Question

Instructions- Calculate this ANOVA by hand. Check your work by also calculating the ANOVA in SPSS or EXCEL. Be sure to do any multiple comparison tests and interpret the results.

	Control GroupA1	PsychoanalysisA2	CBTA3
X1	7	7	3
X2	10	5	3
X3	6	6	2
X4	8	5	3
X5	10	5	4
X6	10	6	2
X barA1
SX (A1)
SX2

SS_T = SX² – T²/an                            (also called [X] – [T])

SS_T =                                                 (a = number of groups, n = number in each group)

                                

SS_T =

SS_A = SA²-   T²                                (also called [A] – [T])

            n        an

SS_A =

SS_A =

SS_S/A = SX² - SA²                                             (also called [Y] – [A] )

                         n

SS_S/A =

Now divide by its df to change it to variance. Variance is called Mean Square.

Complete ANOVA summary table on next page.

ANOVA Summary Table

Source                                   SS                   df                     MS                  F

A-group                    

S/A (error)                

Total                                      

Find F-critical.

Are the groups significantly different?

Write the results statistically.

What do you do next? Why?

Which groups are different?

Answer 1

	Control GroupA1	PsychoanalysisA2	CBTA3
X1	7	7	3
X2	10	5	3
X3	6	6	2
X4	8	5	3
X5	10	5	4
X6	10	6	2	Total
sum	51	34	17	102
sum of squares	449	196	51	696

H₁: At least one of the mean is different from the others

SS_T = SX² – T²/an                            (also called [X] – [T])

SS_T =118

                                           (a =3 number of groups, n = 6 number in each group)

                                

SSE= SST-SSG    = 118-96.3333 = 21.6667

Now divide by its df to change it to variance. Variance is called Mean Square.

Complete ANOVA summary table on next page.

ANOVA table
Source	SS	df	MS	F
A-group	96.3333	2	48.1667	33.3462
Error	21.6667	15	1.4444
Total	118.0000	17

Find F-critical. = 3.68

Are the groups significantly different?

Calculated F= 33.3462 > critical F 3.68. Ho is rejected.

The three groups are significantly different

Write the results statistically.

There is enough evidence in the data that the three groups are significantly different at 0.05 level of significance.

What do you do next? Why?

Since ANOVA is significant, we have to do post hoc test to idendify which pairs of groups are significant.

Which groups are different?

Tukey test shows that all the three groups are significantly different.

SPSS output:

Descriptives
score
	N	Mean	Std. Deviation	Std. Error	95% Confidence Interval for Mean		Minimum	Maximum
					Lower Bound	Upper Bound
1.00	6	8.5000	1.76068	.71880	6.6523	10.3477	6.00	10.00
2.00	6	5.6667	.81650	.33333	4.8098	6.5235	5.00	7.00
3.00	6	2.8333	.75277	.30732	2.0433	3.6233	2.00	4.00
Total	18	5.6667	2.63461	.62098	4.3565	6.9768	2.00	10.00

ANOVA
score
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	96.333	2	48.167	33.346	.000
Within Groups	21.667	15	1.444
Total	118.000	17

Multiple Comparisons
Dependent Variable:   score
Tukey HSD
(I) Group	(J) Group	Mean Difference (I-J)	Std. Error	Sig.	95% Confidence Interval
					Lower Bound	Upper Bound
1.00	2.00	2.83333*	.69389	.003	1.0310	4.6357
	3.00	5.66667*	.69389	.000	3.8643	7.4690
2.00	1.00	-2.83333*	.69389	.003	-4.6357	-1.0310
	3.00	2.83333*	.69389	.003	1.0310	4.6357
3.00	1.00	-5.66667*	.69389	.000	-7.4690	-3.8643
	2.00	-2.83333*	.69389	.003	-4.6357	-1.0310
*. The mean difference is significant at the 0.05 level.