Testing significance of effects in a 2^k factorial experiment

1. Given the average values (y₁, y₂, ...) and sample standard deviations (S₁, S₂, ...) among the replicates at each combination of test conditions. (There will be 2^k combinations in a full factorial experiment with k variables, each at 2 levels. Each combination has r replications, and hence each standard deviation has r-1 degrees of freedom.)
2. Calculate the effects using Yates algorithm
3. Calculate the pooled variance S_p² = (S₁² + S₂² + ...)/2^k (this is just the weighted average)
4. Calculate the standard error of an effect S_E = 2S_p/sqrt(r*2^k) (since half the factorial points are at the high level and half at the low level)
5. Calculate the signal-to-noise ratios for each effect t_E = Effect/S_E
6. Calculate the total degrees of freedom = (r-1)*2^k which is the sum of the degrees of freedom for the individual variances. Look up t_crit from tables for the chosen level of significance and the total degrees of freedom of the pooled variance.
7. Compare with critical value t_crit to test for significance. If t_E is less than t_crit, then we cannot reject the null hypothesis that there is no effect.

Example:

1. The pooled variance S_p² = (312.5 + 480.5 + ... +338.0)/8 = 326.6
2. The standard error of an effect S_E = 2*18.07/sqrt(16) = 9.036
   

Comparing, we find that A and AC interaction are significant

Reference:

Lawson, John and Erjavec, John, "Modern Statistics for Engineering and Quality Improvement", Thomson - Duxbury, 2001. pp.217-220

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