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SPH 247 Statistical Analysis of Laboratory Data 1 April 2, 2013 SPH 247 Statistical Analysis of Laboratory Data

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Basic Principles of Experimental Investigation Sequential Experimentation Comparison Manipulation Randomization Blocking Simultaneous variation of factors Main effects and interactions Sources of variability April 2, 2013SPH 247 Statistical Analysis of Laboratory Data2

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Sequential Experimentation No single experiment is definitive Each experimental result suggests other experiments Scientific investigation is iterative. “No experiment can do everything; every experiment should do something,” George Box. April 2, 2013SPH 247 Statistical Analysis of Laboratory Data3

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April 2, 2013SPH 247 Statistical Analysis of Laboratory Data4 Plan Experiment Perform Experiment Analyze Data from Experiment

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Comparison Usually absolute data are meaningless, only comparative data are meaningful The level of mRNA in a sample of liver cells is not meaningful The comparison of the mRNA levels in samples from normal and diseased liver cells is meaningful April 2, 2013SPH 247 Statistical Analysis of Laboratory Data5

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Internal vs. External Comparison Comparison of an experimental results with historical results is likely to mislead Many factors that can influence results other than the intended treatment Best to include controls or other comparisons in each experiment April 2, 2013SPH 247 Statistical Analysis of Laboratory Data6

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Manipulation Different experimental conditions need to be imposed by the experimenters, not just observed, if at all possible The rate of complications in cardiac artery bypass graft surgery may depend on many factors which are not controlled (for example, characteristics of the patient), and may be hard to measure April 2, 2013SPH 247 Statistical Analysis of Laboratory Data7

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April 2, 2013SPH 247 Statistical Analysis of Laboratory Data8

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Randomization Randomization limits the difference between groups that are due to irrelevant factors Such differences will still exist, but can be quantified by analyzing the randomization This is a method of controlling for unknown confounding factors April 2, 2013SPH 247 Statistical Analysis of Laboratory Data9

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Suppose that 50% of a patient population is female A sample of 100 patients will not generally have exactly 50% females Numbers of females between 40 and 60 would not be surprising In two groups of 100, the disparity between the number of females in the two groups can be as big as 20% simply by chance, but not much larger This also holds for factors we don’t know about April 2, 2013SPH 247 Statistical Analysis of Laboratory Data10

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Randomization does not exactly balance against any specific factor To do that one should employ blocking Instead it provides a way of quantifying possible imbalance even of unknown factors Randomization even provides an automatic method of analysis that depends on the design and randomization technique. April 2, 2013SPH 247 Statistical Analysis of Laboratory Data11

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The Farmer from Whidbey Island Visited the University of Washington with a Whalebone water douser 10 Dixie cups, 5 with water, 5 empty, each covered with plywood Placed in a random order defined by generating 10 random numbers and sorting the cups by the random number If he gets all 10 right, is chance a reasonable explanation? April 2, 2013SPH 247 Statistical Analysis of Laboratory Data12

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The randomness is produced by the process of randomly choosing which 5 of the 10 are to contain water There are no other assumptions April 2, 2013 SPH 247 Statistical Analysis of Laboratory Data13

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If the randomization had been to flip a coin for each of the 10 cups, then the probability of getting all 10 right by chance is different There are 2 10 = 1024 ways for the randomization to come out, only one of which is corresponds to the choices, so the chance is 1/1024 =.001 The method of randomization matters If the farmer could observe condensation on the cups, then this is still evidence of non-randomness, but not of the effectiveness of dousing! April 2, 2013SPH 247 Statistical Analysis of Laboratory Data14

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Randomization Inference 20 tomato plants are divided 10 groups of 2 placed next to each other in the greenhouse (to control for temperature and insolation) In each group of 2, one is chosen using a random number table to receive fertilizer A; the other receives fertilizer B The yield of each plant in pounds of tomatoesis measured The null hypothesis is that the fertilizers are equal in promoting tomato growth April 2, 2013SPH 247 Statistical Analysis of Laboratory Data15

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April 2, 2013SPH 247 Statistical Analysis of Laboratory Data16 12345678910 A13282109143107669510888133 B14088112142118649811393136 diff86311-23553 Pounds of yield of tomatoes for 20 plants

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The average yield for fertilizer A is 106.3 pounds The average yield for fertilizer B is 110.4 pounds The average difference is 4.1 Could this have happened by chance? Is it statistically significant? If A and B do not differ in their effects (null hypothesis is true), then the plants’ yields would have been the same either whether A or B is applied The difference would be the negative of what it was if the coin flip had come out the other way April 2, 2013SPH 247 Statistical Analysis of Laboratory Data17

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April 2, 2013SPH 247 Statistical Analysis of Laboratory Data18 132 lb140 lb 132 lb Fert A Fert B ActualHypothetical ∆ = 8∆ = −8

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In pair 1, the yields were 132 and 140. The difference was 8, but it could have been −8 With 10 coin flips, there are 2 10 = 1024 possible outcomes of + or − on the difference These outcomes are possible outcomes from our action of randomization, and carry no assumptions The measurements don’t have to be normally distributed or have the same variance April 2, 2013SPH 247 Statistical Analysis of Laboratory Data19

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Of the 1024 possible outcomes that are all equally likely under the null hypothesis, only 3 had greater values of the average difference, and only four (including the one observed) had the same value of the average difference The likelihood of this happening by chance is [3+4/2]/1024 =.005 This does not depend on any assumptions other than that the randomization was correctly done April 2, 2013SPH 247 Statistical Analysis of Laboratory Data20

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April 2, 2013SPH 247 Statistical Analysis of Laboratory Data21 12345678910 A13282109143107669510888133 B14088112142118649811393136 diff86311-23553

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April 2, 2013SPH 247 Statistical Analysis of Laboratory Data22 Paired t-test

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Randomization in practice Whenever there is a choice, it should be made using a formal randomization procedure, such as Excel’s rand() function. This protects against unexpected sources of variability such as day, time of day, operator, reagent, etc. April 2, 2013SPH 247 Statistical Analysis of Laboratory Data23

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April 2, 2013SPH 247 Statistical Analysis of Laboratory Data24 Pair NumberFirst Sample Treatment 1A or B? 2 3 4 5 6 7 8 9 10A or B?

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April 2, 2013SPH 247 Statistical Analysis of Laboratory Data25 Pair Num First Sample Treatment random number 1A or B?0.871413 2A or B?0.786036 3A or B?0.889785 4A or B?0.081120 5A or B?0.297614 6A or B?0.540483 7A or B?0.824491 8A or B?0.624133 9A or B?0.913187 10A or B?0.001599

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=rand() in first cell Copy down the column Highlight entire column ^c (Edit/Copy) Edit/Paste Special/Values This fixes the random numbers so they do not recompute each time =IF(C3<0.5,"A","B") goes in cell C2, then copy down the column April 2, 2013SPH 247 Statistical Analysis of Laboratory Data26

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April 2, 2013SPH 247 Statistical Analysis of Laboratory Data27 Plant Pair First Plant Treatment random number 1B0.871413 2B0.786036 3B0.889785 4A0.081120 5A0.297614 6B0.540483 7B0.824491 8B0.624133 9B0.913187 10A0.001599

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To randomize run order, insert a column of random numbers, then sort on that column More complex randomizations require more care, but this is quite important and worth the trouble Randomization can be done in Excel, R, or anything that can generate random numbers April 2, 2013SPH 247 Statistical Analysis of Laboratory Data28

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Randomization in R rand1 <- runif(10) treat <- rep("",10) rand2 <- order(rand1) < 5.5 treat[rand2] <- "Treatment A" treat[!rand2] <- "Treatment B" > rand1 [1] 0.23459799 0.18243579 0.07706528 0.68511653 0.70065774 0.59058980 [7] 0.84561795 0.96164966 0.20475362 0.49222996 > order(rand1) [1] 3 2 9 1 10 6 4 5 7 8 > rand2 [1] TRUE TRUE FALSE TRUE FALSE FALSE TRUE TRUE FALSE FALSE > treat [1] "Treatment A" "Treatment A" "Treatment B" "Treatment A" "Treatment B" [6] "Treatment B" "Treatment A" "Treatment A" "Treatment B" "Treatment B" April 2, 2013SPH 247 Statistical Analysis of Laboratory Data29

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Blocking If some factor may interfere with the experimental results by introducing unwanted variability, one can block on that factor In agricultural field trials, soil and other location effects can be important, so plots of land are subdivided to test the different treatments. This is the origin of the idea April 2, 2013SPH 247 Statistical Analysis of Laboratory Data30

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If we are comparing treatments, the more alike the units are to which we apply the treatment, the more sensitive the comparison. Within blocks, treatments should be randomized Paired comparisons are a simple example of randomized blocks as in the tomato plant example April 2, 2013SPH 247 Statistical Analysis of Laboratory Data31

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Simultaneous Variation of Factors The simplistic idea of “science” is to hold all things constant except for one experimental factor, and then vary that one thing This misses interactions and can be statistically inefficient Multi-factor designs are often preferable April 2, 2013SPH 247 Statistical Analysis of Laboratory Data32

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Interactions Sometimes (often) the effect of one variable depends on the levels of another one This cannot be detected by one-factor-at-a-time experiments These interactions are often scientifically the most important April 2, 2013SPH 247 Statistical Analysis of Laboratory Data33

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Experiment 1. I compare the room before and after I drop a liter of gasoline on the desk. Result: we all leave because of the odor. April 2, 2013SPH 247 Statistical Analysis of Laboratory Data34

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Experiment 1. I compare the room before and after I drop a liter of gasoline on the desk. Result: we all leave because of the odor. Experiment 2. I compare the room before and after I drop a lighted match on the desk. Result: no effect other than a small scorch mark. April 2, 2013SPH 247 Statistical Analysis of Laboratory Data35

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Experiment 1. I compare the room before and after I drop a liter of gasoline on the desk. Result: we all leave because of the odor. Experiment 2. I compare the room before and after I drop a lighted match on the desk. Result: no effect other than a small scorch mark. Experiment 3. I compare all four of ±gasoline and ±match. Result: we are all killed. Large Interaction effect April 2, 2013SPH 247 Statistical Analysis of Laboratory Data36

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Statistical Efficiency Suppose I compare the expression of a gene in a cell culture of either keratinocytes or fibroblasts, confluent and nonconfluent, with or without a possibly stimulating hormone, with 2 cultures in each condition, requiring 16 cultures April 2, 2013SPH 247 Statistical Analysis of Laboratory Data37

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I can compare the cell types as an average of 8 cultures vs. 8 cultures I can do the same with the other two factors This is more efficient than 3 separate experiments with the same controls, using 48 cultures Can also see if cell types react differently to hormone application (interaction) April 2, 2013SPH 247 Statistical Analysis of Laboratory Data38

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Fractional Factorial Designs When it is not known which of many factors may be important, fractional factorial designs can be helpful With 7 factors each at 2 levels, ordinarily this would require 2 7 = 128 experiments This can be done in 8 experiments instead! Each two factors form a replicated two-by-two Some sets of three factors form an unrelicated two-by- two-by-two April 2, 2013SPH 247 Statistical Analysis of Laboratory Data39

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April 2, 2013SPH 247 Statistical Analysis of Laboratory Data40 F1F2F3F4F5F6F7 1HHHHHHH 2HHLHLLL 3HLHLHLL 4HLLLLHH 5LHHLLHL 6LHLLHLH 7LLHHLLH 8LLLHHHL

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April 2, 2013SPH 247 Statistical Analysis of Laboratory Data41 F1F2F3F4F5F6F7 1HHHHHHH 2HHLHLLL 3HLHLHLL 4HLLLLHH 5LHHLLHL 6LHLLHLH 7LLHHLLH 8LLLHHHL

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April 2, 2013SPH 247 Statistical Analysis of Laboratory Data42 F1F2F3F4F5F6F7 1HHHHHHH 2HHLHLLL 3HLHLHLL 4HLLLLHH 5LHHLLHL 6LHLLHLH 7LLHHLLH 8LLLHHHL

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April 2, 2013SPH 247 Statistical Analysis of Laboratory Data43 F1F2F3F4F5F6F7 1HHHHHHH 2HHLHLLL 3HLHLHLL 4HLLLLHH 5LHHLLHL 6LHLLHLH 7LLHHLLH 8LLLHHHL

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Main Effects and Interactions Factors Cell Type (C), State (S), Hormone (H) Response is expression of a gene The main effect C of cell type is the difference in average gene expression level between cell types April 2, 2013SPH 247 Statistical Analysis of Laboratory Data44

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For the interaction between cell type and state, compute the difference in average gene expression between cell types separately for confluent and nonconfluent cultures. The difference of these differences is the interaction. The three-way interaction CSH is the difference in the two way interactions with and without the hormone stimulant. April 2, 2013SPH 247 Statistical Analysis of Laboratory Data45

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Sources of Variability in Laboratory Analysis Intentional sources of variability are treatments and blocks There are many other sources of variability Biological variability between organisms or within an organism Technical variability of procedures like RNA extraction, labeling, hybridization, chips, etc. April 2, 2013SPH 247 Statistical Analysis of Laboratory Data46

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Replication Almost always, biological variability is larger than technical variability, so most replicates should be biologically different, not just replicate analyses of the same samples (technical replicates) However, this can depend on the cost of the experiment vs. the cost of the sample 2D gels are so variable that replication is required Expression arrays, PCR, RNA-Seq, Mass Spect and others do not usually require replication April 2, 2013SPH 247 Statistical Analysis of Laboratory Data47

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Quality Control It is usually a good idea to identify factors that contribute to unwanted variability A study can be done in a given lab that examines the effects of day, time of day, operator, reagents, etc. This is almost always useful in starting with a new technology or in a new lab April 2, 2013SPH 247 Statistical Analysis of Laboratory Data48

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Possible QC Design Possible factors: day, time of day, operator, reagent batch At two levels each, this is 16 experiments to be done over two days, with 4 each in morning and afternoon, with two operators and two reagent batches Analysis determines contributions to overall variability from each factor April 2, 2013SPH 247 Statistical Analysis of Laboratory Data49

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References Statistics for Experimenters, Box, Hunter, and Hunter, John Wiley April 2, 2013SPH 247 Statistical Analysis of Laboratory Data50

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Exercise 1 You have a clinical study in which 10 patients will either get the standard treatment or a new treatment Randomize which 5 of the 10 get the new treatment so that all possible combinations can result. Use Excel or R or another formal randomization method. Instead, randomize so that in each pair of patients entered by date, one has the standard and one the new treatment (blocked randomization). What are the advantages of each method? Why is randomization important? April 2, 2013SPH 247 Statistical Analysis of Laboratory Data51

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Course Website http://dmrocke.ucdavis.edu April 2, 2013SPH 247 Statistical Analysis of Laboratory Data52

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