Statistics

ANOVA

ANalysis Of VAriance (ANOVA) is a collection of statistical models in which observed variance is partitioned into components due to different explanatory variables. So a simply form ANOVA gives a statistical test of weather means of several groups are all equal.

Logic

Partitioning of the sum of squares

ANOVA is based on the following formula of the Sum of Squares

SSTotal = SSError + SSTreatments

So, the number of degrees of freedom (abbreviated df) can be partitioned in a similar way and specifies the chi-square distribution which describes the associated sums of squares:

dƒTotal = dƒError + dƒTreatments

the F-Test

The F-test is used for comparisons of the components of the total deviation. In a one-way or single-factor ANOVA, statistical significance is tested for by comparing the F test statistic:

          variance of the group means         MSTR 
    F = ----------------------------------  = ----  
        mean of the within-group variances    MSE

where:      SSTR
    MSTR = ------ . l = number of treatments
            I - 1

and:        SSE
    MSE  = ----- . nt = total number of cases
           nt - I
            

to the F-distribution with I-1,n_T-I degrees of freedom. Using the F-distribution is a natural candidate because the test statistic is the quotient of two mean sums of squares which have a chi-square distribution.

Example

Students are getting 3 types of sounds during the their study for an exam. They are divided into 3 groups of 8 students. Here are there results:

#	Group description	Test scores
1	Constant sound	7	4	6	8	6	6	2	9
2	Random sound	5	5	3	5	5	7	2
3	No sound	2	4	7	1	2	1	5	5

x1	x12	x2	x22	x3	x32
7	49	5	25	2	4
4	16	5	25	4	16
6	36	3	9	7	49
8	64	4	16	1	1
6	36	4	16	2	4
6	36	7	49	1	1
2	4	2	4	5	25
9	81	2	4	5	25
Sx1 = 48	Sx12 = 322	Sx2 = 32	Sx22 = 148	Sx3 = 27	Sx32 = 125
(Sx1)2 = 2304		(Sx2)2 = 1024		(Sx3)2 = 729
M1 = 6		M2 = 4		M3 = 3.375
SStotal = (322 + 148 + 125) - ( ( 48 + 32 + 27 )2 / 24 ) = 595 - 477.04 = 117.96
SSamong = ( (2304/8) + (1024/8) + (792/8) ) - 477.04 = 507.13 - 477.04 = 30.08
SSwithin = 117.96 - 30.08 = 87.88

So:

Source	SS	dƒ	MS	F
Among	30.08	2	15.04	3.59
Within	87.88	21	4.18

F (dƒ= 2, 21), F must be at least 3.4668 to reach p < 0.05, so F score is statistically significant.

• H₀: Students learn more with constant music.
• H₁: Random or No-sound learn better.

The group with constant music (x₁ has the highest score. However, the signficant F only indicates that at least two means are signficantly different from one another, but unkown is which specific mean pairs significantly differ. To find that a post-hoc analysis (e.g., Tukey's HSD) is needed.

Tukey's HSD

Logic

                      M1 - M2  
  Tukey's HSD = -------------------
                SQR( MSw ( 1 / n) )
Where:
M : Treatment/group mean
n : Number of treatment/group

1. Calculate an analysis of variance (e.g., One-way between-subjects ANOVA).
2. Select two means and note the relevant variables (Means, Mean Square Within, and number per condition/group)
3. Calculate Tukey's test for each mean comparison
4. Check to see if Tukey's score is statistically significant with Tukey's probability/critical value table taking into account appropriate dƒ_within and number of treatments.

Example

From the example above:

                 6 - 4                    2  
M1 vs M2 = -------------------- = ----------------- = 2.767
           SQR( 4.18 ( 1 / 8) )   SQR(4.18 x 0.125) 

           6 - 3.375
M1 vs M3 = --------- =  3.6315 [*]
              0.72

           4 - 3.375
M2 vs M3 = --------- =  0.864648
             0.72

[*] According to the Tukey's sig/probability table, taking into account (dƒ_within = 21 and treatments = 3, p < 0.05) = 3.58, the mean comparison between means 1 and 3 is statistically signficant, but not the other comparisons. (Use the Tukey Calculator: Psychology World Tukeys Calculator).

Trend Analysis

Mann-Kendall test is applicable is cases when the data values X_i of a time series can be assumed to obey the model:

xi = ƒ(ti) + εi   ƒ(t): continuous monotonic increasing or decreasing function of time.  εi: Residuals with the same distribution and a mean of zero.

And is calculated using:

       n-1   n
   S = Σ     Σ sgn(Xj - Xk)
       k=1   j=k+1

Where:
               = +1 ... if Xj-Xk > 0
  sgn(Xj - Xk) =  0 ... if Xj-Xk = 0
               = -1 ... if Xj-Xk < 0

See also

• Psychology World Virtual Statistician of Richard Hall.
• Mann Kendall Test, Detecting Trends of Annual Values of Atmospheric Polllutants by the Man-Kendall Test and Sen's Slope Estimates (Excel Template Application MakeSens).