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BASICS OF DESIGNING EXPERIMENTS Thursday, October 24, 2013 5:00pm - 7:00pm GLC Room G

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About Me Graduate student in Virginia Tech Department of Statistics Enrolled in Master’s program Expected Graduation Date: December 2013 Future: Job in industry Lead Collaborator in LISA On-campus consulting group, led by Dr. Eric Vance and Dr. Chris Franck, with administrative specialist Tonya Pruitt

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About What? Laboratory for Interdisciplinary Statistical Analysis Why? Mission: to provide statistical advice, analysis, and education to Virginia Tech researchers How? Collaboration requests, Walk-in Consulting, Short Courses Where? Walk-in Consulting in GLC and various other locations Collaboration meetings typically held in Sandy 312 Who? Graduate students and faculty members in VT statistics department

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Requesting a LISA Meeting Go to www.lisa.stat.vt.eduwww.lisa.stat.vt.edu Click link for “Collaboration Request Form” Sign into the website using VT PID and password Enter your information (email, college, etc.) Describe your project (project title, research goals, specific research questions, if you have already collected data, special requests, etc.) Contact assigned LISA collaborators as soon as possible to schedule a meeting

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Agenda Introduction to Designing Experiments 3 Main Principles Randomization Replication Blocking (Local Control of Error) EX: Nozzle design and water jet performance EX: Treatment and leukemia cell gene expression Factorial experiments

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INTRODUCTION to Experimental Design

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Why is Experimental Design important? MAXIMIZE… Probability of having a successful experiment Information gain from results of an experiment MINIMIZE… Unwanted effects from other sources of variation Cost of experiment if resources are limited

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Experiment vs. Observational OBSERVATIONAL STUDY Researcher observes the response of interest under natural conditions EX: Surveys, weather patterns EXPERIMENT Researcher controls variables that have a potential effect on the response of interest Which one helps establish cause-and-effect relationships better?

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Correlation ≠ Causation

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EXAMPLE: Impact of Exercise Intensity on Resting Heart Rate Researcher surveys a sample of individuals to glean information about their intensity of exercise each week and their resting heart rate What type of study is this? Reported Intensity of Exercise each week Resting Heart Rate Ptp 1 Ptp 2 Ptp 3 …

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EXAMPLE: Impact of Exercise Intensity on Resting Heart Rate Researcher finds a sample of individuals, enrolls groups in exercise programs of different intensity levels, and then measures before/after heart rates TreatmentBaseline RHR Post Program RHR Ptp 1 Ptp 2 Ptp 3 ……

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EXAMPLE: Impact of Exercise Intensity on Resting Heart Rate What are some factors the experimental study can account for that the observational study cannot?

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Sources of variation Sources of variation are anything that could cause an observation to be different from another observation What are some reasons that measurements of resting heart rate could differ from person to person?

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Sources of variation There are two main types: Gender and age are what are known as nuisance factors: we are not interested in their effects on RHR, but they are hard to control What we are interested in is the effect of the intensity of exercise: this source is known as a treatment factor

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Sources of variation Good rule of thumb: list major and minor sources of variation before collecting data We want our design to minimize the impact of minor sources of variation, and to be able to separate effects of nuisance factors from treatment factors We want the majority of the variability of the data to be explained by the treatment factors

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Designing the experiment: The Bare Minimum Response: Resting heart rate (beats per minute) Treatment: Exercise Program Low intensity Moderate Intensity High Intensity

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Designing the experiment: The Bare Minimum Some assumptions We will be monitoring the participants’ diet and exercise throughout the study (not relying on self-reporting) We will only enroll participants with high (i.e. unhealthy) resting heart rates so that there is ample room for improvement Participants’ resting heart rate is all measured in the same manner, at the same time (upon waking up)

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Designing the experiment: The Bare Minimum Basic Design 36 participants: 24 males, 12 females Every person is assigned to one of the three 8- week exercise programs Resting heart rate is measured at the beginning and end of the 8 weeks What other considerations should we make in designing the experiment?

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THREE BASIC PRINCIPLES OF DOE: Randomization

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Randomization What? Random assignment of experimental treatments and order of runs Why? Often we assume an independent, random distribution of observations and errors – randomization validates this assumption Averages out the effects of extraneous/lurking variables Reduces bias and accusations of bias How? Depends on the type of experiment

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Exercise Example 36 participants are randomly assigned to one of the three programs 12 in low intensity, 12 in moderate intensity, 12 in high intensity Like drawing names from a hat to fall into each group Oftentimes computer programs can randomize participants for an experiment

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Exercise Example What if we did not randomize? Suppose there is some reason behind who comes to volunteer for the study first versus later If we assigned first third to one intensity, second third to another, and so forth, it would be hard to separate the effects of the “early volunteers” and their assigned intensity level Run12345678… EX111111111… EX213231213…

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Completely Randomized Design (CRD) What we just came up with is called a completely randomized design Note that in our case, treatments were assigned randomly, but in some experiments where there are a sequence of runs performed, the order of runs need to be randomized as well

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Summary Randomizing the assignment of treatments and/or order of runs accounts for known and unknown differences between subjects It does not matter if what occurs does not “looks random” (i.e. appears to have some pattern), as long as the order was generated using a proper randomization device

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THREE BASIC PRINCIPLES OF DOE: Replication

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Replication What? Independent repeat runs of each treatment Why? Improves precision of effect estimation Allows for estimation of error variation and background noise Check against aberrant results that could result in misleading conclusions EX: One person for each treatment. What could go wrong?

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Experimental Units (EUs) We now introduce the term “Experimental Unit” (EU) EU is the “material” to which treatment factors are assigned In our case, each person is an EU This is different from an “Observational Unit” (OU) OU is part of an EU that is measured Multiple OUs within an EU here would be if we took each person’s pulse at his/her neck, at the wrist, etc. and reported these observations

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Replication Extension to EU Thus, a treatment is only replicated if it is assigned to a new EU Taking multiple observations on one EU (i.e. creating more OUs) does not count as replication – this is known as subsampling Note that treating subsampling as replicating increases the chance of incorrect conclusions (psuedoreplication) Variability in multiple measurements is measurement error, rather than experimental error PTP12345678… Wrist RHR8069938877897479… Neck RHR8465928681867774…

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Consequences of Pseudoreplication Is it bad to take multiple OUs on each EU then? No, often the solution here is to average the measurements of from the OUs and treat it as one observation What if we don’t do this? We severely underestimate error We potentially overexaggerate the true treatment differences What if measurement error is high? Try to improve measurement process Revisit the experiment and assess the homogeneity of the EUs, thinking of potential covariates

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Exercise Example

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Unbalanced consequences? Suppose the following: This would lead to better estimation of the high intensity treatment over the other two Thus if you have equal interest in estimating the treatments, try to equally replicate the number of treatment assignments TreatmentLowModerateHigh # Participants9 reps 18 reps

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Summary The number of replications is the number of experimental units to which a treatment is assigned Replicating in an experiment helps us decrease variance and increase precision in estimating treatment effects

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THREE BASIC PRINCIPLES OF DOE: Blocking (or Local Control of Error)

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Local Control of Error What? Any means of improving accuracy of measuring treatment effects in design Why? Removes sources of nuisance experimental variability Improves precision with which comparisons among factors are made How? Often through use of blocking (or ANCOVA)

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Blocking What? A block is a set of relatively homogeneous experimental conditions EX: block on time, proximity of experimental units, or characteristics of experimental units How? Separate randomizations for each block Account for differences in blocks and then compare the treatments

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Exercise Example Block on gender? This assumes that males and females have different responses to exercise intensity Would have the following (balanced) design: Here, after the participants are blocked into male/female groups, they are then randomly assigned into one of three treatment conditions BLOCK 1 24 MALES BLOCK 2 12 FEMALES 8 low4 low 8 moderate4 moderate 8 high4 high

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Exercise Example Block on age? This assumes that age may influence the effect exercise intensity has on resting heart rate Would have the following (balanced) design: Here, after the participants are blocked into respective age groups, they are then randomly assigned into one of three treatment conditions BLOCK 1 18-24 years (24 ptps) BLOCK 2 24-35 years (6 ptps) BLOCK 3 35-50 years (6 ptps) 8 low2 low 8 moderate2 moderate 8 high2 high

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Randomized Complete Block Design (RCBD) This design is called Generalized RCBD Generalized merely means there are replications involved Here, each treatment appears in each block an equal number of times Benefits of RCBD We can compare the performance of the three treatments (exercise programs) We can account for the variability in gender that might otherwise obscure the treatment effects

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Summary Blocking is separating EUs into groups with similar characteristics It allows us to remove a source of nuisance variability, and increase our ability to detect treatment differences Randomization is conducted within each block Note that we cannot make causal inferences about blocks– only treatment effects!

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EXAMPLE: Gene Expression in Leukemia Cells

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Leukemia Cells Background Suppose we are interested in how different treatment groups affect gene expression in human leukemia cells There are three treatment groups: MP only MP with low dose MTX MP with high dose MTX Each treatment group has 10 obs What type of design is this?

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CRD Assumptions and Background The simplest design assumes that all the EUs are similar and the only major source of variation is the treatments Recall: A CRD randomizes all treatment-EU assignments for the specified number of treatment replications Recall: We want to aim to have a balanced experiment, i.e. equal replications of each treatment

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Leukemia Cells As before, we want to randomize which subjects receive which of the three treatments The data looks as follows: TreatmentsObservations MP ONLY 334.531.670141.261.269.667.566.6120.7881.9 MP + HDMTX 919.4404.2102454.162.8671.6882.1354.2321.991.1 MP + LDMTX 108.426.1240.8191.169.7242.862.7396.923.6290.4

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Leukemia Cells – Pre randomization These EUs should be similar MP only MP + LDMTX MP + HDMTX

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Leukemia Cells – Post randomization

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Leukemia Cells in JMP We want to enter the data such that each response has its own row, with the corresponding treatment type We then choose Analyze Fit Y by X

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Leukemia Cells in JMP Choose “GeneExp” for Y, Response Choose “Treatment” for X, factor

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Leukemia Cells Visual Analysis General comments Treatment 3 has a smaller spread of data than the other two Treatment 2 has the highest average “gene expression”, followed by Treatment 1, then Treatment 3 Are the differences substantial? What do you see from this graph (to the left) here?

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Leukemia Cells Summary of Fit In more technical terms, it is the percent of variation in response (gene expression) that can be explained by our predictor (treatment group). Based on this first glance at the summary of fit, what would you conclude? R-square is a measure of fit. If it is close to 1, a good model is indicated. If it is close to 0, a poor model is indicated

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Leukemia Cells ANOVA Null hypothesis: The treatments have the same means Test: Is there at least one treatment effect that is different from the rest? SStotal=SStrt + SSError Variance of all observations from the mean of all the data Variance of treatment means from overall mean Variance of observations from their respective treatment means

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Leukemia Cells ANOVA SSTotal is the variance visualized from this plot Each of these groups has its own mean. SSTrt compares these means to the overall mean SSError compares each observation to the treatment means

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Leukemia Cells: ANOVA If the treatments have a similar effect, then SSTrt will be small (since treatment means are close to overall mean) If the treatments are different, then SSTrt will be large (since more of SSTotal comes from SSTrt, i.e. treatment differences are explaining the variance) ANOVA Table calculates these values and gives us a test statistic (F Ratio) to test for treatment effects

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Leukemia Cells: ANOVA Under our null hypothesis, F= MSTrt/MSError follows an F-distribution; from this we obtain our p-value Here Prob > F = 0.0544, which is just over the typical α=0.05 cutoff

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Summary of Leukemia Example Our ANOVA test failed to reject the null hypothesis that the treatment means are the same (p-value =0.0544) It seems that although the treatment means appeared to be very different (237.58, 478.62, and 165.20 for treatments 1, 2, and 3 respectively), the variation of observations from their respective treatment means was so large that not enough of the variation in SSTotal could be attributed to treatment differences

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Take a 10 minute break!

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EXAMPLE: Nozzle Designs and Shape Factor

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Nozzles & Shapes Background Suppose we are interested in how nozzle design (5 types here) affects the shape factor in the performance of turbulent water jets However, the jet efflux velocity has been known to influence the shape factor in a way that is hard to control. What is this called? What can we do to account for this source of variation?

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Nozzles & Shapes Runs Suppose we only have five nozzles total, one of each type of design. Here is a case where we would randomize run order (rather than treatment) Jet Efflux Velocity (m/s) Block 1Block 2Block 3Block 4Block 5Block 6 Nozzle Design Run Order 232154 351441 124513 415322 543235

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Nozzles & Shapes Data The data looks as follows: How many replicates do we have per treatment? What type of design is this? Nozzle Design Jet Efflux Velocity (m/s) 11.7314.3716.5920.4323.4628.74 10.780.800.810.750.770.78 20.85 0.920.860.810.83 30.930.920.950.89 0.83 41.140.970.980.880.860.83 50.970.860.780.76 0.75

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RCBD Background Given t treatments and b blocks, a RCBD has one observation per treatment in each block If we have multiple observations per treatment in each block (replicates), this is a generalized RCBD Is our nozzle example a RCBD or GRCBD? In a (G)RCBD, Blocks represent a restriction on randomization We want to randomize treatment order within each block We are essentially running separate CRDs for each block!

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Nozzles & Shapes in JMP To analyze in JMP, we want to enter the data such that each response is lined up in a different row, with its associated characteristics in the same row Note: make sure Nozzle Design and Jet Efflux are listed as nominal variables! Again, choose Analyze Fit Y by X

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Nozzles & Shapes in JMP Choose “Shape Factor” for Y, Response Choose “Nozzle Design” for X, factor Choose “Jet Efflux” for Block

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Nozzles & Shapes Visual Analysis Always look for a visual pattern first. What do you see in this graph of shape factor against nozzle design? It appears that nozzle design 4 has the highest shape factor, followed by design 3, design 2, design 5, then design 1.

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Nozzles & Shapes Analysis From our earlier discussion on R-square values and ANOVA tests, what is your first intuition here?

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Nozzles & Shapes ANOVA The p-value of Nozzle Design is significant. What does that mean? We have an additional Sum of Squares here. What is it? Are we interested in its effects? The p-value of Jet Efflux is significant. What does that mean?

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Nozzles & Shapes ANOVA The p-value of Jet Efflux indicates how much we reduced experimental error this means that blocking was a good idea! Can we do a CRD analysis if we find out blocking was a bad idea? (i.e. p-value of Jet Efflux is high?) No. Because we did not design the experiment using CRD protocol we cannot conduct the analysis this way. What do you think our next steps should be?

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Contrasts Given v treatments and the treatment means τ 1… τ v : Note: Here, we have 5 treatments, so we would just have our three treatment means τ 1, τ 2,τ 3, τ 4 and τ 5 A contrast is a specific linear combination of these means For example, if we were comparing treatments 1 and 2, we would have contrast : (1) τ 1 + (-1) τ 2 + (0) τ 3 + (0)τ 4 + (0) τ 5 τ 1 - τ 2

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Contrasts The most important contrasts include: Pairwise treatment comparisons Group average comparisons

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Nozzles & Shapes Means Comparison ANOVA only tells us if there is at least one pair of nozzle means that differ conduct pairwise comparisons τ 1 – τ 2 τ 1 – τ 3 τ 1 – τ 4 τ 1 – τ 5 τ 2 – τ 3 τ 2 – τ 4 τ 2 – τ 5 τ 3 – τ 4 τ 3 – τ 5 τ 4 – τ 5

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Nozzles & Shapes Means Comparison We find Nozzle 4 has a higher shape factor than Nozzles 5 and 1 Nozzle 3 has a higher shape factor than only Nozzle 1

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Summary of Nozzles & Shapes In this case, blocking was key in reducing experimental error, allowing us to better distinguish whether at least one of the nozzle designs differed from another (ANOVA Test) This means differences in jet efflux velocity were causing significant variation in shape factor responses Tukey’s pairwise comparisons test allowed us to see which specific nozzle designs differed. We found that: Shape nozz4 > Shape nozz1, Shape nozz5 Shape nozz3 > Shape nozz1

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Introduction to: Factorial Designs

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CRD Extension: More than one factor Suppose we have two or more factors, each with 2+ levels/settings, that we want to investigate to see how they affect the response What are some ways we can conduct an experiment? “Best guess” experiments: researchers have practical and theoretical knowledge they use to “set levels” OFAT experiments: vary each factor individually while holding other factors constant at baseline levels Factorial experiments: Factors are varied together, and response of interest is observed at each combination of levels What are the PROs and CONs of each method?

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CRD Extension: More than one factor Best Guess Experiments PRO: experimenters have a good idea of what might work CON: Can lead to guessing for a long time without guarantee of success OFAT Experiments PRO: easy to interpret, and used extensively in practice CON: Can be inefficient, may not reach optimum solution, and fails to consider interactions (will discuss later) Factorial Experiment PRO: Efficient, can detect interactions

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CRD Extension: Factorial Experiments In a factorial experiment, treatments are a combination of multiple factors with different levels (i.e. settings) There can be as few as two (common) and as many as desired (though this severely complicates the design) EX: In the Leukemia example, we could alter the experiment to low and high doses of MP and MTX so that there are now four “treatments” MTX level MP levelLowHigh Low High

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Leukemia Cells – Factorial Design MP Low MTX Low MP Low MTX High MP High MTX Low MP High MTX High Remember to still randomize these treatments across participants!

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Leukemia Cells – Factorial Design Data is collected as follows: How do we analyze a Factorial Experiment? Factor A (MP) Factor B (MTX) -- +- -+ ++ Treatment Combo A low, B low A high, B low A low, B high A high, B high Rep I Rep II Rep III 88.612291.2 145.2171.8178.9 163200.2169.3 460.4492.3483.1

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Factorial ANOVA Given two factors, A and B, with varying number of levels, what do we want to examine to see how A and B affect the response? Overall mean (of all the data) Cell Means (mean for each treatment combo) Factor A and B level means We use the same ANOVA approach, but further decompose SSTrt into pieces for different factors SSTrt=SSA+SSB+SSAB

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Factorial ANOVA Visualize in contingency table: MTX level MP levelLowHigh Low Cell mean MP Low Factor Mean High Cell mean MP High Factor Mean MTX Low Factor Mean MTX High Factor MeanOverall Mean

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Factorial ANOVA Visualize in contingency table: MTX level MP levelLowHigh Low100.6177.5139.05 High165.3478.6321.95 132.95328.05230.5

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Factorial ANOVA Let’s break down SSTrt into its respective pieces: SSTrt = SSA + SSB + SSAB SSTrt: Compares cell means to overall mean SSA: Compares A level means to overall mean SSB: Compares B level means to overall mean SSA and SSB test for main effects of factors A and B Main effect: average effect of changing from one level of the factor to another, averaging over all levels of the other factors

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Factorial ANOVA Let’s break down SSTrt into its respective pieces: SSTrt = SSA + SSB + SSAB SSAB: Tests for interaction between A and B Interaction: When how factor A affects the response depends on the level of factor B

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Interaction How to determine an interaction? Look at behavior of the means as the levels vary

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Main Effects & Interaction Main effects and interactions are specific types of important contrasts Recall from our discussion of contrasts that group average contrasts are common: Let’s suppose the treatment means are as such: MTX level MP levelLowHigh Low τ1τ1 τ2τ2 High τ3τ3 τ4τ4

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Main Effects & Interaction Main Effects Interaction MP: ½ (τ 3 + τ 4 – τ 1 – τ 2 )MP*MTX ½ (τ 4 + τ 1 – τ 2 – τ 3 ) MTX: ½ (τ 2 + τ 4 – τ 1 – τ 3 ) MTX level MP levelLowHigh Low τ1τ1 τ2τ2 High τ3τ3 τ4τ4

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Factorial Design: Summary One treatment is combination of multiple factors Efficient way to test effect of multiple treatment factors We may extend to more than two factors, but the number of EUs necessarily grows rapidly! Use an interaction plot to help visualize effects Main effects and interactions can be represented through group average contrasts

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Wrap-Up: Conclusions & Questions

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Summary of the Short Course Remember to randomize! Randomize run order, and treatments Remember to replicate! Use multiple EUs for each treatment– it will help you be more accurate in estimating your effects Remember to block! In the case where you suspect some inherent quality of your experimental units may be causing variation in your response, arrange your experimental units into groups based on similarity in that quality Remember to contact LISA! For short questions, attend our Walk-in Consulting hours For research, come before you collect your data for design help

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The End!

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References Cheok, M. H., Yang, W., Pui, C. H., Downing, J. R., Cheng, C., Naeve, C. W.,... Evans, W. E. (2003). Treatment-specific changes in gene expression discriminate in vivo drug response in human leukemia cells. Nature Genetics, 34(1). Theobald, C. (1981). The effect of Nozzle design on the stability and performance of turbulent water jets. Fire Safety Journal, 4(1). http://www.floiminter.net/?page_id=286 http://io9.com/5928595/researchers-identify-the-kinds-of-exercise-that-help- you-live-longer http://io9.com/5928595/researchers-identify-the-kinds-of-exercise-that-help- you-live-longer http://www.webmd.com/heart/taking-a-pulse-heart-rate http://www.docofdiets.com/exercise-3.htm http://en.wikipedia.org/wiki/Leukemia http://www.wash-safe.com/wash_safe_en/bulls-eye-garden-hose-nozzle http://www.rgbstock.com/photo/n3eIwlU/ http://www.operationsports.com/forums/madden-nfl-football-sliders/573238- m13-killer-sliders-offline-ccm-guaranteed-cpu-run-game-92.html http://www.operationsports.com/forums/madden-nfl-football-sliders/573238- m13-killer-sliders-offline-ccm-guaranteed-cpu-run-game-92.html http://www.123rf.com/photo_4819916_heart-rate-vector-design-element.html

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