Statistics and Experimental Design for Animal Research: A Gentle IntroductionRobert J. TempelmanDepartment of Animal ScienceMichigan State University

CANR Statistical Consulting Center

http://www.fw.msu.edu/orgs/canr_big/SCC.htm

What is statistics???Ott and Longnecker (2001): "science of learning from data

Biometry/Biostatistics:Statistics applied to biologyDouble meaning of biometrics

Biological research involves data!!1) Collecting DataExperimental Design2) Summarizing DataSimple numerical and graphical descriptions 3) Analyzing DataFormal statistical methods for hypothesis testing and estimation4) Communicating ResultsDiscussion and Interpretation

Biological data will always be noisy

Why?In a study, data is collected from a finite sample randomly drawn from a conceptually large population.Every subject responds to the same treatment differently (experimental error).Even the same subject might not respond the same way to the same treatment from one day to the next (measurement error).Therefore, ALWAYS an element of uncertainty in drawing conclusions from the statistical analysis of data from finite samples.

Experimental Design

Definition: Plan for assigning experimental units to treatments.Simplest experimental design:Completely Randomized Design (CRD)In a CRD, experimental units are :

1) Randomly chosen from a representative populationthen 2) Randomly assigned to one of several treatments

experimental units should be as homogeneous as possible; otherwise consider blocking (see later)

- Randomization is essential to remove systematic bias!

Blood cholestorol exampleExperimental study with randomization:6 rats assigned at random to one of 2 treatments (n=3 rats per treatment)Blood Cholestorol Data collected (mg/dl):

Would you conclude the treatments lead to different mean blood cholestorol levels ????

Ave yA = 146Ave yB = 152A1A2A3B1B2B3

A MEAN TREATMENT DIFFERENCE IS FOUND!But is itDue to mere chance (biological noise) ??? OrThe real thing (beyond reasonable doubt)?

Statistical inferenceStatistical analysis for previous slide: consider a regular t-test

Significance and PowerPractical significance vs. statistical significanceStatistically significant results may not be practically important.Statistical Power issues!Was the study large enough to allow a reasonable chance of definitively concluding a treatment differenceif one truly existed?

Scientific Methoda) Review and Research the problem

b) Formulate Hypothesis

c) Design experiment that will allow test of hypothesis

d) Evaluate the hypothesis

e) Draw Conclusions

How does statistical inference work? Infer upon the characteristics of a large population based on data from a finite random sample Mechanics (Part of the Scientific Method): Design and collect data from an experiment (e.g blood pressure). Assess the probability of getting the experimental results assuming a true null hypothesis (status quo knowledge..e.g. no treatment difference ).Common investigator objective: disprove the status quo in favor of an alternative hypothesis (there is a treatment effect).Conclusions never made with absolute certaintymust establish proof beyond a reasonable doubt.

Terminology: Sample versus specimenBiologist draws blood from 20 peopleA biologist might state that he/she has 20 samples of blood.Statistician would state that the biologist has one sample of 20 glucuse measurements.20 specimens or 20 experimental units rather than 20 samples.

(Humans)Sample vs. Population SamplePopulationTarget pop'n RandomJudicious inferenceSAMPLE VERSUS POPULATION(Humans)(All rats)Actual versus target population differences could be far more subtle

Variables, Variables!!! Quantitative variables:Due to a true numerical measurement.Ratio scale (e.g. weight) versus Interval Scale (e.g. temperature)Discrete (countable) versus Continuous

Qualitative variables:Nominal scale (classification or group) GENOTYPE, SEXOrdinal scale (ranked variables..small, medium, large)

Dependent (response variables e.g. weight) as a function of Independent variables (GENOTYPE, SEX)

Discrete (quantitative) versus ordinal (qualitative) variables1) Litter size (discrete quantitative)

Discrete (quantitative) versus ordinal (qualitative) variables (contd)2) Calving ease scores (ordinal 1-5 scale in Holsteins) 1 = unassisted5 = Caesarean

Other variable issues (contd)Dont confusediscrete variables with truly continuous variables i.e. some variables appear to be discrete (integers) because of data recording round-off.e.g. age of cattle recorded to the nearest month is a continuous variable.Number of mastitis cases within a lactation is a discrete variable

Parameters versus statisticsPopulation:Characterized by parametersSample: characterized by statisticsSize = NSize = nStatistical inferenceRandom selectionA sample statistic is an estimator of a population parametere.g mean: me.g mean: y

Usual distributional assumption for continuous responses: NormalityDistribution of weight gains of 100 baby chicks over specified time period.

Data does not have to be perfectly normally distributed to use common statistical procedures (t-tests, ANOVA)

GAIN MIDPOINT

3.7

3.9

4.1

4.3

4.5

4.7

4.9

0

10

20

30

Pseudo replication (an obvious example from Gill, 1978)Suppose 6 rats per treatment, each measured twice (stimulus response to drugs on two different occasions).e.g. Rat A1 had response 33 1st time, then 35How much replication?Biological versus technical replication (subsampling)One rat per treatment: no replicationA1A2A3A4A5A6A1A2A3A4A5A6n = 6not 12

Rat Number Within TreatmentTreatment ATreatment B133,3537,33239,3831,30329,3143,45441,4136,38534,3630,39626,2338,39

Remedy? Average each experimental units responsesTreat each experimental units average as the responsethen do regular t-test.Note: there are still benefits to subsampling: controls measurement error. -> but always better off increasing number of rats per treatment than number of measurements per rat.

Rat Number Within TreatmentTreatment ATreatment B13435238.530.5330444413753534.5624.538.5

Animal in pensSuppose you have two pens/litters of pigs:Each pen has four pigsAll pigs in Pen # 1 receive Diet AAll pigs in Pen # 2 receive Diet BDo you have replication?Pen # 1 -> Diet APen # 2 -> Diet B

Animals in pens (contd)Answer to question on previous slide: NO! n = 1Pens/Litters are the experimental units for dietsPigs within pens are merely pseudoreplicates!Need several pens per diets in order to have a valid study.Wainwright, Patricia E. 1998. Issues of Design and Analysis Relating to the Use of Multiparous Species in Developmental Nutrition Studies. Journal of Nutrition 128:661-663.See also

Instructions to the Authors (Journal of Dairy Science, 2007)

The experimental unit is the smallest unit to which an individual treatment is imposed. For group-fed animals, the group of animals in the pen or the paddock is the experimental unit; therefore, groups must be replicated.

i.e. must have more than 1 pen per treatment (and 2 might not be nearly enough!)

Basic design conceptsRandomizationExperimental units need to be randomly assigned to treatments!ReplicationSeveral experimental units per treatment needed to assess experimental errorPower: Having sufficiently large enough sample size (experimental units) to detect a mean differenceif one truly exists BlockingSimilar experimental units could be blocked together and randomization of units to treatments conducted within each block

Suppose the size of each litter is standardized to two pigs.We randomly assign one piglet within each litter to Treatment A and the remaining piglet to Treatment B.This is an example of a randomized complete block design (RCBD)! e.g. you wish to test a new diet supplement (Treatment B) versus a control diet (Treatment A) for growth in piglets

established that there are known litter/pen effects on growth.-> then consider blocking on litters!

BLOCKING

Draw random sample of littersTrt A

Trt B

Trt B

Trt A

Trt A

Trt B

Litter 1Litter 2Litter nRandomly assign treatments to piglets within littersPopulation of litters of size 2The Randomized Complete Block Design (RCBD)

Why block?Remove block (e.g. litter) as a source of variabilityGreater statistical power since treatment comparisons conducted WITHIN each litterBasis of paired t-test when block size = 2.Remove litter as a potential source of biasOther examples of blocking?Identical twins "Before and after" treatment on same subject

Crossover designsWhere animals are blocks for treatments2 period crossover:etc.Design is balanced with respect to diets and periods.

e.g. not good idea to always feed Diet A in Period 1 and Diet B in Period 2otherwise Diets and Periods are confounded with each other

Cow 1

Cow 2

Cow 3

Cow 4

Period 1

Diet A

Diet B

Diet A

Diet B

Period 2

Diet B

Diet A

Diet B

Diet A

Typical experimental designs exploiting blocking and used for animal science.Variants of crossover designs exploiting power of within-animal comparisons have been chosen for comparing two treatmentsConstructed to be balanced with respect to periods4-period crossovers double reversal half of animals -> A B A B other half -> B A B A3-period crossover switchback half of animals -> A B A & other half -> B A B 2-period crossover simple crossover/Latin square half of animals -> A B & other half -> B A

Table 1 from Cox (1980)Cox, D.R. 1980. Design and analysis in nutritional and physiological experimentation. Journal of Dairy Science 63:313-321Table 1. Classification of 24 weight gains measured on 12 cows fed in two pens (2 period crossover)

PeriodPen 1 containingPen 2 containingCows 1 to 6Cows 7 to 12

OneDiet A used and Diet B used and(4 wks)6 gains recorded6 gains recorded

TwoDiet B used and 6 Diet A used and 6(4 wks)gains recordedgains recorded

The experimental units in this situation were pens of animals in a given period no way to separate effects of diet from all other possible factors.PSEUDOREPLICATION ISSUE!!!

Coxs internal tormentOne merely could assume that effects of pens were negligible. This is equivalent to assuming that, when feeding a single diet, the gains of two animals in the same pen are no more alike than the gains of two animals each in different pensBUTPen-to-pen variation has been important too often to make such an assumption credible.

How many animals do I need for a study!It depends on:Your design (blocking versus not blocking, size of experimental unit -> pen vs. animal)True mean difference that you hope to detect!D = mA mBRelative amount of variability (s)Between responses within same animal (se measurement error)Between animals (sa innate or biological variability)Between litters/pens (sp where applicable)Type I error rate: Probability of concluding a treatment effect when one doesnt exist (typically set 80%)

Well, how do we specify some of this stuff?Literature and educated guessingUniformity trials or existing data on subjects in current facility under regular management conditions.Range approximation s 1/4 x Range of responsesse 1/4 x range of responses within same subject and treatmentsa 1/4 x range of responses between subjects within same treatment.sp 1/4 x range of average responses between pens within same treatment.What would be a practically important specification for D = mA mB

Dairy Example: Relationship between within-cow variance s2e (kg2) and DIM (by parity)From Jensen, J. 2001. Genetic evaluation of dairy cattle using test-day models. Journal of Dairy Science. 84:2803-2812.Reasonable assumption:s2e = s2a Early lactationLate lactation

CRD power for individual animal study as function of n and Ds2e + s2a= 4 kg2s2e + s2e= 20 kg2Late lactationEarly lactationPowerPowernn

Two period crossover trial for individual animal trialss2e = 2 kg2s2e = 10 kg2Late lactationEarly lactation

Binary data?Yes or no responses e.g. mastitis, conception rate, Consider comparison of Trt A versus Trt BIncidence rate for Trt A = 5%Incidence rate for Trt B = 7.5%, 10%, 12.5%,., 25%CRD Power to conclude a difference in incidence rates between Trt A (0.05) and Trt B (0.075 to 0.250)

Power calculators

Questions?

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