Nuisance factor on analysis of variance

Question

I had this problem,it simply to see that the variable "Day" can be viewed as nuisance factor.This is my anova table:

  Factor coding (-1; 0; +1)
        Factor Information
        Factor Type Levels Values
        Temperature Fixed 3 -1; 0; 1
        Time Fixed 3 -1; 0; 1
        Day Fixed 2 1; 2

        Analysis of Variance

        Source         DF Adj SS Adj MS F-Value P-Value
           Temperature 2 1985,33 992,67  8,39    0,011
           Time        2 2908,33 1454,17 12,28   0,004
           Day         1 40,50   40,50   0,34    0,575
      Temperature*Time 4 459,33  114,83  0,97    0,474

        S       R-sq   R-sq(adj) R-sq(pred)
        10,8800 85,06% 68,26%    24,39%`

a) The nuisance factor, that is not significative , can be removed? b) is right make the first analysis without interaction (Time*Temperature) to see that the nuisance factor is not significative and , after, add the interaction on the second model (without nuisance factor) and analyze it?

Answer 1

Unless you have some reason to believe that 'Days' were different in some way that matters (e.g, different humidity, different technicians, etc.) my first analysis would be to consider days as replications in each of the nine cells.

Doing that I got the following table using Minitab. (I hope I typed all the data correctly, but wouldn't want to bet on that.)

ANOVA: Thk versus Tmp, Sec 

Factor  Type   Levels  Values
Tmp     fixed       3  -1,  0,  1
Sec     fixed       3  -1,  0,  1

Analysis of Variance for Thk

Source   DF      SS      MS      F      P
Tmp       2  1985.3   992.7   9.05  0.007
Sec       2  2908.3  1454.2  13.25  0.002
Tmp*Sec   4   459.3   114.8   1.05  0.435
Error     9   987.5   109.7
Total    17  6340.5

S = 10.4748   R-Sq = 84.43%   R-Sq(adj) = 70.58%

I would leave the nonsignificant Tmp*Sec interaction in the model, since the two main effects are highly significan. I would have a look at residuals vs. fits to see if variances are stable, and do a Q-Q plot of residuals to see if there is any remarkable departure from normality.

Then I would include 'Day' into the model as a random effect to see what happens:

ANOVA: Thk versus Tmp, Sec, Day 

Factor  Type    Levels  Values
Tmp     fixed        3  -1,  0,  1
Sec     fixed        3  -1,  0,  1
Day     random       2  1, 2

Analysis of Variance for Thk

Source   DF       SS       MS      F      P
Tmp       2  1985.33   992.67   5.45  0.155
Sec       2  2908.33  1454.17  15.78  0.060
Day       1    40.50    40.50   0.23  0.698 x
Tmp*Sec   4   459.33   114.83   1.15  0.447
Tmp*Day   2   364.00   182.00   1.83  0.273
Sec*Day   2   184.33    92.17   0.92  0.468
Error     4   398.67    99.67
Total    17  6340.50

x Not an exact F-test.

S = 9.98332   R-Sq = 93.71%   R-Sq(adj) = 73.28%

So including a random Day effect is not helpful. Furthermore residuals for Day 1 vs Day 2 show nothing interesting. In short, I would find the same things you did, but perhaps in a different order.

In a consulting report of the analysis, I would mention in a footnote that I looked at Days as a random factor without interesting findings, but I would not show the ANOVA table. [I have not seen the word 'nuisance' used to refer to a potential factor in an ANOVA; I don't think I would use that word in a professional report. (I have seen a parameter, such as normal $\sigma,$ called a 'nuisance' parameter, if the goal is to estimate $\mu.$ I think that terminology is quite standard.)]