I. One-Way ANOVA (Video Lesson 15 I) (YouTube version)

If you are testing hypotheses that require the comparison of three or more samples, then you need to use the analysis of variance (ANOVA) statistic.  You can not simply use multiple t-tests because that will increase the Type I error to the point where your statistical analysis will become invalid. A One-Way ANOVA is used if your experimental design has only one independent variable. If you have two independent variables you are forced to use a Two-Way ANOVA.

II. Main Effects

The Main Effect is the effect that an independent variable has on the dependent variable. In a One-Way ANOVA, there is only one Main Effect since there is only one independent variables or Factors of the experiment.

 

    The null hypothesis for the Main Effect of the is:

 

        H₀ : µ₁ = µ ₂ = . . . µ_k

 

    The research hypothesis for the Main Effect of the is:

        H₁ : At least one of the sample means comes from a different population distribution than the others 

III. Source Table

An excellent place to start your calculations of a One-Way ANOVA is the Source Table.  The Source Table consists of Sums of Squares (SS), degrees of freedom (df), Mean Squares (MS) and an F ratio. In a One-Way ANOVA there is a measure of variance Between Groups, Within Groups and for Total Groups.

IV. Calculation of One-Way ANOVA 

The ultimate goal in ANOVA is to solve for your F ratio. Let's look at the formulas in the Source Table to see how we can get to this important F ratio. I think it is best to look at the calculations in reverse order. Let's start with the formula for the F ratio:

 

7. F ratio formula:

or

F = MS_bg / MS_wg

The formula reads: F equals the Mean Square of the between group divided by the Mean Square of the within group.

6. MS_bg formula:

MS_bg = SS_bg / df_bg

The formula reads: Mean Squares of the between group equals the Sum of Squares of the between group divided by the degrees of freedom of the between group.

5. MS_wg formula:

MS_wg = SS_wg / df_wg

The formula reads: Mean Squares of the within group equals the Sum of Squares of the within group divided by the degrees of freedom of the within group.

4. df_bg formula:

df_bg = k - 1

The formula reads: degrees of freedom of the between group equals k or the number of independent variable levels minus 1.

3. df_wg formula:

df_wg = (n₁ - 1) + (n₂ - 1) + . . . + (n_k - 1)

The formula reads: degrees of freedom of the within group equals the sample size of the first level minus 1 plus the sample size of the second level minus 1 plus and continuously until the final level is reached then plus the sample size of the final level minus 1.

2. SS_bg formula:

or

SS_bg = [ (ΣX₁)² / n₁ ) + (ΣX₂)² / n₂ ) + . . . + (ΣX_k)² / n_k ) ] - [ (ΣX₁  + ΣX₂  + . . . + ΣX_k )² / N_total ]

The formula reads: Sum of Squares of the between group equals the sum of X1 then squared divided by n1 plus the sum of X2 then squared divided by n2 plus the sum of Xk then squared divided by nk then take all of that and subtract the following: the sum of X1 plus the sum of X2 plus the sum of X3 then the total squared and then divided by the total n.

1. SS_wg formula:

or

SS_wg = [ (ΣX²₁  + ΣX²₂  + . . . + ΣX²_k ) ] - [ (ΣX₁)² / n₁ ) + (ΣX₂)² / n₂ ) + . . . + (ΣX_k)² / n_k ) ]

The formula reads: Sum of Squares of the within group equals the sum of X1 squared plus the sum of X2 squared plus the sum of X3 squared then subtract the following: the sum of X1 then squared divided by n1 plus the sum of X2 then squared divided by n2 plus the sum of Xk then squared divided by nk.

V. Example Calculation (Video Lesson 15 V) (YouTube version) (One-Way ANOVA Calculation - YouTube version) (mp4 version)

A young psychotherapy intern is interested in the differences between the available SSRIs on the market to treat depression. She conducts a survey using patients diagnosed with Major Depressive Disorder. She has 24 patients with identical Beck Inventory depression scores and 3 months to conduct her study. She starts off by randomly assigning the 24 patients into three groups of 8. The first group (X₁) will receive no treatment at all (they are the control group). The second group (X₂) will receive the standard dose of Paxil for the three months. The third group (X₃) will receive the same dose of Prozac for the three months. At the end of the three month period she uses the same Beck Inventory to measure all of the patients depression scores. The data she collected are presented in the table below:

Step 1: Generate a Source Table

 

Variance Source	Sum of Squares	Degrees of Freedom	Mean Square	F ratio
Between
Within

 

Step 2: Start Filling in the Source Table

 

1. Calculate SS_bg:

SS_bg = [ (ΣX₁)² / n₁ ) + (ΣX₂)² / n₂ ) + . . . + (ΣX_k)² / n_k ) ] - [ (ΣX₁  + ΣX₂  + . . . + ΣX_k )² / N_total ]

We first need to add some rows and columns to the data table and do some calculations:

Then we fill our new values into the formula:

SS_bg = [ (57)² / 8 ) + (33)² / 8 ) + (34)² / 8) ] - [ (57  + 33  + 34 )² / 24 ]

Now we can finish the calculation:

SS_bg = [ (3249 / 8 ) + (1089 / 8 ) + (1156 / 8) ] - [ (124 )²/ 24 ]

SS_bg = [ (406.125) + (136.125 ) + (144.5) ] - [ 15376/ 24  ]

SS_bg = 686.75 - 640.667

SS_bg = 46.083

Variance Source	Sum of Squares	Degrees of Freedom	Mean Square	F ratio
Between	46.083
Within

 

2. Calculate SS_wg:

SS_wg = [ (ΣX²₁  + ΣX²₂  + . . . + ΣX²_k ) ] - [ (ΣX₁)² / n₁ ) + (ΣX₂)² / n₂ ) + . . . + (ΣX_k)² / n_k ) ]

We can use the same values we calculated earlier to solve SS_wg.

	Control (X1)	(X1)2	Paxil (X2)	(X2)2	Prozac (X3)	(X3)2
	8	64	2	4	7	49
	9	81	5	25	5	25
	5	25	6	36	2	4
	7	49	7	49	1	1
	6	36	5	25	7	49
	10	100	4	16	6	36
	8	64	3	9	4	16
	4	16	1	1	2	4
ΣX =	57		33		34
ΣX2 =		435		165		184
n =	8		8		8

 SS_wg = [ 435  + 165  + 184  ] - [ (57)² / 8 ) + (33)² / 8 ) + (34)² / 8) ]

SS_wg = ( 784 ) - (686.75)

SS_wg = 97.25

3. Calculate df_wg:

df_wg = (n₁ - 1) + (n₂ - 1) + . . . +  (n_k - 1)

 

df_wg = (8 - 1) + (8- 1) + (8 - 1)

 

df_wg = (7) + (7) + (7)

 

df_wg = 21

 

Variance Source	Sum of Squares	Degrees of Freedom	Mean Square	F ratio
Between	46.083
Within	97.25	21

4. Calculate df_bg:

df_bg = k - 1

df_bg = 3 - 1

df_bg = 2

5. MS_wg formula:

MS_wg = SS_wg / df_wg

 

MS_wg = 97.25 / 21 

 

MS_wg = 4.631 

 

Variance Source	Sum of Squares	Degrees of Freedom	Mean Square	F ratio
Between	46.083	2
Within	97.25	21	4.631

     

6. MS_bg formula:

MS_bg = SS_bg / df_bg

 

MS_bg = 46.083 / 2

 

MS_bg = 23.042

 

Variance Source	Sum of Squares	Degrees of Freedom	Mean Square	F ratio
Between	46.083	2	23.042
Within	97.25	21	4.631

 

7. F ratio formula:

F = MS_bg / MS_wg

F = 23.042 / 4.631 

F = 4.976

VI. F Table (Video Lesson 15 VI) (YouTube version)

Just as with the R Table and the T Table, we use the F Table to find the critical value of the F ratio. All critical values are calculated for p < .05.  The F Table uses the df_bg to find the proper column (shown as df_N  to designate the Numerator of the F ratio) and uses the df_wg to find the proper row (shown as df_D to designate the Denominator of the F ratio).  Where they meet is the critical value of F.  If your calculated F value is larger than the critical F then the results are significant and we can reject the null hypothesis (p < .05).  If your calculated F value is smaller than the critical F then the results are not significant and we fail to reject the null hypothesis.

 

For the data above our degrees of freedom for the between and within groups are 2 and 21 respectively. Therefore we can go to the F Table and go to column 2 and row 21 and determine that our critical F value is 3.47. Since our calculated F value was 4.976 we can say with confidence that our results are significant (at p < .05) and we can reject the null hypothesis.

 

 

Additional Links about the Concept that might help:

One-Way ANOVA

One-Way ANOVA

Calculating a One-Way ANOVA

One-Way ANOVA

F Table