Simple Method for One Way Analysis of Variance

Data Table
Factor Observations Factor Total No of observations
Factor 1 y11 y12 y13 ... y1n A1 n1
Factor 2 y21 y22 y23 ... y2n A2 n2
Factor ... ... ... ... ... ... ... ...
Factor k yk1 yk2 yk3 ... ykn Ak nk

  1. Determine the grand total of all observations, G
  2. Determine the total number of all observations, N
  3. Determine the crude sum of squares of all the observations SS
  4. Determine the correction factor G2/N
  5. Determine the corrected sum of squares CSST = SS-G2/N
  6. Determine the total for each factor. Square it and divide by the number of observations comprising the total. Add all these quantities and subtract the correction factor to get CSSF, the between factor corrected sum of squares
  7. The error sum of squares CSSE = CSST-CSSF
  8. The mean squares are obtained by dividing the corrected sum of squares by their corresponding degrees of freedom. The mean square factor MSF = CSSF/(k-1).
  9. Similarly, the mean square error MSE = CSSE/(N-k).
  10. Calculate the value of the observed test statistic FO = MSF/MSE.
  11. Compare FO with the critical value chosen of F, Fcrit for the test (with k-1 degrees of freedom for the numerator and N-k for the denominator). If FO is greater, then we reject the null hypothesis that all factor means are equal. That is, at least one of the factor means is significantly different. Otherwise, we cannot reject the null hypothesis.

Example:

Data Table
Factor Observations Factor Total No of observations
Factor 1 62 60 63 59         244 4
Factor 2 63 67 71 64 65 66     396 6
Factor 3 68 66 71 67 68 68     408 6
Factor 4 56 62 60 61 63 64 63 59 488 8

  1. G = 1536
  2. N = 24
  3. SS = 98644
  4. G2/N = 98304
  5. CSST = 340
  6. CSSF = (2442/4 + 3962/6 + 4082/6 + 4882/8) -98304 = 98532 - 98304 = 228
  7. CSSE = 340 -228 = 112
  8. MSF = 228 / 3 = 76
  9. MSE = 112 / 20 = 5.6
  10. FO = 13.57
  11. Fcrit = F0.05,3,20 = 3.10. Since FO > Fcrit, we reject the null hypothesis that all factor means are equal. In other words, at least one of the factors is significantly different.

    Reference:

    Box, George E. P., Hunter, William, G., and Hunter, Stuart J., "Statistics for Experimenters - An Introduction to Design, Data Analysis, and Model Building", John Wiley & Sons, New York, 1978.


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