is a repeated measures arrangement So here we have a design in which independent and repeated measures are mixed. The name split plot comes from the fact that this design is extensively used in agricultural research.

ANOVA mixed split plot designs -formula

As in all ANOVA, the formula for these designs is: $$F={{MS_{between}} \over {MS_{within}}}$$ We read this as follows: Mean square between over mean square within. What is mean square, you ask. It is the mean of squares. What is squares, you ask. Squares is the statistical term for squared deviations (of squared differences) of each score X from the mean. What are the squared differences, you ask. Remember the formula for variance? $$s^2 ={\sum{({X}-{\bar{X})}}^2 \over {n}}$$ Look at the numerator $${\sum({X}-\bar{X})}^2$$ These are the squared differences or summed. To complete our reasoning, we go back to where we started, the F formula, or F ratio, the formula for ANOVA. Why mean sums of squares? Simple because like all average, we divide by the number of scores. If you are observant, you will notice that the F formula is a modified t formula.

FORMAT OF ANOVA MIXED SPIT PLOT SUMMARY TABLE

SOURCE	SS	df	MS	F	p
Between Independent
B	*	*	*	*	*
Error	*	*	*
Total	*	*
Between repeated measures
A	*	*	*	*	*
AxB	*	*	*		*
Error	*	*	*
Total		*
TOTAL		*

HOW TO CALCULATE df OF ANOVA MIXED SPIT PLOT SUMMARY TABLE

SOURCE	SS	df	MS	F	p
Between Independent
B		number of independent groups minus 1	*	*	*
Error		total number of subjects minus the number of independent groups	*
Total		total number of subjects minus 1
Between repeated measures
A	*	number of repetitions minus 1	*	*	*
AxB	*	df A x df B	*		*
Error	*	error between independent x number of repetitions	*
Total		*
TOTAL		total number of scores minus 1

ANOVA mixed split plot- practice examples

ANOVA mixed split plot- practice example 1

An experimenter wanted to test drugs (factor A), Drug 1 (A1) and Drug 2 (A2) for their effect on serotonin level in the blood of patients (factor B) suffering from depression (B1) and schizophrenia (B2) . He randomly selected six patients suffering from depression and gave them Drug 1. He waited for one hour and then he measured the level of serotonin in nano grams per liter (ng/lt) of each subject. He recorded the data. One week later he gave these subjects Drug 2. He waited for one hour and measured the level of serotonin of each subject. He also randomly selected six patients suffering from schizophrenia and repeated the same experiment that he performed with the depressive patients. The data are presented in the table below.

	Subjects	A1 Drug 1	A2 Drug 2
B1	DEPRESSIVE 1 2 3 4 5 6	50 55 56 50 56 54	68 63 65 67 69 68
B2	SCHIZOPHRENIC 7 8 9 10 11 12	99 100 110 90 105 115	122 125 130 135 140 131

ANOVA MIXED SPIT PLOT SUMMARY TABLE

SOURCE	SS	df	MS	F	p
Between Independent
B		1
Error		10
Total		11
Between repeated measures
A		1
AxB		1
Error		10
Total		12
TOTAL		23

ANOVA mixed split plot- practice example 2

An experimenter wanted to test two new drugs for their effect on body temperature in three age groups: Young (20-30), Middle age (40-50), Old (70-80. He randomly selected 5 patients from each of these age groups. He gave Drug 1 to each of the three groups. He waited for one hour and then he measured the body temperature of each subject using a Celsius thermometer. He recorded the data. One week later he gave these subjects Drug 2. He waited for one hour and measured the temperature of each subject.. The data are presented in the table below.

	Subjects	A1 Drug 1	A2 Drug 2
B1	YOUNG 1 2 3 4 5	37.0 37.1 37.3 37.4 37.4	37.6 37.5 37.4 37.4 37.6
B2	MIDDLE AGE 6 7 8 9 10	37.1 37.2 37.4 37,6 375	37.8 37.7 37.6 37.8 37.8
B3	OLD 11 12 13 14 15	37.7 37.8 37.8 37.8 37.9	38.1 38.3 37.9 38.0 38.5

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