The Procedure of ANOVA

The steps below explain the procedure of Anova. The general logic of analysis of variance is the same as that of significance tests. That is, you assume...

Discuss the procedure of ANOVA.

How to calculate ANOVA?

In analysis of variance, a continuous response variable, known as a dependent variable, is measured under experimental conditions identified by classification variables, known as independent variables. The variation in the response is assumed to be due to effects in the classification, with random error accounting for the remaining variation.

The steps below explain the procedure of ANOVA. The general logic of analysis of variance is the same as that of significance tests. That is, you assume H0 to be true and then determine whether the obtained sample result is rare enough to raise doubts about H0.
To do this, convert the sample result into a test statistic (F, in this case). Then locate it in the theoretical sampling distribution (the F distribution). If the test statistic falls in the region of rejection, H0 is rejected; if not, H0 is retained. The only new twist is that if H0 is rejected, follow-up testing is required to identify the specific source(s) of significance.

Step 1 Formulate the statistical hypotheses and select a level of significance.

Ex: Let’s assume the statistical hypotheses to be: Ho → µ1 = µ2 = µ3. Let α=.05 be the level of significance.

Step 2 Determine the desired sample size and select the sample.

After Determining the desired sample size, we need to select the sample.

Step 3 Calculate the necessary sample statistics.

  1. Correction term: 𝐶𝑥=((∑〖𝑥𝑖 )^2 〗)/𝑁
  2. total sum of squares and degrees of freedom: 𝑆𝑆_𝑡𝑜𝑡𝑎𝑙 = ∑〖(∑〖𝑥_𝑖 )^2 〗〗−𝐶_𝑥 = (∑𝑋^2 +∑〖𝑌^2+∑〖𝑍^2)〗〗−𝐶_𝑥
  3. between-groups sum of squares, degrees of freedom, and variance estimate: 𝑆𝑆_𝑏𝑒𝑡𝑤𝑒𝑒𝑛=(∑〖(∑〖𝑥_𝑖 )^2 〗〗)/𝑁−𝐶_𝑥=((∑〖𝑋)^2 〗+(∑〖𝑌)^2+(∑〖𝑍)^2 〗〗)/𝑁−𝐶_𝑥
    Degrees of freedom = k - 1
    Mean square (variance estimate) = 𝑠_𝑏𝑒𝑡𝑤𝑒𝑒𝑛^2 = (𝑆𝑆_𝑏𝑒𝑡𝑤𝑒𝑒𝑛)/(𝑑𝑓_𝑏𝑒𝑡𝑤𝑒𝑒𝑛 )
  4. within-groups sum of squares, degrees of freedom, and variance estimate: 𝑆𝑆_𝑤𝑖𝑡ℎ𝑖𝑛=𝑆𝑆_𝑡𝑜𝑡𝑎𝑙−𝑆𝑆_𝑏𝑒𝑡𝑤𝑒𝑒𝑛
    Degrees of freedom = N – k
    Mean square (variance estimate) = 𝑠_𝑤𝑖𝑡ℎ𝑖𝑛^2=(𝑆𝑆_𝑤𝑖𝑡ℎ𝑖𝑛)/(𝑑𝑓_𝑤𝑖𝑡ℎ𝑖𝑛 )
  5. F ratio: 𝐹=(𝑠_𝑏𝑒𝑡𝑤𝑒𝑒𝑛^2)/(𝑠_𝑤𝑖𝑡ℎ𝑖𝑛^2 )

    Summary of Anova

    Source of variance df SS S2 F ratio
    Between groups k – 1 SSbetween SSbetween / (k-1) 2between / s2 within
    Within groups N – k SSwithin SSwithin/ (N-k)
    Total N - 1 79.14

Step 4 Identify the region of rejection.

Let's say df between = 2 and df within = 6. The critical value of F is 5.14 (using tables). This is the value of F beyond which the most extreme 5% of sample outcomes will fall when H0 is true.

Step 5 Make the statistical decision and form conclusions.

  • Because the sample F ratio falls in the rejection region (i.e., 7.75 > 5.14), we reject the Null Hypothesis. 
  • The overall F ratio is statistically significant (α = .05), so we conclude that the population means differ in some way. 
  • Next, conduct post hoc comparisons to determine the specific source(s) of the statistical significance.

Step 6 Conduct Tukey’s HSD test.

  • Calculate HSD: 𝐻𝑆𝐷=𝑞×√((𝑠_𝑤𝑖𝑡ℎ𝑖𝑛^2)/𝑛_𝑔𝑟𝑜𝑢𝑝 )
  • Compare HSD with each difference between sample means.
  • Make the statistical decisions and form conclusions: Ex: Group 1 is significantly different from Groups 2 and 3; the difference between Groups 2 and 3 is not significant. 

* * *

Above is the procedure of Analysis of Variance.


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