Hello, I am working with data from a balanced designed experiment and I'm unsure of how to model it to get the answers I'm seeking.  Two Detergents (current vs. experimental) are being compared against 6 different types of fabric Soils (motor oil, lipstick, etc).  Thus Soil and Detergent have fixed effects.  Each Detergent was tested on 3 different Loads of laundry. A single laundry Load consists of 3 "Backers"  where a Backer is a fabric sample that's been partitioned into a grid of 6 areas.  Each area contains one of the 6 soils.  All experiments were conducted in a lab using the same washing machine.  Soil Removal is the response measurement.
Whole plots:  Loads    --- 3 nested within each Detergent
Whole plot factor:  Detergent   --- 2 levels
Split Plot:  Backer   --- 3 nested within each Load
Split Plot factor:  Soil   --- 6 levels
Here's my question:  I have somehow gotten the impression that for balanced data, the p-values from a mixed effects analysis can be trusted.  However, the lmer function doesn't output p-values regardless of the case.  I'd like to do multiple comparisons to find the Soil:Detergent combinations that stand out as statistically significant.  Does anyone have advice for accomplishing this? 
Ultimately, my purpose is to provide a method to this chemist who is studying detergents, because he wants do the analysis himself.  Admirable, wouldn't you agree?    I want to avoid a discussion of 'p-values aren't that important, just make a qualitative comparison using the t-values'.  That would be a mixed message considering our statistics group's attempts to get the scientists to do less eyeballing and make decisions more objectively, i.e. consider statistical significance.
I asked a similar question a few months ago, and got a reasonable answer that the blocking factors such as Load and Backer could be modeled as fixed effects.  However, if I do that for this nested case, the Soil effects are conditional on the Load and the Backer.  I apologize if I'm not providing enough information.  Please let me know if I could add anything.  Thank you for taking the time to consider my request.
Regards, Paul
====================================================
> str(SoilOut)
'data.frame': 108 obs. of 6 variables:
$ Soil : Factor w/ 6 levels "DSB.P/C","EMPA.101",..: 1 1 1 2 2 2 3 3 3 4 ...
$ Detergent : Factor w/ 2 levels "EXP2","Xtra": 2 2 2 2 2 2 2 2 2 2 ...
$ X.SoilRemoval: num 76 76 76.5 41.2 34.8 ...
$ Load : Factor w/ 6 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 1 1 ...
$ Backer : Factor w/ 18 levels "1","2","3","4",..: 1 1 1 1 1 1 2 2 2 2 ... 
==============================================
m2<-lmer(X.SoilRemoval~Detergent*Soil + (1 | Load/Backer), SoilOut) 
> summary (m2)
Fixed effects:
                                   Estimate Std. Error t value
(Intercept)                         74.1804     1.7712   41.88
Detergent[T.Xtra]                   -3.2044     2.5049   -1.28
Soil[T.EMPA.101]                   -35.1601     2.0543  -17.12
Soil[T.EMPA.104]                   -26.2447     2.5049  -10.48
Soil[T.EMPA.106]                   -35.4014     2.5049  -14.13
Soil[T.LIPSTICK]                   -12.1608     2.5049   -4.85
Soil[T.MAKE-UP]                    -38.1436     2.5049  -15.23
Detergent[T.Xtra]:Soil[T.EMPA.101]   0.8651     2.9053    0.30
Detergent[T.Xtra]:Soil[T.EMPA.104]  -4.0126     3.5425   -1.13
Detergent[T.Xtra]:Soil[T.EMPA.106]   0.6405     3.5425    0.18
Detergent[T.Xtra]:Soil[T.LIPSTICK]  -0.9912     3.5425   -0.28
Detergent[T.Xtra]:Soil[T.MAKE-UP]    2.9547     3.5425    0.83

Paul Prew   ▪  Statistician
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