overview: completely randomized designs (crds) factors...

Overview:Completely randomized designs (CRDs)

Factors, Factorials, and Blocking

STAT:5201

Week 2: Lecture 1

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Completely Randomized Design (CRD)

Simplest design set-upEUs are are randomly assigned to treatmentsEasiest to doEasiest to analyzeOften sufficient for the goals

Example (One-factor study CRD)

DayLength (short/long) is the only factor in the study. We have eighthamsters (EUs). We use a random number table to assign the shortDayLength to 4 hamsters and the long DayLength to 4 hamsters. NIEnzyme level is recorded at the end of the study.

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A completely randomized design (CRD) has

* N units* g different treatments* ni observations in each treatment where

∑ni = N

− If all groups have an equal n observations it is a balanced design* Completely random assignment of EUs to treatments

Completely random assignment means that every possible group ofunits into g groups with the given sample sizes is equally likely.

Example (One-factor study CRD)

In this balanced single factor study, hamsters have equal probability ofbeing assigned to either of the 2 treatments which are the levels of theDayLength factor or long or short... g = 2, n1 = n2 = 4,N = 8.

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Example (CRD single factor experiment)

The number of times a rod was used to remove entrapped air from aconcrete sample was used as the design variable in an experiment. Theresponse variable was compressive strength of the concrete. Three runswere done on each of 4 levels of the factor Rodding Level (10,15,20,25).This was a CRD as the 12 runs were randomly assigned to the treatmentsin a balanced fashion, run shown in parentheses.

RoddingLevel Compressive Strength

10 1530(1) 1530(4) 1440(9)15 1610(3) 1650(7) 1500(8)20 1560(6) 1730(10) 1530(12)25 1500(2) 1490(5) 1510(11)

Montgomery, Applied Stat and Prob Engrs (2011)

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Example (CRD single factor experiment - SAS)

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SAS plot using coding from Dean et al. page 54

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Another option for plotting

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Diagnostic plots (automatically generated from model fit):1) check constant variance [violated]2) check normality

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In the single factor experiment, the usual items of interest...

Is there evidence that some means are different?

If at least some are different, which are different from each other?

Any pattern in the differences?

Estimates/confidence intervals of means and differences.

In some special cases, variability may be of interest.

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A completely randomized design can have more than one factor.

Example (CRD two-factor experiment)

Besides DayLength (short/long), researchers are interested in a Climate(cold/warm) effect. The combination of these two factors give fourtreatment groups to this study. As a CRD, we will randomly assign the 8hamsters to the 4 treatment groups (placing 2 hamsters in each treatmentgroup).

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It can be useful to perceive a CRD two-factor study as a single factorstudy where the single factor, or “superfactor”, has levels coincidingwith the crossing of the two factors.

Example (CRD two-factor experiment)

This two-factor CRD study could also be perceived as a single“superfactor” experiment...

You might think of it this way when you’re doing the randomization, or forreasons of convenience that may come-up later.

In CRDs, there is no blocking or nesting. Given the treatment group,the observations are independent.

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Multifactor Experiments

In an experiment, factors can be either...

* controlled* controlled for* left uncontrolled* held fixed

If we have the ability to choose and set the levels of a factor, thenthis factor can be controlled.

* Choosing dosage levels of a drug (10ml, 20ml, 30ml,...)* Choosing the temperature at which to run a process (250◦, 300◦,...)* Choosing day length exposure (short, long)

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Sometimes we want to include a factor in our study because it islikely to be a large source of variation, but we don’t have the powerto “assign” the levels. The, we instead control for the factor.

* Sex* Age* Genetic background, family group* Income level

Only factors having relatively small effects on the response should beleft uncontrolled. Randomization should take care of these smalleffects in that our results won’t be biased. And obviously, factorshaving small effects that we are unaware of are left uncontrolled.

Holding a factor fixed is an option, but it means you’ve narrowed thescope of your experiment.

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Factors of interest can be called “Primary factors”.

Other factors may be included in the study as they are known to be alarge source of variation in the response, but not of primary interest.These factors can be called “Nuisance Factors”.

For example, we are interested in comparing two drugs or drug brand(primary factor) but we know that age group (nuisance factor) mayalso be related to the response, but I’m not really interested indetecting an age effect or estimating an age effect.

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When deciding if a factor is a primary factor or not, I might ask aclient:

Do you want a formal p-value comparing the different levels ofof the factor? In other words, do you want to be able to say...‘We found Drug A to be significantly different than Drug B...’If so, then drug effect is a primary factor.

Only nuisance factors, not primary factors, are used as blockingfactors. This is because we’re not interested in how the responsechanges from one “nuisance block’ to the next. We’re reallyinterested in how the response changes from one treatment to thenext within a block.

A factor that is used as a blocking factor is usually confounded withother nuisance factors, and that means any observed differencesbetween the blocks could be due to something other than the blockitself (e.g. what ‘looked’ like an age effect was actually a day effect).

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Factorial Exeriments

Factorials are the simplest kind of multifactor experiment.* design consists of two or more factors* there is no blocking* there is no nesting* CRD set-up, assigning treatments to EUs

Example (Two-factor factorial, 2x2 factorial)

Revisiting our earlier example, we have 4 treatments from thecombinations of DayLength (short/long) and Climate (cold/warm) and8 EUs, with 2 EUs randomly assigned to each treatment as a CRD.

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Interaction Plots

Interaction Plots or Profile Plots

Constructed by plotting the cell means for each combination offactors, such that the levels of one factor are shown on the horizontalaxis and the levels of the other factor (the trace factor) arerepresented by separate lines.

When you have two factors, either one can be used as the factoralong the x-axis. Often, one of the two possible plots seems better forinterpretation purposes.

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Interaction Plots


Interaction plots make it easy to see information in the data quickly...*Which treatment gives the highest response? Lowest response?

If the lines are parallel, then the effects of the two factors are said tobe ‘additive’ or ‘main effect only’ and there is no interaction.

If the lines are not parallel, then we say the two factors interact orthere is interaction between the factors. So, there’s a morecomplicated story there.

If we have additive effects (no interaction present), then the effects ofa factor are the same for all levels of the other factor.

If there is interaction, then the effects of a factor are different atdiffering levels of the other factor.

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Possible Interaction Plots - two factors


Suppose we have two factors A and B and each has two levels aslow and high in a factorial CRD experiment.

There are a number of possible observed interaction plots.

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Parallel lines. No interaction is present.

The effect of factor B is essentially the same for all levels of factor A.

We say the effects of these factors are additive effects.21 / 44



Non-parallel lines. Interaction is present.

The effect of factor B depends on the level of factor A.On the left, the effect of factor B is much larger when factor A is setat ‘high’.On the right, the effect of factor B is not only larger when factor A isset at ‘high’, but it’s in the opposite direction!

NOTE: The cell means give us an idea about interaction, but we need to formallytest for an interaction in our modeling, and not rely on a graphic suggestion. 22 / 44

Interaction Plots - three factors

Higher orders of interaction

The previous discussion has focused on 2-way interaction. But youcan have a 3-way, 4-way, 5-way, ... interaction as well.

These interactions quickly become difficult to interpret, and difficultto deal with (without partitioning the data into subsets).

Often, we hope (or maybe just assume) that these higher orderinteractions are not present. But if we are able to test for them, weshould.

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Three-way interaction exists if the two-way interaction differs acrossthe levels of a third variable.

Example (Three factor factorial, 2x2x2 or 23 factorial)

Consider a three-factor factorial CRD with factors A, B, and C each with alow or high level. There are 8 treatments.

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Example (Three factor factorial)

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3-way 25 / 44

Blocking

As we move away from completely randomized designs to morecomplex designs, the first design we will consider is the randomizedcomplete block design (RCBD).

To block an experiment is to divide, or partition, the experimentalunits into groups called blocks.

A block of units is a set of units that are homogeneous in some sense.To form blocks, we organize EUs into groups having similarcharacteristics.

EUs Possible Blocking factor

patients age (11-20, 21-30, ..., 81-90)patients familypatients the hospital giving the carefactory workers shift (early, late, night)field plots location/soil compositionskin patches two patches from same person

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Blocking

Other ways to form a block:* Physically divide an object into parts, like a manufacturing setting.* Repeat testing of the same object under the different conditions

(like when you give both drugs to the same person).

Best case scenario is when we have enough EUs in a block to observeall treatments (complete block). Otherwise, we have an incompleteblock.

A RCBD utilizes restricted randomization, where we randomlyassign the EUs to treatments within a block. Thus, if there are rblocks, then we will do r restricted randomizations, one for eachblock, to assign the EUs to the treatments within block.

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Blocking

Example (Randomized Complete Block Design)

Drugs A, B, C, and D are to be compared. We have formed blocks basedon age, and we have 4 patients in each of 6 age blocks. We are notinterested in testing for an age effect (it is a nuisance factor) as we aremost interested in comparing treatments. Within each block, we randomlyassign a unique drug to the 4 patients.

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Blocking

Remember, don’t use a primary factor of interest for blocking. Ablocking factor should be a nuisance factor. Something that is asource of variation for the response but is not of great interest.

We don’t use a factor of interest as a block because we confoundnumerous nuisance factors together in a block.

EXAMPLE: Possible nuisance factors are lab assistant & day of week.

Plan - Complete all asst. 1 runs on Monday, complete all asst. 2 runson Tuesday, etc. If there was a large block effect, then blocking wasuseful, and I’m not interested in knowing if it was the assistant or theday (or something else) that had a big impact.

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Blocking

Blocked designs are not completely randomized designs. They use arestricted randomization.

Blocking is a variance reduction technique.

Block-to-block variability is still in the data, but we essentially removethis variability when comparing treatments (because we see alltreatments within a block).

Blocking is most useful when there is wide variability across blocks.We don’t usually test for a block effect because we EXPECT a largedifference across blocks, that’s exactly why we’re using it, and it’s anuisance factor anyway.

In general, we assume there is no interaction between the block andthe treatment. We assume the treatment effect (i.e. differencesbetween the treatments) is the same for all blocks.

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Blocking

By comparing treatments within a block, we remove theblock-to-block variability from our treatment comparison analysis.

Blocking is a powerful tool and should be used if possible to controlfor any ‘nuisance’ variation that is thought to be large.

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Randomized Complete Block Design (RCBD)

RCBD...

Uses ‘restricted randomization’, performed within each block.

* g treatments* g EUs per block* r blocks* rg = N total units

It’s like r single-replication CRDs glued together.

The RCBD is used to increase power and precision of an experimentby decreasing the error variance used in testing.

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RCBD

Example (Randomized Complete Block Design)

Here, we revisit the golden hamster example and perceive the 4 treatmentsas a single ‘superfactor’ created from the combination of DayLength andClimate, and we include a nuisance factor Litter (similar to ‘family’) tobe used as a blocking factor.

We expect large litter-to-litter variability due to genetics. From each of Llitters, we have 4 hamsters.

Treatments: A) cold/short B) cold/long C) warm/short D) warm/long

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RCBD

Example (CRD vs. RCBD)

See handout on CRD and RCBD for litter example.

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Fixed effects vs. Random effects

Sometimes the levels of a factor are random. Then this is a randomfactor and it has random effects.

For example, when we randomly choose the litters in our hamsterexperiment, the factor Litter has random levels, usually numbered as1,2,3,... and they were chosen from a large population of possiblelitters.

If we repeated the hamster experiment, and again randomly choselitters, the litters from the first experiment would be different thanthe litters from the second experiment (again, random levels).

When the levels of a factor are fixed values then it is a fixed factorand has fixed effects. For example, when you have levels of ‘circle’and ‘square’ for the factor Shape, if you repeated the experiment, youwould have the same two shapes (they were not randomly chosen).

Primary factors usually have fixed effects.

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Random effects

Example (Random factor ‘Litter’)

The litters in the experiment are a random draw from the large populationof litters available.

Example (Random factor ‘Day’)

The days in your experiment are a random draw (in theory) from the largepopulation of days available.

The variability among the litters or the days in the examples aboveare meant to represent the general variability among these units in thegiven population.

We model random effects and fixed effects differently. Again, primaryfactors (i.e. factors of interest) are usually fixed effects.

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Fixed effects vs. Random effects

The name of the factor does not tell you whether or not it is hasrandom effects. Consider the factor called School.

Example (School as a fixed effect)

There are 3 schools in a study labeled A,B,C. These are the only schoolspresently of interest. We want to know if the response is significantlydifferent between these schools (A vs B, A vs C, B vs C). If we repeatedthe experiment, we would use these same three schools again.

Example (School as a random effect)

Three schools are randomly chosen from all available schools. We willlabel them as s1, s2, and s3. The variability in the response among s1, s2,and s3 is meant to represent the variability among all schools. If werepeated the experiment and randomly chose schools again, we would notuse these same three schools.

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Random effects

If we have a random factor with random effects, what do we wish toestimate?

For the school random effect, we want to make a statement about allschools using a random sample of school.

Typically, we want to estimate the general variability among units,such as σ2school .

Blocking factors, in general, have random effects, but we will startthe course by considering them as fixed, but change this later as wemove into the random effects topic.

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Real Experiment - Cotton Spinning Experiment

Example (Dean, Section 2.3, p. 13)

Objective - To understand how the degree of twist (i.e. turns per inch) andthe type of guide (i.e. ‘flyer’) affect breakage.

Factors of interest:1) twist (1.63, 1.69, 1.78, 1.90)2) flyer (1=ordinary, 2=special)

Response variable: breaks per pound

The full crossing of the factors of interest would include 8 treatmentgroups, but two were thrown-out based on results from a pilot study(ordinary/1.63, and special/1.90).

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What are some other sources of variation inthe breakage rate?

1) Quality of material (roving below)2) Environmental conditions3) Operator4) Machine

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What design might they use?• Completely randomized design• Randomized complete block design (chosen design, data below)• Latin square

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The data is available in a SAS data file from the author’s website.

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In this analysis, I first created more meaningful names for the treatmentgroups by creating a new variable called ‘trt’ as below.

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Then I generated a plot specifying a unique symbol for each of the 13blocks.

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overview: completely randomized designs (crds) factors...

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