Kempthorne uses the randomization-distribution and the assumption of * unit treatment additivity* to produce a * derived linear model* , very similar to the textbook model discussed previously. [30] The test statistics of this derived linear model are closely approximated by the test statistics of an appropriate normal linear model, according to approximation theorems and simulation studies. [31] However, there are differences. For example, the randomization-based analysis results in a small but (strictly) negative correlation between the observations. [32] [33] In the randomization-based analysis, there is * no assumption* of a * normal* distribution and certainly * no assumption* of * independence* . On the contrary, * the observations are dependent* !

In the post, I show examples of interaction plots with two factors and another with two continuous variables. However, you can certainly create an interaction plot for a factor * continuous variable. For your model, this type of graph will display two lines–one for each level of the age factor. Because you already know the interaction term is significant, the difference between the two slopes is statistically significant. (If the main effect had been significant, the interaction plot would have included it in the calculations as well–but it is fine that it’s not significant.)

where * n* Op is the number of operators, * n* Rep is the number of replicate measurements of each part by each operator, * n* Part is the number of parts, * x̄* is the grand mean, * x̄ i.. * is the mean for each part, * x̄* ·* j* · is the mean for each operator, * x* * ijk'* is each observation and * x̄* * ij* is the mean for each factor level. When following the spreadsheet method of calculation the * n* terms are not explicitly required since each squared difference is automatically repeated across the rows for the number of measurements meeting each condition.

Bentler, P. & Satorra, A. (2010). Testing Model Nesting and Equivalence . Psychol Methods. 2010 Jun; 15(2): 111–123. Retrieved 9/19/2016 from http:///pmc/articles/PMC2929578/.

Carriquiry, A. Multiple Regression. Retrieved 9/19/2016 from: http:///~alicia/stat328/Multiple%20regression%20-%20nested%

Doncaster & Davey (2007). Analysis of Variance and Covariance: How to Choose and Construct Models for the Life Sciences. Cambridge University Press.

Purdue Statistics. A Look at Nested Factors. Retrieved September 17, 2017 from: http:///~bacraig/notes1/

Rigdon, . (1999). Using the Friedman method of ranks for model comparison in structural equation modeling. Structural equation modeling, 6(3), 219-232

Bentler, P. & Satorra, A. (2010). Testing Model Nesting and Equivalence . Psychol Methods. 2010 Jun; 15(2): 111–123. Retrieved 9/19/2016 from http:///pmc/articles/PMC2929578/.

Carriquiry, A. Multiple Regression. Retrieved 9/19/2016 from: http:///~alicia/stat328/Multiple%20regression%20-%20nested%

Doncaster & Davey (2007). Analysis of Variance and Covariance: How to Choose and Construct Models for the Life Sciences. Cambridge University Press.

Purdue Statistics. A Look at Nested Factors. Retrieved September 17, 2017 from: http:///~bacraig/notes1/

Rigdon, . (1999). Using the Friedman method of ranks for model comparison in structural equation modeling. Structural equation modeling, 6(3), 219-232