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Row-column experiment with two crossed blocking factors

A situation that is frequently encountered in field experiments is where there are two crossed blocking factors e.g. "day" and "time in the day" e.g. 4 different time intervals in a day. This is a so called row-column experiment. Suppose one has to check 7 effects, how can this row-column DOE set up with the custom designer in JMP? 

DOERe: Row-column experiment with two crossed blocking factors

Use two blocking factors:

 

 

Here is the JSL code to make this illustrative example:

 

DOE(
 Custom Design,
 {Add Response( Maximize, "Y", ., ., . ), Add Factor( Blocking, 8, "Day" ),
 Add Factor( Blocking, 2, "Time of Day" ),
 Add Factor( Continuous, -1, 1, "X3", 0 ),
 Add Factor( Continuous, -1, 1, "X4", 0 ), Set Random Seed( 353 ),
 Add Term( {1, 0} ), Add Term( {3, 1} ), Add Term( {4, 1} ), Add Term( {1, 1} ),
 Add Term( {2, 1} ), Add Alias Term( {3, 1}, {4, 1} ), Set Sample Size( 32 ),
 Simulate Responses( 0 ), Save X Matrix( 0 )}
);

Re: Row-column experiment with two crossed blocking factors

So for 4 time intervals/day in Time of Day I need to turn the labels 5 6 7 8 and 9 10 11 12 and 13 14 15 16 to 1 2 3 4 ? And in the table the column Day must be ordenened 1 to 4 and then in each day block ordening Time in Day 1 to 4 I got the final DOE?

 

 

 

Re: Row-column experiment with two crossed blocking factors

You have an interesting situation.  One could argue there is no way the time intervals within a day are crossed with day because the within day time intervals are nested within day.  So the numbering is "in question".  When you use the coding 1-4, 1-4 for the time intervals (for 2 days), the software recognizes the 1 for time interval for day 1 to be "identical" to the time interval 1 for day 2.  While the time interval is similar, it can not be identical.  I would call it systematic.  I could argue (both ways) the systematic pattern within day is close enough to the same to investigate the interaction or I could argue that it is definitely not the same, hence it is nested.  I think it extremely important that you have hypotheses to support the time intervals within day are similar enough to justify the crossed nature of the effects.

Re: Row-column experiment with two crossed blocking factors

The naming of the block levels will work either way. If you change the labels for time of day to mean first, second. and so on, then you must indicate that the level is nested in the day factor for the time of day effect in the model.

 

The ordering of the block levels is a convenience to you. It is not necessary for the analysis.

Re: Row-column experiment with two crossed blocking factors

Adding to @statman, the Day and Time of Day effects should probably be defined as random effects in the model.

Re: Row-column experiment with two crossed blocking factors

Thanks for reply. Indeed the levels of Time of day are nested in the day blocking factor; it is an experiment where several effects (quite a umber of runs are required) have to be tested during a number of days because only 4 runs/day are possible, however also during one day temperature and weather conditions can change so each day block needs to be split into 4 different time intervals (in each time interval conditions are  assumed to be +/- stable) --> how do I indicate this nesting in custom design? 

Re: Row-column experiment with two crossed blocking factors

You cannot indicate nesting or random effects in Custom Design. You modify the model in the Fit Model launch dialog. You can save the updated model to the data table for future use.

Re: Row-column experiment with two crossed blocking factors

The key to proper analysis is the model you create.  The numbering in @markbailey's post would work for nesting (sequential, non-repeating).

 

The concept of a block is "theoretically" to hold the noise variables constant within the block thereby increasing the precision of detecting factor effects and allow (or manipulate) those noise variables so they change between blocks, thereby increasing the inference space while not negatively effecting the precision. You are in a situation, where some of the noise variables are changing within the block.  This "theoretically" will increase the error within the block reducing the precision of detecting factor effects.  Can you choose your blocks differently?  Is the within day variation in ambient conditions greater than the between day? Some other options:

1. Measure ambient conditions and treat those measurements as a covariate (e.g., dew point)

2. Use a center point treatment (current conditions?) or perhaps an identical treatment combination to replicate over the course of the experiment.  Analyze this and use the mean square of this replicated treatment as a basis for statistical tests (error estimate).

Re: Row-column experiment with two crossed blocking factors

This covariate approach is interesting! Over the entire field expeiment identical concitons are nearly imossible so center point replication is no option also there may be lot of vaiation between the daily time intervals. Thanks for inputs I will now try to work out a DOE with seven fixed effects using Marc's nesting approach. 

Re: Row-column experiment with two crossed blocking factors

Just another thought I had, was do you really need 4 time periods within day?  Could you reduce this to 2 (morning and afternoon)?  This might be more efficient.

Re: Row-column experiment with two crossed blocking factorsor randomly select 2 time periods within day and treat that as a random effect.Re: Row-column experiment with two crossed blocking factors

I fully agree, 2 time periods would simplify a lot however engineers decided to go for 4 time periods (quite some uncontrolled time variation is expected during one day) and have one DOE run for each so 4 runs per day block, So total testing will take 6 full days. I believe this time period always is a random effect no matter how many periods/day are specified? 

Re: Row-column experiment with two crossed blocking factorsIf you have hypotheses as to what is changing between the time periods and why it would effect the response, then by all means keep track of those time periods (at least systematically). To me a random effect is what Shewhart would call unassignable. I believe if you sample it systematically, you can assign it (but not everyone would agree with me). I often assign the block and block-by-factor effects when I am confident I understand what variables are involved and have hypotheses to back them up.Re: Row-column experiment with two crossed blocking factors

Hi statman, great to read your valuable comments! There is lot of food for thought about the random factor issue and as such great input for interesting discussions on what is random/unassignable and what is systematic/assignable, is not black &white.. I consider time variation witin a certain interval (day, hours, maybe even minutes..) as random when in a given time period things are occuring beyond our control although they can be systematic such as unnatural variation in Shewhart control charts indicating unstable experimental conditions. It indeed is necessary, during the experiment, to monitor environmental effects such as temperature, moisture, operator changes and other ucontrollable parameters that may have an impact on the model responses; these effects are considered as co-variates, potential lurking variables to be followed up closely and taken up in the final model; indication of systematic covariate variation in control charts may mess up the final model particularely when there is collinearity with the specified DOE effects. Remark: would it not be an idea to generate a DOE with  "day" and "time of day" being specified as covariate effects as for the other experimental covariates above?

Re: Row-column experiment with two crossed blocking factors

My thoughts on covariates:

1. You have to consider additional measurement errors

2. Since they are not manipulated, this is more likely evidence of correlation, not causation.  In addition, you have multicollinearity issues.

3. You have to choose what "value" to put in for each treatment even though the variables are "wandering around" (you can only put one in).  How representative are the values you use compared with the natural variation of the covariate?

4. Potential lag effects must be considered

5. You need to either give up a degree of freedom from the design factors you are manipulating (for each covariate) or run random replicates to accommodate the degrees of freedom for analysis

6. You'll likely miss out on interaction effects

7. You are limited on the number of covariates you can consider at any point in the analysis

8. They must be continuous variables, so not variables like Day or time of day, IMHO.

 

There are a number of what I think are more effective and efficient means of handling noise in experiment situations than covariates.  The idea of a block, where noise is somewhat constant within the block and changes between blocks allows you to group or chunk a number of noise variables simultaneously. This is very efficient in the early stages of investigation. For robust design and processes, it is extremely important to be able to assign and understand noise-by-factor interactions.  Of course, split-plot designs and cross-product arrays are also quite useful when you have more specifically identified the number of influential noise variables.

 

Re: Row-column experiment with two crossed blocking factors

I agree that for robust DOE it is important to study the noise-factor interaction effects but these noise effects need to be taken up as inputs in the DOE at carefully specified levels the way that accurate estimation and modeling is possible. I don't see a reason why covariates must be continuous? I remember that a well known DOE expert presented a succesful DOE study with categorical covariates. Also I got some DOE proposals with categorical covariates. I don't know why it would not work here here may be pure theoretical statistical arguments (?) but in DOE one needs to be pragmatic.

Re: Row-column experiment with two crossed blocking factors

Sorry, my italicized must for covariates being continuous was meant to mean there are other options.  Even as categorical variables, they are treated as continuous in ANCOVA.  

 

I'm not sure what you mean by this: "these noise effects need to be taken up as inputs in the DOE at carefully specified levels the way that accurate estimation and modeling is possible".  I have run >100's of experiments successfully using RCBD to estimate noise-by-factor interactions and find factors or levels of factors that are robust to noise.

 

Not sure what you are insinuating from your last comments?  I suppose I have outstayed my welcome.

Re: Row-column experiment with two crossed blocking factors

In DOE studies #runs are most 10 - 30, purpose is to design experiments the way effects can be estimated with maximal precision. So if your goal is robustness  then accurate estimation of the noise - control factors is crucial and a careful specification of the noise effects and their levels is necessary. So in first place one needs do know these noise factors and in lab experiments they can be controlled, in field effects this is nearly impossible so they should be taken up as covariates and carefully monitored during the experimental runs, hoping afterwards that there is not too much collinearity the way estimation of their interactions is possible, before analysis a careful DOE evaluation is necessary.